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 LIBRARY

'

Unrveuo‘lty

J

This is to certify that the

dissertation entitled

REPRESENTATIONS OF TIME AND SPACE
IN THE PLAYER PIANO STUDIES
OF CONLON NANCARROW

presented by

Julie Anne Scrivener

has been accepted towards fulfillment
of the requirements for

degree in Music  

Ph . D .

5m;5 aim/UM

Major professor

Wei? Oct" 2002,

MSU is an Affirmative Action/Equal Opportunity Institution

0- 12771

 
  
 
PLACE IN RETURN Box to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

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6/01 cJCIRCIDatoDuopBS-sz

 
 
 
 
 
 
 
 
 
 
  
 
 
  
 
REPRESENTATIONS OF TIME AND SPACE
IN THE PLAYER PIANO STUDIES
OF CONLON NANCARRow

By

Julie Anne Scrivener

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

School of Music

2002

ABSTRACT

REPRESENTATIONS OF TIME AND SPACE
IN THE PLAYER PIANO STUDIES
OF CONLON NANCARROW

By

Julie Anne Scrivener

The Studies for Player Piano by Conlon Nancarrow (1912—1997) have been studied

for their innovations in rhythm and tempo, including the use of isorhythm, different

techniques of acceleration and deceleration, and most particularly, simultaneous use of

different tempos in sometimes very complex mathematical relationships and textures.

The present study examines Specific structural features native to the Studies, including

the pervasive use of mathematical ratios, the structural use of convergence points in the

canonic Studies, and the presence of fractal characteristics (especially forms).

The first chapter contains a brief biography of Nancarrow, a review of the literature,

and introduces the reader to general characteristics of the Studies. In Chapter 2, the use of

ratios to control various musical parameters are identified, including: tempo relationships,

rhythmic motives, melodic and harmonic materials, and structure. The use of ratios is the

focus of a comprehensive analysis of Study No. 34.

Chapters 3 and 4 are concerned with features of “tempo canons”: a new form

created by Nancarrow in which individual voices present canonic material at different

speeds. Chapter 3 examines the structural use of convergence points in the canonic

Studies. The structural impact of the convergence points, and techniques for emphasizing

and de-emphasizing them, are the focus for this chapter. The chapter concludes with an

analysis of Study No. 27.

Chapter 4 identifies fractal features in Nancarrow’s tempo canons. The defining

fractal characteristics of self-iteration, scaling, and space-filling are identified as they

correspond to formal features in which the same musical material is presented at different

tempos to create large-scale “fractal forms.” Study No. 32, an example of a fractal form,

is analyzed in detail.

The fifth chapter is devoted to analysis of two of Nancarrow’s later works that

contain some of his more mature compositional techniques: Studies No. 25 and 45. The

analysis of No. 25 focuses on the use of the layering technique, while the analysis of No.

45 draws comparisons with a much earlier piece (Study No. 3) and particularly examines

the presence of multidimensionality.

The final chapter summarizes the findings of the study and offers suggestions for

future study of this compelling twentieth-century music.

 
 
ACKNOWLEDGMENTS

During these past seven years Spent pursuing this degree program, I have come to

the inescapable recognition that an effort of this magnitude can only be accomplished

through the encouragement, kindnesses, and forbearances of many people. I have been

truly fortunate to be surrounded by people who have kept me steady and given me what I

needed, and I would like to acknowledge a few of them here.

My guidance committee and dissertation chair, Dr. Bruce Campbell, provided

excellent guidance and was always a font of encouragement and sound advice, both in

terms of guiding my academic program and fostering my career. I appreciate the efforts

of the members of my dissertation committee, Dr. Charles Ruggiero, Dr. Gordon Sly, and

Dr. Dale Bonge, who were responsive when I needed feedback and offered valuable

insights.

I thank the Dean’s office of the College and Arts and Letters at Michigan State

University, particularly Dr. Patrick McConeghy, for financial assistance in the form of a

Dissertation Completion Fellowship. The award came at a crucial time and was instru-

mental in allowing me to complete the writing. Dr. Frederick Tims, the School of Music

graduate advisor, provided critical last-minute assistance in completing the application

and getting it to the college on time.

My research on Nancarrow would never have gotten off the ground without timely

assistance and guidance from two pioneers in the field, Dr. Kyle Gann and Dr. Margaret

Thomas, both of whom were helpful and encouraging correspondents and generous in

their offers of assistance. I am also grateful for the assistance I received from James

Tenney in locating unpublished scores.

I could never have completed this degree program or this dissertation while

working full-time at Western Michigan University without the forbearance and flexibility

iv

 
 
of my co-workers and, most especially, my supervisors. Special thanks are extended to

Dr. Rollin Douma and Dr. Shirley Scott of The Graduate College, and Dr. Thomas Seiler

and Dr. Paul Szarmach of The Medieval Institute.

The people of Gull Lake United Methodist Church in Richland, where I have spent

the past 16 years as choir and worship accompanist, organist, choir director, and now

music director, have been my rock. I especially owe a debt of gratitude to my friends in

the chancel choir, some of whom I am sure are convinced that references in this work to

“spastic rhythm” and “temporal dissonance” refer to them! My other musical friends at

Gull Lake UMC—Betty LeRoy, John and Martha Kuch, John Hill, Allison Hendrix, and

Craig and Joan Schroeder—have brightened my life with their beautiful music, and I

have been blessed in my many musical collaborations with them.

A great gift that I have received as a result of this degree program has been the

friendship of fellow doctoral students Diane deVries and Andrea Dykstra. We shared the

trials of being “non-traditional” commuter students, trying to balance work and home

responsibilities with the rigors of a doctoral program. My journey was enriched by the

treasure of their friendship. Thank you, Diane and Andrea!

Last, but certainly not least, my family—my parents, Leslie and Sharon Gummert

and my brothers, Randy Gummert and Eric Gummert—were always interested in my

progress and proud of my endeavors. My children, Garrett and Taryn Scrivener, who

were only 5 and 2 when I began working on this degree, did not always understand why I

had to work so much or even wanted to, but have now seen the fruits of this intense labor

and will hopefully realize in their own lives that hard work really does pay off. Now that

this is finished, they can call me “Dr. Mom.”

Julie Anne Scrivener

   
 
REPRINT PERMISSION ACKNOWLEDGMENTS

I gratefully acknowledge the following permissions for the inclusion in this work of

the following excerpts:

Nancarrow Conlon Collected Studies for Player Piano, Vol. 5: Studies 2, 6, 7,
14, 20, 21, 24, 26, and 33 for Player Piano
© 1984 by Conlon Nancarrow
© 1988 assigned to Schott Musik International GmbH & Co.
All Rights Reserved
Used by permission of European American Music Distributors LLC, sole US and
Canadian agent for Schott Musik International GmbH & Co.

Nancarrow Conlon Study No. 3 for Player Piano
© 1983 by Conlon Nancarrow
© 1988 assigned to Schott Musik International GmbH & Co.
All Rights Reserved
Used by permission of European American Music Distributors LLC, sole US and
Canadian agent for Schott Musik International GmbH & Co.

Nancarrow Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18 for Player Piano
(Volume 6)
© 1985 by Conlon Nancarrow
© 1988 assigned to Schott Musik International GmbH & Co.
All Rights Reserved
Used by permission of European American Music Distributors LLC, sole US and
Canadian agent for Schott Musik International GmbH & Co.

Nancarrow Study No. 37 for Player Piano
© 1985 by Conlon Nancarrow
© 1988 assigned to Schott Musik International GmbH & Co.
All Rights Reserved
Used by permission of European American Music Distributors LLC, sole US and
Canadian agent for Schott Musik International GmbH & Co.

vi

 
 
TABLE OF CONTENTS

LIST OF TABLES .............................................................................................................. xi

LIST OF FIGURES ......................................................................................................... xiii

CHAPTER 1:
INTRODUCTION AND REVIEW OF THE LITERATURE............................................. 1

Biographical Summary and Review of Existing Literature ........................................4

The Player Piano Studies ...................................................................................6

Musical Characteristics of Nancarrow’s Studies ......................................................29

Rhythmic Features ...........................................................................................29

Pitch and Melodic Features ..............................................................................36

Harmonic Features ...........................................................................................38

Structural Features ...........................................................................................42

Texture, Timbre, and Dynamics ......................................................................46

Comparison of Analyses of Selected Studies ...........................................................47

Summary of the Literature ........................................................................................62

CHAPTER 2:
THE USE OF RATIOS IN NANCARROW’S STUDIES ................................................64

Ratios in Other Twentieth-Century Music ................................................................68

Use of Ratios in Nancarrow’s Music ........................................................................73

Tempo Ratios ...................................................................................................74

“Scales” of Tempos..........................................................................................78

Rhythm/Pitch Analogues ................................................................................. 81

Structural Uses of Ratios.................................................................................. 84

vii

 
 
Analysis of Study No. 34: “Canon 4/5/6/ 4/5/6/ 4/5/6 ” ............................................. 87

9/10/11

Overview .......................................................................................................... 87

Themes .............................................................................................................91

Rests ................................................................................................................. 99

Summary and Conclusions............................................................................. 100

CHAPTER 3:
CONVERGENCE POINTS IN NANCARROW’S TEMPO CANONS ......................... 106

Tempo Canon Terminology and Features ............................................................... 108

Rhythmic and Metric Convergence ........................................................................ 119

Convergence Points in Nancarrow’s Tempo Canons ............................................. 121

Tempo Canons With One (or Almost One) CP ............................................. 121

Tempo Canons With More Than One CP ...................................................... 139

Analysis of Study No. 27: “Canon 5%/6%/8%/11%” ............................................ 157

General Observations ..................................................................................... 158

Overall Structure ............................................................................................ 160

Sections of the Study...................................................................................... 164

Summary and Conclusions............................................................................. 180

CHAPTER 4:
FRACTAL FORMAL FEATURES IN THE TEMPO CANONS................................... 182

Introduction to Fractals ........................................................................................... 182

Fractals in Sound and Music: A Review of the Literature...................................... 185

Measuring Dimension in Fractals ........................................................................... 191

Fractal Characteristics in Nancarrow’s Music ........................................................ 195

Measuring Fractal Dimension in the Tempo Canons..................................... 198

viii

 
Analysis of Study No. 32: “Canon 5/6/7/8”............................................................204

Canon’s Section 1 .......................................................................................... 210

Section 2.........................................................................................................213

Section 3 .........................................................................................................215

Effect of All Voices Together ........................................................................216

CHAPTER 5:
ANALYSIS OF STUDIES NO. 25 AND 45 ...................................................................223

Chronology..............................................................................................................223

Analysis of Study No. 25 ........................................................................................225

Structure .........................................................................................................230

Rhythm and Pulse .......................................................................................... 234

Sectional Characteristics in Study No. 25 ...................................................... 236

Progression of Pitch Centers in No. 25 ..........................................................247

Connections Between Studies No. 25 and 27 ................................................248

Analysis of Study No. 45 ........................................................................................250

Multidimensionality and Temporal Dissonance in Study No. 3 .................... 251

Multidimensionality in No. 45 ....................................................................... 258

No. 45a ...........................................................................................................263

No. 45b...........................................................................................................269

No. 45c ...........................................................................................................275

Concluding Remarks ...................................................................................... 282

CHAPTER 6:
SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FUTURE STUDY ........286

Research Issues .......................................................................................................289

ix

Score Inaccuracies.......................................................................................... 289

Final Thoughts ............................................................................................... 292

APPENDICES

A. ERRATA TO SCORES OF THE PLAYER PIANO STUDIES ............................294

B.

C.

D.

AVAILABLE SCORES, RECORDINGS, AND WRITINGS FOR EACH
STUDY ...................................................................................................................303

INDEX OF REFERENCES TO SPECIFIC STUDIES IN THOMAS (1996) .......322

TRANSCRIBED P. 9 OF STUDY NO. 34 ............................................................324

BIBLIOGRAPHY ............................................................................................................326

LIST OF TABLES

1-1.

Levels of Imitation in Selected Studies ...................................................................41

1-2.

Symmetrical Levels of Imitation in Selected Canons of Study No. 27...................45

2-1.

Relationship of Canon Ratios and Tempo Markings ..............................................75

2-2.

Scalar Acceleration in Study No. 28 ....................................................................... 80

2-3.

Features of Themes in Study No. 34 .......................................................................98

2-4.

Rest Measurements Between Theme Sections in Study No. 34............................ 100

3-1.

Description of Canons and Convergence Points in Study No. 33 ......................... 148

3-2.

Description of Canons and Convergence Points in Study No. 24 ......................... 151

3-3.

Multiplication Factors for Acceleration and Deceleration in Study No. 27.......... 164

3-4.

Calculated and Actual Duration Measurements in Canon 2 of Study No. 27 ....... 168

4-1.

Descriptions of Motives in Study No. 32’s Canon Melody ..................................208

4-2.

Relative Tempos of Durations Occurring in Study No. 32 (after Thomas
1996, 245)..............................................................................................................216

5-1.

Gann’s Chronology of Composition of Nancarrow’s Works................................224

5-2.

12-tone Row Matrix for Study No. 25 (Gann 1995, 242) ..................................... 226

5-3.

Derivation of Palindromic Interval Pattern Between Arpeggio Triads in
Section 3 ................................................................................................................233

5-4.

Approximate Tempos Represented in Study No. 25 .............................................235

5-5.

Clusters of Durations in Section 7.........................................................................236

5-6.

Range of Arpeggios in Each Voice of Section 6, Study No. 25 ...........................243

5-7.

Harmonic Patterns and Bass Lines in Studies No. 3 and 45 .................................252

5-8.

Possible Derivation of Rhythmic Values in No. 45’s “Spastic Rhythm” .............262

5-9.

Tonics of Glissandi Stated in No. 45a ...................................................................268

5-10. Five Patterns of Transposition in Bass Line at End of Study No. 45b ..................273

xi

LIST OF FIGURES

Figure in Thomas (1996, p. 189) illustrating the 3:2 (perfect fifth) ratio
applied to: (a) rhythm, (b) meter, and (c) tempo ..................................................... 19

Segment from Study No. 9 (as renotated by Thomas, 1996, p. 260) exhibiting
a 5:3 rhythmic ratio in which potential simultaneities are avoided. Actual
simultaneities are indicated by a vertical broken line, while potential
simultaneities are marked with an asterisk.............................................................. 23

Proportional notation in “Study No. 8,” p. 18 .........................................................31

7
8 against 4 ostinato with conflicting meters in Study No. 1 (p. 9). The

4

symbol J is a five-division note, in this case a five-thirty-second note ................31

1—5-

1-6.

1-7-

1-8.

1-9.

1-10.

1-1 1

1-12-

1-13-

1-14.

The 5-division symbol used to express a 5:3 division of a half note in Study
No. 3b, p. 16 ............................................................................................................ 32

Melodic writing “around the beat” in Study No. 2 (p. 4) ........................................ 33

The “countdown” technique at the closing of Study No. 12 ................................... 33

The opening of Study No. 2 .................................................................................... 34

First system of Study No. 6, showing ostinato created by alternating different
eighth-note divisions of the same measure length...................................................35

Derivation of resultant rhythm in Study No. 14 (Scrivener 2000, 189) ..................36

Opening of Study No. 31 showing prevalence of melodic “partitioned minor
thir ”........................................................................................................................37

Blues scale patterns in Study No. 45b (p. 7). The notation Nancarrow uses is
“exploded,” which is used when the space on the staff is insufficient for the
number of notes; the given notes take place on the staff below in the time
frame indicated by the dotted lines..........................................................................37

Major ascending/minor descending scalar passages from: (top) Study No. 6
(p. 5) and (bottom) Study No. 7 (p. l—major/minor passage is combined
with 5 + 4 + 2 + 3 + 4 duration series) .................................................................... 38

Entrance of top five voices (of twelve) in third canon of Study No. 37 (p. 9) ........ 39

xii

1-15.

Ostinato line of four chromatically contiguous pitches in Study No. 27 and
progression of its harmonic manipulations .............................................................40

1-16.

1-17.

Interior V—I cadences in Study No. 12: (a) p. 28, first system; (b) p. 28,
second system; (c) p. 29, second system .................................................................4O

Thomas’s (1996, p. 186) taxonomy of canons: (a) conventional; (b) con-
verging; (c) diverging; (d) converging-diverging (arch); and (e) diverging-
converging ...............................................................................................................42

1-18.

Gann’s tempo canon terminology: convergence point, convergence period,
tempo switch, and echo distance (Gann 1995, 20) ..................................................43

1- 19.

Mirrored rhythm in Study No. 35 (Carlsen 1988, 56) .............................................44

1-20.

Symmetrical deployment of canonic voices in (a) Study No. 19 and (b) Study
No. 36 (Carlsen 1988, 6, 24) ...................................................................................44

1-21.

Symmetrical melodic structures in (a) Study No. 9 (p. 5) and (b) Study No.
25 (p. 60) .................................................................................................................46

. (a) Carlsen’s (p. 35) and (b) Gann’s (p. 149) re-notations of the opening

rhythm in Study No. 8. (Carlsen’s example begins with the score’s second

system and corresponds to the pick-up to Gann’s 32 measure.) .............................48

19

1-23.

1-24-

Gann’s three isorhythmic series identified in Study No. 8: (a) section 1, with
the series expressed as multiples of the background durational unit (p. 150);
(b) section 2, expressed in actual measurements in millimeters (p. 151); and
(c) section 3, expressed as multiples of the background unit (p. 152); plus (d)
Carlsen’s graphic representation of the series in section 1 (p. 37) ..........................50

Carlsen’s summary of the canonic trio in the third and final section in Study
No. 8 (p. 45) ............................................................................................................51

Thomas’s identification of triadic structures in the trio canon of the third
section of Study No. 8 (p. 250). (This point corresponds to the entrance of the
third voice in Figure 1-24).......................................................................................52

- Heterophony as projected through four rhythmic layers in Study No. 19
(Thomas 1996, 247). Note the omission of some notes in the lower and
slower layers (E, for example, is projected only in the top voice). The
number series at the left of each staff represents the note durations in eighth
notes according to the formula {11 — 1, n, n + 1, n} ................................................. 53

1-27 .

1-28

Triadic structures in Study No. 19, systems 7-8 (Carlsen 1988, 14) ......................54

Duration series in Study No. 23 (Carlsen 1988,49) ............................................... 56

xiii

1 -29.

Moments of near convergence of attacks in second section of Study No. 23
(Thomas, p. 287) .....................................................................................................57

1 -30.

Carlsen’s (1988, p. 53) diagram illustrating the plotting of pitches in systems
54—55 of Study No. 23. The diagram below the staff represents the piano roll,
on which pitches are ordered lowest to highest from top to bottom .......................59

. Carlsen’s formal diagram for Study No. 35 (1988, p. 65) Showing seven

sections and proportional relationships (A, B, etc.) ................................................61

(a) Cowell’s justly-tuned chromatic scale, with all interval ratios in relation
to C (Cowell 1930, 107); the right column’s parenthetical figures indicate
size of half-steps between adjacent pitches. (b) Interval ratios and cent sizes
in just and Pythagorean intonation (Abraham and von Hombostel 1994, 451) ......66

The overtone series and its numbered harmonics (Jorgensen 1977, 16) .................67

Diagram showing 3:4:5 temporal relationship from Cowell’s New Musical
Resources (cited in Gann 1995, 6). Note how Cowell uses different shaped
noteheads for different divisions of the measure.....................................................69

Stages of composition of the first movement of Cowell’s Quartet Romantic:
(a) Chorale theme and its resultant pitch ratios; (b) first page of the score,
showing the conversion of the Chorale theme’s pitch ratios to rhythmic ratios
(Thomas 1996, 203—04) .......................................................................................... 7O

Tempos and relative ratios in Elliott Carter’s String Quartet No. 2, movement
2 (Thomas 1996, 45): (a) MM speeds used (MM=525 is missing) and ratios
of selected pairs of tempos; (b) combinations of speeds and ratios ........................71

Study No. 46, 3:4:5 polytempo ostinato as renotated by Gann (p. 264) .................77

(a) Formal diagram of Study No. 17, “Canon 12/15/20” (Gann 1995, 22).
Section “A” states the 15-tempo, section “B” the 12-tempo, and section “C”
the 20-tempo. (b) The “duration ratio” resulting from a 12:15:20 tempo ratio
is 3:4:5. This is the ratio which describes the relative durations of sections C,
A, and B ...................................................................................................................78

First page of score to Study No. 37, showing rhythmic expression of
Cowell’s justly-tuned scale. The rhythmic “interval” between each pair of
voices represents a half-step....................................................................................79

2-5-

2-6-

2-7-

2-9.

Excerpt from Study No. 23, p. 4. (In this piece, as in many others, Nancarrow
uses proportional notation to indicate duration—that is, note duration is
directly proportional to the space between notes on the staff.) ............................... 82

2-10.

Score segment from Study No. 29 (p. 6) showing repeated notes F#, A, and B
and rate of note reiteration approximately related to pitch ..................................... 82

xiv

2-11.

Three-note cluster from Study No. 28 (p. 3), exhibiting 15:16:17 tempo and
pitch ratio. Rearticulations in the middle note are regular but some
rearticulations of the upper and lower notes are omitted by Nancarrow ................ 83

2-12.

Opening of Study No. 5, showing 7:5 rhythmic ratio and opening tritone
interval between voices ........................................................................................... 84

2-13.

2-14.

2-15.

Concluding texture of Study No. 3a (score, p. 35). Ostinato lines 2, 4, and 7
express a 2:3:5 rhythmic ratio, while the entire texture expresses a 3:5 ratio
(the three ostinato lines against five lines of irregularly-spaced chords [lines
1, 3, 5a/b, 6, and 8])................................................................................................. 86

Proportional diagram of entrance of themes and levels of imitation in Study
No. 34...................................................................................................................... 88

Succession of tempos in Study No. 34 (based on Gann 1995, 131). Shaded
areas indicate 9:10:11 tempo complexes; recurring 9:10:11 tempo complexes
are indicated by ovals. Dotted lines between adjacent tempos in the top row
indicate breaks in acceleration. Note that: (a) tempos in all A statements
adhere to the 9:10:11 ratio; (b) tempos peak in all voices in section A6, with
steady deceleration beginning in section G; and (c) the piece ends slower
than it began ............................................................................................................90

2-16. Theme A of Study No. 34 .......................................................................................92

2-17. Theme B of Study No. 34........................................................................................93

2-1 8. Theme c of Study No. 34........................................................................................93

2--l 9. Theme D of Study No. 34 .......................................................................................93

2-20. Theme E of Study No. 34 ........................................................................................94

2-2 1. Theme F of Study No. 34 ........................................................................................95

2-22. Theme G of Study No. 34 .......................................................................................95

2‘23. Chromatic movement about the main structural interval, E—G, in themes C

and D .......................................................................................................................98

2‘24. Entrance of theme E in each of the three voices (Study No. 34, p. 8 and

transcribed p. 9) ..................................................................................................... 101

. Passages of near-rhythmic and temporal unanimity between voices: (a) p. 4,

top system (similar rhythm, 15:16 tempo ratio); (b) p. 6, second system
(identical rhythm, 44:45 tempo ratio); and (c) p. 7, second system (identical
rhythm, 10:11 tempo ratio).................................................................................... 102

XV

2-26.

Graph depicting areas of Study No. 34 where all three voices are stating the
same thematic material .......................................................................................... 102

2-27.

Dynamics and relative scale of acceleration and deceleration in bottom voice
of Study No. 34 ..................................................................................................... 104

2-28.

Concluding system of Study No. 34...................................................................... 105

3-1.

Convergence point at middle of Study No. 14 (score, p. 3). Canon is in two
voices (bottom voice is two staves in bass clef, top voice is two staves in
treble clef); CP takes place at beginning of second system. Arrows illustrate
selected echo distances. Dotted lines show potential points of simultaneity;
the convergence period is the distance between potential simultaneities.
Notice that in the fifth convergence period before and after the CP, the echo
distance equals the length of the convergence period—a distance from the
CP that is functionally related to the ratio 4:5 ....................................................... 109

Beginning of convergence period (at shared barline) in Study No. 36 (p. 4 of
score) ..................................................................................................................... l 12

(a) Beginning and (b) ending (score, p. 5) of Study No. 14; eighth-note beats
of note attacks are numbered................................................................................. 113

Beginning (a) and ending (b) of Study No. 36; half-note beats are marked ......... 114

Placement of interior CPs in relation to canonic midpoint, and the role of
potential points of simultaneity. In Study No. 14 (top), canonic midpoint
coincides with simultaneous beat in both voices, allowing CP to coincide
with midpoint; in Study No. 36 (bottom), canonic midpoint does not coincide
with simultaneous beat in all voices and canon is constructed so that CP is
delayed to a later beat. Simultaneities noted are potential points of simul-
taneity as there is not necessarily an attack in each voice at these points ............. 116

Diagram of a tempo overlap at a 3:2 ratio in a diverging-converging canon........ 118

Structure of Study No. 17 (“Canon 12/15/20”) showing the relative duration
of tempo overlap sections. The duration ratio is 3:4:5 between sections C, A,
and B, respectively. The first overlap is between A sections in the top and
bottom voices, and the second overlap is between B sections in the same
voices. There is no overlap involving C, the Shortest section ............................... 118

Three degrees of temporal dissonance at 4:5 tempo ratio as shown in Thomas
(1996, p. 253): (a) all shared downbeats are metrically significant in both
voices for minimal temporal dissonance; (b) only some shared downbeats are
metrically significant in both voices for moderate temporal dissonance; and
(c) no shared downbeats are metrically significant in both voices for
maximum temporal dissonance ............................................................................. 120

3-6.
3-7-

xvi

3-9.

Relationship of simultaneous beats and convergence periods between two
ostinato lines in “Rhythm Study No. 1” ................................................................ 121

3-10.

Line diagrams describing Nancarrow’s tempo canons with only one CP
(Gann 1995, 22, 23, 25, and 26) ............................................................................ 122

. Convergence point from Study No. 4 (score, p. 10) in a passage exhibiting a

loosely interpreted 2:3:5 rhythmic ratio ................................................................ 122

3-12.

Structural diagram of Study No. 14 (“Canon 4/5”), an arch canon....................... 123

3-13.

3-14.

3-15.

3-16.

3~l7.

3-18.

Diagram of Study No. 36 (“Canon 17/18/19/20”), an arch canon with the CP
slightly past the midpoint (top); table of elapsed half-note beats to entrances
of higher voices in comparison to the first (lowest) voice (middle); and
structural diagram of the canon, showing entrance points of recurring theme
and relationship of theme statements to CP (bottom) ........................................... 125

Half-note beats associated with beginnings of convergence periods at the
beginning of Study No. 36. The difference between beats in each voice at
each point is equal to the number of convergence periods remaining to the
CP—for instance, where the difference between beat values is 20, the
distance to the CP is 20 convergence periods ....................................................... 126

The CP in Study No. 36 (“Canon 17/18/19/20”; beginning of second system
shown; from p. 23 of score)................................................................................... 127

Section of Study No. 36 halfway between beginning of “mega-glissando”
and the CP (score, p. 21) where original pitch levels are restated in all four
voices ..................................................................................................................... 129

Area of tempo crossing in Study No. 21 (score, p. 7). The lower voice is
accelerating and the upper voice decelerating. Bracketed area is identified by
Thomas as representing “nearly simultaneous motion” (Thomas 1996, 126) ...... 130

Convergence point in Study No. 21 (score, p. 14). The arrow indicates the
CP, where the canon starts over in both voices, a registral shift occurs in the
lower voice, and the texture in the upper voice (top two staves) changes from
triple octaves to quadruple octaves ....................................................................... 131

Structural diagram of Study No. 18 (“Canon 3/4”), a converging canon in
which the entrance of the second voice coincides with the beginning of a
convergence period................................................................................................ 133

Structural diagram of Study No. 19 (“Canon 12/15/20”) ...................................... 133

. Conclusion of Study No. 19 (“Canon 12/15/20”; score, p. 7), showing the

convergence point and alteration of final pitches to create V-I final cadence...... 134

xvii

3-22.

Structural diagram of Study No. 32 (“Canon 5/6/7/8”)......................................... 135

3-23.

Convergence point of Study No. 32 (score, p. 11): the CP is the final cut-off ..... 135

3—24.

(a) Convergence of arpeggio figures in Study No. 48A (p. 43 of score); (b)
diagram of proportions in Study No. 48 (movement C is movements A and
B played together) ................................................................................................. 137

3—25.

Structural diagram of Study No. 31 (“Canon 21/24/25”), a converging canon
that ends before the CP is reached......................................................................... 138

3-26.

Conclusion of Study No. 31 (“Canon 21/24/25”; score, pp. 8—9) ......................... 138

3-27.

Line diagrams describing structures of Nancarrow’s tempo canons with more
than one CP (Gann 1995, 22, 23, and 25) ............................................................. 140

3-28.

3-29.

Structure of Study No. 15 (“Canon 3/4”), a diverging-converging canon in
two voices.............................................................................................................. 140

Structural highlights of Study No. 15: cadential figures at beginning and end
of canon statements in each voice ......................................................................... 141

3-30.

Conjectural example of a diverging-converging canon in four voices and

tempos, where the tempo ratio is 10:12: 15:20 and the duration ratio is

3:4:5z6. There are no tempo overlaps involving A (the fastest tempo),

overlaps of 143 + 143 = 1/9 of the total piece involving B, 443 = 39 involving

C, and 6/18 = )3 involving D (the slowest tempo) ................................................. 142

3‘3 l. (a) Beginning and (b) ending (score, p. 12) of Study No. 17 (“Canon

12/15/20”). Circled notes in (b) are notes altered to create the cadence ............... 143

3‘32. Brief areas of harmonic convergence in Study No. 17: (a) entrance of section
C (J = 230) in middle voice (p. 5); and (b) entrance of section B (J = 138) in
bottom voice (p. 7) ................................................................................................ 143

3‘33. Score sections from Study No. 43 showing approach to first CP, tempo

switch, and final CP............................................................................................... 145

3‘34. “Collective effects” in Study No. 43 (score, p. 11) ............................................... 147

3‘35 . Convergence points in Study No. 33 (“Canon V2/2”): (a) CP 4, (b) CP 6, and

(c) CP 7 (pp. 12, 25, and 52, respectively, of the score) ....................................... 149

3‘36.

3-37.

Structure of Study No. 24 (“Canon 14/15/ 16”; modified from Gann 1995, 23)...150

Study No. 24 (portions of second and third systems of p. 12), showing CPS 8
and 9 ...................................................................................................................... 152

xviii

3-38.

Structure of rhythmic convergence area in middle of Canon 10, Study No. 24152

3-39.

Structural diagram of Study No. 37 (modified from Gann 1995, 26), showing
the twelve canons and five convergence points .................................................... 153

3-40.

CP4 from Study No. 37 (p. 40 of score). End of canon 6 is shown, where

tempos are arranged in groupings analogous to interlocking diminished

seventh chords. Canon 7 begins at CP4, which is a sixty-fourth rest at

beginning of 4 measure; a tempo switch in some voices also takes place at

3-41.

3-42.

this CP ................................................................................................................... 155

First section of Study No. 8, showing convergence points (vertical
connecting lines), whether durational series are accelerating or decelerating,
and pitches at CPS.................................................................................................. 156

Formal diagram of Study No. 27 (modified from Gann 1995, 161).
Percentages of acceleration (A) or deceleration (R, for ritardando) are given
for each voice. Gann’s diagram has been modified to show convergence
points and the locations of the eleven canons; considerable adjustments have
been made to voice entrances and endings to better reflect relative relation-
ship of voices ......................................................................................................... 159

3-43.

The levels of imitation between adjacent canonic voices and order of
entrances of the eleven canons of Study No. 27.................................................... 161

. The first three lines of the clock line in Study No. 27, showing limited pitch

content and proportional notation of the tempo .................................................... 162

3-45.

3-46.

Canon 1 of Study No. 27 in the lowest (A6%IR6%) voice, in rhythmic
notation showing relative rhythmic values............................................................ 165

Section from Canon 1 of Study No. 27 (p. 3), showing two inaudible CPS and
their leader-follower switches, pitch symmetry about these switches, and
near-synchrony at second switch. (Rests in bottom Staff of middle and
bottom systems have been added after notes FI and El as the onrission of
these rests were errors in the score.) The longest echo distance is reached
halfway between the two leader-follower switches; selected echo distances
are shown by circled notes connected with a line ................................................. 166

. Pitch content of Canon 2, Study No. 27. Chord at CP is symmetrical about

the D# in the clock line.......................................................................................... 167

3‘48

. Canonic line of Canon 3, Study No. 27 ................................................................. 169

349,

Canon 4 of Study No. 27, from lowest voice (A11%/R11%) ............................... 170

xix

3-50.

Segment of Canon 4, Study No. 27 (p. 16, second system), at entrance of 5%
voice. Measurements from the score are given in millimeters for each five-
note grouping (the example is reduced from the score) ........................................ 171

3-51.

Passage of near temporal convergence in Canon 5 of Study No. 27 (p. 22,
second system). At R11% (top line), tempo is ca. 880 while at R8% (fifth
line), tempo is ca. 440 ........................................................................................... 172

3-52.

Canon 6 of Study No. 27 in the lowest (R8%/A8%) voice ................................... 173

3-53.

3-54.

3-55.

3-56.

4-1.

4-2.

4-3.

44.

45.

Portions of 8% and 11% voices, Canon 7 of Study No. 27 (pp. 32—34 of
score), showing compound melody consisting of repetition of extreme
pitches and chromatic descent/ ascent. (Both lines are doubled in the
score—the figure shows the registral extremes: the top line of the 8%
voice and the bottom line of the 11% voice.) ........................................................ 175

Melody of Canon 8 from lowest voice, Study No. 27, rewritten in rhythmic
notation .................................................................................................................. 176

Melody of Canon 9 from lowest voice, Study No. 27, rewritten in rhythmic
notation .................................................................................................................. 177

Conclusion of Study No. 27 (p. 55). Order of voices from top to bottom of
each system is: clock line (two staves), A11%, A8%, A6%, and A5% ................ 179

(a) A fern frond, a naturally-occurring fractal shape (Solomon 1998); three
different scalings of the frond shape are highlighted, and even smaller
iterations of the shape are present; (b) “fem-like” fractal art (Sprott 1996,
105)........................................................................................................................ 183

The Menger sponge: a fractal in three dimensions, or what Mandelbrot calls a
“spatial universal curve” (Mandelbrot 1977/1982, 145) ....................................... 184

A Peano curve, which illustrates the space-filling property of fractals (Mason
and Saffle 1994, 31). With each further iteration of the curve, the length of
the line drawing the curve approaches infinity ..................................................... 184

“Space-filling curves” from Bamsley (1993), p. 242............................................ 185

A self-similar waveform in which frequency doubling results in no change of
pitch (Schroeder 1991, 96) .................................................................................... 186

Three iterations of a “fractalized” sine wave (Mackenzie 1996, 237) .................. 187

XX

4-7.

4-8.

4-9.

4-10.

4-11.

Typical patterns of white noise, “l/f [pink] noise,” and “Brownian noise”
(Gardner 1978, 21). White noise (spectral density = l/f’) exhibits properties
of extreme randomness within a limited range, with no correlation between
events. “Brownian noise” (spectral density = l/f) exhibits properties of
extreme correlation between events and a tendency to “wander” over the
spectrum. “l/fnoise” (spectral density = 1/f) is moderately correlated ................ 188

Construction of a “right-angle canon” from Lindenmayer curves (Mason and
Saffle 1994, 32—33). The method involves reading the horizontal lines of
curves as durations of notes and the vertical lines as pitch intervals between
notes. In this example, the smallest line segment represents either one scale
step (if vertical) or one sixteenth note (if horizontal). If the curve begins with
a horizontal line, the pitch of the first note is assumed to be the first note of
the chosen scale or mode (C major in this example); only the duration for the
first note is taken from the curve. If the curve begins with a vertical line, the
first note can be read as the number of forward moves up or down from the
first note of the given scale or mode. Curve (b) is a 90° counterclockwise
rotation of (a) ......................................................................................................... 189

Properties of self-iteration, scaling, and space~filling in a musical segment
(Dodge and Bahn 1996, 190)................................................................................. 190

(a) A Cantor comb, and (b) a musical phrase division that corresponds to the
Cantor comb (Solomon 1998) ............................................................................... 190

Illustration of derivation of dimension via self-similarity (Connors 1994). In
the line segment in (a), four self-similar pieces are each ’3 the size of the
original, i.e., 4 = 41 pieces; the square shown in (b) consists of 16 self-similar
pieces with sides ’1 the size of the original, i.e., 16 = 42; and the cube in (c)
consists of 64 self-similar pieces with sides ’3 the size of the original, i.e.,
64 = 43 pieces. In each of these simple cases, the exponent gives the
dimension; thus, the dimension of (a) = 1, of (b) = 2, and of (c) = 3 .................... 192

4-12.

The Sierpinski triangle, DH = 1.58 (Frame 1996, 39)............................................ 193

4~13

. The Cantor gasket, DH = 1.89 (Schroeder 1991, 179) ........................................... 194

4‘14.

Construction of the Cantor comb (after Barnsley 1993, 44). The line in each
previous iteration is divided into thirds, with the middle third left blank; the
scaling factor is always 3....................................................................................... 195

4~ls

. Diagram of Study No. 14 (“Canon 4/5”), showing pitch ranges of two

canonic voices and tempo/duration relationship. Voice 2 is compressed in
time but not in pitch............................................................................................... 197

xxi

4-16.

(a) Hypothetical tempo canons of duration ratio 2:3:4:5:6 (left side of
diagram) and 3:4:5:6 (right side); fractal dimension (D) is calculated by
dividing log of number of scaled objects (including the generator) by the log
of the total of the relative dimensions. (b) D = 1.26 [log(4)/10g(3)] also
describes the Koch curve, in which each line segment is replaced by a
segment 4/3 the size of the original ....................................................................... 199

4—17.

Diagram of Study No. 18 (“Canon 3/4”); DH = log(2)/log(1.75) = 1.24 ............... 200

4-18.

Diagram of Study No. 19 (“Canon 12/15/20”); DH = log(3)/log(2.4) = 1.25 ........201

4-19.

4-20.

4-21.

4-22.

Diagram of Study No. 36 (“Canon 17/18/19/20”). Recall that in this canon
the CP is slightly past the canonic midpoint (see Figures 3-5 and 3-13). DH =
log(4)/log(3.69) = 1.06 ..........................................................................................202

Diagram of Study No. 31 (“Canon 21/24/25”). The canon melody is divided
into parts A, B, and C, each separated by eight measures; interval of
imitation in section A is a perfect fifth, and in sections B and C a perfect
twelfth. The duration ratio in each section is 168:175:200; DH =
log(3)/log(2.72) = 1.10 .......................................................................................... 203

Structural diagram of Study No. 32 (“Canon 5/6/7/8”); DH = log(4)/log(3. 17)
= 1.20..................................................................................................................... 205

Proportional dimensions of Study No. 32 (duration ratio = 105: 120: 140: 168)
showing sections 1, 2, and 3 in each voice and beat delay to entrance of later
voices. Entrance of section 2 in third voice nearly coincides with beginning
of section 3 in first (bottom) voice (shown by dotted line) ................................... 207

4-23.

Structural diagram of Study No. 32 (“Canon 5/6/7/8”) from Gann (1995, p.

185). The alignment indicated at Q) does not really occur.................................207

4~24.

Section 1 (mm. 1—74) of Study No. 32’s canon melody in first (lowest) voice,
identifying motives a throughf, division into three registers, and location of
entrance of second voice .......................................................................................211

4~25

. Canon melody in section 2 (mm. 76—107) ............................................................ 213

4‘26.

Canon melody in section 3 (mm. 109—95) showing registral separation on
three staves ............................................................................................................214

4~27

. Tonal areas represented at entrances of new voices: (a) entrance of voice 2
(p. 1), A major; (b) entrance of voice 3 (p. 2), B major; and (c) entrance of
voice 4 (p. 3), B minor/major ................................................................................ 217

xxii

4-28. Bottom system of p. 8 and top system of p. 9 from score of Study No. 32,

showing extended section of G major beginning in bottom voice with
restatement of a .....................................................................................................219

4-29. Tonal area movement in Study No. 32, from beginning of piece to bottom of
p. 9. Occurrences of motive a (descending major triad) from sections 1 and 3
are plotted (open notes represent rhythmic augmentation) and key areas
related to an enlarged expression of motive e (ascending “partitioned minor
third”). (Proportions are approximate.) .................................................................220

4-30. Motivic combinations using contrary motion: (a) motive b rotating about

pitch D#/FI, and motive e rotating about pitch A (p. 3, top system); (b) voice
exchange involving motives a and b (p. 3, bottom system, top two voices);
and (c) voice exchange involving motive b in contrary motion (p. 4, middle
system, bottom two voices) ................................................................................... 221

5-1.

Description of layers in Study No. 25 (section durations proportionate)..............227

5-2.

5‘3.

Study No. 25, top system of p. 2 from score, showing (IV)—V—I cadence at
second statement of P0 in bottom voice. Vertical dotted lines indicate
arpeggios articulating breaks in the sustained notes of bottom voice. Figure a
is a 16-note arpeggio based on the overtone series, while figure a’ is a 15-
note arpeggio in which the fundamental of the overtone series is stated in the
sustained bottom voice ..........................................................................................228

Figures found in the arpeggio/glissando layer, Study No. 25: (a) figure com-
bining two different triads (score, p. 18, top system); (b) figure of seven
different consecutive triads (p. 34, bottom system); (c) figure of alternating
dominant and major seventh chords (p. 40, bottom system); (d) simultaneous
ascending major and descending Locrian glissando (p. 36, bottom system);
and (e) longer version of same figure (p. 60, top system). All of the examples
use the “exploded drawing” technique where the bracket indicates where on
the home staff the figure belongs (bracket and home staff not shown in [e]) .......230

-

Palindromic organization of sections 1 and 4: (a) opening of Study No. 25 (p.
1, top system); (b) end of section 4 (pp. 32—33). Arpeggios in section 4 are
stated at half the tempo of section 1. Location of convergence point in
section 4’s tempo canon is shown at the beginning of first system in (b).
Notice that arpeggios in (b) are stated at half the tempo of those in (a) ............... 232

5‘5 -

End of section 1, Study No. 25 (score, p. 7, top system). Arpeggio/glissando
and sustained notes layers achieve a level of harmonic coordination prior to
and at the cadence..................................................................................................237

5‘6-

Score segment from section 3 (p. 18, bottom system), showing the second of
two convergences between arpeggio and staccato chord layers............................ 239

xxiii

Rhythmic scheme underlying section 3; convergence period is 7 beats in top
layer (arpeggios) against 10 beats in bottom layer (staccato chords). Only
two of five possible convergences are articulated in both voices ......................... 240

Segment from section 4 (p. 28) where same figure appears in both voices,
giving impression that upper voice is leading although it is not ........................... 241

5-9.

Study No. 25 (p. 48), showing multiple convergences in 9:10: 12:15 tempo
canon in section 6. Rhythmic convergences are circled, and dissonant
intervals within rhythmic convergences identified. Lines indicate the same
major triad stated in different voices ..................................................................... 243

5-10.

Arpeggio figures of chains of major seventh chords, section 7: (a)
chromatically rising roots (p. 57); (b) diatonically rising roots (p. 59) ................. 244

5-11.

Two rapid juxtapositions of keyboard extremes in Study No. 25, section 9:
(a) p. 74, top system; and (b) p. 74, bottom system ..............................................246

5-12.

Lengthy arpeggio/glissando figure from section 5 (p. 39, top of figure)
repeated almost verbatim in section 9 (p. 73, bottom of figure) ........................... 247

5-13.

Comparison of alternating major (in rectangles) and pentatonic (in ovals)
pitch patterns in simultaneous glissando figures from (a) p. 6 and (b) pp.
76—77 of Study No. 25 .......................................................................................... 248

5-14.

Diagram of pitch center progression in Study No. 25 (sections not
proportional) .......................................................................................................... 249

5-15.

5—16.

5-17.

Temporal dissonance in Study No. 3b: use of 8 and 16 meters in upper

6

12

.
three voices against 2 meter in bass ostinato vorce (score, p. 7) .........................256

4

.

.

Five-layer canonic texture from Study No. 3b (score, p. 21). Numeric
patterns indicate sixteenth note rhythmic groupings ............................................. 256

Recurring melodic fragments in Study No. 3e, showing use of major/minor

tonic: (a) score, p. 4; (b) score, p. 7 (this segment has a 8 rhythmic pattern

in the middle line)..................................................................................................258

5‘18.

Rhythmic disintegration in bass line and progressively slower melodic
statement at end of Study No. 3e: (a) p. 21; (b) p. 23; and (c) p. 25 .....................259

xxiv

5-19.

5-20.

5-21.

Passages in No. 45a’s melodic layer that emulate metered rhythm: (top) p. 2,
bottom system; (bottom) p. 3, top system (note the rhythmic coordination
between the top two layers). Rhythmic approximations are shown between
the staves; dotted lines in bass layer indicate divisions of the 15-note rhythm.
In the top example there appears to be a rhythmic simultaneity between the
layers on the third note of both statements in the top layer. In the bottom
example, an area of additive deceleration in the top layers is indicated by the
rectangle ................................................................................................................264

Acceleration between glissandi in final section of No. 45a (score, p. 19). Mi]-
limeter measurements between glissando attacks are given. Acceleration rate
ranges from 4.75% to 6.5% ...................................................................................265

End of tempo canon in No. 45a, stopping just short of convergence (score, p.
14). Circled area indicates break in broken octave pattern of bass line. The
dotted lines between layers indicate an apparent rhythmic simultaneity,
which happens to immediately precede the arrival at 1’3 dominant. Arrows
indicate echo distance............................................................................................ 266

5-22.

Earlier section of 5:7 tempo canon in No. 45a (score, p. 13). Arrow shows
echo distance. Canon is effectively constructed so that layers switch between
triads and single pitches ........................................................................................266

5-23.

Extension of V/V tonal area beyond arrival at V area in No. 453 (score,
p. 11)......................................................................................................................267

5-24.

5-25.

5~26.

5-27.

5-28.

Section in No. 45a’s section 7 (score, p. 15) where upper two melodic layers
are closely coordinated tonally but not tightly coordinated with changes of
transposition level in the bass layer. Rectangle identifies repeating melodic
pattern ....................................................................................................................270

Two instances of a 4-note ostinato against thirds aligning with every third
note (the short note value in each 2 + 3 + 3 rhythmic pattern): (a) ascending
pattern in No. 45b, p. 3; (b) descending pattern in No. 45b, p. 4 ..........................271

Same melodic pattern from Figure 5-25b rhythmically augmented to 3 + 5 +
5 (score, p. 6) ......................................................................................................... 272

Syncopation between two melodic layers in 45b (score, p. 12). Ire refers to
the sixteenth time through the I bass pattern. Note the low incidence of
simultaneities between the bass layer and upper layers ........................................ 272

Alternate transpositions in No. 45b’s bass line pattern: (a) movement to key a
perfect fourth higher; (b) movement to key a major second higher; (c)
movement to key a minor third higher; (d) movement to key a minor third
lower; and (e) movement to key a perfect fourth lower ........................................274

XXV

5-29.

The end of Study No. 45b (score, pp. 17—18). From the GI, area to the end
there is complete tonal coordination among all layers. In the last four key
areas (F, C, A, D) there is a melodic emphasis on the dominant seventh,
culminating in a D9 chord that becomes the dominant to the first harmony of
No. 45c ..................................................................................................................276

5-30.

Melodic uses of the boogie-woogie bass pattern in No. 45c: (a) score, p. 2;
(b) score, p. 6 .........................................................................................................277

5-31.

Recurring melodic motive in No. 45c: (a) first appearance, as a 3-note
pattern, in interlocking form (ascending triads below and descending single
notes above), score, p. 1; (b) expansion to 4-note form, repeating the second
note (as triads and octaves), score, p. 3; (c) as the highest pitch in series of
glissandi followed by sustained polychord (score, pp. 5—6); ((1) preceded by
chromatic glissandi (score, p. 16) ..........................................................................278

5-32.

Division of the “spastic rhythm” among all layers in section 8 of No. 45c
(score, p. 21). The 15-note segments demarcated by the vertical dotted lines
are all the same length. This section is preceded by the last of three tempo
canons .................................................................................................................... 279

5-33.

Dominant seventh polychords in each of No. 45c’s first three phrases: (a) p.
1, first system; (b) p. 1, third and fourth systems; and (c) p. 2, first system .........280

5-34.

Ending of No. 45c (score, p. 23), showing near-coincidence of shifts in tonal
centers with beginning of “spastic rhythm” statements (demarcated by
vertical dotted lines—rhythmic values are still being divided among the
layers) and parallel triadic harmony, culminating in a G dominant polychord
(G major, B major, and D major triads). Circled chords indicate where tonal
shifts occur. Staggered V—I cadences are shown in D, F, and G...........................281

Questions about score accuracy in Study No. 25: (a) first C in scalar passage
is flatted while three others are not (from bottom system, p. 75); (b) scalar
passage apparently in B major in which sharps appear to be missing in
circled area (from top system, p. 76) .....................................................................290

xxvi

CHAPTER 1

INTRODUCTION AND REVIEW OF THE LITERATURE

Conlon Nancarrow (1912—1997) considered time “the last frontier in music”

(Garland 1982, 185). Like a pioneer forging his way across untested territory, Nancarrow

faced numerous obstacles in his struggle to find a performing medium that would faith-

fully reproduce his bold musical concepts without the limitations of human performers.

And, like most pioneers, Nancarrow seemed guided by two old proverbs—“where there’s

a will, there’s a way” and “necessity is the mother of invention”—which speak to the re-

sourcefulness that ultimately led him to the seemingly odd combination of the player

piano with an esoteric experimental music rich in rhythmic complexity. The results have

irrevocably stretched the boundaries of human perception and musical thinking, and

forever altered the landscape of time in music.

Nancarrow’s almost fifty Studies for player piano represent an extraordinary body

0f work in their focus on a single performing medium (the player piano), in their ex-

Ploitation of that medium’s unique expressive abilities, and in their tremendous diversity

that belies a long evolutionary development over time. I can think of no other body of

WOrk, musical or otherwise, that consumed an artist for so long in so thoroughly ex-

Ploring the expressive limits of that medium, and so absorbed his attention that he

Completely turned away from writing for other instruments until very late in life. Close to

fifty years were devoted to these almost fifty pieces, involving hundreds of thousands of

hOIes punched laboriously by hand on large rolls of paper to create works that fly by in a

tflatter of minutes—all to realize the composer’s intentions in the purest way possible,

directly from his hands to the player piano with no performer in between. The

achievement is even more extraordinary considering the almost complete isolation in

which the Studies were produced; after moving to Mexico in 1940, Nancarrow remained

(by choice) very much an “underground” composer until his reputation reached the

mainstream in the 19805.

In some ways it is understandable that Nancarrow’s work has not received the

attention that it deserves. It is perceived by some as being melodically and harmonically

unsophisticated (for example, by Pierre Boulez, who found Nancarrow’s pitch vocabulary

disappointing compared to his innovative rhythmic structures [Gann 1995, 10—11]). Also,

the unusual nature of its performance, in which no live performers are involved, has

limited the programming of this music on contemporary music concerts. In comparison to

other “performerless” music such as electronic music, music for player piano has been

even more rarely programmed. Few people have heard a “live” performance of a Nan-

carrow player-piano piece, and his music has been mostly disseminated through

recordings.

James Tenney, writing in 1977, found the continuing obscurity of Nancarrow’s

work to be “nothing short of scandalous” (Tenney 1977, 41). In 1986, Carlsen observed

(in the first dissertation devoted to Nancarrow’s music) that the critical literature on the

Studies for player piano remained very small. Although Nancarrow’s work has become

itlCreasingly well-known (due partially to his selection in 1982 as one of the first class of

l'fi‘rCipients of the MacArthur Foundation “genius grants,” and the release in 1988 and

1989 of most of the Studies on recordings by Wergo), this deficit in the critical literature

has been only partially addressed with the publication of a major analytical book by Kyle

Gann (1995), a 1996 dissertation by Margaret Thomas, and a small number of articles

alld reviews.

The present dissertation is intended to expand the critical literature by examining

Specific structural features native to the Studies. These features include the pervasive use

Of mathematical ratios, the structural use of convergence points in the canonic Studies,

 
and  the  presence  of  “fractal  forms”  in  certain  of  the  canonic  Studies.  In  Chapter  2,  the 

examination  of  the  use  of  ratios  in  the  Studies  identifies  Nancarrow’s  favored  ratios  and 

their  relationship  to  the  Fibonacci  series  and  pitch  ratios  in  the  justly-tuned  chromatic 

scale.  Uses  of  ratios  to  control  various  parameters  are  identified,  including:  tempo  rela- 

tionships,  rhythmic  motives,  melodic  and  harmonic  materials,  and  structure.  The  presence 

of  ratios  is  the  focus  of  a  comprehensive  analysis  of  Study  No.  34. 

The  next  two  chapters  are  concerned  with  features  of  Nancarrow’s  “tempo  canons”: 

canonic  pieces  in  which  the  various  voices  present  canonic  material  at  proportionally- 

related  speeds.  Chapter  3  examines  the  structural  use  of  convergence  points  in  the 

canonic  Studies;  convergence  points  are  a  prominent  feature  of  these  Studies,  because  the 

canonic  voices  are  either  converging  or  diverging  most  of  the  time.  The  structural  impact 

of  the  convergence  points,  and  techniques  for  emphasizing  and  de-emphasizing  them,  are 

the  focus  for  this  chapter.  The  chapter  concludes  with  an  analysis  of Study  No.  27. 

Chapter  4  identifies  fractal  features  in  Nancarrow’s  tempo  canons,  particularly 

formal  features  in  which  the  same  musical  material  is  presented  at  different  tempos  to 

create  large-scale  “fractal  forms.”  Perhaps  the  most  compelling  canonic  fractal  forms  are 

created  in  the  canons  whose  overall  form  is  either  converging  (where  just  one  con- 

vergence  point  occurs  at  the  very  end  of  the  piece)  or  converging-diverging  (an  arch  form 

where  a  single  convergence  point  occurs  somewhere  in  the  middle  of  the  piece).  Study 

No.  32,  an  example  of  the  former,  is  analyzed  in  detail. 

Chapter  5  is  devoted  to  the  analysis  of  two  of  Nancarrow’s  later  works  that  contain 

some  of  his  more  mature  compositional  techniques:  Studies  No.  25  and  45.  The  analysis 

of  No.  25  focuses  on  the  use  of  the  layering  technique,  while  the  analysis  of  No.  45  draws 

comparisons  with  a  much  earlier  piece  (Study  No.  3)  and  particularly  examines  the 

presence  of  multidimensionality. 

Chapter  6,  the  final  chapter,  presents  conclusions  and  offers  suggestions  for  further 

research  into  the  Studies. 

The  remainder  of  Chapter  1  continues  with  a  brief  biographical  summary  and  an 

examination  of  the  four  major  analytical  sources  on  Nancarrow’s  Studies:  Carlsen,  Gann, 

Thomas,  and  CD  liner  notes  and  an  article  written  by  James  Tenney.  The  chapter  then 

introduces  the  reader  to  general  characteristics  of  the  Studies,  including  rhythm,  pitch, 

harmony,  structure,  and  texture;  notational  features  unique  to  Nancarrow’s  Studies  are 

also  examined.  Finally,  the  analytical  treatments  of  four  Studies  (Nos.  8,  19,  23,  and  35) 

found  in  each  of  the  four  major  sources  are  compared. 

Biographical  Summary  and  Review  of  Existing  Literature 

Conlon  Nancarrow  was  born  in  Texarkana,  Arkansas  on  October  27,  1912,  and  died 

August  10,  1997  at  his  home  in  Mexico  City.  His  childhood  was  musically  unexceptional, 

although  it  is  noteworthy  that  the  home  contained  a  player  piano  (about  which  Nancarrow 

said,  “I  was  fascinated  by  this  thing  that  would  play  all  of  these  fantastic  things  by  itself” 

[Gagne  and  Caras  1982,  292]).  Young  Conlon  took  trumpet  lessons,  and  by  the  time  he 

completed  high  school  he  was  good  enough  to  get  steady  work  as  a  jazz  trumpet  player. 

Nancarrow’s  advanced  music  education  consisted  of  a  single  semester  (not  four 

years  as  is  sometimes  reported)  at  Cincinnati  College-Conservatory  of  Music,  where  he 

ostensibly  studied  composition  but  neither  made  an  exceptional  mark  nor  was  profoundly 

influenced.  It  was  here,  however,  that  he  first  heard  Stravinsky’s  Rite  of  Spring,  which 

made  a  deep  and  lasting  impression  on  him.  Years  later  he  pointed  to  this  experience  as 

life-altering  and  identified  Stravinsky  and  Bach  as  his  favorite  compasers  (Gagne  and 

Caras  1982,  282-83).  Nancarrow’s  exposure  to  Bach  apparently  came  about  through  his 

private  study  in  counterpoint  with  Roger  Sessions  in  Boston,  where  he  moved  after 

leaving  Cincinnati.  This  was  unquestionably  his  most  rigorous  compositional  training. 

While  in  Boston,  Nancarrow  also  became  acquainted  with  Walter  Piston  and  Nicolas 

Slonimsky.  These  relationships  began  under  the  guise  of  studying  composition,  but  ac- 

cording  to  Nancarrow  neither  man  taught  him  much  about  composition.  As  Nancarrow 

said about Slonimsky,

Well, with him it wasn’t really studying. I went to see him a few times,
showed him a few pieces of music, and he commented on them. It wasn’t
studying because he didn’t teach; he was just saying, “Well, this is good, this
is bad.” .

.

.

This “studying” didn’t go on for very long. I finally got to be friendly with
him, and I would go to his house for dinner, and the studying sort of drifted
off into the background. (Gagne and Caras 1982, 283—84)

Nancarrow’s experience in working with Piston was similar in that he shared a few

of his pieces with him but they eventually became just occasional dining companions.

It was in Boston that Nancarrow became affiliated with and joined the Communist

Party. In an extraordinary turn of events, Nancarrow (as one of few musicians in the

Party) was asked to organize a Lenin Memorial Concert; what is truly extraordinary is

that the Party was able to book Boston Symphony Hall for this event and Nancarrow

managed to convince Piston to not only program his “Sonata for Oboe and Piano” but to

perform the piano part. (This gesture is quite a testament to Piston’s regard for Nan-

carrow; the administrative powers at Harvard let Piston know in no uncertain terms that

he should never do such a thing again.)

In 1936 Nancarrow pulled out his trumpet and embarked upon a two-month playing

tour in Europe. He returned briefly to the United States before signing up with the

Abraham Lincoln Brigade to spend two years fighting the Fascist Franco regime in the

Spanish Civil War. Upon his return to the United States in 1939, Nancarrow began to

experience governmental harassment for his political affiliation, and as a result of this he

moved to Mexico City in 1940. He became a Mexican citizen in 1956. In interviews in

his later years he continued to express exasperation with the US. government; for a

number of years after Nancarrow became a Mexican citizen he was considered an “un-

desirable alien” and unable to get a US. visa. He was finally able to travel to the US. in

1981, and at last began hearing his music performed at new music festivals and receiving

Commissions and requests to serve as composer-in-residence. The receipt of the

 
MacArthur grant in 1982 solidified his new-found fame (as well as easing his financial

worries), and during the last fifteen years of his life he travelled quite frequently to the

States.

Even before he moved to Mexico, Nancarrow had become personally acquainted

with or known to a number of important twentieth-century musical figures. Besides

Sessions, Piston, and Slonimsky, Nancarrow’s music was known as early as the 19308 by

Elliott Carter, Aaron Copland, and Henry Cowell. It was Slonimsky who was responsible

for introducing Nancarrow’s earlier music to a larger audience. While Nancarrow was

away fighting in Spain, Slonimsky—without Nancarrow’s knowledge—submitted several

of his pieces (Toccata for violin and piano and two pieces for piano: Prelude for Piano,

and Blues) to New Music, which published them in 1938. Nancarrow’s first player piano

piece, Rhythm Study No. I, was also published without his knowledge. At some point he

had given a copy to Carter, and in 1951 Carter arranged for the piece to be professionally

copied and subrrritted it to New Music. Nancarrow learned of this by chance some five

years later!

The Player Piano Studies

Nancarrow’s player piano output includes forty-nine numbered studies.1 The most

comprehensive source to date on Nancarrow’s music is Kyle Gann’s The Music of

Conlon Nancarrow (1995). The other major sources of analytic information on Nan-

carrow’s Studies are the liner notes by James Tenney for the five volumes of recordings

produced by Wergo, and dissertations by Philip Carlsen (1986) and Margaret Elida

Thomas (1996). These are the main sources that will be reviewed and compared in this

discussion of the literature.

1

n

e Music 0

onlo Nancarrow. Gann’s book includes information

Ll‘he Studies are actually numbered through 51, but Nos. 38 and 39 were re-numbered by Nancarrow as
Nos. 43 and 48, respectively.

 
 
based on personal interviews with Conlon Nancarrow in Mexico City from the late 19803

until 1993, where he was also given access to Nancarrow’s studio with the original

punching scores2 and the piano rolls. Gann also was able to interview or contact a

number of other people in Nancarrow’s life, including his first and second wives and his

brother. With this information he is able to paint a vivid and illuminating portrait of the

composer’s early life and influences. In addition, examination of both the punching

scores and the piano rolls provided Gann with highly useful information about the con-

struction of each Study, and allowed him to analyze each in considerable detail without

relying completely on Nancarrow’s hand-drawn scores.

Gann looks at each piece in Nancarrow’s entire output (including his smaller output

of “live performer” pieces) and analyzes each in brief analytical synopses that range from

somewhat superficial to fairly comprehensive. Prior to this, however, the book contains

useful chapters discussing the general characteristics of Nancarrow’s music (a general

discussion of these characteristics will be given in the second part of this chapter) and a

well-informed biography. Gann’s discussion of musical characteristics gives the reader a

thorough understanding of the relationships in Nancarrow’s music between tempo, rhyth-

mic ratios, pitch ratios, and the harmonic series, and the terminology associated with

tempo canon. A particularly useful feature of the first chapter is the “Morphology of

Nancarrow’s canons” on pp. 22—27, which graphically illustrates the formal structures of

most of the canons (except for No. 43, which apparently was inadvertently omitted). He

also discusses melodic and harmonic tendencies and the special kinds of notation used in

Nancarrow’s scores.

Because “there is no wholly satisfactory order in which to discuss the Studies for

Player Piano,” Gann discusses the works by creating the following taxonomy, which is

“based on the primary compositional technique of each study” (p. 69):

2The “punching score” is a graphically drawn score on which Nancarrow plotted out a piece before
punching the piano roll. He described it as “pretty much illegible for anyone else" (Duckworth 1995, 47).

 
 
 
 
1. “Blues years: the ostinato studies” covers the early Studies No. 1, 2a, 2b, 3 (a—e),

5, and 9. Gann notes that:

When Nancarrow began experimenting with polytempo, he turned first to a
simple device that would Show it to striking effect: ostinato, a phrase persis—
tently repeated without variation. It was a natural choice, for ostinato was
much in the air in modernist music of the 19205 and 303. (pp. 69—70)

Gann points to Stravinsky (especially Rite of Spring) and Barték as being partic-

ularly influential on these early pieces, which are also heavily influenced by Nancarrow’s

background as a jazz trumpeter. The I—IV-V harmonic and phrase structure is pervasive

here, and Nancarrow builds several of these pieces by transposing large sections of

musical material to the subdominant and donrinant levels in an obvious blues pattern. He

also already begins exploiting the ability of the player piano to perform polytempo by

building in underlying rhythms that are in ratios such as 4:7 (Study No. 1), 4:5, 4:7, 5:7,

5:9, and 10:12:15z20 (Study No. 2a), and 2:3:5 (Study No. 3). Further, Nancarrow begins

creating series of events that use arithmetical acceleration or deceleration: for instance, a

series of notes with durations of 3, 4, 5, 6, 7, 8, and 9 beats to effect a ritardando. This is

a technique that will recur in many later studies.

2. “Isorhythm: the numbers game” examines Studies No. 6, 7, 10, 11, 12, and 20.

Gann points out that in the 19408 and 503 Nancarrow was collecting Indian music, and

“analyses of the isorhythmic works will make Nancarrow’s affinity for Indian rhythmic

thinking obvious, whether he drew on the influence consciously or not” (p. 85). In ad-

vancing from ostinato to isorhythm, it is important to note, as Gann says, that:

There is a fine line between ostinato and isorhythm, since an ostinato re-
iterates the same rhythm as well as the same pitch, and the studies discussed
here form an admittedly heterogeneous group. What they do have in common
is some internal dissociation of rhythm and pitch [emphasis mine], a use of
talea independent of color.3 (pp. 86—87)

3In the terminology associated with isorhythm, talea refers to the isorhythm‘s repeating rhythmic pattern,
while color is the repeating melodic pattern. They are often of different lengths and consequently out-of-
phase with each other.

 
 
Gann’s discussions of the internal workings of these pieces are richer in detail than

those in the first section—the longest discussion of the blues-ostinato pieces is that con-

cerning Study No. 3, which covers the five movements of this study in just over five

pages; the discussions of the isorhythm studies range from 2 1/2 to 7 pages for each. Gann

focuses much attention on the isorhythmic structures of the studies and their applications

to themes and structures. In several places, he acknowledges the medieval origins of iso-

rhythm by comparing Nancarrow’s technique to that of composers of that time period, as

he does for Study No. 11, and uses terminology more consistent with that time period,

such as the word “chant” in the discussion of Study No. 12 to refer to modally-inflected

themes. Interestingly (and somewhat disappointingly), no further mention is made of

Indian music or its relation to the studies discussed in this section.

3. “Canon: phase 1” concerns itself with Studies No. 4, 13, 14, 15, 16, 17, 18, 19,

26, 31, 34, 44, 49, and 50. Gann points to these studies as being “the ones in which con-

trast of perceptibly different tempos is the primary focus” (p. 111). The studies discussed

in this section combine the use of isorhythm with another old and familiar device: canon.

Continuing the development of polytempo, Nancarrow creates a new kind of “tempo

canon” by establishing increasingly complex tempo ratios between the canonic voices.

Gann continues from the previous chapter the analogy to medieval and Renaissance

music, tracing Nancarrow’s tempo canons back to the prolation canons of composers

such as Josquin des Prez, but identifies Nancarrow as “the first explorer of the true tempo

canon” (p. 1 1 1).

In this section, Gann begins a practice—continued throughout the book—of using

pitch analogies to relate tempos to one another, such as referring to a tempo that is twice

as fast as another tempo as being an “octave” or multiple of the first tempo.4 Gann

thoroughly documents here the rich diversity of tempo ratios created by Nancarrow in

4Readers who are not thoroughly conversant with the nature of pitch ratios in the just intonation system
will find much helpful information on Kyle Gann’s Web site at <http://home.earthlink.net/~kgann/tuning.
html>.

 
these Studies.

The pieces discussed in this section abound with numerical series such as 2 + 3 +

4 + 5 + 4 + 3 (Study No. 4) and the formula {n - l, n, n + 1, n}, which is expressed in

various manifestations in Studies No. 13-19. The series are used rhythmically in two

ways: to construct isorhythms in individual voices, and the shorter patterns such as

{n- 1, n, n + 1, n} are used to create larger rhythmic resultants, such as the 336-note

resultant used in most of Studies No. 13—19. Gann’s identification of occurrences of

numerical series in these pieces is quite thorough.

As is to be expected, Gann also devotes much attention in this section to identifying

canons, their echo distances and pitch intervals of imitation. He also notes the location of

convergence points and summarizes the canonic forms, which range from the standard

types identified by Thomas (such as converging, diverging, converging-diverging [arch],

etc—see Figure 1-17, below) to the more unusual forms that result from tempo switches,

such as that in Study No. 17.

4. “Stretching time: the acceleration studies” deals with Studies No. 8, 21, 22, 23,

27, 28, 29, and 30 and begins with this wonderfully perceptive comment: “None of Nan—

carrow’s achievements is more original, more uniquely his own, than the sense of curved

time [my emphasis] he has created by means of long, slow, smooth acceleration and

ritardando” (p. 146). Gann begins this section by introducing and comparing the two dif-

ferent types of acceleration/deceleration: arithmetical and geometric.5

Gann identifies the presence of acceleration/deceleration in two ways, depending on

whether the process is arithmetical or geometric. For arithmetical changes, either the note

value changes incrementally (e.g., effecting an acceleration by decreasing each note value

by the same note value—say a sixteenth note—from the previous value), or, in the cases

SNancarrow did not have geometric acceleration available to him until Study No. 21. Until that point, his
punching machine had a notched mechanism that allowed only certain increments to be punched on the
roll: Nancarrow had the machine rebuilt to allow any increment to be punched. See Carlsen (1988) p. 4 and
Duckworth (1995) pp. 47—48.

10

 
 
 
where Nancarrow uses proportional notation rather than note values, either the same

invariable measured unit is added or subtracted, or the change is created through a

numerical series such as 10, 9, 8, .

.

.

, 2,

l (a “countdown”). Geometric accel-

eration/deceleration is always notated with proportional notation, because it cannot be

represented with conventional note values. Whenever proportional notation is involved,

Gann identifies the rates of change by carefully measuring distances on the score(s)

and/or piano rolls, usually in millimeters. Geometric acceleration in Nancarrow’s Studies

is often indicated by a percentage value; for instance, the subtitle to Study No. 27 is

“Canon 5%/6%/8%/11%.”

Although all of the Studies in this section feature acceleration/deceleration in some

way, many continue to exhibit features discussed in earlier sections—such as isorhythm

and canon—and Gann continues to point out these features. But the focus here is on rates

of change.

5. “Beyond counterpoint: the sound-mass canons” concentrates on Nancarrow’s

later canons: Studies No. 24, 32, 33, 36, 37, 40, 41, 43, and 48. AS the chapter name

implies, these canons are notable for achieving large sound effects, and Gann suggests

that one is more likely to hear mass effects and structural points in these canons than

individual voices. In this chapter he notes several differences between the earlier and later

canons:

(a) The later canons exhibit considerable concern for structural placement of con-

vergence points (CPS) and their climactic effects, whereas in the earlier canons CPS more

often amounted to just a “collision of voices” (p. 173). Gann points out the tendency in

the later canons for Nancarrow to use longer phrases further from the CPS and pro-

gressively shorter gestures as the CP approaches, and for the placement of CPS to affect a

greater variety of details throughout the canon.

(b) In the later canons, Nancarrow seems less concerned about clear audible distinc-

tions between extremely close ratios. Whereas the early canons primarily were restricted

11

to ratios involving 3, 4, and 5, these later canons use ratios as close as 24:25 (No. 43) and

60:61 (No. 48) and even irrational numbers such as ‘12:2 (No. 33) and em (No. 40). Gann

suggests that it was no longer necessary to create audible tempo distinctions because

“here Nancarrow is not illustrating tempo differences, he is using subtleties of tempo to

create forms and textures that had never been heard before, and which could have been

created no other way” (p. 175).

(c) In the later canons (specifically Nos. 40, 41, 43, and 48), Nancarrow favors a

major tenth level of imitation in two-voice canons, that—coupled with his preference for

the melodic minor third—creates a “bittersweet and ambiguous major/minor harmony”

(p.203).

This chapter contains some of Gann’s most lengthy and involved analyses; he

devotes seven pages each to Nos. 24, 37 and 40, eleven pages to No. 41 (the piece that is

also the most extensively analyzed by Thomas), and—in his most extensive review—

sixteen pages to No. 48 (a 3-movement piece in which the third movement is the first two

movements performed together). In his analyses of these canons, Gann continues to

identify occurrences of ratios between voices and pays special attention to the placement

of CPS. About Study No. 37, he notes that “Convergence points are brilliantly de-

emphasized in this study, for this is the work in which Nancarrow learned how to create

beautiful effects with convergence points by omitting them” (p. 195). Later, about Study

No. 41a, he points out the presence of a false CP, something highly unusual for

Nancarrow.

6. “Synthesizing a language,” the final section, covers Studies No. 25, 35, 42, 45,

46, 47, and 51. These are the pieces in which Gann believes Nancarrow is “synthesizing a

heterogeneous language, reaping the benefits of decades of experimentation, and starting

to combine structural ideas into new hybrids” (p. 240). Of the first piece discussed, Study

No. 25, Gann says “no other composition is such a treasure chest of every type of idea

Nancarrow has worked with” (p. 241).

12

 
 '

L
—

Although Nancarrow continued to make new discoveries and use new techniques in

these pieces (e.g., the use of a tone row and the sustain pedal in Study No. 25, the

“spastic” irrationally-derived ostinato rhythm in Studies No. 45-47), he relies heavily on

the techniques of canon, isorhythm, tempo ratios, and acceleration that have served so

well. Gann points out that, while none of the works in this chapter are canons, most of

them contain canons (p. 240).

In these works Nancarrow is at the height of his creative powers and in complete

command of his compositional tools—Gann remarks that “By now he is so expert at his

favorite devices that isorhythnric canons, accelerating canons, isorhythmic ostinatos, even

isorhythmic canons of arpeggiated ostinatos, appear almost effortlessly” (p. 240). These

devices reach unprecedented levels of complexity, with tempo ratios as complex as the

144:182:351:468:585:624:819:936:1638 ratio that results in Study No. 46, and iso—

rhythms reaching extreme lengths, such as the 99-note recurring isorhythm built entirely

of eighth notes and dotted eighths (values of 2 and 3 in mathematical terms) in Study

No. 47.

Phili

arlsen The Flu er—Piano Music 0

onlon Nancarrow: A Anal sis 0

Selected Studies. Prior to Gann, the most thorough treatments of any of Nancarrow’s

Studies were to be found in Carlsen’s dissertation, which was completed in 1986 with a

monograph of the same title published in 1988. Unless otherwise noted, all discussion in

this study of Carlsen’s contributions will refer to the 1988 monograph.

Carlsen analyzes five Studies selected from the first published collection: Conlon

Nancarrow: Selected Studies for Player Piano, published by Soundings Press in Spring—

Summer 1977 as Soundings, Book 4. The Studies he chooses to examine are Nos. 8, 19,

23, 35, and 36. His treatment of four of these (Nos. 8, 19, 23, and 35) will be reviewed

more thoroughly in the last section in this chapter, where the four major sources are com-

pared and contrasted. Thus, unlike Gann and Thomas, Carlsen establishes no taxonomy

l3

 
 
for the review of the Studies because the five works selected comprise a “convenience

sample.”

In Carlsen’s preface and introductory chapter he includes much useful background

information. He characterizes Nancarrow’s scores as “extremely accurate in their spatial

representation of rhythm and tempo” (p. ix), and points out that even the scores that are in

metered rather than proportional notation use Spatial representation of elapsed time: i.e.,

note attacks are carefully placed on the staff to represent the exact time at which the

attack occurs. He comments on Nancarrow’s view toward performers and how he

actively sought to eliminate the performer after enduring several disastrous performances

of his music. The player piano itself is examined in terms of how it was popularly per-

ceived—particularly by Stravinsky and Hindemith, who wrote for it, and by Cowell, who

saw great potential in it—and how the mechanical characteristics of this instrument

affected Nancarrow’s compositions for it. For instance, the paper roll uptake mechanism

of the player piano introduces a naturally gradual accelerando as a piece progresses and

the amount of paper on the uptake roll increases. Nancarrow actually did not mind this

effect and likened it to the effect of “long, rhythmic African drum performances” (p. 69,

note 15).

Carlsen structures his dissertation to focus most thoroughly on Study No. 19:

The present study does not attempt to survey Nancarrow’s entire output or, for
that matter, to deal comprehensively with more than a handful of his compo-
sitions. A large portion of the analysis will be devoted to a Single short work,
Study No. 19. One may argue that it is not among the greatest of Nancarrow’s
compositions, but it provides many insights into his overall compositional
style. Analysis of Study No. 19 will set the stage for examination, in slightly
lesser detail, of several other works, concentrating on their unique features but
relating them to other studies. (p. 5)

Carlsen’s analyses are more comprehensive than most of Gann’s and focus on a

number of key areas. He calls canon “the predominant structural device in [Nancarrow’s]

music” (p. 18), yet only one of the Studies that Carlsen selects (No. 19) is truly canonic,

so this is necessarily a limited topic of discussion in his analyses. He expresses surprise

l4

 
that the literature to that point had virtually ignored pitch elements in Nancarrow’s music;

it is not unexpected, then, that Carlsen himself examines this aspect of each Study. In the

process he identifies recurrent motives (with the two most prominent involving the

ascending fourth and the “partitioned” minor third, i.e., a minor third partitioned into a

whole step and a half step) and even melodic segments that appear in more than one

work. Long-range harmonic goals are also identified in some pieces, and Carlsen points

out that the pitch levels representing 1, IV, and V are frequently present.

Symmetry of design on different levels is a frequent topic for Carlsen’s observa-

tions. In two of the Studies (19 and 36), Carlsen observes that the player piano’s central

note, E4, was the axis of symmetry for the deployment of the voices. Small-scale sym-

metry is observed in mirrored rhythms (No. 35) and in mirrored pitch patterns (No. 8). In

Study No. 35 he found formal symmetry among the sections of the piece. Carlsen also

identifies correlations between the partitioning of musical “space” and ratios found in

common intervals. In Study No. 36, he observes the prevalence of the major seventh

chord as a structural device—its presence in determining the levels of imitation calls

attention to the major seventh chord itself as a symmetrical structure, consisting of major

thirds on the top and bottom separated by a minor third in the middle. Carlsen also refers

in this same Study to a theme whose “second appearance is twice as fast as the first; it

occurs at a point .

.

. exactly halfway between the beginning and the canonic midpoint,

providing an obvious parallel with the well-known acoustical fact that the halves of a

string vibrate twice as fast as the whole” (p. 30). In his analysis of No. 8, Carlsen makes

the following observation:

When the second voice enters, it does so at a point exactly halfway through
the first voice’s isorhythmic pattern; the third voice enters exactly three-
quarters of the way through the pattern .
. The positioning of these en-
trances is specifically related to the divisions of successive intervals in the
overtone series. Thus, the entrance of the second voice exactly halfway
through the pattern parallels, in the overtone series, the division of the octave
(2:4) into fifth (2:3) and fourth (3:4); by the same token, the further sub-
division of that half into quarters parallels the subdivision of the fifth (4:6)

.

.

15

 
into major and minor thirds (4:5 and 5:6). Obviously, it is no coincidence that
the bass voice is characterized by octaves, the next by fifths, and the highest
by minor thirds:

fifth

minor third

fl /—\
6
4
2

3

5

:

:

:

:

octave

fifth

(p- 43)

Carlsen obviously had great confidence in the intentionality with which these structures

were created, and the arguments he puts forth in support of this view are both intriguing

and persuasive.

In Carlsen’s final chapter, he reflects in broad terms on what he calls “the moti—

vating principles behind the music” (p. 66). The player-piano medium itself is one of

these. Carlsen observes a fortuitous compatibility between the instrument and Nancar-

row’s interest in matters numerical and spatial:

The [piano] roll is highly conducive to the use of exact measurements, tem-
plates, replications, and geometric constructions; thus, to a certain extent, it
exerts an influence that is purely visual and mathematical. Nancarrow, with
his mathematical and structural turn of mind, is quite receptive to such
influences. (p. 66)

Margaret Elida Thomas, Conlon Nancarrow’s “Temporal Dissonance ”: Rhythmic

T xt r

tr ti cation in the

tudies or Pl

er Piano. Thomas completed her dis-

sertation in 1996, for which She had access to both Carlsen’s 1986 dissertation andthe

1988 monograph based on it, and unpublished drafts of portions of Gann’s book (Thomas

1996, ii). Thomas sets out to explore “the rhythmic and textural techniques” of the

Studies, and “temporal dissonance in particular” (abstract). The term “temporal disso-

nance” is traced back to Nancarrow himself, who said in an interview with Roger

Reynolds that he had an “interest in temporally dissonant relationships” (Reynolds 1984,

5). In the course of this dissertation, Thomas makes at least brief mention of all the

Studies except Nos. 13, 22, 26, 28, 29, 30, 34, 43, 44, 46, 47, 48, 49, and 50 (see

Appendix C for a complete index to Studies referenced in Thomas).

16

 
A key part of Thomas’s discussion is the development of a concept she calls “multi-

dimensionality.” This is a somewhat elusive concept that is presented in terms of its rela—

tionship to the techniques of accumulation (progressive addition of musical layers),

canon, and heterophony that she identifies in Nancarrow’s works. Thomas’s definition of

heterophony includes not only line doubling at the octave, but other varieties of parallel

melodic movement (including triads); even lines which are only loosely coordinated

melodically are considered a heterophonic texture if “the heterophonic pitch relationship

causes the voices to lose their individual identities” (p. 90). (See Figure 1-26 for such a

texture.) Thomas further States that multi-dimensionality is neutralized in the presence of

heterophony. Finally, Thomas examines multidimensionality in relationship to temporal

dissonance, for which she States multidimensionality is a prerequisite.

Thomas’s development of the concept of multidimensionality and its relationship to

temporal dissonance is somewhat compromised by repeated references to a lack of

coordination or relationship between the temporally dissonant layers, yet an insistence

later on that in order for a temporal conflict to exist, there must be a discernible rela-

tionship between the musical layers—and indeed, throughout the dissertation there is a

somewhat uneasy balance between these conflicting views. In the abstract, Thomas refers

to “Nancarrow’s aesthetic goal of ‘temporal dissonance’: two or more seemingly un-

coordinated streams of music .

.

. presented simultaneously” (emphasis mine). Later, the

qualification of seeming uncoordination between the layers is not so carefully observed

and Thomas begins instead to equate asynchronous with unrelated. On p. 3 she States

that:

My particular interest in Nancarrow’s studies centers on the extreme form of
polyphony created by his simultaneous use of uncoordinated, independent
streams of music, and the philosophy of musical rhythm and time embodied
therein. .
. the multidimensionality of the works [consists of] the stratification
of their musical time into a number of asynchronous layers. (emphasis mine).

.

Further, she notes that “the works feature several discrete, concurrent temporal processes,

17

 
so that at any given moment a study has a relatively expansive spatial quality owing to

the irreconcilability of its layers” (pp. 3—4, emphasis urine). Finally, on p. 9 She allows

that “it is not mere independence between layers for which Nancarrow strives with his

rhythmic techniques, but real conflict, which is crucial to his overriding aesthetic goal of

creating an effect he calls temporal dissonance.” Yet it still seems that for a conflict to

exist, there must be some discernible relationship. This matter is not finally settled until

Thomas states on p. 114 that “the clearer the conflicting relationship between [the] voices

the more easily we can perceive them as [temporally] dissonant.” And,

Textural density does not translate into temporal dissonance. Instead, disso-
nance seems to rely heavily on the perceptibility of the relationship of simul-
taneous voices, whether in terms of their tempo proportions, their metrically
significant simultaneous articulations (or lack thereof), or their act of con-
verging. (p. 122)

Thomas begins the dissertation with a brief survey of Nancarrow’s compositional

techniques and characteristics of the Studies; her observations are incorporated into the

section on “Musical Characteristics of Nancarrow’s Studies” presented later in this

chapter.

In a very useful section on musical influences, Thomas examines two important

influences on Nancarrow’s music: Henry Cowell, and jazz. Cowell’s New Musical Re-

sources of 1930, which sets forth his groundbreaking theories on rhythm and metric

relationships, had a profound influence on Nancarrow. Cowell espoused the use of pro-

POrtional rhythmic relationships that relate directly to interval ratios. Thomas notes that

such a relationship could be expressed in three different ways (see Figure 1-1): rhyth-

miCally, metrically, and temporally. By far the most common expression of this concept

In Nancarrow’s Studies is the tempo ratio.

18

 
”I

3:2 generates

3

(a)

(b)

(C) J = 72   

f

l' r

Figure 1-1. Figure in Thomas (1996, p. 189) illustrating the 3:2 (perfect fifth) ratio applied to: (a)
rhythm, (b) meter, and (c) tempo.

Thomas notes that, whereas Nancarrow applied Cowell’s ideas on rhythmic propor-

tions in the Studies for Player Piano, his musical goals were different than Cowell’s:

The contrasting simultaneous rhythms, meters, and tempos he [Cowell] dis-
cusses are related by proportions. The parallel to Nancarrow’s asynchronous
layers is unmistakable, of course, particularly in his use of proportionally-
related tempos. There is a fundamental difference in the intended effect of
those proportions, however: whereas Nancarrow strives for temporal disso-
nance, Cowell uses ratios derived from the overtone series to create what he
called “rhythmic harmony.” (Thomas 1996, 21)

Cowell’s ideas were also influential on the form of Nancarrow’s Studies. In

Cowell’s system, large groupings equivalent to formal sections could be articulated by

the arrival of voices in conflicting tempos at a Simultaneous downbeat. These “con-

vergence points” are indeed of great interest in the Studies, particularly in the tempo

canons, and will be the subject of Chapter 3 in this dissertation.

The influence of jazz on Nancarrow is just as pervasive. For a time, Nancarrow per-

formed as a jazz trumpeter, and his lifelong love of jazz emerges in his music in many

ways. Thomas identifies the influence of boogie-woogie in Nancarrow’s early Studies,

and standard blues progressions, “walking” bass lines, and jazz accompaniment patterns

are used to great effect in a number of Nancarrow’s pieces. Rhythmic influences of jazz

19

 
 
include swung (long-short) rhythmic divisions and the evocation of improvisational prac-

tice by placing rhythmic attacks “around the beat.” Thomas also points out that poly-

rhythm and polymeter are typical features of jazz. Nancarrow also used a great many jazz

harmonic idioms, including seventh, ninth, and substitution chords, as well as blues

scales.

Thomas’s second chapter focuses on specific works by four other twentieth-century

composers who, like Nancarrow, had an interest in “exploring the possibility of pre-

senting simultaneous layers of music that are related temporally in unconventional ways”

(p. 29; note the assertion here that the layers are temporally related). Thomas looks at

these works “in order to see how differently temporal stratification can function” (p. 29),

and identifies in each a different genesis for temporal conflict. In the first work examined,

Charles Ives’s Washington’s Birthday (1909), the genesis of the temporal conflict is

Ives’s desire to recreate his actual musical experience of simultaneous performing forces.

Thomas begins to examine here the concept of a “perceptibility threshold” (p. 34, note

12) at which individual layers of sound coalesce into a composite mass.

The next piece examined is the first movement of Henry Cowell’s Quartet

Romantic. Here the genesis of temporal conflict is, of course, Cowell’s theory of

temporal ratios. Thomas explains Cowell’s compositional method, in which he converts

the pitch ratios of simple four-voice chorales to rhythmic ratios (a “harmony of rhythm”)

with the result being that “the effect of the movement is correspondingly progression-

like, in that the rhythms regularly proceed from one set of ratios to the next” (p. 41).

The second movement of Elliott Carter’s “String Quartet No. 2” features a constant

pulsation in the second violin, which functions as the time keeper while the other instru-

ments develop unique rhythmic personalities. AS in Cowell, the genesis of temporal con-

flict in this piece is a variety of tempo ratios related by simple integers. Here, however,

the ratios were not arrived at by conversion from a harmonic system, but through a series

of ratios all relating to the basic tempo of J = 140 and expressed through Carter’s

20

 
technique of “metric modulation.”

Thomas’s final example is GyOrgy Ligeti’s “Automne a Varsovie” from Piano

Etudes, Book One. Thomas notes that Ligeti faced a greater challenge in “articulating a

stratified texture with just one instrument” (p. 47). Ligeti creates temporal conflict be—

tween the pianist’s two hands by articulating almost continuous sixteenth notes through-

out the piece and eventually requiring the performer to articulate every fifth pulse with

the right hand and every third with the left to form a 5:3 ratio.6

Chapter HI continues the development of the concept of multidimensionality, and

concentrates on canon and textural accumulation as means of achieving it. Heterophony

is also examined as a non-multidimensional texture that is created by coordination of

pitch-class content among musical layers. Thomas further identifies factors such as

register, pitch class, and rhythmic, metric, and temporal organization as being influential

on the individuality of layers.

In examining processes of textural accumulation, Thomas identifies some Studies

that clearly cross “the perceptibility threshold for multidimensionality” (p. 63) due to

extreme textural density. She classifies canon in Nancarrow’s Studies as a special kind of

accumulation, and comments that “Indeed, as we will see, the technique of canon does

not necessarily produce a clarity of layers. Sometimes canon seems not to be the aesthetic

point of a Study at all, but rather a tool for creating a texture in which voices that are

theoretically independent are not perceptually distinct” (p. 73). In one instance, while

discussing Study No. 31, she makes a point regarding difficulties in perception when

canonic entrances are widely separated, but rrristakenly calculates the gap between the

opening of the study and the second voice as 38 14 seconds when the gap is really only

6There is a slight miscalculation here regarding the tempo of the 5-grouping in the right hand: the quarter-
note pulse at the opening is MM 144; when the 5-grouping begins. creating a 5:4 ratio against the pulse
(one pulse every 5 sixteenth notes rather than every 4), Thomas calculates this new pulse at MM 113 when
it is actually closer to MM 115 (14404/5 = 115.2).

21

 
about 12 1’2 seconds.7 In the course of this discussion, and later in regard to Study No. 20,

Thomas performs some motivic analysis, which is unusual for this dissertation.

The fourth chapter is devoted to an exploration of temporal dissonance in the

Studies, particularly ways in which Nancarrow controls levels of temporal dissonance.

Thomas notes (p. 93) that this quality is pervasive in Nancarrow’s music, most noticeably

in the Studies in which tempo proportions are used, but it is present even in those Studies

that do not contain multiple Simultaneous tempos. There are, of course, differing levels of

temporal dissonance, and Thomas observes that moments of relative consonance often

serve a formal function in the Studies (p. 96). Absent other mitigating factors, the most

extreme levels of temporal dissonance can be found in the Studies which use irrational

number proportions, since there is a total lack of coordination between the voices, and the

potential for points of simultaneity is virtually non—existent.

Thomas suggests that “one of the clearest compositional strategies in which to hear

proportions is canon” (p. 97), and then proceeds to examine levels of temporal disso-

nance in individual Studies. In this section, she makes many observations about the

perceptual levels of dissonance attendant to techniques such as ostinato/isorhythm and

canon in an environment where proportional tempos are operating. She identifies many

instances in which the levels of temporal dissonance are exaggerated when Nancarrow

subverts potential points of simultaneity by one of two techniques: (1) shifting meters

(i.e., moving the downbeat around), and (2) avoiding the placement of note attacks on

potential points of simultaneity by using notes longer than the basic pulse or using rests.

In illustrating the second technique, Thomas cites an example from Study No. 9 (see

Figure 1-2) in which two voices share an eighth-note pulse at a ratio of 5:3; this, then,

sets up the possibility of a simultaneity every fifth note in the faster voice and every third

7This is easily verified on the recording. The tempo is J: 105 for 22 measures of 4 time. There is a simple

2

math error here: Thomas multiplied the 22 measures by 105/60 rather than 60/105 (each measure elapses in

.57 seconds [60/105] and 60/105 x 22 = 12.57 seconds).

22

 
 
 
note in the slower. However, due to the use of rests in the lower voice, five of nine

possible simultaneities are avoided (significantly, all three of the simultaneities that occur

on mutual downbeats—the first, fifth, and ninth—are retained).

Figure 1-2. Segment from Study No. 9 (as renotated by Thomas, 1996, p. 260) exhibiting a 5:3
rhythmic ratio in which potential simultaneities are avoided. Actual simultaneities are indicated
by a vertical broken line, while potential simultaneities are marked with an asterisk.

Thomas points to one additional factor that contributes to temporal dissonance, and

that is the mixing together in the Studies of three “time types”: (1) material that uses

regular meters and non-changing tempos; (2) non-metered, proportionally notated ma-

terial; and (3) material that gradually changes speeds. She comments that “These dif-

ferent time types raise new issues with regard to an examination of temporal dissonance:

tempo ratios and simultaneous articulations generally no longer serve as markers for

dissonance” (p. 125).

For Thomas’s fifth chapter she selected Study No. 41 for a comprehensive analysis.

This Study is one of Nancarrow’s longest and most complex. It is in three “movements”:

both movements 41A and 41B are two-voice converging-diverging (arch) canons based

on irrational ratios, and movement 41C is the two previous movements played together.

The subtitle for 41A is 7117/V2/3, while 4lB’S subtitle is WlE/N 13/16. Thomas’s

analysis treats A, B, and C separately. For A she identifies six “families” of motives or

23

 
!
?
n
i

l
l
i

melodic gestures and the implied rhythmic patterns that they represent. She notes also

several pc sets that are prominent in some of the motives, and observes that the single

convergence point that occurs about 70% of the way through the movement is the only

“clear formal division” (p. 147), although she points to a prominent moment of silence

that occurs a Short time later as being significant. In B she notes the sirrrilar formal arch

shape, although in B the prominent moment of silence is placed directly after the con-

vergence point and Thomas emphasizes the support given to the arch Shape by respective

processes of accelerando and ritardando. Five motivic families are identified, and their

implied rhythms analyzed, but this time no pitch analysis in the form of pc sets takes

place. Only one pitch is identified as Structural, and that is B, which ends both move-

ments (and consequently 41C) and begins 41B, forming a bridge from 41A. The analysis

of 41C focuses on the “truly staggering complex” (p. 160) that is created when the two

movements come together; Thomas examines the complementarity of the formal shapes

of the first two movements, and certain similarities that exist among the motivic gestures

of 41A and 41B.

In her conclusion, Thomas summarizes the types of textures and textural devices

that have been discussed, and acknowledges that “Because my analysis of temporal

dissonance has (necessarily) been so contextually based, issues of perception have been

prominent in my discourse” (p. 166). She then focuses on the canonic technique, ques-

tioning why Nancarrow uses it when

the canonic process is highly obscured by the complexity of the canonic line,
the number of voices, the Speed at which it progresses, and so on. This raises
the question of why canon is used in these cases: if we cannot hear it, what is
the point of the canon? (p. 167)

Thomas’s final comments concern the importance of the player piano in realizing Nan-

carrow’s goals of temporal dissonance and in imparting a “superhuman” quality to his

music.

24

‘——--A

James Tenney. Gann, Carlsen, and Thomas all make occasional reference to the

liner notes prepared for the Wergo recordings by James Tenney and the 1977 article in

Soundings, Book 4. The 1977 article covers all the Studies through No. 41, and the

Wergo liner notes through No. 41 are almost exactly reproduced from this article (with

the exception of No. 30, which is not included in the Wergo recordings). Tenney’s

comments about Studies No. 42—50 were apparently added in 1988 and 1989. Those

Studies about which Tenney wrote for both the 1977 article and the Wergo liner notes

(i.e., Nos. 1 through 41) are sometimes treated in more detail in the liner notes. In a few

unusual cases, there is considerably more detail—usually about formal matters—in the

liner notes; two good examples are the analyses for Nos. 7 and 9. Yet other Studies

receive only very cursory treatment in both writings.

Tenney laments in his introduction that “the continuing obscurity of Nancarrow’s

work is nothing short of scandalous” (1977, p. 41). Still, he confidently predicts that

21st—century historians will rank Conlon Nancarrow’s Studies for Player
Piano with the most innovative works of Ives, Schoenberg, Stravinsky,
Webem, Varese, Partch, Cage, Xenakis—and perhaps a very few others—as
the most significant works composed Since 1900. This prediction may seem
extravagant to some, but I am convinced that, when Nancarrow’s music is as
accessible and widely known as that of his contemporaries and immediate
predecessors, its importance will be just as widely recognized, and there will
remain no room for doubt. (Wergo Vols. I/II, liner notes, p. 1)

Tenney identifies several terms that he finds critical to a discussion of Nancarrow’s

music, and these include: (a) aggregate, “complex sounds and sound-configurations

which are perceived as singular textural elements” (1977, p. 44); (b) Mn}, “a com-

plex but perceptually Singular ‘1ayer’ or stratum in the polyphonic texture” (1977, p. 44);

and (c) compound-mlyphonic, a term used frequently by Tenney to refer to the texture

created when aggregates and/or resultants are used (but also used in a way as to be easily

confused with heterophonic: “each voice is compound-monophonic, its melodic lines

generally doubled in octaves or filled out by triads” [Wergo Vols. III/IV, liner notes, p.

20]).

25

_ I

 
In writing about the Studies, Tenney creates the following roughly chronological

taxonomy:

1. The “early group,” consisting of Studies 1 through 12. Tenney characterizes these

works as clearly tonal (or modal) and notated metrically except for No. 8. Many of them

have origins in blues, ragtime, or other jazz styles, and while ostinatos are frequently

present, strict canonic writing is rarely used. In the individual analyses, Tenney observes

numerous tempo ratios in effect and the frequent use of rhythmic series, either separately

or as part of an isorhythm. Different versions of the trochaic rhythm pattern (long-short)

are observed, sometimes in the same piece (e.g., No. 2), and these include 3:1, 2:1, 3:2,

and 5:3. Tenney also mentions several times a “rubato-effect” which he mentioned earlier

(1977, p. 44) as one of six rhythmic procedures but never really defines. This effect is

observed by Tenney in Studies No. 6, 10, and 12. In No. 6, it refers to an ostinato that

regularly alternates between measures of the same length that are divided into four and

five eighth notes; in No. 10, a “freely evolving melodic line” is created from numerous

arithmetical accelerations (“countdowns”) to form a “kind of ‘written out’ rubato” (1977,

pp. 50—51); and in No. 12, several countdown devices are evident that impart a feeling of

rubato.

2. The next group is Studies No. 14 through 19, which were part of a set of Seven

Canonic Studies (including No. 13, for which Nancarrow never released the score) all

using tempo ratios based on the numbers 3, 4, and 5. Tenney characterizes these as

having a more abstract Style, with no hints of blues or ragtime; using canon as a formal

organizing procedure; and showing Nancarrow’s increasing interest in tempo as a

structural parameter. Tenney’s analyses include a structural diagram for each piece

showing the number of canonic voices and how they relate to one another registrally and

temporally in terms of tempos used, entrance points for the voices, and any convergence

points that occur. Melodic elements are analyzed with respect to tonal centers, presence

of mixed meters, and use of various doublings. Tempo switches are noted in Nos. 15 and

26

—. —~—-d 

17, and in No. 16 four duration series are identified that contribute to the two resultant

strata.

3. Tenney’s next group is called “The Middle Period,” and includes Studies No. 20

through 32. This is by no means a homogeneous group, and in his “Overview” section

Tenney actually identifies three groups among these works: Nos. 20—27, characterized by

“precisely controlled, very gradual changes of tempo, and a fairly consistent use of a non-

metrical rhythmic notation for the scores” (1977, p. 45); Nos. 28—30, Similar to 20—27 but

distinguished from them by being originally planned for prepared player piano; and Nos.

31—32, which return to metrical notation and fixed rational tempo ratios. The analyses in

this section are at times highly detailed. Tenney observes that Study No. 21 could be

perceived not as the beginning of a new group, but the end of an earlier period. This

Study (“Canon X”) features a gradual acceleration in one voice against a gradual

ritardando in the other, and it is the last piece Nancarrow punched with a roll-punching

machine that punched in fixed incremental units rather than on a continuous time scale.

Tenney also comments that “Beginning with Study #21, Nancarrow’s harmonic-melodic

language becomes more and more difficult to relate to traditional tonality” (1977, p. 55).

Other noteworthy features of these Studies include the two pieces (Nos. 22 and 27) that

Nancarrow punched with the new roll-punching machine, which have percentages in their

subtitles to indicate the rates of tempo change; the “exploded” notation technique Nancar-

row began using in No. 25 to accommodate very fast note fragments; the difficulties

Nancarrow encountered in writing pieces for “prepared” player piano (Nos. 28—30); and

the seeming return in Nos. 31—32 to the simpler textures and rhythmic procedures as seen

in Nos. 14—19.

4. Last in Tenney’s 1977 taxonomy are Studies No. 33—41, which are marked by the

use of higher-order (but still rational) tempo relations; the introduction of irrational

tempo ratios (e.g., Nos. 33 and 40); the use of two player pianos simultaneously; and

richer textures and a greater variety of aggregate structures. Study No. 34 is the one piece

27

in this group in which Tenney finds simple textures and a minimum of aggregates being

used, and he surmises that this is to allow the rather complex tempo relationships to be

clearly heard. Study No. 36 prorrrinently features linear aggregates (grace-note appoggi-

aturas, triadic arpeggios, and chromatic glissandos), while in the remaining Studies ver-

tical structures predominate. In Studies No. 40 and 41, Tenney finds that the texture is

almost completely composed of aggregates:

each stratum consists of a succession of glissandos, chords, repeated-notes,
trills and tremolos—nearly every basic type of aggregate ever used in earlier
Studies, and more different types than in any one previous Study. These ag-
gregates are varied (in duration and/or pitch-register)—and juxtaposed in
time—in a way which suggests the way in which the object sonore is treated
in musique concrete. (1977, p. 64)

5. After Tenney’s 1977 article appeared he wrote of Studies No. 42—50 for the

Wergo liner notes. For these he classified the most recent group as Nos. 40—50, creating a

slight overlap with the previous group. He notes that these Studies primarily use propor-

tional notation, that references to jazz and Spanish music appear again, and that some

new elements appear, such as the use of aleatoric procedures (No. 44) and an introduction

of live performers in the Concerto for Pianola that was the basis for No. 49. The analyses

in this group are, by far, the most detailed that Tenney produced. Several of the pieces

(Nos. 40, 41, 45, 48, and 49) are in multiple movements, and three of these (40, 41, and

48, along with No. 44) involve simultaneous use of two pianos. Tenney continues to

focus on the presence of aggregates, which ones are used in a piece, and how the col-

lections of aggregates might change in the course of a piece. For example, about Study

No. 41 he identifies “glissandos, arpeggios, and other, miscellaneous ‘running’ figures,

sometimes alone, but often leading to, passing ‘through’, or seeming to arise out of,

sustained tones and chords” (Wergo Vols. I/II, pp. 8—9); and about No. 48 he observes

“the great variety of linear aggregate types used, including trills, tremolos, arpeggios,

glissandos, and miscellaneous melodic figures, all at dizzyingly high speed” (Wergo Vol.

5, liner notes, p. 10). His analysis of the three movements of Study No. 45 is perhaps the

28

most thorough of all. He reviews each movement separately in terms of repeating

harmonic-melodic progressions and ostinato bass-lines (largely based on blues), the

presence of canons, and textural changes. (He does not recognize, as Gann does, an

irrational number derivation of the “peculiarly uneven rhythm” [Wergo Vol. V, liner

notes, p. 7] that is the basis of the bass lines in the first and third movements). No. 45 is

one of several jazz-inflected works in this group that prompt Tenney to observe, “In fact,

it now seems to me quite possible that much of what I have called Nancarrow’s ‘abstract’

style might better be understood as simply a new form of jazz” (Wergo Vol. V, liner

notes, p. 11).

Musical Characteristics of Nancarrow’s Studies

General observations about characteristics and features of Nancarrow’s Studies are

given specific attention in Gann (pp. 1-35), Thomas (pp. 7—19), in various parts of

Carlsen’s discussion of individual Studies, and in a brief overview in Tenney (Soundings,

Book 4, pp. 43—46). Particular attention is paid by these sources to matters of notation,

which is an important element of Nancarrow’s work. This summary of the literature will

make note of notational features as they relate to other features.

Rh t

' F atures

Tenney points to rhythmic procedures as one of the two most important character-

istics of Nancarrow’s works (along with polyphonic texture), and makes the following

observation:

Rhythmic independence is essential to real polyphony. Conversely, the most
thorough exploration of rhythmic possibilities must include Simultaneous as
well as successive rhythrrric relations. So it is not surprising that a music
inspired primarily by an interest in rhythm would be polyphonic in its very
essence. (1977, p. 43)

Nancarrow himself commented on the primacy of rhythm to his compositional method by

29

 
remarking that “the thematic notions [in my music] are usually rhythmical rather than

melodic” (Reynolds 1984, 10).

Nancarrow uses two primary types of notation to convey rhythm in his Studies:

conventional metered notation and proportional notation. Conventional rhythmic notation

usually involves the use of meter and bar-lines, even when each voice is in a different

tempo (such as in Figure 1-2). In proportional notation, the passage of time is carefully

measured on the scores; sometimes Nancarrow will indicate this with a notation such as:

 :

:=120

while at other times no such notation is given. As Carlsen is careful to point out, all of

Nancarrow’s scores use spatial representation of time, even when meters and regular note

values are used:

throughout a study, a certain amount of horizontal space on the score page
represents a specific amount of time, and the spaces between notes in the
score are the same—or proportionally the same—as those on the roll. .
. This
exact relationship between space and time is true even of the studies notated
with time signatures, barlines, and the Standard symbols for durations and
rests. (Carlsen 1988, 5)

.

The length of a sustained note in proportional notation is indicated with a thick line

extending to the right of the note for its duration. Thomas notes (1996, pp. 8, 19) that in

his proportional notation Nancarrow generally used quarter notes for sustained notes and

eighth notes for staccato articulations (see Figure 1-3). Notes longer than a quarter note

are unusual in the Studies, primarily because a sustained note on the player piano neces-

sitates that a long row of holes be laboriously punched on the roll. Also, like the

perforations on a Sheet of postage stamps, such long rows of holes made the roll paper

vulnerable to tearing, so Nancarrow took considerable care in placement of sustained

I'IOICS .

30

_

P—c-

—— — ———-—

——

Figure 1-3. Proportional notation in “Study No. 8,” p. 18.8

Perhaps the most characteristic hallmark of Nancarrow’s Studies is the simulta-

neous use of different rhythms, meters, or beat divisions, sometimes in addition to dif-

ferent tempos. This is evident already in the first Study, where an ostinato in which the

l
x
-
r
-
:
:
-
-
t
—
n

same measure length is divided into both 8 and 4 time is pitted against a variety of dif-

7

4

ferent tempos and meters (see Figure 1-4).

(Jada)

(9‘5).

Figure 1-4. 8 against 4 ostinato with conflicting meters in Study No. l (p. 9). The symbol J is

7

4

a five-division note, in this case a five-thirty-second note.

8In this example, two other characteristics of Nancarrow‘s score notation are evident: (1) the entire length
of the staff, from beginning to end (about 165 millimeters), is used for purposes of notating elapsed time,
and (2) to accommodate this, the beginning clefs are always written to the left of the staff, as are any
accidentals that apply to the first note or chord. Nancarrow also tended to avoid writing meter changes on
the staff and usually wrote them above the Staff instead.

31

—‘_LA 

 
 
  
 
Jazz  rhythmic  features  abound  in  Nancarrow’s  Studies.  There  are  many  variations 

on  the  trochaic  (long-short)  “swung”  rhythm,  sometimes  even  within  the  same  piece.  For 

example,  in  Study  No.  3b  both  the  rhythmic  divisions  3:2  and  5:3  are  used.  As  Gann 

remarks, 

Nancarrow  has  from  the  very  beginning  used  the  player  piano  to  recreate 
rhythmic  liberties  taken  in  performance  that  no  notation  could  convey.  In  the 
studies  based  on  [stride  piano]  (Nos.  3,  4,  10,  45),  he  has  implicitly  acknowl- 
edged  that  jazz  pianists  hardly  ever  play  a  dotted  rhythm  in 
ratio; 
instead,  Nancarrow  often  divides  his  beats  into  ratios  of  3:2,  5:3,  or  8:5,  all 
divisions  based  on  the  Fibonacci  series,  related  to  the  intuitively  pleasing 
Golden  Section  as  well  as  closer  to  live  performance  practice.  (p.  9) 

a  3:1 

The  5-division  notational  symbol  (  ]  shown  in  Figure  1-4  above  also  found 

frequent  usage  in  beat  divisions  of  5:3,  in  which  it  complements  a  3-division  note  such  as 

dotted  eighth  or  sixteenth  notes.  See  Figure  1-5. 

an 

q 

¥ 

ee 

qe 

a7 

; 

rv" 

ree 

T 

oa 

|  L— 

nt 

‘Wnt 

fro  ttens 

4
1
0
+

y
r

~~ 

Figure  [-5.  The  5-division  symbol  used  to  express  a  5:3  division  of  a  half  note  in  Study  No.  3b, 
p.  16. 

Earlier,  in  the  discussion  by  Thomas  on  the  influence  of  jazz  on  Nancarrow,  Nan- 

carrow’s  tendency  to  place  notes  “around  the  beat”  was  noted,  and  this  is  another  char- 

acteristic  rhythmic  feature  in  his  music  (see  Figure  1-6).  Other  features  that  have  already 

been  mentioned  include  the  “countdown”  accelerative  technique  (progressively  smaller 

units  between  attacks;  see  Figure  1-7  for  a  metric  countdown  used  to 

particularly 

dramatic  effect  to  close  Study  No.  12),  and  other  types  of  numeric  series  that  often 

involve  adjacent  numbers.  (See  Figure  1-23  for  a  series  that  alternately  decelerates  and 

accelerates.) 

32 

 
 
wn 

ye 

Cd 
—  rae 
I 
’ 
J 
Z 
¥ 

h—» 

T 
T 
—— 

Ba  : 
jae 
ry 
T 

P  ———— 

ee 

eS 

A  a 

T 

ye  Te 

& 

ana 

JD 

eo 

ri 

o—+-} 
tte 
+ 

een  1 

“Yt 
r 

= 
ja 
. 

—2 
m 
: 

wn  7 
a  Se, 

7 

Peat  Ye  le  ~ 

z 

rete  He  7 

bp  cS 

Figure  1-6.  Melodic  writing  “around  the  beat”  in  Study  No.  2  (p.  4). 

a 
al 
+ 

a 

Se 

es  i 

dl  2S  -~ 

7 

v¥  ohne age 

fe 

=  shih  aE  _  ae  2p  rts  ciety,  oa)  —  mm 

—- 
: 
— 

ce 

= 

“mgara  : 

tt 

+f 

ety 

ine 

— 
F4.  ob,  Ph 

beh, 

* 
audia 

7 

i 

¥ 

fe" 

2 

~  a.  Lo  — 

. 

D 

eens  os  ae 
z| 
, 

Mb 

alt 

— 

uF 
J 

. 

3 

2 

\ 
a 
ae 

f 
ty 

yaad 

T 

An 
> 
ZZ" 

ae, | 

“y 
L 

& 

5 

ne 
vs 

a 
y 

AK 

Z  min 

rs 

S 
>  a  a 

a 
—> 
/_  yey 
le  A 
a 

tal 

in 

) 

d 

rf 
Lo  maar  2 
t 

n 
vi 
rm 
i— 

y~ 

= 

a] 

#. 
Sur 

a 
—— 
y= 

4 

, 
uy 
2 

=~ 
- 
oa 
“¢ 

Ss 
tft 
. 

a 

) 

me 
z 

4 
a 
* 

v 

+ 

4 

TI 
a 
y 
¥ 

CA 

m 

are:  a  ‘2 

—— 

= 

ra 

r 

At. 

——_—~ 

d 

5 
i 

+. 
y  a 

= 
+ 

2 

—¥# 

__? 

= 
ork 
—¥4 

J 

yy 

im 
x 

v1 

‘Ses 
5 
x 
32 
az 
=~ 
a 
pSeey 
> 
~~ 

y 

ay 
z 

a 
i 

r, 

Se 

ti 
we 

Figure  1-7.  The  “countdown”  technique  at  the  closing  of  Study  No.  12. 

Rhythmic  ostinati  and  isorhythms  are  a  significant  component  of  Nancarrow’s 

thythmic  landscape.  In  the  literature  reviewed  here  the  terms  ostinato  and  isorhythm  are 

sometimes  used  interchangeably,  but  a  distinction  should  be  made  between  them.  At 

times  the  authors  seem  to  insist  that  an  isorhythm  involves  both  repeating  rhythmic  and 

pitch  elements  which  are  often  of  different  lengths  and  out  of  phase  with  each  other: 

a  fine  line  between  ostinato  and  isorhythm,  since  an  ostinato  reit- 
There  is 
erates  the  same  rhythm  as  well  as  the  same  pitch,  . 
.  [while  isorhythm 
involves]  some  internal  dissociation  of  rhythm  and  pitch,  a  use  of  talea 
independent  of  color.  (Gann  1995,  85-86) 

. 

. 

.  duration  series  are  often  paired  with  pitch  series,  which  may  be  of 

. 
different  lengths,  creating  isorhythm.  (Thomas  1996,  11-12) 

33 

[Nancarrow]  revived  the  medieval  technique  of  isorhythm  (though  inspired  by 
a  strong  interest  in  the  tala  structure  of  Indian  music),  employing  multiple 
repetition  of  the  same  rhythm  against  different  pitch  sequences.  (Gann  2001, 
606) 

Gann,  however,  often  refers  to  rhythmic  patterns  alone  as  isorhythms.  A  slightly 

different  phrase  is  used  by  Carlsen,  who  discusses  the  four  closely-related  repeating 

rhythmic  patterns  in  Study  No.  19  based  on  the  series  {n  —  1,  n,  n  + 

1,  n}  (the  same 

patterns  are  shown  in  Figure  1-11  below)  as  “isorhythmic  patterns,”  which  he  later  calls 

“talea”  (Carlsen  1988,  7).  Thus,  for  Gann  and  Carlsen,  at  least,  an  isorhythm  does  not 

necessarily  require  a  repeating  pitch  element. 

The  term  ostinato  is  just  as  ambiguous.  Gann’s  definition  of  ostinato  as  “a  phrase 

persistently  repeated  without  variation”  (p.  69)  does  not  distinguish  between  a  rhythmic 

ostinato  or  one  that  combines  melody  with  rhythm.  I  think  it  is  safe  to  say,  however,  that 

the  authors  reviewed  here  always  choose  the  term  ostinato  over  isorhythm  when  the 

rhythm  involves  the  same  rhythmic  unit  repeated  over  and  over,  rather  than  a  variety  of 

rhythmic  values—e.g.,  a  steady  repetition  of  eighth  notes  or  eighth  notes  separated  by 

eighth  rests  would  be  considered  an  ostinato,  even  if  a  repeating  pitch  element  is  present. 

Thus  the  term  isorhythm  should  be  held  to  the  more  exacting  standard  of  involving 

varying  rhythmic  values.  The  repeating  pattern  in  an  ostinato  also  tends  to  be  of  brief 

length  whereas  isorhythmic  patterns  are  generally  longer  and  more  complex.  In  Figure  1- 

8,  for  instance,  Thomas  states  that  the  lower  voice  in  the  opening  of  Study  No.  2  is  “a 

small-scale  example  of  isorhythm”  (p.  12);  she  refers  to  the  two  lines  operating  together 

as  an  ostinato  (as  does  Tenney  [Wergo  Vols.  III/IV,  p.  7]).  Gann  refers  to  each  of  the 

Ja? 

2. 

T 
fim  3 
x. 
* 

* 
te 
ja 
7 

p 

Pr 
7 
ry 

mT 
vA 

1 

1 

Ge 
r 

a 
Z. 
Ls 

x. 
4 

é 

p 

S 

sy 
de 
r 

a 

La 

. 

~ 
va 
hl 

rt 

——meny 
ny 
HH 
1 
i" 
*. 

3 
A 

+ 
M 

ws 
Ld 

- 
DY 
va 

i 
a 

i 
i 

* 

> 
a 

z 

z. 

i 

sy 
é 

ra 

A
¥ 

1 
r 
r 
ri 
‘ 

bt 
7 
jpg. 
t 

“5 
Wl 

] 

sS 
- 

T 
H 
y 

aa 

—_ 

ne, 

‘ 

-- 

et 

eal 

Figure  1-8.  The  opening  of  Study  No.  2. 

34 

 
lines in this segment as ostinati, while the bottom line is additionally an “isorhythm—

within-an-ostinato” (p. 73).

Nancarrow sometimes creates resultant rhythms out of constituent numerical series

and other components. An early and transparent example of a rhythmic ostinato, in Study

No. 6, involves the continuous alternation between measures of g and 3 time in which

all measures elapse at the same rate (1 measure per second); see Figure 1-9.

Gummert: ~.-. 60   

B.

r /

—"

Figure 1-9. First system of Study No. 6, showing ostinato created by alternating different eighth-
note divisions of the same measure length.

Several other Studies provide clearer examples of what the writers consider iso-

rhythm. In Study No. 7, Gann (p. 88) identifies the following three isorhythms with

lengths of 18, 24, and 30 eighth notes (a 3:4:5 ratio):

18 isorhythm = 5 +4 + 2 + 3 + 4 (= 5 notes)

24isorhythm=5+5+2+4+3+2+3(=7notes)

30isorhythm=3+2+2+3+2+3+3+2+3+2+2+3(=12notes,
palindromic)

Thomas refers to these patterns merely as “durational series” (p. 91) and Tenney

calls them “duration series” (Wergo Vols. III/IV, p. 9). In Study No. 11, Gann and

Thomas both identify the following accelerating eighth note pattern of 60 beats divided

into three segments:

5564|55343|543332

(4 notes)

(5 notes)

(6 notes)

Thomas calls this a “duration series” (p. 115) while Gann notes that this isorhythm is part

of a 120-note “isorhythmic melody” (p. 97). Note the gentle acceleration built into the

35

series.

In an example using a numerical series, Gann identified the additive formula {11 — l,

n, n + 1, n} as the basis for all the rhythms in Studies No. 13—19. In Study No. 14, the

series is applbd to four voices tint create the resultant rhythm shown in Figure 1-10. Later,

in Studies No. 45, 46, and 47, Nancarrow creates a rhythm “not susceptible to rational

notation” (Gann 1995, 257) by taking a collage of templates representing different tempo

relations, and putting them all together.9

5

7

3

2

6

9

10

4
J

8
J)

1
Ieighth-notebeats
J,
voice #1 (3+4+5+4)
voice #2 (4+5+o+5)
J
voice #3 (5+6+7+6) J)
voice#4 (6+7+8+7)
J,J
D Mic!
 summon. |31
lgri D

J,J
ilgl

13
J

J)

J.J

JJ

J.J

12

14

1

11

15

l6

l7]
J

JJ’

D|§J

Figure 1-10. Derivation of resultant rhythm in Study No. 14 (Scrivener 2000, 189).

Finally, Gann mentions (pp. 9—10) Nancarrow’s somewhat unique ability to com-

bine both additive and divisive rhythms in his Studies, paying homage to Stravinsky’s

legacy in the use of additive rhythms and Cowell’s ideals in the use of divisive rhythm-

particularly the different division of the same unit of time in different voices. Mani-

festations of additive rhythm include the use of short repeating rhythmic patterns that are

often grouped into larger “hyperrneasures.”

Pi

Mel

ic Features

Carlsen observes that:

the pitch organization in much of Nancarrow’s music .
. is based on tradi-
tional tonality. This is true even in the later studies, which are generally more
chromatic or freely atonal than the earlier ones. Tonal centers, even if not

.

.

9
The precrse tempo relations used are unknown because Nancarrow could not recall what templates he had
used to construct this rhythm. See Chapter 5 for a conjectural explanation for the derivation of this rhythm.

.

36

 
 
 
explicit,  are  frequently  implied.  Close-spaced,  root-position  major  triads  are 
ubiquitous,  and  nearly  all  of  the  studies  end  with  some  sort  of  V-I  cadence 
(usually  in  the  form  of  an  ascending  fourth).  (pp.  19-20) 

The  real  significance  of  this  is,  according  to  Carlsen,  that  “the  prevailing  diatonic  context 

makes  a  perfect  foil  for  the  rhythmic  complexities”  (p.  20). 

" 

There  are  several  prevailing  melodic  figures  that  have  been  identified  in  Nancar- 

row’s  Studies,  including  the  ascending  perfect  fourth;  the  minor  third,  particularly  the 

“partitioned  minor  third”  (minor  third  divided  into  a  whole  step  and  half  step;  see  Figure 

1-11);  and  the  melodic  pattern  of  a  whole  step  followed  by  a  minor  third  (5  -  6  -T) 

which  is  variously  described  as  pentatonic  and  derived  from  blues  (an  example  can  be 

seen  in  the  lower  voice  of  Figure  1-6).  (This  melodic  pattern  is  also  prevalent  in  nine- 

teenth-century  hymnody  from  the  southeastern  United  States,  although  this  was  not  likely 

a  significant  influence  on  Nancarrow.) 

V3 

y* 

= 
h
e

x 

ee, 
Pi 
ra 

. 

t 

i
N

r 

. 

t 

y
V

oars 

. 

——— 

a 
ro 
me 

y 

: 

aay 
yr 
z 

-- 
r 

. 

nn 

Figure  1-11.  Opening  of  Study  No.  31  showing  prevalence  of  melodic  “partitioned  minor  third.” 

The  influence  of  blues  on  Nancarrow’s  melodic  writing  can  be  seen  in  passages 

such  as  the  one  in  Figure  1-12  showing  blues  scales  used  in  Study  No.  45b. 

(} 

. 

tT 

try 
ae 
+ 

2 

+ 

~ 

, 
I 
}__f 
T.f 
rc] 

“+ 

| 

| 
| 
L  be 

v3 

< 

Fi 

7 

7 

f 

Pave  4 
a= 

q 

s 

N 

‘\ 

: 

a 

;  al 
alt  7 
eee 

1 

4 
Pe  ee 
rs 

T  {fT 

wa  a7) 

FE} 

7 

( 

7).  The  notation  Nancarrow  uses  is 
Figure  1-12.  Blues  scale  patterns  in  Study  No.  45b  (p. 
“exploded,”  which  is  used  when  the  space  on  the  staff  is  insufficient  for  the  number  of  notes;  the 
given  notes  take  place  on  the  staff  below  in  the  time  frame  indicated  by  the  dotted  lines. 

An  especially  characteristic  melodic  tendency  is  scalar  patterns  that  are  in  the  major 

form  ascending  and  minor  form  descending.  See  Figure  1-13  (see  also  the  top  two  lines 

37 

 
 
 
of Figure 2-6 [p. 71]).

.

fi

2

I

D

1

5

n

I

IV

n

I; A
'

-
' """fi
4 Mini" 4

I

I

U

I 9"”

I

_" t..-

1' i'i'iv

h 1)
: ”eerie: —eii.':‘v
~t#=

n

n

D

L I

11'

L

I

X

Figure 1-13. Major ascending/minor descending scalar passages from: (top) Study No. 6 (p. 5)
and (bottom) Study No. 7 (p. l—major/minor passage is combined with 5 + 4 + 2 + 3 + 4
duration series).

Harmonic Features

This discussion has already noted Nancarrow’s preference for the elements of

traditional tonality, including harmony consisting of prevalent thirds, Sixths, octaves, and

root-position triads; implied tonal centers; traditional cadence structures; and major and

minor scale structures. He also frequently used jazz harmonies such as seventh and ninth

chords (including those with raised fifths and flatted ninths) and major/minor triads, and

blues scales with flatted sevenths and simultaneous raised and lowered thirds. Despite his

rigorous counterpoint training with Sessions, however, Nancarrow claimed to have little

melodic or harmonic imagination.10 Figure 1-14 shows a fairly typical triadic texture in

Nancarrow. In this example, the third canon in Study No. 37, the top five of twelve

voices enter with a level of imitation each time of a fourth higher.

Parallel harmonies like those in Figures 1-13 and 1-14 are indeed quite common in

Nancarrow’s Studies. The presence of parallelism and other non-functional harmonic

structures is what prompts Thomas to comment that “Coordination of pitch class content

among layers, in fact, creates a non-multidimensional texture type called heterophony”

(p. 53). Gann points out that “chords appear not as products of voice-leading or tonal

function, but almost always in parallel, as textural extensions of a Single line” (p. 12). He

lo“One of the main reasons I suppose that I concentrate more on tempo relationships is that l have little
melodic or harmonic invention” (Reynolds 1984, 6).

38

 
 
   
 
   
 
 
 
  
 
Figure 1-14. Entrance of top five voices (of twelve) in third canon of Study No. 37 (p. 9).

goes on to suggest that Nancarrow uses the following harmonic Structures “in increasing

order of emphasis” to fortify a melodic line:

—
V
N
h
t
O

thirds
octaves
parallel major triads
parallel seventh or ninth chords
larger chords generated from a single interval, or
chords which, when condensed within an octave, fill out a contiguous diatonic
or chromatic scale segment. (Gann 1995, 12-13)

These parallel structures are often built up through the accumulation process. For

instance, in Study No. 27 Nancarrow begins with an unharrrronized ostinato and progres-

sively harrnonizes it with thicker harmonic structures. Figure 1-15 shows the four notes

of the original ostinato and the harmonic manipulations to which it is subjected.

As already noted by Carlsen, many of Nancarrow’s Studies end with a V-1 final

cadence, and interior V—I cadences are also plentiful. In the case of canons in which

imitations are at intervals other than the octave (the vast majority). adjustments would be

39

 
  
 
 
  
needed at the end to effect a V-1 ending; Study No. 19 includes such a final cadence

(perhaps because it is the concluding piece in the set of pieces Nos. 13-19). However,

many endings have no harmonic adjustment and conclude straightforwardly in multiple

key areas. Study No. 12 includes a 3-voice canon which establishes three key areas

related by fifths: D, A, and E. The voices cadence at the end on complete triads in these

same three key areas (see Figure 1-7). There is also a number of interior V—I cadences in

tln's canon that are characterized by extended harmony (see Figure 1-16).

pp. 1-18

pp. 18-24

pp. 24-29

pp. 29-42

pp. 42-47

pp. 47-51

pp. 53-55 (final two notes
an adjustment for cadence)

Figure 1-15. Ostinato line of four chromatically contiguous pitches in Study No. 27 and pro-
gression of its harmonic manipulations.

     9

9"

7

-4

A.

(a)

(b)

(C)

Figure 1-16. Interior V—I cadences in Study No. 12: (a) p. 28, first system; (b) p. 28, second
system; (c) p. 29, second system.

40

     
     
 
The influence of blues extended to Nancarrow’s use in several pieces of blues

harmonic progressions (e.g., I—IV-I—V—I) as the basis for ostinati on which an entire

piece is built and also for levels of transposition of large sections in some cases. This was

particularly common in his earlier pieces, such as Nos. 2, 3 (the “Boogie-Woogie Suite”),

and 11, but he returns to this harmonic pattern in some later pieces such as No. 45.

Finally, the levels of imitation used in Nancarrow’s tempo canons have a significant

impact on the harmonic environment. In the earlier tempo canons and other canonic

pieces the level of imitation tended to be at the octave or fourth/fifth; gradually Nan-

carrow began to move to major thirds/tenths to establish what Gann called a “bittersweet

and ambiguous major/minor harmony” (p. 203) in two-voice canons such as No. 43.

Table 1-1 provides a summary of levels of imitation in some of Nancarrow’s Studies.

Table 1-1

Levels of Imitation in Selected Studies

Study No.

No. of voices

Levels of imitation

4

8

12

14

15

18

19

24

32

36

43

49a

49b

3

3

3

2

2

2

3

3

4

4

2

3

3

2 octaves, 2 octaves

2 octaves, 2 octaves

perfect fifth, perfect fifth

2 octaves + perfect fifth

3 octaves

octave

perfect eleventh, perfect eleventh

contains series of canons with levels of imitation
ranging from root position triads to first
inversion triads to open fifths and octaves

perfect fifth between all voices

major tenth, minor tenth, and major tenth
(creating a widely-spaced major seventh chord)

major third

2 octaves + perfect fourth,
3 octaves + perfect fifth

perfect twelfth, perfect twelfth

41

 
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
 
Strucmral thures

Canon is such a pervasive technique in Nancarrow’s Studies that it is treated at con-

siderable length in the four sources. Tenney Speaks quite eloquently of the importance of

this device in Nancarrow’s Studies:

In his frequent use of canon as an organizing device, Nancarrow has revived
an ancient tradition that perhaps began with Machaut in the 14th century, and
reached a point of culmination in the Musikalisches Opfer of Johann Sebastian
Bach before lapsing into a period of neglect as a formal procedure. But no
survey of the past history of the canon could have given us any idea of the
incredible wealth of utterly new possibilities hidden within this humble
device. The word itself has had an intriguing history: deriving from the Greek
karma meaning Simply a length of reed or cane, it became kanon—model,
measure, law, or rule. But like some elegant mathematical equation, what it
“models” is an infinite variety of particular manifestations, becoming the very
“measure” of musical/experiential time. (Tenney, liner notes to Wergo Vols.
III/IV, p. 19)

Thomas creates a taxonomy of canons showing the conventional “1:1” canon plus four

possibilities involving voices moving at different speeds (Figure 1-17). Nancarrow wrote

tempo canons representing each of the four basic types except diverging, as well as

numerous canons that combine various aspects of the basic types.

Figure 1-17. Thomas’s (1996, p. 186) taxonomy of canons: (a) conventional; (b) converging; (c)
diverging; (d) converging-diverging (arch); and (e) diverging-converging.

Gann’s section on tempo canon terminology (pp. 19—28) introduces several

42

 
 
 
  
 
 
 
important terms: convergence point, convergence period, tempo switch, and echo dis-

tance. The convergence point is “the infinitesimal moment at which all lines have

reached identical points in the material they are playing” (p. 21). Convergence points

(CPS) are so critical to the structure of tempo canons that Chapter 3 of this work will be

devoted to them. The convergence period is “the hypermeasure that exists between

(potential) Simultaneous attacks in voices moving at different tempos” (p. 21), which

Gann specifies as being important to keep track of in canons in which meters change

frequently. Not all canons contain a tempo switch, “a device in which Nancarrow

switches the [faster] line to the [slower] tempo and vice versa” (p. 21), but when they

appear they always occur halfway between CPS involving the affected voices. Finally, the

echo distance is “the temporal gap between an event in one voice and its corresponding

recurrence in another” (p. 21); as voices approach a CP the echo distance decreases, and

as voices move away from a CP the echo distances increases. Gann’s illustration of these

terms is Shown in Figure 1-18.

Convergence period
r
 J=80

I  

t

1

EChO distance

L____J
50110 distance

Convergence
point

Tempo
switch

Convergence
point

Figure 1-18. Gann’s tempo canon terminology: convergence point, convergence period, tempo
switch, and echo distance (Gann 1995, 20).

43

  
 
  
 
Symmetry  of  various  kinds  is  also  structurally  significant  in  Nancarrow’s  Studies. 

The  arch-shaped  converging-diverging  tempo  canons  exhibit  symmetry  about  the  central 

CP,  Other  non-canonic  pieces  are  virtually  palindromic,  including  Studies  No.  1  and  43. 

Palindromic  isorhythms,  such  as  the  3+2+2+34+2+34+3424+34+2+2+3  jso- 

rhythm  found  in  Study  No.  7,  abound  in  the  Studies;  several  other  lengthy  examples  from 

Study  No.  8  are  shown  in  Figure  1-23.  Also,  Figure  1-24  shows  a  complex  series  of 

isorhythms  that  is  symmetrical.  An  example  of  a  mirrored  rhythm  is  located  by  Carlsen  in 

Study  No.  35  (see  Figure  1-19). 

Mirror  rhythm  (systems  6-7) 

a 

Figure  1-19,  Mirrored  rhythm  in  Study  No.  35  (Carlsen  1988,  56). 

In  Study  No.  19,  a  3-voice  tempo  canon  subtitled  “Canon  12/15/20,”  Nancarrow 

symmetrically  partitioned  the  keyboard  into  overlapping  4-octave  segments  about  the 

player  piano’s  middle  note,  and  he  employed  a  similar  procedure  in  Study  No.  36  (see 

Figure  1-20). 

CPiayo'S 
CENTRAL  Nore) 

XX 

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CPano's 
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ra 
rae 

be 
7. 

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ns 

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£ 

= 
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(b) 

Figure  1-20.  Symmetrical  deployment  of  canonic  voices  in  (a)  Study  No.  19  and  (b)  Study  No.  36 
(Carlsen  1988,  6,  24). 

Symmetries  also  occur  in  some  of  the  Studies’  harmonic  structures.  The  major 

seventh  chord—a  symmetrical  structure  with  two  major  thirds  separated  by  a  minor 

third—defines  the  levels  of  imitation  in  Study  No.  36  and  pervades  that  piece’s  harmonic 

44 

environment (including the final chord). In Study No. 27, a series of eleven 4-voice

canons, the levels of imitation are symmetrical in a number of the canons (see Table 1-2).

Table 1-2

Symmetrical Levels of Imitation in Selected Canons of Study No. 27

Canon #

Levels of imitation (highest to lowest voices)

1

2

3

4

8

octave + major sixth
octave + minor Sixth
octave + major sixth
octave + perfect fifth
2 octaves + major second
octave + perfect fifth
octave + minor sixth
2 8ves + major second
octave + minor sixth
octave + minor sixth
2 octaves
octave + minor sixth
octave + major third
2 octaves + major seventh
octave + major third

Melodic and pitch inversions are also fairly frequent, such as the second ostinato in

Study No. 9 and mirrored glissandi in Study No. 25 (see Figure 1-21).

Mathematical ratios also play a significant structural role in Nancarrow’s Studies;

this topic will be explored more thoroughly in Chapter 2. In particular, ratios based on

pitch intervals of the justly-tuned scale are prominently used. Besides being used de-

liberately in establishing relative Speeds between voices in tempo canons, mathematical

ratios are used by Nancarrow in a number of ways, such as creating relationships between

pitch ratios and melodic and harmonic features; and to control such factors as the levels

of imitation between voices, positioning of voice entrances, and other structural concerns.

45

 
 
 
 
 
  
 
 
 
a 

; 

1 

— 
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(octave  displacement) 

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Figure  1-21.  Symmetrical  melodic  structures  in  (a)  Study  No.  9  (p.  5)  and  (b)  Study  No.  25  (p. 
60). 

Texture,  Timbre,  and 

Dynamics 

Several  textural  features  of  the  Studies  can  be  traced  to  limitations  of  the  player 

piano  medium.  The  first  is  the  prevalence  of  staccato  notes—Nancarrow  reported  to 

Reynolds  (1984,  p.  20)  that  this  was  because  punching  sustained  notes  was  simply  more 

work,  as  a  line  of  holes  must  be  punched  for  sustained  notes  whereas  a  single  punch 

produced  a  staccato  note.  (Eventually  Nancarrow  had  his  punching  machine  altered  so 

that  he  could  punch  a 

line  of  four  holes  at  once,  and  this  alleviated  the  problem  some- 

what.)  The  other  textural  feature  affected  by  the  player  piano  was  the  number  of 

simultaneous  notes  that  could  be  sounded—due  to  potential  loss  of  air  pressure  from  the 

sounding  mechanism  and  the  possibility  that  too  many  holes  punched  too  close  together 

could  cause  the  roll  to  perforate,  the  player  piano  is  limited  to  playing  about  sixteen  notes 

at  a  time  (Carlsen  1988,  71  note  7).  The  player  piano  has  a  sustain  pedal,  but  Nancarrow 

used  it  in  only  one  piece  (No.  25),  so  the  overall  texture  in  the  Studies  tends  to  be 

somewhat  dry  and  spare. 

46 

 
 
 
 
Carlsen notes Nancarrow’s tendency to assign faster tempos to higher voices,

adding that:

Compelling acoustical reasons have of course also made this true of much
other music; one could cite, as Cowell does, the model of the overtone series,
where the higher notes vibrate at faster frequencies than the lower ones. But it
is interesting that, in most cases, the only concession Nancarrow makes to the
differences in tone quality between high and low registers is to assign them
different tempos. In other respects, high and low are treated equally; Nan-
carrow does not usually compensate for the quick decay and light sound of
notes in the upper registers. (Carlsen 1988, 23)

Finally, despite the fact that gradual crescendos and decrescendos are possible on

the player piano, Nancarrow has a decided preference for terraced dynamics. Gann ex-

plains the mechanics behind the process this way:

At one edge of the piano roll is a place for up to three holes that determine
how much air pressure the machine uses, thus how much volume it produces.
For whatever reason, these holes are designated 2, 4, and 6. Eight dynamic
levels are possible, ranging from no holes to all three. In Nancarrow’s
punching scores, he notates those levels as (—0—), (—2—), (-4—), (-6—),
(4—2), (6—2), (6—4), and (6—4—2). He rarely seems to need all eight levels in
one work, however. Study No. 48a contains six dynamic levels, and in the
final score Nancarrow notates (6—4—2) and (6—4) as fi‘, (—6—) as f, (—4—) as
turf, (—2—) as p, and (—0—) as pp. (Gann 1995, 29)

Comparison of Analyses of Selected Studies

It is instructive at this point to compare the analytical treatment of the Studies in the

various sources. The most comprehensive sources on the individual Studies are Gann and

Tenney. As we have seen, Gann reviews each of the Studies, and Tenney’s liner notes for

the Wergo recordings and 1977 article in Soundings, Book 4 cover all of them, as well,

although in considerably less detail. However, Carlsen’s monograph and Thomas’s

dissertation offer more detailed analyses of fewer, selected Studies. There are four

Studies—Nos. 8, 19, 23, and 35—that are reviewed at least to some extent by all four

sources, and these provide a useful sample for comparison of the four major analytical

sources.

47

 
Study No. 8 is discussed by Gann on pp. 148-58, by Carlsen on pp. 34—46, by

Thomas on pp. 89—91, and in Tenney’s liner notes to Wergo Vols. III/IV on p. 11. There

is general agreement that non-strict canonic imitation occurs in much of the piece, which

is in four voices, and both Carlsen and Gann identify three sections to the study and agree

on their locations (neither Thomas nor Tenney comments on the form). Gann and Tenney

observe that this is the first of the Studies to use proportional notation, and Gann calls

this Nancarrow’s first experiment in acceleration; Tenney is characteristically succinct in

commenting only on the notation and the “gradual fluctuations in tempo, alternately

accelerating and decelerating, in a way that appears to be quite independent of the other

voices, and unrelated to any sense of meter” (p. 11). The acceleration is arithmetical and

accomplished through using multiples of an invariant background unit, although Gann

credits Nancarrow for developing some numerical series in the work that approximate

geometric acceleration. The accelerating rhythm of the opening line is discussed by both

Gann and Carlsen, but Gann’s “conjectural notation” portrays the acceleration of the

melody while Carlsen’s re-notation portrays only its triple time feel (see Figure 1-22).

(b)

Figure 1-22. (a) Carlsen’s (p. 35) and (b) Gann’s (p. 149) re-notations of the opening rhythm in

Study No. 8. (Carlsen’s example begins with the score’s second system and corresponds to the

pick-up to Gann’s 32 measure.)

9

48

 
 
  
  
 
 
 
  
 
 
 
 
 
 
  
 
 
  
The accelerative process illustrated in Figure 1-22 is actually isorhythmic, and both

Gann and Carlsen devote the majority of their attention to identifying the rhythmic series

used and their various manifestations. Gann identifies three accelerating/decelerating iso-

rhythmic series used in the Study while Carlsen refers to “accelerating succession[s] of

durations” (p. 35). Because of the non-metric proportional notation, identifying the rhyth-

mic series requires that the length of time between notes be measured on the score in

millimeters; however, Carlsen and Gann convert these measurements for purposes of

analysis in different ways. Carlsen identifies the number of separate elements in the series

and then assigns each value an ordinal number. Gann identifies an “invariant background

durational unit” (p. 148) in the rhythmic series, and expresses each duration in the first

series as a multiple of that background unit. For instance, both Carlsen and Gann describe

the first rhythmic series as consisting of nineteen durations, with 41 millimeters being the

longest and 10 being the shortest: Carlsen expresses that series in values from 1 (longest)

to 19 (shortest); Gann divides each measurement by the background unit (ca. 1.7 mm.) to

arrive at a series that progresses from 24 (longest) down to 6 (shortest). Note that

Carlsen’s system creates an inverse relationship with respect to length of the note while

Gann’s does not.

Gann’s three series correspond to the three sections of the work, although he notes

that the first series appears in all three sections. Strangely, he does not follow through

with division by the background unit in expressing the second series. Figure 1-23 shows

Gann’s three series and Carlsen’s graphic representation of the first series; the values

expressed in Gann’s three series are not directly comparable, because the second series is

not expressed in multiples of the background unit.

Carlsen, too, devotes a considerable portion of his analysis to examining all the

duration series, particularly their symmetrical applications. The symmetrical structures of

the series are readily evident in Figure 1-23. Other symmetrical structures identified by

Carlsen include the canonic subject of the second section, a 35-note palindromic line

49

.-

Ritard
6

7

8

9

1O

Hit.
9

10

10

11

11

12
Accel.

11

12

13

6

7

8

Hit.
9

10

10

9

10

11

12

11

11

12
Acoei.

13

14

14

15
Acoel.

(a)

7
9 9

9

9

17
24 24 24

1717 17
24

24

12

12 12 12

17

12 12 12
17

12

17 1717

17

max 3
43

43
55

24

24 24 24

24

m max
43
43
(551

2

3

3

4

4 4
5

4

5 5 5
s

e

6

7

(C)

4

5 5 5

6

3

4 4 4

5

(b)

(2)

3

((0

Figure 1-23. Gann’s three isorhythmic series identified in Study No. 8: (a) section 1, with the
series expressed as multiples of the background durational unit (p. 150); (b) section 2, expressed
in actual measurements in millimeters (p. 151); and (c) section 3, expressed as multiples of the
background unit (p. 152); plus ((1) Carlsen’s graphic representation of the series in section 1
(p. 37).

consisting of only three pitches; and the pitch sequence of the top voice in this same

section, an ascending scalar movement in the treble from F# to B and back that is mir-

rored in a descending scalar movement in the bass from F# to C#. When these materials

are combined with the symmetrical rhythrrric series that abound in this piece, it is clear

that the work is as much about symmetry as it is about acceleration.

The work’s third section introduces a different idea—that of heterophony among

three canonic voices. As shown in Figure 1-24 from Carlsen, the canon is duplicated at

alternate notes first at the octave above in the bottom voice, at the fifth above in the

50

 
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51

 
 
 
 
         
 
   
 
   
 
 
 
 
 
 
middle voice, and at the minor third in the top voice. The result is a consistent texture of

major triads expressing the symmetrical rhythmic series of this section.

Thomas chooses to focus on this last section in her discussion on heterophony as an

example of heterophony involving parallel movement of voices in intervals other than

octaves. She disagrees with Gann and Carlsen by stating that “this is a rhythmic but not

pitch canon” (p. 90); it is not clear why She would not comment on the canon, except that

she seems to be much more focused on the creation of vertical triadic sonorities than on

the linear element. She believes that there is so much coordination among the voices that

“the heterophonic pitch relationship causes the voices to lose their individual identities”

(p. 90). At the point that the third voice comes in, she analyzes on the score the triadic

structures that result, as Shown in Figure 1-25 (Gann performs a similar analysis on p.

152).

Figure 1-25. Thomas’s identification of triadic structures in the trio canon of the third section of
Study No. 8 (p. 250). (This point corresponds to the entrance of the third voice in Figure 1-24).

52

  
 
 
 
Smdy No. 12 (Canon IQISZZQ) is discussed by Gann on pp. 114-15 and 119-20, by

Carlsen on pp. 11—28, by Thomas on pp. 87—88, and by Tenney in the liner notes to

Wergo Vols. III/IV on p. 24. The form of this piece is rather transparent: as the title indi-

cates, it is a 3-voice canon with the tempo ratio of 12:15:20. It is also a converging canon,

with the three voices converging on the very last note of the piece. As is typical in Nan-

carrow, the slowest/lowest voice enters first, followed by a higher/faster voice and finally

the highest/fastest voice.

Tenney refers to the voices as “strata,” which is an apt term since each is a com-

posite of four rhythmic layers. Much attention is focused on the rhythmic resultant, with

Carlsen, Gann, and Thomas all detailing the following information: (a) the additive for-

mula {n — 1, n, n + 1, n} is used to establish the following rhythmic patterns (in eighth-

notebeats) forthehighest to lowest voices:3 + 4 + 5 +4,4 +5 + 6+ 5,5 + 6+ 7+ 6, and

6 + 7 + 8 + 7; (b) the same melody is projected in all four voices, with the lower voices

occasionally dropping notes along the way in order to keep up. The result is, as Thomas

notes, an “incipient form of heterophony” (p. 87), which can be seen in Figure 1-26.

<3454>

<5654>

<5676>

<6787> 

Figure 1-26. Heterophony as projected through four rhythmic layers in Study No. 19 (Thomas
1996, 247). Note the omission of some notes in the lower and slower layers (E,, for example, is
projected only in the top voice). The number series at the left of each staff represents the note
durations in eighth notes according to the formula { n — l, n, n + 1, n}.

53

The registral placement of the voices is of interest to Carlsen and Thomas. Carlsen

notes the symmetrical placement of the voices about the player piano’s middle note (E,)

into overlapping four-octave segments, with the interval of imitation between the voices

being an eleventh. To Thomas, this—in tandem with the heterophony displayed in each

canonic layer—results in a lack of registral separation: “the potential for multidimen-

sionality inherent in the registral polyphonization of the single line is denied by its

heterophony” (p. 88).

Carlsen devotes the most attention and speculation to the use of tonality in the work

and attendant harmonic planning and motion. Tenney, Carlsen, and Gann all note the ad-

justment in the canon at the very end so that all voices end on F# and effect essentially a

V—1 cadence, and Gann calls this “the most traditionally tonal of Nancarrow’s tempo

canons” (p. 120). Carlsen asserts that Nancarrow’s compositional method was to first plot

out the rhythmic relationships, then carefully select melodic pitches that would create the

desired harmonies Since “most of the harmonies in Study No. 19 are major and rrrinor

triads or other traditional chords, connected moreover by quite elegant voice-leading” (p.

10). Figure 1-27 is representative of the strong tonal implications that result when all

three voices are part of the texture.

 ~ -

I

5‘--.‘

‘.’I

Figure 1-27. Triadic structures in Study No. 19, systems 7-8 (Carlsen 1988, 14).

54

Carlsen then turns his attention to debating the specific tonality of the piece: he

makes a strong case for B being the most prominent tonality, but finds support for C# and

F# (and a long—range tonal motion between the two) as well.

With all of the above information in mind, Carlsen includes in his analysis an in-

ferred “underlying set of precompositional rules for rhythmic organization” (p. 9), which

I include here in its entirety:

(1) Establish four closely related isorhythmic patterns (taleae).

(2) Give a different pattern to each of four octaves constituting the range of a
voice, assigning the fastest pattern to the highest octave, the next-fastest to
the second octave down, and so on.

(3) Determine the length of the study by finding that point after the beginning

at which notes in all four patterns will coincide.

(4) Replicate the rhythmic patterns for two other voices after deciding on
tempo ratios. Assign the fastest tempo to the highest voice. Carefully
measure and draw these patterns on a piano roll or on music manuscript
paper, working backwards from a simultaneous ending. At this point, the
specific number of notes in the study, the rhythms, and the relationships
between voices are set—the only thing that remains is to choose pitches.
(It is perhaps worth reemphasizing that there are twelve lines of rhythmic
activity—four for each voice.)

(5) Determine the interval of imitation by symmetrically partitioning the

piano into overlapping four-octave segments, one for each voice.

(6) Write a melody for the soprano octave of the lowest voice. Fill in the
pitches for the alto, tenor, and bass octaves by referring to the soprano
line. (Notes in a lower octave must be of the same pitch-class as Simul-
taneous or proximate soprano notes.)

(7) Now that the pitches are all set for the lowest voice, use the interval of
imitation (an octave plus a fourth) to derive all the pitches for the other
two voices. Make a slight adjustment in notes at the very end to achieve a
sort of V-I cadence (C# to F#) and the piece is finished. (p. 9)

Smgy No. 23 is discussed by Gann on pp. 156—59, by Carlsen on pp. 47—54, by

Thomas on pp. 128—32, and by Tenney on p. 14 of the liner notes to Wergo Vols. III/1V.

Carlsen, Gann, and Thomas all agree that the piece is divided into four primary sections,

and they are in agreement as to where the sections divide. Carlsen and Gann both com-

ment on the total lack of canonic imitation in this piece. The sources are in general con-

sensus that there is little relationship between the two voices and, as Carlsen (1988) says,

55

the  “two  voices  remain  quite  independent  of  one  another  from  beginning  to  end,  and  it 

appears  that  one  of  Nancarrow’s  purposes  here  is  simply  to  exploit  the  differences 

between  them”  (p.  47). 

Thomas  points  to  this  piece  as  one  involving  different  “time  types,”  with  one  of  the 

two  voices  being  metered  most  of  the  way  through  and  the  other  voice  being  pro- 

portionally  notated  and  highly  variable  in  speed.  The  metered  voice  frequently  changes 

meters,  but  a  steady  eighth-note  pulse  is  maintained.  One  of  the  more  interesting  effects 

in  the  second  section  occurs  in  the  unmetered  voice:  a  countdown  device  is  combined 

with  a  duration  series  in  which  durations  are  tied  directly  to  pitch  level.  The  71  notes 

from  B,  to  A,  are  assigned  a  gradually  decreasing  duration  ranging  in  measurement  from 

59  millimeters  to  %  millimeter.  Carlsen  points  out  that  durations  that  are  related  by  the 

ratio  of  2:1  are  a  major  ninth  apart  (see  Figure  1-28).  This  same  rhythmic  series  was 

actually  established  in  the  first  section  as  part  of  a  71-note  accelerando  in  which  the 

pitches  are  not  stated  in  chromatic  order.  Both  Carlsen  and  Thomas  correctly  note  that  in 

|  ya 

w
o

m
n

e
o

t
o

e
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w
o

=

e
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n
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7

d
n
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d
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t

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t
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o

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i 
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tg  OE 

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eat 

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+. 

—  te 

&va-3  #& 

Figure  1-28.  Duration  series  in  Study  No.  23  (Carlsen  1988,  49). 

56 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
the second section the durations are ordered from lowest to highest, while in the opening

accelerando they are not.

Thomas focuses on the second section’s countdown device in which the lower voice

states pitches first in quintuplet groups, then quadruplets, triplets, and finally duplets

(Carlsen enumerates 23 quintuplets, 28 quadruplets, 44 triplets, and 71 duplets). For each

grouping, the duration of each note is the same and is determined by the duration as-

signed to the first pitch of the grouping. Additionally, at each point where the size of the

groupings decreases, the tempo of the upper voice increases. Thomas identifies several

passages in this section where there are near convergences in attacks between the lower

voice and the metered upper voice; one such example is shown in Figure 1-29. In the first

Figure 1-29. Moments of near convergence of attacks in second section of Study No. 23 (Thomas,
p. 287). Broken underlines beneath second staff indicate near convergences.

57

 
  
 
  
   
 
 
 
  
 
 
 
 
 
 
 
 
  
     
   
 
 
   
 
 
 
 
   
 
 
 
 
  
case (from the first to the second systems) the bottom voice nearly converges with the

quarter note pulse in the upper voice, and in the second case (second to third systems) it

nearly converges with the eighth note.

Tenney does not delve into the underlying system of this section, but comments on

the effect of the faster, higher figures that are intermittently stated: “Occasionally inter-

jected into this texture we hear fast, high-pitched four- or five-note figures. These are

notated on the same staff as the lower voice, but they suggest to the ear the presence of an

intermittent third voice in the texture” (p. 14).

Carlsen, Gann, and Thomas all comment on the structure of the metered voice,

particularly its pitch content (its rhythm is generally agreed to consist of somewhat

random meter changes with a steady eighth-note pulse.) In the opening section and

through the statement of quintuplets in the lower voice, the upper voice consists of only

combinations of dyads and single notes in which Thomas identifies the prevalence of set

classes 3-1 [012] and 3-2 [013]. Carlsen astutely observes that in the second section, as

the groupings decrease in size, the chords in the upper voice at first become thicker:

where the lower voice has quadruplets, the dyads in the upper voice have been replaced

by trichords; and with triplets in the lower voice, the upper voice states four-note chords;

the upper voice returns to dyads where the lower voice has duplets.

The third and fourth sections are given the fullest treatment by Carlsen. The third

section does not include the metered voice and features an upper voice with a gradually

accelerating sequence and a lower voice of seemingly random sustained pitches that use

the same 71-note duration series already seen. Carlsen makes an extraordinary case that

Nancarrow arrived at the pitches in the lower voice:

by drawing diagonal lines on the roll through the beginnings or ends of notes
that have already been plotted, using the intersections of those diagonal lines
with various other lines to determine the pitches of later notes. He thereby
establishes a long chain of interrelationships between the pitches and temporal
placements of almost all the notes in the passage. (p. 52)

58

Figure 1-30 shows his attempt to verify this method. Gann was quite Skeptical that Nan-

carrow would have worked this way, and indeed says that “Nancarrow denies that he ever

did such a thing, or that he ever let the visual appearance of the roll influence a musical

decision. Indeed, it strikes me that such a non-musical strategy would be foreign to his

thinking” (p. 293, note 8).

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Figure 1-30. Carlsen’s (1988, p. 53) diagram illustrating the plotting of pitches in systems 54—55
of Study No. 23. The diagram below the staff represents the piano roll, on which pitches are
ordered lowest to highest from top to bottom.

Study No. 35 is covered by Gann on pp. 248—53; Tenney very briefly on p. 16 of

the notes to Vols. I—H and on p. 62 of Soundings, Book 4; Thomas on pp. 63-65; and

Carlsen (1988) on pp. 55—65. All four sources note that this piece has canonic features

but that exact canonic imitation is only hinted at and rarely used. They also make note of

the large number of different tempos used (eleven), and Carlsen points out how unusual it

is for Nancarrow to use more discrete tempo markings than the maximum number of

59

      
       
 
  
 
 
sounding voices (five in this case). Gann makes an interesting observation in pointing out

that the fastest five tempos are “octave multiples” of slower tempos, “leaving only Six

real tempos” (Gann 1995, 249).

Tenney’s description is, as usual, very cursory—it does not identify the number of

formal sections, nor much about the pitch elements except to say that there are “different

aggregate-types used, which include octaves, 3rds, triads, and miscellaneous chords.”

There is also no mention of the jazz elements (both rhythmic and melodic) that are

identified by the other sources as being quite prominent. Carlsen notes, for instance, the

trochaic (“swung”) rhythm that characterizes the main melodies, and points to the con-

stant evolution of the melodies as suggesting improvisation; he also characterizes some

accompanimental patterns as “boom-chick” and notes a call-and-response pattern be-

tween voices that mimics jazz performance. Gann also finds similarities to jazz in the g

syncopated melody and the call-and-response pattern.

There is slight disagreement between Gann and Carlsen on the form of the piece

(Thomas discusses only the first nine pages of the work and does not comment on the

form). Gann identifies eight sections in the study and discusses each in turn. He goes into

considerable detail for each section about the progression of two similar “basic melodic

patterns” that he labels A and B, and notes for each section the tempo ratios that result

with the various combinations of the eleven different tempos. Gann identifies the fifth

section as a “brief bridge,” but in Carlsen’s formal diagram this section belongs to the

previous section, giving a total of seven sections in his analysis (see Figure 1—31, in

which the unbroken vertical lines delineate the seven sections). Carlsen’s formal diagram

corroborates his assertion that there is a sectional “acceleration” where each section is

shorter than the previous section, and he astutely identifies a “grand substructure” in

which various combinations of overlapping sections (A, B, C, and D in the example) are

exactly the same length (measured in millimeters since the score is notated in “propor-

tional notation” in which elapsed time is conveyed on the horizontal axis).

60

,A
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119
113’l3
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79V:

Figure 1-31. Carlsen’s formal diagram for Study No. 35 (1988, p. 65) Showing seven sections and
proportional relationships (A, B, etc.).

Gann, Carlsen, and Thomas all characterize the melodic material in the first section

as going through a three-phase evolutionary process, and Gann and Carlsen use similar

diagrams to present the progression of the five voices through this process. This section is

the only one discussed by Thomas, and it is interesting that she chooses this quasi-

canonic piece to lead into her discussion on canon. Thomas discusses this piece as an

example of accumulating layers that maintain their textural identity, and points to the

progressively higher pitch and faster tempo of each successive voice as contributing to

this perception: “There is no pitch imitation, but this broader form of imitation is vivid,

and, in fact, is partially responsible for the perceptibility of the individual layers in the

accumulation, since we can follow a layer’s functional progression” (Thomas 1996, 65).

Like Gann (and, to some extent, Carlsen), Thomas comprehensively follows the pro-

gression of the melody and accompaniment patterns through each of the five voices.

Gann and Tenney both note the presence in this study of isorhythms—or, more

appropriately, recurring rhythmic patterns. The main melody incorporates a rhythmic

pattern (in eighth note durations) of 2 + 3 + 3 + 2 + 4 + 3 + 4 + 2 that is stated three

times, the find time in retrograde. This pattern recurs in later sections, and other isorhyth-

mic patterns are used as well with several of them used together near the end of the piece.

61

 
 
 
 
 
         
   
     
 
Little mention is made by any of the writers on the pitch content of the piece,

despite the overwhelming presence of [025] in the main melody and its progressive

versions. Gann, Carlsen, and Thomas do note the importance of the perfect fourth and

minor third as motivic material, but structural pitches are rarely identified. The formal

analyses by Gann and Carlsen include no mention of structural pitches or large-scale

movement around pitch centers, although such assertions could certainly be made.

Summary of the Literature

The analyses by Gann, Thomas, Carlsen, and Tenney offer varying viewpoints on

Nancarrow’s Studies and focus on different aspects of them. Thomas focuses her

discussion most intently on the layering technique and the factors that contribute to or

detract from multidimensionality; she is less likely than some of the other sources to

comment on form or perform melodic analysis.

The two authors who create a classification system for the Studies, Gann and

Tenney, use completely different approaches. Gann classifies all the Studies into six

groups based on the compositional techniques used, along with formal features; the cate-

gories are well-defined and provide for useful discussion of compositional techniques

such as isorhythm, canon, acceleration, and use of ratios, and the forms that result.

Tenney’s system is chronological, which works fairly well for some periods in Nan-

carrow’s career but not as well in his middle period; Tenney’s remarks tend to focus on

the textures created by the interaction of the musical layers.

Carlsen analyzes only five Studies selected from book 4 of the Soundings series,

focusing especially on symmetry of design at different levels and the use of ratios in

Nancarrow’s forms. He chooses for his most thorough analysis Study No. 19, a 3-voice

converging tempo canon, as an example of a work especially providing insights into

Nancarrow’s “overall compositional style.”

My intent in the present work is to build on the work of these four authors by

62

presenting chapters focused on specific structural attributes of Nancarrow’s Studies: the

use of ratios, the presence and placement of convergence points, and fractal formal fea-

tures. Each of these three chapters will include an analysis of a work representative of the

chapter topic. A final analytical chapter (Chapter 5) will be devoted to analyses of two

works—Nos. 25 and 45—that are representative of Nancarrow’s mature compositional

style.

Our look at these structural features begins in Chapter 2 with the use of ratios in

Nancarrow’s Studies.

63

CHAPTER 2

THE USE OF RATIOS 1N NANCARRow’S STUDIES

Nancarrow’s Studies are pervaded by the deliberate use of ratios, particularly ratios

based on intervals found in just intonation and the overtone series. Nancarrow has

deployed ratios in his compositions in remarkably inventive ways to control many of his

musical materials. Besides the most obvious—the deliberate use of ratios in the establish-

ment of tempo proportions in the tempo canons—Nancarrow used ratios to create iso-

rhythmic and melodic patterns, to establish relationships between pitch materials and

other details, and in certain structural features. Sometimes the same ratio is used within a

piece in a number of different ways to create clever but subtle connections between

musical features.

Ratios have been classified as far back as the ancient Greek mathematicians into the

following five types (Barbera 1984, 194—95; all examples given here are whole-integer

ratios representative of pitch intervals):

(1)Multiple: the greater term is some multiple of the lesser term, e.g., 3:1.1

(2) Suxgparticular: the greater term is exactly one larger than the lesser (with the

lesser term greater than one in order to exclude the ratio 2:1, which is defined as a mul-

tiple), e.g., 4:3, 25:24.

lPitch interval ratios can be presented both with the lesser term first (2:3) or the reverse (3:2), but in both
cases the same interval (in this case, the perfect fifth) is represented. The latter system is more pre-
dominantly used today and will generally be preferred in this study to refer to pitch ratios, although Cowell
used the first system and Nancarrow followed suit. It is useful to be conversant in both systems. The first
system (e.g., 2:3) relates the pitch interval to the portion of a string length that will create the purely-tuned
interval-—thus, for the perfect fifth, the interval is created between a string’s full length and the String
shortened to 2/3 of its original length. However, the second system (e.g., 3:2) has the advantage of placing
all the intervals between the unison and the octave on a logarhythmic scale between 1 and 2, with 1/2
representing the octave’s halfway point (the tritone).

64

 
(3) Sumrpanient: both terms are greater than one and the greater term contains the

lesser plus more than one part (but less than the whole) of the lesser—i.e., the greater

term is something less than twice the lesser. The mathematical expression is (x + m):x,

x > m > 1; examples include 5:3, 8:5, 16:9.

(4) Multiple superparticular: the greater term is one greater than some multiple of

the lesser term, e.g., 7:2, 7:3, 10:3.

(5) Multiple summartient: the greater term contains the lesser more than once plus

more than one part (but less than the whole) of the lesser, e.g., 8:3, 24:7.

The most common types of ratios representing pitch intervals, and thus used in

Nancarrow’s music, are multiple and superparticular ratios. Additionally, it is important

to note that ratios consisting of adjacent pairs in the Fibonacci series 1, 1, 2, 3, 5, 8, .

. .,

in which the greater term is at least 5 (e.g., 5:3, 8:5, 13:8) are superpartient ratios; ratios

consisting of pairs of Fibonacci numbers in which a number is skipped in between and

the greater term is at least 5 are multiple superpartients, e.g., 5:2, 8:3, 13:5. Since a

number of Fibonacci-related ratios describe pitch intervals (e.g., 5:3 = major sixth, 8:5 =

minor sixth, 8:3 = perfect eleventh), these ratios Show up with some frequency in

Nancarrow’s work. Multiple superparticular ratios are probably the least commonly

represented.

In this chapter, references to interval equivalents of ratios will follow either Cowell

(1930) or Abraham and von Hombostel (1994); see Figure 2-1. Figure 2-la identifies the

four different sizes of half-step available in Cowell’s chromatic scale, ranging in size

from a 4.2% increment (25:24) to a 7.1% increment (15:14); Figure 2-1b relates intervals

in the just and Pythagorean systems and shows some less common interval ratios.

The relationship of pure interval ratios to the harmonics of the overtone series

Should not be overlooked. Figure 2-2 illustrates the overtone series based on C.

65

12-tone
tempera-
ment

Just

Pythago—
rean

Ratios

Minor second
(semitone)

Major second
(whole tone)

100

200

Minor third

300

Major third

400

Fourth

Tritone

.
Fifth

.

.

Minor Sixth

.

.

Major Sixth

500

600

700

800

900

Minor seventh

1000

Major seventh

Octave

l 100

1200

90

24:25
2432256

204

408

498

702

15:16

9:10

8:9
7:8

6:7

5:6

4:5

64:81

3:4

5:7
32:45

2:3

5:8
3:5

906

16:27

996

4:7
9:16

3:15

1110
1200

128:243
1:2

70

l 12

182

204
231

267

316

386

498

583
590

702

814
884

969
996

1088

1200

(b)

(15:14)

(21:20)

(16:15)

(25:24)

(16:15)

(21 :20)

(15314)
(16:15)

(25:24)

(21:20)

(15:14)

(16:15)

C

1:1

C#

15:14

I)

E1,

E

F

F#

G

At,

A

B1,

B

C

9:8

6:5

5:4

4:3

7:5

3:2

8:5

5:3

7:4

15:8

2:1

(21)

Figure 2-1. (a) Cowell’s justly-tuned chromatic scale, with all interval ratios in relation to C
(Cowell 1930, 107); the right column’s parenthetical figures indicate size of half-steps between
adjacent pitches. (b) Interval ratios and cent sizes in just and Pythagorean intonation (Abraham
and von Hombostel 1994, 451).

66

 
 
 
 
 
    
 
 
    
Humanist

1 36-

41112111.»

Tent

Figure 2-2. The overtone series and its numbered harmonics (Jorgensen 1977, 16).

Based on the overtone series, Jorgensen makes the following observations about

interval ratios and harmonics:

Intervals taken from consecutive harmonics [superparticular ratios]—

Ratio2
1/2
2/3
3/4
4/5
5/6
6/7
7/8
8/9
9/ 10
10/1 1
11/12

15/ 16

Name

octave
fifth
fourth
major third
minor third
small minor third
maximum tone
major tone
minor tone
minimum tone
3/4 tone

Size in Cents
1200
702
498
386
316
267
231
204
182
165
151

just diatonic semitone

1 12

Intervals taken from every second harmonic [superpartient ratios]—

Ratio
3/5
5/7
7/9
9/1 1

Name
major sixth
small tritone
large major third
neutral third

Size in Cents
884
583
435
347

Intervals taken from every third harmonic [superpartient ratios]—

Ratio
4/7
5/8
7/10

Name

harmonic minor seventh
minor sixth
large tritone

Size in Cents
969
814
617

(Jorgensen 1977, 17)

2Notice the use of the virgule, rather than the colon, in Jorgensen’s discussion to separate the terms of the
pitch ratio. Nancarrow also uses the virgule in describing his tempo ratios.

67

 
 
The importance of the overtone series in musical materials other than pitch was a

major theme in Cowell’s New Musical Resources. In its “Introduction” he writes:

The purpose of New Musical Resources is not to attempt to explain the
materials of contemporary music, many of which are not included in its dis-
cussion, but to point out the influence the overtone series has exerted on
music throughout its history, how many musical materials of all ages are
related to it, and how, by various means of applying its principles in many
.
different manners, a large palette of musical materials can be assembled. .
.
The very fact that such materials are built on the overtone series, which is the
greatest factor in musical relationship, Shows that they probably have
potential musical use and value. .

.

.

The result of a study of overtones is to find the importance of relation-
ships in music and to find the measure by which every interval and chord may
be related. .
. It is also discovered that rhythm and tone, which have been
thought to be entirely separate musical fundamentals (and Still may be con-
sidered so in many ways) are definitely related through overtone ratios.
(Cowell 1930, x-xi)

.

Ratios in Other Twentieth-Century Music

One of Cowell’s primary ideas in New Musical Resources that greatly intrigued and

motivated Nancarrow was the creation of rhythmic complexes based on pure interval

ratios. Cowell’s book contains a number of diagrams, like that in Figure 2-3, Showing

equal measure lengths divided according to a common pitch ratio such as the 3:4:5

second-inversion major triad.

AS has been frequently noted in the literature, Cowell remarked on the potential

utility of the player piano in realizing his complex tempo relationships as well as the

general under-utilization of the capabilities of the player piano:

Some of the rhythms developed through the present acoustical investigation
could not be played by any living performer; but these highly engrossing
rhythmical complexes could easily be cut on a player-piano roll. This would
give a real reason for writing music Specially for player-piano, such as music
written for it at present does not seem to have, because almost any of it could
be played instead by two good pianists at the keyboard. (Cowell 1930, 64—65)

68

L!“- J

ll 5
EM: } Vibration ratio

1)

.

.

.

0

.

Figure 2-3. Diagram showing 3:4:5 temporal relationship from Cowell’s New Musical Resources
(cited in Gann 1995, 6). Note how Cowell uses different shaped noteheads for different divisions
of the measure.

Cowell himself applied these ideas to the composition of live performance pieces

such as Quartet Romantic, one of the pieces discussed by Thomas (1996). Cowell devel-

oped the rhythmic framework for the first movement of this piece in several stages: (1) he

composed several Chorale-like themes in four-part harmony (see Figure 2-4a); (2) the

pitch ratios of the Chorale chords were converted to rhythmic ratios, which then provided

the rhythm for the movement; and (3) the pitch content was changed to virtually elimi-

nate the correlations between the chorale’s pitches and the final rhythm (see Figure 2—4b).

Thomas comments perceptively about Cowell’s motivation for changing the pitch

content in order to dissociate it from the original Chorale themes:

It may seem odd that once Cowell has derived the rhythmic ratios from
his themes, he completely changes the pitch content, so that there is virtually
no correlation between the ratios of the actual pitches in the music and their
rhythms. Is not such a correlation the basis of his entire theory of rhythm, after
all? But what Cowell has actually done is taken the theory one Step further: by
negating the explicit association of pitch with rhythm, he forces the ‘harmony’
of rhythm to stand on its own. Because he has composed the rhythm with the
aid of traditional four-voice chorales, he has ensured that the contrasting
simultaneous rhythms, as well as juxtaposed rhythms, as they change, will
comply to what he considers a ‘natural’ progression, just as the chords of the
chorales comply to tonal progressions. Furthermore, the rhythms change on a
fairly periodic basis, since they are governed by the original harmonic rhythm
of the chorales, and the effect of the movement is correspondingly progres-
sion-like, in that the rhythms regularly proceed from one set of ratios to the
next. (Thomas 1996, 41)

69

 
 
 
  
m
m

hm
m

(a)

1
1
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6
5
4
2

m
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m
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6

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6
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Violin

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r
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Figure 2-4. Stages of composition of the first movement of Cowell’s Quartet Romantic: (a)
chorale theme and its resultant pitch ratios; (b) first page of the score. showing the conversion of
the chorale theme’s pitch ratios to rhythmic ratios (Thomas 1996

. 203—04).

70

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
Elliott Carter was another composer who created complexes of rhythmic relation-

ships using Cowell’s ideas about interval ratios. Carter’s technique of metric modulation

is based on setting up a system of tempos in which simple ratio relationships are preva-

lent and speeds are changed by changing the division of the beat. Thomas examined the

second movement of Carter’s String Quartet No. 2 for its tempo relationships, which are

shown in Figure 2-5. The majority of the paired tempo relationships identified in Figure

2-5a involve superparticular ratios.

M

5.6

23.3

195.

9:8
4:3

5:3

5:2

4:3 f> 9:8

5:4
5.
°

115

233.13

(a)

525| 315 233.3l4
5
175|51
I3
4
140|4

175

140]
3'15
93.3]52 10
l6
[3
56

175|5
140|4
105|3

140
9|9
|
124.4 48|8
IS
3
93.3

43
°

(b)

Figure 2-5. Tempos and relative ratios in Elliott Carter’s String Quartet No. 2, movement 2
(Thomas 1996, 45): (a) MM speeds used (MM=525 is not represented) and ratios of selected
pairs of tempos; (b) combinations of speeds and ratios.

Claims have been made for other composers that they may have used ratios, partic-

ularly for structural proportions, but there is some question as to how much a role intu-

ition played in the presence of the proportions that have been identified and how much

was due to deliberate application of ratios. Kramer (1988) has examined structural pro-

portion in Stravinsky’s music and claims that in many of his compositions, Stravinsky

used the same “proportional ratio consistently throughout a piece” (p. 303). For instance,

Kramer identifies the ratio 3:2 as describing many of the structural proportions in Sym-

phonies of Wind Instruments, although he allows that “Stravinsky’s proportional consis-

tencies are never exact, which implies that he did not consciously calculate sectional

71

 
durations” (p. 289); indeed, in his analysis Kramer allows deviation from 3:2 within a

range of 7%, or from 1.40 to 1.61 as being “equivalent” to 3:2. In Stravinsky’s late ballet

Agon, Kramer’s analysis identifies the piece’s structural ratio as being 1.19:1, which he

says is “not as strange as it may at first seem, as it is really 4‘12zl” (p. 298)!

Kramer also treats the topic of golden-section proportions in music, noting that

[t]heorists and analysts have been studying the use of the ratio 1.62:1 by com-
posers as diverse as Barber, Hindemith, Schoenberg, Barték, Webem, Berg,
Prokofiev, Debussy, Ravel, Delius, Rachmaninoff, Fauré, Scriabin, Saint-
Saéns, MacDowell, Dvorak, Wolf, Brahms, Wagner, Tchaikovsky, Schu—
mann, Mendelssohn, Chopin, Schubert, Beethoven, Mozart, Haydn, Bach,
Handel, Sermisy, Jannequin, Gibbons, Binchois, Dunstable, Ockeghem,
Obrecht, Dufay, and Machaut. (Kramer 1988, 303—04)

Thus, just as Kramer may have been “reaching” in asserting a structural ratio of

4\/2:1 in a piece of Stravinsky’s, there is evidence that many analyses concerning the

presence of the golden section in the music of the composers listed above use approx-

imations or even ranges of deviation (as Kramer has done) to prove the point. Still, the

quest to find golden-section proportions in music is strongly rooted in its ubiquitous

presence all around us, as Kramer notes quite convincingly:

.

It should not be surprising that golden-mean proportions and Fibonacci
numbers appear in music. Numerous mathematical properties of the Fibonacci
series have appealed to artists and scientists for centuries, and golden-section
proportions are frequently found in nature, human or otherwise. There is ex-
perimental evidence, for example, that the golden mean determines the ratio
. There is also experi-
of people’s positive to negative value judgments. .
mental evidence that rectangles (cards, mirrors, pictures, etc.) proportioned
according to the golden section (ratio of the longer to the shorter Side is 1.62)
appeal to our sense of symmetry. .
. Fibonacci numbers are used in certain
electrical networks, and they are approximated in the structures of atomic and
subatorrric particles. The ratios of the distances of the satellites of Jupiter,
Saturn, and Uranus from their parent planets approximate 1.62, as do (in a
weaker approximation) the distances of the planets (including the asteroid
belt) from the sun. .
. Fibonacci numbers relate to the numbers of years in
cycles between peaks and peaks, peaks and lows, and lows and lows of the
stock market. It has even been suggested that Fibonacci numbers determine
the lengths of cycles of grasshopper abundance, automobile factory sales, the
ratio of male to female conceptions, advertising effectiveness, sunspots, tree
ring size, rainfall in India, Nile floods, financial panics, and furniture produc-
tion! (Kramer 1988, 305)

.

.

72

Uses of Ratios in Nancarrow’s Music

While Henry Cowell led the way in showing how pitch ratios and the overtone

series could be used to create new musical textures, no other composer ventured further

than Nancarrow did in using ratios to govern so many different musical parameters. Nan-

carrow extended Cowell’s ideas to go beyond the simpler pitch ratios of just intonation to

ratios as close as 60:61 and even ratios involving irrational numbers such as ‘12, and he

further extended Cowell’s work by using ratios not only to control rhythmic parameters

but to establish relationships among features such as pitch, duration, and Structural

elements.

The player piano medium itself, of course, played a crucial role in the realization of

these ideas on proportions. Nancarrow shared with Reynolds the following insights about

his working process:

I mark out on a blank roll of paper all of the proportional relationships of
tempo, using what I think is going to be the smallest (fastest) note value as the
unit of measure. Of course, occasionally, if I have to use something even
faster, I just go over the roll and put in the smaller values, showing the
relations to the basic scale in the score. I mark the whole thing out from
beginning to end on the blank player piano roll. .
. Then I take the marked
proportions from the roll onto the music paper. It is not as exact as the roll, but
it’s fairly accurate so that the vertical relationships of tempo units will be
more or less what I see graphically on the paper. (Reynolds 1984, 13)

.

The graphical and visual nature of the process of creating the rolls lent itself ex-

tremely well to the types of calculations and alignment procedures that were a necessary

part of Nancarrow’s composition process. Carlsen acknowledged this when he said:

Some of the most characteristic and innovative elements of Nancarrow’s
music were born right on the blank surface of the unpunched paper roll. .
.
The roll is highly conducive to the use of exact measurements, templates,
replications, and geometric constructions; thus, to a certain extent, it exerts an
influence that is purely visual and mathematical. (Carlsen 1988, 66)

.

73

Temm Ratios

The most obvious and deliberate use of ratios in Nancarrow’s Studies occurs in the

pieces known as “tempo canons,” or those pieces in which canonic material is played

Simultaneously at proportionally-related tempos. Nancarrow’s tempo canons are repre-

sentative of large-scale polytempo as envisioned by Cowell.

The ratios given in the subtitles of Nancarrow’s tempo canons relate directly to the

tempo markings for the individual voices; the number of components in the ratio indi-

cates the number of canonic voices in the study (it Should be noted that a canonic “voice”

is not necessarily a monophonic [unharrnonized] musical line, but is more likely a poly-

phonic musical stratum—sometimes quite dense—that is repeated at different pitch levels

in successive voices). Table 2-1 describes most of Nancarrow’s tempo canons, including

the tempo ratios and the tempo markings used in the scores.

Nancarrow’s stated purpose in writing these canons

was [to explore] my interest in temporally dissonant relationships. Temporal
dissonance is as hard to define as tonal dissonance. I certainly would not
define a temporal relation of 1 to 2 as dissonant, but I would call a 2 to 3
relation mildly dissonant, and more and more so up to the extreme of the
irrational ones (Reynolds 1984, 5).

Gann recognizes that, in the earlier Studies through No. 19, Nancarrow restricted

his focus primarily to ratios involving 3, 4, and 5 (No. 19’s ratio, 12:15:20, is actually a

combination of a 3:4 ratio and 4:5 ratio). Starting with No. 24 Nancarrow began to climb

higher in the overtone series for his ratios, and in Study No. 33, with its ratio of \/2/2, he

used his first irrational ratio to reach a new extreme of temporal dissonance. The irra-

tional number ‘12 (1.4142136...) represents the equal-tempered tritone (among the most

dissonant and unstable intervals in tonal music), which is commonly given the ratio 7:5.

Nancarrow casts attention in this piece on this irrational number by using a variety of

ratios that are rational approximations of 1/2. Gann (p. 187) identifies the ratios 7:5 and

10:7 as being used in the final section, where in one voice only note values of five and

74

Table 2-1

Relationship of Canon Ratios and Tempo Markings

Study

No. 14, “Canon 4/5”
No. 15, “Canon 374”
No. 17, “Canon 12/15/20”
No. 18, “Canon 3/4"
No. 19, “Canon 12/15/20”

No. 24, “Canon 14/ 15/16”

No. 31, “Canon 21/24/25”

No. 32, “Canon 5/6/7/8”

Tempo Markings

J = 88, J: 110
J: 165, J: 220
J: 138, J: 172.5, J:230
J: 168, J: 224
J:144,J:180,J:240
J: 149%,,J : 160, J. : 170%,;

J : 224, J. : 240, J. : 256
J: 105, J: 120, J: 125
J: 85, J. : 102, J. : 119, J. : 136
J: 140x12, J: 280
the three voices generally exhibit a
9:10:11 tempo ratio—the openin
markings are J : 72, J: 80, and : 88; the
application of the 4:5:6 ratio results in a
succession of progressivel
in each voice, peaking at = 264 in the
fastest voice
o=85,o=90,o=95,o=100
J:150,J:160?,,J:1683/,,J=180,

faster tempos

 No. 33, “Canon «1272”

No. 34, “Canon 4/5/6] 4/5/6/ 4/5/6 ”

9

/

10 /

11

No. 36, “Canon 17/18/19/20”

187 V, /200/210/225/240/250/262 '/2 /281 1:, ”

 No. 40, “Canon e/tt”

 No. 37, “Canon 150/160 77/168 3/4/180/

   No. 48, “Canon 60/61”
   tempo markings not used—uses

J: 187%,J:200,J:210,J:225,
J:240,J:250,J:262l’2,J:281}:,
tempo markings not used—Nancarrow
notates a simple diagram which gives
timings that establish a relationship of
roughly e to It
Nancarrow does not give tempo markings;
the canons are intended to be played
together, and he establishes the following
relationships using total playing times:
7.1_

4’35”; 3113/16 =6’10”
o=12CLo=125

_1_
112 = 5’00”; 1/2/3 =7’40”; 3‘17:

No. 4113, “Canon 3% AVE/l6”

No. 41A, “Canon 717/1127 ”

proportional notation

No. 434 “Canon 24/25”

I

1

75

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
seven sixteenth notes are used while the other voice consists only of values of seven and

ten sixteenth notes (recall, from earlier in this chapter, Jorgensen’s description of 7:5 as

the “small tritone” and 10:7 as the “large tritone”).

After reaching extremes of irrational tempo ratios in Studies No. 40 and 41,

Nancarrow returned to the use of simpler ratios. His later pieces Nos. 45, 46, and 47,

although not nominally tempo canons, contain some tempo canon sections that have

ratios using lower numbers such as 3, 4, 5, and 6.

Despite the concentration on smaller numbers in many of the Studies’ tempo rela-

tionships, changing beat divisions and the progression of tempo canons can and do result

in fairly large “tempo complexes,” or multiple tempo ratios. Gann identifies tempo com-

plexes of 2:3:5:8:14 in Study No. 1, 21:30:36z50z60 in Study No. 35, 27:32:36z42 and

450:525:588:700 in Study No. 37, and 144:182:351:468:585:624:819:936:1638 in Study

No.46.

Although Nancarrow’s first named tempo canon is No. 14, many of the pieces prior

to No. 14 use tempo ratios in various ways. Nancarrow began exploring different simul-

taneous tempos based on interval ratios in his first player piano piece, “Rhythm Study

No. 1,” in which the two tempos—120 and 210—are related by a ratio of 4:7 (minor

seventh). Within this tempo relationship Nancarrow uses at least twelve durations that

create a duration range from 2 to 35 on what Gann calls a “quasi-logarithmic scale” (p.

70). A portion of this Study is Shown in Figure 1-4 (p. 31).

Tempo ratios are often expressed, especially in the earlier pieces, on a smaller scale

as ostinati or isorhythms. Study No. 2 (Figure 1-8, p. 34) has already been shown as an

example of an ostinato combining divisions of 5 and 6. In Study No. 7, Nancarrow

creates his major themes out of three isorhythrrrs consisting of 18, 24, and 30 beats,

respectively (a 3:4:5 ratio). The ratio 3:4:5 (representing the second-inversion major triad

as shown in Figure 2-3) seems to be a favorite of Nancarrow’s. He uses it in numerous

other studies, including No. 9 where the rhythmic relationships among the three main

76

themes are permeated by this ratio, and in Study No. 46, where a “polytempo ostinato”

projects the ratio 3:4:5 (see Figure 2-6 for Gann’s renotation of this proportionally-

notated study).

Figure 2-6. Study No. 46, 3:4:5 polytempo ostinato as renotated by Gann (p. 264).

V

In Study No. 11, Nancarrow creates a 60-beat duration series divided into three 20-

beat segments:

5564|55343|543332

(4 notes)

(5 notes)

(6 notes)

The progressively smaller note values create an acceleration effect, and the three seg-

ments exhibit a kind of 4:5:6 ratio in the numbers of division of the 20-beat segments.

A different type of ostinato is shown in Figure 1-9 (p. 35), where, in Study No. 6, a

single line alternates between 4- and 5-divisions of a measure.

Another matter for consideration in Nancarrow’s tempo canons is the relative dura-

tion of the sections resulting from the application of the tempo ratio. Tempo and duration

have an inverse relationship, and the “duration ratio” is the inverse of the tempo ratio. In

certain of Nancarrow’s tempo canons with three or more voices this inverse proportion

has a significant effect on the relative proportions of the piece. Study No. 17 offers an apt

example. As shown in Figure 2-7a, each voice states each of the three sections once, in

direct succession. The duration lengths of the sections do not relate by 12:15:20, how-

ever, but by the inverse of this ratio. As Shown in Figure 2-7b, the inverse of the ratio is

77

   
determined by maintaining the relationship of the extreme terms and reversing the rela-

tionship of the interior ratios (12:15 [4:5] and 15:20 [3:41) The result is the ratio

12:16:20, or 3:4:5, and this is the ratio by which sections C, A, and B, respectively, relate

to each other in duration.

Study No. 17 - Canon12115120

1 A
I

I

l

I B
l

1 C
f

l

l

l

l

(a)

1
l

'

1

4.5

3:4

AA
12 : 15 : 20

12:16:20=3:4:5

3:4

4:5

(b)

Figure 2-7. (a) Formal diagram of Study No. 17, “Canon 12/15/20” (Gann 1995, 22). Section “A”
states the lS-tempo, section “B” the 12-tempo, and section “C” the 20-tempo. (b) The “duration
ratio” resulting from a 12:15:20 tempo ratio is 3:4:5. This is the ratio which describes the relative
durations of sections C, A, and B.

In a two-voice canon, then, say “Canon 4/5,” the ratio inverts upon itself so that the

duration of the 4-voice is 5 and of the 5-voice it is 4.

“Scales” of Tempps

In addition to the numerous tempo canons in which the same tempo ratios are heard

between the voices, Nancarrow composed several pieces in which a succession of

tempos—often based on interval ratios—is used.

Technically, Study No. 37 does not meet this definition as, rather than being a

scalar succession of tempos, the simultaneous relative speeds of its twelve voices are

defined by the justly-tuned scale. However, since its tempos exactly relate to the interval

ratios of Cowell’s chromatic scale (see Figure 2-1a) it seems best to discuss it in this

section. One of the fascinating aspects of Study No. 37 is the way in which the “scale of

tempos” aurally illustrates the slightly varying sizes of half steps in Cowell’s scale (see

Figure 2-8 ). Note, for instance, the relative smallness of intervals between the 262V -

78

 
 
 
 
8'“! 

*249+
i!

'bl.

4‘!

(5 7*)

LP-
31‘"

Figure 2-8. First page of score to Study No. 37, showing rhythmic expression of Cowell’s justly-
tuned scale. The rhythmic “interval” between each pair of voices represents a half-step.

79

 
voice and the 250-voice (a 21:20 ratio) and between the 250-voice and the 240-voice (a

25:24 ratio), especially as compared to a 15:14 interval such as that between the 225-

voice and the 210-voice.

Study No. 28, a non-canonic piece based on large-scale acceleration, finds the com-

poser again using the ratios of the justly-tuned chromatic scale to establish tempo rela-

tionships, this time to establish a “scale” of acceleration. Gann (p. 164) describes how an

incremental acceleration in this study from J = 252 to J: 864 adheres to a “scalar”

progression as shown in Table 2-2.

Table 2-2

Scalar Acceleration in Study No. 28

ijpo
( =)

Ratio“

Corresponding
no_t_e

Te po
( =)

Ratio"

Corresponding

£21£__.

252

270

288

300
315

7:10

3:4

4:5

5:6
7:8

It:

F#

G

A1»

A
at

.

,

‘

J

480

504

540

576
600
630
675
720

765

810
864

4:3

7:5

3:2

8:5
5:3
7:4
15:8
2:1

17:8

9:4
12:5

F

F#

G

AL
A
131
B
C

Cit

D
Pi

*Ratios are in relationship to C, representing 1:1 and the “tonic tempo” at J = 360

As has already been observed about Cowell’s chromatic scale (Figure 2-la), there

are four different interval ratios for half steps in this scale and they are represented in

Study No. 28 as follows: the largest increment, 15:14 (increment of 7.1%), occurs be-

tween F#—G and E—B; the ratio 16:15 (6.7%) occurs between G—Al’, B—C, C—C#, D—3,

and E—F; the ratio 21:20 (5%) occurs between A—BL and F—F#; and the smallest

80

 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
  
 
increment, 25:24 (4.2%) occurs between Ab—A and B—E.3 The resulting overall accel-

eration (J = 252 to J: 864) is equivalent to the pitch interval of an octave plus a

diminished seventh, or the pitch ratio 24:7 (10:7 [interval from F# to C] + 12:5 [interval

from C to E» the octave above] = 24:7).

The final section of this chapter will analyze in detail Study No. 34, a piece in

which the two ratios 9:10:11 and 4:5:6 combine to create a long succession of tempos.

flymflflitch Analogues

A number of Nancarrow’s Studies contain ratios simultaneously expressing both

pitch and rhythm or duration. Gann credits Nancarrow’s work in associating pitch with

duration as placing him among

the great innovators of the 19505, at least for those whose conception of that
period assumes a European model. For Study No. 23 is a pioneering work—
Nos. 28 and 29 are others—in the dependent association of frequency with
duration, an idea that was very much “in the air” between 1949 and 1960.
(Gann 1995, 156)

In the first work cited above by Gann, Study No. 23, Nancarrow uses the pitch ratio

9:4 (major ninth) to determine the relative length of 71 consecutive pitches from B0 to A6.

For each major ninth higher the pitch progresses in this series, the note duration decreases

by half (Carlsen 1988, 49—refer again to Figure 1-28, p. 56). In Figure 2-9 from this

Study, pitches are arranged in five-note groupings, with the duration of all notes (written

in proportional notation) in each group determined by the duration of its first note. Note

the progressively longer notes as the first pitch of the group becomes lower.

3Inexplicably, Nancarrow uses two different ratios for the two occurrences of C#. and neither of them is the
15:14 ratio used by Cowell (Figure 2-1). The 16: 15 ratio used for the first occurrence of C# is actually what
Cowell specifies for D (Cowell 1930, 99). That does not explain, however, why Nancarrow’s second
occurrence of C# (17:8) is not an octave of the first (and nowhere does Cowell use the lower octave of this
ratio, 17:16, for either C# or D); the second ratio would have to be 32:15 rather than 17:8 to be an octave
duplication of 16:15. As a result, the two intervals C#—D differ: the first C#—D interval is an increment of
5.5%, or the rather unwieldy ratio of 135:128, while the second (“higher”) C#—D interval is a 5.9% incre-
ment, ratio 18: 17—the only two occurrences of these intervals in this Study.

81

 
 
Figure 2-9. Excerpt from Study No. 23, p. 4. (In this piece, as in many others, Nancarrow uses
proportional notation to indicate duration—that is, note duration is directly proportional to the
space between notes on the staff.)

Gann claims that Study No. 29 also relates pitch to duration in a series of repeated

notes, but this is more difficult to see as the proportions are not as exact as in No. 23, and

in No. 29 the octave of the pitch is irrelevant to its duration. Gann cites the following

one-octave chromatic scale, from D to C# (p. 166):

pitch
C#
C
B
A#
A
G#
G
F#
F
E
D#
D

mm.
11.9
12.5
13.6
14.3?
15.3
16.1
17.0
18.2
19.1
20.4
21.1
22.7

tempo
384
367.7
337.2
320
300.5
285.6
270
252.3
240
224.4
217.4
202.1

Figure 2-10 shows a section with repeated notes F#, A, and B in which these approximate

proportions can be observed. Note that, despite its low octave, B is still the most rapidly

reiterated pitch.

Figure 2-10. Score segment from Study No. 29 (p. 6) showing repeated notes F#, A, and B and
rate of note reiteration approximately related to pitch.

\

In Study No. 28, discussed earlier for its “scale of tempos,” there are eight clusters

82

 
 
of three pitches a half-step apart that are repeatedly articulated but in an irregular fashion.

Gann claims that these clusters of pitches, which can be represented by a pitch ratio of

15:16:17 (two adjacent half-steps), also exhibit a 15:16:17 speed ratio, with the highest

note in each cluster being rearticulated 17: 16 as fast as the middle note and the lower note

rearticulated 15:16 the speed of the middle note. This is difficult to verify due to the

nature of measuring duration in Nancarrow’s proportional notation, and arriving at these

numbers involves working with approximations and averages. What Gann has done is to

measure in millimeters the distance between rearticulated notes and determine an

average. In the note cluster shown in Figure 2-11, my own measurements show that the

average distance between successive articulations of the note A is about 25.8 mm in the

score (measurements in Figure 2-11 are reduced). To effect a 15:16:17 ratio with this

measurement as the middle figure, the measurements for the faster (upper) voice would

need to be about 24.3 mm and the slower (lower) voice about 27.5 mm. The actual

average measurements are 24.0 mm for the upper voice and 27.5 mm for the lower voice,

which is remarkably close to the ratio in question. One must keep in mind that Nan-

carrow’s hand-written scores represent only a rough approximation of the piano rolls,

which are more precise.

at

LL
fr
1 1_
it}

1
I
I
I
1
I
1
I
n; I
1' I 411
71"

I

. T
1.1
1'

A
I
'
1

1
I
I
A
”4"

1

1
A

l
'
I
V

I
1 4 I
I
I
l
A
7 #1

fl"
“fi—
1'
J ' T
V

A
I

Figure 2-11. Three-note cluster from Study No. 28 (p. 3), exhibiting 15:16:17 tempo and pitch
ratio. Rearticulations in the middle note are regular but some rearticulations of the upper and
lower notes are omitted by Nancarrow.

The next example involves metric notation and is even more overt. In Study No. 5,

two rhythmic ostinatos are established which are related by the ratio 7:5 (the “small tri-

tone”), and the opening interval between these two voices is the tritone C—F# (see Figure

2-12). The two lines share a 16 meter signature, but the distance between notes in the top

35

line is multiples of seven sixteenth notes and in the bottom line multiples of five sixteenth

notes (thus the least common multiple, 35, in the meter signature). Although the tonal

83

 
 
 
 
 
implications of these two lines are actually C major and B pentatonic, the harmonic use

of the tritone at the very beginning of this rhythmic context is a striking parallel to the

rhythmic relationship.

brackets indicate 11 values
7

14

'

1 4

'

I     

I

|

5‘1"“ 5‘":- Sempi’;

L

15

ll

1

5

‘F""

7’

25

Figure 2-12. Opening of Study No. 5, showing 7:5 rhythmic ratio and opening tritone interval
between voices.

Like the example cited earlier in this section where Gann identifies the ratio

15:16:17 in both pitch and rhythm, Study No. 31 (“Canon 21/24/25”) exhibits a relation-

ship between the tempo ratio of its title and its opening pitches (see Figure 1-1 1, p. 37).

The opening pitches, G—A-Bt, relate to 21:24:25 in this way: 8:7 (24:21) is a large whole

tone in just intonation (231 cents—see Figure 2-1b); 25:24 is the smallest minor second

in just intonation (just 70 cents); and 25:21 is one of five possible minor thirds in

Cowell’s chromatic scale (Gann 1995, 199—although the actual ratio for the minor third

from G to B1, in Cowell’s scale is 7:6). Of course, the pitch associations with just into—

nation (here and elsewhere) are merely academic as Nancarrow’s pieces are all performed

on pianos tuned in equal temperament.

Sgcmral uses of Ratios

Occasionally ratios appear in Nancarrow’s Studies in structural roles. Levels of imi-

tation (see Table 1-1, p. 41) would seem an area ripe for exploitation in this manner, but

there is little use of ratios to connect levels of imitation in canonic pieces with other

84

  
   
   
   
 
elements. One significant exception to this is Study No. 34, where the ratio 4:5:6 both

determines the succession of tempos and describes the level of imitation, which is a

major triad. This Study will be examined more fully at the end of this chapter.

Carlsen uncovers an extraordinary case of proportional structural integrity in Study

No. 8. The final section is a 3-voice canon in which the lowest voice first states the canon

heterophonically in octaves, the middle voice in fifths, and the upper voice in minor

thirds, creating a texture of root-position triads. The positioning of the entrances in this

canon is carefully worked out so that:

.

When the second voice enters, it does so at a point exactly halfway through
the first voice’s isorhythmic pattern; the third voice enters exactly three-
. The positioning of these entrances
quarters of the way through the pattern. .
is specifically related to the divisions of successive intervals in the overtone
series. Thus, the entrance of the second voice exactly halfway through the
pattern parallels, in the overtone series, the division of the octave (2:4) into
fifth (2:3) and fourth (3:4); by the same token, the further subdivision of that
half into quarters parallels the subdivision of the fifth (4:6) into major and
minor thirds (4:5 and 5:6). Obviously, it is no coincidence that the bass voice
is characterized by octaves, the next by fifths, and the highest by minor thirds:

fifth
l—\
3
:
2

minor third
l—\
6
:
5

:

4

:

octave

fifth

(Carlsen 1988, 43)

The structural application of these ratios is vividly illustrated in Figure 1-24 (p. 51).

In another particularly fine example of a ratio being used structurally in different

ways, Study No. 33 (from the “Boogie- Woogie Suite”) uses a 3:5 ratio on two structural

levels. On a smaller level Nancarrow creates a 2:3:5 rhythmic ratio between three

ostinatos; see Figure 2-13, where the ostinatos are in lines 2, 4, and 7 (recall that the note

symbol 1' in line 2 is Nancarrow’s own symbol for a 5-division note, in this case in the

meter 21!: ). The piece concludes with an eight-voice texture (also Figure 2-13) in which

five voices are expressing irregularly-spaced staccato chords against the three ostinato

lines, thus creating a textural expression of the 3:5 ratio.

85

 
Figure 2-13. Concluding texture of Study No. 3a (score, p. 35). Ostinato lines 2, 4, and 7 express
a 2:3:5 rhythmic ratio, while the entire texture expresses a 3:5 ratio (the three ostinato lines
against five lines of irregularly-spaced chords [lines 1, 3, 5a/b, 6, and 8]).

86

 
 
  
 
    
 
  
    
  
  
Analysis of Study No. 34: “Canon 4/5/67/4/1536/4/26 ”

This chapter concludes with an analysis of Study No. 34, focusing particularly on

the use of ratios in its composition. The copy of the score to No. 34 that I was able to

obtain was, unfortunately, missing page 9. When multiple attempts to obtain the missing

page were unsuccessful, I ultimately resorted to transcribing the missing page myself by

extrapolating themes from elsewhere in the work and, as needed, determining trans-

position levels and rests between themes from the taped performance. The transcribed

page emulates Nancarrow’s proportional notation and the 177-millimeter staff length of

my copy to facilitate accurate measurements of elapsed time. The transcribed page is

shown in Appendix D.

verview

Study No. 34 is a three-voice converging tempo canon with a single convergence

point on the last attack. Its melody is multi-thematic, with a rondo-like A theme separated

by themes B, C, .

. ., G, although Tenney describes it as “a fugue, with thematic material

in each voice alternating between ‘expositions’ of a fairly long ‘subject’ (heard seven

times in each of the three voices) and six ‘episodes’ involving related but distinctly

different melodic material” (Wergo Vols. HI/IV, liner notes, p. 26). Figure 2-14 is a

proportional representation of the progression of themes (A1, B, A2, .

. ., A7) in each

voice and shows the levels of imitation between the voices.

The 4:5:6 ratio of the title describes the major triad, and as shown in Figure 2-14 the

levels of imitation for all A statements happen to spell out major triads (except in A7,

which is imitated at the double octave), creating a structural application of the ratio. In

contrast, none of the non-A sections imitates voices at the major triad; the actual levels of

imitation in these sections (ignoring octave displacements) are:

87

  
Beginning with A5 there is no
overlap between the entrance
of a new theme and the final
entrance of a previous theme

A1

8
bloco

AZ

C

A4
8va_

E
bloco

A5
8va.

F

8V3-

G

A7

A6
8va.

8va.

p __....— -

8b-’9°9_

‘

-

_-

__

'_A

’000
f

P

8b.
loco
ffL— -p

one score system = ca. 3 millimeters

Figure 2-14. Proportional diagram of entrance ofthemes and levels of im'tation in Study No. 34.

section B = ascending (enharmonically spelled) first inversion minor triad

section C = ascending (enharmonically spelled) root position minor triad

section D = descending cluster of major seconds

section B = ascending cluster of minor seconds

section F = ascending perfect fifths

section G = ascending octaves

In effect, the levels of imitation are contracting through section B before expanding

dramatically in sections F and G. The cluster of major seconds in section D could be

construed as another representation of the 9:10:11 ratio. The major second is usually ex-

pressed by the ratios 9:8 or 10:9; however, 11:10 could be a very small major second, and

Nancarrow used a similar technique in Study No. 28 when he used a 15:16:17 ratio to

express both a rhythm and a melodic figure of contiguous half-steps (see Figure 2-11).

An important structural point occurs at the entrance of A5 (see Figure 2-14). This is

the first thematic section in which the first entrance of the theme (always in the lowest

voice) follows the last entrance of the previous theme (in the highest voice). From this

88

      
point on there is no longer overlapping of entrances of new themes with entrances of old

themes. The significance of this point in the piece is emphasized by the arrival at the

loudest dynamic in the piece thus far (f), and by the division of the piece at A5 into the

first eight sections and the last five sections, or the golden section (8:5).4

Besides defining the levels of imitation (and thus, to some extent, the harmonic

content) of the A sections in the piece, the 4:5:6 ratio is represented in No. 34’s tempo

scheme. The entire meaning of the enigmatic title refers to a basic 9:10:11 tempo ratio

between the three voices, with the A sections of each voice then subjected to a series of

4:5:6 operations to create an accelerating “scale of tempos” in the A sections of each

voice. For instance, the progression of tempos in the “9-voice” (lowest voice)5 begins

with the following tempos in the A sections (Gann 1995, 130):

tempo:
ratio:

72
4

90
5

:

108
6

:

144
8

:

180
10

:

216
12

:

In other words, the ratio 9:10:11 can be thought of as applying to the vertical axis

(between the voices) and 4:5:6 to the horizontal axis (time) to create an accelerating

series of tempos. Gann calls the result an acceleration canon, but “the acceleration is by

steps rather than gradual” (Gann 1995, 130).6 Figure 215 shows the succession of

tempos that results from the application of the two ratios in all three voices.

About the tempo structure, Tenney remarks that “Study #34 is in some ways the

most elaborate of all the Studies in its use of rational tempo relations as an integral struc-

tural parameter” (Tenney, Wergo Vols. III/IV, liner notes, p. 26) while Gann says “With

its ratios within ratios, this is one of Nancarrow’s most experimental studies, and it is his

last canon in which no strong connection is made between form and content” (Gann

4I make this assertion while keenly aware of the tendency—as expressed by Kramer (l988)-—to “find” the

olden section to fit one’s momentary purpose.
As is usual in Nancarrow’s tempo canons, in No. 34 the lowest voice is the slowest and the fastest voice
the highest.
Gann uses the term “acceleration canon” to refer to canons containing either or both accelerating and
decelerating processes. This canon accelerates only to section A6 and then decelerates to the end of the
piece.

89

 
Section:

A1

B

A2

C

ll-voice:
lO-voice:
9-voice:

88
80
\72

 /
10633
110
117)“3
117.13
100
96
96--- 90/ 10633

A3

132
120
108

D

A4

E

A5

F

133,13
-...120
120

Section:

A6

G (deceleration begins) ----------

A7 (incomplete statement) -

ll-voice:

10—voice:
9-voice:

264

24o
216

23493

213/‘3
192

2
72?
200
180

177-.5146?3 Q 11mm 783F732
66%
10693
160
60
96
144

133213
1201

120
108

100
9o

80
72

(returns to original tempos)

Figure 2-15. Succession of tempos in Study No. 34 (based on Gann 1995, 131). Shaded rec-
tangles indicate 9:10:11 tempo complexes; recurring 9:10:11 tempo complexes are indicated by
ovals. Dotted lines between adjacent tempos in the top row indicate breaks in acceleration. Note
that: (a) tempos in all A statements adhere to the 9:10:11 ratio; (b) tempos peak in all voices in
section A6, with steady deceleration beginning in section G; and (c) the piece ends slower than it
began.

1995, 132). As shown in Figure 2-15, the 9: 10:11 tempo ratio between the voices is con-

sistent across all the A sections and in sections F and G, but a different process takes

place in sections B, C, D, and B. As a result, the acceleration is not constant through

section A6 and sections E and F are actually decelerations. The 9:10:11 ratio is still

operational in these sections, however, but it has effectively been doubled. If one traces

from the lowest voice of section B, up to the lO-voice and ll-voice, then from the bottom

voice of section C and up that column, and so on in sections D and B, one ends up with

the following series of tempos: 96 96 10633 10693 11713

117*3

120 120 133/‘3

133/V3

146;;

14633. The starting point, 96, creates a 9:10:11:12 ratio with the three

tempos in section A1. Omitting the duplications, the series is revealed as a succession of

9: 10: 1 l-related numbers as shown below:

90

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    
 
 
(4:5:6)

96
1062/3
\1/

117'/3

120

13393

14633

144

(9:10:11)

(9:10:11)

The next number in the series, the tempo 144 in the bottom voice of section F, does

not seem to fit the steadily accelerating pattern until one applies the ratio 4:5:6 to the

numbers 96 and 120 (the first terms of the two 9:10:11 series) to yield 96:120:l44. Thus

144 becomes the lowest term in section F, where a consistent 9:10:11 relationship

between the voices is established and maintained to the end of the piece. The result of all

these procedures is nineteen tempo changes in each voice, with a general acceleration

through section A6 and continuous deceleration in sections G and A7.

Changes in dynamic generally parallel changes in relative speed. The dynamic in-

creases steadily from p at the beginning to f at section A5 (unaffected by the decel-

eration in section B). At section F, where the tempo slows again, the dynamic drops

suddenly to p again for another crescendo to ff at A6 (the piece’s fastest and loudest

section), and from there the dynamic grows softer to the end of the piece. The result is a

most unusual ending for Nancarrow as the decrescendo is accompanied by a gradual

ritardando that winds the piece down to slower than it began. Gann describes this ending

as “coy” and notes that no other of Nancarrow’s tempo canons “ends simultaneously on a

ritard” (Gann 1995, 132); the effect is appreciably heightened by the simultaneous

decrescendo and is quaintly reminiscent of the winding down of a child’s music box.

Themes

Themes A through G of Study No. 34 are shown in Figures 2-16 through 2-22. (All

themes shown are from the lowest voice.)

91

 
The seven themes share three important motives: motive a, a pitch motive, is the

same pitch repeated three or four times with rests in between (motive a’ is a two-note

pitch repetition); motive b, a rhythmic motive, is two consecutive eighth notes; and

motive c, a pitch motive, is a broken major triad (another manifestation of the ratio

4:5:6). Motive b is often combined with one of the pitch motives, and its pitch content is

dominated by the interval of a minor second (34 occurrences out of 58 total), followed by

the P4/P5 interval which occurs twelve times. The only other intervals stated are the

minor third (seven times) and major second (five times).

As is typical for Nancarrow, in this piece there are very few long notes: the longest

is a quarter note, and the majority of notes are staccato eighth notes. Nancarrow fre-

quently changes the meter; the time signature always has an eighth-note denominator and

meters range from 141 to g . Only three measures do not begin with an attack. Isorhythms

or other obvious rhythmic patterns are difficult to ascertain, although there are isolated

numeric sequences affiliated with motive a such as the 2 + 3 + 2 + 1 rhythmic pattern in

mm. 7—8 of theme C (Figure 2-18).

Theme A (1 20 )3) - first statement in lowest voice

piece ends here
in statement A 7

‘  

‘ ‘~.‘

b

“. A
'
~

.

I

mfifiififlmfllflnm

Jun-m

I

’

;

'

.

I

‘~.

b

b

\‘x.

2‘

1

A

.

.
I.
' QH 517i rim-.1 ll gh- )* ‘n' :. "1 11' (iv
y HOW-E T]:
111m

e.g.,, II Key II o '1‘ , V
ii;

IIIDIAUITI

111’ m
”

I

.

-

1

0

a

C

r“ a

or

u
n

37111 at ij-c:__J

‘

entire theme
tracesicl

Figure 2-16. Theme A of Study No. 34.

92

 
 
 
  
  
  
 
 
  
 
 
   
 
 
    
 
 
Theme B (40 ."s) -- first statement in lowest voice (system 6)

-------- rhythmic sequence 1 - - - - - — - - -
~

c  

C

sequence 2 - - -
c

.

‘ ~ ‘

-------- rhythmic sequence 3 ---------
C

c

b

Figure 2-17. Theme B of Study No. 34.

Theme C (67 . 's) -- first statement in lowest voice (system 13)

“m:---rhyt::':nicbsequence 1 ----

----rh¥)thmic:sequence2----

c

3'

Figure 2-18. Theme C of Study No. 34.

2 + 3+2-i-l

entire theme

traces it'l

Thane D (103 J 's) -- first statement in lowest voicg (system 20)

(8b throughout) x“ finsttwnphraeer ,,.

Htrateitl -'

.5

.———1._.

b

‘ ~ . ‘

entire theme

regard

Figure 2-19. Theme D of Study No. 34.

93

   
  
  
  
  
    
  
 
 
  
 
    
 
   
 
 
 
 
   
 
The A theme (Figure 2-16) bears a notable resemblance to the opening of No. 31

(Figure 1-11, p. 37). No. 31 states motives a and b (a half step) immediately in the first

three measures, with the same articulations as No. 34’s A theme and clearly outlining G

minor (the “partitioned minor thir ” of which Nancarrow was so fond). No. 34’s opening

theme hints at E minor but it is not until mm. 5—6 that the first three scale degrees are

clearly stated, and until then the tonal environment is quite ambiguous. The note E

articulated at the beginning of theme A is an important tonal anchor, however.7 It

reappears in the next statement of a at mm. 8—9 (which, in the abbreviated A7, is the

ending of the piece with the E stated in octaves in all three voices). At m. 17, the motive

a is stated on G a minor third higher than the opening E; this G will be another important

tonal anchor, especially in the middle themes. Motive c has two isolated appearances in

Theme E (138 ."s) -- first statement in lowest voice (system 27)

.1

l

a'

1

F

h

l

t

b
l 4' I

'd
1

.'

- - sequenge 1b -

I—1

.
m-‘I‘_.‘-' -'m|- .-‘.‘—- _I--'

.

g

l

"zl—I-II-flmr—fl ---:--li-'-—'

:

'

1

'

ence 2a - - - -'

- - - - sequence 2b - - -'-

V

a'

b

b

:L’

.

I M

I. ”J —

----

”mi-sequence 3a: -

- - sequence 3b- -

v

_

b

arr—7:93.31 3*? r 3 57:31:,” :3 i

'.',’ H 3.535% ,‘Ta—g:“f1.51

. "‘ [’95 3;? I _.

3, __'__—_;t\-~

b

I.

_'._> E? g ._'

bdfi

.

"”*"T’ i_ _D

' W -

’

’ "

'

A

Figure 2-20. Theme E of Study No. 34.

.

.

..

7All references to specific pitches in themes will refer to the bottom voice, from which Figures 2-16
through 2-22 are taken. It is the bottom voice’s E tonal center that is affirmed in the A7 statement when all
voices are imitating in octaves.

94

 
  
   
   
 
 
 
    
 
 
    
 
meme F (130 .‘ 's) -- first statement in lowest voice (system 35)

\ P5 ([_\lf(/.\ ttm t’

eqmminn nf"u"

b

Figure 2-21. Theme F of Study No. 34.

Theme G (50 + 62 .'s) -- first statement in lowest voice (system 41)

1
11‘ 2 593'2':311”3":1,2119 1%.: Sarah‘slllllC 3

(b‘

1—1

5

I use. #.~eprf H U. bV/jI a“: l 3‘ 1111f; W / 211 u

b
r——Ir~.
$1164.; 1511-63le

a, __.: _ 11

fl

,

7 “ V2 7

}: 7: 7

L-

I

,7

¢‘

__..

* = tempo changes occur

a

‘

Pitt/('3 with
beginning (if/‘7
at original pitt'lt
plus upper m'tm'e

Figure 2-22. Theme G of Study No. 34.

95

 
 
 
 
  
 
  
 
 
 
 
 
 
 
 
 
  
 
 
 
  
  
 
    
 
 
 
theme A. The entire theme traverses the interval of a half step (icl), and this interval class

shapes several of the themes. The theme is metrically diverse but begins with nine meas-

ures of 3 time.

Theme B (Figure 2-17), at 40 D5 in length, is the shortest of the themes; the pre-

dominant meter is

173

. The theme is dominated by motive c, while motive a is notably

absent. Theme B is much more organic than theme A; it consists of three phrases, each

beginning with two contiguous statements of c followed by eighth notes and rests. The

three phrases are rhythmically identical for the first eight notes. The second phrase begins

a major third higher than the first, with the third phrase beginning on the same E that

began the first phrase. The theme concludes with its only statement of motive b, stating a

half step.

Theme C (Figure 2-18) begins with two sequential phrases that combine all three

motives, each ending with a’. The three statements of motive a’ are related by thirds. The

predominant meter of theme C is g , and there is an arithmetical rhythmic series (2 + 3 +

2 + l) in the 4-note statement of motive a in mm. 7—8. Like theme A, theme C traverses

icl from beginning to end.

Theme D (Figure 2-19) is dominated by 3 meter and also traverses icl from be-

ginning to end. Its first two phrases also trace icl, and the predominant melodic move-

ment in this theme is by half step; chromatic scale passages occur at mm. 12-13 and

15—17. There are also several examples of Nancarrow’s tendency to use a major form of a

scale ascending and a minor form descending, such as m. 8-10 and 12—14. Except for

one statement of a’, motive a is virtually absent, and there are no occurrences of motive

c.

Theme E (Figure 2-20) is the longest theme (138 Is) and is metrically diverse,

although 15 of the 26 measures are in g . Only motives a/a’ and b appear, and all but one

statements of the b motive state a half step. This theme abounds with sequential phrases,

labeled sequences la/b, 2a/b, and 3a/b; sequence 2 changes pitch direction slightly, but is

96

similar enough to sound sequential. The statement of motive a in mm. 12—15 marks the

halfway point of the theme and is highlighted in importance by its four-note repetition

and its arithmetical deceleration (3 + 4 + 5 eighth-note beats). Deceleration of the a

motive is also a feature of sequence lafb (note that section B represented a deceleration

within the general acceleration that occurs through section A6).

Theme F (Figure 2-21) is also metrically diverse but tends toward the longer

measures such as Z!

, g , and 3 ; there are no 14! measures. A new feature that appears

here for the first time is pitch dyads of either perfect fifths or minor thirds. Only motives

a’ and b are present, although the first four iterations of the perfect fifth G—D (mm. 2, 7,

8, and 11) trace an expansion of motive a. The remaining three P5 dyads trace a

chromatic descent from the G in m. 11 to F# in m. 15 and F at the very end; the descent

concludes at the beginning of theme G with the note E.

Theme G (Figure 2-22) is metrically diverse, is stated entirely in octaves, and shares

some features with Theme A: quarter notes appear in mm. 17, 18, and 19, and like theme

A, statements of the b motive tend to be perfect fourths and fifths. There is only one

statement of motive a but it is the same pitch (G) and rhythmically the same as the state-

ment in mm. 12—15 of theme E; motive c returns here after a long absence. Theme G is

divided into two parts with an unmeasured rest in between; the second half of the theme

contains four tempo changes, all decelerations (making it the only theme with internal

tempo changes with the exception of the last statement of A). The end of the theme elides

with the beginning of A7, at the A theme’s original pitch level plus the upper octave.

It is noteworthy that, in sections F and G/A7, the perfect fifth dyads in F and octave

dyads in G/A7 are identical to the levels of imitation in these sections.

The features of the individual themes are summarized in Table 2-3.

Examination of the seven themes reveals that the most important structural pitches

in Study No. 34’s canonic subject are affiliated with appearances of motive a. The first

such pitch is the opening note E expressed by the first statement of a in Theme A and

97

 
reiterated by the second statement at mm. 8—9. The next statement of a, at mm. 17-18,

affirms the second structural pitch, G. The next statements of a are on E], (theme A, m.

23) and F (theme C, mm. 7-8).

Table 2-3

Features of Themes in Study No. 34

Theme:

A

B

C

D

Contains pitch motive a/a’

Contains rhythmic motive b

Contains pitch motive c

Contains quarter notes

Traces icl from beginning to end

Traces icl within phrases

Contains dyads

-

-

.

0

0

Contains rhythmic or melodic sequences

Contains arithmetical rhythmic series in
motive a

Contains internal tempo changes

|

.

.

-

.

.

.

0

.

.

.

.

o

o

F

.

.

0

-

E

.

.

.

.

G

.

.

.

.

.

.

-

Although the marked absence of motive a in theme D would seem to indicate little

of significance going on there structurally, themes C and D together trace a series of

chromatic pitches about the main structural interval E—G as shown in Figure 2-23. The

same six pitches shown in Figure 2-23—E1,, E, F, F#, G, and G#—are also the starting

pitches of themes Al through A6 in the lowest voice (see Figure 2-14 [p. 88]).

major structural

interval of Study No. 34

interval traced
by theme C

interval traced by
first phrase of
theme D

interval traced by
second phrase of

theme D

L——J

interval traced
by theme D

Figure 2-23. Chromatic movement about the main structural interval, E—G, in themes C and D.

98

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
     
Theme E has two statements of motive a, both stating the pitch G. The second state-

ment, as noted earlier, marks the halfway point of this theme and its rhythm is an arith-

metical deceleration. The pitch G continues to be dominant in theme F; although this

theme contains no statements of motive a, the motive is traced through the perfect fifth

dyads G—D which then chromatically trace the interval from G back down to the note E

that begins theme G. The first half of theme G again traces the interval from E to G again,

and the second half of the theme contains a 4-note version of motive a on G that is

rhythmically the same as the statement in the middle of theme E; the second half of

theme G concludes with a return to the opening pitch E, which becomes the beginning of

section A7.

Rests

The themes are connected by a series of rests that are either metrically notated or

unmeasured. Including the rests, the entire lengths of the canonic subjects are approxi-

mately 7912 millimeters in the lowest voice, 7103 millimeters in the middle voice, and

6440 millimeters in the highest voice, or roughly a 9:10:11 ratio. Because of this one

might expect, perhaps, that Nancarrow would maintain the 9:10:11 relationship between

the voices in the rest sections as well, but this is not the case. Only a few of the rests

between themes maintain 9: 10: 1 l, and Nancarrow’s system of determining the amount of

rest between themes seems somewhat arbitrary at times. Certain relationships between

voices are simple to ascertain while others are baffling.

The rest measurements between sections, as measured either in millimeters on the

score or in measured rests, are shown in Table 2-4.

The liberties that seem to be taken with some of the rests and the resultant entrances

of themes can, at least in some cases, possibly be attributed to Nancarrow placing theme

entrances to coincide with events in other theme statements. Theme E provides a useful

example. Each of its three entrances aligns almost perfectly with an attack in another

99

 
voice, as shown in Figure 2-24.

Table 2-4

Rest Measurements Between Theme Sections in Study No. 34

Rests between sections:

Al-B fiB-AZ

A2-C

C-A3

A3-D

D-A4

A4-E

E-AS

AS-F

F-A6

A6-G

High

Voice

3 @ '

21

t @

39

J=88

mm. ' J=110

mm.

54

mm.

Middle

3@

20

$@

78

:r 62

Voice

Low

Voice

J=80

mm. 1 J=100

mm.

3 @

J=72

52

mm.

t @

J=90

94

mm.

mm.

63

mm.

*30

mm.

25

mm.

94

mm.

*76

15

mm; mm.

*82

mm.

84

mm.

83

1 *101

mm. - mm.

186

mm.

207

mm.

228

mm.

115

mm.

128

mm.

142

mm.

51

mm.

74

mm.

112

mm.

2:5

9: 10: ll

9: 10: 11

Approx.
  Ratio:
Note: Shaded areas denote measurements that are similar enough to be considered equiv-
alent for the purpose of determining ratios.

9: 10:1 1

9: 10: 11

2:425

6:7

-

-

-

4:6:9

* These measurements are at least partially taken on transcribed p. 9 and are more vul-
tremble to error.

Summ

d

onclusions

The net effect of the temporal procedures in this piece is gradually accelerating and

then rapidly decelerating sets of closely related tempos that are usually slightly out of

phase with each other. However, a few places of near unanimity between voices emerge

from the texture in stark relief. Figure 2-25 shows several such passages. Although the

constant shift in tonal areas within the seven themes makes coordination of tonal areas

between voices quite haphazard, these three passages show a remarkable degree of tonal

coherence in addition to near-rhythmic unity: in passage (a), a coordinated shift from D,

to C; in (b), coordination around G minor; and, in (c), arrival at a C major triad followed

by movement to an octave 131,. The prevalence of motive a/a’ in all three segments is

worth noting. The limited rhythmic vocabulary of the themes increases the likelihood of

rhythmic similarity as seen in these passages. Passages (a) and (b) show that, because of

100

 
 
 
 
 
 
 
 
 
Entrance of theme E in bottom voice:

Entrance of theme E in top voice:

(top voice also J = 133-1/3)

i...

5»

2/335

(Via =74627g ‘—

9

61
(middle voice also J = 146-2/3)
3) .

K;

*

Entrance of theme E in middle voice:

(top voice J = 176)

Figure 2-24. Entrance of theme E in each of the three voices (Study No. 34, p. 8 and transcribed
p. 9).

the progression of the tempo canon, ratios larger than the piece’s basic 9:10:11 ratio are

possible. In addition, passages (b) and (c) are taken from two-voice textures [and passage

(a) is opposite a significant rest in the other voice], adding greater emphasis to the sense

of coordination in those passages.

Overall, there are three ways in which the three voices of Study No. 34 can move in

and out of phase with each other. We have already looked at two: tonal, in which frequent

changes in tonal center left few opportunities for coherence; and rhythmic/temporal, in

which the simultaneous combination of similar rhythms and closely-related tempos also

occurred infrequently. The other feature that changes among the voices is which of the

101

 
 
  
   
  
    
   
   
 
 
 
(b)

Figure 2-25. Passages of near-rhythmic and temporal unanimity between voices: (a) p. 4, top
system (similar rhythm, 15:16 tempo ratio); (b) p. 6, second system (identical rhythm, 44:45
tempo ratio); and (c) p. 7, second system (identical rhythm, 10:11 tempo ratio).

(C)

seven themes (A through G) is being stated in each voice. Although the same sequence of

themes occurs in each voice, because of the converging canon there is a fairly limited

number of places where all three voices are stating the same theme. Figure 2-26 graph-

ically represents areas in Study No. 34 where all three voices are stating the same

thematic material.

Theme:

AA

FAGA

p.#:l

3

5

7

9

11

13

Figure 2-26. Graph depicting areas of Study No. 34 where all three voices are stating the same
thematic material.

102

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
As shown in the graph, at the beginning of the piece only theme A is stated simul-

taneously in all three voices, and this is only because of the rondo-like recurrences of

theme A. For instance, in the first shaded A section on the graph, section A2 is being

stated in the bottom voice while the middle and top voices are stating section Al; the

second shaded section includes A2 in the bottom and middle voices and A1 in the top

voice; and so on (compare to Figure 2-14). The converging nature of the canon, with its

significant delays in the entrance of the middle and top voices, creates enough distance

between the B, C, D, and E sections that they are never occurring at the same time in all

three voices (the brevity of themes B and C also contributes to their separation between

voices). Once section F arrives, however, the canon has converged enough so that sec-

tions F and G and the last two A sections, A6 and A7, have more significant portions of

simultaneity taking place.

Several processes occur in this Study that create points of structural significance.

The first such point is the beginning of section A5, which is the arrival point of a

crescendo that began at the beginning of the piece and is also the point at which emerging

themes no longer overlap with beginnings of prior themes; this section is also the ninth

theme section of 13 overall, dividing the piece into parts of eight and five (the golden

section). Furthermore, A5 is the point at which levels of imitation in non-A sections stop

contracting (reaching minor seconds in section B) and begin expanding (to perfect fifths

in F and octaves in G).

At section F there is a marked lull in the action and extreme textural break. Besides

the abrupt change in dynamic, this theme is preceded by the longest rest in the entire

piece. Like section B, section F’s tempos decelerate from the previous section, and this is

where the two tempo schemes shown in Figure 2-15 “collide”; from this point on all

tempo complexes are related by the 9:10:11 ratio. Section F introduces another new

textural component: dyads of perfect fifths and minor thirds; the P5 dyads match the level

of imitation in this section, and this will continue into section G—A7 where the dyads

103

change to octaves. The roots of the P5 dyads trace an expansion of motive a on G, the

upper pitch of the major structural interval E—G. Also beginning in section F, simul-

taneous statements of the same theme in all voices can be heard again (see Figure 2-26).

The fastest and loudest point in the piece is reached in section A6. From here on,

tempos rapidly decelerate; as sections G and A7 are the only sections in which there are

multiple tempo changes (there are five tempos in section G and four in section A7—see

Figure 2-15), the deceleration to the end is much more rapid (and steeper, since the piece

ends slower than it began) than the somewhat halting acceleration at the beginning of the

piece; see Figure 2-27, which illustrates this feature using the three extreme tempos from

the bottom voice.

Temios/

3:1 ratio

72 ”(V

g

,

'

I
.. /.;/“

/-«:::/-::::'r->

‘ P

/

Figure 2-27. Dynamics and relative scale of acceleration and deceleration in bottom voice of
Study No. 34.

As shown in the example, the study reaches its fastest (and loudest) point almost 6/7

of the way through, with the tempo increasing to three times its beginning level. During

the last roughly 1/10 of the piece, the tempo decelerates to 5/ 18 of its fastest with tempo

Changes occurring at the very end of the piece with increasing rapidity: the final seven

tempo changes, from the middle of section G to the end, occur every 14, 21, 15, 12, 10,

1 l, and 7 eighth-note beats, respectively. The final tempo, J: 60, applies to only the final

three notes, which are a statement of motive a on the original pitch, E. Figure 2—28 shows

the final system of the Study, which includes the convergence to the final E in triple

Octaves. The piece’s overall structural interval, E—G, is the final interval in all three

Voices.

 
Figure 2-28. Concluding system of Study No. 34.

105

 
 
 
 
   
 
 
CHAPTER 3

CONVERGENCE POINTS 1N NANCARROW’S TEMPO CANONS

Chapters 3 and 4 will concentrate on two aspects of Nancarrow’s tempo canons:

convergence points, discussed in this chapter; and fractal formal features, discussed in

Chapter 4. The most important structural feature of the tempo canons is the convergence

point, or “the infinitesimal moment at which all lines have reached identical points in the

material they are playing” (Gann 1995, 21). Convergence points (CPS) serve varying

purposes in Nancarrow’s Studies: sometimes they seem to be the raison d’étre for an

entire piece (particularly in his earlier tempo canons with just one CP, such as Nos. 14,

18, and 19), sometimes they vary considerably in significance within the same piece

(such as the thirteen CPs in No. 24), and sometimes they serve to mark the most struc-

turally significant points in a piece (such as the three CPS in No. 27).

Unlike conventional canons in which the voices proceed at the same tempo, tempo

canons present the possibility for one or more CPs between the canonic voices because

the voices are either converging or diverging most of the time. Recall, from Chapter 1,

Margaret Thomas’s classification of tempo canons into four basic types (Figure 1-17,

p. 42): (l) the converging canon, with one CP at the end; (2) the diverging canon, which

begins with a CP; (3) the converging-diverging (arch) canon, in which a single CP is

somewhere in the middle; and (4) the diverging-converging canon, which begins and

ends with a CP. Nancarrow wrote examples of each basic type (although he did not write

any complete diverging canons—there is, after all, not much aural interest in a piece that

106

diverges immediately away from a beginning point of convergence, never to return 1).

This chapter will examine the presence and placement of CPS in Nancarrow’s

tempo canons and their significance as related to the overall structure. CPS will be exam-

ined in their more conventional uses in the canons, and other devices examined will

include tempo switches and overlaps (which can be used to create additional CPS) and

techniques for emphasizing and de-emphasizing CPS. Several of Nancarrow’s more com-

plex works feature many CPS, and further complexity is possible through the addition of

more voices to a canon, providing opportunities for interior CPS among smaller groups of

voices. A special sub-type of Nancarrow’s tempo canons is the “acceleration canon,” in

which voices are accelerating or decelerating, usually by established percentages. Nan-

carrow wrote several such pieces, and this chapter will conclude with an analysis of one

of them: Study No. 27 (“Canon 5%/6%/8%/11%”), a four-voice acceleration canon with

three CPS.

The effect of convergence in Nancarrow’s tempo canons is perceptually heightened

because it involves not only temporal convergence, but convergence of canonic material.

The choice of canon as a means of portraying temporal proportions is quite natural.

Thomas notes the aptness of this technique when she says “One of the clearest compo-

sitional strategies in which to hear proportions is canon” (Thomas 1996, 97); she adds

further that in Nancarrow’s tempo canons “there is a palpable sense of the voices being in

different places at the same time, of gradually moving closer together, of a brief moment

of coordination, and then a departure” (p. 66). This “brief moment of coordination” is the

CP, or Gann’s “infinitesimal moment” at which temporally divergent voices converge.

A CP is an excellent example of what Jonathan Kramer refers to as a “timepoint,”

1Perhaps a good real-life analogy to the perception of the diverging-converging process is airplane travel. A
passenger on an airplane has a far more accurate perception of when the plane is going to meet the ground
during the landing process (i.e., convergence) than of when the plane is going to reach cruising altitude
after takeoff (i.e., divergence). Also, the passenger’s perception of exactly when the landing will occur
becomes more and more accurate as the plane approaches the point of “convergence” with the ground and
is much more acute than the perception of when cruising altitude will be reached.

107

 
or “an instant, analogous to a geometrical point in space” (Kramer 1988, 454). A time-

point has no dimension, as explained by Kramer:

But what is a timepoint? Whereas a timespan is a specific duration (whether
of a note, chord, silence, motive, or whatever), a timepoint really has no dura-
tion. We hear events that start or stop at timepoints, but we cannot hear the
timepoints themselves [emphasis mine]. A timepoint is thus analogous to a
point in geometric Space. By definition, a point has no size: It is not a dot on
the page, although a dot may be used to represent a point. Similarly, a stac-
cato note or the attack of a longer note necessarily falls on and thus may
represent a timepoint, but a timepoint in music is as inaudible as a geometric
point is invisible. A point in space has zero dimensions; Space itself is three-
dimensional; a plane has two dimensions; a line has one; and a point has none.
(Kramer 1988, 82—83)

Like one-dimensional geometric lines in space, Nancarrow’s canonic lines converge

at this dimensionless point in time, the timepoint. Although “we cannot hear the time-

points themselves,” the CP is an audible event that takes place at this point. Eytan

Agmon, in his discussion of musical durations as mathematical intervals, noted that

“every common musical duration is uniquely associated with a Single moment in time,

namely its attack” (Agmon 1997, 48). Indeed, most of Nancarrow’s CPS do occur on a

coincident beat attack; in a number of cases, however, CPS occur on rests; and, in several

cases (the end of Studies No. 32 and 37), a CP is at the end of a long held note. But,

whether articulated or not, a CP always represents a timepoint.

The process of temporal convergence, and the CPs which result from it, is virtually

undiscussed in the literature as it is a phenomenon that occurs in the music of very few

composers (with Charles Ives being probably the other most significant practitioner of

large-scale temporal divergence). A search for the term “convergence point” turned up

nothing in the literature; looking for the term “convergence” itself turned up a small

number of articles, but none dealt with temporal convergence. There is, of course, a con-

siderable body of literature on rhythmic and metric divergence (e.g., Harald Krebs,

Edward Cone, Kramer) but true temporal divergence seems to be a relatively unexplored

area.

108

 
Tempo Canon Terminology and Features

At this point, review of a number of terms relevant to tempo canons will be useful.

Recall from Chapter I (pp. 42—43) that Gann defined the terms convergence point, con-

vergence period, tempo switch, and echo distance. These terms were all illustrated in

Figure 1-18 (p.43). An additional illustration directly from Nancarrow’s scores will be

given here to further elucidate these features and introduce some additional concepts.

Figure 3-1 shows a portion of Study No. 14 (“Canon 4/5”) at the point where the CP

At ® the echo distance equals the convergence
period (5 beats in top voice, 4 beats in bottom)

—-

I'- — — _-

/-:-

 3‘1.

Figure 3-1. Convergence point at middle of Study No. 14 (score, p. 3). Canon is in two voices
(bottom voice is two staves in bass clef, top voice is two staves in treble clef); CP takes place at
beginning of second system. Arrows illustrate selected echo distances. Dotted lines Show poten-
tial points of simultaneity; the convergence period is the distance between potential Simulta-
neities. Notice that in the fifth convergence period before and after the CP, the echo distance
equals the length of the convergence period—a distance from the CP that is functionally related
to the ratio 4:5.

109

 
 
 
 
 
 
 
occurs, and this example will illustrate the relationships among the terms convergence

point, convergence period, echo distance, and potential points of simultaneity. Study No.

14 is a two-voice arch (converging-diverging) canon in which the CP occurs exactly in

the middle of the piece.

Echo distances are shown in Figure 3-1 by arrows. Gann pointed out that the echo

distance “will grow shorter and shorter as the convergence point is approached, reach

zero at the convergence point, then grow progressively longer as it moves away” (Gann

1995, 21), and this is clearly seen in this example. Thomas alludes to the aural effect of

the echo distance when she discusses the relative degree of temporal dissonance near

what she calls the “point of synchrony”:

The notable and increasingly proportionately significant changes in the gaps,
and the attendant modifications in temporal dissonance, are responsible for the
intense perceptual focus a point of synchrony achieves in a tempo canon,
whether that synchrony occupies the begirming of a diverging canon, the ending
of a converging canon, or the middle of a converging-diverging canon. Al-
though it is clear that voices at different tempos are gradually and contin-
uously drawing near to or away from one another, their degree of dissonance
can seem nearly uniform for a good portion of a canon. Only near the point of
synchrony does the fast approaching/departing [temporal] consonance claim
perceptual prominence. (Thomas 1996, 138)

Gann generalizes that as a CP approaches in a piece and the echo distance decreases

that motives tend to Splinter and become briefer, and that the opposite occurs as a CP re-

cedes and the echo distance increases:

Typically, in the late canons, the following motion occurs: immediately fol-
lowing any convergence point, a quick echoing of brief figures creates ex-
citement Signalling the entrance to a new section of the piece. Usually voices
lose their distinguishability in a bristling texture, then slowly separate. Short
figures are gradually displaced by longer and longer motives which sound
calmer (by virtue of their temporal stability) but also more complex, even
developmental. As a new convergence point is approached, the echo-tempo
picks up again, and figures splinter into ever briefer motives, creating a
deliciously gradual feeling of cumulative climax. (Gann 1995, 173-74)

Another event (or series of events) that can signal the approach of a CP is what

Gann calls “collective effects,” or a place in the music where the individual voices

110

coalesce to form a single gesture. Glissandos, arpeggios, and scalar passages are fre-

quently used by Nancarrow in this way, as are other patterns that may at first be quite

separated in the voices before merging together.

Recall that the convergence period is the distance between potential simultaneities.

Gann points out that in canons involving superparticular ratios the echo distance “will

approximate It beats at a point It convergence periods from a convergence point” (Gann

1995, 21). In Figure 3-1, this is obscured somewhat by rests but can be most clearly seen

at the points marked 8), where the echo distance is equal to the convergence period of

five beats in the top voice and four beats in the bottom. At these points it is easiest to see

that the respective lengths of the echo distance (in beats) in each voice are proportional to

the operational tempo ratio at that point; for example, at a point in this 4:5 canon where

the echo distance is 2 beats in the top voice, it will be 1.6 beats in the bottom voice.

Gann defines the convergence period as “the hypermeasure that exists between

(potential) simultaneous attacks in voices moving at different tempos” (1995, p. 21).

Kramer defines hypermeasure as a “group of measures that functions on a deep hier—

archic level much as does a measure on the surface” (Kramer 1988, 453). Gann’s use of

the word hypermeasure, however, unfortunately implies a metrical hierarchy that may or

may not be present, as the beginning of a convergence period does not necessarily coin-

cide with a metric accent. Also, a convergence period in Nancarrow’s tempo canons

could be as short as three or four beats in those canons with smaller ratios, certainly too

brief to be considered a true hypermeasure.

In some of Nancarrow’s canons where the meters do not change, the location of

some convergence periods is clearly marked by shared barlines. Also, for as long as the

metric pulse does not change, the length of the convergence period remains the same.

Shared barlines in Nancarrow’s scores are important markers of convergence and some-

times the only way one can accurately measure the location of events elsewhere in the

Score (such as the number of beats elapsed to the entrance of a later voice).

111

Figure 3-2 shows the beginning of a convergence period from Study No. 36

(“Canon 17/18/19/20”), where a convergence occurs every 17 half notes in the bottom

voice, every 18 half notes in the second lowest voice, every 19 half notes in the second

highest voice, and every 20 half notes in the highest voice. However, since the meter Sig-

nature is : there is a shared barline only every 17/18/19/20 measures, with the beginning

of another convergence period occurring exactly halfway between the shared barlines.2

3.  

_—   _———*

        ‘-

Figure 3-2. Beginning of convergence period (at shared barline) in Study No. 36 (p. 4 of score).

Two additional terms, timespan and point of simultaneity, are important in making

determinations about the placement of CPS and the amount of delay that is needed in

later-entering voices to make a CP occur at a certain point. Kramer defines the timespan

as the “interval between two timepoints” (Kramer 1988, 454). For determining the time-

span of Nancarrow’s canonic subjects, the critical question becomes the location of the

first and last timepoints. The first timepoint is clearly the first beat attack. There are gen-

erally two possibilities, however, for the location of the final timepoint: coinciding with

either the attack of the final note, or the release of the final note (which can be said to

coincide with the attack of the hypothetical beat that would follow the end of the final

note). The most important criteria for making the determination of which should be the

final timepoint are usually how the canon ends metrically and its prevailing metric pulse.

2There are also convergence periods every 17/18/19/20 quarter notes, eighth notes, sixteenth notes, etc., but
these are not viable metric pulses.

112

    
 
Study No. 14 (which was the subject of Figure 3-1) will serve as an example of a

canonic subject in which the final timepoint coincides with the final attack. Gann notes

that there are 337 total eighth-note beats in the subject of Study No. 14 (Gann 1995, 117).

I concur with this assessment because of the concluding nature of the final attack (with

the remainder of the final measure filled with rests) and because the final note matches

the prevailing eighth-note metric pulse (see Figure 3-3). By measuring from the attack of

beat 1 to the attack of beat 337, the distance between these two timepoints, or the time-

span, is 337 — 1 = 336 beats. From that measurement we can determine that the midpoint

of the canon, where the CP occurs, is the attack of beat 169, with 168 beats occurring

prior to this point and 168 beats after. Beat 169 is, indeed, the location of the CP.

1:83   

i.

.
1

6

7

8

4

5
(a)

J
330 331

328

.

333

336 337

(b)

Figure 3-3. (a) Beginning and (b) ending (score, p. 5) of Study No. 14; eighth-note beats of note
attacks are numbered. (See also Figure 1-10, p. 36, regarding derivation of Study No. 14’s
rhythm.)

A different Situation occurs in Study No. 36 (“Canon 17/18/19/20”), a four-voice

arch canon where the final timepoint is the release of the final note and not the final

attack. Figure 3-4 shows the beginning and end of Study No. 36. Gann quite correctly

identifies the half-note beat as the pulse,3 but incorrectly states that there is a total of 842

such beats. There are actually 843 beats, consisting of the following: 105 measures of i

(210 Js) + 1 measure of 2 (3 J3) + 534 measures of i (534 Js) + 48 measures of i (96

Js) = 843 half-note beats (see also Figure 3-13).

3Although the tempo markings state the whole-note beat, the half note is the smallest reasonable pulse

because there is an extended section of 4 time and also a measure of 4 .

2

6

113

      
 
(a)

.1.
.
[511* ‘ Law-- an, x x __ #35
L_____._
___.~._______\__...._..

a.

1

1

2

3

4

5

6

of: -

83_o__

_g z;

34

333

__
835

\\_,__ .____, _~ __ _ __ _ -__

(b)

(\L:

i

I

w

I

:‘eaeLw-i ‘__837“‘”‘”":8'38WWW __34‘
\¥

__

"

" 1“”‘.'_;

:1"; _. __ __.

_l

842

843

Figure 3-4. Beginning (a) and ending (b) of Study No. 36; half-note beats are marked.

Agmon considers the question of how the duration of a complete musical compo-

sition relates to its final attack, and posits as a basic principle that “the end of a piece of

music is never its last attack. A composition’s last attack cannot be an end because it

necessarily initiates a musical duration—specifically, the duration of the musical event

that immediately follows” (Agmon 1997, 46). He concludes that “the duration of a piece

of music constitutes a closed interval [x,y], where x is (usually) the piece’s first attack,

and y is (practically always) a mentally supplied terminal beat Shortly following the last

attack” (p. 51). Although determining the timespan of Nancarrow’s canonic subjects is

Similar to but not quite the same as determining the rhythmic close of a piece, Agmon’s

observations offer some helpful ideas about measuring the timespan beyond a final

attack. In order to calculate the timespan in cases such as these, the attack of the down-

beat that follows the release of the final attack can serve as the final timepoint (in

essence, the mentally supplied terminal beat). Thus, in our example of Study No. 36, the

final attack (beat 832) is obviously not the final timepoint. Because the last note is held

114

 
 
 
 
   
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
  
  
 
 
for considerably longer than the metric pulse, and extends through the end of the final

measure, the attack of hypothetical beat 844 serves as the final timepoint, confirming the

timespan as 843 half-note beats.

Measuring the timespan is somewhat Simpler when the CP occurs at the end of the

canon (i.e., converging and diverging-converging canons) because, in these pieces, the

final CP defines the concluding timepoint. Most of the time (e.g., Studies No. 18, 19, and

24) the CP occurs on the final attack and the timespan would be measured from the first

attack to the final attack. Occasionally, however (e.g, Study No. 32), the CP occurs after

the final attack and the timespan must be measured to a timepoint that follows the CP

(usually a mentally-supplied terminal beat).

Potential points of simultaneity (i.e., the beginning of a convergence period) have

tremendous importance in the structure of tempo canons. Obviously, a CP can occur only

at a point of simultaneity in all sounding voices. A piece may have a large number of

potential points of simultaneity, if the tempo ratio involves fairly small numbers, or it

may theoretically have only one point of simultaneity (especially in those tempo canons

whose proportions involve irrational numbers—although, practically Speaking, there may

still be a number of places in these pieces where virtual simultaneities occur that are

beneath the threshold of perceptual discrimination, or “just noticeable difference”). At the

same time, many potential points of simultaneity are de—emphasized through the use of

rests or held notes (for instance, see Figure 1-2, p. 23, as well as Figure 3-1).

The importance of the location of potential points of simultaneity to the structure of

tempo canons can be illustrated as shown in Figure 3-5. This example compares the two

Studies just presented, Nos. 14 and 36, both arch canons with a single CP in the middle.

Nancarrow composed Study No. 14 so that its CP would fall exactly in the middle of its

337-beat theme—that is, on beat 169. In Study No. 36, the midpoint of the canon’s 843-

beat timespan falls between beats 421 and 422, and is obviously not an appropriate spot

for an audible CP. Nancarrow thus moves the CP slightly later to beat 427; this Slight

115

shift means that 426 half-note beats occur before the CP, and 417 occur after.

Study No. 14 (“Canon 4/5”)

midpo'nt/CP

1

[...]

159

164

169

174

179

[...]

337

Jlbeats

1

[

]

161

165

169

173

177

[...]

337

Study No. 36 (“Canon 17/18/19/20”)

CP

1

J beats

l [...] 387

407

427

447

467 [...] (844)

1

[ .

.

. ]

l

[ .

.

. ]

l

[...]

389

391

393

408

409

410

427

427

427

446

445

444

465

463

461

[ .

.

. ]

(844)

[ .

.

. ]

(844)

[. . .]

(844)

midpoint = beat 421.5

Figure 3-5. Placement of interior CPS in relation to canonic midpoint, and the role of potential
points of simultaneity. In Study No. 14 (top), canonic midpoint coincides with Simultaneous beat
in both voices, allowing CP to coincide with midpoint; in Study No. 36 (bottom), canonic mid-
point does not coincide with simultaneous beat in all voices and canon is constructed so that CP is
delayed to a later beat. Simultaneities noted are potential points of simultaneity as there is not
necessarily an attack in each voice at these points.

One can visualize, in Figure 3-5, the separate voices in each piece being represented

by proportionally-sized blocks of wood in which a notch in the middle represents the

midpoint. The shorter blocks (representing the faster, later-entering voices) are slid into

position above the longest block. In Study No. 14, the notches representing the midpoints

in each voice would be perfectly aligned; in Study No. 36, they would be slightly

staggered.

If it does not matter to the composer whether or not the entrances of later-entering

voices coincide with beat attacks in voices that are already sounding (i.e., with the begin-

ning of a convergence period)—and for Nancarrow it often did not—then the CP can

116

 
 
 
 
 
 
 
 
  
 
 
essentially be placed wherever the composer chooses. Convergence periods will then pro-

ceed both backward and forward from the CP, depending on its location. In the two

Studies shown in Figure 3-5, the entrance of the second voice in No. 14 is between beats

33 and 34 of the first voice, and none of the later voices in No. 36 enters on a beat attack

in an earlier voice. In fact, in most of Nancarrow’s canons later entrances do not coincide

with beat attacks. One clear exception is Study No. 18 (“Canon 3/4”), a two-voice con-

verging canon with the CP on the final attack. Since the 1,680 half-note-beat timespan of

this canon is divisible evenly by both factors in the canon’s ratio (3 and 4), the faster

voice in this canon enters after 420 half-note beats have elapsed, coincident with the

421st beat attack of the slower voice. Thus, in this Study the second voice enters at the

beginning of a convergence period, and this is confirmed in the score by a Shared barline

at the entrance of the second voice.

Finally, two tempo change devices are used with some frequency in Nancarrow’s

tempo canons: the tempo switch and the tempo overlap. The tempo switch is defined by

Gann:

Another major, but less audible, event is the tempo switch, a device in which
Nancarrow switches the fastest line to the Slowest tempo and vice versa, so
that the line that has been lagging catches up with the one that has been
pulling ahead. By mathematical necessity, the tempo switch always occurs
halfway between two convergence points. (Gann 1995, 21)

Tempo switches are useful in creating additional CPS. They are a Significant part of Study

No. 24 (“Canon 14/ 15/ 16”), where they are used to create thirteen CPS which delineate

twelve sections.

Occasionally a tempo changes in only one voice to create a tempo overlap, where

the same tempo is occurring in more than one voice. Figure 3-6 shows how two tempos

can form a tempo overlap in a diverging-converging canon. This is, in fact, exactly how

Study No. 15 (“Canon 3/4”) is configured. The tempo overlap section in a two-voice

canon will consist of a portion of the entire canon that is equal to m + n , where m and n

1

117

are the two elements of the tempo ratio. For instance, in Figure 3-6 the two tempos (and

the sectional durations) are related by the ratio 3:2, so the tempo overlap section occupies

one—fifth of the entire canon.

Tempo 1

Tempo 2

Tempo 2

Tempo 1

1

r

tempo overlap

Figure 3-6. Diagram of a tempo overlap at a 3:2 ratio in a diverging-converging canon.

Another Study with tempo overlaps is No. 17, whose structural diagram was shown

in Figure 2-7 (p. 78). Tempo overlaps occur between section A in the top and bottom

voices, and between section B in the top and bottom voices (see Figure 3-7). Recall that

the “duration ratio” of this 12:15:20 tempo canon is 3:4:5, which is how the durations of

sections C, A, and B, respectively, relate to each other. In general, in a three-voice tempo

canon where the elements of the duration ratio (not the tempo ratio) are x:y:z (with x

being the smallest element and z the largest), the difference between the durations of x

and y related to the length of the entire piece is x+y+z , between y and 2 it is x+y+z ,

A=4

l

3:55

C=3

l

I

i

5

1

[1,12

3:5:

I

C=3}

I

A=4

I

1

1/6

I

I

E

C=3

A=4

:B=5

Figure 3-7. Structure of Study No. 17 (“Canon 12/15/20”) showing the relative duration of tempo
overlap sections. The duration ratio is 3:4:5 between sections C, A, and B, respectively. The first
overlap is between A sections in the top and bottom voices, and the second overlap is between B
sections in the same voices. There is no overlap involving C, the shortest section.

118

 
  
 
 
  
 
 
  
 
 
   
  
and between x and 2 it is x+y+z. Thus, in No. 17 the sum of the duration ratio’s ele-

z — x

ments, x:y:z, is 12, and the durational relationship between contiguous elements in the

ratio (3:4 and 4:5) is one-twelfth of the whole piece while between the outer elements

(3:5) it is one-sixth. As shown in Figure 3-7, the tempo overlap involving section A

occupies one-twelfth of the piece, and that involving section B occupies one-Sixth of the

piece. The section B overlap is long enough to be bisected by a tempo change in the

middle voice.

Rhythmic and Metric Convergence

Although the primary focus of this chapter is canonic convergence in tempo canons,

the topic of rhythmic and metric convergence on a smaller scale in some of Nancarrow’s

other Studies Should be briefly considered. Thomas discusses the metrically different

ways in which contrasting tempos can be placed against each other (see Figure 3-8). She

identifies three degrees of temporal dissonance based on the number of “metrically sig-

nificant” downbeats shared by the two voices, which in this example exhibit a 4:5 ratio:

(a) in this scenario, metric accents occur in both voices at all convergences for minimal

temporal dissonance; (b) here, both voices share the same meter for a metric accent every

fourth eighth note, but the number of shared metric accents is reduced to once every fifth

downbeat in the faster voice and fourth downbeat in the slower voice—this is mild

temporal dissonance; and (c) in this scenario, there are no metric accents in common for

maximum temporal dissonance.

These rhythmically- and metrically-based periods of convergence are often the

result of rhythmic ostinatos, and even in Studies that are not tempo canons these can

reach extremes of complexity. In Nancarrow’s first player piano Study, “Rhythm Study

No. 1 for Player Piano,” two lines in 17; and 44 meters create an ostinato that runs through

the entire piece (see Figure 3-9). Each line declaims a chord at regular intervals, and the

interval of iteration 1S generally tw10e as fast in the 8 vorce to create a 7:4 tempo ratio

119

l

4b- “0

113110

(c)

Figure 3—8. Three degrees of temporal dissonance at 4:5 tempo ratio as shown in Thomas (1996,
p. 253): (a) all Shared downbeats are metrically significant in both voices for minimal temporal
dissonance; (b) only some Shared downbeats are metrically significant in both voices for
moderate temporal dissonance; and (c) no shared downbeats are metrically significant in both
voices for maximum temporal dissonance.

between the voices most of the time. Nancarrow creates an elegantly balanced arch-like

Structure, with the first 14 measures and the last 13 measures containing no Simultaneities

(maximum temporal dissonance somewhat as illustrated in Figure 3—8c, although here the

dissonance is even more extreme because there are no Shared articulations—accented or

otherwise—at all). The rate of iteration increases incrementally and proportionately to the

beginning of measure 21, which begins a section of convergences on every other down-

beat in both parts of the ostinato (closer to Thomas’s minimal temporal dissonance as

shown in Figure 3-8a). The 3-bar convergence periods of mm. 15—21 are then balanced

120

 
 
 
by the same in mm. 81-87, and then at m. 87 Nancarrow again creates increased temporal

dissonance by thwarting all Simultaneities. In this section, after the final simultaneity at

m. 87, the rate of iteration now incrementally decreases to return to the original rates; this

time, however, the changes in the rates of iteration are staggered in the two voices to

create additional ratios of 7:3 and 7:5.

measure no. (not a proportional representation)
1

1a

15

21

9

81

87

88.5

89-1/7

92-4/7

93.5

(no simu

rate of
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13
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392°32’8- 52m“ periods or
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1.
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1.1

13

37:3

J
3).-7:4

13.3
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7:5

U.
t1

3:7:4

first simultaneity

last simultaneity

Figure 3-9. Relationship of simultaneous beats and convergence periods between two ostinato
lines in “Rhythm Study No. 1.”

Convergence Points in Nancarrow’s Tempo Canons

1:me Canons With One (0; Almost Qne) QP

Two of the basic types of tempo canon classified by Thomas (Figure 1-17, p. 42)

have just one CP: the converging-diverging (arch) canon, and the converging canon.4 In

the arch canon the CP falls somewhere in the middle, and in the converging canon it falls

at the end. Figure 3-10 shows line diagrams of Nancarrow’s tempo canons with just one

CP.

Some of Nancarrow’s earlier pieces have interesting examples of CPS. Study No. 4

is a loosely canonic piece and an early precursor of the tempo canon technique; a crude

variety of tempo canon is effected by notating the canon in different voices in different

time values, e.g., five Sixteenth notes tied together in one voice, dotted eighth notes in

4The diverging canon type also has a single CP, but Nancarrow did not write any complete examples of
these.

121

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Study No. 14 - Canon 4,5

4,
1
I

Study No. 18 - Canon 3/4
———-l
#1
l

Study No. 19 - Canon 12/15/20

41
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Study No. 31 - Canon 21/24/25

Study No. 32 - Canon 5/6/7/8

Study No. 21 - Canon X

>< A

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B

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B

c

C

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[vergence pount
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Study No. 36 - Canon 17/18/19/20

S‘UdlgNO- 48 - Canon 60/61

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No. 48C

No. 493

No. we - Canons 4/5/8

Study No. 49A
—-l

—i—-
—4——1————1

———)

Figure 3-10. Line diagrams describing Nancarrow’s tempo canons with only one CP (Gann 1995,
22, 23, 25, and 26).

 0‘0

Figure 3-11. Convergence point from Study No. 4 (score, p. 10) in a passage exhibiting a loosely
interpreted 2:3:5 rhythmic ratio.

another, and eighth notes in another (a 2:3:5 ratio). This is the ratio in effect in the

passage shown in Figure 3-1 1, which shows the one CP in this piece involving all three

voices (two systems prior to this point is a CP involving two of the voices). Note,

however, the inconstancy with which the tempo/rhythmic ratio is applied in that pitches

that are sixteenth notes are the same value in all voices; this procedure is consistent with

122

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
the free rhythmic imitation that takes place in the rest of the Study.

Arch Canons. Nancarrow’s arch canons vary widely in ambition and scope. Study

No. 14 has its CP exactly at its canonic midpoint (see Figure 3-1), and the realization of

this structure seems to be the primary goal for the piece. Study No. 36 has one of the

most spectacular CPs in all of Nancarrow’s oeuvre in terms of its preparation and

realization. Study No. 21 has one true canonic CP in addition to an area of “crossing

tempos” and a loosely canonic CP on the last note. Study No. 498 returns to a simpler

texture but also has a rather arresting CP.

As has already been noted, Study No. 14 (the first that Nancarrow subtitled

“Canon”) exhibits the basic arch canon form with the CP exactly in the middle of its 337-

beat theme (see Figure 3-12). With a tempo ratio of 4:5, the faster voice occupies 80% of

the duration of the slower voice, with the remaining 20% of the timespan split so that

33.6 beats elapse prior to that voice’s entrance, and 33.6 beats after it concludes; thus, the

second voice does not begin at the beginning of a convergence period with the lower

voice, but rather three-fifths of the way between the lower voice’s beats 33 and 34.

convergence point (beat 169)

33.6 beats elapse before upper voice enters

l

4

I J=110 (follower)

J=88 (leader)

4

(leader)

(follower)

(timespan from attack of beat 1 to attack of beat 337 = 336)

337 total beats in each voice

33.6 beats
1

I

i

3

>

Figure 3-12. Structural diagram of Study No. 14 (“Canon 4/5”), an arch canon.

Another important feature of arch canons, and any tempo canons with interior CPS,

is the changing leader-follower relationship between the voices that takes place at interior

CPs (see the CP in Figure 3-1). In fact, in Study No. 14 there is so little else calling

attention to the CP that the changing leader-follower relationship is thrust into relief and

123

 
   
 
 
 
 
is what most immediately captures the listener’s attention. Gann points out (p. 118) that

Nancarrow exerted so much rhythmic control in the creation of his canonic subject in No.

14 that melodic and harmonic coordination between the voices is virtually absent, and

this lack of coordination does little to prepare and support the CP. There is also a fairly

low incidence, in general, of rhythmic coordination between the voices due to the numer-

ous rests; both Gann and Thomas note that the fairly small 4:5 ratio between the voices

would potentially allow for convergence every four to five beats, but actual simultaneous

attacks are avoided more often than realized. Thus, in Study No. 14 the CP’s placement at

the exact midpoint defines the structure, but Nancarrow’s highly contrived rhythmic

scheme precludes a high incidence of convergence at the metric level.

The formal sophistication of Study No. 36 (“Canon 17/18/19/20”) makes it a stark

contrast to the somewhat humble beginnings of Study No. 14. It is an arch canon in

which the canonic midpoint falls between beats because the timespan is an odd number:

843 half-note beats, placing the canonic midpoint between beats 421 and 422. In this

canon, then, it is not possible for the CP and midpoint to coincide on a beat attack as they

did in Study No. 14 (see again Figure 3-5). Nancarrow instead delays the CP to beat 427,

thus creating two sections consisting of 426 and 417 beats (see Figure 3-13).

Philip Carlsen notes that the placement of the CP on beat 427 creates a particular

relationship of the elapsed time between statements of a recurring theme (which he calls a

rondo theme) within the canon and the CP (see Figure 3-13, bottom diagram):

The rondo theme’s second appearance [at m. 107] is twice as fast as the first;
it occurs at a point (system 36 in the bass voice) exactly halfway between the
beginning and the canonic midpoint, providing an obvious parallel with the
well-known acoustical fact that the halves of a string vibrate twice as fast as
the whole. (Carlsen 1988, 30)

Carlsen slightly rnisspeaks here by noting that the relationship is with the canonic

midpoint when it is actually with the CP, as shown in Figure 3-13. Carlsen’s observation

concerning the acoustical parallel can be carried further by noting that the first three

124

statements of the theme occur on CZ, F#z, and C3 so that the second statement “bisects”

not only the elapsed time to the CP but the octave between the first and third statements.

Such connections are quite common in Nancarrow.

427m J = CP (m. 320)

20

follower 1  follower 2

19

17

fl

follower 3

timespan = 843 J '5

>

Jelapse needed before and
after faster voices for CP to
occur at midpoint (between
beats 422 and 423)

Actual J elapse before
entrance of faster voices

Actual J elapse after
conclusion of faster voices

18-voice

ZO-voice I

23.4(I/180f42l.5)
44.4 (2/19 of 421.5)
63.2 (3/20 of 421.5)

23.7(l/l8of42_6)
44.8 (2/l9 of 42_6)
63.9 Q/ZO of 42_6)

23.2(1/180f4l7)
43.9 (2/19 of4l7)
62.6 (3/20 of 417)

GP 0 m. 320

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first statement of
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Figure 3-13. Diagram of Study No. 36 (“Canon 17/18/19/20"), an arch canon with the CP slightly
past the midpoint (top); table of elapsed half-note beats to entrances of higher voices in com-
parison to the first (lowest) voice (middle); and structural diagram of the canon, showing entrance
points of recurring theme and relationship of theme statements to CP (bottom).

The table of elapsed beats in Figure 3-13 describes the number of beats elapsed to

125

  
      
    
 
   
 
 
 
 
 
   
 
 
 
 
 
 
the entrance and from the drop-out point of the top three voices, in relation to the first (or

lowest) voice. The first column shows the beat elapse needed to place the CP exactly at

the canonic midpoint; the second and third columns show the beat elapses that actually

occur prior to and after, respectively, the statements in the top three voices.

None of the voices enters at the beginning of a convergence period, as shown in

Figure 3-14. In fact, even the first voice begins on the last beat of a convergence period,

as half-note beat 2 marks the beginning of the first complete convergence period. The

second voice’s 23.7-beat elapse places its entrance within the second convergence period.

Figure 3-14 also shows how the echo distance decreases by one beat with the passing of

each convergence period; at the beginning of each period, the number of beats between

the voices is equal to the number of convergence periods remaining to the CP.

beat number of J beats at beginning of convergence period

20-Iine

19-Iine

18—line

17-Iine

2

25

1 9

24

9

31

53

1 3

36

23

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7

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49

70

21

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67

87

47

66

85

1 04

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103

121

20

19

18

Number of convergence periods to GP

Figure 3-14. Half-note beats associated with beginnings of convergence periods at the beginning
of Study No. 36. The difference between beats in each voice at each point is equal to the number
of convergence periods remaining to the CP—for instance, where the difference between beat
values is 20, the distance to the CP is 20 convergence periods.

The stunning CP in Study No. 36 (Figure 3-15) is one of Nancarrow’s finest mo-

ments. Even the composer was surprised, upon hearing the passage where the CP occurs,

to hear the effect of what Reynolds refers to as “low-register difference tones” (Reynolds

1984, 16) generated by the extremely rapid iteration of notes in all the voices. Carlsen

describes the passage this way:

126

 
 
  
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In this passage, the four voices sweep up to their point of convergence with
rapid chromatic glissandos, arriving simultaneously on their highest notes—
Bb3, D5, F6, and A7? Each high note is then rapidly repeated at the speed of a
quarter-note in its own voice’s particular tempo. . .. Even in the slowest
voice, the reiterated high notes fly by at a rate of more than 340 per minute (5
per second). But that is not all: the spaces between high notes are filled in with
ascending thirty-second-note glissandos. At such speeds, the perception of
individual pitches is impossible. The thirty-second-notes (which, in the bass
voice, move at over 3000 per minute or 50 per second) are fast enough to
theoretically start generating additional low pitches with frequencies in the
range of approximately 50—60 Hz. (Carlsen 1988, 25)

The convergence point is preceded by what Gann identifies as a “mega-glissando”

that begins 80 measures before the CP, creating a fine example of a “collective effect”

that is an extraordinarily long preparation to the CP. Figure 3-16 shows a section in the

middle of this passage. Exactly halfway between the beginning of the mega-glissando

and the CP, the CI half note in the lowest voice begins a restatement of the opening

transposition levels, which spell out a widely spaced major seventh chord. The last voice

reaches its restatement at the beginning of the next convergence period; two measures

prior to this, the lowest voice starts another restatement, this time an octave higher.

This section of rising glissandos creates a tremendous build-up to the CP. The glis-

sandos emphasize to the listener the leader-follower relationship of the four voices, and

Nancarrow very cleverly manipulates the decreasing echo distance, by using steadily de-

creasing gaps between the glissandos, to eventually create long, continuous glissandos

through the four voices. After the CP, the glissandos descend through the voices and the

listener is once again acutely aware of how the leader-follower relationship has changed

at the CP.

An unusual and unique canon is Study No. 21 (“Canon X”), an acceleration canon

whose subtitle is derived from the manner in which the tempos of the two voices “cross”

in the course of the piece. Study No. 21 is unlike any of the other tempo canons, with its

gradually changing tempos (accelerating in one voice while the other voice decelerates)

SThe levels of imitation in Study No. 36 spell out a major seventh chord.

128

 
beginning of new convergence period

Figure 3-16. Section of Study No. 36 halfway between beginning of “mega-glissando” and the CP
(score, p. 21) where original pitch levels are restated in all four voices.

that almost imperceptibly “cross” somewhere in the middle of the piece. Gann reports

that in punching this piece, Nancarrow “went through a tedious process of placing the

[roll punching] mechanism one half or one quarter of the way between notches” in order

to effect geometric acceleration and deceleration; the process was so frustrating that Nan-

carrow “had the machine altered so that, instead of punching holes at points indicated by

notches on the mechanism, it would move along a smooth continuum, capable of placing

a hole at any point” (Gann 1995, 106).

In No. 21’s proportionally-notated score, there is no meter and no shared downbeats

are notated, but it appears that there are three significant convergences. This Study thus

129

  
 
 
 
 
 
 
 
 
 
fits somewhat uneasily into this section on arch canons with one interior CP; however,

only one of the convergences is a true canonic CP, and it does appear in the interior of the

piece. The first convergence, just over one-third of the way through the piece where the

two voices cross tempos, is not a canonic convergence point but more of a convergence

area (see Figure 3-17). Thomas (1996, pp. 126, 281) identifies the bracketed area in the

figure as a passage of “nearly simultaneous motion.” However, the tempos of the two

lines actually coincide somewhere near the beginning of the next system, where the note

articulations are nearly evenly staggered between the voices. The crossing of the tempos

is thus not marked by a convergence point.

.   

area identified by Thomas
(1996) as “nearly simul-
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Figure 3-17. Area of tempo crossing in Study No. 21 (score, p. 7). The lower voice is accelerating
and the upper voice decelerating. Bracketed area is identified by Thomas as representing “nearly
simultaneous motion” (Thomas 1996, 126).

At just over two-thirds of the way through the piece, an actual canonic CP does

occur (see Figure 3-18). Its significance is enhanced by several factors: the 54-note row

(B-C—Al .

. .) that comprises the canon begins again in each voice; there is a registral

shift upward in the lower voice; and the texture of the upper voice changes from triple

130

 
   
    
 
 
 
 
 
 
 
octaves to quadruple octaves. Thus, this CP is both the conclusion of a converging canon

and the beginning of a diverging canon. At this point in the piece there are about 16 notes

in the bottom voice between successive articulations in the upper voice.

Study No. 21 concludes with a non-canonic CP on the final note; at this point there

are about 48 notes in the lower voice between articulations in the upper voice. The canon

is altered at the end so the piece ends on a V—I cadence with a quintuple C octave.

J.

i

ii

.l.

3

Figure 3-18. Convergence point in Study No. 21 (score, p. 14). The arrow indicates the CP, where
the canon starts over in both voices, a registral shift occurs in the lower voice, and the texture in
the upper voice (top two staves) changes from triple octaves to quadruple octaves.

The final arch canon shown in Figure 3-10 is Study No. 498, the middle of three

movements based on the tempo ratio 4:5:6. This author was not able to obtain a copy of

the score, and according to Gann (1995, p. 136) there are considerable discrepancies be-

tween the score and the Wergo recording. Analysis of this piece here is thus somewhat

limited.

No. 49B’s three voices are monophonic throughout, and like all of Nancarrow’s

tempo canons the piece begins with the bottom voice, followed by the faster middle

voice, and then the fastest top voice. The voices appear to be symmetrically arranged

about the CP, which occurs just past the midpoint of the piece (at 1’23”, or 52% of the

way through the piece’s total 2’40”6). The two most striking features about the CP are the

6The natural acceleration of the piano roll may account for the slightly longer duration of the first half.

131

  
   
  
  
 
way in which all three voices accelerate (not just become more rhythmically active, but

actually accelerate the beat) in the 10 seconds leading to the CP; and the convergence of

the three voices on a sustained major triad. Since the voices originally entered at a level

of imitation at the twelfth between all voices, at some point prior to the CP the levels of

imitation change.

After the CP, the voices are only loosely canonic and certain motives from earlier in

the piece become more affiliated with specific voices. A second convergence occurs at

about 2’04”, where first the middle voice, then the lower and top voices state an ascend-

ing and descending melodic minor scale, with the three voices converging on octaves.

Thus, through the course of the piece the levels of imitation have converged from

twelfths, to spelling out a major triad, to octaves. Also, the expected leader-follower rela-

tionship of voices after the CP, or top-middle-bottom, is not strictly adhered to in stating

canonic material.

Converging Canons. The next group of canons to be examined is the converging

canons shown in Figure 3-10 (p. 122), which include Nos. l8, 19, 31, 32, 34, 48, 49A,

and 49C.

One of the distinguishing characteristics of converging canons is that the leader-

follower relationship of voices is maintained to the final CP. This is easily seen in a

structural diagram for Study No. 18 (“Canon 3/4”), which was mentioned earlier as an

example of a canon where the entrance of a later voice coincides with the beginning of a

convergence period (see Figure 3-19). This is made possible because the canon’s time-

span (1,680 eighth-note beats) is divisible by both 3 and 4 and the first voice begins at the

beginning of a convergence period.7

7It should further be noted that 1,680 is also divisible by 5; had Nancarrow wanted to make this a 3-voice
canon with a 3:4:5 ratio, the entrance of the third voice would also have coincided with the beginning of a
convergence period—specifically, after elapse of 672 beats in the bottom voice and 336 in the middle. In a
way it is unfortunate that he did not write such a canon, as none of his converging canons is structured so

132

 
convergence point (beat 1,681)

420 beats elapse before upper voice enters

T

I

J J=224 (follower)

l

J=l68 (leader)

4

1,681 total beats in each voice
(timespan from attack of beat 1 to attack of beat 1,681 = 1,680)

P

Figure 3-19. Structural diagram of Study No. 18 (“Canon 3/4”), a converging canon in which the
entrance of the second voice coincides with the beginning of a convergence period.

In Study No. 19 (“Canon 12/15/20”), like most of Nancarrow’s tempo canons, later-

entering voices do not enter at the beginning of a convergence period (Figure 3-20). The

duration ratio of the three sections is 3:4:5 (see Figure 2-7, p. 78), and the bottom voice

relates to the middle voice by the ratio 4:5 and to the top voice by the ratio 3:5. The time-

span of 336 eighth-note beats is divisible by 3 and 4, but not by 5; thus, since both the

middle and top voices relate to the bottom voice in ratios containing the factor 5, neither

of these voices enters at the beginning of a convergence period with the bottom voice.

The middle voice enters after 20% of the bottom voice has elapsed, or one-fifth of the

way between beats 67 and 68. In the top voice, the tempo relationship with the bottom

convergence point (beat 337)

134-4 beats

duration = 3

  J=240 (follower 2)

  67.2 beats

 l J=180 (follower 1)

duration = 4

J=l44 (leader)

duration = 5

1:

337 total beats in each voice
(timesparfrom attack of beat 1 to attack of beat 337 = 336)

V

Figure 3-20. Structural diagram of Study No. 19 (“Canon 12/15/20”).

that all voices enter at the beginning of a convergence period and this would have been an impressive
effect.

133

 
 
  
 
 
 
 
 
 
 
 
 
 
voice is 3:5 and it enters after 40% of the bottom voice has elapsed, or two-fifths of the

way between beats 134 and 135.

The CP at the end of Study No. 19 is noteworthy because Nancarrow somewhat

uncharacteristically alters the interval of imitation between the voices (from elevenths to

double octaves) to create a unison V—I cadence (see Figure 3-21). I surmise that he may

have done this to convey an additional sense of finality because this canon concludes the

six-canon set of Nos. 13—19, all of which are based on the rhythmic series {n - 1, n, n +

Figure 3-21. Conclusion of Study No. 19 (“Canon 12/15/20”; score, p. 7), showing the con-
vergence point and alteration of final pitches to create V—I final cadence.

interval of imitation changed
from elevenths to double octaves

Study No. 32 (“Canon 5/6/7/8”) is a straightforward four-voice converging canon

whose structure is shown in Figure 3-22. The duration ratio for the four voices is 105:

120:1401168. No. 32’s final convergence point (see Figure 3-23) is unusual in that it

occurs at the final cut-off—that is, at the hypothetical downbeat that follows the final

beat. There are 431 J. beats in the canon, and the timespan is the same number because

134

 
the CP actually occurs on hypothetical beat 432. The second voice enters five-sixths of

the way between beats 71 and 72 in the lowest voice; the third voice enters one-seventh

of the way between beats 123 and 124; and the top voice enters five-eighths of the way

between beats 161 and 162. This canon will be the subject of more thorough analysis in

Chapter 4.

convergence point (beat
432—occurs at final cut-off)

161.63 beats

J; 136 (follower 3)

duration = 105

123.14 beats

J; 119(f0110wef 2)

duration = 120

71.83 beats

0'; 102 (follower 1)

duration = 140

J; 85 (leader)

duration = 168

<

P 
431 total beats in each voice
(timespan from attack of beat 1 to attack of beat 432 = 431)

Figure 3-22. Structural diagram of Study No. 32 (“Canon 5/6/7/8”).

Figure 3-23. Convergence point of Study No. 32 (score, p. l 1): the CP is the final cut-off.

135

 
 
 
 
 
  
 
 
  
11
In Study No. 34 (“Canon 4/5/67 4/5/6/ 4/5/6”), which was the analytical subject of

10 /

9

/

Chapter 2 (see, in particular, Figures 2-14 [p. 88] and 2-15 [p. 90]), measuring entrances

to later-entering voices is complicated by the almost complete lack of noted shared down-

beats in the score, the multiple sections of the canon, and the unmetered rests that occur

between many of the sections (making it impossible to establish a length for the canon’s

timespan; see Table 2-4, p. 100). There is even an unmetered rest within section G (al-

though it appears to relate to the 9:10:11 ratio effective in that section). The CP (Figure

2-28, p. 105) occurs on the attack of the final beat (this is, in fact, the only downbeat in

the entire score shared by all three voices).

Study No. 48 (“Canon 60/61”) is a remarkable achievement in convergence. It

consists of three movements: movements A and B are each 60:61 canons, which are then

played together to create movement C. The performance of C involves two carefully

coordinated player pianos. Movement A has a major tenth level of imitation, with an echo

distance that begins at about eight seconds (the score is proportionally notated and in-

cludes additive acceleration in all four voices, thus measurements are inexact), dwindling

to zero over the course of some 127 pages. There is a minimal level of convergence

between the two voices built into the canon; the most pervasive is regularly-recurring

pairs of arpeggios in which the second of the pair in the bottom voice aligns with the first

in the top voice (see Figure 3-24a).

Movement B is slightly shorter than movement A, as, during the performance of C,

it should enter after movement A’s 61-voice (see Figure 3-24b). From Nancarrow’s score

it is not clear what is the intended ratio between the two movements; it is not 60:61, but

closer to 34:35 (based on page count measurements), but in the actual performance of C

the B voices enter about a page earlier than indicated in the score, so this makes the

actual ratio closer to approaching 60:61. Movement B’s interval of imitation is the perfect

fifth; this allows the final chord of movement C to spell out a B; major seventh chord at

the extraordinary moment of coordination where all four voices converge. Dynamics play

136

_w

   61-voice of mvt. A

B (

121.7

.
PP )

(

123.7

PP )
.

A

(125.3

.)

PP

(127.3

.)PP

CF

61

6O

61

60

60-voice of mvt. A

__. __.._ J

(a)

(b)

Figure 3-24. (a) Convergence of arpeggio figures in Study No. 48A (p. 43 of score); (b) diagram
of proportions in Study No. 48 (movement C is movements A and B played together).

a critical role in the preparation for this convergence; Nancarrow brilliantly creates an

almost complete coordination of rapidly changing and highly contrasting dynamics

(mostly between ff and pp) for the final 28 pages of the piece, creating a convergence

of dynamics long before the temporal convergence.

Studies Nos. 49A and 49C are both converging canons in the ratio 4:5:6 (which

gives them a duration ratio of 10:12:15). Again, due to the lack of a score these remarks

are somewhat limited. The entrance timings of later voices in both movements conform

to the 4:5:6 tempo ratio. The CP in both movements is preceded by a section of glissandi

that coalesce in all three voices into larger, sweeping glissandi. The section which con-

cludes movement C is more extensive and the sweeping glissandi cover a wider range for

a more satisfying effect. This movement also contains an earlier converging glissando

section about one-third of the way through the piece.

137

  
 
 
 
 
   
 
  
  
Study No. 31 (“Canon 21/24/25”) is an unusual case. It is a converging canon with

no CP, as the canon is truncated before the CP is reached. As shown in Figure 3-25, the

canon’s actual length is 669 eighth-note beats, which is 36 beats shy of the 705 needed to

reach convergence (the timespan is 704 beats) at a ratio of 21:24:25. The lack of a

temporal convergence is seemingly underscored by the three key areas a fifth apart to

create what Thomas (1996, p. 67) refers to as an “unresolved asynchronicity” (see the

canon’s conclusion in Figure 3-26). The canon is in three distinct sections, each separated

by an eight-measure rest, and the third section is both registrally enlarged (the level of

imitation increases from a fifth between voices to an octave plus a fifth) and the canonic

line reinforced by octaves and full of melodic leaps. Based on these features, Thomas

offers a very perceptive and plausible reason for the lack of convergence at the end:

I___________ I

25 113.64 beats

l‘ ,-

-

_

, C

duration = 168

£69 (rears, fl 3 , -- 3- -3 >r36

24 I________ |
88 beats

l‘

duration = 175
669 beats

_

|
_ bl 36

_

I

1

I

I
2 1 I

<1 1
4

~

1

duration = 200
669 beats
1

7

,

,

,

,

I
F36 5115:" i

s

A, b

b 

timespan from attack ofbeat 1 to attack ofbeat 705 = 704 beats

_

-

Figure 3-25. Structural diagram of Study No. 31 (“Canon 21/24/25”), a converging canon that
ends before the CP is reached.

113—J'-    

,Vrf‘ff '1':

151;-

‘1“

o

.814:

.-

Figure 3-26. Conclusion of Study No. 31 (“Canon 21/24/25”; score, pp. 8—9).

138

 
 
 
    
  
It is in the final section of the study that the canon becomes most difficult to
perceive, and in this lies what may have been a potential reason for ending the
study before it can converge on a point of synchrony. Each single voice is
difficult to follow in and of itself in this section because of its extreme
registral leaps and because the articulation types are limited exclusively to
. Since it is difficult to follow a single voice, it is nearly impos-
staccato. .
sible to follow the canonic relationship among voices. The staggered ending
thus reorients us to the canonic nature of the passage: as each voice drops
from the texture in quick succession we are reminded retrospectively of them
as linear entities, and we hear in their concluding gestures their shared
material and imitative relationship. (Thomas 1996, 69—70)

.

As shown in Figure 3-25, the entrance of the middle voice in Study No. 31 con-

verges with the eighty-ninth beat in the bottom voice, and the top voice enters 1’62’s of the

way between the bottom voice’s beats 113 and 114. Although the timespan of 704 is not

evenly divisible by either 7 or 8 (the two factors to which 21 and 24 reduce), the entrance

of the middle voice coincides with a beat attack (beat 89) in the bottom voice because the

bottom voice does not start at the beginning of a convergence period; the first complete

convergence period begins at beat 5 in the bottom voice, which does allow convergence

at beat 89 when the second voice enters (89 - 5 = 84, a number evenly divisible by 7).

Tempo Canons With More Than One CP

Canons with more than one CP are of the following types: diverging-converging

canons (which begin and end with a CP), canons with interior tempo switches, and

special cases such as acceleration canons. Figure 3-27 shows the simpler canonic struc-

tures that contain more than one CP.

One of the basic types of tempo canon, the diverging-converging canon, has two

CPs (one at the beginning and one at the end). Nancarrow wrote two examples of this

type of canon: Studies No. 15 and 17 (see Figure 3-27). Canons of the diverging-con-

verging type have at least one tempo overlap (the number depends on the number of

voices and tempos in the canon). Study No. 15 contains a simple example of a tempo

overlap. This two-voice Study has a tempo and duration ratio of 3:4 and a canonic subject

139

.

I

3

7

u

- 

?
..‘52
'3
WH

...
p_.
r;
—.
0-D

‘1‘.
.
I
.

.
,
.
,
,
.
.

I
"

..-

’
3
'

,l—

‘
5

9

l

t
.

'4

‘Ql

.
"

.

.

Study No. 15 - Canon 3/4
A
A
w‘
‘r
i
l

i

A

A

i

B

41

Study No. 17 - Canon 12/15/20
A
leg.
i
1
‘1
l 
l
P

,
__.—(___+__|___
-—l—

Study No. 22 - Canon 1°/o/1-1/2°/o/2-1/4°/o
,8

i
:3

1”

1C

lA

A

.L

A

B

I

I

I

Study No. 33 - Canon an

A

A

B
—i—-
B

C

C

D

l D
1

i
L -
l

X

i

l
l

Study No. 43 - Canon 24/25

25 follower

24 leader

leader

24

follower x

25

follower

leader

Figure 3-27. Line diagrams describing structures of Nancarrow’s tempo canons with more than
one CP (Gann 1995, 22, 23, and 25).

of 336 eighth-note beats; the canon is stated twice in each voice, once at the slower

tempo and once at the faster tempo. When the faster voice repeats the canon at the slower

tempo, both voices are stating the slower tempo for a period of 84 eighth notes (one-

fourth of the canon’s length), and this section comprises the middle one-seventh of the

entire length of the piece (see Figure 3-28). An interesting by-product of this structure is

that the top voice, which initially states the faster tempo, remains the leader in stating the

canon throughout the piece. The piece is firmly entrenched in G major, and Nancarrow

composes the canon so that a G cadence occurs three-fourths of the way through, allow-

ing the two voices to coordinate in a cadence on G at the beginning of each statement; the

canon concludes with a V7—I cadence (see Figure 3-29).

4——— 336 D's ———>e

336 11's

A (J = 220): leader
A (J=165): follower

A (J=165): leader

A (J=220): follower

L——l
tempo overlap @ J=165 (84 D’s)

b

A

V-I cadence in
both voices

V7—I cadence
in both voices

Figure 3-28. Structure of Study No. 15 (“Canon 3/4”), a diverging-converging canon in two
voices.

140

  
 
 
 
 
 
 
 
 
  
  
 
 
 
   
 
q];

l_ — I _ "I“
A

Beginning;

Jezzo

(galore
J2”;

Beginning of canon’s repeat in top voice
and of tempo overlap section:

I

3:145“

8Q _

0G0   

= 165)

     1.4

0"“ "‘ l‘ "°

V-l cadence in (3 built in
3/4 of way through canon

Beginning of canon’s repeat in bottom voice
an end of tempo overlap section:

Final convergence point:

,.
g:

\ Jews)   
A: 210    l 900

V7

-

I

Figure 3-29. Structural highlights of Study No. 15: cadential figures at beginning and end of
canon statements in each voice.

Another important structural feature of diverging-converging canons is that each

voice successively states every tempo, with each voice stating the tempos in a different

order and each tempo uniquely associated with a canonic section that is the same length

in each voice. This can be clearly seen in the structure of Study No. 17 (“Canon

12/15/20”), which was shown in Figure 3-7 (p. 118). The three sections are stated once in

each voice, with the A section associated with the tempo J = 172.5 and a duration of 4,

the B section associated with the tempo J = 138 and a duration of 5, and the C section

associated with the tempo J = 230 and a duration of 3. As shown in Figure 3-7, there are

no tempo overlaps involving C, the shortest section. The longest section, B, has the

longest overlap, with the A section having an overlap half that length.

The same principles are applied in Figure 3-30, which shows the structure of a con-

jectural four-voice diverging-converging canon and the associated tempo overlaps.

141

    
     
  
  
 
 
 
   
Figure 3-30. Conjectural example of a diverging-converging canon in four voices and tempos,

where the tempo ratio is 10:12:15:20 and the duration ratio is 3:4:5:6. There are no tempo

1

l

1

overlaps involving A (the fastest tempo), overlaps of /18 + 48 = /9 of the total piece involving

4

6

B, /18 = % involving C, and /18 = J"3 involving D (the slowest tempo).

Study No. 17 does not exhibit as much harmonic convergence at section changes as

No. 15 does, and, with almost continuous eighth-note motion in three different (and quite

fast) tempos, it would be difficult to hear such coordination anyway. The CP at the begin-

ning of No. 17 coordinates on a F major triad while the final CP features a cadence on A

(see Figure 3-31). Throughout the piece, the A section always begins on the note C, the B

section on A, and the C section on F. Nancarrow keeps to a limited number of frequently

changing key areas (mostly F, A minor, C, G, and D, with their affiliated accidentals

B1,, F#, C#, and G#), which allows some brief and random areas of harmonic convergence

to occur. Two such areas are shown in Figure 3-32; the first example shows an area of F

major in the bottom two voices where the middle voice begins section C; and the second

example shows an area of A minor in all three voices at the point where section B starts

in the bottom voice. The melodic construction is highly organic, and the majority of the

melodic patterns are shown in these examples: the melodic movement is all in eighth

notes in groupings of one to four notes, mostly monophonic with occasional single

octaves, and a preponderance of movement by half steps.

142

 
 
 
 
 
 
 
  
 
1‘ {‘32 = iii/L: 5381:1110 -:

v-J-
w—

4:

(a)

(b)

Figure 3-31. (a) Beginning and (b) ending (score, p. 12) of Study No. 17 (“Canon 12/15/20”).
Circled notes in (b) are notes altered to create the cadence.

F’

84

i
T

M
1

‘
l
l
fi

m I

t

Y
‘
A
H
L

]
3
.
1
7

h
t
-
L
H
H
V

(a)

(b)

Figure 3-32. Brief areas of harmonic convergence in Study No. 17: (a) entrance of section C (J =
230) in middle voice (p. 5); and (b) entrance of section B (J = 138) in bottom voice (p. 7).

Several Studies (Nos. 24, 33, and 43) use internal tempo switches to create addi-

tional CPs. The structures of Nos. 33 and 43 are shown in Figure 3-27; No. 24’s structure

is shown in Figure 3-36 (p. 150). Study No. 43 (“Canon 24/25”), with its single tempo

switch and two CPs, provides a simple structural framework for looking at the tempo

143

   
 
 
  
 
 
 
  
 
 
   
switch; its two CPs and the tempo switch are shown in Figure 3-33. The two voices of

Study No. 43 are not only very close in tempo ratio (24:25) but also in their interval of

imitation, a major third. Gann describes the proportions of the Study in relation to the

CPs as follows (p. 219):

Exposition

CP

CP

Dénouement

Convergence Periods*:

Beats:

*given in quarter notes

50
1 198

24
588

68.5
1643

Thus, the tempo switch occurs exactly halfway between the two CPs, after twelve

convergence periods at a point where the echo distance is twelve quarter-note beats (this

is shown by the arrow connecting the 2 bars in the middle example of Figure 3-33). The

timespan to the first CP is 1198 quarter-note beats, and the higher voice enters 1323 of the

way between beats 47 and 48 in the lower voice. At the end of the piece, the lower voice

drops out with 65.72 (65 1/82’s ) beats remaining in the higher voice.

As he did in Study No. 36, Nancarrow precedes the first CP with ascending move-

ment. The second CP, which is actually a rest, is followed by descending movement in

sixteenth notes (the most rhythmically active section of the piece). The CPs are used to

set off a middle section in the palindromic structure of the piece; Gann describes that

structure in this way:

This 24/25 canon is more nearly palindromic than No. 36; no extravagant
post-CP event appears to throw the symmetry off balance. Instead, the tex-
tures and motives of the first half of the piece return in almost exact reverse
order in the second half, though varied, some of them more elaborately de-
veloped, others shortened.

.

.

. No. 43 is palindromic in spirit, although there are no actual retro-

grade passages. (Gann 1995, 218)

The “palindromic spirit” and arch shape of the piece are also reinforced by

increases and then decreases in note density and dynamics before and after the CP

section. Gann’s generalization about the splintering of motives that become briefer as a

CP approaches somewhat holds true here, as some of the motives prior to the first CP are

144

Approach to first CP (p. 14):

l?"
,.

I

 LL..___/

8"}

Tempo switch (p. 18):

0212-0

Second CP (p. 21 ):

cp

 1- -J

L-

we

erg

Figure 3-33. Score sections from Study No. 43 showing approach to first CP, tempo switch, and
final CP.

145

 
 
subjected to rhythmic compression, and in the section leading up to the CP there is a

countdown extending from 12 quarter notes down to l.

The section between the two CPs consists entirely of a 3-note motive, expressing

the first three degrees of the minor scale in broken octaves. This motive appears earlier,

in measure 150 (all references to locations in the score will refer to the bottom voice)

where the pitches are Et-F-Gl. At the first CP (see Figure 3-33) both voices are stating the

motive in their treble registers, with the bottom voice stating C-D-Et while the top voice

states E-F#-G. At measure 199, the bass register enters in the bottom voice on F -G-Al,

and in the top voice on ABC, creating the hexachords F-G-At—C-D-FJ, and A-B-C-E-F#-

G which are suggestive of the Dorian mode. At the tempo switch, only the bottom tri-

chord is being stated in each voice, and shortly thereafter tonal centers begin shifting,

first every three measures to as often as every three beats. Five measures before the

second CP, the original hexachords return (see second CP in Figure 3-33).

Due to the frequently changing meters in Study No. 43 there are few shared bar-

lines, but there is one area about 40 measures before the first CP where “collective

effects” are clearly illustrated within a single convergence period.8 As shown in Figure 3-

34, a collective glissando begins in the bottom voice prior to the first shared barline,

followed by a pattern of a chord, broken octave, and lengthy rest that are alternated

between the voices. The second iteration of this pattern in the lower voice coincides with

the beginning of the next convergence period.

Study No. 33, with its irrational tempo ratio of V2z2, presents special challenges in

measuring and calculating distances and durations. The structure of the Study is shown in

Figure 3-27. There are four distinct canons (which Gann labels A through D in the dia-

gram, but refers to as Canons 1 through 4 in his narrative): Canon 1 is a diverging-

converging canon with a tempo switch in the middle; Canon 2 is an arch canon with a CP

8

.

.

.

.

.

.

.

.

The number of beats in each v0lce wrthln thls convergence perlod would seem to indicate a score error:
there are 26 beats in the top voice to the bottom voice’s 25, instead of the 25 and 24 that would be indicated
by the tempo ratio.

146

 
n1

gt!

“collective ” glissando begins here

    LL' _. _ _ _J

iteration 1

—_ “‘C—_——

__.

3

iteration 1

iteration 2

Figure 3-34. “Collective effects” in Study No. 43 (score, p. 11).

in the middle; Canon 3 is another diverging-converging canon; and Canon 4 is also a

diverging-converging canon, but it begins with a short interlude prior to the canon’s first

CP. All told, there are seven CPs. Only the first one takes place on a note attack in both

voices; most of the remainder occur either on held notes or rests. Even the final CP does

not take place on a coincident beat attack. See Table 3-1 for a description of the canons

and convergence points in Study No. 33.

Due to the irrational ratio, tempo switches do not occur at beat attacks. For instance,

Canon 1 is 46 measures (184 quarter-note beats) in length and the tempo switch is no-

tated as taking place near the beginning of measure 20 in the bottom (slower) voice and

measure 28 in the top (faster) voice. A duration of 46 measures divided according to the

ratio \/2:2 results in a split of approximately 26.95 and 19.05 measures (‘12x + 2x = 46;

x = 3.414); thus the tempo switch actually takes place just prior to the completion of

147

 
  
 
measure 27 in the top voice and just after the beginning of measure 20 in the bottom

voice.

Table 3-1

Description of Canons and Convergence Points in Study No. 33

Event

Description

Canon 1

diverging-converging canon, tempo switch in the middle

Canon 2

Canon 3

Canon 4

CP 1

articulated, coincides with initial attack of Canon 1 in both
voices

CP 2

unarticulated, is the final cutoff of Canon 1

arch canon, no tempo switch
in middle of Canon 3; unarticulated, takes place in middle of
measure on a held note—changing leader-follower
relationship is clearly audible following the rest

diverging-converging canon, tempo switch in middle
at beginning of Canon 3; unarticulated, begins with a rest
closes Canon 3 and elides with the beginning of Canon 4
where it is articulated in bottom voice only

diverging-converging canon, begins with a brief interlude
before CP 6; tempo switch between CPs 6 and 7
unarticulated, between two quarter note rests

unarticulated, occurs just after last note attack in lower voice

CP 3

CP 4
CP 5

CP 6

CP 7

Figure 3-35 shows three of the unarticulated CPs in Study No. 33. CP 4 marks the

beginning of Canon 3, and begins on a rest. CP 6 is technically the beginning of the

diverging-converging canon in Canon 4, but it also occurs at a rest. CP 7 is an enigma—

the final beat attacks in both voices are almost coincident, but not quite, as indicated by

the lack of a shared barline at the final measure. The effect, then, is that the top voice

finishes slightly ahead of the bottom voice to create a sort of V—I cadence in D1,. Un-

fortunately, it is impossible to calculate exactly where the final CP falls in the last

measure. Due to a shared barline, the exact location of CP 6 is discernible, but because

the exact measurement from CP 6 to the tempo switch between CP 6 and CP 7 is not

known or calculable, it is not possible to use the measurement of that half of Canon 4 to

148

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
determine the full canon’s length and project the exact ending point of the canon. If it

were possible to calculate the exact measurement of the first half of the canon to the

tempo switch, the exact location of CP 7 would be calculable. Thus its location can only

be approximated.

6 CP

lal.—

CP

(3)

(b)

(C)

Figure 3-35. Convergence points in Study No. 33 (“Canon ‘12/2”): (a) CP 4, (b) CP 6, and (c) CP
7 (pp. 12, 25, and 52, respectively, of the score).

Study No. 24 (“Canon 14/ 15/ 15”) has twelve canonic sections demarcated by thir-

teen CPs (see canon’s structure in Figure 3-36). Each of the twelve canons is of the

diverging-converging type, with a tempo switch between the outer voices (i.e., the fastest

and slowest voices) in the middle of each canon; the middle voice carries the middle

tempo throughout the piece. Canon 10, however, has a tempo overlap section in the

middle before the tempos switch. Also, in addition to the tempo switches in each of the

other eleven canons, an additional tempo switch is inserted at the beginning of Canon 9.

Thus, there are twelve tempo switches and one tempo overlap in this Study.

Table 3-2 lists and describes the canons and CPs that occur in Study No. 24. As the

table shows, the majority of the CPs elide the end of one canon with the beginning of

149

 
 
 
 
 
 
 
 
 
 
 
  
         
  
  
 
      
Study No. 24 - Canon 14/15/16

E] slow. pp E] fast. 1‘f
(circled sections begin with an attack in all three voices on the CP)
59°“°"@
Treble 16

® ©
:3

5. 3’

2

6(2) 69*
3

W
A

Middle 15

Bass 14

Convergence 27 (18 x 1.5)
Periods

3
" , V

’;:‘~. ..‘.;’.;
"4.11 '»-“E'-..
if. 1
{“47 3'
1.1.1»
‘2‘
12

A

18

10

* An additional tempo switch occurs
at the beginning of Canon 9

i e,
:z
.7." '1'
3.1
-
6

9

G)

12

Figure 3-36. Structure of Study No. 24 (“Canon 14/15/ 16”; modified from Gann 1995, 23).

tempo overlap
about the 15-voice

another; the exceptions are CPs 4, 7, and 10. The table also shows that CPs 1, 4, 5, 7, 8,

11, and 13 are articulated in all voices while the others are unarticulated (i.e., take place

on rests or held notes).

CPs and tempo switches sometimes occur with startling rapidity in this Study, par-

ticularly in the area of Canons 6—9 where (as shown in Figure 3-36) CPs occur every 3, 4,

l, and 2 convergence periods, respectively. Figure 3-37 shows Canon 8 (the briefest

canon, consisting of only one convergence period of 15 sixteenth notes) and the begin-

ning of Canon 9. As mentioned earlier, an additional tempo switch is inserted at the

beginning of Canon 9. Normally the leader-follower relationship between the top and

bottom voices reverses at each CP; at Canon 9, however, the additional tempo switch

allows the bottom voice to remain the leader for both canons.

At Canon 10 (by far the longest canon, comprising 74 convergence periods), Nan-

carrow inserts a tempo overlap on the middle tempo (J. = 240) that lasts for 60 sixteenth

notes (four convergence periods). The tempo overlap allows for a rare rhythmic converg-

ence to be created in all three parts by repeating the same rhythmic pattern three times, as

150

      
 
 
   
 
 
 
 
  
 
   
 
 
 
 
 
Table 3-2

Description of Canons and Convergence Points in Study No. 24

Event

Canons l—12

CP 1

CP 2

CP 3

CP 4

CP 5

CP 6

CP 7

CP 8

CP 9

CP 10

CP 11

CP 12

Description

every canon is a diverging-converging canon; Canon 10 is the only
canon which has a tempo overlap rather than a tempo switch in the
middle

articulated in all voices at beginning of Canon 1

unarticulated (occurs at dotted quarter rest), elides with beginning
of Canon 2

unarticulated (occurs at half rest), elides with beginning of Canon 3

articulated in all voices, Canon 3 extends one 3 measure beyond
this CP before Canon 4 begins

9

articulated in all voices, elides with beginning of Canon 5

unarticulated (occurs at eighth rest), elides with beginning of Canon
6

articulated in all voices, occurs one measure beyond beginning of
Canon 7

articulated in all voices, elides with beginning of Canon 8

unarticulated (occurs at quarter rest), elides with beginning of
Canon 9 (there is also a tempo switch at this CP)

unarticulated (occurs near end of long held note in all voices),
precedes beginning of Canon 10 by five sixteenth notes

articulated in all voices, elides with beginning of Canon 11

unarticulated (occurs at seven eighth rests), elides with beginning
of Canon 12

CP 13

articulated on last attack of Canon 12

shown in Figure 3-38. After this rhythmic convergence, the top and bottom voices switch

to the tempo opposite the one they had before the convergence.

Study No. 37, whose structure is shown in Figure 3-39, is another canon with mul-

tiple CPS. Its twelve voices state the tempo ratio of the justly-tuned scale (see again

Figure 2-la, p. 66), which seems imposing but allows Nancarrow the ability to create

widely-ranging textures and to combine the tempos in different groupings (comparable to

151

 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
CP at beginning
01 section 8

t

- h
empiswnc

“silent” CP at
beginning of section 9

H 35:6

..

/

1.: 33'“

"
_ fig

  3.:3'40) \

silo: 3.3.:

__..

..

F"

_ _

v
1’ng _.

_:

_

Figure 3-37. Study No. 24 (portions of second and third systems of p. 12), showing CPs 8 and 9.

section 9 has tempo
switch at beginning
as well as in middle

J. = 256W 2nd §tat§mgnt

J. = 240
§rd statement

J = 224

J_ = 240

J. = 224

1st statement

2nd statement

3rd statement

1st stgtement

2nd statement

3rd statement

. = 256

Figure 3-38. Structure of rhythmic convergence area in middle of Canon 10, Study No. 24.

chordal structures) that are no longer rigidly bound by his practice of assigning the slow-

est tempo to the lowest voice, etc. Gann identifies twelve canons and five CPs in this

Study, and although Figure 3-39 does not very accurately portray this, different tempos

are consistently applied to different voices. In Canon 1, the tempos are in order from

fastest in the top voice to slowest in the bottom voice and expressing a rhythmic analogue

of just intonation (see Figure 2-8, p. 79); in Canon 2, the order of tempos is reversed. In

later sections of the Study, Nancarrow creates “chordal” groupings of tempos, such as in

Canon 6 where the tempo order from bottom to top is 150 187 )3 240 1605/7 200 250

152

   
   
     
 
   
  
    
 
  
  
 
 
 
  
 
  
 
168 3/4 210 262 1/2 180 225 281 1/4 (see Figure 340, left edge of score); as shown by

the bold and italic type, the tempos are arranged into three interlocking “diminished

seventh chord” groupings. Gann has plotted the tempo relationships of all of No. 37’s

canons on a musical staff by converting each tempo to its pitch analogue, and compared

this to the transposition levels of the voices (Gann 1995, 196), allowing one to see the

considerable variety Nancarrow brings to the canonic process.

Study No.37 - Canon 150116051711663/4113011871122001210/225/2401250/2621121281114

Canon: 1

)
Q

5

1
0

3
0

N

:
a

   CP

nme: 0:00

op
CP
0:23 0:46 1:07

0
(

_

s o 

6:19

7:02

7:36

8:10

10:22

Figure 3-39. Structural diagram of Study No. 37 (modified from Gann 1995, 26), showing the
twelve canons and five convergence points.

Convergences between voices are clearly important here as in many places Nancar-

row has marked with vertical broken lines where entrances of new voices coincide with

beats in other voices (something he could have marked in many other scores, but did not),

even when the voices are widely separated in the score and the convergence is with an

interior beat rather than a downbeat. These convergences between voices are more

common than might be supposed from the unwieldy tempo ratios. In several of the later

canon sections (e.g., canons 8, 9, and 10) where tempos are in “minor third” groupings,

simpler tempo ratios such as 6:5 and 7:6 are common between voices. These middle and

153

 
 
 
 
    
later canonic sections, then, are more temporally consonant, which is a virtual necessity

for perceptual purposes because the melodic material here is much more active than in

the earlier sections.

Gann says about the convergence points in No. 37, “Convergence points are bril~

liantly de-emphasized in this study, for this is the work in which Nancarrow learned how

to create beautiful effects with convergence points by omitting them” (p. 195). The CPs

are not really omitted, of course, but unarticulated, as not one of the five CPS occurs at a

note attack. CPs 1 and 2 occur on a rest, with the voices entering in a staggered fashion

just after the CP. CP3 also takes place at a rest, just after the conclusion of canon 3. The

most climactic CP is unquestionably CP4 at the beginning of canon 7 (Figures 3-39 and

3-40). This CP marks the most rhythmically active section of the piece and comes at a

place where the texture thickens to all twelve voices after an extended section in canons

4, 5, and 6 where the voice saturation is considerably thinner. In comparison, the con-

cluding CPS is notably anti-climactic, occurring at a whole note in all twelve voices that

is held over from twelve tied half notes and marked with a fermata.

The processes of acceleration and deceleration create special challenges for plan-

ning and placement of convergence points, and for this we look at Studies No. 8 and 22.

Recall that Nancarrow used two varieties of acceleration: arithmetical and geometric (see

pp. 10—11), where arithmetical tempo changes involve adding or subtracting the same

invariable rhythmic unit to the previous value, and geometric changes involve changing

note values by an invariable percentage. Arithmetical acceleration creates a constantly

increasing or decreasing rate of change, but can be notated in conventional notation (see,

for example, the “countdown” at the end of Study No. 12, Figure 1-7, p. 33); geometric

acceleration creates a smooth continuum of changing speed that must be notated in

proportional notation.

Although it predates the first tempo canons, Study No. 8, an early example of

Nancarrow’s acceleration technique, makes considerable use of convergence points. Its

154

sz’u._ __ _ __ __ _
(r —————   

gnaw":

'

———-—-

Jal?}\/t

fl 1.“."07’

1315‘0

H lew

\
\

19‘

Figure 3-40. CP4 from Study No. 37 (p. 40 of score). End of canon 6 is shown, where tempos are

arranged in groupings analogous to interlocking diminished seventh chords. Canon 7 begins at

CP4, which is a sixty-fourth rest at beginning of 4 measure; a tempo switch in some voices also

4

takes place at this CP.

155

   
first section, called by Carlsen the “trio” because it is in three voices, consists of canonic

lines that alternately accelerate and decelerate within a series of nineteen durations, each

of which consists of a pair of notes in a 2:1 rhythmic relationship (an eighth note fol-

lowed by a quarter note with a sustain line—see Figure 1-3, p. 31); the 2:1 rhythmic

pattern creates a loping effect that particularly elucidates the processes of acceleration

and deceleration. Although the notation is proportional, the process of tempo change is

arithmetical as an invariable background unit is continually added or subtracted to

determine the next note values (see again pp. 48—51; note in Figure 1-22, p. 48, the two

possibilities posed by Carlsen and Gann for metric notation in this section).

Nancarrow imposes strict order on the form of this section by bringing in each new

voice at a convergence point as shown in Figure 3-41. As Carlsen notes, the main pitch in

this section is G, and all CPs take place on G octaves or triadic intervals involving G. No

two voices are ever stating the same rate of change at the same time, even if they are

using the same tempo process. For instance, in the fourth phrase of this section, where

both the first and second voices are in accelerating-decelerating patterns, the patterns

begin and end at different places in the nineteen-duration series and also change speed

direction at different places. One of the interesting effects of this procedure is that the

echo distance between voices is constantly different, and not steadily increasing or de-

creasing as would normally be the case.

Vorce
3

2

1

G
G
G

G
G
G

G
G E
(.3 G

G
B»

decal. - arxal.

decal. - accal. -

decal. - - - accel. -

-

- -

decal. - - aocal. -

-

accel.

accel. - decal.

accal. - decal.

decal. - .

- accal. - - - -

decal. > - - eccal. - - -

accel.

decal.

accel.

accel. - decal.

decal. - - accel. -

- decal. -

- accel. -

-

-

G
8'1.

+

V—I cadence in
voices 1 and 2

E
G

G
B,

Figure 341. First section of Study No. 8, showing convergence points (vertical connecting lines),
whether durational series are accelerating or decelerating, and pitches at CPs. (Diagram is not
strictly proportional.)

156

  
     
 
   
 
 
 
 
 
Despite this variety in rate of change within the voices, Nancarrow is able to ensure

convergences at regular time intervals by keeping the total number of background units

the same in each voice within each phrase. There are also occasional near-convergences

that pop out of the texture due to the use of the same limited number of durations within

each voice; also, the short note in each durational pair casts such nearly-aligned attacks,

when they do occur, in stark relief.

In contrast, Study No. 22 (“Canon l%/1l/2%/2l/4 %"9), with its geometric accelera-

tion, has only three widely-spaced convergence points (see Figure 3-27, p. 140): the first

concludes Canon A, a converging canon; the second is in the middle of canon B, an arch

canon; and the third begins Canon A’, a diverging canon. Study No. 22 is a strict palin-

drome in both rhythm and pitch; CP2, which takes place at a rest, is thus the rhythmic

line of symmetry in the piece.

In Canon A, the first note of each voice is the same length (75 millimeters in the

score’s proportional notation), but by the CP the different rates of acceleration (with the

2’3% line accelerating 125% as fast as the 1% line) have resulted in very different

lengths for the last note before the CP: 7.5 mm. in the fastest voice, or a duration 10% as

long as at the beginning; 17 mm. in the middle voice, or about 23% as long as at the

beginning; and 30 mm. in the slowest voice, or about 40% as long as at the beginning.

Thus, at the CP the fastest voice has sped up to four times the speed of the slowest voice.

As a result, as the CP approaches one hears the exponential change in the echo distance

between the voices.

Analysis of Study No. 27: “Canon 5%/6%/8%/11%”

This chapter concludes with an analysis of Study No. 27, one of two “acceleration

canons” written by Nancarrow (the other being No. 22). Like Study No. 22, Study No. 27

has three widely—spaced convergence points. However, whereas No. 22 consists of three

’Each contiguous pair of percentages relates to the same ratio—3:2.

157

 
of the basic canon types (converging, converging-diverging, and diverging), Thomas

notes that No. 27 is unlike any of the four basic canon types and instead “incorporate[s]

the technique of gradually changing speeds, so that the relationship between the voices is

constantly changing” (p. 13).

General Qbservations

The canonic material is primarily in four voices (sometimes doubled) representing

the four acceleration/deceleration percentages of the subtitle. Due to the geometric accel-

eration technique, the Study is notated entirely in proportional notation. In the middle of

the canonic texture is a line of non-varying tempo that is commonly referred to as the

“clock line,” representing what Nancarrow referred to as the ticking of an “ontological

clock” (Reynolds 1984, 9), or a time—keeping device against which one can detect the

constantly changing speeds of the other voices. According to Jan Jarvlepp (1983/84,

219), Nancarrow considered the clock line to be an ostinato even though it seems to have

no recurring pattern of pitch or rhythm. Thomas’s description of the clock line (refer back

to Figure 1-15, p. 40) emphasizes its function as an ostinato:

.

It is presented in staccato articulations at intervals equivalent to either quarter
or eighth notes. .
. The basic rhythmic and pitch material of this layer is
limited, to be sure, but it is subjected to perpetual, random reorderings. As a
result, it has no recurring patterns nor metrical implications. Nonetheless, the
layer functions contextually as an ostinato because of its constant presence
and its relatively steady character: although its articulations are not regular, its
tempo is unchanging, while all the other layers in the study are continually
subjected to accelerandi and ritardandi. (Thomas 1996, 14)

Figure 342 shows Gann’s formal diagram of Study No. 27. It identifies the relative

positions of the clock line and the four canonic voices, with the percentage of accel-

eration or deceleration indicated for each voice. It also identifies Gann’s eight sections;

these eight sections correspond to the changing textures of the clock line as shown in

Figure 1-15 (p. 40). Section 1 contains four different canons (identified separately by

Gann because the levels of imitation do not remain the same), and I have modified Figure

158

Study No. 27 - Canon 5%/6°/o/8°/o/ 11%
Section 1:

Canon 1

Canon ép

Canon 3

Canon 4

Clock.

[ER—
E %A

|S%R |5°/.A

l

[6%A [67.9]
m-

|5°/.R

[11%/x lime ] [11%AJ11‘ARJ

Section 2:

Section 3:

Section 4:

Canon 5

Canon 6

OF

‘/.A

Canon 7

_11%R‘
-%n ‘

m :- -

m -1

[8%A

[8%R b9“ page ]
-E-

-ma1m-

Section 5:

Section 6:

Section 7:

Section 8:

CP

%A

8°/.R

Ease

|

596A

Canon 9
Canon 8
Durations not strictly proportional

Canon 10

Canon 11

Figure 3-42. Formal diagram of Study No. 27 (modified from Gann 1995, 161). Percentages of
acceleration (A) or deceleration (R, for ritardando) are given for each voice. Gann's diagram has
been modified to show convergence points and the locations of the eleven canons; comiderable
adjustments have been made to voice entrances and endings to better reflect relative relationship
of voices.

3-42 to identify these four canons; each of the other sections has one canon, for a total of

eleven. CPS (at the end of Canons 2, 5, and 11) are also identified, and I have modified

the diagram to more accurately reflect the relative relationships between canons and

 
 
 
   
  
  
 
 
 
where voices enter and drop out. AS Shown in the diagram, within each canon each

percentage is uniquely affiliated with a specific voice that does not change within that

canon; in some canons both acceleration and deceleration are used, and in some only one

process is used.

Overall Structure

The importance of convergence points in this Study seems a matter for some dis-

agreement. Gann asserts (p. 163) that “convergence point is not an important issue in this

study.” Thomas, on the other hand, points to the first CP as a place where “the process of

convergence is very plainly the point of this particular canon” (p. 135), and, indeed, it

seems that with the kind of mathematical manipulations that were required to create this

and the other CPS it could hardly be otherwise. If convergence is not the main point of

this Study, it certainly is at least a significant one. The first two CPS involve the canonic

voices but not the clock line, while the CP that concludes the piece involves all the

voices.

Thomas characterizes this Study as “decidedly non-tonal” (p. 17), and, although

like many of Nancarrow’s pieces it contains numerous tonal elements such aS scales and

triadic structures, there is little evidence of functional tonality or long-range tonal motion.

These tonal elements tend to be used together in a manner suggestive of what Tenney

refers to as “aggregates”——for instance, the seventh chords in the clock line in Canon 9

(see Figure l-15, p. 40).

Carlsen points to this Study (along with Nos. 19 and 36) as one in which “It is

Significant that the exact midpoint on Nancarrow’s instruments is the note E4. .

.

. This

has a direct bearing on the symmetrical Structures Nancarrow sets up using every note on

the instrument” (p. 5). The clock line itself, consisting of the pitches D#,,, 134, F4, and GL4,

is almost evenly deployed about the piano’s central note, and thus it not only contin—

uously marks time in the piece, but also constantly articulates the vertical axis about the

160

center of the player piano keyboard. Symmetries also exist in the levels of imitation

within some canons, as was Shown in Table 1-2 (p. 45). Figure 3-43 shows the levels of

imitation in each of the canons; these intervals form vertically symmetrical structures in

Canons 1, 2, 3, 4, 6, 8, 9, and 11, while Canons 5, 7, and 10 are not quite symmetrical.

Q

Canon 1

Canon 2

Cl

3

Canon 4

-

l’

Canon 5

_

8w + M6

live + P5

811‘ + mh

8w + m6

live + M6

clock line

I

I

f 2 8ves + M2

live + m6 

,’ 2 8ves + M2

8vc + m7 ‘.

live + m6

the + P5

’

Canon 6

8va .

ii Canon 7

Canon 8

Canon 10

Canon 111,,”

8vc + M3

m2 - m2 - mZ - m2

2 8ves + M7

the + M3

|‘

(in 5 ocruvcsj’lp

8

Figure 3-43. The levels of imitation between adjacent canonic voices and order of entrances of
the eleven canons of Study No. 27.

CP

The first three lines of the ubiquitous clock line are shown in Figure 3-44. The

161

 
 
clock line consists of two rhythmic values related 2:1, with the longer value articulating

the Stated 220 tempo. Thomas points out that the clock line never approaches temporal

consonance with the other voices (p. 134), and since the rhythmic values in the canonic

voices are always changing there are only random, non-consecutive, cases where these

values match the clock line’s.

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Figure 3-44. The first three lines of the clock line in Study No. 27, showing limited pitch content
and proportional notation of the tempo.

All references to the clock line in the main analytical sources comment on its appar-

ent total randomness of both pitch and rhythm throughout the piece, and any rhythmic or

melodic patterns or non-musical associations that Nancarrow may have incorporated into

this line are so well hidden as to be indecipherable. I even entertained the possibility that

Nancarrow might have created the rhythm by using a system such as Morse code to

encode a text in this line; however, in Morse code the duration of a dash relates to a dot

by the ratio 3:1 and the system includes breaks equal to a dot between elements of a

character, the length of a dash between characters, and the length of seven dots between

words.lo Since the only relative durations in Nancarrow’s clock lines are lengths of 2 and

1, this system could not have been used. Short repeated pitches appear fairly frequently in

the clock line, with the first set of three repeated notes articulating E4, the piano’s central

"’See www.soton.ac.uk/~scp93ch/morse/morse.html.

162

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
note. The longest series of Short notes in the line is nine, and of long notes it is five.

Despite the overall appearance of randomness in both pitch and rhythm in this ever-

present line, it still seems likely that some type of patterning must have been involved in

its creation. For one thing, as Pickover (1995) has noted, it is quite difficult for humans to

generate truly random sequences:

Sit down at a computer keyboard .
. and randomly press the 1 and 0 keys.
Try to make the String of numbers as “pattemless” as possible. In other words,
try to generate random numbers. [Pickover then gives a 6—line example of 1 ’s
and 0’s that he generated]

.

.

.

. I have tested dozens of supposedly random sequences typed by col-
leagues, and found out that it is curiously difficult for humans to type pattern-
less sequences. To start with, we would expect about a 50 percent occurrence
of each digit, and I did amazingly well. The [example I typed] had 49.3
percent 1’s and 50.7 percent 0’s. Next, we would expect 25 percent occur-
rences of the following pairs: 00, 11, 01, 10. In fact, doing my best to make a
random sequence, I produced 15 percent, 13 percent, 36 percent, and 36 per-
cent occurrences, respectively. Apparently my fingers preferred to oscillate
rather than producing doublets such as 00 and 11. Perhaps I was trying to
avoid clumpiness of digits, when in fact Strings of identical digits should exist
in a truly random sequence. (Pickover 1995, 233)

In creating the clock line, then, Nancarrow would not only have been faced with this dif-

ficulty in creating random rhythmic values, but randomness among the four pitches. Per-

haps he created a 4 x 2 matrix (four pitches by two rhythmic values) and threw darts at it!

The percentages in the Study’s subtitle are used to create both accelerating and de-

celerating series of rhythmic values. Table 3-3 shows the multiplication factors needed to

derive successive values of the series. The factors can be applied over a series of acceler-

ating or decelerating values by using successive multiplication; for example, an accel-

erating series of twenty values (i.e., a series of nineteen value changes) at 6% that begins

with a rhythmic value of 100 millimeters will end with a value of 100 0 0.9434”, or 33.1

millimeters, while a decelerating series of twenty values at 6% that begins with a rhyth-

mic value of 20 millimeters will end with a value of 20 0 1.06”, or 60.5 millimeters.

163

Multiplication Factors for Acceleration and Deceleration in Study No. 27

Table 3-3

Percentage

Agcelergtign

Dgelggatjgn

Multiplication Factors

5%

6%

8%

l 1%

0.9524

0.9434

0.9259

0.9009

1.05

1.06

1.08

1.1 1

Sections of the Study

Qangns 1 and 2: Study No. 27’s first CP concludes the section consisting of these

two canons. Canon 1 is one of only a few canons in the piece in which more than one

basic rhythmic value is used, and in this way it forms a rhythmic parallel to the clock

line. Canon 2 forms a pitch parallel to the clock line in that each of the four canonic lines,

like the clock line, consists of four contiguous chromatic pitches. Both Canons 1 and 2

(as well as 3 and 4—5ee Figure 3-43) have symmetrical patterns of levels of imitation

between the voices. In Canon 1, the first two voices (A6%/R6% and A1 1%/R1 1%) finish

stating the canon before the other two voices enter; in Canon 2, the four voices enter in

succession so that the canon iS eventually being stated in four voices at once.

Thomas asserts that “the canonic and ostinato layers seem completely unrelated and

unrelatable” (p. 133); however, in the first canon rhythmic values are sometimes halved

and doubled to create a 4:2:1 rhythmic duration relationship, and this partially parallels

the 2:1 relationship of the values in the clock line. Because both accelerative and deceler-

ative processes are used in this canon, Jarvlepp (1983-84) astutely points out (p. 220) the

simultaneous existence of three different kinds of tempo layers: steady tempo, accel-

erating, and decelerating.

The pitch content and relative rhythm of the lowest voice of Canon 1 are Shown in

164

 
 
 
 
 
 
 
 
 
  
Figure 3—45. The first six notes—with the first three in the equivalent of quarter notes and

the next three in eighth notes—along with the first rest, serve to set up a pattern that

divides the line primarily into trichords. This trichordal division is maintained throughout

much of the canon through rests and changing note values. The most prevalent pitch-class

set represented is 3-4 [015], while sets 3-2 ([013], Carlsen’s “partitioned minor third”), 3-

3 [014], and 3-9 [027] are presented three times each; melodic movement by half steps

and P4/P5 prevails. The canon’s concluding pentachord, 5-218, is a subset of both the

opening hexachord 6—210 and the hexachord that precedes it, 6-Zl 1.

A6% (29 notes)
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R6% (35 notes)
[0H] “1015110271
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Figure 3-45. Canon 1 of Study No. 27 in the lowest (A6%/R6%) voice, in rhythmic notation
Showing relative rhythmic values.

Figure 3-46 Shows a section from Canon 1 where the canon is in two voices. This

passage is a rare instance in Study No. 27 where the voices are close enough together in

Stating the canon that the echo distance is audible. There are two leader-follower Switches

on this page, and these are made more clearly audible because of the rests that accom-

pany the switches. The passage is especially interesting because of the symmetry of the

intervals between the voices at these points.

Canon 2’s pitch content is, like the clock line, limited to four contiguous chromatic

pitches, as Shown in Figure 3-47. The canon concludes with a CP on the attack of the

165

 
"“

A
V

I

.
L
H
N

l

.

H
N

i

A

1

H

~
I

I

L

4

;
s
t
i

.m“

CP/leader-follower switch

longest echo distance

in

(near-synchrony)

CP/leader-follower switch

   —

Figure 3-46. Section from Canon 1 of Study No. 27 (p. 3), showing two inaudible CPs and their
leader-follower switches, pitch symmetry about these switches, and near-synchrony at second
switch. (Rests in bottom staff of middle and bottom systems have been added after notes FI and
E. as the omission of these rests were errors in the score.) The longest echo distance is reached
halfway between the two leader-follower switches; selected echo distances are shown by circled
notes connected with a line.

final pitch, which is the same as the first pitch of the canon’s pitch cluster (a minor 7th

chord on E); these pitches, as shown in the figure, create a symmetrical pitch structure

about the D# in the clock line.

Thomas finds the CP at the end of this canon to be remarkable in its swift move-

ment from extreme temporal dissonance to consonance:

166

        
    
   
   
 
What is so remarkable about this canon is its single-minded pursuit of con-
vergence, and its rapid attainment of it. Within the span of approximately
eighteen seconds the four related but temporally clashing canonic voices
resolve their dissonance through their convergence upon a simultaneity, which
represents consonance. Because the voices not only start at different speeds
but also proceed at different rates of change, the convergence is dramatically
faster than in a tempo proportion canon: the pace of the process itself is mag-
nified through the use of gradually changing speeds. (Thomas 1996, 135—36)

(levels of imitation shown between stoves)
*A-
**-'“-“——~—‘-m

-- -~ ‘- ' ‘— " -. '_“ *

'-

r
:&____ -i_ -2 .._..- __-...“ -.--. n... 1-_. --:._ -- _-+-...-.----- . f-.- . 2-.--..“ _l “__. - __.--.____ -
- .-
,_ -. __ -.__

--_. ‘fh1‘j' ‘r ——“# -__._-_.---,-._.--I. E H-~—~'—‘I
__.. -_. _____..--.3
_,._- __-._- ___, j

-- .--__-. .-__________-_. __~ -

. ,_ ._. __.r-_2--

_ .-. .._...- __.. _ __---

-._ _. ._.A_..-____.

--

_. -_

-._

-_

__

'

.-

__

-

8.89; In. he

COP

8ve + P5

Figure 3-47. Pitch content of Canon 2, Study No. 27. Chord at CP is symmetrical about the D# in
the clock line.

As Thomas notes, the four voices in the canon begin at “different speeds” (i.e., with

different durations); however, the length of the final duration (the CP) is the same in all

voices. Table 3-4 reports the actual and calculated measurements in millimeters of the

twenty-three decelerating pitches in Canon 2; the final note in the series represents the

CP, which occurs at the attack of the final note in the canon. Within a certain amount of

tolerance, there is generally good agreement between the actual and calculated measure-

ments. The canon’s final pitch, the CP, is 108 millimeters long in all four voices, and this

value happens to relate to only one of the voices: it is roughly double the value in the

series at that point in the 5% voice.

It is interesting to speculate how the CP at the end of Canon 2 might have repre-

sented a greater achievement in convergence had the opening values in each voice been

167

 
 
 
 
    
Table 3-4

Calculated and Actual Duration Measurements in Canon 2 of Study No. 27

Rit. 5%

Rit. 6%

Rit. 8%

Rit. 11%

ale. AM @; A3121!
18.7
18.5
19.6
18.5
20.6
19.5
21.6
20.5
22.7
22
23.8
23
25.0
(25)
26.3
25.5
27.6
26.5
29.0
28
30.4
30
31.9
31
33.5
623)
35.2
35
37.0
36
38.5
38.8
40.7 @ 33.6
35.6
41.5
42.8
37.8
45
44.9
40.1
45.5
47.2
42.5
51
49.5
*45
52
*52
—
108
(54.6)

Mimi
alc. m CA9.
3
4.1
7.5
8.3
12
13.2
4
4.6
8
9.0
13
14.0
4
5.1
8.5
9.7
14
14.9
4.5
5.7
10
10.5
15
15.8
5
6.3
@913
11.3
16
16.7
6
7.0
11
12.3
17
17.7
7
7.7
12
13.2
18
18.8
7
8.6
14
14.3
19
19.9
9.5
9.5
15
21.1
15.4
19.5
8.5
10.6
15.5
22.4 ® 16.7
10.5
11.7
16.5
18.0
22.5
23.7
(25'
13.0
19.5
19.4
25
25.1
13—
14.5
19
21.0
25.5
26.6
14.5
16.1
22
22.7
27
28.2
16.5
17.8
Q49
29.9
24.5
29.5
18.5
19.8
25.5
31.7 @ 26.5
21
22.0
27.5
28.6
33
24
24.4
30.5
30.9
35
26.5
27.1
33
33.3
37
30.0 @
36.0
40
35.5
34.5
38.9 @ 33.3
42.5
37
*37
42
*42
45
108
—
108
—
108

Calculation of opening values required in all voices to result in final value of 108:
——
— |
18.7
|
108
108
[ —

— 1
108

— |
108

10.0
| —

15.2
| —

5.5
| —

|
]

Score measurements are to the nearest half millimeter.

Circled measurements are taken over a system break and more subject to inaccuracy.
*These are actual measurements from which the increasingly smaller values are calcu-
lated (5% values multiplied by 0.9524, 6% values by 0.9434, 8% values by 0.9259, and
11% values by 0.9009). The largest measurement was used to calculate smaller values
because it would be less subject to inaccuracy.

168

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
such that 108 was double the value in the series at the CP. The bottom part of Table 3-4

shows the calculated values for opening durations that would have resulted in this “dura-

tional convergence” on the value 108 at the CP. The resulting values in the 6%, 8%, and

11% voices are closest to the values that occur in the third note of each series (shaded

area in table). It would have been a simple matter for Nancarrow to begin the 6%, 8%,

and 11% voices with these values in order to facilitate this durational convergence, and it

is unclear why he did not do so.

Canons 3 and 4. These two canons conclude the first section, in which the clock line

is still iterating single pitches (see Figure 1-15, p. 40). In Canon 3, the voices above the

clock line (8% and 5%) first decelerate and then accelerate, while the voices below the

clock line (6% and 11%) do the opposite. In Canon 4, each voice first accelerates and

then decelerates, with the fastest-changing voice (11%) at the bottom and the slowest

(5%) at the top.

Figure 348 shows Canon 3’s 23-note canon as it appears in the first voice

(A6%IR6%). As Gann notes (p. 160), the canon has a narrow pitch focus of a major sixth.

The canon’s opening contour is similar to that of Canon 1, and there is a fairly strong

implication of first E major and then B minor in the last half of the canon.

halfway point
I

-._.- F _ _ __ descending dim.triad _-

j
[041;j ‘g:?§--:{.--_11.12;.-1.;1L _.at: 4:19;,-

--..___-__ ..

*4 .

191

r:1-~1.:12.77.324-51?

l--- ImpIILsE major --------------------------- l

1 implies 8 minor -—1

ascending dim. triad

l

Figure 3-48. Canonic line of Canon 3, Study No. 27.

The halfway point of the canon, the twelfth note, is where the A6%/R6% voice is

notated to switch from acceleration to deceleration, but in the score the R5%IA5% and

169

 
 
 
  
   
 
All%/Rll% voices are marked to switch on the note prior to this, and the R8%/A8%

voice on the note following the halfway point. Besides the fact that this seems a violation

of the spirit of the canonic process, there are other reasons to believe that all the voices

should be marked in the score to switch at this halfway point. Based on measurements in

the score, it appears that the tempo change process occurs as follows: ten duration

changes take place in one direction (either accelerating or decelerating), with the duration

of the eleventh note used again for the twelfth note, followed by ten duration changes in

the opposite direction, with the final (23rd) note a longer duration which is apparently un-

related to the prevailing duration series. If the switch takes place on the middle note and

the same number of duration changes is taking place on both sides of this point, the

opening value and the value preceding the long concluding note should be roughly the

same. Indeed, this does seem to be the case with all four voices, leading to the reasonable

conclusion that the switch in each voice should be marked at the twelfth note of the

canon.

Canon 4 unites an organic principle of organization in both pitch and rhythmic

duration, organizing both around five-note groupings separated by barlines. The melody’s

five-note pattern is a sort of major/minor upper trill, including a total of 24 groupings and

16 iterations of the trill figure as shown in Figure 3-49; groupings of one to four of these

“trills” are followed by a single short note. Fourths/fifths and triadic structures appear to

..

.._

~M-.--..T____-‘ _.

~4—I4 lb-oI—H—1—
21,22.2».22221’—

F?211411221..112- 122,2a....22J’“{.,332;,13;,2H::2 

N111111!

.1'11m1'pi111112.1 111 1I111 L Iim

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1
‘17-: 211-
--1-
2 2.; 2:,
1?}1271131122‘“JfLLJT'vLWb.

.......

2 2—-

ij"

!

Figure 349. Canon 4 of Study No. 27, from lowest voice (A1 l%/Rl 1%).

170

 
  
 
 
 
   
  
dominate the melodic movement between the first notes of each “measure”; the trill in the

fourth and fifth measures spells out the clock line’s tetrachord, and several other group-

ings include the chromatic tetrachord. The deceleration portion of the canon is two

groupings longer than the acceleration portion.

Like the pitch changes, the rhythmic durations of this canon change with each five-

note grouping. Figure 3-50 shows a representative portion of the score and duration

measurements from this section.

f coca - f?‘

,

_____

__ _. -

Ibfi

:E—vi: if-

3‘.

;

fi1#*-IP'-'A

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40.5

36.7

3&5

55.0

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i
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27.0

r U r

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l LfifjJe—W
1
292l

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31,5

3‘40!

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(R130)

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— ’435—

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r
if

48.3

J1
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L: #2:Mi:-

l

3'

,1

l
55.3

Figure 3—50. Segment of Canon 4, Study No. 27 (p. 16, second system), at entrance of 5% voice.
Measurements from the score are given in millimeters for each five-note grouping (the example is
reduced from the score).

gallon 5. Canon 5 introduces two related texture changes: the canon includes many

half-step trills, and the ostinato line is reinforced by octaves and registrally divided with

the bottom half—step pair (D#IE) in the bass register and the top pair (F/GL) in the treble.

Like Canon 4, this canon’s melody is arranged with variable groupings of trills followed

by short notes.

Figure 3-51 shows a passage from Canon 5 where there is a near temporal

171

 
 
  
  
 
   
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
convergence among the 11% and 8% voices and the ostinato. The switch to “rit.” is

almost simultaneous in these voices (although in the 11% voice the switch is marked one

measure earlier than in any of the other voices). At the ritardando, the 11% voice’s tempo

approximates 880 and the 8% voice’s 440, while the ostinato is maintaining its steady

220 tempo, for an overall tempo ratio of nearly 4:2: 1.

— 1—3"—_*etc-igizn-

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Figure 3-51. Passage of near temporal convergence in Canon 5 of Study No. 27 (p. 22, second
system). At R11% (top line), tempo is ca. 880 while at R8% (fifth line), tempo is ca. 440.

Canon 5 concludes with the Study’s second CP, with the convergence taking place

at the release of the final note. The convergence at the note release is made the more

striking by the sudden change in dynamic from f to p and the change back to a single

note texture in the clock line immediately following the CP. The final values in each

voice at the CP are not equivalent and thus not a “durational convergence,” but in each

voice the durations of the notes at the CP appear to be close to the next value in the decel-

eration series taking place at that point (although in some places, particularly in the 11%

172

 
 
 
  
 
 
 
 
 
  
 
 
 
 
 
  
 
 
 
 
 
voice, Nancarrow’s score measurements do not match the series very well).

Canon 6. At Canon 6, the ostinato line returns to its original single-note texture, and

the dynamic level drops from f to p. As shown in Figure 342, each voice first decel-

erates in the bass register below the clock line, and then, in tempo, leaps five octaves plus

a minor sixth to the treble register above the clock line for its accelerative portion. This

idea of registral separation is an extension of the texture used for the clock line in Canon

5.

Canon 6’s melodic line is shown in Figure 3-52. Thanks to the registral separation

within the canon, it covers the widest range in the Study—in fact, the lowest note of the

lowest voice (8%) occurs at the last note of the ritardando portion and is the lowest note

on the player piano keyboard (B0), and the highest note of the highest voice (11%) is at

the end of the acceleration and is the piano’s highest note, A,. Nancarrow has thus in-

cluded a rough parallel between frequency and speed—the slowing pitches sink to the

lowest register and the accelerating pitches climb to the top of the keyboard. Combined

with the very close intervals of imitation (M2 - m2 — M2), this texture is an extreme

contrast to any heard in the Study thus far.

rit. (3! notes)      

rade inversion

last 13 notes

L2?THE \
”5:

2 2 .
In.»—

lowest note
on keyboard

be

33:: . «.._—l,
2222»? 2227/12

b0

L,

r8va ---------------------
‘

* interval between bass and
treble portions = 5 8ves + m6

retrograde inversion offirst 13 notes

highest note on keyboard (A7)

in highest canonic voice (11%)

Figure 3-52. Canon 6 of Study No. 27 in the lowest (R8%/A8%) voice.

173

  
  
 
 
   
 
 
 
 
 
  
 
 
 
The canon opens with a chromatic descending line in which the intervals progres-

sively widen (five m2’s, two M2’s, three m3’s, and a M3) in a manner reminiscent of the

overtone series in retrograde. The first and last thirteen notes of the canon are the same

melodic series in retrograde inversion; between these extremes, melodic movement again

favors intervals of fourths and fifths. This canon’s use of retrograde inversion at the be-

ginning and end of the melody and the use of the entire range of the keyboard highlight

the significance of Canon 6 as the midpoint of the Study.

The tempo change scheme in Canon 6 is very straightforward, and each voice

begins and ends with the same durational value.

Qagon 7. Canon 7’s melody consists of sustained and staccato notes, all of which

remain in tempo. The texture thickens dramatically in this canon. The clock line expands

to major and minor thirds, and the canonic lines are doubled five octaves apart—a simul-

taneous (vertical) expression of the registral separation idea that was sequential (hori-

zontal) in Canon 6. And, like Canon 6, which covered the entire range of the player piano

keyboard, Canon 7 covers almost the entire range (minus the very top note). The extreme

pitches, the B0 in the lowest octave of the 11% voice and the Gib/Al,7 in the highest

octave of the 8% voice, are reiterated over and over again, particularly in a passage of

compound melody near the middle of the canon. The compound melody consists of the

extreme registral pitch plus a descending chromatic line; these two extreme lines of

compound melody are shown in Figure 3-53. Note that the extreme repeated pitch is a

long note and the other notes are staccato during the chromatic descent, while the

repeated pitch is a staccato note (and its spelling changes) and the others sustained during

the chromatic ascent section.

As shown in Figure 3-42, the tempo change pattern in the 8% and 11% voices is

accel.—rit.—accel.—rit., while in the 5% and 6% voices it is just accel.—rit.; this is the only

canon in this piece in which a different number of changes takes place in different voices.

174

persistent repetition

ofhigh AMG#

compound melody consisting of repeated high pitch and
chromatic descent and ascent through a tritone

.._-1‘...

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A1 1 %

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lowest note on keyboard

_. Er

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2

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chromatic tritone descent (all staccato notesf

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compound melody with chromatic
descent to lowest note

   \

|

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T
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3'

Figure 3-53. Portions of 8% and 11% voices, Canon 7 of Study No. 27 (pp. 32—34 of score),
showing compound melody consisting of repetition of extreme pitches and chromatic descent/
ascent. (Both lines are doubled in the score—the figure shows the registral extremes: the top line
of the 8% voice and the bottom line of the 11% voice.)

The 98 pitches of the canon are divided differently in each voice between acceleration

and deceleration, with the pitches divided as follows:

8% voice:

A=36 R=34 A=17 R=ll / =98 total

5% voice:

A = 46 R = 52

6% voice:

A = 49 R = 49

/ = 98 total

/ = 98 total

11% voice:

A = 24 R = 27 A = 25 R = 22

/ = 98 total

Canon 8. From here to the end of the piece, the percentages stay in voice order: first

175

 
  
      
   
   
  
 
 
 
   
   
 
 
 
 
 
 
 
   
  
 
 
 
 
 
 
  
  
 
 
 
from slowest change (5%) in the bottom voice to fastest change (11%) in the top voice in

Canons 8 and 9, reversing in Canon 10, and reversing back in Canon 11.

This canon’s main textural idea is the minor/major triad. The clock line here ex-

pands to major and minor triads: the D#4 and E4 are the fifths of the G#lAl, minor/major

and A minor/major triads, while the F4 and F#lGL4 are the roots of F minor/major and

F#/G[, minor/major triads, forming a complex of eight triads in the clock line. The minor/

major triad is also spelled out by the levels of imitation (B, D, D#, and F#—see Figure 3-

43). Additionally, occasional half-step trills take place in the canon melody on the third

of the triad to create alternate minor and major triads (except for the final trill, which

alternates between the major third and the fourth scale degree).

Figure 3-54 shows the canon melody from the bottom line (R5%IA5%). Canon 8

contains two rhythmic values in the ratio 2:1, a feature that was previously used only in

Canon 1. The lowest note on the keyboard, B0, that was prominent in both Canons 6 and

7 is just as much so here: the lowest voice enters first, on a B major triad built on this

note that is repeated twice more. The melody in the lowest voice is tonally organized

around B, with the first section (to the point where the tempo direction changes) con-

cluding on a supertonic triad, which near the middle of the second half of the melody

lowest note on the key-
board in the lowest voice

tempo direction changes take
place here (except in highest
voice, where it is one beat later)

_bt

l

,

_

r

b.
L
A 1"

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22222:?

VN-V

altemately:'ii and VN

this trill does not create
the minor/major triad

Figure 3-54. Melody of Canon 8 from lowest voice, Study No. 27, rewritten in rhythmic notation.

176

  
  
 
 
  
becomes a minor/major triad and alternates in function between the supertonic and the

V/V, finally becoming (enharmonically) VN moving to V at the very end. In the highest

voice (11%), the keyboard’s highest note A7 is stated four times.

Canon 2. Canon 9 is rhythmically the most complex canon, encompassing note

values related by the ratio 423:2:1. Figure 3-55 shows the canon melody as it would

appear in rhythmic notation in the lowest voice; the treble and bass registers in each voice

are rhythmically coordinated but there is never a simultaneous note attack between them.

The trill in some triads in the treble register involving the chord third and the fourth

degree is retained from the end of Canon 8.

,

.

_

.n

.

I: _p_ £;-;:-'—::_—1:§3%§,Cff;;:;_:s;—éi5_,~531:4::11,,:+;Tii":_~;:fg:—_— Piggsg‘q]

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-

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‘0‘

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.

.

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Figure 3-55. Melody of Canon 9 from lowest voice, Study No. 27, rewritten in rhythmic notation.

The clock line in Canon 9 states various kinds of seventh chords, again with D#4

and E4 at the top of the chord structures containing those notes, and F4 and Fit/Gr,4 at the

bottom of chords containing them. The levels of imitation in Canon 9 are the same as the

pattern in the clock line: m2 - m2 — m2 — m2, descending in the bass from the highest

voice’s D1 to the keyboard’s lowest note, B0, in the lowest voice, thus filling out the chro-

matic tetrachord below the clock line. The most extreme high pitch in the treble of the

177

 
 
     
  
  
  
     
highest voice is A27, just below the top of the keyboard.

Like an arch canon, Canon 9’s voices are placed almost symmetrically about the

durational center of the canon, but, unlike the arch canon in the middle of Study No. 22,

the symmetrical placement of the voices here does not allow for a convergence in the

middle because of the constantly changing tempos.

Canons 10 and 11. The final two canons are each very brief and both use only

acceleration. Canon 10’s 21-note melody is an ascending chromatic scale extending an

octave plus a minor sixth, against a clock line that has returned to its original form of four

chromatic pitches. The intervals of imitation progressively increase by a minor second:

the interval between the top two voices is a perfect fourth, between the second and third

voices an augmented fourth, and between the third and fourth voices a perfect fifth (see

Figure 3-43). The beginning pitches of the four voices—C#, C#, D, and G—add two

notes above and below the chromatic space occupied by the clock line, and the A that

concludes the first voice and the 13;, that concludes the third voice add two more pitches to

the chromatic space.

At the conclusion of Canon 10, both the clock line and the lowest voice of the

canon conclude on the note 15,, which is then expanded to four octaves and repeated four

times in the clock line; the clock line states only the major second E—F in Canon 11 (with

the exception of the final two notes—see Figures 3-42 and 343).

Canon 11 is the most tonally-oriented canon in the Study; all the intervals of imi-

tation are octaves, and the pitch content is entirely G major. The major second that is

being expressed in the clock line is imitated in the canonic lines, where every note is

expressing a major second trill (thus, no trills involving the intervals B-C and F#-G

appear in the canon). The final CP (see Figure 3-56), unlike the first two, incorporates the

clock line and occurs on the final staccato attack; thus there is no opportunity for a

“durational convergence” as there was in the first CPs.

178

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Summm and Conclusions

In Study No. 27, Nancarrow was able to express the same ideas in several different

contexts, sometimes sequentially and sometimes simultaneously. The clock line’s chro-

matic tetrachord, for instance, is manifested in the pitch content of individual voices in

Canon 2, in the opening pitches of the canonic voices in Canon 6, and in the levels of

imitation in Canon 9. As another example, the concept of registral expansion moves

through three consecutive canons: first in the clock line in Canon 5, then in the canonic

voices of Canon 6, and then a simultaneous (vertical) expression of this idea in Canon 7.

Thomas found Study No. 27 to be “decidedly non-tonal” (p. 17), but several tonal

areas do achieve some significance. The final tonal goal is clearly G, the note imme-

diately above the chromatic tetrachord of the clock line; G major is stated unequivocally

in all the voices of Canon 11 and this canon is a shocking contrast to the tonal aim-

lessness and ambiguity of the rest of the Study. G is established almost immediately in

Canon 1, being the first pitch in the lowest voice, and the lowest voice in Canon 2 begins

on this same pitch. By the conclusion of Canon 4, however, with its F#—B final interval in

the lowest voice, B plays a more prominent role—particularly with the emphasis on B0 in

the bass, which reaches a peak in the compound melody section of Canon 7. The filling

out of the chromatic tetrachord below the clock line (D, C#, C, and B) in the bass register

of the four canonic voices of Canon 9 provides a final emphasis on B before a G tonal

center is reasserted in Canons 10 and 11.

Study No. 27’s convergence points do little to support the establishment of these

tonal centers. Each of the CPs is structured differently. The first two CPs involve only the

canonic voices, take place at the end of a decelerating section, and end with sustained

notes; the first CP occurs at the beginning of this note, and the second CP is at the end of

the sustained note. The first CP has a durational convergence on a value related to only

one of the voices while the second CP’s final sustained note is of different durations and

180

represents the next series value in each voice. The Study’s final CP, which concludes the

piece, takes place in all the voices (including the clock line) at the end of an accelerating

passage on the attack of the final staccato note; this CP also includes a pitch convergence

on G.

It is somewhat surprising to note in Study No. 27 how little control Nancarrow

seemed to exert in areas where he had typically done so in many of his other tempo

canons: for instance, in the establishment of tonal centers, areas of harmonic conver-

gence, and various kinds of symmetry. There is also very little in this Study that could be

considered a “collective effect” other than the similar gestures in the canonic voices;

these gestures are generally quite disjointed and rarely achieve a level of interdependence

except at the CPs.

When considering both Studies No. 22 and 27 together, it is apparent that Nan-

carrow had far less success in these two geometric acceleration canons in creating col-

lective effects, symmetry, and harmonic convergence than he did in a piece such as Study

No. 8, which is based on arithmetical acceleration. The arithmetical acceleration tech-

nique allows for a common background unit, while geometric acceleration does not.

Without the background unit, it is nearly impossible to overcome the independence of the

voices to create any sense of interdependence. Nancarrow must have felt this, and one is

left to wonder how satisfied he was with the results of the acceleration canon technique

when considering Gann’s observation (p. 163) that Nancarrow wrote no more accelera-

tion canons after Study No. 27.

181

CHAPTER 4

FRACTAL FORMAL FEATURES IN THE TEMPO CANONS

In addition to convergence points, another interesting feature of some of Nancar-

row’s tempo canons is that their formal structures exhibit properties characteristic of

fractals. The following sections will introduce the reader to fractals, recount the ways in

which they have been observed in sound and music, and demonstrate how they are repre-

sented in Nancarrow’s tempo canons.

Introduction to Fractals

Fractal geometry is a relatively newly-described branch of mathematics based on

the 1977 work of Benoit Mandelbrot, in which elements of self-iteration, scaling, and

space-filling are recognized in a variety of naturally-occurring objects as widely diverse

as coastlines, plants, and bodily structures such as the brain and bronchial lobes. Manfred

Schroeder (1991) describes the prevalence and nature of fractals, first by noting their

close relationship to chaotic systems:

But no matter how chaotic life gets, with all regularity gone to bits,
another fundamental bulwark often remains unshaken, rising above the turbu-
lent chaos: self-similarity, an invariance with respect to scaling; in other
words, invariance not with additive translations, but invariance with multipli-
cative changes of scale. In short, a self-similar object appears unchanged after
increasing or shrinking its size. Indeed, in turbulent flows, large eddies beget
smaller ones, and these spawn smaller ones still—and so on ad infinitum
(almost). In general, one of the conspicuous consequences of self-similarity is
the appearance of exceedingly fine-grained structures, now generally called
fractals after Benoit B. Mandelbrot, the father of fractals. (Schroeder 1991,
1—2)

The term fractal escapes easy definition. Mandelbrot himself deliberately avoided

182

 
defining the term, but he did venture so far as to identify a “family of shapes I call

fractals,” further noting that “Some fractal sets are curves or surfaces, others are dis-

connected ‘dusts,’ and yet others are so oddly shaped that there are no good terms for

them” (Mandelbrot 1977/1982, 1).

As the study of fractals has advanced, several properties have been acknowledged

as characteristic. The first two such properties, as mentioned above by Schroeder, are

related: the properties of self-similarity and scaling. According to Larry Solomon (1998),

“Perhaps the most important defining property of fractals is self similarity on many dif-

ferent scales; i.e., they have self-iterating geometric structures that repeat in different

sizes” (no page). These properties can be seen in both natural and representational

objects. Solomon uses the example of a fem frond (Figure 4-1a), a naturally-occurring

object in which the same characteristic shape is iterated on a number of different scales;

Figure 4—1b shows fractal art that resembles the shape of the fern frond.

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scalings of the frond shape are highlighted, and even smaller iterations of the shape are present;
(b) “fem-like" fractal art (Sprott 1996, 105).

Fractals can exist in one, two, or three dimensions; the Menger sponge, a three-

dimensional fractal, is shown in Figure 4-2.

183

 
Figure 4—2. The Menger sponge: a fractal in three dimensions, or what Mandelbrot calls a “spatial
universal curve” (Mandelbrot 1977/1982, 145).

Another important property of fractals is that of space-filling. Consider the line

fractal known as a Peano curve (Figure 4-3). With each new iteration of the generating

shape, the surface space is more tightly filled and the length of the line drawing the curve

increases. The number of iterations and the length of the line can approach infinity,

moving toward filling the space but never completely doing so (since in geometry 3 line,

by definition, has no breadth). The iterative process that results in the generation of the

fractal shape is known as an “iterated function system,” or IFS (see Schroeder 1991,

Bamsley 1996, and Frame 1996). The IFS consists of an initiator shape and a generating

shape, or the function that is repeatedly applied to create the figure.

Figure 4-3. A Peano curve, which illustrates the space-filling property of fractals (Mason and
Saffle 1994, 31). With each further iteration of the curve, the length of the line drawing the curve
approaches infinity.

184

 
 
 
A spectacular example of a space-filling curve, or what Bamsley calls “a sequence

of curves ‘converging to’ a space-filling curve,” is shown in Bamsley (1993, p. 241). The

figure is not reproduced here because the limits of resolution will not do it justice. Two

similar examples from Bamsley are shown in Figure 4-4.

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Fractals in Sound and Music: A Review of the Literature

The close relationship between music and geometry goes back thousands of years to

the Greek quadrivium. Mandelbrot’s theories on fractals began to be applied to music and

sound beginning in 1978, with fractal structures being identified in the nature of sound

itself (Gardner 1978; Schroeder 1991), in melodies (Mason and Saffle 1994; Chesnut

1996), and in the phrase structures of binary and ternary forms (Solomon 1998). Fractals

have also been explored as a means of composing music (Gardner 1978; Dodge and Bahn

1986) and creating new self-similar scales (Pierce, in Schroeder 1991). Among the fractal

structures that have been related to musical structures are the Cantor comb, Sierpinski's

triangle, Peano curves, and the Koch snowflake.

Mason and Saffle began their 1994 article with the following observations:

185

 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
 
 
 
 
 
  
 
 
  
 
Symmetries and self-similarities abound in the world of geometry, and a great
deal of music also shares these properties. Throughout the nineteenth and
twentieth centuries, historians and theorists have discovered musical self-
similarities of various kinds. These include the micro- and macrolevel “bar”
forms that Alfred Lorenz discovered in Wagner’s music dramas; the recurrent
melodic and harmonic background patterns in European art music of the
eighteenth, nineteenth and early twentieth centuries discovered by Heinrich
Schenker and other Schenkerian theorists; rhythmic patterns discovered by
Leonard Meyer and his colleague Grosvener Cooper; and generative organiza-
tional patterns in tonal music, discovered by Fred Lerdahl and Ray
Jackendoff. (p. 31)

The nature of sound itself has been found to display fractal characteristics. In 1964,

before fractals had even been described, Roger Shepard reported on a special class of

sounds in which the property of scaling is actually present in the waveform itself. These

sounds, which Mandelbrot later termed “scaling noises,” have the fascinating property

that the pitch decreases as the frequency increases (Schroeder 1991, 95). Schroeder

(1991) gives an example of another self-similar waveform in which frequency doubling

results in no change in pitch (Figure 4-5). Synthetic waveforms have also been produced

based on fractal properties, such as Mackenzie’s “fractalized” sine wave shown in Figure

4-6.

Figure 4-5. A self-similar waveform in which frequency doubling results in no change of pitch
(Schroeder 1991, 96).

Richard Voss focused on how the fractal nature of sound relates to the construction

of pleasing melodies. Gardner (1978) describes how Voss identifies the frequency spectra

for three types of “noise”—white, l/f (“pink”), and Brownian (see Figure 4—7)——and

demonstrates how the properties of these different waveforms could be interpreted as

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melodies incorporating elements of randomness (white noise), correlation (Brownian

noise), or both (pink noise) as exhibited in the sound patterns. It turns out that 1/f(“pink”)

noise exhibits fractal self-similarity whereas white and Brownian noise do not, and it is

the melodies based on moderately correlated llf noise that most people in a test audience

found most appealing, based on the melodies’ effective balance between complete ran-

domness (surprise) and extreme correlation (expectation).

Fractal properties of melodic structures have been further studied by Mason and

Saffle (1994), who showed how right-angle drawings known as Lindenmayer (L-system)

curves could be used to create melodies—albeit of questionable musical value. Melodies

are created from the curves by interpreting horizontal line segments as durations and

vertical line segments as pitches (Figure 4-8). Mason and Saffle also assert that many

existing melodies can be shown to have strong correlations with L-system curves,

although their work in 1994 is very preliminary. They did, however, identify L-system

curves that “generate tunes that are similar or even identical to hundreds of existing

melodies by classical and popular composers” (p. 35), and they go on to discuss several

187

 
 
     
 
 
 
 
  
 
 
 
    
 
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1978, 21). White noise (spectral density = l/f’) exhibits properties of extreme randomness within
a limited range, with no correlation between events. “Brownian noise” (spectral density = l/f)
exhibits properties of extreme correlation between events and a tendency to “wander” over the
spectrum. “llfnoise” (spectral density = II)‘) is moderately correlated.

such tunes. Mason and Saffle’s work with the melodic associations of these curves

prompted them to observe that:

Aspects of certain theories about the origins and fundamental structures of
melodies suggest that much—perhaps all—beautiful music is, in some essen-
tial sense, fractal in its melodic material and internal self-similarity. (p. 35)

Schroeder (1991, pp. 99—101) described the self-similar structures in the equal-

tempered scale, with the equal-tempered semi-tone functioning as the unit of self-

188

 
 
 
 
 
 
 
 
 
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Figure 4-8. Construction of a “right-angle canon” from Lindenmayer curves (Mason and Saffle
1994, 32—33). The method involves reading the horizontal lines of curves as durations of notes
and the vertical lines as pitch intervals between notes. In this example, the smallest line segment
represents either one scale step (if vertical) or one sixteenth note (if horizontal). If the curve
begins with a horizontal line, the pitch of the first note is assumed to be the first note of the
chosen scale or mode (C major in this example); only the duration for the first note is taken from
the curve. If the curve begins with a vertical line, the first note can be read as the number of
forward moves up or down from the first note of the given scale or mode. Curve (b) is a 90°
counterclockwise rotation of (a).

similarity in creating a musical scale. The near equality of 3:2'2 (z 129.7, or twelve

perfect fifths) to 27 (= 128, or seven octaves) creates a nearly-closed system. Schroeder

then described the “Pierce scale” (p. 102) in which John R. Pierce attempted to generate a

self-similar scale like the equal-tempered scale in which the octave (2:1) is replaced by

the “tritave” (3:1). The “semi-tone” that results is 3y”, with the result that the Pierce

system is as nearly well-closed as the equal-tempered scale, although Schroeder reports

that the musical utility of this scale is “Open to debate” (p. 102).

Dodge and Bahn (1996) applied the properties of self-iteration, scaling, and space-

filling to the composition of a musical phrase and related the results to the fractal known

as the Koch snowflake, as shown in Figure 4-9.

189

   
 
 VVV

(a) The Koch snow-
flake.

(b) A “generating motif” of music composed of a few intervals
and durations (analogous to the largest triangle in the Koch
snowflake).

(c) The first and second layers of a polyphonic musical composition based on the generating
motif. The first layer (bottom voice) is the original motif, while the second layer (upper
voice) is merely a faster (and transposed) repetition of that motif added to each of the original
motif’ 3 notes (analogous to the smaller triangles attached to the larger triangles).

Figure 4-9. Properties of self-iteration, scaling, and space-filling in a musical segment (Dodge
and Bahn 1996, 190).

Solomon (1998) demonstrated how fractal structures can be identified in musical

phrase structures. A simple example is the Cantor comb (Figure 4-103), which Solomon

(21)

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comb (Solomon 1998).

190

 
 
 
 
 
correlated to phrase divisions in a typical binary form (Figure 4-10b). The Cantor comb

itself is defined by subsequent iterations of ternary division (of which the middle third is

blank); still, it is useful to see how its structure somewhat correlates with phrase struc-

tures of subsequent binary divisions.

Measuring Dimension in Fractals

Every fractal can be described in terms of its fractal dimension:

How big is a fractal? When are two fractals similar to one another in
some sense? What experimental measurements might we make to tell if two
different fractals may be metrically equivalent? .

.

.

There are various numbers, associated with fractals, which can be used
to compare them. They are generally referred to as fractal dimensions. They
are attempts to quantify a subjective feeling we have about how densely the
fractal occupies the metric space in which it lies. Fractal dimensions provide
an objective means for comparing fractals.

Fractal dimensions are important because they can be defined in con-
nection with real-world data, and they can be measured approximately by
means of experiments. For example, one can measure the “fractal dimension”
of the coastline of Great Britain; its value is about 1.2. Fractal dimensions can
be attached to clouds, trees, coastlines, feathers, networks of neurons in the
body, dust in the air at an instant in time, the clothes you are wearing, the dis-
tribution of frequencies of light reflected by a flower, the colors emitted by the
sun, and the wrinkled surface of the sea during a storm. (Bamsley 1993, 171)

As Pickover (1995) describes it, “the fractal dimension .

.

. characterizes the size-

scaling behavior of the pattern. This value gives an indication of the degree to which the

pattern fills the space” (p. 204). The fractal dimension further serves to describe whether

the fractal is between the topological dimension of a point (0) and a line (1), whether it is

two-dimensional (a dimension between 1 and 2), or three-dimensional (a dimension be-

tween 2 and 3); Frame (1996) notes that the dimensions of a line segment = 1, a filled-in

square = 2, and a solid cube = 3 (p. 38).

The most commonly used fractal dimension is called the Hausdorff dimension (DH;

Schroeder 1991, p. 15 ff.). The Hausdorff dimension describes the relationship between

the number of self-similar objects that are created between successive iterations and the

191

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size of these objects compared to the original. Figure 4-11 shows a simple derivation of

this dimension based on self-similarity in lines, squares, and cubes. Connors (1994) gives

the formula N = S”, where N is the number of self-similar pieces in the divided figure, S

is the scaling factor (the factor by which each smaller piece relates to the original—in

each case here, S = 4)‘, and D (the power to which S is raised) is the dimension. Solving

the equation for D yields:

SD = N

log SD = log N

D log 8 = log N

DH = lOg(N)/log(S)

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//J/4)

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(a)

(b)

(C)

Figure 4-11. Illustration of derivation of dimension via self-similarity (Connors 1994). In the line

segment in (a), four self-similar pieces are each ’3 the size of the original, i.e., 4 = 4' pieces; the

square shown in (b) consists of 16 self-similar pieces with sides ’3 the size of the original, i.e., 16
= 42; and the cube in (c) consists of 64 self-similar pieces with sides ’1 the size of the original,
i.e., 64 = 43 pieces. In each of these simple cases, the exponent gives the dimension; thus, the

dimension of (a) = 1, of (b) = 2, and of (c) = 3.

Some examples follow to illustrate the calculation of this fractal dimension using

this formula, DH = log(N)/log(S). Let us first consider the Cantor comb shown in Figure

4-lOa. The first iteration of the generating function—to remove the middle third of the

line’s length—results in two line segments one-third the size of the original, and this

continues through successive iterations potentially ad infinitum. The dimension of the

1The scaling factor refers to how the side length of the scaled object compares to the original, not its area or
volume in the case of two- or three-dimensional figures.

192

 
 
 
 
 
  
 
 
   
    
 
 
 
 
 
 
 
 
   
 
 
 
Cantor comb is thus described:

DH 2 log(2)/log(3) = 0.63

Thus, the Cantor comb is a one-dimensional fractal with its dimension being between that

of a point and a line.

Now let us look at some two-dimensional fractals and their dimensions. The Sier-

pinski triangle (Figure 4-12) divides the initiating shape into three copies of itself, each of

which has a line length one-half the size of the original (i.e., each of the three sides of the

initiating triangle is bisected to form the generating shape, a blank inner triangle). The di-

mension is calculated as:

DH = log(3)/log(2) = 1.58

(0. 1)  

(0. 0)

(l)

(1 . 0)

(b)

‘

.

Figure 4-12. The Sierpinski triangle, DH = 1.58 (Frame 1996, 39).

Another two-dimensional fractal is the Cantor gasket, shown in Figure 4-13. It

divides the initiating shape into eight copies of itself, each with a line length one-third the

size of the original. The dimension is thus:

DH = log(8)/Iog(3) = 1.89

At this point, in comparing Figures 4-12 and 4-13, one will notice that as the fractal

dimension approaches 2, the space is more solidly filled. Thus, in a sense, the fractal

dimension describes how much of the metric space in which the fractal exists is “carved

193

 
 
 
 
  
 
out” to create the fractal figure, with a dimension closer to 1 indicating that more is

carved out than in a figure with a dimension closer to 2.

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The Menger sponge that was shown in Figure 4-2 will illustrate dimensions in

three-dimensional fractals. A comparison with Figure 4-13 reveals that this fractal is the

three-dimensional equivalent of the Cantor gasket. Determining its dimension necessi-

tates being able to determine how many cubes, of side-length one-third, are created by

each iteration of the generating function. There are 20 such cubes (eight on the top layer,

eight on the bottom layer, and the four comers of the middle layer). Thus the fractal

dimension is:

which confirms that the figure is three-dimensional.2

DH = log(20)/log(3) = 2.73

Fractals are also described in terms of whether the fractal is totally disconnected,

“just-touching,” or overlapping. Fractals such as the Cantor comb (Figure 4-10a), or

“dust” fractals, are totally disconnected; their successive iterative generations are not

touching any previous generations. In “just-touching” fractals such as Sierpinski’s tri-

angle (Figure 4-12) and the Cantor gasket (Figure 4-13), successive iterations are wholly

2A curiosity about the dimension relationship between the Cantor gasket and the Menger sponge is that the
decimal part of their dimensions (.89 for the Cantor gasket and .73 for the Menger sponge) is not equivalent
as one would expect them to be. Note that if the Menger sponge contained 24 self-similar cubes rather than
20, the decimal portion of its DH would be exactly the same as the Cantor gasket’s [log(24)/log(3) = 2.89].
The reason for the discrepancy is that the cube of side length one-third at the very center of the Menger
sponge appears in all six of the cuts from one side of the figure to the other; this overlapping cube can be
subtracted only once from the possible total of 27, with the other six eliminated cubes coming from the
center of each of the main cube’s faces to result in 20.

194

  
 
     
 
 
 
contained within the initiating shape and the points which define the extremes of the gen-

erating shape create smaller figures that are just touching. Fractal shapes can also over-

lap, and these are the kinds of fractals we will see in Nancarrow’s tempo canons.

Fractal Characteristics in Nancarrow’s Music

Although Nancarrow’s tempo canons do exhibit the fractal characteristics of self-

similarity, scaling, and space-filling, they exhibit as well some important departures from

the fractals we have observed so far.

The first (and most important) difference is that the percentage of scaling between

successive iterations in tempo canons involving more than two voices is not the same, as

it would be in standard fractals based on iterated function systems (IFS). For instance, let

us look at the procedure used in the construction of the Cantor comb from Figure 4-10a.

Figure 4-14 shows the successive division of a line extending from O to 1 into one-third

segments. Note that between any pair of successive iterations, the same ratio of division

is in effect; the scaling factor is always 3.

O

0

0

.1.
3

3
9

l.
9

2
9

l

1

l

2
3

6
9

l
9

8
9

"1 '7.

27 27 27

a. a

27 27 27 27

5:2 2:11

27 27 27 27

2T2: 2?,

27 27 27

Figure 4-14. Construction of the Cantor comb (after Bamsley 1993, 44). The line in each previous
iteration is divided into thirds, with the middle third left blank; the scaling factor is always 3.

Nancarrow’s tempo canons, on the other hand, usually exhibit tempo (or duration)

relationships based on superparticular ratios, which in the case of 3- or more element

ratios ensures that the scaling between different pairs of adjacent voices will not be the

195

 
  
same. For example, in a 12:15:20 tempo canon, which has a duration ratio of 3:4:5, the

scaling between 3:4 and 4:5 is not equivalent. In a duration ratio such as 4:6:9, on the

other hand, both adjacent pairs are related by the same ratio; such a ratio is symmetrical

in that the tempo ratio and its inversion, the duration ratio, are the same. Nancarrow did

not normally use such symmetrical tempo ratios, however, with perhaps the only example

being the ratio 1%/1 (’2 %/22; % of Study No. 22, an acceleration canon in which the ratio

between both pairs of adjacent voices is 3:2. Thomas cites Nancarrow’s avoidance of

geometric ratios when she says “The ratios are geared toward creating the effect of inde-

pendence between the layers, and to that end Nancarrow avoids geometric ratios (such as

3:6:12) which would create tempos potentially subordinate to a common, broad rate of

pulsation” (Thomas 1996, 9).

Another way in which Nancarrow’s tempo canons differ from typical fractals is that

the scaling compression is in only one dimension (time/tempo) as melodic intervals (and

therefore pitch ranges) are not compressed. Figure 4-15 represents as an example the

structure of Study No. 14, a two-voice arch canon, in terms of the pitch range of the two

canonic voices and their tempo/duration relationship. The interval of imitation is two

octaves plus a perfect fifth, and both voices have a total pitch range of 45 pitches. Thus

the scaling of the second and faster voice involves only the tempo and not the pitch

range. (Note that this was also the casein the example from Dodge and Bahn, Figure 4-9,

which has a scaling factor of 2). This discrepancy will be ignored in this study when

determining fractal dimensions of the tempo canons and dimensions will be based only

on scaling relationships of the durations of the canonic voices.

Finally, a third difference between Nancarrow’s tempo canons and typical fractals

is their self-limiting nature. Most of the tempo canons discussed here contain only two or

three scaled generations while the most complex (Study No. 36, “Canon 17/18/19/20”)

has four. Meanwhile, the number of generations in a geometric fractal is potentially in-

finite.

196

I

'

"

_
1"“ ,7

.

It

Total pitch range =
E] - G7
(76 pitches)

Voice 2 = B3 - G7
(45 pitches)

Duration = 4

Pitch overlap =
B3 - C5
(14 pitches)

Voice 1 = E1 - C5
(45 pitches)

Duration = 5

C7 -

C6 ..

C5

C4

C3

C2

C1

I

I

I

CP

Figure 4-15. Diagram of Study No. 14 (“Canon 4/5"), showing pitch ranges of two canonic voices
and tempo/duration relationship. Voice 2 is compressed in time but not in pitch.

197

 
 
 
 
 
 
  
 
Measuring Fractal Dimension in the Tempo Canons

These and other factors lead to complications in attempting to derive the fractal di-

mensions of the tempo canons. Besides the fact that the scaling is not the same across all

iterative generations, in Nancarrow’s tempo canons each iterative generation of the “gen-

erating shape” (the canonic subject) results in just one scaled copy of the subject rather

than multiple copies as in all of the fractal shapes that we have observed. Further, the

“fractalized” voices in Nancarrow’s tempo canons overlap each other, and the Hausdorff

equation does not account for overlapping in fractals. Clearly another method is needed

to determine the fractal dimension of these canons.

In consideration of the fact that each successive generation in the tempo canons is

scaled by a different amount and results in only one new copy, I propose that the dimen-

sion equation DH = log(N)/log(S) be modified for calculating the fractal dimension of

tempo canons so that N represents the total number of scaled voices (including the

original voice, or the “generating shape”) and S represents the total relative scaling of the

duration of these voices. (This equation will not, as already noted, take into account the

overlapping of the scaled objects.) Figure 4-16a will serve as an example. The figure

simultaneously illustrates two hypothetical tempo canons: on the left side a canon with

the duration ratio 2:3:4:5:6, and on the right side a duration ratio of 3:4:5:6. For the canon

with the duration ratio 2:3:4:5:6, N = 5 and S is the sum of the five scaled durations, 1 +

0.83 + 0.67 + 0.5 + 0.33, more precisely summed up as

2+3+4+5+6
6

29

= 6 = 3.33.

Thus, DH = log(5)/log(3.33) = 1.34. For the canon with the duration ratio 3:4:5:6, N = 4

and S =

3+4+5+6
6

m

= 6 = 3; DH = log(4)/Iog(3) = 1.26. [Log(4)/log(3) also happens

to describe a fractal known as the Koch curve, in which the middle third of a line segment

is repeatedly raised to the point of an equilateral triangle; see Figure 4-l6b.]

This definition of fractal dimension will now be applied to several of Nancarrow’s

tempo canons. The diagrams in this section will show the relationships between canonic

198

2+3+4+5+6=_2_0_=3.33

6

6

= log(S) = 1.34

log(3.33)

|

3+4+5+6=1_8_=3

6

6

D = log(4 = 1.26
log(3)

J

3

.

J

.
Q

I N

I

=
= 4

= 5

.._...

(a)

(b)

Figure 4—16. (a) Hypothetical tempo canons of duration ratio 2:3:4:5:6 (left side of diagram) and
3:4:5:6 (right side); fractal dimension (D) is calculated by dividing log of number of scaled
objects (including the generator) by the log of the total of the relative dimensions. (b) D = 1.26
[log(4)/log(3)] also describes the Koch curve, in which each line segment is replaced by a
segment 4/3 the size of the original.

voices in terms of duration, range, and overlap of voices. The horizontal x axis in the dia-

grams will represent the duration of each Study (which will, of course, vary from Study

to Study), while the vertical y axis will represent the player piano’s entire pitch range

from B0 to A7.

Figure 4-15 shows the fractal formal structure for Study No. 14. This tempo canon

consists of just two scaled voices, with relative scaled durations of 1 and 0.8; therefore,

DH = log(2)/log (1.8) = 1.18.

Figure 4-17 shows the structure for Study No. 18 (“Canon 3/4”). In this Study, the

second voice of the tempo canon consists of only the treble melody. For these two scaled

lines, the relative scaled durations are 1 and 0.75, and DH = log(2)/log(1.75) = 1.24.

Figure 4—18 shows the structure for Study No. 19 (“Canon 12/15/20”). Its duration

ratio is 3:4:5, and it consists of three scaled voices with a total scaled duration of
3+4+5
5

= 2.4; thus, DH = log(3)/log(2.4) = 1.25.

199

 
 
 
  
 
   
 
 
 
  
 
 
  
  
 
 
  
  
Voice 2 = F5 - F7*
(25 pitches)

Duration = 3

Pitch overlap =
F5 - F6

(13 pitches)

Total pitch range
Bo - F7
(79 pitches)

C7

C6

C5

C4

C3

Top part of
Voice 1 = F4 - F6
(25 pitches)

Duration =

C2

   Bottom part of

Voice 1 = B0 - F4
(43 pitches)

Duration = 4

C1

*Voice 2 restates only the top part of Voice

.
.
.
0

0
'

Figure 4-17. Diagram of Study No. 18 (“Canon 3/4”); DH = Iog(2)/log(l .75) = 1.24.

200

 
 
 
   
 
   
 
  
  
 
 
  
 
   
  
 
   
   
 
   
  
  
    
 
.- Voice 3 = A3 - A7
' (49 pitches)

Duration = 3

1

"

a?“

41“ ‘

Lil};

:31

1

.

"

.

,.
'

1

Z' 91

,
Pitchoverlap of
‘Voices2and3=

.
A3- E6 (32 pitches)
t:

i E"

.Vé‘

Z"

-.
>3

'.

..

”"21 '_l:(. ‘ .1"

,

..

" “1.3!
,q;‘‘

r.

g

_
..t .J; M: ,
t,”it" 7'*1 fm) ' {‘5'

n;-
1.

7,

Pitch overlap ofall7Total pitch range--
3 voices—- A3- B4 ‘30- A 7
,é'ffl5pitches)
c“ *7“ .

1: (83 pitches)
3‘.

. ”it”?T‘ZJZ:2

‘

Voice 2 = Q - 56
(49 pitches)

Duration = 4

Pitch overlap of
Voices I and 2 =
E2 - B4 (32 pitches)

Voice 1 = H) — B4
(49 pitches)

Duration = 5

C7

‘1

C6 -

C5-

C4

C3

C2

C1

-
-
-
!
O
1

Figure 4-18. Diagram of Study No. 19 (“Canon 12/15/20"); DH = log(3)/log(2.4) = 1.25.

Figure 4-19 presents the structure for Study No. 36 (“Canon 17/18/19/20”). This

canon’s duration ratio is 2907:3060:3230:3420; its four scaled voices have a total scaled

duration of

2907 + 3060 + 3230 + 3420
3420

1__2617

= 3420 = 3.69. Thus, DH: log(4)/log(3.69) =

201

 
 
   
    
 
 
 
   
L

Voice 4 = 3,4 - A7
(36 pitches)

Duration = 2907

r.

1

(20 pitches)  
I 

 ,-'
3 '11 "5“ L‘
;.,;4' 5.:
li. Voice 3 = Q3 - F6 0:}th first...»
1'-
'
‘ ‘r': (36 pitches)

Pitch overlap be-
tween contiguous
voices : BM - F6

Duration = 3060 x"
179%;
i
1
S

1.1.4:...3 r!

txrwfiéz). 1.

M -

.

’

-

.

'

,‘

,

15‘.) t." 5. ”72

‘V'

Pitch overlap be-
tween contiguous
voices = 01.3 - D5
(21 pitches)
, Fifi}!- '

'

1.;

:1

.1

‘.

I“

.

r,

L‘s-5‘ .-

1' i

i.) Z ‘4

"31>:- 'f i. .1 1I*1.'.Zl’il'~

Voice 2 = E2 - D5
(36 pitches)

Duration = 3230

Voice 1 = Bo - 3,3
(36 pitches)

Duration = 3420

Pitch overlap be-
tween contiguous
voices : £12 - 81,3
(20 pitches)

Total pitch range =
30 - A 7
(83 pitches)

C7-

C6-

C5— C4-

C3

C2

C1

arrow indicates canonic midpoint

:
I
CP

Figure 4—19. Diagram of Study No. 36 (“Canon 17/18/19/20”). Recall that in this canon the CP is
slightly past the canonic midpoint (see Figures 3-5 and 3-13). D, = log(4)/log(3.69) = 1.06.

Figure 4-20 presents the structural diagram for Study No. 31 (“Canon 21/24/25”).

202

 
 
 
  
 
The duration ratio here is 168:175:200; the three scaled voices have a combined scaled

168 + 175 + 200 E

duration of

200

= 200 = 2.72. Therefore, DH = log(3)/log(2.72) = 1.10.

.

‘

-

 Voice 3 (sec. A) =
F#6 - C7 (6 pitches)
Duration = 168

Voice 3 (sec. B) =

I
C‘
I
L

C4 ‘

Voice 1 (sec. A) =
E5 - 31,5 (6 pitches)
Duration = 200

Voice 2 (sec. A) =
85 - F6 (6 pitches)
Duration = 175

F#6 - G#7 (15 pitches)   
C1 -    

pitch overlap
between adjacent
voices = 26 notes
—— Voice 2 (sec. C) =
pitch overlap ofall
three voices = Cal! -

V0106 1 (sec. B) =
E3 - F#4 (15 pitches)

Voice 3 (sec. C) =
C#4 - A7 (44 prtches)

Voice 1 (sec. C) =
Bo - G4 (44 pitches)

“*2 ' 1% (44 pitches)

G4 (7 notes)

C2 "

3 a

C

.

.

CP

Figure 4-20. Diagram of Study No. 31 (“Canon 21/24/25”). The canon melody is divided into
parts A, B, and C, each separated by eight measures; interval of imitation in section A is a perfect
fifth, and in sections B and C a perfect twelfth. The duration ratio in each section is 168:175:200;
DH = log(3)/log(2.72) = 1.10.

203

 
   
   
  
 
  
 
 
 
 
 
A review of the fractal dimensions determined for these tempo canons reveals the

following tendencies:

l. The fractal dimension is generally higher in tempo canons with more voices (for

example, the hypothetical 2:3:4:5:6 and 3:4:5:6 canons shown in Figure 4-16a), espe-

cially when the following is true:

2. The fractal dimension tends to be higher as the total scaled duration decreases; in

other words, the smaller the duration of the scaled voices compared to the original, the

higher the fractal dimension is. See, for example, Studies No. 18 (Figure 4-17) and 19

(Figure 4-18) and the two hypothetical canons in Figure 4-16(a); compare to the mini-

mally-scaled voices in Studies No. 36 (Figure 4-19) and No. 31 (Figure 4-20). Consider

also the example given by Dodge and Bahn (Figure 4-9) in which the scaled copy is half

the duration of the original; in this case, DH = log(2)/log(1.5) = 1.71.

Analysis of Study No. 32: “Canon 5/6/7/8”

This chapter now concludes with an analysis of Study No. 32, a four-voice con-

verging canon in which the final cut-off is the CP (see Figure 3-23, p. 135). The deri-

vation of Study No. 32’s duration ratio from the tempo ratio 5:6:7:8 is shown below:

5 : 6 : 7 : 8
|—-—l |-—-l l——l

l'_ll-'-Il'—'I
105 :120:l40: 168

This duration ratio and the elapsed number of beats in the first voice to the entrances of

later voices were shown in Figure 3-22 (p. 135). The canon’s timespan is 431 dotted

quarter-note beats (the canon is written entirely in g and 3 meters) and the CP occurs on

the hypothetical downbeat of beat 432.

The fractal formal structure of Study No. 32 is shown in Figure 4-21. As shown in

the figure, the interval of imitation between all voices is the perfect fifth, and the four

204

scaled voices have a total scaled duration of

105 + 120+ 140 +168
168

533

= W = 3.17; thus

DH = log(4)/log(3.l7) = 1.20.

Total pitch range 2
Bo - Al7
(82 pitches)

pitch overlap between
. contiguous voices = 48 pitches

55 pitches; duration = 168 

Voice 4 = G#Q - C#5 + G#s - A1,?
55 pitches; duration = 105

V0ice2=F¥1l -B3+F#I-Q,6
55 pitches; duration = 140

amtw 1 Voice 3 = CiQ - F#4 + c115 $13.7” .1

Voice1=Bo-E3+B3-Q,6

55 pitches; diration = 120

"

Figure 4-21. Structural diagram of Study No. 32 (“Canon 5/6/7/8”); DH = log(4)/log(3.l7) = 1.20.

I
CP

205

Although our analysis of fractal dimension does not take into account pitch dimen-

sion, it is apparent—in comparing Figure 4-21 with Figures 4-15, 4-17, 4-18, 4-19, and 4-

20—that Study No. 32 has the largest pitch overlap (in both absolute and relative terms)

of any of these pieces. Study No. 14 (Figure 4-15) has a pitch overlap of 31% (14 of 45

pitches) between adjacent voices, Study No. 18 (Figure 4-17) an overlap of 52% (13 of

25 pitches), Study No. 19 (Figure 4-18) an overlap of 65% (32 of 49 pitches), Study No.

36 (Figure 4-19) an overlap of between 56% and 58% (20 or 21 of 36 pitches), and Study

No. 31 (Figure 4-20) an overlap in its third section of 59% (26 of 44 pitches), while

Study No. 32’s pitch overlap between adjacent voices is 87% (48 of 55 pitches). This

considerable overlap of pitch between voices is an intrinsic part of the characteristic

sound of this piece, as will be revealed shortly.

Gann identifies three sections in the canonic melody. Section 1 is comprised of 164

beats, or 38% of the canon’s total length; section 2 is 77 beats, or 18%; and section 3 is

190 beats, or 44%. The structural proportions of Study No. 32’s sectional divisions are

shown in Figure 4-22. There is one near-coincidence of section beginnings, as shown in

the figure by the dashed line between the first and third voices. The beginning of section

2 in the third voice is within one beat of the beginning of section 3 in the first (bottom)

voice: the first two sections of the bottom voice comprise 164 + 77 = 241 beats, while the

third voice’s beat delay to its entrance (123 1‘1) plus the duration of its first section (164 0

5/7 = 117’?) adds up to 2402/7.

In another near-coincidence, the entrance of the fourth (top) voice is just over two

beats prior to the beginning of the first voice’s section 2. The nearness of this entrance to

the beginning of the first voice’s section 2 is indicated by comparing the number of

elapsed beats to the entrance of the fourth voice (161.63 beats) and the number of beats in

the canon’s first section (164). Also, as indicated by the arrow in Figure 4-22, the begin-

ning of section 2 in the fourth voice happens to coincide very neatly with the piece’s

golden section.

206

golden section: 161.63 + 102.5 = 264.13
(264.13/431 = 0.612)

Ratio
factor:
105

Delay to later voices
(compared to voice 1)

161.63beats

120

123.14beats

140

71.83 beats

168

4

sec. 1
(l64beats)

'

sec. 2
(77 beats)

431 beats

sec. 3
(190 beats)

b

Figure 4-22. Proportional dimensions of Study No. 32 (duration ratio = 105:120:140:168) show-
ing sections 1, 2, and 3 in each voice and beat delay to entrance of later voices. Entrance of
section 2 in third voice nearly coincides with beginning of section 3 in first (bottom) voice
(shown by dotted line).

Gann’s diagram of the structure of No. 32 (Figure 4-23) is apparently based on the

tempo ratio 5:6:7:8 rather than the duration ratio (although its proportions at the left seem

to indicate the correct ratios—compare with Figure 4-22); it incorrectly implies a coinci-

dence between the beginning of section 2 in the fourth voice and the beginning of section

3 in the second voice.3 In the score, this relationship is actually off by a whole score sys-

tem: in the top voice the elapsed beat count is 161 5/8 beats to its entrance plus 102.5 beats

(164 - 543) in section 1 for a total of 2641/8; in the second voice, section 3 begins after

715/6 beats prior to its entrance, plus section 1’s 13633 beats (164 0 545) and section 2’s

[A
l

IA}
I
T

1

18
l

IA
I

I3
l

[C
l

[C

i8.
1:!
I

[C
l

[A
r—

18
l

[C
l

J
1

_j
l

l
l

I

Figure 4-23. Structural diagram of Study No. 32 (“Canon 5/6/7/8”) from Gann (1995, p. 185).

The alignment indicated at C) does not really occur.

3Gann’s diagram labels the sections A. B, and C although in his text he refers to them as l, 2, and 3.

207

  
 
 
 
 
 
  
 
 
  
 
 
  
 
 
64I/6 beats (77 0

54)) for a total of 271 3’3 beats. Gann’s diagram does, however, correctly

identify the near-coincidence of the entrance of the fourth voice with the beginning of

section 2 in the bottom voice.

Study No. 32’s highly organic canon melody is monophonic throughout and con-

tains the seven motives identified in Table 4-1. The table shows the motives in their most

common rhythmic forms as well as showing examples of rhythmic augmentations and

diminutions that occur. Each motive is classified as having a primary beat division of

either the quarter note or dotted quarter note; this classification is generally determined

by the most common rhythmic relationship between the motive’s first two notes. Motives

a, c, and e are based on the quarter note division, and motives b, b2, (1, and fon the dotted

quarter note.

Table 4-1

Descriptions of Motives in Study No. 32’s Canon Melody

My;

Description and comments W

a

descending major triad; motive does not appear in sec. 2
Rhythmic forms:

J

most common form (m. 1)

q
—
t
l
—
h
«
—
o
—
b

f.

—l

augmented form (m. 156)

b

ascending or descending minor third

J

most common form (m. 3) 

augmented form (m. 9)

diminished form (m. 115)

208

 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
  
Table 4-1 (cont’d.)

b2

two successive interlocking ascending minor thirds;
motive does not appear in sec. 2
Rhythmic forms:
1

-r— j
WM” lFF’L‘T—fmml

A

I

J

most common form (m. 24)

W413i; %””1;r

C

diminished form (m. 66)

c

ascending or descending half step
Rhythmic forms:

J

;;:;;7J

most common form (m. 13)

m: 1 _
1
’-
‘J 9 L
\, .

l

r“

.

.

minimally augmented form (m. 6)

‘

fliiigzfii—L—ii maximally augmented form (m. 46)

V

d

ascending or descending octave (or multi-octave) leap;
largest leap = 4 octaves (e.g., mm. 6—7)
Rhythmic forms;

J

W‘fiwmflt 3:17;?

:E

most common form (m. 17)

augmented form (m. 106)

diminished form (m. 31)

e

ascending or descending “partitioned minor thir ” (3-
note figure consisting of major second and minor
second); motive does not appear in sec. 2

J

Rhythmic forms:

J1
v

a
‘
:

r
h

most common form (m. 50)

augmented form (m. 182)

209

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
Table 4-1 (cont’d.)

 f

ascending or descending perfect fourth; most commonly
in descending form in sec. 2

J

Rhythmic fgrms:

  most common form (m. 26)

Eggh , mid—— iv Pg} __«1

augmented form (m. 79) 

Canon’s Section 1

Section 1 (164 beats, mm. 1—74) introduces all seven of the motives shown in Table

4—1. Figure 4-24 shows Study No. 32’s melodic line as it appears first in the lowest voice.

The canon is written in the figure on three staves to clarify its compound nature as noted

by Gann (1995, p. 182): a high melodic line of staccato and sustained notes; a line of

sustained notes in the bottom area of the bass clef; and mostly staccato notes in the

extreme bass register.

Immediately, the canon establishes a conflict between the J rhythmic division of a

and the J. division of b, and by m. 10 the J. division is added. The last half of section 1

(after the second voice enters in m. 34) is less rhythmically active, contains numerous

rests, and has few notes in the top register, all of which allow the beginning of the second

voice to be heard more clearly.

The beginning of the canon is subdivided by three statements of motive a: first in A

major, then F# major (mm. 18—19), and D major (mm. 27—28). The statements of the

motive in A and D are the most similar, with each subsection stating the a, b, and c

motives in that order. The dominant note, E, is prominent throughout section 1, and is

traced in a slow chromatic descent (F#—Fl+-E) in the deepest bass register prior to the

statement of the F# major triad as well as being a melodic goal in mm. 38—52. The half-

step motive c is twice used in this section in both ascending and descending forms to lead

210

 
 
 
 
 
 
 
Figure 4-24 (continued on next page). Section 1 (mm. 1-74) of Study No. 32’s canon melody in
first (lowest) voice, identifying motives a through f, division into three registers, and location of
entrance of second voice.

211

 
 
 
 
  
 
 
 
  
   
  
Figure 4-24 (cont’d.)

to the same pitch in different registers: at mm. 13—14 and 16 to lead first up and then

down to the pitch E, and at m. 4648 and 53—55 to lead first up and then down to the

pitch C. The pitch E also represents the tonal center of an extended passage from measure

38 to measure 52.

Thomas (1996) identifies the clear registral separation in this canon as contributing

to its extreme sense of multidimensionality: “internal registral gaps can make an osten-

sibly single line sound like more than one line, thereby extending and possibly also con-

fusing a work’s multidimensionality” (p. 84). This extreme sense of registral separation

212

 
 
  
 
 
also occurred in Canon 7 of Study No. 27 (see Figure 3-53, p. 175), where canonic lines

are separated by five octaves and compound melody is also established.

S_e§ti2n_2

Several motives (a, b2, and e) rather conspicuously drop out in the canon’s second

section (77 beats, mm. 76—107), which is shown in Figure 4-25. Section 1’s upper

register is also not present in this section, and the fastest rhythmic movement is by dotted

half note; thus the quarter-note beat division is entirely absent. Gann describes this as the

“get out of the way section” (p. 183), and that is an apt description of the effect of its

condensed low register, elongated rhythmic movement, and motivic restriction. Section 2

is framed on both ends by motive d (the only occurrences of this motive in this section),

and the section’s five phrases each end with a staccato quarter note followed by a rest.

    9_—_:_jlll“fi1b._2.;'_;;“ 3. J

LL.

f

—‘——*“

Figure 4-25. Canon melody in section 2 (mm. 76—107).

8Vb

213

  
 
_ ,__:,. ‘9; I?!

ab

” Ellj':it§f§ {:31"i iii

wI?!._“9'::li_.:if'
‘ gas:EnE—ufi1;:~

   it}: 13;": Li ’i:

' iii— "2:9:E:tifijgagfl

(displaced by octave due to range restriction)

2 :4 3- [I iii 1*:_ 18 1

C

Figure 4—26 (continued on next page). Canon melody in section 3 (mm. 109-95) showing registral
separation on three staves.

214

 
 
 
 
   
 
 
 
 
  
Figure 4-26 (cont’d.)

172

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188

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Section 3

Section 3, the longest of the sections (190 beats, mm. 109—95), is shown in Figure

4-26. The separation into three registers returns, and all seven motives are used again,

and rhythmic augmentation is a significant feature of this section, particularly in those

motives that in section 1 were based on the quarter note division. The restatement of

motive a in dotted half notes at m. 156 establishes a particularly significant harmonic

goal.

Section 3 concludes with two statements of e, one ascending and one descending,

that appear in rhythmically augmented form a few measures before the stacks are stated.

The two statements rotate about the note C#z first D—C#—B in mm. 176—77, followed by

C#—D#—E in mm. 182—84. This section actually begins at m. 172 with the note B0 initi-

ating a 4-octave leap up to an interrupted ascending statement on e on B4; there is then a

215

    
 
      
  
   
 
 
4-octave leap back down from D5 to D1 and an inverted statement of e. The C#, in m. 181

initiates another ascending 4-octave leap into the ascending statement of e on C#; the C#3

beginning in m. 184 bisects the 4-octave leap.

Effect of All Voices Together

Thomas (1996) points to this piece as exhibiting extreme multidimensionality in

two ways: first, in the separation of the canonic voice into three registral layers which,

when all four voices are sounding, obscures the canon; and second, the variety of

rhythmic durations that, when combined with the tempo ratio 5:6:7:8, creates a variety of

closely-related durations. Table 4-2 shows the rhythmic durations cited by Thomas

(1996, p. 245); Thomas points out the duplication of the tempo value 204 (which is also

related to tempos 102 and 306) and the closeness among values such as 119, 127.5, and

136, and between 170 and 178.5.

Table 4-2

Relative Tempos of Durations Occurring in Study No. 32
(after Thomas 1996, 245)

MiG—OI
M1192
8
7
6
5

J.
J.: 136
J: 119
J: 102
J. : 85

,

Rhythmic Duration
J.
J.:272
J.:238
J.:204
J. : 170

J
- J:204
J: 178.5
J: 153
J: 127.5

--

J
J:4os
J:357
L306"
J: 255

Nancarrow seems to have written the canon so that convergence of tonal areas

occurred to varying degrees at entrances of later voices. Figure 4-27 shows the entrance

of voices 2, 3, and 4. At the entrance of voice 2 (Figure 4-27a), an area of A major (of

which the first triad in voice 2 is V) is maintained for several measures; at the entrance of

216

 
 
 
 
  
 
 
 
 
  
I
-
a
r
I

voice 3 (Figure 4-27b), a brief period of B major takes place; and at the entrance of voice

4 (Figure 4-27c), the key area is alternately B minor and major.

J_:/5'2J

.1.

(a)

(b)

(C)

(2 Ti:

(6»

i

Figure 4-27. Tonal areas represented at entrances of new voices: (a) entrance of voice 2 (p. 1), A
maJOY; (b) entrance of voice 3 (p. 2), B major; and (c) entrance of voice 4 (p. 3), B minor/major.

There is another extended area of tonal convergence involving all four voices, and

this occurs on pp. 8—9 of the score, where first a section of predominantly F major (first

217

 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
   
 
 
 
 
 
 
 
 
  
 
 
 
 
 
   
 
   
 
 
two systems of p. 8) is followed by a section of G major which begins with a statement of

motive a on a C major triad in the bottom voice, followed immediately by the same triad

in rhythmically augmented form. There is a remarkable degree of tonal homogeneity in

the G major section, broken only by an El. in the bottom voice and a C# in the second

voice. This section, placed so near the end of the piece, is quite prominent due to the

length of time it stays in the same tonal area and because of the augmented statement of

a. It is also significant, I believe, that it is approximately at the first augmented statement

of a (measure 156) in the lowest voice that the listener really begins to ascertain the echo

distance between the voices and anticipate the approaching convergence point. The pro-

file of the augmented a statement is so prominent that it is recognized immediately as it is

restated at progressively shorter time intervals in succeeding voices. The fact that the

rhythmically augmented major triad is immediately preceded by the same triad in the

original note values also serves to heighten the listener’s awareness of the augmented

statement.

The rhythmically-augmented statement of a in the passage shown in Figure 4-28 is

the beginning of the arrival at Study No. 32’s most significant tonal goal. Figure 4—29

shows the piece’s large-scale movement between tonal centers, from the A major triad

expressed in motive a at the opening to the rhythmically—augmented A major triad two

octaves higher at m. 156 in the top voice. With this movement, the piece completes a fifth

stack below A to complement the fifth stack above A represented by the interval of

imitation between the voices. The figure identifies the two major key areas established by

the statements of a in the entrance of the four voices (A major in the first two voices, and

B major in the last two), and how these two key areas plus the C major triad at m. 156 in

the bottom voice project motive e. The figure also shows how the movement in the open-

ing two key areas from the tonic tn'ad to the triad statement in the subdominant is

answered in key areas G and A at the end of this large section by movement from the

subdominant to the tonic.

218

 
 
 
  
  >

Figure 4-28. Bottom system of p. 8 and top system of p. 9 from score of Study No. 32, showing
extended section of G major beginning in bottom voice with restatement of a.

The piece concludes with two fifth stacks: one sustained for 23 dotted half notes

and built on C#3 in the lowest voice (beginning at m. 184), and the other a dotted quarter

note in m. 190 (six measures from the end) built on B0. In all four voices, the two stacks

together spell out the hexachord B-F#—C#—G#—D#—A#, and the roots of these two stacks

are actually based in the two rhythmically-augmented statements of e that conclude the

canon.

Finally, the organicity of the motives in Study No. 32 (especially motives b and e,

which both cover the interval of a minor third, along with a, which contains a minor

third) and the fact that motives like b, c, andf involve inversion of motion (i.e., ascending

for descending or vice versa) rather than inversion of interval, allow for effects such as

219

 
 
 
 
 
a is displaced by octave
due to range restriction

motive e

Figure 4-29. Tonal area movement in Study No. 32, from beginning of piece to bottom of p. 9.
Occurrences of motive a (descending major triad) from sections 1 and 3 are plotted (open notes
represent rhythmic augmentation) and key areas related to an enlarged expression of motive e
(ascending “partitioned minor thir ”). (Proportions are approximate.)

voice exchanges to occur with some frequency. Figure 4-30 shows several examples:

example (a) shows two instances of the same motive inverted and rotating about the same

pitch, and examples (b) and (c) are voice exchanges involving the minor third interval.

With its monophonic canon, highly organic motivic content, highly regular metric

notation, and consonant level of imitation between voices, Study No. 32 seems at first to

be rather unadventurous. However, Nancarrow injects the piece with several elements

that introduce different layers of complexity: the widely separated registral layers within

220

 
 
 
 
  
 
    
  
   
  
 
 
 
  
 
   
 
 
 
 
 
   
 
the canon, and the conflicting (and sometimes changing) beat division of quarter note and

dotted quarter note, which—when combined with the tempo ratio of 5:6:7:8—create a

carefully modulated texture where the distinctness of the individual elements is blurred to

varying degrees. As Thomas (1996) notes, once all four voices are active the precise

identity of any particular layer is extremely difficult to ascertain, and Tenney (notes to

Wergo Vols. I/II) speaks of a “perceptual realignment between phrases” (p. 15).

(
61' TTTTT}

b

(a)

(b)

(C)

.J

\—.

l»)-

; ~

Figure 4-30. Motivic combinations using contrary motion: (a) motive b rotating about pitch Dill/E;
and motive e rotating about pitch A (p. 3, top system); (b) voice exchange involving motives a
and b (p. 3, bottom system, top two voices); and (c) voice exchange involving motive b in
contrary motion (p. 4, middle system, bottom two voices).

Ultimately, however, the strong profile of the a motive anchors the listener through

this morass, and its reappearance in rhythmically-augmented form (particularly after its

absence in section 2) at a point near the end of the canon where the echo distance begins

to be clearly heard begins to impart a welcome clarity to the piece. The moment of clarity

notwithstanding, once A major seems to be affirmed with the augmented statement of a

in the top voice, the piece ends without tonal or rhythmic closure: the final two stacks of

221

 
  
 
 
      
  
 
open fifths a step apart shift the piece to a B major tonal area, and the last note attack

occurs five measures before the inaudible CP. It is a curious twist, allowing the piece’s

tonal momentum to be overstepped while its rhythmic momentum stops short of com-

pletion.

1
2
3
.
5
-
r
e
r
a
r

222

 
 
CHAPTER 5

ANALYSIS OF STUDIES No. 25 AND 45

This chapter will present analyses of two of Nancarrow’s more substantial compo-

sitions that employ a variety of compositional techniques: Studies No. 25 and 45. This

examination will particularly seek to reveal “connecting threads” between these and other

Studies, with the analysis of Study No. 25 focusing on its use of a twelve-tone row, palin-

dromic features, underlying rhythmic pulse, use of the layering technique, and connec-

tions to Study No. 27. The analysis of No. 45, meanwhile, will concentrate on drawing

comparisons between it and a much earlier Study, No. 3.

Chronology

Before considering the analyses of Studies No. 25 and 45, it is helpful to make a

few remarks about the chronology of the Studies. It is not possible to pinpoint even the

year of composition for most of Nancarrow’s Studies, especially the earlier ones. Accord-

ing to Gann (p. 68), Nancarrow refused to put dates on any of his Studies, and by the time

he was interviewed about specific works it was often impossible for him to provide much

detail about date of composition. That, plus Nancarrow’s isolation in Mexico and the fact

that many of his pieces were not published until years after their composition, makes the

construction of a chronology for the Studies difficult to do with any accuracy or

precision.

Gann constructs a chronology of the composition of Nancarrow’s Studies based on

the best information available to him, including interviews with the composer and inspec-

tion of piano rolls and punching scores. His chronology, in part, is as shown in Table 5-1

223

(compiled from Gann 1995, 69 and “Conlon Nancarrow: List of Works”).

Table 5-1

Gann’s Chronology of Composition of Nancarrow’s Works

Studies

Year completed

Year begun

Year published

3
2a
2b
1
4—30

1948
late 19403
ca. 1950
by 1951
1948—1960

31, 32, 33, 37

1965—1969

1983
1984

1951 in New Music
No. 25 in 1975; Nos. 8, 19,
23, and 27 in 1977; Nos. 4,
5, 6, 7, 9, 10, 11, 12, 14,
15, l6, 17, 18, 20, 21, 24,
and 26 in 1984
No. 31 in 1977; No. 37 in
1982; No. 33 in 1984

34, 35, 36

1965—1969 —

34, 35, 36, 40, 41

1969—1977

43, 48
42—50

1977—1988

Nos. 35, 36, and 40 in
1977; No.41 in 1981

1969—1977 —

Note: Nos. 38 and 39 were renumbered as Nos. 43 and 48, respectively.

From this chronology, one can make the following observations. Study No. 1, the

first of Nancarrow’s published Studies, was not the first to be composed; that distinction

belongs to Study No. 3, the “Boogie-Woogie Suite.” Also, the chronology reflects a five-

year hiatus from composing from about 1960 to 1965; Gann refers to this time period

when he characterizes Study No. 31 as Nancarrow’s attempt to “get the old compositional

pitching arm back in shape” (p. 129). Further, it can be seen that there is a considerable

lag time between composition dates and publication, particularly for the Studies written

in the late 19408. Finally, a number of Studies remain unpublished, including 13 and 30

(both of which Nancarrow was unsatisfied with and “withdrew” the scores), 32, 34, and

42—50.

224

 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
Analysis of Study No. 25

Study No. 25 conveniently marks the approximate halfway point in the body of

Nancarrow’s player piano compositions, and its description by Gann as a “treasure chest

of every type of idea Nancarrow has worked with” (p. 241) highlights why it makes a

suitable subject for study in this chapter. Duckworth (1995, p. 48) described the piece in

an interview with Nancarrow as “the most spectacular of those early pieces,” to which

Nancarrow replied, “Well, it is sort of dramatic. But there’s nothing special; it’s just a lot

of flurry.” AS shown in Table 5-1, Study No. 25 was only the second of the Studies to be

published, appearing in print 24 years after the publication of “Rhythm Study No. 1.”

There are several characteristics that distinguish Study No. 25. It is the only one of

Nancarrow’s Studies that uses a twelve-tone row in the traditional sense, and it is the only

Study in which the player piano’s sustain pedal is used. The row matrix, as identified by

Gann (p. 242), is shown in Table 5-2; forms of the row that appear in this piece are

shaded in the table.

Study No. 25 is written entirely in Nancarrow’s typical style of proportional nota-

tion, with sustained notes written as quarter notes with thick extension lines and staccato

notes as eighth notes. The piece’s structure, which is shown in Figure 5-1, is based on

five layers of distinct types of figures that drop in and out of the texture. Figure 5-1

shows the piece divided into nine sectionsI (proportionally represented in the figure)

based on textural changes. The figure also describes the following activity taking place in

Study No. 25: appearances of the five musical layers; appearances of the twelve-tone row

beginning on D#/E1, and its transformations; and various tempo pulses that occur and the

ratios between layers expressing different tempos.

Gann drvrdes No. 25 into eight sections, combining what I have identified as sections 7 and 8 into a Single
section; I felt that the introduction of the acceleration technique and changes in the pitch content of the
chords layer at page 60 merited a separate section designation.

225

Table 5-2

12-tone Row Matrix for Study No. 25 (Gann 1995, 242)

 10
I1
A#
E

110

111

D#

G#

C#

F#

P0

I9

I4

18

13

15

17

I6

12

G

A

D

C

B

F

P2

P9

P6

P1

Pl 1

P10

P8

P3

P4

P7

P5

F

C

A

E

D

C#

B

F#

G

A#

G#

D#

B},

G#

D#

G

D

C

B

A

E

F

G#

F#

C

G

F

E

D

A

B1

C#

B

B

F#

D#

B;

A},

G

F

C

D1

E

D

E

G#

D#

C#

C

B1,

F

G;

A

G

F#

C#

A#

F

D#

D

C

G

A;

B

A

G

D

B

F#

E

D#

C#

A1,

A

C

B},

A

E

C#

G#

F#

F

D#

B1,

B

D

C

D

A

F#

C#

B

A#

G#

D#

E

G

F

C#

G#

F

C

By

A

G

D

D#

F#

E

B1,

F

D

A

G

F#

E

B

C

D#

C#

C

G

E

B

A

G#

F#

C#

D

F

D#

Nancarrow builds Study No. 25 around forms of the row that begin or end with

D#/EJ,, most notably P0/R0 and P5, and this pitch is the goal reached at the end of sec-

tions 1, 2, 4, and 5. Also, by ending the row with A#/B{,, Nancarrow sets up a possibility

of a V—1 cadence when P0 is immediately restated, as it is a total of seven times in the

first section (pp. l—7). At the first restatement of P0 in section 1 this V—I formula is

already quite evident. Figure 5-2 shows the second page of the score, where the first

restatement of P0 occurs; V-I is clearly stated and reinforced by the bottom notes of the

arpeggios in the top layer. Further, the appearance in the row of G#/A; prior to A#/Bl,

adds a suggestion of IV, although this pitch is not reinforced by its accompanying

arpeggio

Gann points out the major second interval on both ends of the row, which allows

Nancarrow to overlap some row statements in section 2 by two notes. The row‘s two

opening pitches, D#—C#, later become the level of imitation Stated in the two-voice

226

 
 
 
  
9
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1 I1
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fundamental

f
l

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i
fundamental |

I

T
l

    T

  T

(IV) ------- v ------------------------ r

{H «I

l g“:

Figure 5-2. Study No. 25, top system of p. 2 from score, showing (IV)—V—I cadence at second
statement of P0 in bottom voice. Vertical dotted lines indicate arpeggios articulating breaks in the
sustained notes of bottom voice. Figure a is a 16—note arpeggio based on the overtone series,
while figure a’ is a 15-note arpeggio in which the fundamental of the overtone series is stated in
the sustained bottom voice.

tempo canon of section 4, and row P10, which is used throughout section 8, begins with

the second pitch of this important structural interval, C#.

The score segment shown in Figure 5-2 also illustrates several other features of this

section of the work: an arpeggio figure a/a’ (a being a 16-note arpeggio stating the first

sixteen harrrronics of the bottom pitch, while a’ is a similar lS-note arpeggio in which the

fundamental is stated in the sustained voice and harmonics 2 through 16 are stated in the

arpeggio), which is prominent throughout the piece; the use of scalar glissandos (here a B

major scale) within the arpeggio layer; and the use of arpeggios and glissandos in the top

layer to punctuate breaks in the sustained line in the bottom layer. Later in the piece’s

first section, the 16-note overtone series arpeggio figure is divided so that the top note is

in the sustained layer.

Nancarrow’s most prominent uses of the layering technique prior to Study No. 25

were in Nos. 3a and 5, both of which were examples of cumulative layering where new

layers are continually added to the musical texture and sustained to the end of the piece.

Study No. 25 represents a different type of layering—one in which, as Shown in Figure 5-

228

 
 
 
   
 
 
 
 
    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
   
   
l, the layers intermittently drop in and out of the texture. Another difference is that, while

the layers in previous Studies tended to be registrally distinct (a concept carried to the

extreme in No. 5, in which the entire keyboard is distributed among thirteen layers with

no pitch overlap), there is considerable registral overlap and changing of registers among

the layers in No. 25.

All of the layers in No. 25 are established by the third section. As Figure 5-1 Shows,

the arpeggiated layer is the only one appearing throughout the Study, and in sections 6

and 9 it is the only layer (although in section 6 there are four different arpeggiated layers

forming a 9:10:12:15 tempo canon). Tenney comments about the arpeggio/glissando

layer (liner notes to Vols. III/IV, p. 15) that “Fast arpeggios and/or glissandos had pre-

viously been used in a similar way only in Studies #5 and 12.”

The arpeggio/glissando layer contains the following kinds of figures: the 15- and

l6-note harmonic overtone arpeggios (figures 3 and a’ Shown in Figure 5-2, and there is

also a similar lS-note figure later in section 1 in which the fundamental is present in the

arpeggio and the sixteenth harmonic appears in the sustained layer), always ascending

through section 3, changing to descending figures in section 4, then a mixture of both

ascending and descending; major scale glissandos, both ascending and descending, of up

to 3y2 octaves in length; arpeggios of major triads and seventh chords, often in combi-

nations; and what Gann calls a “simultaneous glissando” (p. 243) figure, a lengthy glis-

sando combining a major scale going in one direction with the Locrian mode in the other

direction (see Figure 5-3). This layer must be what Thomas was referring to when she

spoke of this Study as having a high “concentration of tonal figures in a non-tonal

setting” (Thomas 1996, 17).

The remaining non-arpeggio layers are nearly constantly stating the row (or por-

tions of it) in one way or another, with the exception of the staccato notes layer in section

2. The layers of sustained single pitches and sustained chords are Similar in function and

presentation, with the layer of single pitches usually progressing to chords. In the

229

 
 
sustained and staccato chords layers, the row is embedded in a number of different chord

structures, including triads and dominant seventh chords (section 1), 8-note stacks of

19
\—

§?__é

thirds (section 2), lO-note chromatic chords (section 3), 7-note chords containing all the

   .

pitches of the Lydian mode (sections 5 and 7), and pairs of unrelated triads (section 8).

‘1   ‘ "'t

Peal.

(d)

(6)

‘|—~.—_._ __.—.._,

a;

Figure 5-3. Figures found in the arpeggio/glissando layer, Study No. 25: (a) figure combining two
different triads (score, p. 18, top system); (b) figure of seven different consecutive triads (p. 34,
bottom system); (c) figure of alternating dominant and major seventh chords (p. 40, bottom
system); (d) simultaneous ascending major and descending Locrian glissando (p. 36, bottom
system); and (e) longer version of same figure (p. 60, top system). All of the examples use the
“exploded drawing” technique where the bracket indicates where on the home Staff the figure
belongs (bracket and home staff not shown in [e]).

Sflcgre

Gann comments that “Never before or since [No. 25] has Nancarrow used so many

devices in a single movement, and yet the piece holds together like a rock through the

230

  
 
  
 
 
 
 
 
 
 
   
 
 
 
 
 
 
 
 
 
 
 
 
audacity of its timbral gesture” (p. 242). It is, however, more than timbral gesture that

holds No. 25 together. Other characteristics that serve to unify the piece include the use

of the tone row and a sense of an underlying pulse.

The nature of 12-tone composition, with its row that can be used forward and back-

wards, is conducive to the incorporation of palindromic elements, and indeed Study No.

25 contains significant palindromic organization as a result of the presence of the row.

Thomas comments on the significance of palindromes and retrograde in Study No. 25:

Retrograde can also play an important part in the construction of the
studies, as in Study No. 25, for example, which, as the only study using a 12-
tone row, relies more heavily on serial techniques than most. The row
accounts for some but not all of the material in the study, however. When it is
present it is subjected to various transpositions and retrogrades, and the retro-
grades of the row are generally rotated, as well. Some of the other material in
the study is organized palindromically, with non-retrogradable pitch-class
patterns. It is more characteristic to find, as in many of the studies, only small
bits of material subjected to retrograde. (Thomas 1996, 15)

Thomas understates the prevalence of palindromic organization in Study No. 25.

Section 4 is entirely a retrograde of section 1, with the exceptions that the sustained layer

is reduced to single notes rather than the successively larger chord structures found in

section 1, there are no breaks in the sustained layer in section 4 as there are in section 1,

and the arpeggios are stated in section 4 at half the tempo of section 1; these charac-

teristics can be seen in Figure 5-4, which shows the relationship of the beginning of

section 1 to the end of section 4. The reduction of the sustained layer to single notes in

section 4 is a practical matter, since this section is a tempo canon at the major second

below in a very low register (the closest level of imitation ever used by Nancarrow, and

the only one in which the echoing voice is lower than the original voice) and using

section 1’s full chord structures at that close interval would result in a very muddy

texture. The relationship of arpeggios to sustained notes is maintained in section 4, how-

ever, so that arpeggio figures that were punctuating breaks in the sustained line are now

Stated during a sustained note; compare, for example, the placement of the F# arpeggio

and G major scale figures in Figure 5-4. The other effect of the palindrome is that the

231

 
 
Mom-4A - M- -

A# missingis “f“-   

score error:

——

———

————

———I 

l
(:1)pr
.
I 

.-'—i

r

T

1

2

I
j

3

(a)

(b)

Figure 5-4. Palindromic organization of sections 1 and 4: (a) opening of Study No.25 (p. 1, top
system); (b) end of section 4 (pp. 32—33). Arpeggios in section 4 are stated at half the tempo of
section 1. Location of convergence point in section 4’s tempo canon is shown at the beginning of
first system in (b). Notice that arpeggios in (b) are stated at half the tempo of thosein (a).

presentation of row R0 in section 4 instead of P0 eliminates a cadence formula such as

section 1’s IV—V—I (G#—A#-D#) from the end of P0 to the beginning of its restatement.

Another significant example of palindromic structure occurs in section 3, which

contains a layer of lO-note block chords and a layer of arpeggios that contain two

232

 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
'

.L.

-

different major triads, first descending then ascending. The arpeggio layer contains a

palindrome in the intervals between the two triads. Two forms of the row are presented

across both the falling and rising triads, and the two different triads in each figure are

determined according to the pattern shown in Table 5-3.

Table 5-3

Derivation of Palindrorrric Interval Pattern Between Arpeggio Triads in Section 3

Row‘o
P7

Falling
triads:

Rising
triads:

Row:
R11

Row:
R8

Falling
triads:
Rising
triads:

Row:
P0

B1,

D

11

0

G1,

D

L

1

1

A1,

A

0

1

E

F#

2

2

D;

G

1

2

G

A

3

3

E

BI,

2

3

4

A

B

3

4

5

B

F#

4

5

A;

El,

D1,

D

4

E

5

F

6

6

C

E

5

6

C

G

7

7

D

E1

6

7

B1,

C

8

8 91011

5

G

D1,

7

8

F

B

9

G,

A;

8

9

D

A

*

10

EL.

F

9

10

A

BI,

11

'
F :

b
d
-
-
-

C

10

11

B

EL

0

Notes: Dotted line indicates line of symmetry between first and second halves of palindrome;
shaded area indicates where missing accidentals in the score (system 44) obscure the relationship
between the triads.

*According to the row, this pitch should be A1,. Presumably this is a score error.

The reflexive use of both a prime form of the row and a retrograde in the two con-

current patterns (falling and rising triads) and the migration of both patterns up a half Step

in the second half of the palindrome yields the following progression of intervals between

the triads for the first half of the progression, which is mirrored in the second half:

M3 m2 T T M2 P4/5 M3 m2 T M2 M2 P4/5

Note how Nancarrow rotates the rows for the rising triads (the last in each arpeggio)

so that the E, triad of row P0 ends the section.

233

 
 
 
 
 
 
 
 
 
Yet another palindromic structure occurs in section 6, where on p. 45 a fifth voice

joins the four-voice tempo canon, stating ascending 20-note arpeggios of major triads.

The triads, whose roots span the tritone from Er, to A, are stated in this order:

(E1F#A)FE1,A1FGF#E|E1A|EF#GFA1,E1,F

The palindrome begins with the fourth triad and breaks off for two chords after the first

seven triads of the palindrome, before restating the fourth through tenth triads in reverse

order.

Rhythm and Pulse

Gann cites the “audacity of timbral gesture” as primarily holding Study No. 25 to-

gether, but the feeling of a strong underlying metric pulse is also a critical unifying

element in a piece such as No. 25, where such seemingly unrelated and uneven rhythmic

gestures are juxtaposed between and within sections. The most consistent rhythmic ele-

ment throughout the piece is the speed of the notes in the arpeggios, which are stated at

three related speeds: the fastest speed, a blistering tempo of about 10,560 (176 notes/

second), is used in section 1; the arpeggios in section 2 are one-fourth this tempo, or

about 2640; and the remainder of the piece is essentially at half the original tempo, or

5280. Even in section 8, where both the sustained and arpeggio/glissando layers are

accelerating, the 5280 tempo is maintained within the arpeggios although the distance

between their onsets iS decreasing. There is one lengthy glissando in section 7 (p. 54) that

returns to the fastest (10,560) tempo, but otherwise the 5280 tempo is maintained in the

arpeggios from section 3 to the end, and this is the tempo of section 9’s monstrous final

arpeggio/glissando figure.

Figure 5-1 identifies the various tempos that appear in Study No. 25 and in which

sections they appear. Only sections 2, 3, 5, and 6 contain discernible tempos, with the

approximate tempos shown in Table 5-4 represented. AS can be seen in the table, there is

one tempo continuation between sections 2 and 3 but otherwise little clear relationship

234

 
among tempos appearing in different sections; the common tempo, 82, is established at

the beginning of section 2 and then changes to one-half that speed, 41, before both those

values are used in a mixed context in section 3 (essentially J and J values at J = 82).

One other tempo related to tempos 41 and 82 is 123 in section 6.

Table 5-4

Approximate Tempos Represented in Study No. 25

Sec. 2

Sec. 3

Sec. 5

Sec. 6

36

41

66

\ 5 8

l

69

\ 92

103
\‘123

152

201

The variety of note durations in some sections of this piece almost defies any form

of systematization, with durations indeed seeming to be random as Gann suggests (p.

242). In section 7, however, all the note durations tend to coalesce into a limited number

of duration clusters as shown in Table 5-5; because of variability in score measurements,

this leaves open the possibility of durations being derived from multiples of a common

background unit. This possibility seems all the more likely since one of the duration

clusters (96—98) appears in both layers, indicated by bold type in the table. In the sus-

tained layer, durations are approximately divisible by 4, but in the arpeggio layer any

background unit is indiscemible and likely very small (perhaps as small as a single notch

on the punching mechanism?)

235

 
 
 
 
 
 
 
 
 
Table 5-5

Clusters of Durations in Section 7

Duration measurements
in millimeters

Possible
background unit?

 Sustained layer:

9 durations:

Arpeggio layer:

7 durations:

"

_

T

_

7/8
1 1/1 1.5/ 12
15/16/16.5
22.5/23
28/28.5/29
36/37
50/51/51.5/52
67/68/69
96/97/97.5/98

21/21.5
30/30.5
38.5/39/39.5
63/63.5/64
81.5/82/82.5/83
96/97
1205/1215

Sectional

haracteristics in Stud No.25

2
3
4
6?
7
9
13
17
24

division of 4?

_J

'—

.. 

???

Section divisions in No. 25 are clearly demarcated by the diversity of the textural

layers, and sometimes by cadences supported by the tone row. There are also three

obvious pauses in the work, at the end of sections 1, 3, and 4 as shown in Figure 5-1. The

diversity of the combinations of the layers allows each of the nine sections to be dis-

tinctly different. Each section will be briefly described below.

Section 1 (p. 1 through system 13, top of p. 7) establishes immediately the contrast

of the arpeggio/glissando layer against the sustained notes layer (see Figure 5-4a). As

Gann notes (p. 242), the sustained layer states P0 three times as sustained notes, twice as

the root of root position triads gradually rising in register, and twice as the root of domi-

nant seventh chords (although three of the chords at the bottom of p. 6 appear to be

236

 
 
 
 
 
 
 
 
  
missing the requisite notes—see Appendix A).

Through most of the section, the arpeggio/glissando figures punctuate a break in the

sustained layer, although these figures appear elsewhere as well. The tone row does not

appear to be operative in the arpeggio/glissando layer, which is only loosely related har-

monically to the sustained layer. The punctuating arpeggios and glissandos are more

likely to be harmonically related to the notes they reinforce in the sustained layer (see, for

example, Figure 5-2). A number of the overtone series arpeggios are of the 15-note

variety where either the fundamental or the highest harmonic is not stated in the arpeggio

but in the sustained layer (these are marked a’ in Figure 5-2 and in Figure 5-5, below).

By the conclusion of section 1, a considerable level of harmonic coordination between

the two layers is achieved and the section concludes on a strong V-I cadence to E1, as

Shown in Figure 5-5. There is a 1.4" pause between sections 1 and 2.

.t
«  
1 ’----,

V

IV """ V """" I

1.4 second pause

a:

Figure 5—5. End of section 1, Study No. 25 (score, p. 7, top system). Arpeggio/glissando and
sustained notes layers achieve a level of harmonic coordination prior to and at the cadence.

Section 2 (system 14, bottom of p. 7 to middle of system 35, p. 16) presents an

237

 
  
 
 
 
 
I

:
W
E
A
'
F
I

immediate contrast to section 1 in its change of dynamic level from 13‘“ to pp and in its

statement of arpeggios at one-fourth the speed of the first section. The sense of D#/E as a

tonal goal is also completely dissipated. The first layer to enter is a 4-note major triad

staccato arpeggio pulsing at the 82 tempo in the bass; its pitch pattern seems to be

random and includes numerous repeated triads. This layer is soon joined by an 8-note

chain of thirds that is sustained for random durations; Gann points out the presentation of

row forms P5, RIS, R0 and 110 in the roots of these arpeggios. Shortly thereafter a layer

of broken octaves, characterized by Gann as an “erratic stride bass” (p. 245) enters in the

deep bass; the rhythm here appears to be truly random: the ascending order of the broken

octaves never changes, but the distance between the notes of the broken octaves varies

widely from about 2 to 63 millimeters. The section concludes with the sustained arpeggio

layer stating the 41 tempo, while the staccato layer (now expanded to 7-note major triads)

states the pulse of 66, a roughly 5:8 ratio. The statement of row P5 at the end of the

sustained layer allows the section to end on a third stack built on E1.

The two layers in section 3 (middle of system 35, p. 16 to middle of system 46, p.

21) form a rhythmic canon relating by the tempo ratio of 7:10. The top layer expresses

two major triads in each 20-note arpeggio (see Figure 5-3a), stating the row according to

the palindromic pattern shown in Table 53; this layer’s tempo is about 58, with the ar-

peggios expressing the equivalent of quarter and half note durations. The bottom layer of

staccato chords also expresses quarter and half note durations, but in the tempo 82 that

carries over from section 2. Gann analyzes the widely-spaced chords in this layer as

containing “every note of the chromatic scale except for those a minor third and a minor

tenth above the bass” (p. 245), although the missing pitches are actually a minor third and

minor seventh above the bass. The three statements of the row in this layer are found first

in the lowest pitch, then in the second lowest, and finally in the third lowest. About every

two systems, the dynamic level increases by one marking, effecting a crescendo across

the section from pp to fi.

238

There are two convergences between the two layers in this section: one at the

bottom of p. 17, and the other at the bottom of p. 18 (see Figure 5-6). A representation of

the entire rhythmic structure of section 3 is shown in Figure 5-7; as the figure shows, in

addition to the two convergences noted, the final two possible convergences are sub-

verted, as is the first one, which occurs at the third staccato chord.

(P7) ,5:

   r-

2"

9

"1

R11)
8

9’

10

J
(underlying rhythm @ J = 82)

convergence

Figure 5-6. Score segment from section 3 (p. 18, bottom system), showing the second of two
convergences between arpeggio and staccato chord layers.

Section 4 (middle of system 46, p. 21 to middle of system 59, p. 33) is a two-voice

converging-diverging tempo canon at the ratio 21:25. The prime row’s opening dyad,

D#—C#, defines the pitch levels of the two voices of this tempo canon.

As noted earlier, the lower voice is a complete palindrome of section 1 except that

its sustained layer remains in single notes rather than becoming progressively thicker

chords (such as those shown in Figure 5-5) and this section’s arpeggios are stated at half

the speed of section 1’s. There is an interesting moment in the canon where one falsely

239

  
  
w

senses a change in the leader-follower relationship between the voices; as Shown in

Figure 5-8, the same BI, major arpeggio appears first in the upper voice at a place where

the lower voice is still leading.

2: 5'8-

(24 Eventfdt/ ,

i

I

I

l J 61

I61

I

l

*

(p. 17.2)

fig:
I

..I

J

(36 '_'ej-'ents")’£ i l

v

l

I

l

I

l

l

l

l

1

l

I

I

1.1.4.5551_J__LL.J_1_._1_._Iw

(12./8.2)

I

x.

I

WV

.

* = subverted convergence

Figure 5-7. Rhythmic scheme underlying section 3; convergence period is 7 beats in top layer
(arpeggios) against 10 beats in bottom layer (staccato chords). Only two of five possible con-
vergences are articulated in both voices.

The canon’s CP (shown in Figure 5-4b) barely misses being articulated in both

voices and occurs very near the end of the section, just prior to the last four notes of the

sustained layer. The restatement in reverse of the opening material from section 1, bring-

ing the bass line back to the opening D#, and the 1 second pause before section 5 estab-

lish the end of section 4 as the close of the first half of the piece.

Section 5 (middle of system 59, p. 33 through first third of system 77, p. 42) intro-

duces a new texture: sustained arpeggio/glissando figures punctuated by staccato chords.

The chords in this case are 7-note chords, Spelling “a Lydian scale built above the root”

(Gann 1995, 245). It Should be noted that the Lydian mode bears a relationship to bar-

monics 8—16 of the overtone series figure.2 As Gann notes, the chords spell out first P11

2Specifically, the top octave of this figure is what is known as the “overtone scale,” and it contains both a
raised fourth (characteristic of the Lydian mode) and a flatted seventh (characteristic of the Mixolydian
mode).

240

  
 
 
 
  
 
 
 
 
 
 
same figure appearing
first in upper voice
although lower voice
is leading

(‘
q -- - -~.

(é- _
I"

1

echo distance

,

l
I

L
r

v

'T—

H A
T

J-

11.

I

A

I
7

TL
J

l
1

I

Figure 5-8. Segment from section 4 (p. 28) where same figure appears in both voices, giving
impression that upper voice is leading although it is not.

in the bass notes, then P3 in the second-lowest notes. The arpeggio/glissando figures in

this section include the most lengthy and elaborate so far, including figures of 210 notes

(pp. 35-36), 99 notes (p. 37), 91 notes (pp. 38-39), and 153 notes (p. 39).

Gann finds that “beneath the [arpeggio/glissando] figures the chords seem to imply

an underlying beat of about 208, a duration of about 13.7 millimeters in the score” (p.

246). This beat is actually established only very briefly near the end of the section (and

my calculation is that it is closer to 201), stating the following rhythmic equivalent:

241

 
 
 
  
  
 
 
 
JJo

JJJJJJ. ..VJ. JJ

The section ends tonally ambiguous with a sustained chromatic sweep of the entire key-

board (at the 10,560 tempo of the arpeggios in section 1) that is held with the pedal for

almost 10 seconds.

Section 6 (middle third of system 77, p. 42 to beginning of system 90, p. 49) is

another tempo canon, this one in four voices at the ratio 9:10:12:15 with the interval of

imitation between all voices being a perfect twelfth. The canonic voice consists of four-

note major triad arpeggios (a texture Similar to the beginning of section 2) stating a

rhythm of durations related 2:1. The canon is converging but, as Gann points out, it ends

before reaching its CP, an idea that Nancarrow would expand to an entire piece in Study

No. 31.

Near the end of the section, the rhythm changes to just short durations, setting up

more frequent possibilities of convergences among the voices. Figure 5-9 shows the last

full page of the section, with convergences marked. Over the four-voice tempo canon a

fifth voice enters stating l9-note arpeggios at a tempo of 69, creating an overall tempo

complex of 27:30:36:40:45 (Gann, p. 246); the arpeggios in this voice consist of major

triads spread over six octaves, and state the nearly palindromic series noted earlier.

The tone row does not appear in this section at all. In each of the five voices, the

arpeggios stated traverse the interval of a tritone as shown in Table 5-6; each pair of

voices in the tempo canon, then, is contiguous in terms of pitch. As shown in Figure 5-9,

this results in quite a few dissonances at points of rhythmic convergence (and elsewhere):

thus, the section ends with a high degree of rhythmic convergence, but not of tonal con-

vergence. Four of the six rhythmic convergences on this page are at the interval of a

major or minor second (reduced from the compound intervals), and there is little tonal

connection between the 19-note arpeggio line and the tempo canon lines (with the excep-

tion of the A1, arpeggio). The section ends on an F major arpeggio in the 19-note voice,

providing closure with its beginning on an F arpeggio in the first tempo canon voice.

242

 
Figure 59 Study No. 25 (p. 48), showing multiple convergences in 9:10:12:15 tempo canon in
section 6. Rhythnric convergences are circled, and dissonant intervals within rhythmic con-
vergences identified. Lines indicate the same major triad Stated in different voices.

Table 5-6

Range of Arpeggios in Each Voice of Section 6, Study No. 25

19-note arpeggios
tempo canon voice 4
tempo canon voice 3
tempo canon voice 2
tempo canon voice 1

(voices in score order)

F1,—A
D—G#
G—C#
C—F#
F—B

Section 7 (beginning of system 90, p. 49 through two-thirds of system 101, p. 60) is

very similar to section 5’s two layers in the elaborateness of its arpeggios/glissandos and

because it restates the 7-note Lydian chords found in the earlier section. As noted earlier

243

  
 
 
 
 
  
 
 
 
 
 
in Table 5-5, the durations in the sustained layer condense into nine duration clusters,

while the arpeggio layer has seven duration clusters. The sustained layer combines both

sustained and staccato chords.

In addition to arpeggiated triads and major scale glissandos, many of the arpeggio

figures in section 7 consist of chains of major seventh chords; see Figure 5-10.

(a)

(b)

Figure 5-10. Arpeggio figures of chains of major seventh chords, section 7: (a) chromatically
rising roots (p. 57); (b) diatonically rising roots (p. 59).

Section 8 (last third of system 101, p. 60 through first third of system 114, p. 73) is

separated from section 7 by the rising C major/descending C Locrian simultaneous glis-

sando figure, and its texture is very similar to section 7. Several changes do take place,

however: staccato chords disappear from the sustained layer, which is no longer stating

the 7-note Lyd'nn chords but two different triads based on the prime/retrograde pairs P0/R0

and P10/R10 (another reference to the D#—C# dyad that dominated section 4); and the

arpeggio layer is now stating 20-note rising arpeggios consisting of alternating major and

dominant seventh chords, e.g., DA7—E7—FA7—AL7—GA7, all of which begin on either D,,

D, or E1, (another projection of the interval D#—C#). The arpeggios alternate through the

three beginning pitches, but each arpeggio contains a different mixture of seventh chords.

The most important change in this section is the introduction of acceleration. Both

layers accelerate at different rates, and both have an underlying rhythm of long and short

244

 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
durations. The layers are in a rhythmic canon, with the second duration of the arpeggio

layer corresponding to the first duration of the sustained layer. Gann identifies the rates

of acceleration as 6.44% in the chords and 3.22% in the arpeggios but does not supply

any supporting measurements or calculations. My own measurements, taken over a

variety of segments in this section, consistently indicate acceleration rates of approx-

imately 4.75% in the chords and 2.25% in the arpeggios. Here are sample measurements

in the chords layer (the multiplication factor for acceleration by 4.75% is 1/1.0475 =

0.9547), where the first duration iS 62.5 mm. (the opening duration is a short duration and

all measurements for calculating acceleration are to short durations):

62.5 mm. to 43 mm. in 8 steps: 62.5 x (0.9547)8 = 43.1
62.5 mm. to 29.5 mm. in 16 steps: 62.5 x (0.9547)'6 = 29.7
62.5 mm. to 11 mm. in 37 steps: 62.5 x (0.9547)37 = 11.2

Measurements in the arpeggio layer are almost as consistent (the multiplication factor for

acceleration by 2.25% is 1/ 1.0225 = 0.9780). The first duration3 is 80.5 mm. (a long dura-

tion—all subsequent measurements are to long durations):

80.5 mm. to 62.5 in 11 steps: 80.5 x (0.9780)ll = 63.0
80.5 mm. to 46 mm. in 24 steps: 80.5 x (09780)“ = 47
80.5 mm. to 24 mm. in 53 steps: 80.5 x (0.9780)53 = 24.8

This section invites comparisons to two other Studies. The accelerative percentage

of 2.25% was, of course, used in Study No. 22 (“Canon 1%/1 J’2 %/2}2%”). The use of

multiple rhythmic values in an accelerative context is an idea that reappears in Study No.

27 (“Canon 5%/6%/8%/11%”), reaching a peak in Canon 9 of No. 27 where four dif-

ferent rhythmic values are used.

Section 2 (middle third of system 114, p. 73 to end), described by Gann as an “ar-

peggiated cacophony” of “continuous single notes” (p. 248), consists of many of the

figures already seen: major scale passages, major triad arpeggios,4 major and dominant

3In the arpeggio layer, all “durations” given are measurements between the first notes of adjacent arpeg—

ios. The tempo within arpeggios does not change.
In this entire Study, I noticed only one minor triad arpeggio: an F# minor arpeggio presented in the first
six notes of section 9. This seems likely to be an error in the score.

245

 
 
 
seventh chords, and broken octaves (a relic from section 2’s staccato notes layer). The

extremes of the player piano keyboard are rapidly juxtaposed in two places as Shown in

Figure 5-11.

     I?“

(a)

(b)

Figure 5-11. Two rapid juxtapositions of keyboard extremes in Study No. 25, section 9: (a) p. 74,
top system; and (b) p. 74, bottom system.

Despite the seeming stream-of-consciousness derivation of the pitch material in this

section, close examination of the pitch content reveals that several lengthy arpeggio!

glissando passages from earlier in the piece are repeated verbatim in section 9. For

example, on p. 35 of the score there is a 210—note arpeggio whose first 165 notes are

repeated verbatim beginning on the top system of p. 76; and on p. 39 there is a 153-note

arpeggio of which about the middle third is almost entirely repeated in the bottom system

of p. 73 (see Figure 5-12).

The 15- and 16-note overtone series figure does not appear in section 9 (in fact, it

last appears in section 5). The other most recognizable arpeggio/glissando figure from the

first eight sections, the simultaneous glissando figure, does not reappear in section 9 until

the very end. Section 9’s final gesture bears a notable resemblance to this simultaneous

glissando figure as it first appears in section 1. The figure from p. 6 (Figure 5-13a) com-

bines the C major scale and the G; pentatonic scales to fill out the chromatic space; the

scales in the first half of the figure are alternated and split between the bass and treble

registers as marked in the figure. The gesture at the very end of the piece (Figure 5-l3b)

246

 
 
 
 
I
l

  omitted in section 9

Page73:
__.._ __. L. ______

_

..

‘31

.. IE4: 5

l

6___-_
'42:}:

Ac6r

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if?

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L

@i

7'._

K

'

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7ij

l

d

Figure 5-12. Lengthy arpeggio/glissando figure from section 5 (p. 39, top of figure) repeated
almost verbatim in section 9 (p. 73, bottom of figure).

is a shorter simultaneous glissando that combines the F# major scale with the G penta-

tonic scale while maintaining a similar split between the registers; the chromatic Space in

this figure is not quite filled as B appears in both scales, and C does not appear in either.

The gesture ends with an F# major triad arpeggio; the last high C# before the arpeggiated

F# major triad creates a V—1 tonal movement to the final high F#.

Progression of Pitch Centers in No. 25

Study No. 25’s nine sections divide fairly clearly into two parts: the palindrome of

section 1 in section 4 closes out the first part, and the second part consists of sections 5

through 9. The most important pitch centers in the piece trace the first three pitches in P0,

247

 
 
 
 
 
 
 
 
 
 
 
   
       
 
  
   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
 
 
 
 
é

keyboard ’3

hest note

---—

- --fi

C#—-F# (V—I)   

’s lowest note

(a)

(b)

G pentatonic

(‘5, -

Figure 5-13. Comparison of alternating major (in rectangles) and pentatonic (in ovals) pitch
patterns in simultaneous glissando figures from (a) p. 6 and (b) pp. 76—77 of Study No. 25.

 r—r—

D#—C#—F#, with pitch centers D# and C# dominating the piece through section 5, and a

V—1 movement from C# to F# taking place in sections 8 and 9. D#, then, assumes a

submediant role, for which pitches 10 and 11 of the row, G# and A#, briefly create a

secondary cadence formula in section 1. Figure 5-14 summarizes important tonal centers

in the piece.

Connections Between Studies No. 25 and 27

Study No. 25 appears to have more in common with Study No. 27, which pre-

sumably was written at most a few years later, than with any of the other Studies. Several

interesting parallels can be drawn. Two of the parallels involve No. 27’s clock line (see

Figure 3-44, p. 162). The first is a sense of rhythmic randomness that appears in the

“erratic stride bass” of the staccato note layer in No. 25’s section 2; No. 27’s clock line

consists of apparently random long and short durations relating by a 2:1 ratio. The

248

 
 
 
 
 
 
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249

   
  
  
 
 
   
 
 
 
 
  
 
  
concept of randomness was already discussed in Chapter 3 (pp. 162—63), particularly as it

applies to binary decisions, and whenever one encounters seemingly random elements in

Nancarrow’s work it seems presumptuous to dismiss these elements as truly random.

Another parallel with the clock line occurs in No. 25’s sections 3, 5, and 7, where

the row is embedded in different positions within different chord structures, much like the

clock line’s four pitches move to different positions within the structures shown in Figure

1-15 (p. 40). In Study No. 27, however, the clock line pitches occupy varying positions

within the chord Structures, whereas in No. 25 the row pitches usually maintain the same

relative position.

The other parallel concerns the geometric acceleration technique employed in No.

25’s section 8, where values in the series are sometimes either halved or doubled to form

two different accelerating rhythmic values (such as J and J). This same technique was

used briefly in Study No. 22, but was brought to full fruition in No. 27 (see Figures 3-45

and 3-55, pp. 165 and 177).

Analysis of Study No. 45

Study No. 45 is a three-movement work that resembles Study No. 3, the “Boogie-

Woogie Suite,” in its use of a boogie-woogie bass line as the bottom layer of a multi-

layered texture that moves through transpositions based on a I—IV-V harmonic pattern.

Since these two “suites” are based on the same premise and they virtually frame Nan-

carrow’s compositional career,5 the two works offer an almost irresistible opportunity to

observe evolution of Nancarrow’s technique and compositional goals over his career.

Also, since there is not a great deal of substantive information that can be added to Kyle

Gann’s thorough documentation of the materials and formal procedures used in Study

No. 45 (Gann 1995, pp. 256—63), the present analysis will be focused less on that task

5N0. 3, the first piece Nancarrow composed, was written by 1948, and No. 45 was completed between 1977
and 1984, when it was performed in Los Angeles (Gann, p. 48); see Table 5-1.

250

 
 
 
than on elucidating relationships with No. 3 (although material supplementary to Gann’s

analysis will be provided when possible). A particular challenge in analyzing Study No.

45 is the inexactitude of the score, especially in the third movement (of which my copy is

quite faded and at times difficult to read).

Much of this analysis of No. 45 will build upon Thomas’s work on the concept of

multidimensionality and temporal dissonance. The analysis will seek to compare degrees

of multidimensionality within Nos. 3 and 45, particularly in the domains of rhythm (i.e.,

temporal dissonance) and tonality. The analysis will show that No. 45 exhibits a higher

degree of both rhythmic (or temporal) and tonal dissonance. The analysis will further

reveal that, while temporal dissonance is more evident in Study No. 45, multidimen-

sionality is more pronounced in some of the textures of Study No. 3.

The comparative analysis will primarily involve only movements (a), (b), and (e) of

No. 3, as these are the only movements involving a true boogie-woogie bass line. Table

5-7 shows the harmonic pattern and bass line used in each of the movements included in

the analysis. As can be seen in the table, Nancarrow used different harmonic patterns for

every movement, with actual blues patterns being used only in Study No. 3. The patterns

in No. 45 also primarily involve I, IV, and V, but are not traditional 12-bar blues patterns

and very short in comparison; No. 45a also incorporates a VN tonality in the harmonic

pattern. The only bass line pattern that is duplicated is that used for Nos. 36 and 45c

(although in 45c the pattern is used only sporadically along with several other patterns).

Multidimensionality and Temmral Dissonance in Smgy N9. 3

Thomas’s concept of multidimensionality is grounded in the typically multi-layered

texture of Nancarrow’s works:

The texture of Nancarrow’s studies is best described as multilayered: a
work typically contains a number of independent, contemporaneous layers
that are irreconcilable to a single governing musical process, and that, as a
consequence, create an extreme form of polyphony. (Thomas 1996, 52)

251

 
 
Table 5—7

Harmonic Patterns and Bass Lines in Studies No. 3 and 45

St d
d
Mgvzmagnt

Harmonic Pattern

Bass Line

3a

I-I-I-I IV-IV-I-I V-V-I-I

3b

I-IV-I-I IV-IV-I-I V-IV—I-I

$36 Se m f r:

*1

.
l¢5a+a 5'1: N P re

R

FF

3e

I-I-IV-I-V-I

45a

I-IV-V/V-V
(preceded by a 6-note
dominant)

45b

I-IV-V-I

45c

(no continuing pattern)

*5
q L_._._ — ——_ [920

I

Thomas points to a number of musical features that may contribute to a work’s

multidimensionality, including “differences in melodic content, register, articulation,

rhythm, tempo, and/or relative placement within shared melodic material (canon)” (p. 5).

Factors that can increase a work’s multidimensionality include “registral polyphoniza-

tion” (melodic separation of a layer into several layers, similar to compound melody),

252

 
   
   
 
   
 
 
 
 
 
 
 
  
  
 
   
   
 
  
 
 
 
 
 
mixing together different “time types” (e.g., metered rhythm, proportionally-notated

rhythm, and accelerating/decelerating rhythms), and introducing rhythmic variety and

meter changes to reduce the incidence of possible rhythmic simultaneities.

Factors that can diminish a work’s multidimensionality include melodic coordi-

nation of pitch-class content (what Thomas calls “heterophony”), registral overlap, a high

frequency of simultaneous note articulations among layers, and coalescence of the layers

into a sound mass or groups of layers (a phenomenon that occurs once the “perceptibility

threshold” is exceeded and individual layers are no longer perceived as such; common

causes of this are sheer density of sound and extreme note Speed). Multidimensionality

can also be somewhat negated by harmonic coordination among layers, a feature that will

be particularly examined in Study No. 45.

For those works that are considered less multidimensional, Thomas says, “What

sets [some] studies fundamentally apart from those that are multidimensional is the

reconcilability of the layers to a governing process: metrically, temporally, in pitch con-

tent, and/or phrase structure” (p. 54, emphasis mine). In essence, then, Thomas’s concept

of multidimensionality depends on the manner in which two features are achieved in

Nancarrow’s music: differentiation of the layers, and reconcilability of the layers.

Thomas points out further that “There is no absolute boundary between polyphonic and

multidimensional textures, of course, but rather a continuum of the reconcilability of

layers. This reconcilability is principally a matter of the degree of differentiation among

simultaneous layers” (p. 58).

Thomas identifies multidimensionality as being a prerequisite for temporal disso-

nance, but not all multidimensional textures are temporally dissonant:

There iS an intimate link between the concept of multidimensionality .

.
and temporal dissonance. Multidimensionality is, in effect, a prerequisite for
temporal dissonance: in order for two or more voices to have a dissonant re-
lationship they must first be convincingly differentiated and independent. As
we have seen, layers forming a multidimensional texture can be distinguished
through differences in register, melodic material, position in a canon, place-
ment of entrance, tempo, meter, and/or rhythmic design. If their differenti-

.

253

 
 
 
3..“ a_.'-

.. 1 'I

ation involves significant conflicts in the time parameter, then temporal disso-
nance is created. Not all multidimensional textures are temporally dissonant,
however. (Thomas 1996, 94)

Temporal dissonance, as expressed through multidimensional musical textures, was iden-

tified by Nancarrow as his primary compositional goal and led to his development of the

tempo canon.

Thomas points to Study No. 3a as having an “intensely multidimensional” texture

(p. 165). No. 3a’s concluding eight-layer texture, built through the process of accumula-

tion, was Shown in Figure 2-13 (p. 86). Thomas describes the influence of polyrhythm

and polymeter on this texture:

Polyrhythm and polymeter themselves are common in jazz .
. In jazz,
moreover, polyrhythm and polymeter are not often simple affairs, but rather
they can involve quite a number of voices, creating an extended polyphonic
texture. One need only look as far as Study No. 3(a), a boogie-woogie work
with eight discrete layers, to find an analog. (pp. 26—27)

.

.

One of the ways in which Nancarrow increases multidimensionality in this texture

is by expanding the original boogie-woogie bass line (line 7 in Figure 2—13) to two addi-

tional layers (lines 2 and 4) stating the same line in different registers and at different

tempos; here the three layers form a 2:3:5 tempo ratio (essentially, a 3-voice tempo

canon) and there are two octaves between each pair of layers. Another multidimensional

technique used here to differentiate layers is registral partitioning: although there is some

overlap among layers, each layer occupies a distinct registral Space. Harmonically, multi-

dimensionality is mostly expressed through layer 5, which states a F#—C# open fifth in

four octaves (pitches not shown in Figure 2-13) against the prevailing C major of the

other layers. Also, the transpositions to subdonrinant and dominant levels according to

the blues harmonic pattern are slightly Staggered among the different layers according to

where they are in their rhythmic patterns, and this contributes to the differentiation of the

layers and a greater sense of multidimensionality.

Temporal dissonance in No. 3a is also fairly pronounced, especially in the final

texture. Gann identifies, in addition to the 2:3:5 rhythmic ratio between the three ostinato

254

 
 
line layers, five isorhythmic layers which are notated in the same meter as layer 7 but

whose rhythmic patterns are 23, 29, 39, 43, and 47 sixteenth notes in length.‘5 Nowhere in

this concluding texture is there a simultaneity involving all the layers—not even at the

last note, leaving the temporal dissonance unresolved. Thomas finds overall that Study

No. 3a, at its conclusion, reaches a state of such temporal and textural density that it

perches precariously on the perceptibility threshold “beyond which it is difficult to per-

ceive the individual layers and, by extension, the multiple processes and multidimension-

ality they create” (p. 60).

Thomas refers to No. 3b as having a more conventional texture that is not so

extremely multidimensional as No. 3a’s. The movement opens with two layers: the

boogie-woogie bass line and a melodic layer. Both layers are rhythmically coordinated in

a 5:3 swung division of the quarter note beat, and, although the melodic layer’s note

attacks often occur “around the beat,” they are coordinated with the bass layer’s short

notes. The entrance of a third layer, however, introduces an element of multidimen-

sionality in an interesting guise: its straight quarter and eighth note rhythm sounds

dissonant in the prevailing swung-note context. Thomas points specifically to the fourth

layer, with its basic pulse of a dotted quarter note, as “setting up the multidimensional

character of the remainder of the study” (p. 56). As in No. 3a, here Nancarrow uses

numerous meter changes and isorhythmic patterns of varying lengths (including several

arithmetical decelerations, or “countdowns”) to introduce further rhythmic variety (see

Figure 5-15). At the thickest point of the piece there are five distinct layers (see Figure 5-

16) but in this canonic section (this is a traditional canonic texture, not a tempo canon),

rhythmic variety has been reduced to small rhythmic groupings that mostly match the

bass line; the grouping of 3 + 2 + 3 Sixteenth notes, for instance, merely subdivides the

first value of the bass line’s 5 + 3 pattern. Thus, as the textural density in this piece

increases, the temporal dissonance decreases. In the section shown in Figure 5-16, how-

6These patterns are not, as Gann implies (p. 77), based all on prime numbers as 39 is not prime.

255

 
 
 
ever, there is a greater degree of tonal dissonance between the bass and upper layers with

the introduction in the melodic layers of decorative minor seconds.

j. 

Figure 5-15. Temporal dissonance in Study No. 3b: use of 8 and 16 meters in upper three

6

12

voices against 2 meter in bass ostinato voice (score, p. 7).

4

2+ 0

5+3

Figure 5-16. Five-layer canonic texture from Study No. 3b (score, p. 21). Numeric patterns
indicate sixteenth note rhythmic groupings.

256

 
 
 
 
 
 
Thomas says very little about multidimensionality in No. 3e although temporal dis-

sonance is more intensely displayed here than in 3b. This movement is a good example of

a piece in which multidimensionality is confined almost entirely to temporal dissonance;

the temporal dissonance itself is expressed more successively than cumulatively. The

straight eighth notes of the blistering fast bass line are never accompanied by more than

three other layers at a time, but a continuous stream of rhythmic patterns that conflict

with the bass line’s

: meter challenge the listener’s metric orientation. Some of these

patterns, which share the eighth note pulse with the bass line, include a 182 pattern, a :1,

pattern, a 3 pattern, and patterns of 6 + 7 + 4, 4 + 3 + 3, and 7 + 3 sixteenth notes; other

patterns align less frequently with the bass, such as a ..I D (5 + 3) pattern aligning with

the half note pulse in the second statement of the harmonic pattern.

Multidimensionality in No. 3c is tempered by consistent tonal coordination (the

major/minor tonic triad is particularly prominent) and repeated use of the same melodic

motives among the layers (see Figure 5-17). Differentiation of the layers is denied by the

consistent harmonic coordination among the layers; at the transpositions of the bass line

to IV, V, and back to I, the melodic layers are always in harmonic accord with the bass.

The final section of the piece is characterized by a rhythmic disintegration in the

bass layer, with first every ninth eighth note being replaced by a rest, then every eighth,

seventh, and so on (Gann, p. 80). Concurrent with this process is a melody in the top

layer that is progressively lengthened rhythmically, allowing the layers to further separate

(eSpecially when the 16 meter is introduced in the second statement; see Figure 5-18).

15

257

 
(a)  

_
h

“
9
4
-

.
.
2
1

I

(b)

 ‘

Figure 5-17. Recuning melodic fragments in Study No. 3e, Showing use of major/minor tonic: (a)

score, p. 4; (b) score, p. 7 (this segment has a 8 rhythmic pattern in the middle line).

9

Multidimensionality in No. 45

Turning now to Study No. 45, this examination of multidimensionality will focus

Primarily on degrees of accord among the musical layers in terms of rhythm and tonality.

Tonally, No. 45 is somewhat of an anomaly among Nancarrow’s late Studies. Once he

began writing tempo canons, it was unusual for Nancarrow to use the same tonality in all

musical layers at once; see, for instance, the variety of levels of imitation listed in Table

1‘2 (p. 45). As Gann noted, in the later 2-voice tempo canons, beginning with No. 40,

258

 
 
Nancarrow’s favored interval of imitation was a tenth, which imparted a “bittersweet and

ambiguous major/minor harmony” (p. 203).

(a)

(b)

(C) 

Figure 5-18. Rhythmic disintegration in bass line and progressively slower melodic statement at
end of Study No. 3e: (a) p. 21; (b) p. 23; and (c) p. 25.

We begin by examining the background rhythmic scheme in the three movements.

Gann characterizes No. 45 as exhibiting “chaotic rhythmic energy” (p. 10). The

259

 
 
 
 
 
 
difference that one most immediately notices between Nos. 45 and 3 is the change in the

rhythm of the bass line from straight, rapidly pulsing (or consistently swung) eighth notes

in No. 3 to a loping, lopsided rhythm in No. 45 (see Table 5-7). The unevenness of the

bass rhythm in the slower middle movement (45b), in particular, could remind one of a

weaving drunk. Gann points to this uneven rhythm as an “irrational ‘spastic rhythm

9”

(p.

256) of fifteen durations that runs through all three movements of No. 45 as well as Nos.

46 and 47 (all of which were originally part of a six-movement work).7 This rhythmic

pattern is not subject to rational notation and is notated proportionally by Nancarrow (the

fifteen-duration “spastic rhythm” can be seen in Table 5-7 in the entry for No. 45a). All

Nancarrow could recall about the derivation of the rhythmic series was that he “took a

bunch of [tempo] templates and started putting them together, purely intuitively, and

finally came up with that proportion that I liked” (Gann 1995, 257). The exact derivation

of the series has remained a mystery.

As it turns out, a good share of the durations in the series can be deduced in a

systematic way and related to consistent divisions of the time continuum. Gann reports

(p. 257) the measurements of the values in the rhythmic series at its slowest point, ex-

pressed as averages in millimeters on the score. Using Gann’s measurements, I multiplied

each by 10 to convert all to whole numbers and added all the values together to determine

a cumulative value for the series:

378 + 357 + 250+ 220+ 270+ 132 +460 +128 + 318 + 357 + 92 +
171+ 382 + 224 + 435 = 4174

Dividing each duration by the cumulative value allows each to be expressed as a per-

centage of the whole series. An examination of the resulting values reveals that in a

number of cases the values could be represented quite closely by fractions with

7This point is later contradicted when Gann notes that 45b’s bass rhythm is a 24-note ostinato derived from
“a measure divided simultaneously into three, four, five, and fourteen equal divisions” in which the bass
line’s rhythmic values “skip freely back and forth between the 3-, 4-, and 5-tempos” (p. 258). The second
movement does not, in fact, contain the lS-note “spastic rhythm.”

260

 
 
 
aha-ulna... .a' -

denominators of prime numbers, especially ll, 17, 19, 29, and 31. This characteristic

begins with the first value, 378, which is extremely close to 1’n of 4174 (379.5).

Recognizing that Gann’s measurements represent an average approximation from

Nancarrow’s imperfect score, I then created a grid dividing the value of 4174 equally into

11, 17, 19, and 31 parts and looked for close matches with the duration measurements. It

turns out that all but four of the rhythmic values can be related to the resulting values;

two of the remaining values can be related fairly closely to the common fractions J’2 and

-
_
.
.

34:, indicating that Nancarrow may have also used a 4-template; the remaining two values

are not close enough to any of the values in the grid to be related to them with any cer-

tainty (although one is fairly close to 2%9 ). See Table 5-8 for full results.

The ten values that closely match the prime-number-denominator fractions in Table

5-8 are close enough (particularly the values 378, 735, 1475, 1607, 2195, 2962, and

3515), within a reasonable margin of error to accommodate inexactitude of measurements

from the score, to be compelling evidence that these divisions may have been the ones

used by Nancarrow to create the rhythm. Table 5-8 also shows that two contiguous pairs

having the same denominator appear in the table (347 and 447, and 1’13 and u1’9), further

supporting this conclusion. It is particularly interesting to note that Nancarrow settled on

some divisions that have historically been used to create equal-tempered scales, most

particularly the l9-division (proposed by Salinas in 1577) and the 3l-division (proposed

by Vicentino in 1555); see Jorgensen (1977, p. 376). Given Nancarrow’s penchant for use

of references to historical tuning systems in his rhythmic schemes, even though this may

have been a serendipitous accident it is not entirely surprising to find such a connection.

Of course, Nancarrow was aided in the creation of this rhythm by having the benefit

of the modified punching mechanism that allowed him to punch a hole on the piano roll

virtually anywhere on the time continuum, an ability he didn’t have until Study No. 22.

The availability of this technique also created another difference between Nos. 3 and 45

in the constancy of the underlying rhythm. In No. 3 the rhythmic background unit stays

261

 
 
Table 5-8

Possible Derivation of Rhythmic Values in No. 45’s “Spastic Rhythm”

 Step 1: Rhythmic series is given as a series of relative values and running cumulative:

Values:

378 357 250 220

270 132 460

128

318

357

92

171

382

224 435

Running
cumulative: 378 735 985 1205 1475 1607 2067 2195 2513 2870 2962 3133 3515 3739 4174

Step 2: Values from grid dividing 4174 into equal parts of 11, 17, 19, and 31 (one value
is derived from a l3-division) are compared to running cumulative values to identify
closely matching note attacks (the derived value identifying the note attack defines the
end of the previous note):

Individual Value

Cumulative Value

Closest Match

378
357
250
220
270
132

460
128
318
357
92
171

382
224*

378
735
985
1205
1475
1607

2067
2195
2513
2870
2962
3133

3515
3739

1/11 (379.5)
3/ 17 (736.6)
4/ 17 (982.1)
9/31 (1211.8)
6/17 (1473.2)
5/13 (1605.4)
none; value is 0.495 of cumulative total,
suggesting relation to 1/2 (2087)
10/19 (2196.8)
none; value is 0.602
none; value is 0.688 (20/29 = 0.690)
22/31 (2962.2)
none; value is 0.751 of cumulative total,
suggesting relation to 3/4 (3130.5)
16/ 19 (3514.9)
17/19 (3734.6)

*The final value, 435 (435/4174 = 0.104), approximates 3/29 (0.103)

constant throughout a movement,8 while geometric acceleration takes place at the end of

two movements in No. 45. In order to create an effect of tempo changes in No. 3, Nan-

carrow relied on the technique of arithmetical deceleration near the end of 3b. The con-

stancy of the eighth note is eroded at the end of No. 3e by the “dropping out” technique

shown in Figure 5-18 that makes for a very effective winding down of this whirlwind-like

8With the exception of a curious notation “accel. poco a poco” at the beginning of No. 3b, which is un-
explained except for the possibility that it refers to the natural slight acceleration that takes place as the
piano roll dwindles during playback.

262

 
 
 
 
 
 
 
 
 
piece.

No. 45a

Interestingly, Tenney points to the unevenness of the bass octaves as contributing to

a degree of registral polyphonization in the bass layer:

.

There is .
. one additional perceptual result of this rhythmic irregularity
which I could not have predicted before hearing it: the upper and lower notes
of the successive broken octaves seem to separate into two distinct “streams,”
each of which is more internally cohesive—and more clearly segregated from
the other—than seems to be the case with a line of broken octaves in an even,
regular rhythm. Thus, what’s usually heard as a single voice is here split into
two voices .

. (Tenney, Vol. V, p. 8)

.

.

In addition to the inherently uneven bass line rhythm described in the previous

section, No. 45a is structurally uneven in another important way. The 192—note color

(pitch pattern) of the bass line is not evenly distributed among the four harmonic areas of

the harmonic pattern but is divided as follows:

I(C)—->IV (F)—->V/V(D)—->V(G)—-)I(C)

Dbeats:

[6]

54

46

so

42 = 192

Thus the pattern spends the most amount of time in I, and the least amount of time in V.

The color is preceded by a 6-note dominant pitch pattern (as shown in Table 5-7), and

each new transposition level is preceded for six notes by its dominant.

There are eight complete statements of the 192-note color. The smallest common

denominator of the 15-note rhythmic pattern and the 192-note pitch pattern is 960 notes,

or every five statements of the color. The beginning of the 15-note rhythmic pattern

rarely aligns with a change in transposition level, doing so on only six occasions

(numbers in parentheses indicate during which color the alignment occurs): IV“), 1(3),

Vo)’ was» 1(7), and V0)-

Temporal dissonance in No. 45a is achieved in several ways. First, the piece mixes

“time types,” with the result that there are virtually no (if any) simultaneities between the

upper (melodic) layers and the bass. The first two time types are metered rhythm and

263

 
 
 
irregular (proportionally-notated) rhythm. Along with the proportionally-notated bass in

No. 45a there is no actual metered rhythm, but there is a strong sense of meter in some of

the melodic figures of the upper layers, especially earlier in the piece. Some examples are

shown in Figure 5-19.

' v

I

i4):- 3..

__.._- --'--- --___I*
'

(IV).

Figure 5-19. Passages in No. 45a’s melodic layer that emulate metered rhythm: (top) p. 2, bottom
system; (bottom) p. 3, top system (note the rhythmic coordination between the top two layers).
Rhythmic approximations are shown between the staves; dotted lines in bass layer indicate divi-
sions of the lS-note rhythm. In the top example there appears to be a rhythmic simultaneity
between the layers on the third note of both statements in the top layer. In the bottom example, an
area of additive deceleration in the top layers is indicated by the rectangle.

Further temporal dissonance is created by the use of acceleration, the other “time

type” identified by Thomas. A case of additive deceleration is shown in Figure 5-19. The

eighth and final section of No. 45a introduces geometric acceleration in the form of scalar

glissandi inserted within a line of octaves (see Figure 5-20). The distance between the

264

  
 
  
 
 
 
 
first pair of glissandi is 77 millimeters; in 18 steps, this measurement decreases to 25

millimeters, an overall acceleration of about 6.5% (although the amount is variable—for

the first 8—10 steps of the accelerative process the acceleration is closer to 4.75%). Nan-

carrow used this technique in a very similar way once before, with arpeggio figures in

section 8 of Study No. 25. In both cases, the figures themselves are not accelerating, but

the space between their attacks steadily decreases.

‘———--- ‘7

or

Figure 5-20. Acceleration between glissandi in final section of No. 45a (score, p. 19). Millimeter
measurements between glissando attacks are given. Acceleration rate ranges from 4.75% to 6.5%.

The subversion of potential simultaneities also contributes to temporal dissonance,

and, as Gann points out, the presence of converging canons that are cut off just prior to

their convergence is a hallmark of Study No. 45. In 45a, there is one such canon, a two-

voice canon in an approximate 5:7 tempo ratio. The ending of this canon, with its near-

convergence, is shown in Figure 5-21. There appears, however, to be a rhythmic con-

vergence between all three voices on the last note of the canon in the leading voice; also

at this very spot, there is a break in the broken octave pattern of the bass line (the only

such break in this movement). The canon’s ending also neatly coincides with arrival at

the 6-note dominant leading to the return to I.

265

 
   
 
 
 
.d—g- *_._ ..

.

.
1
1
I

Figure 5-21. End of tempo canon in No. 45a, stopping just short of convergence (score, p. 14).
Circled area indicates break in broken octave pattern of bass line. The dotted lines between layers
indicate an apparent rhythmic simultaneity, which happens to immediately precede the arrival at
1’5 dominant. Arrows indicate echo distance.

Earlier in the canon, Nancarrow sets up the canon melody so that the two layers

neatly switch between triads and single pitches; see Figure 5-22.

(V/V)

'

'

Figure 5-22. Earlier section of 5:7 tempo canon in No. 45a (score, p. 13). Arrow shows echo
distance. Canon is effectively constructed so that layers switch between triads and single pitches.

In sum, then, temporal dissonance in No. 45a is achieved through the irregular bass

rhythm, mixing of the three time types, and the subversion of potential simultaneities as

most significantly expressed through the thwarting of the convergence of the tempo

266

 
 
 
canon. As shown in Figures 5-19, 5-20, 5-21, and 5-22, the melodic materials of No. 45a

consist primarily of single pitches, triads (overwhelmingly parallel major triads), octaves,

and glissando scalar figures. The glissando figures (see Figure 5-20) are generally either

one or two octaves long and state the boogie-woogie bass line pattern (without the broken

octaves) beginning on the dominant (i.e., the pattern shown in Table 5-7); the glissandi

are almost always stated in a direction opposite to the direction of the bass (the last two

figures in Figure 5-20 being notable exceptions).

The tonal environment is primarily major with jazz inflections, e.g., fiatted third

and seventh scale degrees appearing along with the same degrees unaltered, as well as

occasional diminished seventh chords. There is considerable tonal coordination among

the layers—with the melodic layers coordinating with the transposition levels in the

bass—until the last few sections of the piece. For instance, for the first seven statements

of the color, the glissandi are overwhelmingly tonally coordinated with the transposition

level being stated, although occasionally a glissando will anticipate the arrival of a tonal

area or extend a tonal area past the arrival of a new area. Table 5-9 identifies the tonics of

the glissandi appearing in each tonal area. In section 8, as shown in the table, the attacks

between glissandi are accelerating, and the increasing numbers of glissandi reach more

remote key areas than have appeared before; note, however, the statement of I—IV—

VN—V in the last four glissandi, all being stated within the V area.

There are places where the melodic layers veer noticeably away from the bass

layers. Such a place occurs just prior to the entrance of the 5:7 tempo canon in section 6.

On the previous page, the melodic layer extends the VN tonal area (D) for at least half a

system beyond the movement to tonal area V; see Figure 5-23.

267

Table 5-9

Tonics of Glissandi Stated in No. 45a

Section

Transposition levels/Glissandi

Comments

I (C): C

2

IV (F):

VN (D): A

  double-octave glissandi

V (G): G

I (C): C, C

IV (Fig, 131,, F
VN (D): F, F, F, D

V (G): G, G

I (C): none

single-octave glissandi

double-octave glissandi 

IV (F): none

VN (D): Bfi
V (Gt/D. D
I (C): C

  double-octave glissandi

IV (F): Bl,

VN (D): A

V (G): G, G

double-octave glissandi 

I (C): D, E, A, Bi,
IV (F): B, C, G, D

V/V (D): A1,, B, F, E, D
v (G): G, A, Br, F, D, G, C

Notes: There are no glissandi in sections 6 and 7. Arrows indicate anticipations of new
tonal area or extensions of previous area. Shaded tonics in last section are stating the final
four transposition levels (IV—VN—V—I).

After the increased differentiation between the two melodic layers of the tempo

canon in section 6, section 7’s two melodic layers are, by contrast, extremely tonally

coordinated with each other but only somewhat loosely with the bass layer; see Figure 5-

24. The middle layer brings back a melody from earlier in the piece (p. 4) which has a

recurring motive that clearly outlines the four transposition levels (the system imme-

diately preceding this example is shown in Figure 5-21).

268

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
  
 
 
 
_____ ______.._-__-._-_-._..___'

Arr 1.9.? l—p LP ,m p
[a . .
r
+1

r'
V

n
..
v "v

v

VN (D major) extends beyond transposition to V

> 

 LL“T_-‘:_-T--:V" ' - 3" l/m

Figure 5-23. Extension of VN tonal area beyond arrival at V area in No. 45a (score, p. 11).

Thus, overall No. 45a exhibits a variety of multidimensional textures in terms of its

tonal cohesion. Thomas asserted that multidimensionality in a piece is rarely static,

instead usually representing a continuum, and that is certainly the casein this movement.

Both temporal dissonance and multidimensionality reach a peak in section 8 (the final

section) where acceleration between the glissandi accentuates the temporal disparity and

the more remote tonal areas expressed in the glissandi and the triad layer separate these

layers tonally from the bass layer.

No. 45b

No. 45b has many of the same multidimensional features as No. 45a. Its rhythm is

more noticeably based on metric configurations, however, and it has a higher frequency

of simultaneous note attacks between layers. Acceleration does occur in places, but when

it does it is arithmetical acceleration rather than the more multidimensional geometric

acceleration that was used in 45a. The upper melodic layers, of which there are only two,

are tightly coordinated both rhythmically and tonally with each other, but less so with the

bass line. The upper layers do faithfully adhere to the tonal transpositions of the bass, but

the changes are often staggered between the bass and upper layers.

269

  
 
 
 
  
 
 
 
 
 
 
 
 
 
 
     
(extension of G)

([L.__-__ - .._; --

-'-r'-1*:._:-.- .._... _:---_-__J

'

---------- IV

Figure 5-24. Section in No. 45a’s section 7 (score, p. 15) where upper two melodic layers are
closely coordinated tonally but not tightly coordinated with changes of transposition level in the
bass layer. Rectangle identifies repeating melodic pattern.

As already noted, the bass line rhythm in this movement is based, according to

Gann, on a background measure (not notated as such in the proportionally-notated score)

simultaneously divided into 3, 4, and 5 parts. This same rhythmic idea, a “polytempo

ostinato,” was used in Study No. 46 (shown in Figure 2-6, p. 77). Gann also asserts that

the upper melodic layers relate to this same background measure as a l4-division of the

measure. Presumably, then, the entire movement could have been written in metric nota-

tion, but in proportional notation the metric relationships among layers are somewhat less

clear. If Gann’s assertions about the 3, 4, 5, and 14 divisions are correct (and I have no

reason to believe they are not), then the top layers and bass layer share a common down-

beat, which presumably would allow for frequent simultaneities among the layers.

270

 
   
       
  
As Gann further notes, the rhythm in the upper melodic layers consists of values

that are multiples of the same background unit, and these multiples are most frequently 2,

3, 5, 8, or 10 units in duration (and thus heavily based on Fibonacci numbers; rhythmic

groupings such as 2 + 3 + 3 and 3 + 5 + 5 also imply Fibonacci). Short ostinati and

isorhythmic passages abound. The two passages shown in Figure 5-25 contain an iso-

rhythmic pattern consisting of a four-note pitch pattern with a 2 + 3 + 3 rhythmic pattern.

(In this section, figures showing score segments from No. 45b will have the relative

values of the durations noted.)

a

s

:

E

:

é

a

i

E

l

'

E

5

(b)

+1

I

E

Figure 5—25. Two instances of a 4-note ostinato against thirds aligning with every third note (the
short note value in each 2 + 3 + 3 rhythmic pattern): (a) ascending pattern in No. 45b, p. 3; (b)
descending pattern in No. 45b, p. 4.

The Fibonacci-related rhythmic values also are used to create rhythmic augmen-

tation. Figure 5-26 shows the same melody shown in Figure 5-25b rhythmically aug-

mented as a 3 + 5 + 5 rhythmic pattern.

The score examples shown in Figures 5-25 and 5-26 illustrate complete rhythmic

271

 
  
 
  
 
 
  
 
 
 
 
 
 
 
 
 
 
    
 
 
 
 
 
 
 
 
 
 
agreement between the melodic layers. Later in the movement, a steady 5 rhythmic value

is established in the lower layer, setting up the syncopation shown in Figure 5-27.

P
I
   '

|

Figure 5—26. Same melodic pattern from Figure 5-25b rhythmically augmented to 3 + 5 + 5
(score, p. 6).

Multidimensionality is heightened in No. 45b by structural imbalances that drive a

wedge between the bass and melodic layers, and deny the listener’s expectations about

regularity in the recurring harmonic pattern. Although the harmonic structure of No. 45b

is based on the pattern I—IV-V—I, there are some inconsistencies in the pattern (occasion-

ally an area appears out of sequence, and on p. 13 the area VN appears twice). Also, the

.
.
.
3
H

.32
.

.

   5
5

(Syncopation) .......

-
u
-
-
-

!
U

I
U

I
U

u
‘

.

l

-
4
\

|

H
‘

L

:
U

~

%
l"

J
J
\

l

j:—

5m"
116

Figure 5-27. Syncopation between two melodic layers in 45b (score, p. 12). I“, refers to the
sixteenth time through the I bass pattern. Note the low incidence of simultaneities between the
bass layer and upper layers.

272

 
 
 
 
  
   
  
 
 
   
  
 
 
 
 
 
     
 
 
  
 
 
 
  
 
 
 
 
 
   
piece ends much the same way as No. 45a did, with a rapid cycling through remote key

areas. Pages 14—18 of the score progress through the following keys:

FALBEABCErALFDGBtEtAtBG;DLALFCAD

The majority of tonal movement is by either minor third (up or down) or perfect fourth]

fifth, and from the C highlighted in the line above, those are the only intervals of move-

ment. Nancarrow moves to these areas by making adjustments in two parts of the bass

pattern: the resolution of the seventh scale degree (the highest note of the pattern) by

either minor second or major second; and by varying which scale degree the resulting

resolution note is in the new key. Only two patterns are used prior to the cycling of keys

beginning on p. 14; thereafter three new patterns (plus two irregular patterns that are each

used only once) are introduced that allow Nancarrow to create the tonal movements

shown in Table 5-10 and Figure 5-28.

Table 5-10

Five Patterns of Transposition in Bass Line at End of Study No. 45b

Transposition

Resolution of Opening
Key’s 7th

Resolution Note’s Scale
Degree in New Key

up a perfect fourth (Fig. 5-28a)

up a major second (Fig. 5-28b)

up a minor third (Fig. 5-28c)

down a minor third (Fig. 5-28d)

down a perfect fourth (Fig. 5-28e)

m2

m2

M2

m2

m2

third

fifth

fourth

octave

second

Like the inconsistent number of notes in No. 45a’s harmonic pattern, the sense of

regularity in No. 45b provided by the repeating harmonic pattern is compromised further

by diSparities in the durational length of the key areas. Because there is no systematic

Coordination of changes to new tonal areas with the downbeat of the background meas-

ure, there seems to be no rhyme or reason to the lengths of the areas. For instance, the

first six key areas have lengths of 96, 107, 111, 121, 121, and 87 millimeters; the range of

273

 
 
 
 
 
  
 
  
 
lengths is from 48 (in the accelerating section on p. 10)9 to 147 millimeters, and adjoining

areas are often of quite different lengths.

m2

t p ”‘3
if
’

fit 11
$1 4‘

1‘.
i‘fi

is
"X?

G

(b)

m2
5:

0

we: .

1

E 3 g "'T' ‘&n

a f

\f—

(C)

((0

m2

F

'1:
Q—h—Hwyw

[xx

.

I
9 Q 1';

8%

F

V

.\ .

at?

C

(6)
Figure 5-28. Alternate transpositions in No. 45b’s bass line pattern: (a) movement to key a perfect
fourth higher; (b) movement to key a major second higher; (c) movement to key a minor third
higher; ((1) movement to key a minor third lower; and (e) movement to key a perfect fourth lower.

Structure is further thrown off balance because the arrival of new sections in No.

45b only rarely coincides with a return to I (in No. 45a, by contrast, all new sections

coincided with an arrival at I). Gann identifies nine sections in No. 45b according to

textural changes; it is, in fact, difficult to ascertain clear section breaks because there is so

little alignment with new textures and arrival at returning key areas.

As mentioned earlier, the upper two melodic layers are fairly coordinated tonally,

but less so with the bass layer. This changes remarkably in the closing section during the

9There is a duration of 45 at the end of p. 11, but that appears to be due to a missing system.

274

  
 
 
  
   
  
   
 
  
 
 
  
 
 
 
  
   
   
 
 
  
 
H
M
.
.
.
.
_
_

rapid cycling through remote key areas. The last three systems of the piece are shown in

Figure 5-29; as shown in the figure, all three layers are tonally coordinated during the last

seven key areas. (Incidentally, the key areas G1,, D1,, Ag, and F are virtually identical in

duration, perhaps indicating the length of a background measure or hypermeasure.) The

last four areas (F, C, A, and D) include a strong melodic emphasis on the note estab-

lishing each as a dominant seventh, and the piece ends in strongly dominant fashion on a

D9 chord. The curious ending on a D dominant chord functions essentially as a half

cadence to lead the listener directly into No. 45c, which begins with a chromatic glis-

sando starting on G, leading to an extended G dominant harmony. Not only does 45b end

on a non-tonic harmony, there are no simultaneities between any of the layers at the end,

further intensifying the lack of resolution.

No. 45c

The structural irregularities in Nos. 45a and 45b pale in comparison to those in

45c—the “spastic rhythm” returns in its full glory, appearing not only in the bass but in

the melodic layers, and the boogie-woogie bass line and the implication of tonal areas I,

IV, and V only loosely undergird the structure. Tenney says about 45c that “the process

of abstraction .

.

. is carried still further, juxtaposing and superimposing fragmentary,

altered elements of a boogie-woogie texture in highly unpredictable ways” (notes to Vol.

V, p. 9).

The boogie-woogie pattern itself (Table 5-7) is used only occasionally, and it

appears not only in the bass but in several melodic statements (see Figure 5-30).

One other melodic motive is prominent throughout the movement: a first 3-note,

then 4-note pattern outlining the “partitioned minor third” so prevalent in Nancarrow’s

work. The motive is presented in a variety of textures, including single notes, octaves,

parallel major triads, and as part of more complex textures (see Figure 5-31). The motive

is actually a “head motive” from a much longer theme (53—54 notes long) that recurs in

275

25

325

325

325 3232325

J

  D

A

Figure 5-29. The end of Study No. 45b (score, pp. 17-18). From the G1. area to the end there is
complete tonal coordination among all layers. In the last four key areas (F, C, A, D) there is a
melodic emphasis on the dominant seventh, culminating in a D9 chord that becomes the dominant
to the first harmony of No. 45c.

276

 
   
  
 
 
 
 
4:7'" ‘" — — _ -— -— ———,‘i:;c..

(b)

 I

“,3________ 4.1

Figure 5-30. Melodic uses of the boogie-woogie bass pattern in No. 45¢: (a) score, p. 2; (b) score,
p. 6. [Note: the score copy to which I had access is quite badly faded in this movement. I have
attempted to heighten certain details in these figures where possible]

several sections, including being the theme for the last (and longest) of three tempo

canons in the movement. The motive is further identifiable by its fairly consistent rhyth-

mic shape: a long note, followed by a shorter note, and a long note (exact proportions

vary).

Unlike movements (a) and (b), No. 45c does not really mix time types as the

rhythms are mostly based on the “spastic rhythm” and everything is proportionally no-

tated. In this movement more than the others, it seems that Nancarrow studiously avoided

simultaneities among layers. To achieve this, Nancarrow introduced a new procedure for

use of the 15-note spastic rhythm that guarantees no simultaneities among layers: in

much of the piece, the rhythm is divided among the layers. The piece’s final section

(what Gann identified as section 8) is perhaps the best example of this; see Figure 5-32.

Although the division of the spastic rhythm among layers does heighten the sense of

temporal dissonance due to the lack of simultaneities, in a way this same procedure intro-

duces a sense of registral polyphonization since all the layers are participating in stating

the same rhythmic pattern. Besides the use of the spastic rhythm to avoid simultaneities

and lack of a semblance of metered rhythm, temporal dissonance is also introduced into

277

 
 
 
 
   
 
(a)

(b)

g

D

i3: 3? if é 

I

W

g/L.__._—.____-_.

(C)

((1)

Figure 5-31. Recurring melodic motive in No. 45c: (a) first appearance, as a 3-note pattern, in
interlocking form (ascending triads below and descending single notes above), score, p. 1; (b)
expansion to 4-note form, repeating the second note (as triads and octaves), score, p. 3; (c) as the
highest pitch in series of glissandi followed by sustained polychord (score, pp. 5—6); (d) preceded
by chromatic glissandi (score, p. 16).

278

 
 
 
 
 
     
 
    
,   

(end of third tempo canon)

__— — —_————_———————— —-—-

Section 8

__fi

Figure 5-32. Division of the “spastic rhythm” among all layers in section 8 of No. 45c (score, p.
21). The 15-note segments demarcated by the vertical dotted lines are all the same length. This
section is preceded by the last of three tempo canons.

No. 45c by three converging tempo canons, all of which stop just short of convergence

and leave the temporal processes unresolved. All of the canons end on G—the first on G

octaves, the second on G major triads, and the third (Figure 5-32) also on G triads. The

tempo canons are, ironically enough, the only places in this movement where occasional

simultaneities occur between layers.

Acceleration, the third time type, is not much of a presence in this movement. The

only instance is one of sectional deceleration, created by an increase in the length of the

lS-note rhythm statements on pp. 15 and 16 (between the second and third tempo canons)

by a 1:2:3 ratio. The boogie-woogie bass pattern is first stated twice at a length of 75

millimeters per rhythmic statement, then twice at 150 millimeters, then—underneath the

third tempo canon—a number of times at ca. 225 millimeters. The effect of the first

deceleration is quite dramatic because it occurs at the highest point of this widely-ranging

line and before the tempo canon voices enter.

As can also be seen in Figure 5-32, No. 45c has sections of thick parallel

279

    
 
 
 
 
harmonies, specifically parallel major triads that introduce a sense of polytonality to the

texture. There are also thick polychords in places, such as the stacks of two minor

seventh chords with roots a major third apart shown in Figure 5-31c. Another prominent

polychordal structure involves stacked dominant seventh chords with roots a fifth apart,

several of which appear at the beginning of the piece: each of the first three phrases

begins with a long chromatic glissando followed by a sustained polychord (see Figure 5-

33). As Figure 5-33 illustrates, the introduction of more remote keys happens much

earlier in No. 45c than it did in 45a or 45b. No. 45c begins conventionally enough, with

the first phrase forming the dominant to the boogie-woogie bass line in C and the second

phrase to F, but the third statement resolves into a B}, tonality in the top voice (shown in

Figure 5-30a) before the head motive enters in Er.

 I- 1:2-:1:----   

(a)

(b)

(C) 

Figure 5-33. Dominant seventh polychords in each of No. 45c’s first three phrases: (a) p. 1, first
system; (b) p. 1, third and fourth systems; and (c) p. 2, first system.

It is this frequent use of parallel major triads in the theme (and elsewhere) and the

280

 
 
 
  
  
  
  
  
  
 
resulting subversion of tonal function within each layer that contributes most signif-

icantly to tonal multidimensionality in this piece. Figure 5-34 shows the very end of No.

45c. Here shifts in tonal center, as detemrined by the bottom pitch of each chord, are

fairly coordinated among the three layers and the shifts in tonal center are in close

proximity to the beginnings of the 15—note rhythmic pattern, as well as being reinforced

by V—I cadences (although the attacks are staggered). However, the constant movement

I

E:

Figure 5-34. Ending of No. 45c (score, p. 23), showing near-coincidence of shifts in tonal centers
with beginning of “spastic rhythm” statements (demarcated by vertical dotted lines—rhythmic
values are still being divided among the layers) and parallel triadic harmony, culminating in a G
dominant polychord (G major, B major. and D major triads). Circled chords indicate where tonal
shifts occur. Staggered V—l cadences are shown in D, F, and G.

281

 
in parallel major triads (and open fifths in the bass), in opposition to functional tonality,

adds significantly to multidimensionality. This is perhaps most striking at the G dominant

polychord that immediately precedes the final phrase. The function is clearly a dominant

to C, but the actual tonal center is not confirmed until the line of broken octaves con-

tinues in the bass. (Nancarrow used a similar polytonal technique in cadences in Study

No. 12—see Figure 1-16, p. 40).

Structurally, the l—IV—V—I harmonic pattern on which 45a and 45b were based has

been reduced in 45c to little more than a touchstone, and, as Tenney’s comment reveals,

there is much fragmentation of the background harmonic pattern. Tonal areas I (C) and

IV (F) do figure prominently in a number of sections (see Figures 5-3ld and 5-32, for

instance, for examples of I), but V (G) functions only in brief sections that are bridges to

I. There are also numerous secondary tonal areas of significance, including Er, on page 2,

A on page 3 (Figure 5-31c), and D on page 4.

Concluding Remarks

The three movements of Study No. 45, described by Gann as a “Boogie-woogie

Suite No. 2” (p. 263), employ a number of different techniques that enhance both tem-

poral and tonal multidimensionality. Some of these techniques were also used in the first

“Boogie-woogie Suite,” Study No. 3, and some evolved as Nancarrow gained composing

experience (particularly through the development of the tempo canon) and access to more

precise punching mechanisms.

Multidimensionality is expressed as temporal dissonance in No. 45 in several ways.

Mixing of time types (metered, proportional, and accelerative) occurs in movements (a)

(Figure 5-19) and (b) (Figure 5-27), but almost not at all in (c). Still, temporal dissonance

seems most intense in 45c, where thelS-note proportional “spastic rhythm” (Tables 5-7

and 5-8) is partitioned among layers (Figure 5-32) and simultaneous attacks are virtually

non-existent. No. 45b is the least dissonant temporally; although its bass line is still

282

 
r
1

I

metrically irregular, the upper melodic layers are in a pseudo-metric rhythm with regu-

larly-recurring duration values (2, 3, 5, 8, etc.) and are often in metric accord, as well as

having a fairly high incidence of simultaneities with the bass line. The use of metric

regularity allows Nancarrow to incorporate syncopation into 45b (Figure 5-27), a feature

non-existent in the other two movements.

No. 45a’s most striking temporal dissonance is achieved through geometric accel-

eration (Figure 5-20), and it is the only movement to use this form of acceleration. Arith-

metical tempo change also occurs in this movement (Figure 5-19), as it does in No. 45b

(Figures 5-25 and 5-26). Further, there is sectional acceleration in 45b and deceleration in

45c.

Temporal dissonance is also imparted to these pieces by the subverting of the

temporal convergence process in several tempo canons that abruptly end just prior to con-

vergence. Such canons occur in 45a (Figure 5-21) and three times in 45c (the end of the

last canon is shown in Figure 5-32).

The potential for tonal multidimensionality in these works is primarily dependent

on two factors: (1) the number of layers operating at any given time, and (2) the harmonic

expectations established by the background harmonic pattern in each movement. Multi-

dimensionality is increased as more layers introduce different tonal areas and as tonal

areas fail to conform to harmonic expectations according to the pattern. In No. 45a, tonal

multidimensionality is heightened where tonal areas in the upper layers either anticipate

or lag behind new areas in the bass (Figure 5-23); where the upper layers are tonally

coordinated with each other but not the bass (Figure 5-24); and where key areas remote to

the original I—IV—V/V—V harmonic pattern are introduced, particularly the rapid cycling

of remote key areas (combined with geometric acceleration) at the piece’s end (Table 5-

9).

Tonal dissonance in No. 45b is more moderate. Its two upper layers are highly

coordinated rhythmically and tonally (Figures 5-25 and 5-26) but less so with the bass

283

 
 
layer (Figure 5-27). Like No. 45a, 45b ends by rapidly cycling through a number of keys

remote to the original harmonic pattern (Table 5-10 and Figure 5-28).

No. 45c has the densest texture of all three movements (Figures 5-32 and 5-34), and

tonal dissonance in this movement is represented in the use of parallel major harmonies

and dense polychords (Figure 5—33). As a result of the parallel harmonies, even passages

in which there is tonal agreement among layers (Figure 5-34) have a high incidence of

extra-tonal pitches. Also, tonal areas remote to the I—IV—V—I harmonic pattern that subtly

underlies this movement are introduced much earlier than in the first two movements.

Because of the frequency with which it strays from the basic harmonic framework,

No. 45c is the most extreme and dramatic example of the sort of structural multidimen-

sionality that exists in Study No. 45. No. 45a is the most structurally regular movement;

although its tonal areas are of different lengths, its sections clearly coincide with returns

to the I tonal area and sections often begin with a simultaneity among layers. However,

its structure begins to disintegrate at the end with the rapid cycling through remote key

areas and the introduction of geometric acceleration. No. 45b introduces further structural

disintegration with its highly variable length of tonal areas (varying by a factor of about

3) and in how its sections so rarely coincide with I or even the beginning of a new tonal

area. It, too, undergoes harmonic disintegration at the end (Figure 5-29).

In comparing Study No. 45 to its early ancestor, Study No. 3, it is the structural

irregularities of No. 45, especially the inherently uneven rhythms of the bass layer, where

this “second boogie-woogie suite” differs most strikingly from Study No. 3. Nonetheless,

the structural disintegration that is so much a part of No. 45 has a predecessor in the

rhythmic disintegration that takes place at the end of No. 3e (Figure 5-18). From there,

this process of disintegration advances to tonal disintegration at the end of 45a and, to a

lesser extent, 45b, and culminates in the structural disintegration of the I—IV—V—I pattern

throughout 45c. It also finds expression in structural irregularities such as the uneven

lengths of tonal areas in movements (a) and (b).

284

 
 
  
Beyond this, multidimensionality is expressed in different ways in Nos. 3 and 45. In

the three movements of No. 3 examined here ([a], [b], and [e]), textural accumulation, or

the gradual addition of differentiated new layers to the texture, is much more in evidence

than in No. 45, which generally has no more than two melodic layers and a bass layer.

Within No. 45’s layers, however, there is more likely to be registral polyphonization than

there is in No. 3.

Another basic difference between the multidimensional textures of the two suites is

in the potential for simultaneities among the layers. Even though portions of No. 3 are

thickly layered and the layers are effectively differentiated, the use of short, recurring

rhythmic patterns and steady ostinati—couched in a variety of tempo ratios—results in a

higher incidence of simultaneities than in No. 45, with its skewed rhythms in the bass that

rarely find accord with the other layers.

Further multidimensionality is introduced into No. 45 by the incorporation of con-

verging tempo canons that don’t quite converge, thick parallel major harmonies (par-

ticularly in movement 45c), and a greater likelihood of anticipation or delay of new tonal

areas between the upper layers and the bass (especially in movements [a] and [b] where

the harmonic patterns are still functionally present in the bass line). A greater mixing of

time types than in No. 3, and the use of geometric acceleration in No. 45a, add further to

the potential for multidimensionality in No. 45. Thus, while Nancarrow achieved a

greater variety of differentiated layers in Study No. 3, in No. 45 he achieved multi-

dimensionality in a greater variety of ways.

285

 
 
 
CHAPTER 6

SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FUTURE STUDY

The music of Conlon Nancarrow has proven increasingly fascinating to students of

twentieth-century music, but has continued to suffer from a lack of serious attention.

Some of the reasons for this include reluctance by some to take seriously Nancarrow’s

performing medium of choice, the player piano, while others (Boulez, for example) have

cited the simplicity of Nancarrow’s tonal materials in dismissing his music. Nancarrow’s

isolation in Mexico and his securing lack of interest in advocating his own music also

contributed to a continuing Obscurity.

At its heart, Nancarrow’s music is characterized by layers and the widely varying

ways in which these layers interact with each other, particularly rhythmically. As Thomas

(1996) showed in her work on multidimensionality and temporal dissonance in Nan-

carrow’s Studies, the continuum of multidimensional complexity in the music is indeed

very broad, extending from pieces that could be described as non-multidimensional (No.

37) to the simpler two-layer pieces with frequent rhythmic simultaneities (such as Nos.

14, 15, and 18) to those that teeter on the brink of the “perceptibility threshold,” such as

Nos. 3a, 41, and 48.

Nancarrow’s favored compositional device of the tempo canon has two features—

the use of tempo ratios, and placement of convergence points—that are intrinsically re-

lated and directly impact the formal interaction of the musical layers. Chapter 2 showed

that ratios in Nancarrow’s music extend beyond just the use of ratios to relate different

tempos to each other; ratios also governed and created connections between pitch pat-

terns, intervals of imitation between voices, and structure. Nancarrow’s use of ratios was

286

 
 
  
heavily influenced by those found in just intonation, but he also sought to use more

complex ratios, up to and including irrational numbers such as 1: and ‘/2. Chapter 2 also

introduced the concept of the duration ratio, the ratio that describes the relative duration

of sections in a tempo canon; the duration ratio becomes an important concept in Chapter

3’s discussion of convergence points.

Chapter 3 discussed the four basic types of tempo canons, of which the most com-

mon in Nancarrow’s music are the converging canon and the converging-diverging (arch)

canon. Considerations for measurement of the canon melody’s timespan and the concept

of potential points of simultaneity were discussed as they related to the placement of CPS.

By introducing the technique of the tempo switch, Nancarrow was able to create more

complex structures with greater numbers of convergence points (CPS), perhaps best

represented by Studies No. 24 and 43. Several of his works also include multiple sections

of Shorter tempo canons (e.g., Study No. 37), allowing further structural freedom in the

placement of CPs.

The tempo canon device also gave birth to a new kind of musical form, one in

which characteristics of fractals are represented. Chapter 4 introduced the characteristics

of fractals (self-iteration, scaling, and Space-filling) as defined in fractal geometry, and

reviewed the ways in which fractals have been identified in music and sound. The con-

cept of fractal dimension was introduced and reviewed as it relates to traditional fractals,

and a modification of the equation for finding the Hausdorff dimension, D,,' = log(N)/

log(S), was proposed for finding fractal dimension in Nancarrow’s tempo canons, which,

unlike traditional theoretical fractals, are self-limiting in the number of iterations and do

not have consistent scaling. The modified equation produces the fractal dimension based

on the number of scaled Objects (e.g., canonic voices) and the average of their scaling

relationships. It was concluded, based on the application of the modified equation to

Studies No. 14, 18, 19, 31, and 36 that:

1. The fractal dimension is generally higher in tempo canons with more voices,

287

  
especially when the following is true:

2. The fractal dimension tends to be higher as the total scaled duration decreases; in

other words, the smaller the duration of the scaled voices compared to the original, the

higher the fractal dimension is. See, for example, Studies No. 18 (Figure 4-17) and 19

(Figure 4-18) and the two hypothetical canons in Figure 4-16a; compare to the mini-

mally-scaled voices in Studies No. 36 (Figure 4-19) and No. 31 (Figure 4-20).

Finally, Chapter 5 was devoted to the analysis of two pieces: Studies No. 25 and

No. 45. Study No. 25 is the only one of Nancarrow’s works to use a 12-tone row, which it

does in conventional ways. Retrograde is found not only in the way the tone row is used,

but in entire sections that are retrogrades of other sections. The work’s five layers include

a layer of arpeggios and glissandos that extends throughout the piece, culminating in the

blistering final section where Nancarrow applies the sustain pedal for the only time in the

Studies. Study No. 25 is found to have interesting parallels with Study No. 27 in the use

Of geometric acceleration, rhythmic randomness, and the embedding of a line (No. 25’s

tone row or No. 27’s “clock line”) within different chord structures.

Study No. 45, a three-movement piece based on boogie-woogie bass lines and short

blues harmonies, is analyzed in view of how it achieves multidimensionality as described

by Thomas (1996), and the results are compared to Study No. 3, the “Boogie-woogie

Suite.” The pieces achieve multidimensionality in different ways. No. 3 relies more

heavily on textural accumulation, or the gradual addition of temporally diverse layers,

reaching a peak in No. 3a where multidimensionality teeters on the brink of the “percep-

tibility threshold.” No. 45 does not reach such extremes of multidimensionality since it

uses fewer layers, but it achieves a wider variety of multidimensional textures by in-

corporating techniques of registral polyphonization, less tonal coordination among layers,

more mixing of time types (including acceleration in some layers against steady tempo in

other layers), and tonal and structural disintegration.

288

 
 
 
 
 
Research Issues

Score Inaccuracies

Future research on Nancarrow’s Studies will have to address the problem of occa-

sional inaccuracy in his scores. Although Nancarrow’s scores are carefully drawn and a

very helpful aid to the analyst, they contain numerous errors and omissions. Appendix A

of this dissertation identifies a number of score inaccuracies found by this author, but the

list is by no means complete. The final authority on Nancarrow’s true intentions in the

Studies is the player piano rolls themselves (and by extension the recordings made from

the rolls), of which the scores represent Nancarrow’s best—but imperfect—representa-

tion.

There are some score problems that will only be resolved by analysis of either the

piano rolls themselves or extremely slow playback. Study No. 25 provides some excellent

examples of apparent score errors that defy verification from listening to full-speed

playback of the recordings. Section 8, the final section of this Study, blitzes by the

listener at a speed of about 88 notes per second, or as Gann describes, “too fast for the ear

to register as individual events” (p. 248). Difficulties in analyzing this section are com-

pounded by the apparent errors in the score. The section contains numerous major key

scalar passages, some several octaves long; the same key area is maintained throughout

each passage. Figure 6-1 identifies two areas in this section where scalar passages appear

to be intended but all the necessary accidentals are not there. In example (a), the passage

at first appears to be a G], major scale, but only the first of four C’s in the passage is

fiatted, leading one to question whether the passage is really in G1, or 1),; and, in example

(b), a 2-octave—plus segment in B major is missing four sharps in the middle. (It turns out

that this latter omission is confirmed by comparison with p. 35, where this passage earlier

occurs.)

289

 
(a)

(b)

Figure 6-1. Questions about score accuracy in Study No. 25: (a) first C in scalar passage is flatted
while three others are not (from bottom system, p. 75); (b) scalar passage apparently in B major
in which Sharps appear to be missing in circled area (from top system, p. 76).

The average listener will not be able to verify such errors by listening to a full-

speed playing of this piece—only very slow playback or examination of the piano rolls

will allow such errors to be corroborated with complete assurance. And, in the case of

this section from Study No. 25, the use of the sustain pedal through this entire passage

exacerbates the difficulty of making this assessment from the recording alone.

In general, when analyzing the Studies from the scores, the analyst must be con-

stantly aware of the compositional processes Nancarrow has used. Maintaining this

awareness will occasionally lead to fruitful questions about score accuracy. Many of the

errors—particularly rhythmic and metric errors, such as missing rests and time signa-

tures—are obvious at once. The plethora of repeating patterns (canons, ostinati, iso-

rhythms, tone rows, repeating chord patterns, blues harmonic patterns, etc.) in Nancar-

row’s music allows one to arrive at all sorts of other questions about score accuracy.

Some of these types of errors include chord spellings, rhythmic and pitch patterns (such

as ostinati and countdown patterns), and other variations to established patterns. Here are

some examples (all are chronicled in Appendix A):

1. Chord spellings. In Study No. 1 (which is not in Nancarrow’s hand but was pro-

fessionally copied for publication in Henry Cowell’s New Music in 1952 [Gann 1995,

70]), the opening section has a repeating series of three-note open fifth chords on Dr, G1,,

and A]; on p. 2 at the end of the second system there is a three-note chord spelled

Dt—Gt—Dt.

290

 
 
  
Another example is in Study No. 11, which is a series of changing textures based on

a blues harmonic pattern and a fifteen-note isorhythm. In the fifth texture (begins p. 7),

there is a major seventh chord (with a missing fifth) in the first and third voices from the

top at the beginning of each 3 measure, with the exception of the last measure in the

second system of p. 7; my belief is that the two notes in the top staff in this measure are

one leger line too high, and should be A and D rather than C and F, making the chord

Bt—A—D.

2. Rhythmic and pitch patterns. In Study No. 12, the meter signature in the partial

measure at the end of the first system on p. 9 should be 1% rather than 1% ; this not only

reflects the actual rhythmic content of the measure, but the metric pattern 5—4—3—4—5—9,

of which the 1% measure is a part, has been stated several times earlier in the piece.

In Study No. 3a (first movement of the “Boogie-woogie Suite”), the entire move-

ment is built over a 12-bar blues harmonic pattern. On p. 16, during the thirteenth state-

ment of the pattern, the F pattern from the end of the second system should only be

repeated once instead of four times. The third and fourth systems should have the follow-

ing harmonic patterns: F—C—C and G—G—C.

Some errors in patterns are more difficult to detect. Study No. 5 contains an

elaborate system of rhythmic countdowns in which the rests between short ostinato

patterns progressively decrease in length by ten sixteenth notes at a time; eventually there

are eight different patterns taking place. Careful monitoring of the countdowns (which

can somewhat easily be done because each system is exactly 70 sixteenth notes long)

reveals that on p. 13 there is an ostinato statement missing in the tenth system from the

top. On p. 19, in the second system the 1% ostinato pattern is placed one sixteenth note

too soon. There are also some inconsistencies in the length of the rests, with some of

them being from one to ten sixteenth notes off according to the established pattern.

All of the examples listed above are from Studies that are metrically notated. Score

problems are more difficult to detect, of course, in scores that are proportionally notated,

291

 
although awareness of patterns and systems such as canon and ostinati continue to be

helpful in spotting score errors in these Studies. In Study No. 27, for instance, it is

evident from observing the canon on p. 3 that rests are missing in the bottom voice after F

in the second system and after B in the third system (in each case, half the value of the

sustained note should be a rest). In other scores such as that to No. 21 (“Canon X”), one

has to wonder how accurate the proportional notation is throughout such a complex score,

especially in the more densely notated sections.

In the Studies where geometric acceleration/deceleration is used, it is sometimes

perilously difficult to determine the percentage of change being used when Nancarrow

has not specified it. Recall, for instance, my disagreement with Gann on the rates of

acceleration in Study No. 25 (Chapter 5) and elsewhere.

Final Thoughts

Conlon Nancarrow’s Studies for Player Piano represent a remarkable compositional

achievement, one in which the composer Spent some fifty years exploring the possibilities

of the player piano as a performing medium that afforded him the compositional freedom

he craved. While not every one of the Studies is a masterpiece, each makes a contribution

in addressing interesting compositional problems set forth by the composer, and most of

these problems involve creating different rhythm and temporal relationships.

Nancarrow’s greatest compositional achievement, I believe, is in the creation of the

genre of the tempo canon, of which Studies No. 36, 40, 41, and 48 represent the most

spectacular examples. It is in these pieces that Nancarrow most successfully creates struc-

tures that capitalize on the dynamic relationship between ratios and placement of con-

vergence points. The fractal forms created in many of the tempo canons also represent a

unique musical achievement.

Nancarrow’s achievements in the use of accelerative techniques must also not be

overlooked. In the ability to create “curved time,” as Gann calls it, Nancarrow found an

292

 
  
exciting new way to constantly alter the dynamic relationship among layers, either grad-

ually through geometrical acceleration or exponentially through additive acceleration.

Where does research on Nancarrow’s Studies for Player Piano go from here? From

my vantage point in 2002, five years after Nancarrow’s death, the only thing that is clear

is that the field is still emerging. A very good start has been made in introducing Nan-

carrow’s music to a wider audience, with Carlsen’s insightful 1986 monograph, Gann’s

comprehensive book of 1995, and Thomas’s groundbreaking analysis of multidimension-

ality and temporal dissonance in 1996. There are many more contributions waiting to be

made, and much more to be found in this music. It is my sincere hope that interest in the

music of this reclusive and innovative composer will continue to build among twenty-

first-century musicians, and that it will increasingly receive the wider critical attention it

so richly deserves.

293

APPENDIX A: ERRATA TO SCORES OF THE PLAYER PIANO STUDIES

Conlon Nancarrow’s hand-written scores, while generally quite precise, contain numer-
ous errors. Most commonly these are errors of omission (e.g., missing rests, accidentals,
dots, ties, meter markings, clef signs, and octave displacement lines); more occasionally,
notes are misplaced on the staff by one line or space. The errors can often be deduced due
to the use of transparent compositional methods such as canon, isorhythm, and other
clearly recurring patterns.

In order to be as profitable as possible to scholars studying Nancarrow’s scores, the fol-
lowing list of score errors includes those that are already noted in the major sources in the
literature as well as those noted by this author during the course of this study. In all cases,
the final authority on the accuracy of the scores is the recordings made from the piano
rolls; all errors listed here have been verified on the recordings whenever possible.

0noted in Margaret Thomas, Conlon Nancarrow ’s ‘Temporal Dissonance Rhythmic and

Textural Stratification in the Studiesfor Player Piano (1996)

+noted in Kyle Gann, The Music of Conlon Nancarrow (1995) or in Gann’s score copies
§noted in Philip Carlsen, The Player Piano Music of Conlon Nancarrow: An Analysis of

Selected Studies (1988)

Study #1 (This is the only published score not in NancarTow’s hand. Apparently Elliott

Carter had the score recopied before publication in New Music. See Gagne and
Caras, 1982, p. 291).

Eb
Ab
p. 2, m. 4, bass clef: chord should be spelled d’
p. 9, third system, bottom staff: bar line is missing after first two notes; sixth note (DI)

Should be a dotted eighth note (dot is missing)

p. 10, second system, second staff: fifth note should be C#6 (sharp Sign is missing)
p. 14, first system, bottom staff: dotted-Sixteenth rest missing at end of first complete

measure

SLIM
p. 4, second system, middle (second) staff: eighth rest missing on seventh eighth-note

beat

p. 8, third system, middle staff: quarter rest missing after each sixteenth note
p. 16, third system, bass staff: ostinato should switch to C pattern in second measure;

also, in last system ostinato should be G pattern for first two measures and become C
pattern in last measure

294

 
 
 
 
t

# b

0p. 1, first system, bass staff: second note in first measure (D93) should be dotted eighth

note (dot is missing)

p. 5: all initial clefs are missing from both systems
p. 5, first system, bottom staff: second note in first measure should be a dotted eighth

note (dot is missing)

p. 5, second system, bottom staff: first note should be B1,l (flat Sign is missing)
p. 8, second system, bottom staff: repeat Sign is missing in second measure
p. 12, bottom system, top staff: in second measure, first and sixth chords should contain

C# octaves (both Css are missing the sharp Sign)

p. 21, first system, third staff: in second measure, fourth chord should be a dotted eighth

note (dots are missing)

p. 24, last system: initial clefs missing (treble-treble-bass)

Study #3c
0p. 3, third system, top staff: first note in second measure (A1,,) should be dotted quarter

note (dot is missing)

Stud # d
0p. 1, fourth system, top staff: second chord of second measure should be D# octave

(sharp Sign is missing on top note)

0p. 1, fourth system, bottom staff: dotted-half note chord on second heat should have flat

signs repeated for notes B13 and D1,,

Study #4
p. 1, second system, treble staff: tie is missing from first note (E16)
p. 1, third system, bass staff: F2 should be tied over for 6 bars rather than change to G2
p. 7, second system, middle staff: first measure contains 8 sixteenth notes and does not fit

1% meter; perhaps dot should be removed from dotted quarter?; also, in the same
measure, the staff above should contain 5 Sixteenth notes—flag should be added to
first note. It is difficult to determine what is intended here s'mce each line of the system
might contain a different (and constantly varying) number of sixteenth-note beats.
p. 8, first system, bass staff: third note from end of that line should be A1,l (flat sign is

missing)

p. 8, second system, top staff: penultimate note should be B, and not G,

p. 8, second system, bass staff: 16 meter srgnature rs rmssrng after 16 measure

5

.

.

.

.

13

5
p. 8, last system, bass staff: in the three 16 measures that follow the 16 measure, the

10

rhythm should be DJ) (flag is missing from first note in each of those measures)

p. 10, second system, bass staff: meter should not change to 16 until third measure

5

p. 10, last system, bass staff: meter signature for first complete measure should be 16 and

5

6
for next measure 16

p. 12, first system, bass staff: last note of that line should be E12 (flat Sign is missing)

295

tud #5

p. 13, tenth staff: 16 ostinato (see ostinato figure on p. 11) should be stated beginning

7

with 47th sixteenth note of that line—figure should be preceded by blank 16 measure

35

21

and 16 measure

p. 19, second staff: ostinato figure should be notated one sixteenth note later than it is

35
p. 20, third staff: time signature 16 should precede last note (C7)

p. 28, fifth staff: three time signatures missing (16 ,16 ,16 )
p. 32, first staff: sharp Sign missing for F#, across whole line
p. 32, fourth staff: flat signs missing from chord B1,5—D1,7 on last note

21 30 21

M
p. 11: all initial clef signs are missing (treble-treble-bass-bass)

Study #7
p. 1, third system, treble staff: tie missing between sixth and seventh measures
p. 1, last system, treble staff: tie missing between fourth and fifth measures

Study #8
p. 1, third system, bottom (bass) staff: first note (G2) should be an eighth note

Study #9

[)6
p. 3, first system, top staff: m. 4 should be Bbs eighth note on beat 1, and present mm. 4—5
Should become mm. 5—6 (in other words, present mm. 4-5 should be notated one
measure later)

p. 14, third system, bottom staff: sharp sign is missing for bottom note of second chord

(should be F# octave)

Study #10

+p. 4, second system, bottom staff: in third measure, chord should change to 62 and be

F3
[)3

tied over to fourth measure

Study #11

p. 7, second system, top staff, last measure: chord should be A6 for correct voicing of

[)7

M7 chord (no 5th) that occurs consistently on each 8 bar that opens a 20 eighth-note
segment

5

Study #12

15
p. 4, second system, top staff: dotted eighth rest needed at beginning of line to fill up 16

measure

296

i-u-u-up G-Ifl-

p. 6, first system, bottom staff, third measure: flag is missing on second note (should be

dotted eighth note)

9

6

.

.

p. 9, first and second systems: trme srgnature at end of first system should be 16 (not 16 )

to be consistent with metric pattern of opening canon the first three times, and

following measure should have 16 signature

6

p. 12, third system: last meter change (16 ) is notated one bar too early

27

Study #16
0p. 7, staff “e”: first note of second measure should be J rather than D

tud #17

p. 4, first system, top staff: first chord on that line should be an octave G

5
p. 4, third system, top staff: tempo change (J = 138) not notated at double bar (8

measure)

p. 8, first system, bottom staff: clef Should be bass and not treble

M
+(Gann notes existence [p. 291, n4] of errors in the score, particularly in rhythmic series)

Study #22
p. 5, second system, middle staff: this staff should begin with a continuation of the rest

that ended the previous line

Study #23
p. 1, third system, bottom staff: 8va. b. line is missing below this staff (without this,

Gann’s assertion [p. 156] that the same note is not used twice in this line is not true)

tud #24

p. 12, third system, bottom staff: chord in first 4 measure should be open fifth stack, and

6

F2 should not be flatted (see Gann, p. 176, transposition levels for Canon 9)

MILLS
p. 1, first system, top staff: third “exploded” arpeggio is based on overtone series, so fifth

note should be A# rather than A

p. 2, first system, bottom staff: extension line after B2 is missing (note Should be

sustained)

p. 5, first system, fourth staff: top note of second chord should be C#., (leger line is

missing)

p. 6, second system, bottom staff: first, sixth, and seventh chords are missing notes in this

context of root position dominant seventh chords

p. 7, first system, bottom two staves: note E1, appears to be missing (context is root

position dominant seventh chords)

p. 8, second system, second staff: extension lines to indicate sustained notes are missing

from final chord in the system

297

p. 20, second system, arpeggio layer: accidentals are missing in second and third

arpeggios; second arpeggio should state first a B1, and then a C triad, while third
arpeggio Should state first an F and then a B triad

p. 21, first system, arpeggio layer: accidentals are incorrect in first arpeggio,which should

state first a D and then an A1, (not A) triad

p. 21, second system: left brace is missing
p. 22, first system, arpeggio layer: in second arpeggio, final note should probably be G3

rather than D, to fit pattern of 16-note arpeggios stating first 16 notes of the overtone
series

p. 23, first system, arpeggio layer: first arpeggio appears to be missing note E1, (to

complete the overtone series)

p. 24, Sixth staff: in third (last) arpeggio 8 b. marking is missing under last five notes
p. 24, bottom Staff: Stems missing from all notes
p. 26, bottom staff: bass clef sign missing between sixth and seventh notes
p. 32, fourth staff: initial bass clef sign is missing
p. 41, first system, arpeggio layer: final 16-note arpeggio is based on G overtone series,

and last two notes should be G3 and G2

+p. 44, second system, top two staves: C3 in first arpeggio is probably an error as it does

not fit E, major arpeggio (see Gann p. 294 n. 5)

p. 50, bottom staff: to fit the row statement (pitch 6 of P11), bottom note of second chord

should be E and not D#

p. 53, fourth staff: extension lines for sustained notes are missing after first chord
p. 59: left brace is missing, and bottom three clefs from top down should be treble, treble,

bass

p. 59, bottom three staves: extension lines from previous page should extend to beginning

of first staccato chord

p. 74, bottom system: left brace is missing, and clefs should be treble, bass
p. 76, top system: under the second 8va. marking, it appears that last five notes are

missing Sharps to fit B major scale

td #26

p. 5, second system: all initial clefs are missing; the order, from top, should be: treble,

treble, treble, treble, treble, bass, bass

tud #27

0p. 3, second system, bottom staff: there should be a brief rest notated after Fl, occupying

half the note value

0p. 3, third system, bottom staff: there should be a brief rest notated after El, occupying

half the note value

Stud #2
p. 2, first system, third staff (first “occupied” staff): octave higher designation is missing

from third note through bass clef

p. 3, second system, third staff (first “occupied” staff): line denoting octave lower is

missing

Study #31

0p. 2, thrrd system: erghth note rests are rmssrng 1n entrre line (meter IS 8 )

.

.

.

.

.

.

4

.

.

298

p. 3, second system, middle (second) staff: rhythm in sixth complete measure should be

7

Study #32

p. 1, fourth system: 3 meter marking is missing in third measure and 3 meter marking in

fourth measure

p. 2, fourth system, middle staff: 8va b. continuation line is missing under first two

measures

p. 3, first system, middle staff: first note of fourth full measure should be F# rather than

Fr

p. 3, second system, bottom staff: 8 meter marking is missing in last measure of the line

6

p. 5, second system, bottom staff: all meter changes are missing in this line: 8 in second

complete measure, 8 1n next measure, and 8 In final measure

6 .

9 .

p. 5, third system, second staff from top: octave higher designation in mm. 4 and 5 is

erroneous

p. 5, third system, bottom staff: 8va b. marking should continue through first note
p. 6, first system, second staff from top: note in last complete measure should be F2 rather

than F3

p. 6, second system, top staff: 3 meter marking is missing in last (incomplete) measure

p. 6, third system, top staff: C11?5 J is missing (tied over from previous system) in

incomplete measure at beginning of line

p. 6, third system, top staff: 8 meter marking is missing in first complete measure
p. 6, third system, top staff: note in seventh complete measure should be C, rather than C3

6

p. 8, first system, bottom Staff: 8 meter marking is missing in first full measure

9

p. 8, second system, bottom staff: 8 meter marking is missing in first full measure

6

p. 9, first system, bottom staff: all meter changes are missing in this line: 8 in second

9

6
complete measure, 8 two measures later

p. 10, third system, top staff: change to treble clef missing in third full measure

Study #33

p. 30, second system, bottom staff: first complete measure should have a 16 meter

5

signature

Study #34

p. 5, last system, middle staff: 3 measure is missing three eighth rests (beats 2, 4, and 7)
p. 7, second system, top staff, fifth full measure: eighth rest missing on second beat
p. 11, second system, bottom staff, third measure: rest on second beat Should be eighth

rest rather than quarter rest

299

p. 11, third system, middle staff, third measure: eighth rests missing on beats 2 and 4
p. 11, third system, bottom staff: articulations missing in entire line (all eighth notes

staccato except those beamed to a following eighth note; beamed eighth notes slurred
together)

Study #3;
+p. 27, last system: tempo designation in bottom two staves should be J = 283/V3 rather

than 204

M

p. 2, third system, top staff: 4 meter signature should appear prior to the entrance of this

4

voice

p. 18, third system, bottom staff: in last measure, top note of half-note chord should be

D1,3 (flat Sign missing)

p. 47, first system, third staff: 4 meter change should be inserted at beginning of third

4

full measure

p. 48, first system, bottom staff: 4 meter change should be inserted at beginning of third

4

measure

Stud #37

p. 60, seventh staff from top: 8 measure is missing an eighth rest at end of measure
p. 67, seventh staff from top: tempo marking (J = 200) is missing at beginning of this

4

staff

Stud #41a
p. 1, first system, top notated staff: change to bass clef is needed before final four notes of

system

p. 3, third (last) system, third staff: line indicating drone note should be on B2 rather than

G:

p. 5, first system, top staff: 23-note run at beginning of system should be marked pp

rather than ff

Stud #41
p. 25, second system: left brace is missing in this system

Stu

#4

p. 1, second system, bottom staff: rests are missing after first quarter note in 4 measure:

II

dotted half rest followed by whole rest

p. 1, third system, third staff: initial treble clef is missing
+(marked in Gann’s copy of the score) p. 1, third system, fourth (lowest) staff: first note

7

of 4 measure should be E#3, not G#3
4

p. 6, second system, bottom staff: first 4 measure is missing a quarter rest on second beat

300

.

.

p. 6, tlurd system, second staff from top: first 4 measure 13 rrussrng a quarter rest on

4

.

.

.

second beat

p. 9, first system, thrrd staff from top: quarter rest rs mrssrng at end of 4 measure
p. 21, second system, second and fourth staves: in measure that begins with shared

.

.

11

.

barline, clef should change from treble to bass

8
p. 35, third system, bottom two staves, and p. 36, first system, bottom two staves: 4

measure that begins on bottom of p. 35 concludes after first half note at top of p. 36;

10

barline and 4 meter signature should be inserted after this half note and the present
10
4 barline deleted

Study #45a
p. 9, top system, middle staff: glissando at beginning of system apparently is missing a

concluding A,

p. 11, top system, middle staff: glissando is missing a concluding E,
+p. 12, bottom system, top staff: tenth note should be a C major triad rather than note G,5
p. 21: piece concludes in the bass layer with a C octave, which is not noted

Study #45b
+p. 2, first system, bass staff: the eighth note (F2) occurs earlier in the recording than

indicated on the score—it occurs before, not after, the grace notes and chord in top
vorce

p. 8, bottom system, middle staff: after fourth eighth note, a B, eighth note is missing (in

the same rhythm as previous four notes)

+p. 9: Gann notes that an entire system (system 23) is apparently missing after top system

on this page

p. 13, bottom system, top staff: the glissando at the end of the line appears to be missing

the note D5

U
'
P
'
P
'
P
'

9
'

. 14, second system, middle staff: 8b. indication missing under first four notes
14, second system, top staff: 8va. indication missing over chord F#—A#—C#
15, middle system, top staff: sustain line from top staff of previous system is missing
15, bottom system, bottom staff: after second C, (near middle of system), note E1, is
missing
17, top system, bottom staff: fourth note (written as F#,) Should be one octave higher
(F#.)

p. 18, top system, bottom staff: eighth note(written as C,) Should be one octave lower

(C3)

Study #4Sg
p. 2, third system, bottom staff: eleventh and twelfth eighth notes should be A1 rather than

A natural

p. 8, fourth system, top staff: A major triad is sustained after third glissando, but is

missing In score

p. 14, first system, second staff: line should end with a major triad eighth note built on F,

301

 
#48

Stu
pp. 3—6: some of the initial clefs are missing on the bottom system (p. 3, treble clef on

third line from bottom, bass clef on second line from bottom; p. 4, same as p. 3; p. 5,
treble clef on third line from bottom; and p. 6, bass clef on second line from bottom)

302

APPENDIX B: AVAILABLE SCORES, RECORDINGS,
AND WRITINGS FOR EACH STUDY

Key:
Study #

52m
Recording
Writings

N.B. Each of the Studies is discussed by Kyle Gann in The Music of Conlon Nancarrow.
(Cambridge: Cambridge University Press, 1995) so that information is not repeated in the
listings below.

#1

#2

_S_co_r_'§: Nancarrow, Conlon. Rhythm Study No. I for Player Piano. 1952.
Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10,11, 12,13,15, 16, 17,18,19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S-l786, 1981.

Writing: Tenney, James. “Studies for Player Piano Vol. 111: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. IH/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Score: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, I4, 20, 21, 24, 26, and 33 for Player Piano . Santa Fe: Soundings Press,
1984.

Rpgordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S-1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. IH: Notes on Studies #1 , 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43. 50.” In Conlon Nancarrow: Studiesfor
Player Piano, Vols. III/IV [compact discs], pp. 7-19. Wergo: WER 60166/67-50,

303

 
   
 
#3

#4

#5

1987.

Thomas, Margaret Elida. Conlon Nancarrow ’3 ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

m: Nancarrow, Conlon. Study No. 3 for Player Piano. Santa Fe: Soundings Press,

1983.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 1 [Studies No. 3, 20,

41]. 1750 Arch: S—1768, 1977.

Writings: Tenney, James. “Studies for Player Piano Vol. 1: Notes on Studies #3 (a—e),
20, 41 (a—c), 44.” In Conlon Nancarrow: Studies for Player Piano, Vols. I/II
[compact discs], pp. 7-11. Wergo: WER 6168-2/6169-2, 1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

m: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recording; Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 4la, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—1777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
I4, 22, 26, 31, 32, 35, 3 7, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/II [compact discs], pp. 7—1 1. Wergo: WER 6168-2/6169-2,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studiesfor Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Score: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/II [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

304

#6

#7

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S-l777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. 11: Notes on Studies #4, 5, 6,
14, 22, 26, 31, 32, 35, 37, Tango .7, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/II [compact discs], pp. 7-1 1. Wergo: WER 6168-2/6169-2,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Sppm: Nancarrow, Conlon. Collected Studiesfor Player Piano, Vol. 5: Studies No. 2,
6, 7, 14, 20, 21 , 24, 26, and 33 for Player Piano . Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. l/II [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S-l777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
14, 22, 26, 31, 32, 35, 37, Tango .7, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/II [compact discs], pp. 7—1 1. Wergo: WER 6168-2/6169-2,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

SEE: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, 14, 20, 21, 24, 26, and 33 for Player Piano. Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. III: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. III/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wamaby, John. “Conlon Nancarrow: Studies for Player Piano; Piece No. 2 for Small

305

 
  
“
.
T

Orchestra; Tango; Toccata; Trio; Sarabande and Scherzo” [review]. Tempo no.
189 (June 1994): 49—50.

#8

S993: Nancarrow, Conlon. Conlon Nancarrow: Selected Studies for Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring—Summer 1977.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—l786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. 111: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studiesfor
Player Piano, Vols. III/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

Carlsen, Philip. The Player Piano Music of Conlon Nancarrow: An Analysis of

Selected Studies. I.S.A.M. Monographs, no. 26. Brooklyn, N.Y.: Institute for
Studies in American Music, 1988.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Spope: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, II, 12, 15, 16, 17, and 18for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, ll, 12, 13, 15, 16, 17, l8, 19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—l798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, I 1,
12, 13, I6, 17, 18, I9, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. III/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11—13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

306

 
#10

m: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, I 1, 12, 15, I6, 17, and 18for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, ll, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studiesfor Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. III: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. III/IV [compact discs], pp. 7-19. Wergo: WER 60166/67-50,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Sm: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, I6, 17, and 18for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11,12, 13,15,16, 17,18,19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studiesfor Player Piano, Vol. 4 [Studies Nos. 9, 11,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—l798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, 16, 17, I8, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. III/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studiesfor Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27-29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#11

#12

m: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, II, 12, I5, 16, 17, and 18for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12,13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studiesfor Player Piano, Vol. 4 [Studies Nos. 9, ll,

12, 13, 16, 17, l8, 19, 27, 28, 29, 34, 36]. 1750 Arch: S-1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, II,

307

  
12, I3, 16, 17, 18, I9, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. III/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High

Fidelity/Musical America 34 (May 1984): 12—13.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance ’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wamaby, John. “Conlon Nancarrow: Studies for Player Piano; Piece No. 2 for Small
Orchestra; Tango; Toccata; Trio; Sarabande and Scherzo” [review]. Tempo no.
189 (June 1994): 49—50.

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#13

Sm: never released by Nancarrow (see Gann, p. 68)
Recprdings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. IH/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17,18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, I6, 17, 18, I9, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. III/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#14 (Canon 4/5)

Spore: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, I4, 20, 21, 24, 26, and 33 for Player Piano. Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. l/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—l777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
14, 22, 26, 31, 32, 35, 37, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/II [compact discs], pp. 7—11. Wergo: WER 6168-2/6169-2,
1987.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High

Fidelity/Musical America 34 (May 1984): 12—13.

308

  
Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#15 (Canon 3/4)

W: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, I5, 16, 17, and 1 8for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. IH/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18,19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—1786, 1981.

Nancarrow, Conlon. Continuum Performs Nancarrow. MusicMasters: 7068-2-C,

1991.

Writings: Tenney, James. “Studies for Player Piano Vol. HI: Notes on Studies #1, 2a,
2b, 7, 8, 10, I5, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. IH/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance ’.° Rhythmic
and Textural Stratification in the Studiesfor Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#16

Sm: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, I7, and I 8for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
NancarTow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, I I,
12, I3, 16, 17, 18, I9, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. IH/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#17 (Canon 12./15/20)

Score: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, I7, and 18for

309

  
Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. IH/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, ll, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, ll,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—l798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, 16, I7, 18, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. IH/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studiesfor Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45-46.

#18 (Canon 3/4)

W: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, I6, 17, and 18for

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, ll, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, I6, I 7, I8, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. III/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance ’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11—13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#19 (Canon 12/15/20)

Scprp: Nancarrow, Conlon. Conlon Nancarrow: Selected Studies for Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring-Summer 1977.

Rgcordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.

310

Nancarrow, Conlon. Complete Studiesfor Player Piano, Vol. 4 [Studies Nos. 9, ll,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, I I,
12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. HI/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Carlsen, Philip. The Player Piano Music of Conlon Nancarrow: An Analysis of

Selected Studies. I.S.A.M. Monographs, no. 26. Brooklyn, N.Y.: Institute for
Studies in American Music, 1988.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance ’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27-29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#20

Score: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, I4, 20, 21, 24, 26, and 33 for Player Piano . Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 1 [Studies No. 3, 20,

41]. 1750 Arch: S—1768, 1977.

Writings: Tenney, James. “Studies for Player Piano Vol. 1: Notes on Studies #3 (a—e),
20, 41 (a—c), 44.” In Conlon Nancarrow: Studiesfor Player Piano, Vols. I/H
[compact discs], pp. 7—11. Wergo: WER 6168-2/6169-2, 1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studiesfor Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#21 (Canon X)

m: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, 14, 20, 21, 24, 26, and 33 for Player Piano . Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. IH: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. IH/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,

311

1987.

Thomas, Margaret Elida. Conlon Nancarrow ’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#22 (Canon 1%Il-1/2%/2-l/4%)

Sgo_r_e_: unpublished
Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—l777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
14, 22, 26, 31, 32, 35, 37, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/H [compact discs], pp. 7-11. Wergo: WER 6168-2/6169-2,
1987.

#23

m: Nancarrow, Conlon. Conlon Nancarrow: Selected Studiesfor Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring—Summer 1977.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, l6, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studiesfor Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. III: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. HI/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

Carlsen, Philip. The Player Piano Music of Conlon Nancarrow: An Analysis of

Selected Studies. I.S.A.M. Monographs, no. 26. Brooklyn, N.Y.: Institute for
Studies in American Music, 1988.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#24 (Canon 14/15/16)

Score: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, I4, 20, 21, 24, 26, and 33 for Player Piano. Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, l7, l8, 19, 21, 23, 24, 25, 27, 28, 29, 33,

312

  
  
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. HI: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. IH/IV [compact discs], pp. 7-19. Wergo: WER 60166/67-50,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#25

Sc_ore: Nancarrow, Conlon. “Study No. 25 for Player Piano.” Soundings 9 (Summer

1975).

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, ll, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S—1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. HI: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. III/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#26 (Canon 1!] [ll/l/l/l/ll)

Sco_re: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, 14, 20, 21, 24, 26, and 33 for Player Piano. Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No.4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—l777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
14, 22, 26, 31, 32, 35, 37, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/II [compact discs], pp. 7-11. Wergo: WER 6168-2/6169-2,
1987.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High

Fidelity/Musical America 34 (May 1984): 12-13.

313

#27 (Canon 5%/6%/8%/11%)

Spgpe: Nancarrow, Conlon. Conlon Nancarrow: Selected Studies for Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring-Summer 1977.

Recprgings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, ll, 12, 13, 15, l6, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, ll,

l2, 13, 16, l7, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—l798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, I I,
12, 13, 16, 17, I8, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. HI/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Jarvlepp, Jan. “Conlon Nancarrow’s Study #2 7for Player Piano Viewed

Analytically.” Perspectives ofNew Music 23/1—2 (Fall-Winter 1983/Spring-
Summer 1984), 218-22.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45-46.

Scpr_e: unpublished
Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S-1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, I6, 17, I8, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. IH/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#28

#29

Sflrpz unpublished
Recprdings: Nancarrow, Conlon. Studies for Player Piano, Vols. IH/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12,13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,

314

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S—l798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. HI/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#30

Score: Nancarrow never produced a score (see Gann, p. 169)
Recordings: No commercially available recordings (see Gann, p. 169)

#31 (Canon 21/24/25)

Sm: Nancarrow, Conlon. Conlon Nancarrow: Selected Studies for Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring—Summer 1977.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—1777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
I4, 22, 26, 31, 32, 35, 37, Tango?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/II [compact discs], pp. 7—1 1. Wergo: WER 6168-2/6169-2,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#32 (Canon 5/6/7/8)

S_ccle: unpublished
Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. IIH [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 408, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studiesfor Player Piano, Vol. 2 [Studies No.4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—1777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
14, 22, 26, 31, 32, 35, 37, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/H [compact discs], pp. 7—1 1. Wergo: WER 6168-2/6169-2,
1987.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High Fidelity/

Musical America 34 (May 1984): 12—13.

Thomas, Margaret Elida. Conlon Nancarrow ’s ‘Temporal Dissonance’: Rhythmic

315

 
and Textural Stratification in the Studiesfor Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#33 (Canon 12/2)

m: Nancarrow, Conlon. Collected Studies for Player Piano, Vol. 5: Studies No. 2,
6, 7, 14, 20, 21 , 24, 26, and 33 for Player Piano. Santa Fe: Soundings Press,
1984.

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. IH/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 3 [Sonatina, Studies

Nos. 1, 2, 7, 8, 10, 15, 21, 23, 24, 25, 33]. 1750 Arch: S-1786, 1981.

Writings: Tenney, James. “Studies for Player Piano Vol. IH: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studiesfor
Player Piano, Vols. III/IV [compact discs], pp. 7—19. Wergo: WER 60166/67—50,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

9
#34 (Canon 4/5/6/ 4/5/6/ 4/5/6)

10 /

11

/

Sgpgg: unpublished; string trio version exists (see Gann, p. 132)
Recprdings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, ll,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S-1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, I I,
12, I3, 16, 17, 18, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. HI/IV [compact discs], pp. 20—30. Wergo: WER
‘
60166/67-50, 1987.

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High FidelityMusical America 34 (November 1984):
45—46.

#35

Sag: Nancarrow, Conlon. Conlon Nancarrow: Selected Studies for Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring-Summer 1977.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. [In [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

316

 
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—l777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
I4, 22, 26, 31, 32, 35, 3 7, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/H [compact discs], pp. 7—1 1. Wergo: WER 6168-2/6169-2,
1987.

Carlsen, Philip. The Player Piano Music of Conlon Nancarrow: An Analysis of

Selected Studies. I.S.A.M. Monographs, no. 26. Brooklyn, N.Y.: Institute for
Studies in American Music, 1988.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#36 (Canon 17/18/19/20)

Spppe: Nancarrow, Conlon. Conlon Nancarrow: Selected Studies for Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring—Summer 1977.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, ll, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,

12, 13, 16, 17, 18, 19, 27, 28, 29, 34, 36]. 1750 Arch: S-1798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, 16, I7, 18, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. III/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

Carlsen, Philip. The Player Piano Music of Conlon Nancarrow: An Analysis of

Selected Studies. I.S.A.M. Monographs, no. 26. Brooklyn, N.Y.: Institute for
Studies in American Music, 1988.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance ’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Wierzbicki, James. “Nancarrow: Studies for Player Piano, Nos. 9, 11-13, 16—19,

27—29, 34, 36” [review]. High Fidelity/Musical America 34 (November 1984):
45—46.

#37 (Canon 150/160 kl [168 3/4 I180/187 1’2 [200/210/225/240/250/262 V2 [281 Kl )

Sm: Nancarrow, Conlon. Study No. 3 7for Player Piano. Santa Fe: Soundings

Press, 1982.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/II [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

317

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—l777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
I4, 22, 26, 31 , 32, 35, 37, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. l/H [compact discs], pp. 7-11. Wergo: WER 6168-2/6169-2,
1987.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

renumbered by Nancarrow as Study No. 43

renumbered by Nancarrow as Study No.48

#38

#39

#40

m: Nancarrow, Conlon. Conlon Nancarrow: Selected Studies for Player Piano
[With Critical Material by Gordon Mumma, Charles Amirkhanian, John Cage,
Roger Reynolds, and James Tenney], edited by Peter Garland. Soundings, Book
4. Berkeley, CA: Soundings Press, Spring-Summer 1977.

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. I/II [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

14, 22, 26, 31, 32, 35, 37, 40]. 1750 Arch: S—1777, 1979.

Writings: Tenney, James. “Studies for Player Piano Vol. H: Notes on Studies #4, 5, 6,
I4, 22, 26, 31, 32, 35, 37, Tango ?, 40 (a, b).” In Conlon Nancarrow: Studies for
Player Piano, Vols. I/H [compact discs], pp. 7—1 1. Wergo: WER 6168-2/6169-2,
1987.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High

Fidelity/Musical America 34 (May 1984): 12—13.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#41

S_cgr_e: Nancarrow, Conlon. Study No. 41 for Player Piano. Santa Fe: Soundings

Press, 1981.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Nancarrow, Conlon. Complete Studiesfor Player Piano, Vol. 1 [Studies No. 3, 20,

41]. 1750 Arch: S—l768, 1977.

Writings: Tenney, James. “Studies for Player Piano Vol. 1: Notes on Studies #3 (a—e),

318

20, 41 (a—c), 44.” In Conlon Nancarrow: Studies for Player Piano, Vols. I/H
[compact discs], pp. 7—1 1. Wergo: WER 6168-2/6169-2, 1987.
Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance

Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#42

Score: unpublished; Gann makes reference to viewing the punching score (p. 34) and
the “original manuscript” (p. 254); no information available on a published score

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vol. V [Studies No. 42,

45a, 45b, 45c, 48a, 48b, 48c, 49a, 49b, 49c]. Compact disc. Wergo: WER 60165-
50, 1988.

Writings: Tenney, James. “Studies for Player Piano Vol. V: Notes on Studies #42, 45,
48, and 49.” In Conlon Nancarrow: Studies for Player Piano, Vol. V [compact
disc], pp. 7—12. Wergo: WER 60165-50, 1988.

Thomas, Margaret Elida. Conlon Nancarrow ’s ‘Temporal Dissonance ’: Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

#43

S995: unpublished
RecordingS: Nancarrow, Conlon. Studiesfor Player Piano, Vols. III/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, l8, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Writings: Tenney, James. “Studies for Player Piano Vol. HI: Notes on Studies #1, 2a,
2b, 7, 8, 10, 15, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studies for
Player Piano, Vols. IH/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

#44

Sm: aleatoric in nature, no real score?
Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a,
3b, 3c, 3d, 3e, 4, 5, 6, 14, 20, 22, 26, 31, 32, 35, 37, 40a, 40b, 41a, 41b, 41c, 44;
Tango?]. Compact discs. Wergo: WER 6168-2/6169-2, 1987.

Writings: Tenney, James. “Studies for Player Piano Vol. 1: Notes on Studies #3 (a—e),
20, 41 (a—c), 44.” In Conlon Nancarrow: Studies for Player Piano, Vols. I/H
[compact discs], pp. 7—11. Wergo: WER 6168-2/6169-2, 1987.

#45

Score: unpublished; Gann makes reference to viewing the punching score (p. 34) and

his Xerox score (p. 257); no information available on a published score

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vol. V [Studies No. 42,

45a, 45b, 45c, 48a, 48b, 48c, 49a, 49b, 49c]. Compact disc. Wergo: WER 60165-
50, 1988.

319

 
 
Writings: Tenney, James. “Studies for Player Piano Vol. V: Notes on Studies #42, 45,
48, and 49.” In Conlon Nancarrow: Studiesfor Player Piano, Vol. V [compact
disc], pp. 7-12. Wergo: WER 60165-50, 1988.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High

Fidelity/Musical America 34 (May 1984): 12-13.

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’ Rhythmic
and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microfilms, 1996 [see Appendix
C of this dissertation for page numbers].

Score: unpublished; Gann makes reference to viewing the punching score (p. 34) and

a final score (p. 263) which lacks barlines (p. 265)

Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33.
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, I I,
12, I3, 16, 17, 18, I9, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. IH/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

#46

#47

S_c_or_e: Gann notes the score was lost and he worked from the roll (pp. 31, 266)
Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, I3, 16, 17, I8, 19, 27, 28, 29, 34, 36, 46, 47.” In Conlon Nancarrow: Studies
for Player Piano, Vols. IH/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

#48 (Canon 60/61)

Score: unpublished; Gann notes that it is not as accurate as the punching score
Recordings: Nancarrow, Conlon. Studiesfor Player Piano, Vol. V [Studies No. 42,

45a, 45b, 45c, 48a, 48b, 48c, 49a, 49b, 49c]. Compact disc. Wergo: WER 60165-
50, 1988.

Writings: Tenney, James. “Studies for Player Piano Vol. V: Notes on Studies #42, 45,
48, and 49.” In Conlon Nancarrow: Studies for Player Piano, Vol. V [compact
disc], pp. 7—12. Wergo: WER 60165-50, 1988.

#49 (Three Canons, 4/5/6)
Score: unpublished
Recordings: Nancarrow, Conlon. Studies for Player Piano, Vol. V [Studies No. 42,

45a, 45b, 45c, 48a, 48b, 48c, 49a, 49b, 49c]. Compact disc. Wergo: WER 60165-
50, 1988.

Writings: Tenney, James. “Studies for Player Piano Vol. V: Notes on Studies #42, 45,

320

 
 
48, and 49.” In Conlon Nancarrow: Studies for Player Piano, Vol. V [compact
disc], pp. 7—12. Wergo: WER 60165-50, 1988.

_5_/l
#50 (Canon 3 )

Score: unpublished
Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9,10,11,12,13,15,16,17,18, 19, 21, 23, 24, 25, 27, 28, 29, 33,
34, 36, 43, 46, 47, and 50]. Compact discs. Wergo: WER 60166/67-50, 1987.
Writings: Tenney, James. “Studies for Player Piano Vol. IH: Notes on Studies #1, 2a,
2b, 7, 8, 10, I5, 21, 23, 24, 25, 33, 43, 50.” In Conlon Nancarrow: Studiesfor
Player Piano, Vols. HI/IV [compact discs], pp. 7—19. Wergo: WER 60166/67-50,
1987.

321

 
APPENDIX C: INDEX OF REFERENCES To SPECIFIC STUDIES IN THOMAS (1996)

Study No

.1—18, 31,168n3

Study No

. 2 — 8, 12, 14, 15, 18, 22, 24, 26, 27, 101—04, 168n3

Study N0

. 3 — 12, 14, 15, 18, 24, 25, 26, 27, 34n12, 53, 54—60, 62, 64, I65, 168n3

Study No

.4 — 22, 24, 86n10

Study NO

.5 — 10, l4, l7, 34n12, 60—63, 64, 168n3

Study No

. 6 — 16, 24, 25, 53, 168n3

Study No

.7 — 12, 16, 17,18, 89, 91—92, 118—21,168n3

Study No

.8 — 8, 10, ll, 12, 13, 19, 24, 86n10, 89—91, 167

Study No

.9 ~—— 7, 8, 13, 104-08, 168n3

Study No.

10 —15,18, 24, 27, 53

Study No.

ll — 6, 7,10,15, 24, 27, 53,115—18, 136, 138-41,151

Study No.

12 — 8, 17, 24, 53, 168n3

Study No.

l4 — l3, 17, 66—67, 97—99, 136—38, 167, 168n3

Study No.

15 — 13, 16, 17, 94, 96, 97, 99—101, 167, 168n3

Study No.

l6 — 122—24

Study No.

17 — 13, 17

Study No.

18 — 13, 96, 168n3

Study No.

l9 — 13, 16, 17, 87—88, 168n3

Study No.

2O — 8, 17, 18, 81—84, 86nlO, 88—89, 165

Study No.

21 — 8, 13,15, 17, 125—26,127

Study No.

23 — 8, 10, 17, 128—32, 134, 151

322

Study No. 24 — l3, 16, 17, 26, 70—74, 76, 84, 167

Study No. 25 — 13, 15, 17, 19, 127—28

Study No.27 — 8, 10, 13, 14, 17, 132-36

Study No. 31 — 8, 13, 17, 66—70, 72, 12—22

Study No. 32 — 7, 13, 84—86

Study No.33 — 13, 17, 34n12, 74—77, 97, 108—10, 114, 151, 167

Study No. 35 — 63-65

Study No. 36 — 13, 19, 96

Study No. 37 — 13, 18, 23, 77—81, 84, 86, 96, 165

Study No.40 — 19, 97, 110—14

Study No.41 — 8, 19, 28, 97, 114, 141, 142—64, 167

Study No. 42 — 24

Study No. 45 — 24, 27

323

APPENDIX D: TRANSCRIBED P. 9 OF STUDY NO. 34

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