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LIBRARY '

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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

Ph . D . degree in Music

 

 

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Major professor

Wei? Oct" 2002,

 

MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771

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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 N ancarrow (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 N ancarrow’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

9/10/11

 

Analysis of Study No. 34: “Canon 4/5/6/ 4/5/6/ 4/5/6 ” ............................................. 87
Overview .......................................................................................................... 87
Themes ............................................................................................................. 91
Rests ................................................................................................................. 99
Summary and Conclusions ............................................................................. 100

CHAPTER 3:

CONVERGENCE POINTS IN NANCARROW’S TEMPO CANON S ......................... 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:

FRACT AL FORMAL FEATURES IN THE TEMPO CANON S ................................... 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. AVAILABLE SCORES, RECORDINGS, AND WRITINGS FOR EACH
STUDY ................................................................................................................... 303
C. INDEX OF REFERENCES TO SPECIFIC STUDIES IN THOMAS (1996) ....... 322
D. 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

1—5-

1-6.
1-7-
1-8.

1-9.

1-10.
1-1 1

1-12-

1-13-

1-14.

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 4

8 against 4 ostinato with conﬂicting meters in Study No. 1 (p. 9). The
symbol J is a ﬁve-division note, in this case a ﬁve-thirty-second note ................ 31

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 N o. 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.

1-16.

1-17.

1-18.

1- 19.
1-20.

1-21.

1-23.

1-24-

1-27 .

1-28

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

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

Gann’s tempo canon terminology: convergence point, convergence period,

tempo switch, and echo distance (Gann 1995, 20) .................................................. 43
Mirrored rhythm in Study No. 35 (Carlsen 1988, 56) ............................................. 44
Symmetrical deployment of canonic voices in (a) Study No. 19 and (b) Study

No. 36 (Carlsen 1988, 6, 24) ................................................................................... 44
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

19
system and corresponds to the pick-up to Gann’s 32 measure.) ............................. 48

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
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.

1 -30.

2-5-

2-6-
2-7-

2-9.

2-10.

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

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

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

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.

2-12.

2-13.

2-14.

2-15.

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

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

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 N o. 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.

2-27.

2-28.
3-1.

3-6.
3-7-

Graph depicting areas of Study No. 34 where all three voices are stating the

same thematic material .......................................................................................... 102
Dynamics and relative scale of acceleration and deceleration in bottom voice

of Study No. 34 ..................................................................................................... 104
Concluding system of Study No. 34 ...................................................................... 105
Convergence point at middle of Study N o. 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

xvi

3-9.

3-10.

3-12.
3-13.

3-14.

3-15.

3-16.

3~l7.

3-18.

Relationship of simultaneous beats and convergence periods between two

ostinato lines in “Rhythm Study No. 1” ................................................................ 121

Line diagrams describing N ancarrow’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

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

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.

3-23.

3—24.

3—25.

3-26.

3-27.

3-28.

3-29.

3-30.

3‘36.
3-37.

Structural diagram of Study No. 32 (“Canon 5/6/7/8”) ......................................... 135
Convergence point of Study No. 32 (score, p. 11): the CP is the ﬁnal cut-off ..... 135

(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
Structural diagram of Study No. 31 (“Canon 21/24/25”), a converging canon
that ends before the CP is reached ......................................................................... 138
Conclusion of Study No. 31 (“Canon 21/24/25”; score, pp. 8—9) ......................... 138
Line diagrams describing structures of Nancarrow’s tempo canons with more

than one CP (Gann 1995, 22, 23, and 25) ............................................................. 140

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

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
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.
3-39.

3-40.

3-41.

3-42.

3-43.

3-45.

3-46.

3‘48

349,

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

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

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

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

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

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 F I 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
. Canonic line of Canon 3, Study No. 27 ................................................................. 169
Canon 4 of Study No. 27, from lowest voice (A11%/R11%) ............................... 170

xix

3-50.

3-51.

3-52.

3-53.

3-54.

3-55.

3-56.

4-1.

4-2.

4-3.

44.
45.

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

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

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

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.

4-12.
4~13

4‘14.

4~ls

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/f noise” (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
The Sierpinski triangle, DH = 1.58 (Frame 1996, 39) ............................................ 193
. The Cantor gasket, DH = 1.89 (Schroeder 1991, 179) ........................................... 194

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

. 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.

4—17.
4-18.

4-19.

4-20.

4-21.

4-22.

4-23.

4~24.

4~25
4‘26.

4~27

(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

Diagram of Study No. 18 (“Canon 3/4”); DH = log(2)/log(1.75) = 1.24 ............... 200
Diagram of Study No. 19 (“Canon 12/15/20”); DH = log(3)/log(2.4) = 1.25 ........ 201

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

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
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

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

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

. 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. 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

5‘3. 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

5-9.

5-10.

5-11.

5-12.

5-13.

5-14.

5-15.

5—16.

5-17.

5‘18.

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

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

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

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

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

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

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

6 12
Temporal dissonance in Study No. 3b: use of 8 and 16 meters in upper

4 . . .
three voices against 2 meter in bass ostinato vorce (score, p. 7) ......................... 256

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

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.

5-22.

5-23.

5-24.

5-25.

5~26.

5-27.

5-28.

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

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

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

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.

5-30.

5-31.

5-32.

5-33.

5-34.

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

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

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

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

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

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, N ancarrow 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” (T enney 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,

 

:ence of “fractal forms” in certain of the canonic Studies. In Chapter 2, the
of the use of ratios in the Studies identiﬁes Nancarrow’s favored ratios and
rship to the Fibonacci series and pitch ratios in the justly-tuned chromatic
of ratios to control various parameters are identiﬁed, including: tempo rela-
ythmic motives, melodic and harmonic materials, and structure. The presence
1e focus of a comprehensive analysis of Study No. 34.
:xt two chapters are concerned with features of Nancarrow’s “tempo canons”:
:es in which the various voices present canonic material at proportionally-
:ds. Chapter 3 examines the structural use of convergence points in the
lies; convergence points are a prominent feature of these Studies, because the
:es are either converging or diverging most of the time. The structural impact
rgence points, and techniques for emphasizing and de-emphasizing them, are
' this chapter. The chapter concludes with an analysis of Study No. 27.
:r 4 identiﬁes fractal features in Nancarrow’s tempo canons, particularly
res in which the same musical material is presented at different tempos to
scale “fractal forms.” Perhaps the most compelling canonic fractal forms are
he canons whose overall form is either converging (where just one con-
int occurs at the very end of the piece) or converging-diverging (an arch form
gle convergence point occurs somewhere in the middle of the piece). Study
sample of the former, is analyzed in detail.
:r 5 is devoted to the analysis of two of N ancarrow’s later works that contain
more mature compositional techniques: Studies No. 25 and 45. The analysis
cuses on the use of the layering technique, while the analysis of No. 45 draws
with a much earlier piece (Study No. 3) and particularly examines the
nultidimensionality.
:r 6, the ﬁnal chapter, presents conclusions and offers suggestions for further

I the Studies.

 

:mainder of Chapter 1 continues with a brief biographical summary and an
I of the four major analytical sources on Nancarrow’s Studies: Carlsen, Gann,
d CD liner notes and an article written by James Tenney. The chapter then
he reader to general characteristics of the Studies, including rhythm, pitch,
.ructure, and texture; notational features unique to Nancarrow’s Studies are
ed. Finally, the analytical treatments of four Studies (Nos. 8, 19, 23, and 35)

:h of the four major sources are compared.

Biographical Summary and Review of Existing Literature

1 N ancarrow was born in Texarkana, Arkansas on October 27, 1912, and died
1997 at his home in Mexico City. His childhood was musically unexceptional,
s noteworthy that the home contained a player piano (about which N ancarrow
fascinated by this thing that would play all of these fantastic things by itself”
Caras 1982, 292]). Young Conlon took trumpet lessons, and by the time he
igh school he was good enough to get steady work as a jazz trumpet player.

rrow’s advanced music education consisted of a single semester (not four
:ometimes reported) at Cincinnati College-Conservatory of Music, where he
:udied composition but neither made an exceptional mark nor was profoundly
[t was here, however, that he ﬁrst heard Stravinsky’s Rite of Spring, which
I and lasting impression on him. Years later he pointed to this experience as
and identiﬁed Stravinsky and Bach as his favorite composers (Gagne and
282—83). Nancarrow’s exposure to Bach apparently came about through his
y in counterpoint with Roger Sessions in Boston, where he moved after
:innati. This was unquestionably his most rigorous compositional training.
)ston, Nancarrow also became acquainted with Walter Piston and Nicolas
These relationships began under the guise of studying composition, but ac-

Jancarrow 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 N ancarrow’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, N ancarrow 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 ‘1 2:2 (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

 

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

 

'—L

 

 

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 N ancarrow’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 N ancarrow’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 /—\
2 : 3 : 4 : 5 : 6
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 N ancarrow’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 N ancarrow’s Studies is the tempo ratio.

18

 

 

3:2 generates

(a) 3 (b) (C) J = 72

    

f l' r ”I

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. N ancarrow 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

 

2
7This is easily veriﬁed on the recording. The tempo is J: 105 for 22 measures of 4 time. There is a simple
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

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

ill in?!

‘——--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 (N o. 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 T enney (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 N ancarrow 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-

n—t --::-r-xl

ferent tempos. This is evident already in the ﬁrst Study, where an ostinato in which the
7 4
same measure length is divided into both 8 and 4 time is pitted against a variety of dif-

ferent tempos and meters (see Figure 1-4).

(Jada)

 

 

 

(9‘5).

7 4
Figure 1-4. 8 against 4 ostinato with conﬂicting meters in Study No. l (p. 9). The symbol J is
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

 

.zz rhythmic features abound in Nancarrow’s Studies. There are many variations
.rochaic (long-Short) “swung” rhythm, sometimes even within the same piece. For

B, in Study No. 3b both the rhythmic divisions 3:2 and 5:3 are used. As Gann

ancarrow has from the very beginning used the player piano to recreate
ythmic liberties taken in performance that no notation could convey. In the
udies based on [stride piano] (Nos. 3, 4, 10, 45), he has implicitly acknowl-
lged that jazz pianists hardly ever play a dotted rhythm in a 3:1 ratio;
stead, Nancarrow often divides his beats into ratios of 3:2, 5:3, or 8:5, all
visions based on the Fibonacci series, related to the intuitively pleasing
olden Section as well as closer to live performance practice. (p. 9)

re 5-division notational symbol [J] Shown in Figure 14 above also found
t usage in beat divisions of 5:3, in which it complements a 3-division note such as

:ighth or sixteenth notes. See Figure 1-5.

 

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

rrlier, in the discussion by Thomas on the inﬂuence of jazz on Nancarrow, Nan-
s tendency to place notes “around the beat” was noted, and this is another char-
0 rhythmic feature in his music (see Figure 1-6). Other features that have already
entioned include the “countdown” accelerative technique (progressively smaller
etween attacks; see Figure 1-7 for a metric countdown used to particularly
3 effect to close Study No. 12), and other types of numeric series that often
adjacent numbers. (See Figure 1-23 for a series that alternately decelerates and

ites.)

32

 

 

7'0
Q—ﬂ

1-6. Melodic writing “around the beat” in Study No. 2 (p. 4).

.f—v- v-l- 'f .
~L—4- "L v u .
4- ' '— :3-Zttf- ’1':-

 

7 ,. it ,

1-7. The “countdown” technique at the closing of Study No. 12.

{hythmic ostinati and isorhythms are a signiﬁcant component of Nancarrow’s
11C landscape. In the literature reviewed here the terms ostinato and isorhythm are
mes used interchangeably, but a distinction Should be made between them. At
the authors seem to insist that an isorhythm involves both repeating rhythmic and

:lements which are often of different lengths and out of phase with each other:

[here is a ﬁne line between ostinato and isorhythm, since an ostinato reit-
:rates the same rhythm as well as the same pitch, . . . [while isorhythm
nvolves] some internal dissociation of rhythm and pitch, a use of talea
ndependent of color. (Gann 1995, 85—86)

. . duration series are often paired with pitch series, which may be of
lifferent lengths, creating isorhythm. (Thomas 1996, 11-12)

33

F

 

ancarrow] revived the medieval technique of isorhythm (though inspired by
;trong interest in the tirla structure of Indian music), employing multiple
)etition of the same rhythm against different pitch sequences. (Gann 2001,
6)

Inn, however, often refers to rhythmic patterns alone as isorhythms. A slightly
: phrase is used by Carlsen, who discusses the four closely-related repeating
: patterns in Study No. 19 based on the series {11 — 1, n, n + 1, n} (the same
are shown in Figure 1-11 below) as “isorhythmic patterns,” which he later calls
(Carlsen 1988, 7). Thus, for Gann and Carlsen, at least, an isorhythm does not
ily require a repeating pitch element.

e term ostinato is just as ambiguous. Gann’s deﬁnition of ostinato as “a phrase
rtly repeated without variation” (p. 69) does not distinguish between a rhythmic
or one that combines melody with rhythm. I think it is safe to say, however, that
ors reviewed here always choose the term ostinato over isorhythm when the
.nvolves the same rhythmic unit repeated over and over, rather than a variety of
: values—e.g., a steady repetition of eighth notes or eighth notes separated by
:sts would be considered an ostinato, even if a repeating pitch element is present.
3 term isorhythm should be held to the more exacting standard of involving
rhythmic values. The repeating pattern in an ostinato also tends to be of brief
hereas isorhythmic patterns are generally longer and more complex. In Figure 1-
stance, Thomas states that the lower voice in the opening of Study No. 2 is “a
ale example of isorhythm” (p. 12); she refers to the two lines operating together

tinato (as does Tenney [Wergo Vols. III/IV, p. 7]). Gann refers to each of the

:47

 
   
  

“.5"
l—_-—/.

8. The opening of Study No. 2.

34

f“

 

 

 

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

    

r / B.
—"

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

 

 

Ieighth-notebeats 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 l6 l7]
voice #1 (3+4+5+4) J, J J) J J
voice #2 (4+5+o+5) J J) JJ JJ’
voice #3 (5+6+7+6) J) J.J J.J
voice#4 (6+7+8+7) J,J J,J

 

summon. |31 lgri D ilgl 1 D Mic! 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)

eal signiﬁcance of this is, according to Carlsen, that “the prevailing diatonic context
S a perfect foil for the rhythmic complexities” (p. 20).

There are several prevailing melodic ﬁgures that have been identiﬁed in Nancar-
; Studies, including the ascending perfect fourth; the minor third, particularly the
itioned minor thir ” (minor third divided into a whole step and half step; see Figure
; and the melodic pattern of a whole step followed by a minor third (3- 6 - I)
1 is variously described as pentatonic and derived from blues (an example can be
in the lower voice of Figure 1-6). (This melodic pattern is also prevalent in nine-
1-century hymnody from the southeastern United States, although this was not likely

1iﬁcant inﬂuence on Nancarrow.)
:: 05—
J i. . . /-. . .....——_.

f

3 1-1 1. 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

as the one in Figure 1-12 Showing blues scales used in Study No. 45b.

I<ii’“"’ -j- -. tr .....

I .31 f . ' f
3 1-12. Blues scale patterns in Study No. 45b (p. 7). The notation Nancarrow uses is

)ded,” which is used when the space on the Staff is insufﬁcient for the number of notes; the
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

ascending and minor form descending. See Figure 1-13 (see also the top two lines

37

 

hm

 

 

of Figure 2-6 [p. 71]).

 

 

 

 

 

 

 

 

 

 

. D I; A n U D n n h 1)
ﬁ 1 I IV I I I L I 11' X L I
' - _" t..- : ”eerie: —eii.':‘v
2 ' """ﬁ 1' i'i'iv ~t#=
I 5 n 4 Mini" 4 I 9"”

 

 

 

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:

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)

OthNV—

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"
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.

7 -4

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 3 2 octaves, 2 octaves
8 3 2 octaves, 2 octaves
12 3 perfect ﬁfth, perfect ﬁfth
14 2 2 octaves + perfect ﬁfth
15 2 3 octaves
18 2 octave
19 3 perfect eleventh, perfect eleventh
24 3 contains series of canons with levels of imitation
ranging from root position triads to ﬁrst
inversion triads to open ﬁfths and octaves
32 4 perfect ﬁfth between all voices
36 4 major tenth, minor tenth, and major tenth
(creating a widely-spaced major seventh chord)
43 2 major third
49a 3 2 octaves + perfect fourth,
3 octaves + perfect ﬁfth
49b 3 perfect twelfth, perfect twelfth

 

 

41

Strucmral thures

Canon is such a pervasive technique in N ancarrow’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). N ancarrow 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

 

J=80 r I

  
  

 

t 1 L____J
EChO distance 50110 distance

  

Tempo Convergence
Convergence switch point
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.
°ch-shaped converging-diverging tempo canons exhibit symmetry about the central
ther non-canonic pieces are virtually palindromic, including Studies No. 1 and 43.
romicisorhythms,suchasthe3+2+2+3+2+3+3+2+3+2+2+3iso-
3 found in Study No. 7, abound in the Studies; several other lengthy examples from
No. 8 are Shown in Figure 1-23. Also, Figure 1-24 Shows a complex series of
thms that is symmetrical. An example of a mirrored rhythm is located by Carlsen in

No. 35 (see Figure 1-19).

 

 

 

 

 

 

 

ror rhythm (systems 6—7)
I L r r r 7 . h _-
'._.__.I_V:_ _. -_I.__ V“? I‘_..-;(:__L;__ , W V

4

1—19. Mirrored rhythm in Study No. 35 (Carlsen 1988, 56).

11 Study No. 19, a 3-voice tempo canon subtitled “Canon 12/15/20,” Nancarrow
etrically partitioned the keyboard into overlapping 4-octave segments about the

piano’s middle note, and he employed a similar procedure in Study No. 36 (see

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: 1-20).
Bur --
(Pianos 4 ‘ ("440" 8‘3. "
GEN-rut. Halt) '2: : CW NOW) “ ,’:
L X _" 1"— l I v 1 “—
; - R— L .- F 4% I I 49:
1' 1 _ ‘1 EIJL II F A v
I ’3 Ir 7 ” If
’ I ’ I
I I ‘ b‘ 1 +1 '
:_L J v Q T‘ 4‘ HT
;‘r l T ’ A— 1’—
I__ ._ ‘1? B—O“
r 11"
--J 8%--..
(a) (b)

1-20. Symmetrical deployment of canonic voices in (a) Study No. 19 and (b) Study No. 36
:n 1988, 6, 24).

lymmetries also occur in some of the Studies’ harmonic structures. The major
h chord—a symmetrical structure with two major thirds separated by a minor

—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)

octave + major sixth

1 octave + minor Sixth
octave + major sixth
octave + perfect ﬁfth

2 2 octaves + major second
octave + perfect ﬁfth
octave + minor sixth

3 2 8ves + major second
octave + minor sixth
octave + minor sixth

4 2 octaves

octave + minor sixth
octave + major third

8 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 a
(octave displacement) / D
_..\~ . ,

I

F?

.1.

l7

   
 

(a)

 

1-21. Symmetrical melodic structures in (a) Study No. 9 (p. 5) and (b) Study No. 25 (p.

:, Timbre, and Dynamics

everal textural features of the Studies can be traced to limitations of the player
nedium. The ﬁrst is the prevalence of staccato notes—Nancarrow reported to
ds (1984, p. 20) that this was because punching sustained notes was simply more
is a line of holes must be punched for sustained notes whereas a single punch
3d a staccato note. (Eventually Nancarrow had his punching machine altered so
could punch a line of four holes at once, and this alleviated the problem some-
The other textural feature affected by the player piano was the number of
neous notes that could be sounded—due to potential loss of air pressure from the
1g mechanism and the possibility that too many holes punched too close together
ause the roll to perforate, the player piano is limited to playing about sixteen notes
1e (Carlsen 1988, 71 note 7). The player piano has a sustain pedal, but Nancarrow
in only one piece (No. 25), so the overall texture in the Studies tends to be

hat 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

9
pick-up to Gann’s 32 measure.)

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 6
7 7
8 Hit. Hit. 8
9 9 9 9
1O 10 10 10 10 10
11 11 11 11 11 11
12 12 12 12
Accel. 13 13 Acoei.

14 14
15
Acoel.

(a)

7

9 9 9 9 .-
12 12 12 12 12 12 12 12

17 1717 17 17 17 17 1717 17
24 24 24 24 24 24 24 24 24 24
max 3 m max
43 43 43 43
55 (551
(b)
2 (2)
3 3 3 3
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
s e 6 6
7
(C) ((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 N ancarrow’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
‘.’I 5‘--.‘

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

‘two voices remain quite independent of one another from beginning to end, and it
an that one of Nancarrow’s purposes here is simply to exploit the differences
'een them” (p. 47).

Thomas points to this piece as one involving different “time types,” with one of the
voices being metered most of the way through and the other voice being pro-
onally notated and highly variable in speed. The metered voice frequently changes
:rs, but a steady eighth-note pulse is maintained. One of the more interesting effects
.e second section occurs in the unmetered voice: a countdown device is combined
a duration series in which durations are tied directly to pitch level. The 71 notes
I B0 to A6 are assigned a gradually decreasing duration ranging in measurement from
rillimeters to 34: millimeter. Carlsen points out that durations that are related by the
of 2:1 are a major ninth apart (see Figure 1-28). This same rhythmic series was
illy established in the ﬁrst section as part of a 71-note accelerando in which the

168 are not stated in chromatic order. Both Carlsen and Thomas correctly note that in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“Hull“mmmillllllllllll1:miui:i......

I : E : :
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'e 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 “N ancarrow 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

 

 

 

 

 

 

   
    

    

  

   
 

3 I I
0 | Tw‘: E
iA i . 1
rsupos RATIOS ,A 4 ; I [C : ﬂ 5
6 E I 'c : ' ' :
2:39]; 58 E i :r : _1:
238 42 ' i i : i
204 36 .
170 30 l ' I I
1412/3 25 t D

119 21

113’l3 20
102 18
85 15
79V: 14

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 N ancarrow’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- Just Pythago— Ratios
ment rean
70 24:25
90 2432256
Minor second 100
(semitone) l 12 15:16
182 9:10
Major second 200
(whole tone) 204 204 8:9
231 7:8
267 6:7
Minor third 300
316 5:6
386 4:5
Major third 400
408 64:81
C 1:1 498 498 3:4
(15:14) Fourth 500
C# 15:14
(21:20) 583 5:7
I) 9:8 590 32:45
(16:15) Tritone 600
E1, 6:5 .
(25:24) Fifth 700
E 5:4 702 702 2:3
(16:15) . .
F 4:3 Minor Sixth 800
(21 :20) 814 5:8
F# 7:5
(15314) . . 884 3:5
G 3:2 Major Sixth 900
(16:15) 906 16:27
At, 8:5
(25:24) 969 4:7
A 5:3 996 996 9:16
(21:20) Minor seventh 1000
B1, 7:4
(15:14) 1088 3:15
B 15:8 Major seventh l 100
(16:15) 1110 128:243
C 2:1 Octave 1200 1200 1200 1:2
(21) (b)

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 Name Size in Cents
1/2 octave 1200
2/3 ﬁfth 702
3/4 fourth 498
4/5 major third 386
5/6 minor third 316
6/7 small minor third 267
7/8 maximum tone 231
8/9 major tone 204
9/ 10 minor tone 182
10/1 1 minimum tone 165
11/12 3/4 tone 151
15/ 16 just diatonic semitone 1 12

Intervals taken from every second harmonic [superpartient ratios]—

Ratio Name Size in Cents
3/5 major sixth 884
5/7 small tritone 583
7/9 large major third 435
9/1 1 neutral third 347
Intervals taken from every third harmonic [superpartient ratios]—
Ratio Name Size in Cents
4/7 harmonic minor seventh 969
5/8 minor sixth 814
7/ 10 large tritone 617

(J orgensen 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
1) ll 5 . . .
EM: } Vibration ratio
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

—___— ___...
__. __. __..r

   

        

  

 

 

S d
.m e
e .w
s r
8
NM _m mm .0
V0 7642 s 2
.nﬂ m /
66 MB m h
dVIO 1 It“.
so 6542 .m
.mm
to
me."

 

 

 

11.6542

 

 

 

h
mm
mm

 

 

 

 

(a)

Violin
Viola

lo~=——nui
r". 3..

\I
b
(

 

. 203—04).

chorale theme and its resultant pitch ratios; (b) ﬁrst page of the score. showing the conversion of
70

Figure 2-4. Stages of composition of the ﬁrst movement of Cowell’s Quartet Romantic: (a)
the chorale theme’s pitch ratios to rhythmic ratios (Thomas 1996

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
5:3
23.3
9:8
5:2 195. 4:3

4:3 f> 9:8

 

5:4

5. 115 525| 315 233.3l4 140] 3'15 175|5 140 | 9|9

° 43 175|51 5 175 I3 93.3]52 10 140|4 124.4 48|8
233.13 ° 140|4 4 56 [3 l6 105|3 93.3 3 IS

(a) (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
of people’s positive to negative value judgments. . . . There is also experi-
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.

N ancarrow’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 Tempo Markings
No. 14, “Canon 4/5” J = 88, J: 110
No. 15, “Canon 374” J: 165, J: 220
No. 17, “Canon 12/15/20” J: 138, J: 172.5, J:230
No. 18, “Canon 3/4" J: 168, J: 224

 

No. 19, “Canon 12/15/20”

J:144,J:180,J:240

 

No. 24, “Canon 14/ 15/ 16”

J: 149%,,J : 160, J. : 170%,;
J : 224, J. : 240, J. : 256

 

No. 31, “Canon 21/24/25”

J: 105, J: 120, J: 125

 

No. 32, “Canon 5/6/7/8”

J: 85, J. : 102, J. : 119, J. : 136

 

No. 33, “Canon «1272”

J: 140x12, J: 280

 

9 / 10 / 11
No. 34, “Canon 4/5/6] 4/5/6/ 4/5/6 ”

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 faster tempos
in each voice, peaking at = 264 in the
fastest voice

 

No. 36, “Canon 17/18/19/20”

o=85,o=90,o=95,o=100

 

No. 37, “Canon 150/160 77/168 3/4/180/
187 V, /200/210/225/240/250/262 '/2 /281 1:, ”

J:150,J:160?,,J:1683/,,J=180,

J: 187%,J:200,J:210,J:225,
J:240,J:250,J:262l’2,J:281}:,

 

No. 40, “Canon e/tt”

tempo markings not used—Nancarrow
notates a simple diagram which gives
timings that establish a relationship of
roughly e to It

 

1
No. 41A, “Canon 717/1127 ”

I
No. 4113, “Canon 3% AVE/l6”

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_ _1_
112 = 5’00”; 1/2/3 =7’40”; 3‘17:
4’35”; 3113/16 =6’10”

 

No. 434 “Canon 24/25”

o=12CLo=125

 

 

No. 48, “Canon 60/61”

 

 

tempo markings not used—uses
proportional notation

 

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 N o. 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).

    

V

Figure 2-6. Study No. 46, 3:4:5 polytempo ostinato as renotated by Gann (p. 264).

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 N ancarrow’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 4.5 3:4
AA
1 A I B 1 C 1 12 : 15 : 20
I l f l
I l l ' 12:16:20=3:4:5
l l l 1 3:4 4:5
(a) (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

 

I;

*

219+
f!

'b

l.
6%!

 

LP-

(5 31) ‘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 N o. 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 Te po Ratio" Corresponding
( =) no_t_e ( =) ng§§_ _.
252 7:10 F# 480 4:3 F
270 3:4 G 504 7:5 F#
288 4:5 Al» 540 3:2 G
300 5:6 A i 576 8:5 AL
315 7:8 ah i 600 5:3 A
. , 630 7:4 Bk
E1: 675 15:8 B
720 2:1 C
765 17:8 C#
810 9:4 D
864 12:5 51

 

 

 

 

 

 

 

 

*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.

ﬂythgyﬂitch 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: l7—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 mm. tempo
C# 11.9 384
C 12.5 367.7
B 13.6 337.2
A# 14.3? 320
A 15.3 300.5
G# 16.1 285.6
G 17.0 270
F# 18.2 252.3
F 19.1 240
E 20.4 224.4
D# 21.1 217.4
D 22.7 202.1

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.

 

 

 

 

 

LL 1 l A
at fr I I L I A I I ' I I ﬂ"
1 1_ 1 1 l . T I I I 1 4 1 “ﬁ—
n; 1 1 1.1 ' I 1 V I l I 1'
it} 1' I 411 1' 1 A A 7 #1 A J ' T
r ”4" V

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

35
2-12). The two lines share a 16 meter signature, but the distance between notes in the top

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 4” values 1 4
14 7 ' '

I | I

     
 

 
    

S‘l'aLr—c 5+9 Snafu

 

  
   

L ll 1 ‘F""

15 5 7’ 25

Figure 2-12. Opening of Study No. 5, showing 7:5 rhythmic ratio and opening tn'tone 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-BL, 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-
quarters of the way through the pattern. . . . The positioning of these entrances
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 minor third
l—\ l—\
2 : 3 : 4 : 5 : 6
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. 3a (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 I" 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 AZ C A4 E A5 F A6 G A7
bloco 8va_ bloco 8va. 8va.

       

8V3- 8va.

8b-’9°9_ ‘ - _- __ L. ’000 8b. loco

f P ffL— -p

p __....— -
one score system = ca. 3 millimeters

Figure 2-14. Proportional diagram of entrance of themes and levels of im'tation in Study No. 34.

section B = ascending (enharrnonically spelled) ﬁrst inversion minor triad

section C = ascending (enharrnonically 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: 72 90 108 144 180 216
ratio: 4 : 5 : 6 : 8 : 10 : 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” (T enney, 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 A3 D A4 E A5 F

 

 

 

/
ll-voice: 88 10633 110 117)“3 132 133,13
lO-voice: 80 96 100 117x"3 120 -...120
9-voice: \72 96--- 90 / 10633 108 120

 

 

 

 

 

 

 

 

 

 

 

 

Section: A6 G (deceleration begins) ---------- A7 (incomplete statement) -

72? 177.; 1463’3 551172355 783F732

ll-voice: 264 23493 2
10—voice: 24o 213/‘3 200 160 133213 120 10693 100 so 66%
9-voice: 216 192 180 144 1201 108 96 9o 72 60

(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 117/”3 117*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 1062/3 117'/3 120 133'/3 14633 144

\1/

(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 96:120:144. 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 E). 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 3 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 ‘ ‘~.‘ b
in statement A 7 ‘

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“. A . '
I ~ ; ' I Jun-m ’
mimﬁiﬁumdﬂnm
. I ‘~.
b b
\‘x. entire theme
A a C ‘ traces t'cl
2‘ . . 1 . I. I - 1,10 m
or r“ a ' gill 517i rim-.1 ll gh- )* ‘n' :. "1 ll" Hy ”
n IIIDIAUITI ii; 111m 0

1 u Eryn, rr Wzy II o '1‘ , V y HOW-E T]: 37111 at ij-c:__J

 

 

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

92

Theme B (40 ."s) -- first statement in lowest voice (system 6)

C

sequence 2 - - -
c

   

-------- rhythmic sequence 1 - - - - - — - - -
~ 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::':nicb sequence 1 ---- ----rh¥)thmic: sequence2----

c 3'

  

 

 

 
 

2 + 3+2-1-l

 

entire theme
traces iel

Figure 2-18. Theme C of Study No. 34.

Thane D (103 J 's) -- first statement in lowest voicg (system 20)

 

 

 

 

 

 

 

 

(8b throughout) x“ ﬁnsttwnphraLfr , ,.
‘.(raret'tl -'

 

 

 

b ‘ ~ , ‘ entire theme

    

15 .———1._. ream”

 

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)
- - sequenge 1b -

a' h b 'd
.1 l 1 F l s l 4' I 1 .' I—1

. l g .
m-‘I‘_.‘-' -'m|- .-‘.‘—- _I- -'
"zl—I-II-ﬂmr—ﬂ ---:--li-'-—'
: ' 1 '

 

 

 
   
       

V

ence 2a - - - -' - - - - sequence 2b - - -'-

 

 

 

 

 

 

    

a' b b
:L’ . I M ~ 9; — ---- ”mi-sequence 3a: - - - sequence 3b- - v _
b b
@fjipgﬂ 3*? r $.1ifii';,i1,::,“ﬁi;,’ H 3;??? 7‘??? “f 1.51
I. _'._> E? g ._' bdﬁ "”*"T’ i_ _D ' W - ’ ’ " ' A . "‘ [’95 1;? I _. ~, __'__—_;t\-~
. . . ..

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(/.\Itl1( e
erpwrxiun of" u" b

 

 

 

Figure 2-21. Theme F of Study No. 34.

Theme G (50 + 62 . 's) -- first statement in lowest voice (system 41)

 

 

 

b
1 (b‘ 5 1—1 r——Ir~.
l,‘ 2 533'2' :Wl ”€731,289 iq': 331%” iiiii C t :21!qu 1211-6:le
1 LL-‘ #.~eprf H U. bV/jI a“: l 3‘ 1111f; W / ,1] u z, __.: _ 11

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ﬂ , 7 “ V2 7 }: 7: 7 L- I ,7 ¢‘ __..
a ! ('Ifdt's with

* = tempo changes occur beginning (if/‘7
at original pitch

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 + 1) 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 171 , 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 N o. 34

 

 

 

 

 

 

 

 

 

 

Theme: A B C D E F G
Contains pitch motive a/a’ - . . . . 6
Contains rhythmic motive b - . . . o . .
Contains pitch motive c . . e .
Contains quarter notes 0 .
Traces icl from beginning to end 0 . .
Traces icl within phrases 0 0
Contains dyads - .
Contains rhythmic or melodic sequences - o .
Contains arithmetical rhythmic series in o . o
motive a
Contains internal tempo changes | -

 

 

 

 

 

 

 

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—EL, 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

 

 

 

 

 

 

L——J
interval traced interval traced by interval traced by interval traced
by theme C ﬁrst phrase of second phrase of by theme D

theme D 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 N ancarrow’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 3 @ ' 21 t @ 39 54 *30 *76 15 186 115 51
Voice al=88 mm. ' J=110 mm. mm. mm. mm; mm. mm. mm. mm.
Middle 3@ 20 $@ 78 :r 62 25 *82 84 207 128 74
Voice J=80 mm. 1 J=100 mm. mm. mm. mm. mm. mm. mm. mm.
Low 3 @ 52 t @ 94 63 94 83 1 *101 228 142 112
Voice J=72 mm. J=90 mm. mm. mm. mm. - mm. mm. mm. mm.
Approx. 9: 10: 11 2:5 9: 10: ll 2:4:5 6:7 - - - 9: 10:1 1 9: 10: 11 4:6:9

 

 

Ratio:

 

Note: Shaded areas denote measurements that are similar enough to be considered equiv-
alent for the purpose of determining ratios.

* 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 E1. 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)

   
  

(Via =Tqb% ‘—
9

   
  

1

(middle voice also J = 146-2/3)

‘) .

    

i...

   

 
 
 

5» 2/335

 

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)

 

(C)

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).

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.

3:1 ratio

Temios/
72 ”(V g , ' r /-«:::/-::::'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 IN 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 1 (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 n beats at a point n 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 Ii 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

ii

   
   

 

        

‘-
_—

   

_———*

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

    

it

. J .
1 4 5 6 7 8 328 330 331 333 336 337
(a) (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 Ii (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
2 6
because there is an extended section of 4 time and also a measure of 4 .

113

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ii: ' E. .
(a) [511* ‘ 12V» 31:1 \ x __ #35
1____.. ___.~._______\__...._..
1 2 3 4 5 6
of: - _‘ z; __ w
83_o__ 333 34 835
\\_,__ .____, _, __ _ __ _ -__
(b) (\L: i 7 r
:‘éss;~~--~ ‘__837“‘”‘”":8'38 WWW __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
Dbeats
1 l ]

 

 

161 165 169 173 177 [...] 337

 

Study No. 36 (“Canon 17/18/19/20”)
CP

1

l [...] 387 407 427 447 467 [...] (844)
1 [ . . . ] 389 408 427 446 465 [ . . . ] (844)
l [ . . . ] 391 409 427 445 463 [ . . . ] (844)
l [...] 393 410 427 444 461 [. . .] (844)

 

J beats

 

 

 

 

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

1
canon will consist of a portion of the entire canon that is equal to m + n , where m and n

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

 

 

 

 

r 11
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 I 3:5: I C=3
3:55 i I C=3} I E A=4
C=3 I 5 A=4 I :3:5
1 [pm 1 1/6 I

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

z — x
and between x and 2 it is x+y+z. Thus, in No. 17 the sum of the duration ratio’S ele-

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 8’ 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 iteratron rs generally twrce as fast 1n the 8 vorce to create a 7:4 tempo ratro

119

it

 

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 (e) 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 9 1a 15 21 81 87 88.5 89-1/7 92-4/7 93.5
2 conv. . 2 conv.

(no simu neities) ----- periods of 392$; 52"” periods of (no at ultaneIties) - - -------------
rate of 3 bars ea. ‘ 3 bars ea.
1 7
33;,“ 8 U»; .1? 1 J J. 1 J. J J Li [.1
3““. 4 17:4 _,P7:4 27:4 17:4 )724 \ 7:4 [.74 7:3 ’;::-7:4 7:5 In
389° 4 «-1 .1 .1. » .1» .1 J/ .. .LJ;

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 Study No. 19 - Canon 12/15/20

 

 

 

 

 

 

Study No. 18 - Canon 3/4 41
4, ———-l i
1 #1 J
I l I
Study No. 31 - Canon 21/24/25 Study No. 32 - Canon 5/6/7/8
Study No. 21 - Canon X A B c

 

 

 

 

 

 

 

I

|(the only canon 1

>< A B 0.....11
[vergence pount I

has beyond rts I

A B C 1]

| temporal frame)

 

 

 

 

 

 

 

 

Study No. 36 - Canon 17/18/19/20 S‘UdlgNO- 48 - Canon 60/61
I l
I B j >No. 488
I A l No. 48C
i n
J A : >No. 48A
I I
Study No. 49A No. 498 No. we - Canons 4/5/6
—-l —i—- ———)

—4——+————+

Figure 3-10. Line diagrams describing N ancarrow’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 33.6 beats
l 1
4 I J=110 (follower) (leader) I i ‘
J=88 (leader) (follower)
4 >

 

 

337 total beats in each voice
(timespan from attack of beat 1 to attack of beat 337 = 336)

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

 
 
 

follower 3

 

 

 

 

 

 

 

 

 

 

 

 

17
ﬂ timespan = 843 J '5 >
Jelapse needed before and
after faster voices for CP to Actual J elapse before Actual J elapse after
occur at midpoint (between entrance of faster voices conclusion of faster voices
beats 422 and 423)
18-voice 23.4(I/180f42l.5) 23.7(l/18 of42_6) 23.2(1/180f4l7)
44.4 (2/19 of 421.5) 44.8 (2/l9 of 42_6) 43.9 (2/19 of4l7)
ZO-voice I 63.2 (3/20 of 421.5) 63.9 Q/ZO of 42_6) 62.6 (3/20 of 417)
GP 0 m. 320
I
mm.1 106107 : 619 641 650 688
4 6 2 i 4 i
4 4 4 I 4 I

 

I I I I I
:<—21a J's—>:<—e18 J's —>:<—299 J's—.4440 J's-b:<—78 .1»

 

first statement of second third fourth
recurring theme statement statement statement
IV 426 J. >:< 417.“; r I

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 7 27 47 67
19-Iine 9 28 47 66 85
18—line 1 3 31 49 67 85 103

17-Iine 2 1 9 36 53 70 87 1 04 121

 

 

 

25 24 23 22 21 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

 

 

 

 

 

 

 

 

 

 

 

 

ATIII II

 

[11 ‘
I!

 

A
[XVIIIIAJ'
-Y Y

.———.
l
I
r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

127

 

 

 

-/;@ar-iw"wdﬁ1;,g

 

. Y‘ I'_ :V ‘;

 

 

 

 

 

 

 

 

 

 

 

L
Y , ’

A

7&71

Inga-3;:
)o

ﬁr
”

V

F:

 

ii -: _1,

 

 

 

1
'AL;
I

 

 

 

 

 

 

 

 

Figure 3-15. The CP in Study No. 36 (“Canon l7/18/19/20”; beginning of second system shown; from

p. 23 of score).

 

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, F 6, 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-
taneous motion ” (p. 126)

 
   
    
 

'.
‘ --
- ‘0
.

    

 

 

 

 

 

c- . — _ — — _ — _ D— a“ a.

«f
/‘ o- A; 1 ; - th-ﬁ . ’2;- {-2'
x ’i
1'

 

I Y. YL ' I I I h L
l A ' I I I - V IY I I
I

1} § i iv:— 1' i § '7 I x J: A *1: .- -'_' . i q
tempos actually coincide near the beginning of this system
1 e 5 - ' . . A

1
I I v v I v 1‘ y I 1‘
l J A -W In; 7 I. V l I... . )1 r
l v L A 1 1' Y‘ 1' V

v

. J 0 HI ' Iv ;

‘17
I- - - - O

*—
——.—.___J

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—At . . .) 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—1 cadence with a quintuple C octave.

.l. J.

  
 
 

 

i
ii

 
 
   

A

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 P

1,681 total beats in each voice
(timespan from attack of beat 1 to attack of beat 1,681 = 1,680)

 

 

 

 

 

 

 

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
1f 7

337 total beats in each voice
(timesparfrom attack of beat 1 to attack of beat 337 = 336)

 

 

 

 

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 +

 

interval of imitation changed
from elevenths to double octaves

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.

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

9 / 10 / 11
In Study No. 34 (“Canon 4/5/67 4/5/6/ 4/5/6”), which was the analytical subject of

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 N ancarrow’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 B1, major seventh chord at

the extraordinary moment of coordination where all four voices converge. Dynamics play

136

  

_w

   

61-voice of mvt. A

 

 

 

 

   

 

 

 

  

CF
121.7 .
B ( PP ) 61
123.7 .
( PP ) 6O
60-voice of mvt. A
(125.3 .)
A PP 61
(127.3 .)
PP 60
__. __.._ 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 f f 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 duration = 168 I 1
25 113.64 beats l‘ V - _ , C 369 (Lean, c- I c -- c- -1 >r36
24 I ________ | duration = 175 | I
88 beats l‘ 669 beats _ _ _ _ bl 36
I duration = 200 I
2 1 I 669 beats r36 beat- 1
<7 7 ~ 7 7 7 , e , , - A, b s
4 b

 

timespan from attack of beat 1 to attack of beat 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.

,Vrf‘ff 1'1

o

la}; ‘1“ 113—J'-

      

444'

  

 

 
 

.-

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
staccato. . . . Since it is difﬁcult to follow a single voice, it is nearly impos-
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

..
.,,,.
l

. 9 "7 .

3

7

u

I

.
..-
‘Ql
...
p_.
r;
—.
0-D
,l—
‘1‘
. .

I
.
'4
?

.
.‘52
'3

.t
W
H

v»! ﬂit ".-

 

Study No. 15 - Canon 3/4 Study No. 17 - Canon 12/15/20 Study No. 22 - Canon 1°/o/1-1/2°/o/2-1/4°/o

 

 

 

 

 

 

A A A B

‘r i n. i 41 leg. A ]8 , lA

A A B A I I I

l i i 1 1C i ‘i __.—(___+__|___
P l" :3 l .L -—l— l

 

Study No. 33 - Canon y'2/2
A B C D
—i—- i i
A B C l D L - X l
1 l l
Study No. 43 - Canon 24/25

25 follower leader 24 follower

24 leader follower x leader
25

 

 

 

 

 

 

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 ﬁ's b
A (J = 220): leader A (J=165): leader
A (J=165): follower A (J=220): follower

 

 

 

L——l A

tempo overlap @ J=165 (84 D’s)

 

V-I cadence in V7—I cadence
both voices in both voices

Figure 3-28. Structure of Study No. 15 (“Canon 3/4”), a diverging-converging canon in two
voices.

140

Beginning; Beginning of canon’s repeat in top voice
q]; and of tempo overlap section:
Jezzo l_ — I _ "I“ 4:185“ 8Q _
A I

 
 
 
 

     

0G0

   

  

(cup-c

J2”; = 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 Final convergence point:
an end of tempo overlap section:

 

\ Jews)

    

 

g:
,.

   

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‘ {‘12 = 119/L: Sen 1::ng -:

v-J- 4:

w—

    

 

(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

Ti
1M
ﬁll‘
m
It

HY

LA‘

 

 

 

 

 

 

 

 

 

VHHL-th

“I

   

(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 N08. 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*: 50 24 68.5
Beats: 1 198 588 1643

* given in quarter notes

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..___l
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 El-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 ln each v0lce Wlthln thls convergence penod 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

l2

 

   

LL' _. _ _ _J
“collective ” glissando begins here

gt!

3 __.

iteration 1

—_ “‘C—_——

    

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
CP 1 articulated, coincides with initial attack of Canon 1 in both
voices
CP 2 unarticulated, is the ﬁnal cutoff of Canon 1
Canon 2 arch canon, no tempo switch
CP 3 in middle of Canon 3; unarticulated, takes place in middle of
measure on 8 held note—changing leader-follower
relationship is clearly audible following the rest
Canon 3 diverging-converging canon, tempo switch in middle
CP 4 at beginning of Canon 3; unarticulated, begins with a rest
CP 5 closes Canon 3 and elides with the beginning of Canon 4
where it is articulated in bottom voice only
Canon 4 diverging-converging canon, begins with a brief interlude
before CP 6; tempo switch between CPs 6 and 7
CP 6 unarticulated, between two quarter note rests
CP 7 unarticulated, occurs just after last note attack in lower voice

 

 

 

 

 

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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

14

l.—
6

 

CP

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°“°"@ 2 3 ® © 6(2) 69*
Treble 16 5. ., .1 ’
W " , V

Middle 15 A ’;:‘~. ..‘.;’.; i e,
"4.11 '»-“E'-.. A L:

4‘ .' if. 1 .7." '1'

Bass 14 ‘2‘ its" - 3.:

Convergence 27 (18 x 1.5) 12 18 6 9
Periods

 

 

     
 

 

 

 

 

 

 

 

 

 

 

* An additional tempo switch occurs

10 at the beginning of Canon 9 G) 12

 

 

 

 

 

 

 

 

tempo overlap
about the 15-voice

Figure 3-36. Structure of Study No. 24 (“Canon 14/15/ 16”; modiﬁed from Gann 1995, 23).

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 Description
Canons l—12 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

CP 1 articulated in all voices at beginning of Canon 1

CP 2 unarticulated (occurs at dotted quarter rest), elides with beginning
of Canon 2

CP 3 unarticulated (occurs at half rest), elides with beginning of Canon 3

9

CP 4 articulated in all voices, Canon 3 extends one 8 measure beyond
this CP before Canon 4 begins

CP 5 articulated in all voices, elides with beginning of Canon 5

CP 6 unarticulated (occurs at eighth rest), elides with beginning of Canon
6

CP 7 articulated in all voices, occurs one measure beyond beginning of
Canon 7

CP 8 articulated in all voices, elides with beginning of Canon 8

CP 9 unarticulated (occurs at quarter rest), elides with beginning of
Canon 9 (there is also a tempo switch at this CP)

CP 10 unarticulated (occurs near end of long held note in all voices),
precedes beginning of Canon 10 by ﬁve sixteenth notes

CP 11 articulated in all voices, elides with beginning of Canon 11

CP 12 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

 

t - h “silent” CP at
CP at beginning emptiswnc beginning of section 9

of section 8

H 35:6 1.: 33'“ "
.. _ ﬁg

/

   
   

      
 
   

  
  

J.:3Ho) \

     
 

silo: 3.3.:

  
 
 

 

F" v

__.. .. _ _ "Wi— _ _: _

section 9 has tempo
switch at beginning
as well as in middle

 

Figure 3-37. Study No. 24 (portions of second and third systems of p. 12), showing CPs 8 and 9.

 

 

 

 

 

 

 

J. = 240
J. = 256 W 2nd §tatgmgnt 3rd §tatement J = 224
J_ = 240 1st statement 2nd statement 3rd statement
J. = 224 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 y: 240 1605/7 200 250

152

168 341 210 262 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 N ancarrow brings to the canonic process.

Study No.37 - Canon 150/160517/1683/4118011871Iz/200f210l2251240/‘250I2621121281114

Q)
5
Oi
03
N
a:

Canon: 1

 

 

 

 

 

 

 

 

 

CP cp CP
nme: 0:00 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

lentI’te._ __ _ __ __ _ éJzﬁe'y"
I (r —————

   
   

———-—-

Ja\?}\/t ﬂ 1.“."07’
13/5‘0 ﬂ 1::le

ll

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
4

CP4, which is a sixty-fourth rest at beginning of 4 measure; a tempo switch in some voices also

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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G G G
G G G E G
Vorce G G (.3 G B»
3 decal. - arxal. decal. - accal. - decal. - - - accel. - - - - decal. - - eccel. - -
2 accel. accel. - decal. eccel. - decal. decal. - . - accal. - - - - decal. > - - accal. - - -
1 accel. decal. accel. accel. - decal. decal. - - accel. - - decal. - - eccal. - - -
G + E G
8'1. G B,

V—I cadence in
voices 1 and 2

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%/ll/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).

Ceneral Cbservations

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
[ER— |5%R |5°/.A I [am [67.9]
Clock. E %A m-
|6°/.R [11%A lime ] [11%A j11°/.RJ
Section 2: Section 3: Section 4:
Canon 5 Canon 6 Canon 7
OF ‘/.A
_11%R‘ [8%A I8%R bot/t 1854.1: ]
-%n ‘ -a:-
m :- -
.7... -j -m11%A-
Section 5: Section 6: Section 7: Section 8: CP

 

%A 8°/.R 5169 | 596A

 

Canon 8 Canon 9 Canon 10 Canon 11
Durations not strictly proportional
Figure 3-42. Formal diagram of Study No. 27 (modiﬁed from Gann 1995, 16]). 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#,,, 13,, F4, and G5,,
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.

Cl
3 - _

Q
Canon 1 Canon 2 Canon 4 l’ Canon 5

the + M6

8w + Mo the + PS the + m6 8w + n16

clock line

I

’ 8vc + m7 ‘.

I
f 2 8ves + M2 ,’ 2 8ves + M2

the + P5 live + m6 the + m6

 

8va .
Canon 6 3: _Canon 7 Canon 8 Canon 10 Canon 11”,.

 

8vc + M3 m2 - m2 - mZ - m2

2 8ves + M7 |‘ (in 5 neutral”?

8

 

the + M3

CP

Figure 3-43. The levels of imitation between adjacent canonic voices and order of entrances of
the eleven canons of Study No. 27.

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.

Rl\ J75 $+accaCTOL AHLL o‘lkcr dmra‘Xichs ism-\‘xccch 51“ I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A R X
J 4 A II 3 f h
T I "I I I I I '—1 j)
E; . LA L . I 7 1
l r 7 iI A A A V' L i A 1 a a f 4‘ 1
f ‘ f - v . v 1‘ * v
’ F A i I 7 i K
1. A A— A %
A! A I A I A ' v 1 A j j ! A
v v V . . - T
_. ~ f—T—i
‘1 A I T J T n :F:_T 7 J I
j I f ' I ﬁf f 1' I I I I
is. A I A a A a j I A A I
L? _A 1 n ' T : t c ‘L V l A_ ? A v V '

 

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, 1 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

Table 3-3

Multiplication Factors for Acceleration and Deceleration in Study N o. 27

 

 

 

 

 

 

Multiplication Factors
Percentage Acceleration Dgelggatjgn
5% 0.9524 1.05
6% 0.9434 1.06
8% 0.9259 1.08
l 1% 0.9009 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—see Figure 3-43) have symmetrical patterns of levels of imitation
between the voices. In Canon 1, the ﬁrst two voices (A6%/R6% and Al 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)

 

_1—1—11:x] ”\10151 1014] 1(1I31 NW5] [0|3]
E -..- 1'11._. 7 _ -.. A. .' , ' ;-_..;:. I V '
I? I #‘ 11;};15111.\ 111’” 1.. ...'#" .1) 1.181.;- " ,1 .1011, i .._ Y 121-1,,
____________________________________________ l 0
|1)|3457|= 6210 l0|2345l= 64
R6% (35 notes)

[0|21 110141 __» 10141 “1015110271 __10371____‘ 10371 1 10151

[?: {1’0ghd 131.“ 0—1'10120 'T} ‘gﬂA—fﬂa 1:10;.3 ALOLITL' d :1

1
_____ _. 10131_ \1 102 71 __ 10151 ”N __ 11mg__ 1
[‘23 I 101*" 1111115?” '11" ”I “7‘" It“
__ltiIiﬁsl-lisfzif" I‘D-“7': 5” _”[.(II_2-—l57|“62’ll _____ 111145712511?

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

         
     
   
    

 

.m“
"“ . i

NHL.
NH
1
I~ H
I
L
4
its;

A

I V l

CP/leader-follower switch

A

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) 8.8%;1" i1: COP
r **-'“-“——~—‘-m -- -~ ‘- ' ‘— " _- ___.._ * -- '- *A- ' __- ‘f—“L‘j' ‘r ——“# Tma‘mwww‘l E H-~—~'—‘I
:&____ -i_ -2 .._..- __-...“ -,-_-. n... 1-_. -::._ -- _-+-...-.---W . 1*-.- . 2-.--..“ _l “__. - __.--.____ - . i- ._. __.r._2.-- __.. -_. _____..--.3
,_ -- __ --__ -_ _. u __ - -- _.- __ .- _ .-. .._..- __.. _ __--- - M .-___-. .-__________-_. __~ - -._ _. ._.A_..-____. _,._- __-._- ___, j

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! alc. m CA9. Mimi
18.7 18.5 13.2 12 8.3 7.5 4.1 3
19.6 18.5 14.0 13 9.0 8 4.6 4
20.6 19.5 14.9 14 9.7 8.5 5.1 4
21.6 20.5 15.8 15 10.5 10 5.7 4.5
22.7 22 16.7 16 11.3 @913 6.3 5
23.8 23 17.7 17 12.3 11 7.0 6
25.0 (25) 18.8 18 13.2 12 7.7 7
26.3 25.5 19.9 19 14.3 14 8.6 7
27.6 26.5 21.1 19.5 15.4 15 9.5 9.5
29.0 28 22.4 ® 16.7 15.5 10.6 8.5
30.4 30 23.7 22.5 18.0 16.5 11.7 10.5
31.9 31 25.1 25 19.4 19.5 13.0 (25'
33.5 623) 26.6 25.5 21.0 19 14.5 13—
35.2 35 28.2 27 22.7 22 16.1 14.5
37.0 36 29.9 29.5 24.5 Q49 17.8 16.5
38.8 38.5 31.7 @ 26.5 25.5 19.8 18.5
40.7 @ 33.6 33 28.6 27.5 22.0 21
42.8 41.5 35.6 35 30.9 30.5 24.4 24
44.9 45 37.8 37 33.3 33 27.1 26.5
47.2 45.5 40.1 40 36.0 35.5 30.0 @
49.5 51 42.5 42.5 38.9 @ 33.3 34.5
*52 52 *45 45 *42 42 *37 37
(54.6) 108 — 108 — 108 — 108
Calculation of opening values required in all voices to result in ﬁnal value of 108:
| 18.7 — 1 15.2 — | 10.0 — | 5.5 —— |
1 — 108 | — 108 | — 108 | — 108 ]

 

 

 

 

 

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
-._2 F _ _ __ descending dim. triad _, *4 . ___ ,. j
[;._;;—;_; ‘g:?§--:{._-_ #L; iii—9 h--— _. 21*: 4:?-

ascending dim. triad l

 

 

 

 

 

 

 

 

191 r: 1- ~ 3.6;, ;;'-”’:,i_1g~+fé-_:~~.fl

l--- Implics E major --------------------------- l 1 implies 8 minor -—1

 

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%/A5% 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

,. .._ ~__._h___-.‘T____-‘ _.
~4—I4 lb-oI—H—

1—
Tu 01.255”wa-

 

 

 

 

 

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rmuzl

F? :11 2; 3222.49 - 1 w T 1, a... 23l'7b'3ﬁ;j-;,ij;?::?l

 

   

 

——mr --1- I la: a n-

! s ,T" _______ I 2.; 2:, T T-
Fglv 52113112 1‘“ —TfLU7'vT'v b.

 

 

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.

 

 

 

 

 

 

 

1" 40.5 3&5 36.7 55.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘93???) Y1+'-(87p) I l l 1 l . 1
91m 3 1i i l U T l %f§LW
27.0 ' I27.0I 292l 31,5 1 ‘ 3‘40! 1
(R130) . .Jﬁ 5 l .1 3' u
9? T; T %—T T 122%.: #2 Mi:-

VW+%+%-—=T,._ 48.3 55.3

 

 

— ’435—
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).

Qanon 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/Gt) 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

f coca - f?‘ , _____ __ _. -
Ibﬁ :E—vi: if- 3‘. ; ﬁ1#*-IP'-'A ‘ 1 -_“_ba_ b.3—

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.

 

 

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9
9

 

 

 

MP‘M‘T‘S “a ‘h- M *V'M‘haw *V'W r3 4;er *L'M +vvm~vv~l
:, ; _.' A 4 x , 41 r -I 11" 3L1 11' i-
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Wizﬂﬂt‘: 21:4 :jiﬂr W" ”kl—‘4 Jr 7- .. IV
'0: A l' ' ' 'II ‘I 3 {TB} 1' ﬁ— ' 1 ‘1‘ #317] i
0
35/. +,,.... MM... +wWA
¥ 11' '11 I I T ‘
:ﬁL 4r ’ 7; "‘ I 7‘”
- I44 1K *1
v 1.1 I A
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__ _ m __..

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

   

T2 THE \

 

 

 

 

2 2 . ”5:
lowest note
on keyboard r8va ---------------------
In L, be

 

  

M 33:: . :22 ‘
FY’22? 2227212

 

 

retrograde inversion of ﬁrst 13 notes

 

* interval between bass and

treble portions = 5 8ves + m6 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.

Qggon 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 G#7/At7 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

of high fit/04‘ compound melody consisting of repeated high pitch and
chromatic descent and ascent through a tritone

 
  

        
 

 
   
 
 

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(5 :l: 41‘}: 7‘" (It) '3'
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persistent repetition of
lowest note on keyboard

HHH
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_. Er -_
m " V? L y-o- Vr—rr— .1.
If V Ir 2 {2 11 1y) 2') J . 2 22 v
5— i
chromatic tritone descent (all staccato note? '
A1196 2. .-(u0/.

    

\

 

l lb—j ~+—l~

é compound melody with chromatic
descent to lowest note

 

 

 

a.“

 

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’r‘ j:

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#2

 

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 / = 98 total

6% voice: A = 49 R = 49 / = 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#lAt, minor/major
and A minor/major triads, while the F4 and F#th,4 are the roots of F minor/major and
F#th 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- tempo direction changes take
board in the lowest voice place here (except in highest
voice, where it is one beat later)

  
  

 

_bt 4 , A b. L K

0
_ A 1" ll A Y

“b, If -- - -W't: “i; “'3 ’ ' _ir ._'_

 

ii

222.222

VN-V

B, ‘leV
Wﬂrnﬁr‘i 23731981? ,WMTTM ~'

altemately:' ti 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 (enharrnonically) 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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T - __.‘f--__ - .-._- 2E3 "F - . In. _- -. - .. . . . . ' . n
l b‘ ‘0‘ b0 6i

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 At”, 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#, G#, 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 Bl, 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 Et—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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179

Summm and Qonclusions

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 N ancarrow’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 fern 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 a 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

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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. “l/f noise” (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

 

 

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(a) The Koch snow- (b) A “generating motif” of music composed of a few intervals
ﬂake. 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)

Period (16 measures)

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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

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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SD = N

log SD = log N

D log S = log N

DH = lOg(N)/log(S)
/’2
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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) = l, 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

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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)/log(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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Figure 4-13. The Cantor gasket, DH = 1.89 (Schroeder 1991, 179).

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:

DH = log(20)/log(3) = 2.73
which conﬁrms that the ﬁgure is three-dimensional.2

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 l
0 .1. 2 1
3 3
0 l. 2 2 6 1 ﬂ 1
9 9 9 9 9 9
"1 '7. a. a 5:2 2:121 23.1 2?,
27 27 27 27 27 27 27 27 27 27 27 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 l%/1 la %/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 N ancarrow 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 N ancarrow’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 case in 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
I I
Voice 2 = B3 - G7
C7 - (45 pitches) ' "
Duration = 4
C6 ..
C5
Pitch overlap =
B3 - C5 _
(14 pitches) 1"“ ,7
C4 I . $1
Total pitch range =
E] - G7
C3 (76 pitches)
C2 Voice 1 = E1 - C5
(45 pitches)
Duration = 5

C1

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 +

2+3+4+5+6 29
0.83 + 0.67 + 0.5 + 0.33, more precisely summed up as 6 = 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
3+4+5+6 m

and S = 6 = 6 = 3; DH = log(4)/log(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

 

 

Q.
II
N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2+3+4+5+6=_2_0_=3.33 = 3+4+5+6=1_8_=3
6 6 = 4 J 6 6
= log(S) = 1.34 = 5 3 D = log(4 = 1.26
log(3.33) . log(3)
| d=6 J
(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

C7

 

Pitch overlap =
F5 - F6
(13 pitches)

C6

  
  

 

 

 
 

Top part of
Voice 1 = F4 - F6
(25 pitches)

Duration =

C5

 
   
  
 

    
   

Total pitch range
Bo - F7
(79 pitches)

C4

 
   
  

C3

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 = log(2)/log(l .75) = 1.24.

200

 

 

 

 

    

 

 

 

   

.- Voice 3 = A3 - A7
C7 ' (49 pitches)
‘1 Duration = 3
1 " 1‘1" 41“ ‘
131;; :31 I .
C6 - . . . . T“
" Pitch overlap of
,. ‘ Voices2and3= .
' A3- E6 (32 pitches)
"" I EM .Vé‘ -. ’1":-
.T ”"21 '_l:,. 1 I." >3 '. , .. 1.
C5- " “1."! ..T . 1:; .IT , _ 7,
4112‘ ‘ r. g i, ”it" 7' *1 fm) ' ‘5;
Pitch overlap of all 1 Total pitch range- -
3 voices— - A3- B4 :‘T‘Bo- A 7
C4 £11U5 pitches) I: (83 pitches)
11’ 17“ . ‘ , "22"” f‘i": .1 ‘L
Voice 2 = Q - 136 Pitch overlap of
C3 (49 pitches) Voices I and 2 =
Duration = 4 E2 - B4 (32 pitches)
C2
Voice 1 = H) — B4
(49 pitches)
Duration = 5
C1

O---
11

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

2907 + 3060 + 3230 + 3420 1__2617
duration of 3420 = 3420 = 3. 69. Thus, DH: log(4)/log(3.69) =

201

 

C7-

C6-

C5—

 

C4-

C3

C2

C1

Voice 4 = 3,4 - A7
(36 pitches)
Duration = 2907

L

 

Pitch overlap be-
tween contiguous
voices : BM - F6
(20 pitches)

 

 

 

 

,-' . ' 1111111'111‘! ;T;;' 5.: 3 '11 "5“ L‘
1’; Voice 3 = Q3 - F6 *v-Lx‘WM 111-131..“ , ' 1'- 1
'T

‘ ‘r‘! (36 pitches)
7 Duration = 3060 x"
M - 7911-11

‘

1.; :1 .. T. I“ . r,

L‘s-5‘ ,- 1' i f. 1 ‘4

Voice 2 = E2 - D5
(36 pitches)
Duration = 3230

Voice 1 = Bo - 3,3
(36 pitches)
Duration = 3420

I
S -

ujrwﬁézy 1.

  

,1

15‘.) 1.51 1‘11

1V"

Pitch overlap be-
tween contiguous
voices = 01.3 - D5
(21 pitches)

, Fiﬁ}!- ' '

)i

 

 

<31»:- ‘f is a “1411!?“-

Pitch overlap be-
tween contiguous
voices : £12 - 81.3
( 20 pitches)

Total pitch range =
30 - A 7
( 83 pitches)

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) =
F#6 - G#7 (15 pitches)

   
   
   
 
 

 

 

 

 

 

Voice 1 (see. A) =
E5 - 31,5 (6 pitches)
Duration = 200

C‘

LII

 

 

Voice 2 (sec. A) =
85 - F6 (6 pitches) .
C4 ‘ Duration = 175 V0106 1 (sec. B) =
E3 - F#4 (15 pitches)

Voice 3 (sec. C) =

 

C .
3 n pitch overlap C#4 - A7 (44 pitches)
between adjacent
voices = 26 notes
—— Voice 2 (sec. C) =
pitch overlap of all F#2 ' 1% (44 pitches)
C2 " three voices = ca -
G4 (7 "01“) Voice 1 (sec. C) =
Bo - G4 (44 pitches)
C1 -

 

 

 

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: 1751200;
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 1__]

r-—H-—II-—-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

105 + 120+ 140 +168 533

scaled voices have a total scaled duration of 168 = W = 3.17; thus

DH = log(4)/log(3.l7) = 1.20.

Total pitch range 2
Bo - At7
(82 pitches)

pitch overlap between
V contiguous voices = 48 pitches

Voice 4 = G#Q - C#5 + G#s - A1,?
55 pitches; duration = 105

mmwm * Voice 3 = ca - F#4 + C315 $13.7” e
55 pitches; diration = 120 '1

V0iceZ=Fﬂ -B3+F#1-Q,6
55 pitches; duration = 140

Voice1=Bo-E3+B3-Q,6
55 pitches; duration = 168

 

I
CP

Figure 4-21. Structural diagram of Study No. 32 (“Canon 5/6/7/8”); DH = log(4)/log(3.17) = 1.20.

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 Delay to later voices
factor: (compared to voice 1)

105 161.63beats

 

 

 

 

120 123.14beats

 

 

140 71.83 beats '

 

 

 

 

 

 

 

168 sec. 1 sec. 2 sec. 3
(l64beats) (77 beats) (190 beats)
4 431 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 I3 [C J
l l l 1
IA} i8. [C _j
T I 1:! l

1 I
IA 18 [C l
I l l l

[A iB [C

r— 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 f on 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 J

Rhythmic forms:

 

 

most common form (m. 1)

 

 

b—o—«h—lt—q

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; J
motive does not appear in sec. 2
Rhythmic forms:
A l I -r— j
WM” lFF’L‘T—fmml most common form (m. 24)
W413i; im” 1; r diminished form (m. 66)
C
c ascending or descending half step J
Rhythmic forms:
;;:;;?IJ most common form (m. 13)
m: i _ l . .
‘J 9 L I- 1 r“ minimally augmented form (m. 6)
\, . ‘
ﬂiiigzﬁi—L—ii maximally augmented form (m. 46)
V
d ascending or descending octave (or multi-octave) leap; J
largest leap = 4 octaves (e.g., mm. 6—7)
Rhythmic forms;
W‘ﬁwmﬂt 3:17;? most common form (m. 17)
:i
augmented form (m. 106)
diminished form (m. 31)
e ascending or descending “partitioned minor thir ” (3- J

note ﬁgure consisting of major second and minor
second); motive does not appear in sec. 2

Rhythmic forms:

J1
v

:‘a
hr

most common form (m. 50)

 

 

 

augmented form (m. 182)

 

 

209

 

 

Table 4-1 (cont’d.)

 

f ascending or descending perfect fourth; most commonly J
in descending form in sec. 2

Rhythmic fgrms:

 

most common form (m. 26)

 

 

 

Eggh , mid—— iv {gin __«i 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 (Fit—Fli-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 —‘——*“

 

 

8Vb

Figure 4-25. Canon melody in section 2 (mm. 76—107).

213

 

ab
_ ,__:,. ‘9; I?!
” Ellj':3g¥f§ {:31" i iii

 

 

wI?!._“9'::li_.:if'
‘ gas: Esuﬁ ,2; ~

 

 

   

 

 

 

 

 

 

 

it}: 13;": Li ’i: ' iii — " 2:9:E: iiﬁjgéﬁgﬂ

(displaced by octave due to range restriction)

 

 

2 :4 3- [I ii i 1*:_ it 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.)

 

   

 

 

 

 

 

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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 Rhythmic Duration
M1192 J. J J. J
8 J.: 136 - J:204 , J.:272 J:4os
7 J: 119 J: 178.5 J.:238 J:357
6 J: 102 J: 153 -- J.:204 L306"
5 J. : 85 J: 127.5 J. : 170 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

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)
(2 Ti: i
(6»
(C)

 

 

 

 

 

 

 

 

 

 

 

 

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

Ira-I

 

 

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 H. 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, and f 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 \—.

 

15- ; ~
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.

222

 

rarer-5.321

 

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 N ancarrow’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 1948 1983

2a late 19403 1984

2b ca. 1950

1 by 1951 1951 in New Music

4—30 1948—1960 No. 25 in 1975; Nos. 8, 19,
23, and 27 in 1977; Nos. 4,
5, 6, 7, 9, 10, 11, 12, 14,
15, 16, 17, 18, 20, 21, 24,
and 26 in 1984
No. 31 in 1977; No. 37 in

31, 32, 33, 37 1965—1969 1982; No. 33 in 1984

34, 35, 36 1965—1969 —
Nos. 35, 36, and 40 in

34, 35, 36, 40, 41 1969—1977 1977; No.41 in 1981

43, 48 1969—1977 —

42—50 1977—1988

 

 

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 N ancarrow’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)

 

 

10 110 13 I6 111 I1 12 14 I9 18 15 17

P0 D# C# F# A D E F G C B G# A#
P2 F D# G# B E F# G A D C# B}, C
P9 C B}, D# F# C# D E A G# F G
P6 A G C D# G# A# B C# F# F D E
P1 E D G B; D# F F# G# C# C A B
Pl 1 D C F A}, C# D# E F# B By G A

P10 C# B E G C D D# F A# A F# G#

P8 B A D F B1, C C# D# G# G E F#

P3 F# E A C F G A1, Bl, D# D B C#
P4 G F B; D; G; A; A B E D# C D
P7 A# G# C# E A B C D G F# D# F

P5 G# F# B D G A B}, C F E C# 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. 1—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

 

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a I a' I
(‘5 I ‘x If I IﬁI i f I:
f; f :' i T
fundamental 1 fundamental | l
{H «t

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 lS-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
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).

19
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. ‘1

   

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Peal.

‘|—~.—_._ __.—.._,
a;

(d) (6)

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

 

 

score error:
A# missing

is “f“-

    

(a)

 

 

——
———
————
———I

 

1
.-'—i T I

 

 

(1::9 I I
l
r . 2 j 3

 

 

 

 

 

 

 

 

(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 those in (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

Mom-4A - M- -

 

 

 

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 Palindromic Interval Pattern Between Arpeggio Triads in Section 3

Row‘o 1 2 3 4 5 6 7 8 91011

 

 

P7 5
Falling '
triads: B; A; D; E A B C D G G; E; F :
Rising
triads: D A G B; B F# E E; D; A; F C

 

---db

Row:
R11 11 0 1 2 3 4 5 6 7 8 9 10
Row:
R8 0 1 2 3 4 5 6 7 8 9 10 11
Falling
triads:

Rising
D F# A *
triads: ; D E F G C B A B; E;

 

G; E G A; E; D; C B; F D A B

 

 

Row:
P0 1 2 3 4 5 6 7 8 9 10 11 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 A;. 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

' .L. -

 

 

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 E; to A, are stated in this order:

(E;F#A)FE;A;FGF#E|E;A|EF#GFA;E;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
\ 5 8
66 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 Possible
in millimeters background unit?

 

Sustained layer:
" 7/8 2
1 1/1 1.5/ 12 3
15/16/16.5 4
22.5/23 6?
9 durations: 28/28.5/29 7 division of 4?
36/37 9
50/51/51.5/52 13
67/68/69 17

_ 96/97/97.5/98 24 _J

 

 

 

Arpeggio layer:
T 21/21.5 '—
30/30.5
38.5/39/39.5
7 durations: 63/63.5/64 ???
81.5/82/82.5/83
96/97

_ 120.5/121.5 ..

 

 

 

 

 

 

 

Sectional haracteristics in Stud No.25

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.

Seggign 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 E;, as

Shown in Figure 5-5. There is a 1.4" pause between sections 1 and 2.

.4
V 1 ’----, «

  
 
  

 

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

 

 

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 E;.

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

IF'AEW: I

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.

R11) 9’
8 10

  
   

   

r-

(P7) ,5: 2"
9 "1

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

senses a change in the leader-follower relationship between the voices; as Shown in
Figure 5-8, the same B; 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: (36 '_'ej-'ents")’£ i l l I l I l l l l 1 l I I
I v

 

 

 

 

 

 

 

 

 

 

 

 

..I J 1_J_J;+IJJ _J__LJ_' SIC—IT! w
(1)./8.2) I >5. 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

w

same ﬁgure appearing

ﬁrst in upper voice

although lower voice (‘ (é- _
is leading q -- - -~. 1' 1

echo distance

 

 

 

 

 

 

 

 

l
, l L I 4L 1
I r A 7 J I
H A
I 11.
v "— J- T

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. OVJ. 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 5-9. Study No. 25 (p. 48), showing multiple convergences in 9:10:12zlS tempo canon in
section 6. Rhythmic 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

 

 

 

 

 

 

 

l9-note arpeggios FJ,—A
tempo canon voice 4 D—G#
tempo canon voice 3 G—C#
tempo canon voice 2 C—F#
tempo canon voice 1 F—B

 

 

(voices in score order)

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 PO/RO
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 (0.9780)24 = 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}/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.

      

r?“

(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 5-13b)

246

 

 

 

 

 

 

 

 

 

 

omitted in section 9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Page73: _ 1
__.._ __. L. ______ 6___-_ .5-—-—I I
.. x31 .. IE4: 5 1 Ac6r '42:}: ‘1 ' .::%|' ,1: J31: l
@i ’Tﬁ ? i if? IL r I!“ :l 7111 l
L d
7';
K

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

(a)

’s lowest note

 

C#—-F# (V—I)

    

 

(b)

 

G pentatonic (‘5, -

 

r—r—

 

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.

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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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 Study 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
Mgvzmaelnt Harmonic Pattern Bass Line
$36 Se m f r:
3a I-I-I-I IV-IV-I-I V-V-I-I
A . R
Radio 5": N P re
FF
3b I-IV-I-I IV-IV-I-I V-IV-I-I
3e I-I-IV-I-V-I
I-IV-V/V-V
45a (preceded by a 6-note
dominant)
45b I-IV-V-I
45c (no continuing pattern)
e I
q L_H_._ — ——_ [9&0

 

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

 

 

 

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 subdominant 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

3..“

«Ho's .. 1 'I

 

 

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.

 

6 12
Figure 5-15. Temporal dissonance in Study No. 3b: use of 8 and 16 meters in upper three

4
voices against 2 meter in bass ostinato voice (score, p. 7).

 

 

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 HI 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

15
(eSpecially when the 16 meter is introduced in the second statement; see Figure 5-18).

257

 

(a)

 

 

 

(b)

 

‘

Figure 5-17. Recurring melodic fragments in Study No. 3e, showing use of major/minor tonic: (a)

9
score, p. 4; (b) score, p. 7 (this segment has a 8 rhythmic pattern in the middle line).

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

h_

 

-49“

I .H.

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

9”

weaving drunk. Gann points to this uneven rhythm as an “irrational ‘spastic rhythm (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

 

 

denominators of prime numbers, especially ll, l7, 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

HEY-hm“,— a' H

.._-

 

 

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 13-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
378 378 1/11 (379.5)
357 735 3/ 17 (736.6)
250 985 4/17 (982.1)
220 1205 9/31 (1211.8)
270 1475 6/17 (1473.2)
132 1607 5/13 (1605.4)
none; value is 0.495 of cumulative total,
460 2067 suggesting relation to 1/2 (2087)
128 2195 10/19 (2196.8)
318 2513 none; value is 0.602
357 2870 none; value is 0.688 (20/29 = 0.690)
92 2962 22/31 (2962.2)
171 3133 none; value is 0.751 of cumulative total,
suggesting relation to 3/4 (3130.5)
382 3515 16/ 19 (3514.9)
224* 3739 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),
V0), 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

 
 

_-._- --'--- .-___13‘
(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.

 

’21):- 3..

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 w

 

   

 

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

 

 

 

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
I’s 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

.d—g- *_._ .. .

I11.

 

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., ﬂatted 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

V (G): G

I (C): C, C

IV (Fig, BL, F

VN (D): F, F, F, D

V (G): G, G

I (C): none

IV (F): none

VN (D): BC

V (Gt/D. D

I (C): C

IV (F): Bi,

VN (D): A

V (G): G, G

I (C): D, E, A, Bi,

IV (F): B, C, G, D

V/V (D): A1,, B, F, E, D
v (G): G, A, B1,, 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).

 

 

 

double-octave glissandi

 

 

 

 

 

single-octave glissandi

 

 

 

double-octave glissandi

 

 

 

 

 

double-octave glissandi

 

 

 

double-octave glissandi

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_____ ______.._-__-._-_-.H.___I
[ELF 1.4.9.1? l—p .l-p N p n
l». . . r' a
r +1 V v v "v

VN (D major) extends beyond transposition to V >

 

 

 

 

 

 

     

 

I/L—Tffuiu'l'v" ' - 3" I/wo

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'-i:._:-.- .._... 3H--_-_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 14-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 : é a
+1 r : i l '
(b) E E E E 5

 

 

 

 

 

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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'IP
|

 

 

 

 

 

 

 

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

.32

1+3...

     

 

   

5 5
(Syncopation) .......

. ‘3’

 

Ur
--Hu-
Ur
Ur
UI

~

\JJ
.
l

 

 

‘u
|

 

 

i

 

 

 

 

%
2"

 

 

‘H
N-

 

 

 

 

 

L

j:—
rm"
116

Figure 5- 27. Syncopation between two melodic layers in 45b (score, p. 12). 1,, 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:
FALBEABCEALFDGBLEALBGLDiAtFCAD
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 Resolution Note’s Scale
Key’s 7th Degree in New Key
up a perfect fourth (Fig. 5-28a) m2 third
up a major second (Fig. 5-28b) m2 ﬁfth
up a minor third (Fig. 5-28c) M2 fourth
down a minor third (Fig. 5-28d) m2 octave
down a perfect fourth (Fig. 5-28e) m2 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
1? p I”: 1‘,
ﬁt 11 if i‘ﬁ is
$1 4‘ ’ "*3“
G
(b)
m2
5:
0 E ‘1'; g ”‘T' ‘&n
We v" .. f
1 \f—
F
(C) (d)
m2
'1: I
. [LI 9 Q 1';
Q—h—Hwy w 2‘ I
8% V at?
F 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

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,, Ar, 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

__....MH

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:;r..

 

 

 

 

 

I 4,3 ________ 4.1

    

(b)

 

Figure 5-30. Melodic uses of the boogie-woogie bass pattern in No. 45c: (a) score, p. 2; (b) score,
p. 6. [Note: the score copy to which 1 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)
i3: 3? if é

 

g D

 

 

     

W

I g/L.__._—.____-_H

(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
15-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 Bi, tonality in the top voice (shown in

Figure 5-30a) before the head motive enters in EL.

 

1. pt::;':"_ ----

    

(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 H:

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 I—IV—V—l 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 El, 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

 

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

1r
I

 

 

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 seeming 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 V2. 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 N ancarrow’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
ﬂatted, 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 A1,; 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 N o. 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 Studies for 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 N ancarrow’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 C58 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 (A14) 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 beat should have ﬂat
signs repeated for notes B13 and D1,4

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'nrce 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 4 and not G4

5 . . . . 13
p. 8, second system, bass staff: 16 meter Signature 1S mrssrng after 16 measure

5 10
p. 8, last system, bass staff: in the three 16 measures that follow the 16 measure, the

rhythm Should be DD (ﬂag is missing from ﬁrst note in each of those measures)

5
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

6
for next measure 16

p. 12, ﬁrst system, bass staff: last note of that line should be E, (ﬂat Sign is missing)

295

tud #5
7
p. 13, tenth staff: 16 ostinato (see ostinato ﬁgure on p. 11) should be stated beginning

35
with 47th sixteenth note of that line—ﬁgure should be preceded by blank 16 measure

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)

21 30 21
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

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 BIS 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

F3

[)3
+p. 4, second system, bottom staff: in third measure, chord should change to 62 and be
tied over to fourth measure

Study #11

[)7
p. 7, second system, top staff, last measure: chord should be A6 for correct voicing of

5
M7 chord (no 5th) that occurs consistently on each 8 bar that opens a 20 eighth-note
segment

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 Signature at end of ﬁrst system should be 16 (not 16 )
to be consistent with metric pattern of Opening canon the ﬁrst three times, and

6
following measure should have 16 signature

27
p. 12, third system: last meter change (16 ) is notated one bar too early

Study #16
0p. 7, staff “e”: ﬁrst note of second measure should be .I rather than li

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

6
p. 12, third system, bottom staff: chord in ﬁrst 4 measure Should be Open ﬁfth stack, and
F2 should not be ﬂatted (see Gann, p. 176, transposition levels for Canon 9)

M

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 D4 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 E6 (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

. . . . . . . . 4
0p. 2, third system: eighth note rests are rmssrng 1n entrre line (meter rs 8 )

298

p. 3, second system, middle (second) staff: rhythm in sixth complete measure should be
'1

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

F1

6
p. 3, second system, bottom staff: 8 meter marking is missing in last measure of the line

p. 5, second system, bottom staff: all meter changes are missing in this line: 8 in second

6 . 9 .
complete measure, 8 1n next measure, and 8 1n ﬁnal measure

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: C111?5 J is missing (tied over from previous system) in
incomplete measure at beginning of line

6
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 C4 rather than C3

9
p. 8, ﬁrst system, bottom staff: 8 meter marking is missing in ﬁrst full measure
6
p. 8, second system, bottom staff: 8 meter marking is missing in ﬁrst full measure

9
p. 9, ﬁrst system, bottom staff: all meter changes are missing in this line: 8 in second

6
complete measure, 8 two measures later

p. 10, third system, top staff: change to treble clef missing in third full measure

Study #33

5
p. 30, second system, bottom staff: ﬁrst complete measure should have a 16 meter
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

4
p. 2, third system, top staff: 4 meter signature should appear prior to the entrance of this
voice
p. 18, third system, bottom staff: in last measure, top note of half-note chord should be
D13 (ﬂat sign missing)

4
p. 47, ﬁrst system, third staff: 4 meter change should be inserted at beginning of third
full measure

4
p. 48, ﬁrst system, bottom staff: 4 meter change should be inserted at beginning of third
measure

Stud #37

4
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
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
11

p. 1, second system, bottom staff: rests are missing after ﬁrst quarter note in 4 measure:
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

. 4 . . .
p. 6, tlurd system, second staff from top: ﬁrst 4 measure 18 mrssrng a quarter rest on
second beat

. . . . 11
p. 9, ﬁrst system, third staff from top: quarter rest 18 mrssrng at end of 4 measure
p. 21, second system, second and fourth staves: in measure that begins with shared

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 A4
p. 11, top system, middle staff: glissando is missing a concluding E4
+p. 12, bottom system, top staff: tenth note should be a C major triad rather than note G6
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 B4 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
. 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 C4 (near middle of system), note E1,4 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 4) should be one octave lower

(C3)
Study #43

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 1n score
p. 14, ﬁrst system, second staff: line should end with a major triad eighth note built on F4

"9'91”:

'9

301

 

Stu #48

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 #
5.2%

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_'p: Nancarrow, Conlon. Rhythm Study No. I for Player Piano. 1952.

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.

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.

Rppordings: N ancarrow, 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: Studies for
Player Piano, Vols. III/IV [compact discs], pp. 7-19. Wergo: WER 60166/67-50,

303

 

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].

#3

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].

#4

Mp: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18 for
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—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/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].

#5
Score: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18 for
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.
N ancarrow, Conlon. Complete Studies for Player Piano, Vol. 2 [Studies No. 4, 5, 6,

304

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].

#6

Sppm: N ancarrow, 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/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,
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].

#7

SEE: N ancarrow, 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: N ancarrow, 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.

N ancarrow, 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. 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. 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 N ancarrow: Studies for Player Piano; Piece No. 2 for Small

305

 

 

 

#8

Orchestra; Tango; Toccata; Trio; Sarabande and Scherzo” [review]. Tempo no.
189 (June 1994): 49—50.

S993: N ancarrow, 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, l7, 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: Studies for
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].

Sppr_e: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, 17, and 18 for
Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: N ancarrow, 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.

N ancarrow, 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.

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 F idelity/Musical America 34 (November 1984):
45—46.

306

T.“

 

#10

m: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, I 1, 12, 15, 16, 17, and 18 for
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 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].

#11

Sm: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, I6, 17, and 18 for
Player Piano (Volume 6 ). Santa Fe: Soundings Press, 1985.

Recordings: N ancarrow, 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. 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 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 F idelity/Musical America 34 (November 1984):
45—46.

#12

m: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, I6, 17, and 18 for
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 Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,
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. HI/IV [compact discs], pp. 20—30. Wergo: WER
60166/67-50, 1987.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High
F idelity/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 F idelity/Musical America 34 (November 1984):
45—46.

#13

Sm: never released by Nancarrow (see Gann, p. 68)

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.

N ancarrow, Conlon. Complete Studies for Player Piano, Vol. 4 [Studies Nos. 9, 11,
12, 13, l6, l7, 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. 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 F idelity/Musical America 34 (November 1984):
45—46.

#14 (Canon 4/5)

Sporp: N ancarrow, 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.

N ancarrow, 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, 15, 16, 17, and 1 8 for
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, 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, 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].

#16

Sm: Nancarrow, Conlon. Studies No. 4, 5, 9, 10, 11, 12, 15, 16, I7, and 1 8 for
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.

N ancarTow, 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 F idelity/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 18 for

309

 

 

Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. IH/IV [Studies N o.
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—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 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 F idelity/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 18 for
Player Piano (Volume 6). Santa Fe: Soundings Press, 1985.

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies N o.
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—l798, 1984.

Writings: Tenney, James. “Studies for Player Piano Vol. IV: Notes on Studies #9, 11,
12, 13, I6, 17, 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.

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 F idelity/Musical America 34 (November 1984):
45—46.

#19 (Canon 12/15/20)

Scprp: N ancarrow, 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. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, l6, l7, 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 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, 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 F idelity/Musical America 34 (November 1984):
45—46.

#20
Score: N ancarrow, 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: N ancarrow, 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/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].

#21 (Canon X)

m: N ancarrow, 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 N o.
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.

N ancarrow, 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%)

Sc_o_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.

N ancarrow, 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 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: N ancarrow, 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.

N ancarrow, 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. 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: N ancarrow, 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: N ancarrow, Conlon. Studies for Player Piano, Vols. HI/IV [Studies No.
1, 2a, 2b, 7, 8, 9, 10, 11, 12, 13, 15, 16, l7, 18, 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_orp: Nancarrow, Conlon. “Study N o. 25 for Player Piano.” Soundings 9 (Summer
1975).

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. 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: N ancarrow, 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—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/II [compact discs], pp. 7-11. Wergo: WER 6168-2/6169-2,
1987.

La Barbara, Joan. “The Remarkable Art of Conlon Nancarrow.” High
F idelity/Musical America 34 (May 1984): 12—13.

313

#27 (Canon 5%/6%/8%/11%)

Spgpe: N ancarrow, 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. Studies for 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, 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 N ancarrow’s Study #2 7 for Player Piano Viewed
Analytically.” Perspectives of New 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 F idelity/Musical America 34 (November 1984):
45-46.

#28

Scpr_e: unpublished

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. III/IV [Studies N o.
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.

N ancarrow, 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, 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 F idelity/Musical America 34 (November 1984):
45—46.

#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—1798, 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 F idelity/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: N ancarrow, 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. 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.

N ancarrow, 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_C(£_CI unpublished

Recordings: Nancarrow, Conlon. Studies for Player Piano, Vols. IIH [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,
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 F idelity/
Musical America 34 (May 1984): 12—13.

Thomas, Margaret Elida. Conlon Nancarrow ’s ‘Temporal Dissonance’: Rhythmic

315

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].

#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. 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.

N ancarrow, 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. HI/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 / 10 / 11
#34 (Canon 4/5/6/ 4/5/6/ 4/5/6)

Spore: unpublished; string trio version exists (see Gann, p. 132)

Recprdings: 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.

N ancarrow, 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, 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 F idelityﬂl'lusical 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, 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, 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 F idelity/Musical America 34 (November 1984):
45—46.

#37 (Canon 150/160 kl [168 3/4 I 180/187 ”2 [200/210/225/240/250/262 ”2 [281 ’4 )
Sm: Nancarrow, Conlon. Study No. 3 7 for 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. 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 Studies for Player Piano [Ph.D. dissertation,
Yale University]. Ann Arbor, Mich.: University Microﬁlms, 1996 [see Appendix
C of this dissertation for page numbers].

#38
renumbered by Nancarrow as Study No. 43

#39
renumbered by Nancarrow as Study No.48

#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. 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.

N ancarrow, 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: N ancarrow, 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 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),

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: N ancarrow, 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. 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.

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: N ancarrow, 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. 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.

 

319

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.

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].

#46

Score: unpublished; Gann makes reference to viewing the punching score (p. 34) and
a ﬁnal score (p. 263) which lacks barlines (p. 265)

Recordings: N ancarrow, 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, 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.

 

#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: N ancarrow, 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.

 

#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: Studies for
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
Study No
Study N 0

Study No
Study No

Study No
Study No
Study No

Study No

Study No.
Study No.
Study No.
Study No.
Study No.
Study No.
Study No.
Study No.
Study No.
Study No.
Study No.

Study No.

.1—18, 31,168n3
. 2 — 8, 12, 14, 15, 18, 22, 24, 26, 27, 101—04, 168n3
. 3 — 12, 14, 15, 18, 24, 25, 26, 27, 34n12, 53, 54—60, 62, 64, 165, 168n3
.4 — 22, 24, 86n10
.5 — 10, l4, l7, 34n12, 60—63, 64, 168n3
. 6 — 16, 24, 25, 53, 168n3
.7 — 12, 16, 17,18, 89, 91—92, 118—21,168n3
.8 — 8, 10, ll, 12, 13, 19, 24, 86n10, 89—91, 167
.9 ~—— 7, 8, 13, 104-08, 168n3
10 —15,18, 24, 27, 53
11— 6, 7,10,15, 24, 27, 53,115—18,136,138-41,151
12 — 8, 17, 24, 53, 168n3
l4 — l3, 17, 66—67, 97—99, 136—38, 167, 168n3
15 — 13, 16, 17, 94, 96, 97, 99—101, 167, 168n3
16 — 122—24
17 — 13, 17
18 — 13, 96, 168n3

l9 — 13, 16, 17, 87—88, 168n3
20 — 8, 17, 18, 81—84, 86n10, 88—89, 165
21 — 8, 13,15, 17, 125—26,127

23 — 8, 10, 17, 128—32, 134, 151

322

Study No. 24 — 13, 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

324

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325

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N ancarrow, Conlon. Rhythm Study No. 1 for Player Piano. 1952.

N ancarrow, 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.

Nancarrow, Conlon. Studies for Player Piano, Vols. I/H [Studies No. 3a, 3b, 3c, 3d, 3e, 4,
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