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RHYTHM IN THE THEORY AND MUSIC OF
PAUL HINDEMITH
BY
Gary Allen Sprague
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Music Theory
School of Music
1 997
ABSTRACT
RHYTHM IN THE THEORY AND MUSIC OF
PAUL HINDEMITH
By
Gary Allen Sprague
This study investigates the similarities and differences between Paul
Hindemith’s theoretical writings on rhythm and the application of rhythmic
principles in his musical compositions. Hindemith’s primary texts address the
function of harmony and melody; the omission of rhythm has been described as
a major flaw in his theory. However, references to rhythm are scattered
throughout Hindemith’s writings. Additional questions address the relationship
of rhythm to melody and harmony according to Hindemith, similarities and
differences between Hindemith’s concept of rhythm and other theories, and the
development of an analytical model for use in studying Hindemith's music.
Chapter One presents biographical information concerning Hindemith and
his major works. A brief survey of rhythmic theory is presented to place
Hindemith’s ideas in historical context. Chapter Two presents Hindemith’s
theories of melody and harmony from The Craft of Musical Composition.
References to rhythm are taken from all three volumes of The Craft Elementary
T_raining for, Musicia_ns_, the two volumes of ‘_I'r_aditional Harmonv. and
Hindemith’s lecture notes. Chapter Three develops an analytical model for
rhythm by exploring five recent models. Chapter Four applies this model in a
rhythmic analysis of the First Piano Sonata (1936) by Paul Hindemith. To
answer the primary research question, a comparison is made between the
rhythmic characteristics of Hindemith’s music and the general features of rhythm
in music of the common practice period.
The major findings of the research are:
1) Hindemith’s understanding of rhythm is orthodox compared to his
alternative theory of harmonic and melodic organization;
2) Hindemith’s writings confirm that rhythm, and especially meter, is the
organizational force that controls the other elements and parameters of
music;
3) Hindemith’s application of rhythm is similar to that of the common practice
period, although he does veer from common practice norms in certain
areas;
4) The proposed model of analysis based on Hindemith’s compositional
procedures and metric organization does not conflict with Hindemith’s
theories of rhythm;
5) Hindemith’s application of rhythmic principles in his music does support his
theoretical references to rhythm.
Copyright by
GARY ALLEN SPRAGUE
1 997
This dissertation is dedicated to my family, who have remained loyal and
supportive through the life of this project: my wife Julie, children Sarah and Ben.
Thank you for your loving patience and support.
ACKNOWLEDGEMENTS
The writer graciously expresses his thanks to Dr. Bruce Campbell, guidance
committee chairman, for his assistance in completing this dissertation project.
Committee members Dr. Charles Ruggiero, Dr. Dale Bonge, and Dr. Jere
Hutchesen, and Dr. Dixie Durr are also thanked for the finalreading of the
manuscript. Thank you to Dr. Cynthia Taggart for much-needed
encouragement.
Dr. David Neumeyer of Indiana University is thanked for his assistance in
acquiring the partial translation of m and the Cleveland lecture notes of
Hindemith. Without his previous research this work could not have been
completed.
Permission was granted from European American Music Distributors
Corporation, sole US. and Canadian agent for Schott & Co., to reproduce
excerpts and examples from the copyrighted works of Paul Hindemith: from
Erste Sonata far Klavier, 1936, © B. Schott’s Soehne, Mainz, 1936, © renewed,
all rights reserved; from The Craft of MM Composition Sock 1: Theory,
©1942 Schott & Co., Ltd., London, copyright renewed, all rights reserved; from
Craft of Musical Composition iook g:_§xercises in 2—Pa_rt Writing, @1939 by B.
Schott’s Soehne, English translation ©1941 by Schott & Co., Ltd., copyright
renewed, all rights reserved.
vi
“Theory cannot do everything: it can only guide. The human element of feeling,
with its imperfections, must be present to move the emotions.â€
C. F. Abdy Williams,
The Rhythms of Modern Music.
(London: Macmillan and Co., Ltd.,
1909: 85.)
vii
TABLE OF CONTENTS
List of Tables ........................................................... x
List of Figures .......................................................... xi
Key to Symbols ...................................................... xiii
CHAPTER ONE ......................................................... 1
Introduction ......................................................... 1
Statement of the Problem ............................................ .2
Background of the Problem ............................................ 4
Early Rhythmic Theories ............................................. .8
Purpose of the Study ............................................... .18
Research Questions ................................ . ................ 19
Organization ....................................................... 20
Scope and Limitations .............................................. 21
Definitions ......................................................... 22
CHAPTER TWO ....................................................... 25
Review of Hindemith’s Theory ........................................ 25
Hindemith’s Theory of Harmony ...................................... 36
Hindemith’s Theory of Melody ........................................ 46
Example of Hindemith’s Method of Analysis ........................... 50
The Role of Rhythm in Hindemith’s Theory ............................ 54
CHAPTER THREE ..................................................... 66
Analytical Procedures .............................................. 66
Development of the Model of Analysis ................................. 68
The Derived Model of Rhythmic Analysis .............................. 85
CHAPTER FOUR ...................................................... 87
Demonstration of the Proposed Method of Analysis ..................... 87
Hindemith’s Piano Music ............................................ 88
Form-Defining Rhythms of Hindemith’s First Piano Sonata ............... 91
The Phrase Rhythms of Movement One .............................. 102
The Unified Groups of Section A .................................... 108
The Unified Groups of Section B .................................... 114
The Unified Groups of the Coda ..................................... 1 17
viii
CHAPTER FIVE ..................................................... 123
Summary of the Research Findings .................................. 123
Answer to Collateral Question 1 .................................... 123
Answer to Collateral Question 2 .................................... 125
Answer to Collateral Question 3 .................................... 127
Answer to Research Question ...................................... 128
Areas of Continuing Research ...................................... 132
APPENDIX A Cooper and Meyer’s The Rhythmic Structure of Music ....... 134
APPENDIX B Structural Theory and Rhythmic Analysis .................. 140
Structural Levels After Schenker .................................... 144
APPENDIX C Figure 29 Analysis of Erste Sonate for Klavier, I ............ 147
APPENDIX D Graphs of the Tonal Structure of Movements 2-5 ............ 157
BIBLIOGRAPHY ..................................................... 1 62
Bibliography ...................................................... 1 62
General References ............................................... 167
LIST OF TABLES
Table 1: Hindemith’s Scale Derivation, Step 1 ............................ 29
Table 2: Hindemith’s Scale Derivation, Step 2 ............................ 30
Table 3 : Table of Chord Groups ..................................... 38-39
Table 4: Nonchord Tones .............................................. 49
Table 5: Duration, Complement, Range, and Hierarchy .................... 80
Table 6: Estimated Durations of the First Piano Sonata .................... 94
Table 7: A Comparison of Performance Times of the First Piano Sonata ..... 97
Table 8: Phrase Durations of Movement 1 .............................. 103
Table 9: Durational Values of Movement 1, Upper Voice of Two-Voice
Framework .................................................. 106
Table 10: A Comparison of Rhythmic Styles ............................. 129
Table 11: Cooper and Meyer’s List of Analysis Symbols .................. 138
LIST OF FIGURES
Figure 1: The Medieval Rhythmic Modes ................................ 12
Figure 2: Notational Mensurations of the French Ars Nova ................. 14
Figure 3: The Overtone Series ......................................... 28
Figure 4: Hindemith’s Scale Compared to Ptolemy and Malcom ........... 32
Figure 5: Series 1 ..................................................... 34
Figure 6: Symbols of Chord Root Relationships .......................... 34
Figure 7: Series 2 .................................................... 34
Figure 8: Cadence Formulas ........................................... 43
Figure 9: Melodic Degree Progression .................................. 47
Figure 10: Melodic Step Progression .................................... 47
Figure 11: Hindemith, E;rste Sonate fur Klavier 1936 ...................... 51
Figure 12: Graph of Formal/1' onal Design ............................. 57-59
Figure 13: Lester’s Model of Rhythmic Analysis ........................... 70
Figure 14: Epstein’s Model of Time Structure in Music ..................... 73
Figure 15: Smither’s Classification of Metric Structures .................... 78
Figure 16: Hindemith, ERSTE SONATA FUR KLAVIER (1936)
Movement 2, m. 1—4 ......................................... 79
Figure 17: Tonal Plan of Piano Sonata Number 1 ......................... 92
Figure 18: Tonal Plan of Movement 1 .................................. 104
Figure 19: Unified Groups of Section A, Measures 1-10 .................. 109
Figure 20: Unifying Patterns of Section A ............................... 1 1 1
Figure 21: Unified Groups of Measures 11-22 ........................... 1 12
xi
Figure 22:
Figure 23:
Figure 24:
Figure 25:
Figure 26:
Figure 27:
Figure 28:
Figure 29:
(List of Figures, continued)
Unified Groups of Section B ................................. 115
Movement 1, Coda ..................................... 1 18-1 19
Unified Groups of Movement 1, Coda ........................ 121
Analysis of Erste Sonata fur Klavier, I ......................... 147
Tonal Plan of Movement 2 ................................... 157
Tonal Plan of Movement 3 ................................... 158
Tonal Plan of Movement 4 ................................... 159
Tonal Plan of Movement 5 ................................... 160
xii
Key to Symbols
Hindemith's Neume er's
Ls hols M
<I> <I>
6‘
9 9
v1 vi
111 III
II\III 111
V\VI V1
11 11
v1\vn VII
xiii
Root
59mm
Tonic
Perfect Fifth
Perfect Fourth
Major Sixth
Major Third
Minor Third
Minor Sixth
Major Second
Minor Seventh
Minor Second
(Upper Leading Tone)
Major Seventh
(Lower Leading Tone)
Tritone
Chapter 1
Introduction
Paul Hindemith (1895-1963) was influential as a performer, conductor, com-
poser, teacher, and music theorist. His musical life has borne much criticism,
from his belief in the usefulness of music (popularly labeled Gebrauchsmusik),
to his formulation of unique theoretical ideas. Kemp has stated that Hindemith
was “the foremost German composer of his generation ;†he was important to
musical developments in both Europe and America preceding and following
World War II.1 Musicians still perform Hindemith’s sonatas while most major
orchestras include Symphony: Mathis der Maler (1934) and Symphonic Meta-
morphosis of Themes by Carl Maria von Weber (1943) in their repertoires.
Other works such as Ludus Tonalis for piano (1942) and the song cycle Qa_s
Marienleben (1922-23, revised version 1948) are interesting from a contrapun-
tal and theoretical viewpoint.
Hindemith’s teaching stresses three criteria for a composer: inspiration,
worthwhile musical ideas, and technique. “When a composer writes he must be
able to do so without any consciousness of technic [flaunthe composer must be
wholly unhampered by mechanical contrivances of any kind.â€2 His books
1 Stanley Sadie, ed. New Grove’s Dictionary of Music and Musicians,
Volume 8, (New York: MacMillan Publishing Co., Ltd., 1980), s. v. “Paul Hinde-
mith,†by Ian Kemp, 573.
2 Paul Hindemith, “Time, Only, Tells,†Etude 57 (October, 1939): 629; Idem,
The Craft of Musical Compgsition, Vol. 1: Theoretical Part, trans. by Arthur Men-
del (New York: Associated Music Publishers, 1942, rev. 1945): 2, 11.
1
2
develop a concise theory of music, emphasizing the development of technique
for the success of the composer.
Statement of the Problem
Hindemith felt that the old system of tertian harmony was useful only as an
historical method,3 so many of his books on music theory provided teaching
materials for his classes at Yale University. The two-volume set A Concentrated
_(_3_o_urse in Traditional Harmony provides teaching materials in condensed form
which introduced the student to the study of traditional harmony in an efficient
yet thorough manner.4 His intent was to move the student quickly from the exer-
cises to what he considered to be “living music.â€
Elementary Tra_ining for Musicians is a rudiments text covering basic funda-
mentals and aural training for the beginning musician.5 Each chapter is divided
into three parts: Action in Time, Action in Space, and Coordinated Action. Exer-
cises develop awareness of rhythm and meter by stressing the activities of clap-
ping or tapping rhythms, singing intervals and melodies, and taking rhythmic
and melodic dictation. A third book, A Comugser’s World: Horizons and Limita-
tions, is based upon the 1949 Norton lectures at Harvard University.6 In this
3 Hindemith, Traditional Harmony. Vol. I, iv.
4 Paul Hindemith, A Concentrated Course In Trflitional Harrnonv. Vojlume I.
(New York: Associated Music Publishers, 1943); Idem, A ConcentrategCOIï¬e;
In TraditionaI_Ha_rmonv. vmme g:_ Exercises for Advanced Students, translated
by Arthur Mendel. (New York: Associated Music Publishers, 1949).
5 Hindemith, Elementagy Training for Musicians. (New York: Associated
Music Publishers, 1946).
6 Hindemith, A Composer’s Worlg: Horizons and Limitations. (New York:
Anchor Books, 1952, rev. 1961).
3
volume Hindemith discusses many aspects of music, including his personal phi-
losophy of music.
The Craft of Musical Commsition is the main source for Hindemith's theory.7
In these three volumes he works out a system of composition and pedagogy
that attempts to explain all music—past, present, and future—and that is also
applicable to music of non-Westem traditions. The Craft series is both a theo-
retical treatise on the nature and order of tones as well as a method of teaching
composition. Hindemith emphasizes the role of harmony and melody in his the-
ory, providing a unique alternative to traditional harmonic and melodic analysis.
The development of his theory in The Craft led him to an increasing reliance on
a type of functional harmony, especially with those works directly aligned with
The Craft theory, such as LudusTonalis for piano (1942).
His devotion to tonality, a melodic style based on the four-phrase sentence
derived from folk-song, chorale, and march, and a metric accentuation drawn
from similar models yielded a personal profile already well-established by
the 1930’s.8
Although many references to rhythm occur in his books and articles, he fails
to include a detailed discussion of the role of rhythm in the creation of a musical
composition.9 When rhythm is mentioned it is usually linked in some way to its
interaction with melody and harmony.
While Hindemith consistently criticizes the world of music theory for being
7 Paul Hindemith, Ina Cgft of Musical Commsition, Vol. 1: Theoretical Part,
trans. by Arthur Mendel (New York: Associated Music Publishers, 1942, rev.
1945); Idem, Vol. 2: Exercises in Two-Part Writipg, trans. by Otto Ortmann (New
York: Associated Music Publishers, 1941); Idem, Ubungsbuch fiJr den dreistim-
migen Satz, ed. by Briner, Meier, and Rubeli. (Mainz: Schott and Co., 1970.)
Hereafter referred to as Craft I II or m.
8 Glenn Watkins, Soundings: Mu_sic in the Twentieth Centug, (New York:
Schirmer Books, Inc., 1988): 348.
9 William Thomson, “Hindemith’s Contribution to Music Theory,†Journal of
Music Theom 9 (1965): 66, 69.
4
unable to explain the working processes of rhythm, he himself formulated only
an incomplete theory of rhythm.10 In light of the fact that Hindemith’s theories
of harmony and melody have been used to analyze much of his music, the spe-
cific role of rhythm in his compositions and theory has been largely neglected.
A comparison of Hindemith’s theoretical concepts of rhythm with the application
of rhythmic principles in his musical compositions needs to be addressed more
completely.
Background of the Problem
Hindemith’s work has been divided into three style periods: his youth to
1923, 1924-1933, and 1933-1963.11 Born on November 16, 1895, his family
moved to Frankfurt, Germany in 1902 where his father was a house painter.
Hindemith began violin lessons at age 9 and four years later received free
admission to the Hoch Conservatory, his teachers being Anna Hegner and
Adolf Rebner. In 1912 he began composition lessons with Arnold Mendelssohn
and Bernard Sekles.
The works of the first period consist primarily of required conservatory
compositions in the late Romantic style of Brahms.12 Representative of his
student pieces are the String Quartet in C major from 1915 (his Opus 2), a
10 Hindemith, Craft I, 179; Idem, Elementagy Training for Musicians, 157,
159; Idem, Craft Ill, 30, Translation, 21 -22; Idem, “Methods of Music Theory,†23-
24.
11Sadie, s.v. Kemp, 575.
12 For a complete chronological listing of Hindemith’s works see David Neu-
meyer, The Music of Paul Hingemith (New Haven: Yale University Press,
1986), 253-283.
5
Concerto in Eb for cello and orchestra (Opus 3 composed in 1916), and many
smaller works for voice, piano, and accompanied instruments. He was the
leader of the Frankfurt opera orchestra and second violinist in the Rebner string
quartet. In 1917 Hindemith was drafted to serve in the German infantry during
World War I where his commanding officer allowed him to form a regimental
band and string quartet. Hindemith continued to compose and perform after the
war, experimenting with different styles and means of expression.
His first important operas are the one-act plays produced during the 1919-21
seasons in Frankfurt. Mbrder, Hoffnung der Frauen (1919) is based on the play
by Kokoshka which ten years earlier had started the expressionist movement in
literature. Das Nusch-Nuschi (1920) mimics Schoenberg’s “Pierot†in the use of
Burmese marionettes. The third opera, Sancta Susanna (1921), is a study of
sexual tension and religious desecration. The scandalous subject matter of
these three operas is reinforced by a very chromatic musical language in the
style of the atonal expressionists. Hindemith was also influenced by American
jazz with its robust rhythms and futuristic, experimental tendencies representa-
tive of the Dadaist movement in art.13
Hindemith received further national recognition as a composer at the 1921
Donaueschingen festival. The Amar quartet, in which he played viola, per-
formed his second String Quartet. The success of Hindemith’s Kammermusik
No. 1 for twelve instruments at the festival the following year established him as
one of Germany’s leading young composers.14 During the 1920’s, Hindemith
concentrated on reading and studying historical music manuscripts and trea-
tises by such authors as Boethius, Zarlino, and Fux. He was also well versed In
the writings of his contemporary theorists Riemann, Schenker, and
‘3 Watkins, 291 -292.
14 Ian Kemp, Hindemith, (London: Oxford University Press, 1970), 10.
Schoenberg.1 5
Neoclassicism is one description of Hindemith’s style, which he may have
emulated from his knowledge of early music.16 Neoclassical tendencies
include the use of smaller, mixed ensembles of instruments, the use of concise
formal structures based on earlier models such as the sonata and dance forms,
and the extensive use of counterpoint.†The music of Bach is especially
influential during his second style period from 1924-1933. The Kammermusi-
ken series for chamber orchestra and various solo instruments reflects the Bar-
oque concerto grosso in instrumentation and contrapuntal technique. These
years have been labeled Hindemith’s neo-Baroque period.18
Two developments are important to Hindemith’s career during his second
period.jThe first is his appointment to teach at the Hochschule fiJr Musik in Ber-
lin in 1927. “His teaching...was based more on practice—composing, singing,
playing instruments—than on theory. He insisted on the writing of many basic
exercises and on practical experience in ensemble performance."19
:3Hindemith quickly fell in love with teaching and eventually developed
several useful books and techniques to explain contemporary music. Hinde-
mith required his students to perform music by singing, playing instruments, and
‘5 David Neumeyer, The Music of Paul Hindemith. (New Haven: Yale Uni-
versity Press, 1986): 24.
16 James Kidd, “Aspects of Mensuration in Hindemith’s Clarinet Sonata,"
The Music Review 38/3 (August, 1977): 211-222. Kidd claims to have discov-
ered features of fourteenth-century mensural rhythmic practices in measures
12-25 of Hindemith’s clarinet sonata by 1) the change of pulse duration oppos-
ing the meter, 2) division from triple to duple, and 3) unusual proportional
groupings such as four half notes in the time of three and three half notes in the
time of six.
17Watkins, Soundings, 3411f.
18Sadie, s.v. Kemp, 573.
19 Marcelle Vernazza, “Paul Hindemith-Music Educator,†The American
Music Teacher (June/July, 1974): 30.
7
composing basic exercises in theory and composition.
The second development is founded in his belief in the practical necessity of
music. He wrote Gebrauchsmusik, or music for use, beginning in the mid-
twenties to 1932. Hindemith tried to bridge the gap between the creator, the
performer, and the listener by writing music that is readily accessible to all.20
He believed that composers had a moral obligation to create music for the good
of society, which includes amateurs as well as professional performers.21 “In
composing music for this series, he used the same practical principles he
insisted on with his composition students, that Is, that the composer keep in
mind who is to play the music and for what kind of an audience�2 These
pieces include, among many others, the last six Kammermusik (1924-1927) for
various soloists and ensembles, the musical Lehrstucke (1929), the children’s
musical WIr bauen eine Stadt (1930), and PIOner Musik tag (1932).
.tjgï¬fl'he change in politics in Germany during the 1930’s is a catalyst for Hinde-
mith’s third style period beginning about 1933. He refused to write music that
directly supported Nazi government policy which promoted German nationalis-
tic music of a more conservative nature. The avante-garde chromatic style of
his early music was not favored by the government. I .He also maintained profes-
sional and personal relationships with Jewish musicians. His opera Mathis der
Mg (1933-35) about Mathias Grflnewald’s struggle during the peasant revolts
of the sixteenth century was construed as representing the government’s
censorship of non- -German artistsTThé National Socialist political party began a
newspaper campaign against Hindemith in the late 1920’ 3. Hindemith strug-
gled to retain artistic individuality, but his music was officially boycotted by the
20 Neumeyer, The Music of Paul Hindemith, 5.
2‘ Hindemith, A Compgser’s World: Horizons and Limitations (New York:
Anchor Books, 1952), 144; Idem, Craft I, 12.
22 Vernazza, 31.
8
Nazis in October of 1936j He was accused “of lacking proper respect for Ger-
man traditions, associating and working with Communist musicians and artists,
writing music simply for profit, and other acts deemed felonious by the Nazis"?3
Folk music is another influence on Hindemith’s style during this time .24 Folk
elements are used extensively in the opera MatLIS der Male; (1933-35), the
viola concerto “Der Schwanendreher" (1935), and his third organ sonata
(1940).25 He also began a series of sonatas for every instrument of the orches-
tra, in keeping with his neoclassical emphasis. Other important works are the
Symphonic Metamorphosis on Themes of Carl Ma_ria von Weber (1943), the
T“
revised opera Cardillac (1926, 1952) and Die Harmonie tier Welt (1957). (Hin-
demith began teaching at Yale University in 1940. From 1953-58 he was pro-
fessor of music at the University of Zurich. He remained active as a composer
and conductor until his death in 1963.)...
Early Rhythmic Theories
Since a primary concern of this study is to explore Hindemith’s ideas about
rhythm, it will be necessary to investigate the history of the word “rhythm.†The
English word “rhythm†has been traced to its ancient Greek origins meaning
“to flow,†as in the action of moving water. Some writers suggest that to study
23 Luther Noss, Paul Hindemith in the @Ited States (Urbana: The Univers-
ity of Illinois Press, 1989), 10-11. See also Heinrich Strobel, Paul Hingemith:
Lestimony in Picturi (Mainz: B. Schott’s Sohne, 1961), 53-55.
24 Watkins, Soundings, 293-94.
25 Albert George Bolitho, “The Organ Sonatas of Paul Hindemith†(PhD.
diss., Michigan State University, 1968).
9
rhythm is to study sounds and silences as they occur in the flow of time.26
Rowell has provided a more complete understanding of the etymology of
“rhythm.â€27 The ancient Greek word rhythmos has several origins that resulted
in a variety of meanings. The older form of the word deals with both the charac-
ter and temper of a person and the daily divisions or events In a person’s life.
Later, the term takes on other more specific meanings, from shaping a cake, to
directing one’s mind, to the motion of a battle line or the scansion of poetry. He
concludes that,
...the basic meaning was m, specifically the form of a moving thing with both
Internal structure and external Iimitation....The Greeks slowly worked out a
problem—how to describe what they perceived as structural movement with
clear limitations. In this process the meaning hold, flow, and pull, as well as
maracter, form, and back-and-forth, were mingled. By the time of Aristides
Quintilianus, rhythm in music was defined as schema (‘shape’) + MS
(‘structure’) + kInesis (‘motion’).28
The problem arises in defining the articulations. What divides the internal struc-
ture and sets the external limitations, whether It be In a building, a sculpture, or
a musical composition? Rowell goes on to explain:
In the theory of Greek music the semeia are the points that mark the extremi-
ties of musical figures, a dancer’s pose, the extremity of a gesture, speech syl-
lables. Both Aristides and Aristoxenus clearly thought of single musical tones
and single musical time units as points-'39
These “points†are directly related to the points of mathematical geometry and
astronomy. Furthermore, ancient Greek music is based upon mathematical pro-
portions. The primary purpose of rhythm is to organize the musical elements
26Allen WII'IOId, “Rhythm in Twentieth-Century Music,†A_spect§_pf Twenti_e;h_
Centugy Music, ed. by Gary Wittlich. (Englewood Cliffs, New Jersey: Prentice-
Hall, Inc., 1975): 209.
27 Lewis Rowell, “The Subconscious Language of Musical Time,†Music
TheoLy SmNm 1 (1979): 96-106.
28 lbid., 99-100.
29 lbid., 102.
10
into comprehensible structures.30 Anything not based on proper proportions
was considered unnatural and therefore unmusical. A definition of rhythm that
is based upon the Greek meaning should also include an idea of the “seams†or
time-points which articulate structure.
In Greek music, the semia, or timepoints, may have been related to dura-
tional emphasis derived from spoken language and poetry. The three primary
types of accent in ancient Greek language, as well as other ancient languages,
are the acutus the circumflex, and the gravis, and indicate high or rising, rising-
falling, and low or falling pitch respectively.31 Greek accents did not Include the
Idea of dynamic stress; the emphasis is one of duration In which syllables with
long vowels are approximately twice as long as the duration of syllables with
short vowels.
Another concept from Greek theater is the arsis and thesis, or upbeat and
downbeat. The leader marks time for the dance and chorus by tapping a series
of beats very lightly with the foot. Poetry which emphasizes the thesis too much
is considered flawed.32 It Is, however, argued that stressed syllables occurring
with the thesis are a natural phenomenon, even as it developed in the Greek
chorus.33 The Greek concept of marked time is quite different than the modern
emphasis on dynamic accent and stress.
Features of rhythm in medieval music coincide with the development of
music notation. Music in the Roman empire maintains many of the concepts of
3° lbid., 103-104.
31 James Morgan Thurmond, Note Grouping A Method for Achieviryq
Expression end Style in Mueipej Performï¬ce, (Camp Hill, PA: JMT Publica-
tions, 1982): 26-27.
32lbid., 28.
33 Curt Sachs, Rhflhm and Temm: A Study in Music Histom, (New York:
WW. Norton, Inc., 1953): 140-143.
11
Greek music theory and rhythm. During the early Christian era music is divided
between the Gregorian chant In the sacred liturgy of the church and mono-
phonic secular music. By the ninth century the system of neumes, used as a
mnemonic aid in the northern countries, is used to show relative pitch
relations.34 Some early manuscripts also show rudimentary rhythmic symbols,
but there is no agreement as to the Interpretation of those symbols.35 Eventu-
ally the four-Iine staff, attributed to Guido d’Arezzo, shows more exact pitch rela-
tionships.
A more precise system of rhythmic notation developed as a result of meas-
ured polyphonic music at Notre Dame and elsewhere In the twelfth century.
The rhythmic modes, probably based upon meters of ancient Greek poetry, pro-
vided a means to notate rhythm as well as pitch (Figure 1). The rhythmic
modes relied on the difference in length between the long and breve and were
used In both sacred and secular music of the time. The modes helped bring
about the idea of the single group of notes heard as a unit, which possibly
Influenced the development of the single measure or bar.
Rhythmic developments of the fourteenth century are an increased freedom
in the application of different durations and the the use of rhythm as a formal
device. The isorhythmic motets of this century use both a repeated melody, the
color, and a repeated metric or rhythmic pattern, the talea, set against a cantus
firmus. Rhythm begins to play a more prominent role In the construction of
musical forms.
34 Richard Hoppin, Mievajï¬lusic, (New York: W. W. Norton, 1978), 57-58.
35 lbid., 89-90.
12
Trochaic _ o —- O J Oh J b
.ambic O _ C, _J)J ï¬J
Dactylic _ O O J J3 7 J
O U _ J) J J.
Spondaic _ _ J. J.
Tribrachic U U O J3 ,h J.)
Figure 1: The Medieval Rhythmic Modes
13
Notational conventions quickly codified musical practices of the Ars Nova.
Increasingly, composers use shorter and shorter note values and mensural
notation eventually replace the old rhythmic modes in practice as well as effect
(Figure 2).36 Triple subdivisions of the longer note values are labeled “perfectâ€
while duple divisions, now allowable, are considered “imperfect.†A system of
time signatures is introduced to assist performers in interpreting the new rhyth-
mic notation. By the end of the fifteenth century notational practices became
more consistent, even though older systems are in use long after new develop-
ments take place. The basis for modern notation is In place by the beginning of
the Renaissance.37 Textual phrases and repetitions determined formal musical
structures of secular monophonic songs such as the ballade, virelai, and ran-
deau.
The notational conventions of the Ars Nova continue to develop into the
Renaissance period. Metric proportions are a regular feature of Renaissance
music. As proportional meters replace the original meter In sections of the com—
position, or even in different vocal parts, the result Is notationally complex and
often conflicting rhythm patterns.
A solution to reading music more fluently Is the addition of barlines to the
score.38 An ultimate result of the stress on the thesis in the old rhythmic modes
is rhythmic grouping around the barline.39 The regular addition of barlines to
the score result in a stronger emphasis on the downbeat and assist in the devel-
opment of four- and eight-bar phrase structures in stylized dance music.
36Hoppin, 354-355.
37Gustave Reese, M_usic in the Middle Ages, (New York: WW. Norton and
Co., 1940): 346.
38 lbid., 257-259.
39Thurmond, 31-33.
14
Perfect time, perfect prolation
@- . . .
31...,» J- J- l.
.__
EIL
.—
«—
EIL
._
.—
SIC
F
Perfect time, Imperfect prolation
O O 0
3J- J J J
:1:
i]:
i]:
Imperfect time, perfect prolation
c. . . .
gJ- J- J.
@—
EIL
F
.—
3..
0——
Impzrfect time, imperfect prolation . I I I I
gJ J J r7 r7
Figure 2: Notational Mensurations of the
French Ars Nova
15
Throughout the Baroque era rhythm is divided between free and strict prac-
tices.40 In general, early Baroque Italian madrigals and the monody and recita-
tive of the first operas require a free, unencumbered rhythmic flow. Instrumental
music, in the form of freely improvised toccatas, adapt the free rhythms and tem-
pos of vocal music. However, compositions which are of more precise formal
construction such as dance movements, fugues, canons, and the da cam aria
require a much stricter tempo and rhythmic flow.
By the end of the Baroque period, the modern system of notation and rhythm
is finalized. Once composers consistently use standardized notational practic-
es, theorists begin to concentrate on matters of form, or the anatomy of musical
composition. Mattheson and Koch, among others, use terms such as para-
graph, sentence, predicate, subject, and comma to describe musical phrases
and period structure. In KIrnberger’s theory, accentual differences of adjacent
measures are prominent. He did not apply the concept of alternating heavy and
light accents to larger formal units, however.41 Concepts of phrase length and
rhythmically-defined harmonic cadences are used to analyze music by the end
of the eighteenth century.42
Some theorists in the nineteenth century looked more broadly at the con-
structions of rhythm. According to Morgan, the theorist Gottfried Weber formu-
lated the first clear hypothesis concerning the function of rhythm on a higher
level than measure to measure.
In Weber’s theory of rhythmic grouping, beats group together to create
measures. These groups then in turn form larger symmetrical groups with the
4° Sachs, 273-82.
41 Robert P. Morgan, “The Theory and Analysis of Tonal Rhythm,†The Musi-
cal Quarterly 64/4 (October, 1978): 436-37.
42 Leonard G. Ratner, “Eighteenth-Century Theories of Musical Period
Structure,†The Musical Quarterly 62:4 (October, 1956): 442.
16
same inherent qualities as the smaller group. These longer groups are then
combined to build even longer rhythmic structures. Three basic assumptions
inherent in Weber’s theory of rhythm permeate rhythmic thought to the present:
(1) accents of sufficient weight and importance assume formal significance;
(2) these accents relate to one another in a manner analogous to the rela-
tion of beats within measures; and (3) larger formal units result from an
accumulation of smaller units through a process of addition (or more preci-
sely, multiplication) into ever larger groups."-3
Hauptmann’s systematized theory of symmetrical rhythmic structures,
which appears later in the nineteenth century, stresses beginning-accented
rhythms which he termed “positive,†and end-accented rhythms which he
labeled “negative.†Hauptmann’s theory is based on a two-element pattern
called the “thesis.†Rhythmic units that appear to be longer than two-element
units are made up of intersections of the shorter two-element units. The three-
element unit is the “antithesis“ and the four element unit (the combination of
two-element units) is the “synthesis.â€44
In Riemann’s theory a motif (a stream of pulses or beats in a weak/strong
relationship) is considered the smallest rhythmic unit. Motifs combine to create
phrases, then periods, and finally whole sections of a composition.45 Phrases
of shorter or longer length are a result of elision, extension, or truncation.46 Rie-
mann’s system of diagraming phrases uses slurs to indicate the various types of
extensions and truncations. He further develops Weber’s original concept of
rhythmic structural symmetry, insisting on the notion that all phrases commence
with an upbeat, the universality of the four-bar phrase, and the relationship of
43Morgan, 437.
44 Ian Bent, Analysis. (New York: WW. Norton and Co., 1987): 30.
45 Bent, Analysis, 90-93.
46Sadie, Volume 1, 375-376.
17
the antecedent and consequent.47 Of Riemann’s treatise, Hindemith says
“what [Riemann] accomplished was the analysis of rhythms, not the discovery of
the principles underlying them.†43
Morgan has divided these early rhythmic theories Into two schools of
thought, both of which arrive at similar conclusions. 49 One school uses the
analogy of language and describes musical rhythm in terms of poetic feet. This
group Includes Westpahl (1880), Wiehmayer, and Cooper and Meyer (1960).
The other school emphasizes meter and the measure and includes Weber,
Hauptmann, and Riemann.5° Both schools begin with small units that are
combined into longer structures organized around a system of accents which
creates a hierarchy of structural levels. Hindemith’s ideas concerning rhythm
fall Into the group concerned with metric organization rather than organization
by poetic feet.
Through analysis of the music of various composers, Winold has identified
twelve characteristics of common practice rhythmic patterns.51 Common
47 Nicolas Slonimsky, ed, Baker's Biographical Dictionary of Musicians, 7th
ed. (New York: Schirmer Books, Inc, 1992): 1,898. Riemann took a different
approach to harmonic theory during his time, which taught chords as part of pat-
terns of progressions according to Rameau’s ideas of the thorough bass. Rie-
mann developed the idea of the functionality of triads in the tonal system. The
secondary chords (II, III, and VI) served as representatives of the primary chords
rather than independent members of a chord progression. Therefore, the II
chord fulfills the function of IV, III of V or I, and VI of l or IV. Functionality of the
triads was presented In great detail in his treatise Vereinfachte Harmonie lehre
oder die Lem: von den ton_a_len Funktionen der Akkorde (1893).
48 Hindemith, “Methods of Music Theory,†24.
49 Morgan, 439.
5° The conflict between language-based rhythms and musically-based
rhythms has existed throughout Western music history, even in ancient Greece.
SeeSachs, 138-140..
51 Allen Winold, “Rhythm in Twentieth-Century Music,†in ï¬pecte of Twen-
tieth Century Muï¬, ed. by Gary WIttIich. (Englewood Cliffs, New Jersey: Pre-
ntice-Hall, Inc., 1975): 216-17, 244.
18
rhythmic features among many tonal compositions include the following:
.‘1 9’91 99°10?
Regular pulses are clearly heard or Implied and are equal-timed;
Pulses are grouped into two’s or three’s;
Pulse groups are maintained throughout most of the composition;
Pulse groups on various levels coincide with pulse groups on higher or
lower levels;
Constant tempo predominates throughout an entire composition or section;
Meter signatures remain constant throughout an entire composition or sec-
tion;
One duration predominates as the basic unit of movement, and very short
durations are rare except in special localized circumstances;
8. Motivic, (“rhythmicâ€) units are primarily based on metric patterns;
9.
A limited number of rhythmic units combine to create longer “rhythmic
gesturesâ€;
10. Upbeat, or anacrustic, beginnings predominate;
11. Repetition and variation of rhythmic gestures is quite common as well as
rhythmic gestures in ternary patterns;
12. The composite rhythm generally confirms the metric structure.
Such a list of common features as this is used as a standard to identify stylistic
traits. It is interesting to see how composers work against these basic features
in their compositions to create their own individual styles.
Purpose of the Study
The brief survey of the history of rhythm serves to show the disparity among
theorists and composers regarding the rhythmic element of music. Finding a
complete and adequate explanation of rhythm may be altogether Impossible, as
agreement among scholars is Inconsistent.52
It has been proposed that Hindemith’s theory presented in The Craft is lack-
ing in one important area—that of an adequate explanation of rhythmic
52 David Epstein, Shaping Time: Mueig the Brain and Performance (New
York: Schirmer Books, 1995): 3-4.
19
principles, yet it will be interesting to see how Hindemith’s views on rhythm fit
into the overall historical debate. Since rhythm is considered to be one of the
primary elements of music, it seems that this omission is a major flaw of Hinde-
mith’s theory.
There are, however, numerous references to rhythm throughout Hindemith’s
writings. This researcher’s purpose is to discover if there are similarities
between Hindemith’s theoretical writings on rhythm and the application of rhyth-
mic principles in his musical composition. Furthermore, The Craft theories are
directly linked to Hindemith’s experiences as a teacher of composition and his
practice as a composer. A rhythmic analysis of some of his music will lend prac-
tical information to the comments from the written literature. The Craft must be
understood in light of its intent: to provide a method of teaching musical compo-
sition. The theories of melody and harmony presented by Hindemith will be
studied to see if Hindemith’s rhythmic ideas are explained equally well.
Research Questions
Research question: Does Hindemith’s use of rhythm In his musical composi-
tions reflect his explanations of rhythm in the theoretical writings?
Collateral question 1: What is the relationship of melody, harmony, and
rhythm In The Craft of Musical Commsition and Hindemith ’3 other writings?
Collateral question 2: What are the similarities and differences between
Hindemith's concept of rhythm and other theories of rhythm?
Collateral question 3: Can a method of rhythmic analysis be developed
which supports Hindemith’s concept of rhythm?
20
Organization
In the Introductory chapter, I have presented necessary background informa-
tion about the topic. The need for the study is stated as a necessary endeavor
which will clarify our understanding of Hindemith’s conception of rhythm. Bio-
graphical information concerning Hindemith and his major works with divisions
of his compositional life has been given as an aid to understanding the devel-
opment of the theory and his growth as a musician. A brief historical survey of
rhythmic theory has been presented to place Hindemith in proper historical per-
spective.
In Chapter 2 I will review Hindemith’s theories of melody and harmony from
The Craft. His ideas of rhythm will be explored from The Craft and other
sources. Chapter 3 will compare five current models of rhythmic analysis. A
method of analysis will be developed which supports Hindemith’s understand-
ing of rhythmic structure in both foreground patterning and background organi-
zation. An application of this model of analysis will be presented In Chapter 4
using Hindemith’s Piano Sonata in A from 1936 as the music sample. Finally,
Chapter 5 will provide a summary of the findings of the research, answering the
stated research questions presented above. Areas of continuing research will
also be proposed and discussed.
21
Scope and Limitations
This study will be limited In its scope to a narrow range of Hindemith’s out-
put. It is believed that the music written at the same time that The Craft theories
were being developed reflects the application of those theories. Therefore, the
sample of music to be analyzed will be chosen from Hindemith’s second style-
period. A second concern Is in the varied genres of Hindemith’s music. The
text of vocal genres may influence the application of certain rhythmic principles
because of the requirements of language. For example, normal structures of
vocal music are dependent upon characteristics of the text such as number of
syllables per line, phrase length, and rhyme scheme, among others. The study
will therefore be limited to instrumental music. The first piano sonata (1936)
falls quite readily into this context and provides a sample that is easily acces-
sible. This study will not directly compare Hindemith’s three style periods.
Traits of his style will be recognized which may be used for comparison to his
earlier and later music. Another limitation of this study will be in the specific
analysis. The project is concerned primarily with rhythm in the theory and music
of Paul Hindemith. Therefore, the analysis will deal with the nature of durations,
from longer sections and movements to individual notes and groups of notes.
The separation of rhythm from the other elements of music is in keeping with the
pedagogical method of The Craft, as will be seen in Chapter 2.
The present study will be profitable in several areas. It will explore the omis-
sion of rhythm in Hindemith’s theory and correlate that omission with current
thought, providing a new perspective on Hindemith’s music which emphasizes
his concepts of rhythmic theory. The study will also provide a summation and
22
review of current approaches to rhythmic analysis and develop a model of anal-
ysis that will be applicable specifically to Hindemith’s music as well as to the
music of other composers.
Definitions
For this study the definition of rhythm will consist of these two parts: from a
formal standpoint rhythm consists of the sectional divisions of a musical compo-
sition articulated by specific musical events. On a more fundamental level
rhythm is the grouping of time into recognizable units by duration and stress.
The New Harvard DictionaLy of Music defines rhythm as “the pattern of
movement In time."53 There are two connotations that this definition entails, the
general idea of rhythm covering all aspects of musical movement as ordered in
time, and the attacks of musical sounds that fall into patterns. These attacks or
changes are perceived against the backdrop of tempo, the rate of speed of the
changes. The patterns within the passage of time are caused by the changes In
musical events. The definition given by The New Grove’s DictionaLy of Music
and Musicians describes rhythm as “the subdivision of a span of time Into sec-
tions perceivable by the senses; the grouping of musical sounds, principally by
means of duration and stress.â€54.
Schachter points out that rhythm must be defined in terms of how we
hear—the perception of musical events. He refers to the Greek idea of flow In
53 Don Michael Randel, ed. The New Hervard DictionaLy of Music,
(Cambridge: The Belknap Press of Harvard University Press, 1986): 700.
54 Sadie, 804-805.
23
defining rhythm “as organized movement in time.â€55 Smith has used a similar
definition of rhythm: “the organization of musical things (events) in time.â€55
There must be some sort of patterning or grouping in effect to aid the listener in
organizing the ebb and flow of rhythmic events; the continuity of rhythmic flow
must be divided and articulated in some way.
Meter is caused by the regularly recurring pulsation of beats organized into
accented and unaccented patterns.57 In Western tonal music meter is the most
fundamental method of grouping. The perception of a stress or accent on the
first beat of a group creates the feeling of meter. The most common metrical
groupings are of duple and triple beats. Rhythm and meter should not be con-
sidered as separate elements, even though rhythms often oppose the effect of
the metric barline. In notation barlines is separate the patterns of meter which
are Identified by the time signature at the beginning of the composition or bar.
Meter can also be Identified by grouping of events into recognizable patterns of
stressed and unstressed beats.
Meter also is hierarchic in nature, that is, It exists on several levels simulta-
neously. Rhythm and meter are established by the interaction of at least two
levels of activity.58 A single stratum cannot induce a sense of rhythm, but must
be understood in relationship to a higher or lower stratum. Therefore, a rhyth-
mic pattern has implications on several different levels.
55 Carl Schachter, “Rhythm and Linear Analysis: A Preliminary Study,†in
The Music Forum, Vol. IV, ed. by Felix Salzer, (New York: Columbia University
Press, 1976): 311-312.
55 Charles Smith, “Rhythm Restratified.†Review of The Stratification of Musi-
cal Rhï¬hm, by Maury Yeston, (Perspectives of New Muelp 16/1, 1977): 150.
57Joel Lester, The Rhythm:s of Tonal Music, (Carbondale, IL: Southern Illi-
nois University Press, 1986): 45.
55 Maury Yeston, The Stratification of Musical Rhflhm, (New Haven: Yale
University Press, 1976), 38.
24
Accent is important for perception of meter; it is the emphasis of a particular
musical event which sets it apart from others.59 There are several types of
accents that occur in music. In the setting of meter an accent Is perceived to
occur on the first beat of a measure. The dynamic accent refers to a tone being
louder than others, the tonic accent is a pitch being higher or lower than others,
and in an agogic accent a pitch has a longer duration.50
Often several types of accent simultaneously reinforce the musical event.
The accented event becomes the focal point around which the unaccented
events are grouped. Furthermore, accents often mark the beginning of a musi-
cal event and therefore can be considered “points of Initiation."61 These points
can be emphasized by a change in any of the elements of music.62 Syncopa-
tion occurs when the expected metric accent is shifted to a weak beat or pulse.
One rarely perceives an individual note unless it is accented in some way,
but even accented notes are understood within a context as a beginning,
middle, or ending note. Notes that are predominantly aligned with the metric
structure are said to be rhythmically consonant. Those that oppose the meter or
are misaligned with the metric structure of a composition are rhythmically dis-
sonant.53
59 Grosvenor W. Cooper and Leonard B. Meyer, The Rhythmic Structure of
Music. (Chicago: The University of Chicago Press, 1960): 8.
6° Randel, 3.
5‘ Lester, 16.
52 Anne Alexandra Pierce, “The Analysis of Rhythm in Tonal Music.†Ph.D.
diss., Brandeis University, 1968, 42.
53 John D. White, The Analysis of Music, 2nd ed. (Metuchen, New Jersey:
The Scarecrow Press, Inc., 1984), 83-84.
Chapter 2
Review of Hindemith’s Theory
Hindemith began teaching at the Hochschule fI'Ir Musik in Berlin in 1927. He
enjoyed teaching from the very beginning and held subsequent positions at
Harvard, Yale, and the University of Zurich. In addition, Hindemith helped to
design a state-sponsored music school In Turkey during the 1930’s.1 He firmly
believed that music theory and composition could be taught systematically. He
wrote,
Music theory investigates, arranges, and explains the working material of
the composer...to comprehend once and for all the whole domain of tone
In all directions and relationships, so that every conceivable sort of music
can be explained....2
But he was displeased with the lack of music textbooks that explained modern
music. The theory of harmony which was formulated from music of the common
practice period was inadequate to explain the new music, according to Hinde-
mith.3 He set out to develop a theory of music that would explain music of the
past, present, and future.
The Craft of Musical Commsition is Hindemith’s solution to the lack of con-
ventional theory textbooks to adequately explain music. More than an apology
for Hindemith’s musical style, it attempts to develop a systematic method for
1 WIlliam D. Pack, P_a_ul Hindemith in Turkey: Some Contributions to Music
Education (Brigham Young University: Ph.D. Diss., 1977).
2 Hindemith, “Methods of Music Theory,†20.
3 Vernazza, “Paul Hindemith—Music Educator,†30.
25
26
teaching composition based on the species exercise.4
Hindemith’s theory of music Is derived from his investigation and perception
of “the laws of nature.†He tries to prove the immutability of a tonal system built
on the major triad derived from the overtone series.5 But this foundation In nat-
ural order has a much broader basis than physical and mathematical principles.
“He felt a moral responsibility In being a composer, and believed a composition
should bring about moral improvement in the listener.â€5 Hindemith believed
strongly in the social obligation of the composer. Even in his mature style he
“remained loyal to his belief that music should be useful and practical, and
should not be a vehicle for self-expression.â€7
Hindemith was familiar with the philosophical and natural laws formulated
by many writers such as Boethius, Zarlino, Tartini, Rameau, Descartes, and his
contemporary Schenker. He tried to rationalize his new theory by an appeal to
a natural order. Hindemith writes:
4 Hindemith, Craft l, 1-2. Neumeyer points out that Hindemith intended to re-
vise The Craft and other writings into a complete method of teaching composi-
tion. The new curriculum would begin with flementarv Treining for Musiciane
(1946), continue with the combined reworking of Craft I end II which would pro-
vide the speculative theory and two-part exercises, Craft III (published
posthumously in 1970) for three-part writing, an unfinished Craft IV which would
include four-part writing as well as incorporating materials from Editional Har-
mony l fltd II, and end with a fifth volume providing a transition to free composi-
tion. Unfortunately Hindemith was unable to complete a revision of such magni-
tude. (Neumeyer, The Mueic of Paiul Hindemith, 21-23).
5 Hindemith, Craft I, 4. See also Norman Cazden, “Hindemith and Nature,â€
IhteMusic Review 15/4 (November, 1954): 288-306 and Victor Landau,
“Hindemith the System Builder, A Critique of His Theory of Harmony,†m
Music Review 22 (1961): 136-151. Clifford Taylor addresses problems with
both Cazden’s and Landau’s criticisms in “The Hindemith Theories, A Revalua-
tion of Premise and Purpose,†The Music Review 44n3-4 (August/November,
1983): 246-262.
5 Sister R. Christine Daehn, “Paul Hindemith: Author, Craftsman, Philoso-
pher.†(unpublished Master’s thesis, Michigan State University, 1974.): 3.
7 Sadie, s.v. Kemp, 579.
27
The teacher will find in this book basic principles of composition, derived
from the natural characteristics of tones, and consequently valid for all
periods. To the counterpoint and harmony he has already Iearned...he
must now add a new technique, which, proceeding from the firm founda-
tion of the laws of nature, will enable him to make expeditions into
domains of composition which have not hitherto been open to orderly
penetration. (emphasis added)9
Hindemith built his theory on his own investigation of these “natural laws.â€
The basic premise with which Hindemith works is the necessity of tonal cen-
tricity. The major triad is the basis of all music because it occurs naturally from
the first six pitches of the overtone series (Figure 3). 9 He claimed that the major
triad is “the most impressive phenomenon of nature."10 The continuation of the
overtone series results in the pitches of the chromatic scale, which in equal tem-
pered tuning divides the octave into twelve equal parts.
Equal tempered tuning is one of the problems Hindemith saw in music. Hin-
demith was displeased with equal temperament because in his cpinion it
lacked luster.
In equal temperament...music performed on keyboard instruments lacks the
fine lustre [sE] of the light that falls at ever-changing angles as it is cast by dif-
ferent generators. It does not have any of that fine inner agitation that arises
from slight variations of pitch.11
Instead he develops a “new†tuning system to derive the twelve semitones of the
octave. 1 2
5 Hindemith, Craft I, 9.
9 Richard Bobbitt, “Hindemith’s Twelve-tone Scale,†Music Review 26
(1963): 105. Bobbitt points out an error In the translation. The term “overtoneâ€
is used incorrectly as this indicates the harmonic partials excluding the funda-
mental. “Partial†is clearer, describing the multiple vibration frequencies of a
composite tone including the fundamental.
10Hindemith, Craft I, 4.
11 lbid., 43; 154-155.
‘2 lbid., 32-41.
28
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According to the theory the most Important partials for scale derivation are
the first six. But instead of using one fundamental pitch as the foundation for the
series, he uses multiple fundamentals. To keep within the limits of the octave
C(64)13 to c (128), the third partial of C(64), g (192), becomes the second partial
of the fundamental tone G (96).14 In other words, “to arrive at each new tone of
the scale, divide the vibration-number [gig] of each [partial] successively by the
order-numbers of the preceding tones in the series.â€15 The process is done In
three steps.
The first step takes the third through sixth partials of C (64) and divides them
by the preceding partial numbers 2, 3, 4, and 5 respectively (Table 1).
Table 1
Hindemith’s Scale Derivation, Step 1
Partial 3: g (192) + 2 = G (96)
Partial 4: c1 (256) -:- 3 = F (85.33)
Partial 5: e1 (320) + 4 = E (80)
Partial 6: 91 (384) + 5 = Eb (76.8)
Step two uses the principle of addition, using the “sum†(up to and including
the sixth partial) by considering the pitches to be higher in a series of
‘3 The number in parentheses refers to the frequency of vibrations per sec-
ond of that particular pitch.
14Several methods of octave Identification are in use. The one employed for
this paper is C1 -C-c-c1-c2 -c'5 -c4-c5(middle c =c1).
15Hindemith, Craft I, 34. The vibration-number is the frequency expressed in
the number of vibrations per second.
30
overtones.16 The third, fourth, and fifth partials of C (64) are used as the fourth,
fifth, and sixth partials, dividing to achieve new fundamentals, then multiplying
the result by 2 In order to keep the tones within the octave C (64) to c (128). In
steps one and two, the first six partials of C (64) were used to derive G (96), F
(85.33), A (106.66), E (80), Eb (76.8), and Air (102.4).
Table 2
Hindemith’s Scale Derivation, Step 2
g (192) +4 = G (48)
g (192 ) -:- 5 = E91 (38.4) x 2 = Eb (76.8)
g(192)+6=C1(32)x2=C(64)
C1 (256) + 5 = C (64)
c1 (256) + 6 = A51 (51.2) x 2 = AI’ (102.4)
81 (320) + 6 = A1 (53.33) x 2 = A (106.66)
In step three the remaining six chromatic tones are derived from “grandchildâ€
fundamentals using the same processes previously described.17 The com-
pleted scale adds 0 (72), BI’ (113.78), ob (68.27), B (120), Gb (91.02), and F!
(90). An inconsistency in Hindemith’s procedure lies in using some of the par-
tials more than once to derive the scale pitches. He also uses several different
mathematical equations to achieve his end results. In looking at the derived
scale from an historical point of view, there is a striking resemblance to two
15 lbid., 35.
17 Hindemith, Craft I, 39-41.
31
earlier models.18
In translating Hindemith’s scale into cents (the octave divided into 1200
cents, each tempered half step equaling 100 cents), Bobbitt discovered that it is
identical to Ptolemy’s diatonic syntonon (AD. 140) and a chromatic scale pro-
duced by Alexander Malcolm in 1721 (Figure 4). Hindemith merely duplicated
earlier scale formations. Bobbitt even explored a simpler method of formulating
Hindemith’s scale, and corrected some minor calculations overlooked by him.
Hindemith’s concept of chromaticism and diatonicism is also different than
earlier theorists. Hindemith believes that tonality includes both the seven diato-
nic scale pitches as well as the chromatic pitches. In traditional harmony the
seven diatonic pitches of the major or minor scale produce triads or chords.
The chords function in relation to the tonic/dominant axis.
In Hindemith’s theory, the Interval, not the chord, Is the basic unit of measure
against the tonic. He says that...
the key and its body of chords is not the natural basis of tonal activity. What
Nature provides is the Intervals. The juxtaposition of Intervals, or of chords,
which are the extensions of intervals, gives riee to the key. We are no longer
prisoners of the key. Rather, we have a free hand to give the tonal relations
whatever aspect we deem fitting.19
The use of the interval in this way is an Important innovation of Hindemith’s the-
ory.20 It allows the composer to break away from the influence of the traditional
harmonic framework but still retain a tonal center.
19 Bobbitt, 111-12. See also Felix von Cube, The Book of me Musical Art-
work: An Interpreta_tion ofihe Musyigaj Theories of Heinrich Schenker, translated
by David Neumeyer, George Boyd, and Scott Harris (Lewiston, New York: The
Edwin Mellin Press, 1988), 66-72.
19 Hindemith, Craft I, 107.
2° \MIliam Thomson, “Hindemith’s Contribution to Music Theory,†Journal of
Music Theogy IX (1965): 55.
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Series 1 is the tonal framework which Hindemith derives from the overtone
series.21 It places the twelve tones of the chromatic scale In a relative relation-
ship to the starting tone. (Figure 5) Neumeyer suggests that the order of tones
in Series 1 is of primary importance to Hindemith’s theory.
It is not only an indication that the tones have a family relationship,
expressed in their connections to the principal tone; it is also an index to
the ranking of these connections....This value-order of the relationships Is
valid under all circumstances.22
The tones to the left in the series have a closer relationship to the starting tone
than those to the right. Thus, beginning with C, the pitches of Series 1 in
decreasing relationship are G, F, A, E, E9 or of, AI’, 0, 8", ob, B, and Ft. Were
the starting pitch other than C, the Intervals would remain the same. Hindemith
provides further insight into the function of Series 1 In Craft III:
A. It does not consist of ‘tones.’ The notation of tones is only a necessary
aid, but each tone really stands for one of the poles of a field of harmonic
force existing between It and the tonic.
B. The tonic, as the central tone of the tonality, stands above and apart
from tonal relationship.
C. The relationships of the series follow one another in steadily decreas-
ing harmonic value. The dominant represents the highest-valued relation-
ship.
D. The most distantly related of all is the tritone. Its connections with the
tonic are so tenuous and ambiguous that it, too, in the opposite sense,
stands apart from (Le. below) the other related tones?3
Series 1 is not to be considered a tone row or a scale, but is used by the
composer to establish control over a root progression, to establish the tonal
center of a composition, and to establish the “best interval†for determination of
chord roots. The best intervals of the series also Identify closely related keys.
2‘ Hindemith, Craft I, 56.
22 Neumeyer, The Music of Paul Hindemith, 54.
23 Hindemith, Unterweisung im Tonsatz Ill: Ubungsbuch fI'Jr den Dreistimmi-
gen Satz (Mainz: B. Schott’s Schne, 1970), 86. Typescript translation of Chapt-
ers 12-16, Yale University and Indiana University libraries, 77.
34
o
Tonic
(Closely related keys)
Figure 5: Series 1
o
Hindemith's Symbols
(1) 6 9 v1 111 mm V\VI II VI\VII j; 1‘ U
Neu meyer‘s Adaptations
¢59me IIIVI IIVIIi’l‘ a
Figure 6: Symbols of Chord Root Relationships
V Root v Root V Root V Root V Root
0
0
P8 P5 P4 M3 MS m3 M6 M2 m7 m2 M7 TT
Figure 7: Series 2 (Hindemith, The Craft of Musical Composition
Book 1:The0[y, 81. ©1942 Schott & Co., Ltd., London. Copyright
renewed. All rights reserved. Used by permission of European
American Music Distributors Corporation, sole US. and Canadian
agent for Schott & Co., Ltd., London.)
35
Symbols given by Hindemith are used to describe analytically the harmonic
relationships.24 These symbols, listed in Figure 6, Identify the relationship of
the chord roots to a given tonic. According to Series 1, the fifth holds the
strongest relationship to the tonic, while the trItone is the furthest removed.
Series 2 Is an ordering of the interval relationships within the chromatic
scale, ranking the intervals from strongest to weakest (Figure 7). Used to iden-
tify the roots of intervals and chords, the members of each pair are not of equal
strength—the root of the Interval is the lower tone of the first and the upper tone
of the second of each pair except for the minor second and major seventh, in
which the root is the upper tone of the first and lower tone of the second. The
octave stands at the beginning of the series because of its strength while the tri-
tone. the only interval without a root, stands alone at the end of Series 2.
Series 2 is derived from another “natural phenomenon,†combination tones.
Unlike overtones, which “are produced in varying numbers by a single tone,
combination tones arise only when two or more tones sound simultaneously.â€25
First order combination tones are those created from the two original tones. The
frequency of the combination tone is the difference in frequency of the two origi-
nal tones. In other words, the c (128) is produced by the sounding together of
c1(256) and 91(384), (384 - 256 = 128). Secondary combination tones are pro-
duced by the difference between the frequency of the first combination tone and
the lower pitch of the original interval. Hindemith claims, however, that the prim-
ary combination tones are only audible with amplification, and that the second-
ary combination tones are even “less intense than the original ones.â€26
24 Hindemith, Craft Ill, 99 (Translation, 87), quoted in Neumeyer, The Music
of Paul Hindemith, 53,55.
25 Hindemith, Craft l, 58.
26lbid., 60-61.
36
Earlier theorists had identified combination tones. In the early eighteenth
century Tartini recommended them as an aid in tuning violin double stops.
They have also been used extensively in organ tuning. Helmholtz, calling them
difference tones, also identified summation tones, equal to the sums of the
frequencies of the original tones.27 Hindemith, however, claims to be the first
theorist to use the combination tones to eXplaIn the “properties of musical mate-
rials and the rules for musical writing,†but gives credit only to organ builders for
his theory of combination tones.28
Hindemith is in agreement with Apel in recognizing that combination tones
are generally Inaudible unless amplified.29 If such is the case, the question
arises as to their importance in actual perception of musical sounds. If the com-
bination tones cannot be heard, does the listener use them to develop a hier-
archy as Hindemith suggests? Series 2, then, is not based so much on the
audibility of the combination tones, but rather on the physical relationship of the
combination tones to the original tones.
Hindemith’s Theory of Harmony
The rationale for Series 1 and Series 2 comprises a large portion of Craft l.
Of more importance are Hindemith’s theories of harmonic and melodic organi-
zation. Hindemith Identifies three alternative criteria for chord analysis:
27 Victor Landau, “Hindemith the System Builder: A Critique of His Theory of
Harmony,†Music Review 22 (1961): 141.
28 Hindemith, Craft I, 59.
29Apel, WlllI, s.v. “Combination tone,†185-186.
37
1. Construction in thirds must no longer be the basic rule for the erec-
tion of chords.
2. We must substitute a more all-embracing principle for that of the
invertibility of chords.
3. We must abandon the thesis that chords are susceptible of a
variety of interpretations.30
Neumeyer also lists several factors from The Craft series which help identify
the root of a chord.31
From Craft l:
1. The chord root is determined by locating the best interval nearest the bass
(the root of this interval is the root of the chord);
2. Doubled notes count only once;
3. If two intervals of the same type appear, the one whose lower note is
nearer the bass has the chord root;
4. Chords in which the bass and root are the same are stronger than those of
the same type in which the bass and root do not coincide;
5. Series 2 determines the root of the best interval;
From Craft Ill:
1. Doubling of the root makes the chord stronger and more clearly defined
than the chord class might suggest;
2. If two or more Intervals share the same root, emphasis on that note is
greatly increased and may overrule the best root;
3. If the chord root is also a member of a tritone interval in the chord, the pos-
sible functions of the tritone determine the true root;
From an unpublished manuscript of a proposed fourth volume of The Craft:
1. For certain chords, the appearance of a familiar sonority in the lower voices
can change the chord root.32
The chord is then found to belong to one of six groups divided Into two types:
those with and those without tritones (T able 3). Those without tritones are con-
sidered the most stable. Several subgroups of chords Involve those containing
3° Hindemith, Craft I, 94-108.
31 Neumeyer, The Musjgaf Paul Hindemith, 56-58.
32 Hindemith, “Ubung 21,†from an incomplete Ubungsbuch fl'Jr gen vierstim-
migen Satz. Photolithographic copy in the Yale Hindemith Collection. See also
Andres Briner, Eeul Hingemith, (Mainz: B. Schotl’s SOhne, 1971).
38
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seconds and sevenths. Hindemith leaves an “indeterminate†group In each cat-
egory for those chords which lack strong root Identity.
Sometimes two voices appear to form an interval instead of a chord. Inter-
vals can also assume a value according to the table. Fifths and thirds belong to
group 11, fourths and sixths belong to group 12, seconds belong to group 1112,
sevenths belong to group 1111, and the tritone to group VI.
In addition to the root of the chord, a guide-tone pitch must also be identified
in chords which contain one or more tritones.33 The guide-tone is one pitch of
the tritone interval “which stands In the best relationship to the root.†If the root
contains one of the pitches, then the other is the guide-tone. Also, In successive
chords which contain tritones tension may be increased by not resolving them.
When a tritone does resolve, however, the guide-tone should move by a good
interval to the root of the chord of resolution.
Hindemith introduces three concepts that assist the composer in setting
tones: the two-voice framework, the degree progression, and harmonic fluctua-
tion.
The two-voice framework results from the contrapuntal Interplay of the bass
voice and the most Important upper voice. Hindemith described the two-voice
framework as a type of scaffolding which guides the composer in constructing
an effective harmonic progression derived from Series One and Two.34 These
two voices are to be written with strict adherence to Hindemith’s contrapuntal
guidelines presented in gm, yet they are not to become more prominent than
the remaining voices.
The degree progression helps determine larger harmonic sections. A good
degree progression emphasizes intervals of the fifth and fourth and avoids the
33 Hindemith, Craft I, 104-105; 126-131
34Hindemith, Craft I, 113-115.
41
tritone, broken chords other than major or minor, minor seconds, and melodic
treatment. Series 1 controls the root movement of the degree progression.35
The degree progression governs the long span of chord progressions and key
relationships; it presents the background organization of the harmony. It simply
states the roots of the chords and identifies the tonal centers of the music. This
simple “melody†created by the root succession represents the background har-
monic structure. It has no melodic or rhythmic relationship to the surface of the
music.
The degree-progressions formed by the tonIcs of successive tonalities do
Indeed combine to make a melodic line; but this line is so remote from any
directly melodic effect that it hardly enters our consciousness as a linear for-
mation. Similarly, the rhythmic structure of such a degree-progression of
tonics (consisting of the proportions of the lengths of its members) remains
vague and obscure, since It belongs to the too distant background.36
In Hindemith’s theory, the tonal center can change at any time in a composi-
tion. A cadence at any specific point in the composition defines the key of that
section. A change in tonal center occurs when a tone usurps its control over all
others by the cadence, its favorable position, the recurring appearance of a
tone, and support by its most closely-related tones in Series 1.37
Hindemith states that In the cadence the three primary elements of music
(harmony, melody, and rhythm) work in cooperation to create a full sense of clo-
sure. In the conclusive nature of the cadence,
the rhythm confines itself to a few clear and unmistakable time-divisions,
the melodic steps proceed directly to their goals, the two-voice framework
employs the simplest intervals, and the harmonic fluctuation exhibits the
most unambiguous progression...38
35Hindemith, Craft I, 142-48.
35 Hindemith, Craft Ill Translation, 78.
37 Hindemith, Traditional Harmony II, 39.
38 Hindemith, Craft I, 138-139.
42
Cadence formulas consist of either three chords of Group A or one chord of
Group B and one of Group A. Hindemith categorizes the cadence types accord-
ing to the interval relationships of the chord roots according to Series 1 and 2.
The four categories are rated from the strongest types to the weakest and are
listed in Figure 8. The strongest cadences are formed by the intervals made up
entirely of a fourth and fifth or a fourth or fifth plus the step of a second. A milder
cadence formula consists of the fifth preceded by an Interval other than the sec-
ond or fourth from the tonic. An Interval other than the fifth preceding the tonic
greatly diminishes the strength of the cadence. The weakest of all cadences
consists of the tritone preceding the tonic.
Cadences define points of tonal stability and are one type of harmonic pillar
chord.39 During passages where the harmony is indeterminate or unstable, the
tonality may be built upon these pillar chords which are placed at “wisely calcu-
lated-out places†In the structure of the composition. Pillar chords, or
,flrmonieche Hauptstfltzpl'lnkt,†provide orientation for the harmonic structure
by providing a resting point for melodic activity and support for the two-voice fra-
mework. Pillar chords also help define the form of the composition and the flow
of motion from phrase to phrase.
In terms of harmonic motion, Hindemith defines harmonic fluctuation as the
increase of tension In the chords as they proceed from those of Group A, includ-
ing subgroups, to those of Group B, including subgroups (See Table 3). It is the
increase and decrease of tension by gradually moving between chord groups.4o
39 Hindemith, Craft Ill, 202-203. See also Neumeyer, The Music; of Peytin-
demith 14, 36, 39. Hindemith borrowed the Idea of the pillar chord from Ernst
Kurth’s book Die Romentlsche Hermonik und ihre Krise in Wegners “Tristep‘:
(1920). Kurth used the term Grundpfeiler, or “basic pillarsâ€, to explain the har-
monic progression in a series of chromatic passing chords in Wagner’s Tristan.
The first and last chords in the progression become the basic pillars of the har-
mony. Lee Rothfarb, Ernst Kurth as Theorist and Analyst, 181-185.
40 Hindemith, Craft I, 115-121.
43
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44
The best application of harmonic fluctuation is one which permits a gradual dis-
sonant crescendo and diminuendo. Hindemith dismisses the idea of harmonic
motion based on traditional triadic root movement. A series of chords of similar
quality such as occur In the diatonic system have only harmonic relationships
but not harmonic fluctuation. The Increase in tension is more important in
creating motion than the root movement or degree progression.“
But the harmonic fluctuation and tonality often remain Independent even
though they coexist in a composition.
Fluctuation and tonal ordering never influence one another. The intimate
tonal relation of harmonies may be very conspicuous, and yet they may show
great differences of value In their fluctuation. On the other hand harmonies of
the same fluctuation value may be very distant from one another so far as
tonal relation is concerned.42
Hindemith describes how harmonic fluctuation in a composition can be planned
to include the smallest detail, so that every detail can be prearranged in refer-
ence to the two-voice framework and degree of harmonic fluctuation. Even
though a composer can control the degree of harmonic fluctuation,
“the principle of harmonic fluctuation is analytic and not prescriptive.
Hindemith did not clearly recommend specific practices in the fluctuation of
chord values; neither did he disapprove others....ln short, it hinges completely
on expressive purpose, which can conceivably justify any plan of fluctuation
no matter how haphazard or arbitrary it may appear to be.â€43
The importance of the two-voice framework and harmonic fluctuation cannot
be understated. In Craft Ill he elaborates on the relationship of these to the
degree progression and musical composition:
4‘ Landau, “Hindemith the System Builder,†151.
42 Hindemith, Craft III 49-50, (Translation, 40).
43 Victor Landau, “Paul Hindemith, A Case Study in Theory and Practice,
The Music Review 21 (1960): 42.
45
Voice-leading, two-part framework, and harmonic fluctuation, although they
are wonderfully helpful in regulating the flow of harmony, nevertheless pro-
vide no absolute guarantee that music constructed according to their laws
will be from the harmonic point of view perfectly convincing. They can truly
perform their functions of organization only when a harmonic principle on a
still higher level regulates their relations with one another and arranges
them with reference to a common denominator that embraces them all. This
common denominator is tonality, and the principle which organizes all other
means in subordination to tonality is to be found in the relations of the
tones....ln doing this we relied on a line of single tones, which consisted of
the roots extracted from all the harmonies in a piece—the degree-
progression. The degree-progression will continue to serve us in the future
as the most obvious and most easily applicable means for handling the rela-
tions of the tones and achieving tonal organization...44
The concept of harmonic fluctuation and the table of chord values is one of
Hindemith’s most important contributions. It provides an alternative method of
defining chord progressions distinct from conventional Roman numeral analy-
sis. Based on the degree of dissonant tonal relationships within a chord, it
gives criteria for judging the strengths and weaknesses of a chord progression
other than those of conventional root movement. However, Searle observed
that Hindemith's table of chord values can be as arbitrary and inflexible as the
system It replaced in that it is difficult to consistently apply his principles to
highly chromatic and strictly non-tonal music.45 Hindemith would classify these
atonal and highly chromatic types as poorly constructed music because they do
not conform to the preconceived strictures of Series 1 and 2.
44 Hindemith, Craft Ill 85, Translation, 76.
45 Humphrey Searle, Twentieth Centugy Countermint (Great Britain: Wll-
Iiams and Norgate, Ltd., 1954), 69.
46
Hindemith’s Theory of Melody
The analysis of melody in Hindemith’s system Is also dependent on Series 1
and 2. Two aspects of melodic motion are important. First, Hindemith believed
that melodies have roots, determined by the interval content of a specific section
of melody in much the same way that the degree progression determines root
movement (Figure 9). Certain melody notes “which can be heard without effort
as a harmonically related group†are enclosed in brackets. 45 Unfortunately he
gives no further criteria for determining these melodic segments.
In addition to determining the Degree Progression of a melody, the Step
Progression is also important (Figure 10). The step progression identifies
step-wise connections in the melody. Major and minor seconds are linked
together to show melodic contour and motion.47 Brackets indicate stepwise
motion within a section of melody. Hindemith declares that intervals of the sec-
ond in melodies measure the content of “brief melodic sections†and also “larger
melodic sections.â€45
45 Hindemith, Craft I, 183.
47 The step progression has been compared to Schenker’s Idea of the Urli-
nie In the “delayed linkeage [gel by steps throughout total melodic life†yet with-
out the imposed stepwise descent. (Thomson, “Hindemith’s Contribution,†67).
A stronger connection, however, can be made to the theory of Ernst Kurth in his
idea of the lug, or “line," proposed in 1917 in the treatise Grundlegen gas line-
aren Kontrapunkte. Lee A. Rothfarb, Ernst Kurthee Theorist and Analyst.
(Philadelphia: University of Pennsylvania Press, 1988): 185; 226; 233fn. See
also Neumeyer, The Music of P_a_lu Hindemith, 47-48n, 67.
45 Hindemith, Craft I, 187.
47
Degree Progression
Figure 9: Melodic D ree Progression. Hindemith, The Creft of
Musical Com sition ok 1:Theorv. 184. ©1942 Schott and Co.,
Ltd., London. Copright renewed. All Rights Reserved. Used by
permission of European American Music Distributors Corporation,
sole US. and Canadian agent for Schott & Co., Ltd., London.
Figure 10: Melodic Step Progression. Hindemith, The Craft of
M_usical Compositpn Book 1: Theory, 194. ©1942 Schott and Co.,
Ltd., London. Copright renewed. All Rights Reserved. Used by
permission of European American Music Distributors Corporation,
sole US. and Canadian agent for Schott & Co., Ltd., London.
48
Stepwise connections of pitches are used on a single level. A tone can
belong to several different step progressions and may occur in immediate suc-
cession or over a longer time span. “The primary law of melodic construction Is
that a smooth and convincing melodic outline is achieved only when these
important points form a progression in seconds.â€49 The “important points†he
defines as being “guide-posts“ of the melody, usually consisting of the highest
and lowest notes, and metrically prominent notes. While the intervals between
the guide-posts are often filled in with major or minor seconds, the guide-posts
themselves should also form seconds. Hindemith uses the melodic degree pro-
gression and step progression to formulate criteria for a well-composed melody,
which should contain a logically developed degree-progression and clear line
in the step progression. He recognizes the existence of non-chord tones and
develops a set of symbols for their identification (T able 4).50 Non-chord tones
are determined by “normal accented rhythm.“ A problem of Interpretation can
occur, however, In a highly chromatic tonal language. Hindemith admits that it
is often difficult to distinguish chord tones from nonchord tones.51
The two-voice framework, degree progression, and step progression provide
harmonic and melodic organization. By introducing these concepts, Hindemith
gives the analyst tools to increase the listener’s awareness of the musical line.
Thomson argues that these are “positive contributions†to musical thought, even
though there are similarities with the concepts of other theorists.52
49lbid., 193.
5° lbid., 164-74.
5‘ lbid., 174. See also Craft Ill, 60, Translation, 70.
52 Thomson, “Hindemith’s Contribution,†66-68.
ME
Changing Tone
(Returning Tone)
Passing tone
Suspension
Unprepared Sus-
pension or neighbor-
ing tone
Neighboring tone
left by leap
Neighboring tone
approached by leap
Anticipation
Unacoented Free
Tone
Accented Free
Tone
$1M
W
[Wechselton]
D
[Durchgang]
‘V
[Vorhalt]
‘N
[Nebenton]
V‘
[Vorausnahme]
49
TABLE 4
Non-chord Tones
ï¬finitiun
occurs when one member of a chord moves
from Its place in the chord by step or skip to another tone
for a short time and then returns
a stepwise transition from one chord tone to
another
In a succession of two Intervals or chords, part of the first
is held over Into the second, where it creates a tension
with the other chord-factors, and is resolved during the
existence of the second interval or chord
atone occurring in a relatively strong rhythmic position,
at the Interval of a second above or below a chord-tone
and resolving to the latter while the rest of the chord
remains
on the final rhythmically weak fraction of the time-value of
a chord, one of Its tones moves up or down a second,
then leaps to a chord tone In the subsequent chord
atone standing at the Interval of a second from one of
the tones of the second chord, but sounding during the
time-value of the first chord which it leaves by leap
one or more tones of the second chord of a progression
are introduced too soon, so that they occur during the
duration of the first chord
atone of slight rhythmic value, in unstressed position,
which Is not part of either of the chords between which it
occurs, similar to ,N
atone in a rhythmically stressed position which Is not part
of either of the chords between which it occurs, similar to
‘N without resolution
50
Example of Hindemith’s Analysis
Hindemith ends Craft I with several analyses based upon his theory of
degree progression, chord values, two-voice framework, and harmonic fluctua-
tion.53 He contends that his method of analysis Is practical for all styles and
forms of music and includes examples ranging from medieval chant to nine-
teenth and twentieth century opera.
In Figure 11, measures 1-4 of the first movement of Hindemith’s Piano
Sonata No. 1 (1936) are given as an example of Hindemith’s analytical tech-
nique.54 The analysis, completed by this writer, is divided into two sections:
Immediately under the piano score are three staves for the melodic analysis
and four for the harmonic analysis. The most prominent voice, in this case the
top line, is notated as it appears in the score on the first staff. The melody is
divided into harmonically related groups by brackets, the root of each group
becoming the degree progression which is notated on the second staff. The
third staff then shows the melodic step progressions located by brackets.
The harmonic analysis consists of Identifying the two—voice framework, fluc-
tuation, degree progression, and tonality. The two-voice framework is notated
on the fourth and fifth staves and serves the purpose of isolating the bass and
melody so that the contrapuntal interplay of these two lines may be examined.
53 Hindemith, Craft I, 202-223.
54 It Is difficult to draw definitive conclusions about Hindemith’s musical style
from such a brief excerpt; my purpose is to demonstrate the procedure involved
In Hindemith‘s analytical method and his application of the theory to analysis.
51
Buhig bewegte Uiertel (J 96)
1111f
A. Melodic Analysis of the Upper Voice
Degree Progression
I—l_l
I__I_l
B. Harmonic Analysis
Two-voice
11 1112 12 1112 [VI 1111 Ilbl 11 12 1111 1111 1112 11
5. Degree Progression
6. Tonality
Figure 11: Hindemith, Erste Sonate fur Klavier 1936.© B. Schott's
Soehne, Mainz, 1936 rene . 9 ts eserved. Used by
permission of European American Music Distributors Corporation,
sole US. and Canadian agent for B. Schott's Soehne, Malnz.
52
The harmonic fluctuation occurs under this and identifies the chords of the
piece as they are found In the Table of Chord Values. Chords are labeled by
determining the root of each chord according to the interval content. The root is
that which occurs in the best interval nearest the bass according to Series 2.
Tritones, seconds, and sevenths then further locate the chord in the Table. The
roots or root representatives of each chord are placed on the staff immediately
under the fluctuation. From these the tonality of each section of the piece Is
identified and written on the lowest staff. The analyst then draws objective con-
clusions based on these observable facts concerning the music.
For example, it is observed that the initial tonality of A is firmly established in
the first two measures by a circular progression, but then the center of tonality
quickly moves to the key of B, the dominant of the chord which begins the fol-
lowing phrase. The harmonic motion Is reinforced by the melodic degree pro-
gression. Also, the two-voice framework helps to establish the shift of tonal cen-
ter: the zenith and nadir of the framework occur in conjunction with each other
and the beginning of the modulation. As far as the fluctuation is concerned, the
chords are relatively stable, the majority being of group I or III without tritones.
The example Is too brief, however, to draw significant conclusions about the
harmonic fluctuation and tonality of the entire movement.
It has been argued that, while good for diatonically based music,
Hindemith’s analytical method fails in Its ability to adequately interpret highly
chromatic or atonal music.55 A piece that consistently uses chord structures
that appear on the low end of Hindemith’s table of chord values does not neces-
sarily mean the music is poorly constructed, as Hindemith suggested. The ana-
lytical examples which Hindemith provided at the end of Craftl are intended to
55 Searle, Twentieth Centum Countemint, 63.
53
demonstrate the applicability of his theory to a wide range of musical styles.55
The rules he applied to various styles of composition are not convincing in
every instance.
One might think Hindemith’s method of analysis best serves his own music,
but he wrote relatively few pieces which were directly based on The Craft theory
of composition. Landau has shown that he was inconsistent in applying his
own rules.57 However, the inconsistency is because Hindemith considered his
theory as a starting point for the composer to learn basic techniques of composi-
tion. After the rudiments are learned, the composer is free to choose his own
way. Furthermore, Hindemith was attempting to show the common ground bet-
ween all styles of music with a consistent method of analyzing music.59 Hinde-
mith’s analyses at the end of petty are little more than descriptions of the musi-
cal events based upon his premises from gaft_l. In that, his theory is no better
nor worse than the one it seeks to replace. The analyses in mm were
intended only to demonstrate the applicability of his theory of harmony rather
than to be definitive interpretations of the music, as is suggested by his
acknowledgement of differing opinions.59 He suggested that analysis is merely
an aid to listening, giving “pleasure in the recognition and judgment of the
Impressions received.â€
53 Hindemith, Craft I, 197-223.
57 Landau, “Paul Hindemith, “A Case Study,†38-54.
58 Neumeyer, The Music of Paul Hindemith, 45.
59 Hindemith, Craft I, 203.
54
The Role of Rhythm in Hindemith’s Theory
The Craft of Musical Commsition Is both a theoretical treatise on the nature
and order of tones as well as a method of teaching composition. Hindemith
emphasizes the role of harmony and melody in his theory. Only scattered refer-
ences to rhythm occur throughout the literature.60 When rhythm Is mentioned it
is usually linked in some way to its Interaction with melody and harmony.
Hindemith was aware of earlier works on rhythm by such theorists as West-
phal, Riemann, and Wiehmayer.61 Even though Hindemith’s views on rhythm
echo the theory of Riemann in the metrical determination of accent, he veers
from Reimann’s theory in the use of asymmetrical structures and long-range for-
mal design.62 Hindemith felt that Riemann’s explanation of the principles of
rhythm was Inadequate.63 He consistently criticizes the world of music theory
for being unable to adequately explain rhythm.64
Hindemith chose not to deal with rhythm in The Craft for pedagogical
reasons.55 By isolating the Individual elements In order to treat each In detail,
the beginning composer would not be encumbered by unnecessary problems.
In keeping with his chosen method of instruction, the species exercise,
5° Wllliam Thomson, “Hindemith’s Contribution,†66, 69.
61Neumeyer, The Music of Qul Hindemith, 2-3.
52 David Neumeyer, “Tonal, Formal, and Proportional Design in Hindemith’s
Music,“ Music Theory Smtrum 9 (Wlnter 1987-88): 95-96.
53 Hindemith, “Methods of Music Theory,†24.
54 Hindemith, Craft l, 179; Elementagy Training for Musicians, 157; Craft Ill,
30, Translation, 21-22.
65 Hindemith, Craft I, 110.
55
Hindemith confines rhythm to specific applications which gradually expand as
the student learns to set tones against a cantus firmus. Hindemith recognizes
this and subsequently omits rhythm from the beginning stages of his theory,
only discussing it as necessary for an understanding of the interaction of the
three elements of music. Hindemith sought the simplest method by which a
student could learn to create music. He was looking for a compositional theory
of rhythm, not an analytical theory.66 He stated that he would attempt an expla-
nation of rhythm at a later time.57
While an analytical method is one by-product of The Craft music analysis is
not the sole purpose of the theory either. Hindemith stresses the compositional
method throughout his writings. In Traditional Harmony 1 and II, meant to be
used as theory textbooks for the “historical†music written between 1600 and
1900, the student is lead Into more advanced exercises of composition, for one
goal of studying music theory is understanding the historical resources of musi-
cal composition.58 The specific aim of Traditional Harmony 1 is to make the
study of music theory swift and practical for the general music student. Stud-
ents complete part-writing exercises demonstrating the harmonic materials of
tonal music. More advanced compositional exercises which reinforced the
same harmonic materials are provided in Traditional Harmony II.
The final chapter of Traditional Harmony II is of Interest to the present study.
In this chapter Hindemith sets forth a working procedure which demonstrates
certain ideas about the interaction of the elements of music on rhythmic formal
structures.59 The student Is to complete a four-movement suite for orchestra,
55 Hindemith, Elementam Training, 157.
57 Hindemith, Craft I, 179. See also Elementag Training for Musicians, 159;
“Methods of Music Theory,†23-24.
63 Hindemith, Traditional Harmony l, iv.
59 Hindemith, Traditional Harmony II, 37-53.
56
the melodic materials of which are supplied. Hindemith presents five represen-
tational graphs for the first movement, plotting out the melodic formal scheme,
the tonal plan, harmonic fluctuation, harmonic density (harmonic rhythm), and
texture. The student merely follows the plan to complete the exercise.
Hindemith taught his students to define the formal characteristics of a new com-
position first. Neumeyer has demonstrated the application of a similar model of
composition which Hindemith seemed to follow consistently.7o
The graphs of the formal and tonal plans of the first movement of the suite
have been duplicated in Figure 12 (combining the two for purposes of space).71
The themes of the composition are identified as A, B, and C and require unam-
biguous tonal areas, in this case the keys of 0‘, F, and C moving through A to
Ff.
Theme areas D, E, and F are transitional in character and therefore require
much less tonal stability as they move from one key area to another. Other
graphs which have not been duplicated here show the harmonic fluctuation,
harmonic rhythm, and texture. Using such a procedure, the composer can
effectively plan the growth of each section of the composition. The composi-
tional design includes four steps: 1. determining the general character and
function of the piece; 2. the formal divisions Including rhythmic character, tem-
po, melodic phrasing, and texture; 3. the overall harmonic plan with areas of
tonal stability and instability; 4. the specific thematic material.72
70 David Neumeyer, “Tonal, Formal, and Proportional Design,†97.
7‘ Hindemith, Traditional Harmony III 38, 40.
72 Neumeyer, “Tonal, Formal, and Proportional Design,†97; Idem, Ih_e_
Music of Paul Hindemith 35-38.
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â€â€™7 Other references to rhythm include chapters from ElementenLTLaining for
Musicians sections of Craft l-Ill notes from a lecture given at the Cleveland
Institute of Music in 1949, and an incomplete chapter from a proposed volume 4
of The Craft called Ubung 19.
In the Cleveland lecture, Hindemith proposes that a theory of music needs to
address both the individual elements of music and their Interaction. For a
theory of rhythm, Hindemith emphasizes that meter is the organizational force of
music and that the basic units of meter are two’s and three's and their com-
pounds. He suggests that a possible theory of form might include accents in
some way and that the establishment of measurements in the temporal propor-
tion of lengths needed to be formulated.73
-4 In the same lecture, Hindemith recognizes that rhythm influences melody by
the placement of accents at specific points and the division of melody Into
motives, phrases, and sections. What he needs to define more completely are
the speed of the development of melody and the comparative lengths between
specific melodic points. In the relationship of harmony to rhythm, Hindemith
tries to develop a theory of form by comparing the lengths of composite sections
in the overall tonal plan of the composition.74 Elsewhere, he stresses that the
three elements of music each have a different function which are combined in
various ways to create a musical composition:
Rhythm determines the duration of the chords, and groups them by division
into stressed and unstressed members of the structure. Melody in voice-
Ieading regulates linear expansion, and in the two-voles framework sets the
pitch limits. In the placing of the harmonic center of gravig and in the regula-
tion of relationships we see harmonic energy at work.75
73 Hindemith, “Old and New Problems of Music Theory,†lecture given at the
Cleveland Institute of Music, March 3, 1947. (Notes from the Paul Hindemith
Institute, Frankfurt am Main.): 9-10.
74Ibid., 13-14.
75 Hindemith, Craft l,109.
61
In the preface to Elementary Tra_Ining Hindemith discusses two distinctions
of rhythm: the length of notes with their placement in a metrical context, and as
the generator of musical form.75 Two primary concepts in rhythmic organization
on the foreground level, then, are metric placement of the accent and durational
patterns of the tones. Hindemith considers rhythm to “represent everything that
takes place in the medium of time: beats as well as the durations of extended
musical forms and their proportions.â€77
Note values and tones of different length are “the most primitive form of tem-
poral action.“79 The exercises in Elementag Training gradually introduce the
different note values and basic principles of notation that affect durational
values, including augmentation dots, ties, and slurs. When the student pro-
gresses to a more advanced level, Hindemith describes rhythm as
...the boundless and continuous stream of time intervals in which our
actions follow one another, the duration of each determined only by Its
character, purpose, speed, and intensity. This corresponds to musical
rhï¬hm, which has countless possibilities of combining tones of various
lengths with melodic lines and harmonic combinations. What character-
izes musical rhythm Is Infinite variety, ruled by higher laws of construction
and determined by the power of esthetic Lsi_c] judgement [ejp] and choice.79
Hindemith expresses the idea that durational values are combined In various
ways to create rhythm patterns depending upon the skill and artistic purposes of
the composer. These patterns are dependent upon pitch for melodic and har-
monic identity. The “higher laws of construction†are what remains Incomplete
In Hindemith’s explanation.
75 Hindemith, Elementary Treining for Musicians, xii.
77 Hindemith, “Methods of Music Theory,†23.
79 Hindemith, Elementapr Treining for Musicians 3.
79 lbid., 93fn. Hindemith might agree with Creston’s working definition of
rhythm: “the organization of duration in ordered movement." Paul Creston, Efl
ciples of Rhflhm, (New York: Franco Columbo, Inc., 1964), 1.
62
In Craft l Hindemith did not emphasize the role of rhythm, but he did say that
it determines the duration of the chords, and “groups them by division Into
stressed and unstressed members of the structure.â€90 Meter Is the factor that
organizes the durational values into identifiable units. “Every step from one
tone to the next involves a durational relationship, and consequently depends
on a regular metric beat as a unit of measure."81
Hindemith recognizes the interaction of rhythm and meter. He defines meter
as the “divisions of time into distinct and proportionally related intervals.â€92 The
notational values interact with the divisions of the meter, either reinforcing the
metric accent or opposing It.
Meter Is dependent upon accent for Its perception. Hindemith defines two
types of accent: the metric accent which is based upon the perception of pulse,
and the dynamic accent which is caused by the application of increased
force.93 The metric accent divides the basic pulse Into either duple or triple
groupings. The barline has one primary function, to mark the place of the prim-
ary metric accent.
The duple and triple groupings form the basis for larger structures. He
related these “smallest particles,“ or rhythmic motives of a basic rhythmic
grouping, to larger formal functions.84 Like Riemann before him, Hindemith
recognized that form is created by “the accumulation of the effects of smaller
constituent parts,†but the effects of rhythmic units longer than the basic two- or
three-beat groupings are only realized in retrospect upon the completion of the
80 Hindemith, Craft l, 109.
51 lbid., 178.
92 Hindemith, Elementag Training, 93fn.
33 lbid., 93-99.
34 lbid., 157-158.
63
longer musical form. In Craft 1 Hindemith recognized the need for the student
composer to “know as much about the form and inner dimensions of motives as
about their number and duration.â€85 In the same paragraph Hindemith dis-
cussed rhythm as the regulator of larger formal units of unequal lengths.
Hindemith was unable to formulate ideas for the measurement of the larger
formal units. He did, however, begin to set down some basic principles in the
investigation of rhythmic form. Four factors which influence rhythm, melody, and
harmony in formal constructions are duration, tempo, relative speed of unfold-
ing, and closeness of texture.86
Hindemith defines the relative speed of unfolding as the “proportional inter-
relationships of the constituent parts,†while the closeness of texture refers to the
“degree of complexity†of the musical elements. Duration and tempo can be
measured by a clock and metronome, but unfolding and texture lacked any
means of measurement at the time of his writing. Proportional schemes of
measurement may be useful in comparing phrase structure, length, and overall
formal design.87
A theory of proportions was not completely worked out by Hindemith. A few
works from the 1930’s use proportional relationships as an element of composi-
tional design; however, the problem he faced was to connect smaller rhythmic
units such as motifs and meter to the longer formal structures.88
Hindemith identifies the effects of repetition, variation, and change in the
construction of formal units.
35 Hindemith, Craft I, 178-179.
85 Hindemith, flementangining, 158-159.
97 Neumeyer, “Tonal, Formal, and Proportional Design,“ 93-116.
88 Neumeyer, 113 Mueic of Peul Higdyemitp, 41. Works which Neumeyer
cites as using proportional designs are Angelic Concert from Mathis der Maler
and the Second Piano Sonata (1936).
64
The principles by which the compound effects are achieved when the sound-
ing material Is cast into molds of temporal construction are three:
(a) Repetition (re-use of one constituent part of the formal entity, on the
same pitch level, or in transposition).
(b) Variation (some of the elements of a constituent part are changed
while others remain unchanged. For instance, the melodic line may be
retained but with different harmonic and rhythmic treatment; or the
rhythmic shape of a motive-or melody, etc. —may be retained, but with
changed melodic outline and new harmonies)
(0) Change (one constituent part gives place to an entirely different one.)89
These three common techniques of composition provide an unlimited range of
possibilities for the composer to create musical forms. The techniques are com-
bined in various ways to create the melodic, harmonic, and formal/rhythmic ele-
ments of a composition.
In many of his references to rhythm, Hindemith emphasizes the relationship
between melody, harmony, and rhythm. All three combine to create the effect of
motion or flow in the composition. The “subordinate elements,†such as
dynamics, tone-color, articulation, etc., are for decorative purposes and affect
only the listener’s impression of the work but not the substance of its construc-
tion.90
A survey of Hindemith’s views of rhythm reveals several basic tenets.
1. Meter Is the basic organizational factor In Hindemith’s music.
2. Accent groups pulses into basic units of two’s and three’s.
3. Durational patterns make up motivic structures which are the basic units
of form.
4. The process of composition begins with determining the rhythmic form
and rhythmic character of each section, then the tonal relationships and
finally the specific melodic material is written.
5. Shorter rhythmic units undergo repetition, transformation, or change as
they are combined to create longer structures.
6. Rhythm will influence melodic and harmonic structures.
7. Rhythmic form can be determined by a comparison of the lengths of the
sections of the tonal framework.
99 Hindemith, Elementagy Training, 158-159.
90 Hindemith, “Methods of Music Theory,†21.
65
Hindemith’s combined references to rhythm do not constitute a well-
developed theory in comparison to his theory of melody and harmony from IE2
gait, But they do show that while he was somewhat conventional in his view-
points on rhythm, he was trying to find a connection between the beat-to-beat
patterns of the surface rhythms and the long-range patterns of formal structures.
One must turn to his music to try to formulate stronger conclusions about Hinde-
mith’s application of rhythmic principles.
Chapter 3
Analytical Procedures
The analytical procedure presented by Hindemith In M demonstrate a
practical application for his theories of melody and harmony. Although it
provides a consistent method of analyzing music in terms of the harmonic and
melodic content, the theory Is compositional rather than analytical in nature.
Hindemith did not prescribe a method of analysis for rhythm, stating elsewhere
that
the temporal material In music, if It is to be used rationally, must be subjected
to measurement, as was the spatial element, harmony....No scientist’s
research, no musician’s intuitive genius, no layman’s common sense has
ever been able to find ways of measuring rhythm, in an attempt to establish a
rational basis for the construction of temporal musical forms...[yet] some
rational, discoverable, and understandable law of construction must exist
which could be put into effective operation.‘
Perhaps an adequate method of measuring rhythm did not exist at the time
Hindemith wrote The Craft of Musical Commsition, but much recent research
has been done on the rhythmic element of music. The purpose of this chapter is
to explore a method of analysis which tries to answer Hindemith’s concerns of
measuring rhythm.
Hindemith mentions In M that rhythmic organization involves three ele-
ments: the tempo of the succession, the durations of the individual
1 Paul Hindemith, “A Commser’s Worldâ€, 86, 88, quoted in David Neumey-
er, “Tonal, Formal and Proportional Design in Hindemith’s Music,†95.
66
67
combinations, and the position of the accent in relation to the barline.2 These
three properties of rhythm (tempo, duration, and metric accent) provide the
basis for understanding the relationships between the individual components of
the composition as well as its formal construction. Hindemith viewed the study
of rhythm as a “top-down†principle, moving from the large to the small In keep-
ing with his method of composition discussed in Chapter Two.
According to Hindemith, a musical composition is like a sentence that is
made up of individual words, the meaning of the sentence determining which
words are used. Individual tones, or even musical motives, are the words of the
sentence; they are not understood separately but as a part of the context in
which they occur. The musical form therefore cannot be fully understood until
its conclusion.
As we have already seen, even among the few successive tones contained
in two or three successive melodic Intervals, relations develop which cannot
be explained simply as sums of individual tones or of the Intervals between
two tones. The lowest form of these melodic entities, superior to the individ-
ual tones and intervals and essentially different from the mere sum of their
effect, we have already recognized in the harmonic cells and fields of mel-
ody on the one hand and in the step-progression on the other. These supe-
rior form-constituents in turn combine to build up still more comprehensive
forms, to which they are related just as their own constituents were to them;
here too the aesthetic effect of the entire form is in no way equal to the sum
of the individual effects of the formal constituents. The latter always produce
a new and superior structural element indispensable to the understanding of
the complete form....lt is the whole, then, that determines the part and In no
way the part whose appearance and special forms determine or develop the
form of the whole.â€3
It would seem, then, that in order for a rhythmic analysis to be compatible
with Hindemith‘s theory, the analysis would have to take into account note-to-
note relationships and how these are built up to create the musical form. An
understanding of temporal relationships is achieved by comparing the
2 Hindemith, Craft Ill 30 (Translation, 20).
3 lbid., 20.
68
Individual rhythmic components and approximate performance times of each
component. These components Include note durations, motives, phrases, peri-
ods, and sections.
Rhythm in this analysis will be separated from the other elements of music
as much as possible, just as Hindemith separated rhythm from his theory of mel-
ody and harmony. However, it is recognized that all musical elements are con-
trolled by rhythm, and therefore the study of rhythm is equivalent with the study
of music.
The study of rhythm is the study of the flow In time of sounds and silenc-
es. If this seems to be a definition of music itself, it may serve to remind us
that the study of rhythm involves some consideration of all the aspects of
music. Duration is the special province of rhythm, but the ordering and
organization of temporal units is the vital core of rhythmic analysis, and
where pitch, harmony, texture, dynamics, and timbre may play as significant
a shaping role as the durations themselves.4
Various writers have proposed several models of analysis that explore the rela-
tionship of rhythm to the various elements of music.
Development of the Model of Analysis
One purpose of music analysis is “the understanding of musical style."5
Analysis identifies differences between style periods, between different com-
posers, and even between different works by one composer. Music analysis
should give the musician assistance in making stylistic performance judgments.
4 Allen Winold, “Rhythm in Twentieth-Century Music,†AmS of Twentieth
Centugy Music, ed. by Gary Wlttlich. (Englewood Cliffs, New Jersey: Prentice-
Hall, Inc., 1975): 209.
5 John D. White, The Analysis of Music, 2nd ed. (Metuchen, New Jersey:
The Scarecrow Press, Inc., 1984): 1.
69
White has proposed one model of analysis. The procedure includes
descriptive analysis of the musical events and synthesis of the data. After
general background information about the music, composer, and time period
are researched, the descriptive analysis consists of three levels.5
The microanalysis identifies the details of melodic, harmonic, and rhythmic
events. It also includes details of orchestration, texture, timbre, dynamics, and
accent. Specifically regarding the rhythmic element of music, the microanalysis
consists of identifying pitch durations, accents, rhythmic motives, harmonic
rhythm, and text setting. The second level, or middle-analysis, involves descrip-
tion at the phrase and section level. The rhythmic elements included In the
middle-analysis are the metric and rhythmic structure of phrases and the rela-
tionships of other formal units. One Important aspect of middle-analysis Is iden-
tifying repetition of musical material, development of motives or figures, varia-
tion on original material, and the use of new material.7 Finally, the macroanaly-
sis consists of describing how the events develop into the total time span of the
composition: the broad harmonic changes of tonality, meters and tempos, the
overall rhythmic style, durations of large sections, and rhythmic and proportional
relationships between movements. Identifying formal structures Is one result of
macroanalysis.
Lester presents a more detailed model of rhythmic analysis. The model is
broken down into four categories: durational patterns, accent and meter, group-
ing or segmentation, and musical continuity (Figure 13).8 The durational pat-
terns consist of Individual note values that combine to create the rhythms of the
melody and other independent parts. Also studying the composite rhythm
5Ibid., 13-18.
7 Hindemith simply called these repetition, variation, and change.
9 Lester, 5-12.
70
identifies changes in texture, density, and combined rhythmic events which help
define formal divisions from phrase level to the entire composition. The rhythm
created by changes in other parameters of music are also noted.
The harmonic rhythm helps establish meter in tonal music. Important points
of harmonic change include cadences, phrase beginnings, and the overall pace
of the harmonic changes. Other musical parameters which can create rhythmic
events Include regular changes In texture, timbre, articulation, and dynamics.
I. Durational patterns
A. of individual parts (the rhythm of a part)
B. of textures (composite rhythm)
C. of changes in
1. harmony (harmonic rhythm or rhythm
of harmonic change)
texture
timbre
articulation
dynamics
. other aspects
ll. Accent and Meter
A. of note-to-note and measure-to-measure
B. of larger levels
1. phrase accentuation and hypermeter
2. accent, meter, and musical form
Ill. Grouping or segmentation (motives, phrasing, form)
IV. Musical continuity and flow
mmeww
Figure 13: Lester’s Model of
Rhythmic Analysis
71
Lester places much emphasis on accent and meter. He defines accent as a
“point of initiation.â€9 Different types of accent include metric, dynamic, long
duration (agogic), harmonic, and textural. A noticeable change In any musical
parameter can create the effect of an accent; however, accents should not all be
considered equal in strength. Familiarity with the music is needed to properly
identify the Importance of different types of accent.1o Furthermore, in music of
the common practice period accents occur within a metric context.
Meter is dependent upon accent for Its identification. In the analysis, It is
important to Identify the factors that help establish the metric pulse and how the
flow of the music either supports the pulse or conflicts with it. The accent gives
organization to the pulses that are measured into groups of two or three
pulses.11
According to Lester, pulses can be established in a number of ways. Two of
the most common are the use of a recurring note value or the subdivision of
longer note values. The subdivision of durations into equal parts establishes
the metric hierarchy. For example, a whole note is subdivided Into two half
notes, a half note is subdivided into two quarter notes, and so on. It is quite
common for subdivisions to change in tonal music, as in an eighth-note accom-
paniment pattern changing to triplets.12 Also, regular changes of harmony help
establish groups of pulses into metric divisions.
The principle factor in establishing form is grouping, Lester’s third level of
analysis. The form of the music Is built up from smaller segments, from motives
to phrases, to periods and sections. An important part of the analysis is to
9 lbid., 16.
‘0 lbid., 40.
‘1 lbid., 45.
‘2 lbid., 48.
72
determine how notes are grouped on the various levels of organization and
how the music flows continuously from one part to another.13 “Grouping is the
operation whereby we organize differentiated notes into units consisting of
several notes (or, at higher levels, several groups of notes)"14 Factors that
influence grouping are proximity of the musical events, similarity or contrast In
the patterns of notes or other musical elements, and duration of the passage.
Generally, groupings are made out of the shortest patterns; the longer the pas-
sage the less likely it will be perceived as a single group.
Unified groups can also be formed by identifying repeated patterns of pitch.
Two common types are ostinatos and melodic sequences. Other factors include
patterns of repeated note values or durations, and patterns caused by changes
In articulations.15 A unified group, then, is a series of notes that is understood
as a single unit. The unified group as a generator of formal structures causes
an accent of longer duration than the metric or dynamic accent. Lester’s final
level of analysis synthesizes the existing data from previous levels into an
explanation of how the various elements of music work in cooperation to create
a sense of musical continuity and flow.
Epstein’s model of rhythmic analysis offers a slightly different explanation of
rhythm than the previous two (Figure 14).15 The general properties of time are
identified as duality, hierarchy, demarcation, and motion. Duality refers to what
Epstein calls the “Chronometric†and “integral†aspects. Chronometric time is
the clock-like organization of time into recognizable divisions; integral time Is
the specific events or experiences within Chronometric time.
‘3 Lester, 218-19.
14 Ethan Haimo, “Rhythmic Theory and Analysis,“ ( In Theory Only 4/1): 18.
15 Smither, “Rhythmic Analysis,†61.
16Epstein, “Shaping Time,†11.
73
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74
Specifically regarding the mechanisms of music, duality refers to the difference
between meter and rhythm, meter being the equivalent of the chronometric time
and rhythm the integral aspect of music.17 A hierarchy of time Is explicit in the
many divisions and subdivisions of chronometric time. In music the mechan-
isms of the hierarchy are displayed by the various levels of meter: the beat or
pulse, measure and motive, hypermeasure and phrase.18
Demarcation refers to the segmentation of time Into recognizable units, such
as days, hours, and minutes, or other boundaries of segmentation. In music
demarcation refers to the events which create grouping or segmentation of the
music. Epstein stresses the importance of accent as the most significant
method of demarcation. He refers to Cooper and Meyer's classic definition of
accent as a point of emphasis that Is more prominent than the encircling
events.19 Accent must exist within a hierarchy of structural events. It differs
from stress in that it is structural and vital to the organization of the music,
whereas stress Is often ornamental, creating varying levels of dynamic intensity
on the surface of the music but not necessary to its underlying structure.
Epstein divides the mechanisms of music into two categories. On the one
hand are those elements which define structure and organization. These
include the various aspects of meter and rhythm: beat and pulse, measure and
motive, hypermeasure and phrase, and larger segments. On the other hand are
the processes of performing music, the elements of tempo such as proportions
of speed, rubato, accelerando and ritardando. All create the sense of motion or
pacing. All affect the listener’s perception of the music.
17Ibid.,10-11.
‘5 lbid., 28-37.
19 lbid., 24ft. Epstein refers specifically to the definition of accent presented
by Cooper and Meyer as “a stimulus which is marked for consciousness in
some way.“ (Cooper and Meyer 1960, 8).
75
Epstein recognizes that tempo helps establish the pacing of musical events
as they unfold in time. Correct and proper tempo is based partly on the Interpre-
tation of the performer, the directions in the notated score, and the tradition of
the music/20 From the notational directions the performer receives Information
regarding how the piece is to be played. Tempo, then, refers specifically to how
all the elements of a work “unfold“ and develop into the total whole. Only refer-
ring to the metronomic speed in measuring the tempo of a piece may be inade-
quate because other factors are left out, such as environment, technical ability,
and maturity of the performer.
Tempo Is one of the most subjective of performance parameters. No two
performers will play in exactly the same tempo and no two performances by the
same person will be the same, either. Rubato, ritardandos, and accelerandos
also cannot be taken under consideration, as these are left to interpretation. Of
course an exacting metronomic performance would be considered unmusical
as well and would leave out the elasticity of rhythm that the earliest definition of
the word requires.21 Therefore only approximate calculations of duration can
be given from the information in the score. Furthermore, the tempo markings in
the score should be considered the extreme fast or slow speeds.22 In choosing
a slower or faster tempo than that Indicated in the score, one should keep the
sections and movements in proper proportions. Nonetheless, the theorist can
gain an understanding of the proportional structure of a piece of music from the
durations of movements and sections measured by the tempo indications.
20 lbid., 100. Epstein refers to the performer’s Interpretation as personal
authority (maturity and reputation) and intuition (musicianship).
21 Refer to Chapter 1 of this paper for a historical study of the word “rhythm.â€
22 Epstein, 99.
76
Two primary emphases can be deduced from these three models of rhythmic
analysis. The first is the identification of patterns of surface elements that define
stylistic features. Some of the features to Identify are the relationships of dura-
tional values of notes to metric placement, prominent pitch classes, and group-
ing of durational values into rhythmic and melodic motives. Other surface ele-
ments that play a prominent role In rhythm are changes in harmony, articula-
tions, and dynamics. Durational values of pitches and the patterns created by
the composer’s grouping of different note values is an element of style analysis.
wlnold and Smither have each developed tools to help identify consistent uses
of durational values and their grouping into motives and phrases.23
Winold divides rhythmic structure into the foreground and background ele-
ments. The background rhythm Is the regular, on-going metric structure upon
which the durational patterns of the foreground are placed.24 The background
controls time and tempo; It provides regularity for the individual patterns of pitch
durations. The rhythmic analysis, according to Winold, must explain the back-
ground rhythmic structure, the foreground durafional patterns, and the relation-
ship between the two.25
Metric structure is measured by the pulse of the meter. There may be
several simultaneously occurring levels of pulse in a composition. The beat
level is given as the pulse that is the basic unit of measure usually identified by
the meter signature or the most common durational value. Pulses can occur on
23 Smither, “Rhythmic Analysis,†54-88; Allen Wlnold, “Rhythm in Twentieth-
Century Music,†In A§pects of Twentieth Centug Music, ed. Gary Wlttlich.
(Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1975): 208-269.
24Wlnold’s concept of background metric structure and foreground rhythm
patterns are similar to many other theorist’s views. Refer specifically to the chro-
nometric and integral divisions of time described earlier by Epstein and Hinde-
mith’s discussion from Elementam Training, 93m.
25 Wlnold, 211.
77
different levels, however. For example, the beat may be subdivided into faster
pulses known as division levels. Conversely, pulses may incorporate two or
three beats. These are called multiple levels. Pulses are distinguished and
grouped by different types of accents. Winold‘s description of common practice
metric structures has been cited in Chapter 1. He continues to define the
differences of twentieth-century metric structures. A more concise method of
identifying metric structures has been offered by Smither (Figure 15).26
Like Winold, Smither identifies the beat, the basic unit of measure, as the
“primary level.“ His secondary level is the duration of two or several beats.
Accent plays a particularly strong role in Smither’s analysis (not unlike those
previously cited), as accent helps define grouping of notes.
In addition to the metric structure on the beat level, division levels, and multi-
ple levels, Wlnold has also proposed a method for analyzing durational patterns
in twentieth-century music.27 The number of Individual durational values are
identified and counted in the duration scale. These are then ranked from short-
est to longest values in the duration complement. The duration range then com-
pares the length of each durational value to the shortest value listed at the top of
the duration complement. The frequency of use of each durational value is
counted in the duration hierarchy. All of the data are then organized in a table.
26Smither, 73-83. Smither gives the following as examples of his metric
classifications: IA. Schoenberg, Third String Quartet, molto moderato; lB. Bar-
tok, Fifth String Quartet, Alla bulgarese; IC Orff, Catulli Carmina, vive; Webem,
String Quartet, Op. 28, Gemachlich; II. Stravinsky, L’Histoire du Soldat; III.
Messiaen, L’Ascension, Quatre Meditations Symphoniques; IVA. Copland,
Short Symphony No. 2; IVB. Blacher, Ornaments fI'Ir Klavier, Op. 37; IVC.
Carter, Second String Quartet.
27 Wlnold, 235-241.
78
I. Metrical rhythm.
A. Equal beats with regular accentuation at the secondary
level.
B. Unequal beats with regular accentuation at the secondary
level.
C. Equal beats predominating with vague or no accentuation
at the secondary level.
II. Polymetrical rhythm.
Two or more Independent patterns of regular accentuation
used simultaneously.
Ill. Metrical-nonmetrical rhythm.
Equal beats with irregular accentuation at the secondary level.
IV. Nonmetrical rhythm.
A. Unequal beats with irregular accentuation at the secondary
level.
B. Unequal beats predominating with vague or no accentua-
tion at the secondary level.
C. Free accentuation which defines neither primary nor sec-
ondary levels.
Figure 15: Smither’s Classification of Metric Structures
For example, Figure 16 presents measures one through four of the second
movement from Hindemith’s Erste Sonate ftir Klavier (1936).28 Table 5 shows
the duration scale, complement, range, and hierarchy for the melody of this sec-
tion. The scale in the left column of Table 5 shows the 7 different durational val-
ues used in the melody arranged from shortest to longest. Note that Hindemith
used a very limited number of note values.
23 For comparison purposes, Wlnold analyzed measures 1-25 of Hinde-
mith’s Piano Sonata No. 2 to demonstrate his method of rhythmic analysis.
79
Im ZcilmaB cines sehr Iangsamcn Marsches (JetwaSO)
Figure 16: Hindemith, ERSTE SONATE FUR KLAVIER (1936),
Movement 2, m. 1-4 (©B. Schott's Soehne, Mainz, 1936
©renewed. All Rights Reserved. Used by permission of European
American Music Distributors Corporation, sole US. and Canadian
agent for B. Schott's Soehne, Mainz.)
80
TABLE 5
Duration Complement, Range, and Hierarchy
Complement Range Hierarchy
1 9
[TIT—>00)
15 1
0—s—
The interesting feature shown in Table 5 is the number of times the thirty-
second note and double-dotted eighth note are used. These two values are
linked to a motive; the thirty-second note always precedes the longer note
which occurs directly on a beat at the first multiple level. Another feature Is that
the longest note values, the tied half-noteldoubIe-dotted eighth and the double-
dotted quarter note, stand in a unique tonal and rhythmic relationship. They
both agogically accent the tonality of the piece. The first represents 051 on the
opening pitch of the melody and the g" exactly one measure later clearly
establishes CIi as the tonality.
81
It might seem pointless to identify each durational value in a melody or tex-
ture of a composition, but the analyst uses such information to provide concrete
data for style analysis. By identifying the most frequently used durational value
in a melody, one can identify the rate of motion of the music, comparing it to the
metric structures at the beat level, multiple, and division levels. The data can
also be compared to the rate of motion In other parameters of the music, such
as the harmonic rhythm. The frequency of certain durational values can be
related to melodic motion, pitch class, dynamic intensity, etc. Certain stylistic
features of the rhythm of a composition are identified for comparison to other
sections, movements, or pieces.
Winold continues the analysis of surface rhythms by identifying rhythmic
units and gestures. A rhythmic unit "occupies a period of time equivalent to a
given unit of the underlying metric structure"?9 Rhythmic units are more tradi-
tionally termed motives, but Winold is primarily concerned with the rhythmic val-
ues rather than the pitch relationships of the units. The rhythmic units can also
be placed In a table showing the unit complement, the ratio of durations from
shortest to longest in each unit, and the number of times, or frequency, that the
unit Is used.
Rhythmic units are categorized according to their relationship with the
underlying metric pulse. Units either reinforce the meter or conflict with it. Four
categories define the relationship of rhythmic units to the pulse:
1. Metric or even-note patterns. The durations of the pattern are identical
with pulses on a given level of the metric structure.
2. lntrametric or confirming patterns. The durations of the pattern are based
upon pulse groups within the metric structure but are not Identical to the
pulses of the metric structure.
3. Contrametric nonconfirming, or syncopated patterns. The durations of
the pattern are Identical to or based upon the pulses of the metric structure
like types 1 and 2, but unlike these, the accents of the pattern do not confirm
or support the accentuation of the metric structure;
29Winold, 237.
82
4. Extrametric or irregular patterns. The durations of the pattern are based
upon pulse groups which are outside the normal pulse groups of the metric
structure and are nonsynchronous with them.30
Wlnold also proposes a method of labeling longer rhythmic units. Rhyth-
mic gestures, more commonly referred to as phrases, are identified by com-
mon features in successive units and are subject to the grouping parameters
mentioned earlier. Gestures are labeled according to the beginning and end-
ing of the gesture, i.e. strong, weak, anacrustic, etc. The three types of begin-
ning gestures are thetic (strong pulse beginning), anacrustic (weak pulse
beginning), and Initial rest (beginning after a rest or tied note). The three
types of endings are strong (usually ending on the first beat of a measure),
weak (not ending on the first beat of a measure), and upbeat (ending on the
final beat of the measure).31
The second focus of attention from the three models of rhythmic analysis
previously discussed is the identification of background rhythmic elements. We
have already discussed the background metric structures Identified by Winold
and Smither that occur just below the surface of the music. Background rhythm
includes patterns which are not readily identified on the metric surface of the
music and involve longer groupings or phrase structures. The relationships of
formal divisions to one another is a prominent consideration of background
structure. The elements that help define the background structure are the tonal
relationships of sections and movements, length or time-span of longer sec-
tions, changes in composite rhythm and meter from section to section, texture
changes including number and instrumentation of voices, polyphonic vs homo-
phonic writing, registral changes, and density or thickness.
Hindemith’s compositional diagrams that he demonstrated in Traditional
3° lbid., 238.
3‘ lbid., 239.
83
Harmony II may be used as a starting point to understanding background struc-
tures of his music.32 The background structure of the music can be charted by
diagraming the phrases onto a graph representing measure numbers (see Fig-
ure 10). In this technique one can readily see the durational relationships of
one section to another. A similar approach has been presented by Neumeyer.
Neumeyer has developed a model of analysis for Hindemith’s music based
on a hierarchy of five stages:
Stage I: Controlling Structure: tonal areas and formal design
Stage II: Pillar chords with voicings and cadential progressions
Stage III: Interpretation of events between pillar chords
Stage IV: Harmonic Activity: Analytic symbols, chord roots, harmonic
fluctuation, tonal relations
Stage V: Melodic Activity- “step progression“ and “arpeggiation'
These stages are derived from Hindemith’s theory In Craft l-lll as well as other
writings. The stages differ from Schenkerian techniques in the lack of a
continuous hierarchy of levels.33 Because Hindemith’s music is non-functional
in terms of traditional harmony, certain adaptations were made: there is no
compliance to a strict background structure, or 915$, since Hindemith’s music
does not conform to traditional harmonic patterns anyway; it admits the element .
of form In Stage I; the Schenkerian principle of “composing out†Is ineffectual
since there is no strict background level; each stage of the graph can be con-
sidered independent of the others. Furthermore, more than one graphic repre-
sentation and Interpretation Is possible-34
The primary emphasis In Stage I is to identify formal divisions in the music.
32 See my description of Hindemith’s compositional method from Traditional
Harmony II in Chapter 2 of this study.
33A description of Schenker's terminology and methodology as well as
developments based on Schenker’s theory for an understanding of rhythm Is
addressed in Appendix B.
34 Neumeyer, The Music of Pa_ul Hingemith, 49,fn1.
84
These can be considered rhythmic divisions in the sense that they span a
certain length of time. The Stage I graph includes the degree progression of the
cadential and other pillar chords and may include the proportional scheme of
each section.35 Other rhythmic principles such as meter and foreground dura-
tional values have been omitted.
In the Stage II graph, the full pillar chords with the voice-leading progression
are represented. Neumeyer suggests that pillar chords outline phrases in
shorter compositions, but in longer compositions may consist primarily of the
boundaries of periods or sections. Hindemith did not give clear instructions in
determining pillar chords, nor does Neumeyer’s graphing procedure give spe-
cific criteria for identifying pillar chords other than harmonic and tonal relation-
ships.35 Certainly cadential formulae are important determinants, as sug-
gested by Hindemith, but other criteria from melodic and rhythmic principles
may exist as well.37 Again, the Stage II graph omits the element of foreground
rhythmic patterning that is crucial to the stylistic interpretation of the music.
Even though each graph of Neumeyer’s five stages can be considered sepa-
rately, there is also a link between them. Stages 1 and Il provide the tonal back-
ground upon which Stages lII, IV, and V are based. These three graphs show
details of tonal relationships nearer the surface of the music. Stage IV provides
a detailed analysis of the harmonic activity of the composition. It fills in the
harmonic progressions between the pillar chords with the degree progression,
voice-leading, and other harmonic activity.
35 lbid., 39, 50.
35 David Neumeyer, Bloomington, Indiana, personal letter via electronic mail
correspondence to the writer, January 3, 1996.
37 Other criteria may include the recurrence of a given harmony, rhythmic
placement, support from surrounding harmonies, and melodic goals. See Hin-
demith, Traditional Harmony ll, 39.
85
Melodic activity independent of the harmonic field is measured In the Stage
V graph. Specific melodic configurations, especially the linkage of the various
step progressions, arpeggiations, and harmonic cells inherent In the melodic
voices, are connected in a graph similar to a Schenkerian middleground reduc-
tion.39 While this type of graphing technique reveals relationships in certain
melodic motions, once again specific foreground rhythmic patterns are not read-
ily defined. Finally, Neumeyer’s Stage III graph pulls all the preceding informa-
tion together in an Interpretation of the most important relationships of the
events In the musical composition.
The Derived Model of Rhythmic Analysis
It would be difficult if not impossible to determine precisely what analytical
method Hindemith would prescribe for the study of rhythm in his music. He felt
very strongly at the time he wrote much of his theory that his understanding of
rhythmic principles was incomplete. At best, one can hope to apply a model of
analysis which Is in agreement with much of Hindemith’s understanding of
rhythm.
The three models presented by White, Lester, and Epstein each emphasize
to some extent the relationship of rhythm to the other elements of music. Musi-
cal events do occur within a temporal boundary, and Hindemith himself rec-
ognized the partnership of melody, harmony, and rhythm. However, In keeping
with the tradition of The Craft, It may be beneficial to separate the rhythmic
element from the other elements of music, at least in an initial analysis.
33 Neumeyer, 39, 66-71 .
86
The three models also recognize the hierarchical element of rhythm. A hier-
archical understanding of rhythm is compatible with Hindemith’s prescribed
method of composition. (Hindemith understood rhythm to take place on two
basic levels: the events of the note durations on the surface of the music, and
the overall structure or form created by melodic and harmonic goals) Using a
top-down approach, the background formal structure will be identified first, then
the temporal relationships of the proportions of each movement and section will
be analyzed and compared. Proportional relationships and tempos between
movements and sections will also be explored using the durations of the tempo
markings indicated in the score. The graphic notation tool to be used will be
similar to Hindemith’s method of planning the rhythmic structure of a composi-
tion. Sectional divisions of the movements will also be Identified in the graph.
For surface rhythms, Hindemith believed rhythm to take place in a metrical
context with pulses organized in two- or three-beat groups organized around
the metric accent. The barline determines the metric accent. Therefore, the
analytical model for foreground patterning to be used will use the metrical
accent as a determinant of structure from the smallest motive to the longest
phrase. The analytical methods presented by Winold and Smither appear to be
compatible with Hindemith’s understanding of meter and rhythm. Tables for
comparing durational values will be used where appropriate. The principles
laid forth by Wlnold and Smither will be used to label and Identify metric struc-
tures and grouping in the foreground rhythmic patterns. These will then be
compared to Wlnold’s list of rhythmic elements of the common practice period
discussed in Chapter 1. The types of metrical structures will be added to the
metric graphs showing the relationship between the type and quality of surface
rhythms to the overall background structure.
Chapter 4
Demonstration of the Method of Analysis
Hindemith’s work In the genre of the sonata seems to address the general
definition of the term sonata, “to sound.†Rather than following the “classical“
formula for sonata form movements (I.e. allegro with exposition!
development/recapitulation, adagio, minuetltrio, allegro) Hindemith seems more
concerned with non-compliance to standard structures.1 The resulting compo-
sitions are extremely individual in character and design. Hindemith said
I want to compose a whole series of sonatas. Each of them is to be com-
pletely different from the preceding ones. I want to see whether I can’t, in a
whole series of such pieces, increase the potentialities (which are not very
great in this type of music and this combination) and extend the horizon.2
In his series of over forty sonatas for virtually all the orchestral instruments, Hin-
demith wrote three for piano solo. In this chapter the relation of the first piano
sonata to Hindemith's other works for piano will be explored. The form of the
first piano sonata will be analyzed in reference to the tonal scheme of the com-
position. A comparison of the proportional lengths of the large formal sections
will be made. The first movement will then be analyzed in detail to reveal char-
acteristic uses of rhythm.
1 Hindemith, Elementm Training, 159.
2 Paul Hindemith, quoted in Melvin Berger, Guide to Sonatas: Music for One
or Two Instruments (New York: Anchor Books, 1991), 111.
87
88
Hindemith's Piano Music
Hindemith wrote many pieces for piano, beginning with a lost sonata from
his student years. His first published piano music Is a set of five Dance Pieces
for Piano Opus 19, completed from 1920 to 1922 and published In 1928. A bet-
ter known piece from this period is the 1922—Suite for Piano. Paul Vlï¬ttgen-
stein commissioned a piano concerto for the left hand in 1923 but never per-
formed the unpublished manuscript.3 The famous Kammermusik Number 12
Opus 36 Number 1, features an obbligato piano part with twelve solo strings
and winds that was featured at the 1924 Donaueschingen festival. Earlier com-
positions in this genre show an increasing reliance on a tonal framework as
Hindemith rejected the extreme influences of his experimental period.4
The Klaviermusik Opus 37 Number 1 and 2 from 1926-7, are a set of three
etudes and thirteen short piano pieces that show the development of a new
reliance on tonal organization that eventually developed into The Craft theory.5
Also from this period are his Opus 40 Number 1 Toccata the Kleine Klavierrnu-
fl (five-note teaching pieces), and his Opus 49, Concert Music fpr Piano, Ten
Brass Instruments, and Two Harps. The three piano sonatas of 1936 represent
that instrument in his series of sonatas, but also in this type are the Sonata for
3 Neumeyer, 260.
4 Melvin Berger, Guide to Sonatas: Musicpr One or Two lnst_rument§,
(London: Anchor Books, 1991), 111. See also Peter Evans, “Hindemith's Key-
board Music,†The Music Times 97 (1956), 573.
5 Denis Matthews, ed. KeyQard Music, (Great Britain: Penguin Books,
1972). 331-332.
89
Piano Four Hands published in 1939 and the Sonata for Two Pianos from
1942. The Ludus Tonalis from 1942 is a set of fugues and Interludes written
specifically to demonstrate his theory of tonal organization from The Craft. A
concerto for piano and orchestra published in 1948 completes his most Import-
ant works for piano.
Hindemith’s musical style, which is highly contrapuntal In nature and
“dependent on clear articulation of the rhythmic pulse,†is clearly reflected in his
piano music.6 Much of the piano music represents his philosophy of
Gebrauschmusik. It is often not technically challenging compared to other com-
posers, but it is musically challenging to create a convincing performance.7
Hindemith regarded writing for the keyboard as a compact and concise means
of expression, with the individual lines of contrapuntal activity being organized
by the harmonic progression.8
The Erste Sonate fur Klavier (1936) is no exception. Hindemith Is con-
cerned more with creating a well-designed formal structure that emphasizes
craftsmanship over expression.9 The sonata was influenced by the poem Q11
Mg by Friedrich HOIderlin. The poem speaks of a singer who Is left without a
homeland. The singer is traveling the world on a noble quest represented by
the temples of Athens, but the thought of the homeland expressed by the river Is
always in the mind of the traveler. Hope is regained as the traveler finds solace
in the company of others as the River Main also joins forces with The Rhine:
5 Matthews, 331.
7 Evans, 574.
3 Hindemith, Craft I, 111.
9 F. E. Kirby, A Short Histom of Kemuard Music, (New York: Schirmer
Books, 1966), 402.
90
The River Main10
True, on this living earth there are many lands
1 long to see, and over the hills at times
My heart runs off, my wishes wander
Seaward, and on to those shores which more than
All others that I know have been glorified;
But far away not one is as dear to me
As that where now the sons of gods lie
Sleeping, the moumful, the Hellenes’ country.
0 once I long to land there, on Sunium’s coast
Once ask my way to your columns, Olympion,
And soon, before the northern gale can
Bury you too In the scattered rubble
Of temples Athens raised, and their imaged gods;
For long now desolate you have stood, O pride
Of worlds that are no more! And 0 you
Lovely Ionian isles, where breezes
Walt coolnm to warm shores from the open sea
While under potent sunbeams the grape matures,
And, oh, where still a golden autumn
Turns Into songs the poor people’s sighing,
Now that their lemon grove, their pomegranate tree
That bends with purple fruit, and sweet wine and drum
And zither to the labyrinthine
Dance have allured them, however troubled-
To you, perhaps, you islands, yet one day shall
A homeless singer come; for he’s driven on
From stranger still to stranger, and the
Earth, the unbounded, alas, must serve him
In place of home and nation his whole life long,
And when he dies—but never, delightful Main,
Shall I forget you or your banks, the
Variously blessed, on my farthest travels.
Hospitably, though proud, you admitted me,
And, smoothly flowing, brightened the stranger’s eye
And taught me gently gliding songs, and
Taught me the strength that’s alive in silence.
O calmly as the stars move, you hmpy one,
You travel from your morning to evening,
Towards your brother, Rhine; then, with him,
Joyfully down to the greater ocean.
10Friederich HOIderlein, Poems and Fragments, trans. by Michael Ham-
burger (Ann Arbor: University of Michigan Press, 1967), 94-97.
91
Hindemith was traveling extensively at the time he wrote the three piano
sonatas, spending several months at a time In Turkey to organize a national
music school there. Perhaps the poem about the River Main In Germany
Inspired his sense of nationalism. Hindemith may have been reflecting upon
his conflict with the German government over freedom of musical and artistic
expression. He eventually left Germany along with many other artists and musi-
cians preceding the advent of the second world war.
Forrn-Defining Rhythms of Hindemith's
First Piano Sonata
The First Piano Sonata of 1936 by Paul Hindemith has been closely asso-
ciated with the compositional procedures of The Craft.11 The proximity of publi-
cation of the sonatas and the German edition of The Craft (1936 and 1937
respectively) might suggest that Hindemith demonstrated the new theory of
composition in concise forms through the genre of the piano sonata. However,
the set of fugues and interludes in Ludus Tonalis from 1942 more specifically
demonstrates the theory.
The first piano sonata is in five movements. Figure 17 shows the tonal plan
of the sonata. The overall tonality is In A (the first and last movements). Move-
ment one is a binary form with Section A in the tonality of A, followed by Section
B in the contrasting key of E. Movement two, a ternary form, uses C5 as the
tonal center. The third movement, the longest of the five, is in BI'. This move-
ment has been described as a “hybrid†rondo form with five main divisions
‘1 Flood, 4-5.
92
followed by a coda.12 The fourth movement repeats the binary structure of
movement one by reversing the two sections, modulating through the tonality of
D and eventually ending in E at the coda. The last movement, a six-part rondo
form, begins and ends in the tonality of A with the other formal sections in con-
trasting keys.
Mvt1 Mvt2 MV13 MV‘I4 Mvt5
lnAch 8 III 111 1'! III 51) <9 (I) <I>
Figure 17: Tonal Plan of Piano Sonata Number 1
The controlling force of Series 1 plays a significant role In the tonal organi-
zation of the sonata. The tonality of each movement sets up a goal of motion
which leads directly into subsequent movements. The three tonalities of the
opening two movements outline an A major sonority. Movement 1 begins In A
and closes in E, the dominant. The second movement begins and ends in C‘, a
third relationship to both tonal areas of the preceding movement. The third rela-
tionship clearly reinforces A as the tonal center of the composition by complet-
ing an A-major triad. Movement three Is harmonically the furthest removed from
‘2 Flood, 84.
93
the tonic; the key of BI! functions as the upper leading tone of A. It also func-
tions In the tritone relationship to the dominant E which closes the first and
fourth movements. The tritone/upper leading tone relationship creates the
greatest tonal tension midway through the composition.
In movement four the tension created by the remote key area of movement
three Is relaxed. The melodic material from the first movement is restated in dif-
ferent keys and in reverse order. Movement four begins in D, the second-most
stable key to the A. In a similar association of movement one to movement two,
movement three and four are related harmonically: the tonality of D stands in a
third relationship to the preceding movement, balancing the tonal motion of the
first four movements. The fourth movement ends in the dominant key E which
prepares for the return of the original tonic in movement five.
The approximate durations of each movement have been measured by
charting the metric structure and comparing the tempos. 13 The estimated dura-
tion is then calculated by determining the number of metric pulses in each sec-
tion and dividing by the number of beats per minute. The remainder is figured
as the percentage of the metronome marking which is multiplied by 60 to give
the remaining time in seconds. For example, movement one has a tempo mark-
ing of quarter note=96. The total number of quarter note pulses in movement
one is 172 at 96 per minute. 172 divided by 96 is 1 with a remainder of 79. The
remainder 79 beats is 82% of 96 beats per minute. 82% of one minute (sixty
seconds) is 49 seconds. The approximate performance time or duration of
movement one Is 1' 49".
A comparison of the approximate durations of each movement is given in
Table 6. The movement, section, measure numbers, metronome marking, and
the number of beats or pulses in each section are given in the first five columns.
13The metric structure of each movement is given in Appendix C.
94
The estimated performance times and percentages of the total are given in the
last three columns of the table.
Table 6
Estimated Durations of the First Piano Sonata
Movement Section Measure # NM Pulses Est. Tlme % % Total
Mvt 1 A 1-22a 96 78 0' 49" 45%
B 22b-39a " 61 0' 39" 36%
Coda 39b-51 " 33 0' 21 " 19%
1' 49" 8.49%
Mvt 2 A 1-26a 50 99 1' 59" 32%
B 26b-598 72 131 1'50" 29%
A' 59b-89 50 1 18 2' 22" 39%
6' 11" 28.91%
Mvt 3 A 0-59 168 179 1' 05" 20%
B 60-1 12 " 1 63 0' 59" 1 9%
C 113-216a 72( 216) 310 1' 27" 27%
Retrans. 21 6b-2558 1 68 1 1 0 0' 40" 12%
A' 255b-296 " 124 0' 45" 14%
Coda 296-321 " 74 0' 27" 8%
5' 23" 25.1 7%
Mvt 4 B‘ 0-23a 96 60 0' 38" 35%
(Mvt 1) A' 23b-468 " 81 0' 51" 46%
Coda 46b-58 " 33 0' 21 " 1 9%
1' 50" 8.57%
Mvt 5 A 1-27 120 79 0' 40" 11%
B 28-45a " 49 0' 25" 7%
C 45b-94 " 121 1' 01" 16%
D 95-155 1 12 172 1' 33" 25%
B' 1 56-1 92 1 20 1 03 0' 52" 1 4%
A' 1 93-237 " 125 1' 03" 1 7%
Coda 238-280 1 68 1 00 0' 36" 1 0%
6' 10" 28.83%
Total: 21 ' 23"
95
The durational relationship of the individual movements to the entire struc-
ture of the sonata is revealed in the estimated performance times. A compari-
son of all five movements reveals a well-proportioned compositional plan for the
entire sonata.
The two shortest movements, one and four, are of nearly equal length:
1’ 49" and 1’ 50" respectively. The similarity is not surprising as they have the
same basic structure: movement four repeats the transposed melodic material
from movement one in reverse order. However this equality is odd seeing that
the total number of measures is 51 for the first and 58 for the second.
The disparity arises from the second theme. In movement one, the B theme
consists of sixty quarter note pulses within 17 measures. Movement four has
the same material with sixty quarter note pulses in 23 measures. The additional
measures are due to the B’ theme stated In alternating 3/4 and 2/4 measures
(creating eight hypermeasures of 5/4). This would normally add eight additional
pulses to the section, however these are offset by omitting nine pulses from the
restatement of the thematic material. (Measure 31-32 and the last beat of meas-
ure 39 from movement one are omitted in movement four.) The result is an
equal number of pulses for the restatement of the B theme in movement four.
Movements one and four are linked to the subsequent movements with the
performance direction nach kurzer Pause anschlieBen: “continue after a slight
pause.“ They serve as preludes to the much longer second and fourth move-
ments. The linking of the outer movements emphasizes the balanced structure
of the sonata. The two pairs of outer movements each have a combined
approximate duration of 8’ 00" taken in the strictest metronomic measurement.
Movement three, the middle of the five movements, is 5’ 23", almost exactly 33%
shorter than each of the combined outer movements, giving a ratio of 1.5: 1: 1.5
(323 seconds into 480 seconds yields 1.486, or 1.5.) In terms of duration, this
96
creates an arch-type temporal structure that should be clearly represented in
performance.
The estimated durations of Table 6 also reveal objective criteria for interpret-
ing or evaluating a performance of the sonata. Even considering Hindemith’s
original markings to be the extreme ranges of tempo, Glenn Gould’s times are
greatly out of proportion with those indicated by the original tempo markings.14
Table 7 compares the estimated durations from the score with the durations
of three recordings. It is seen that Gould’s recording is over seven minutes
slower in total performance time than that required by the score (25’ 40" com-
pared to Gould’s 32’ 31"). Badura-Skoda15 and Roberts15 are within reason-
able performance times.
An extreme example of the disproportionate Interpretation of Gould’s perfor-
mance is in movement three. The marking Etwas ruhiger (more quietly) at the
beginning of the second theme does not necessarily mean a change of tempo
as much as a change of mood (one is neither inclusive nor exclusive of the
other). Hindemith is following the classical rondo form where a lively and brisk
first theme is followed by a more lyrical and melodic second theme. The quiet-
ness Is achieved by the smoother rhythm of the new melodic line. A slight
change of tempo may be called for, but Gould allows too much contrast of
tempo between the two themes.
14Paul Hindemith, Hindemith, The Three Piano Sonatas Glenn Gould, pia-
no, Sony Classical CD SMK 52670. Performance date October 13, 1966.
15“Paul Badura-Skoda Plays Hindemith, Piano Sonatas nos. 1 and 3,â€
Westminster XWN 18200, 1956.
15 Bernard Roberts with David Strong, “Hindemith, Music for One and Two
Pianos,“ Nimbus Records, compact disc NI 5459/60, 1996.
97
Table 7
A Comparison of Performance Times
of the First Piano Sonata
Mvt | Section [ Measure # Est. Time Gould I Badura- l Roberts
from MM Skoda
W1 7 A 1-22a 049" 1'22" 0'51" 051'
B 22b-39a 0' 39" 1' 05" 0' 39" 0' 46"
Coda 39b-51 0' 21" 0' 45" 0' 28" 0' 29"
Total: 1' 49" 3' 42" 1' 58" 2' 06"
Mvt 2 A 1-26a 1' 59" 3' 20" 2' 15" 2' 13"
B 26b-598 1'50" 2' 33" 1' 51" 1' 45"
A' 59b-89 2' 22" 3' 37" 2' 42" 2' 56"
Total: 6' 11" 9' 30" 6' 48" 6' 54"
Mvt3 A 0-59 1' 05" 1' 30" 1' 10" 1' 12"
B 60-112 0'59" 0' 96" 1' 13" 1' 18"
C 113-216a 1' 27" 1'54" 1' 37" 1' 41"
Trans. 216b-255a 0‘ 40' 0' 55" 0' 41" 0' 44"
A' 255b-296 0' 45" 1' 03" 0' 51" 0' 57"
Coda 296-321 0' 27" 0' 54" 0' 38" 0' 44"
Total: 5' 23" 7' 52" 6' 10" 6' 36"
Mvt 4 B' 0-23a 0' 38" 1' 10" 0' 40" 0' 48"
A' 23b-46a 0' 51" 1' 15" 0' 54" 0' 56"
Coda 46b-58 0' 21" 0' 58" 0' 28" 0' 31"
Total: 1' 50" 3' 23" 2' 02" 2' 15"
Mvt 5 A 1-27 0' 40" 0' 44" 0' 45" 0' 47"
B 28-45a 0' 25" 0' 25" 0' 27" 0' 28"
C 45b-94 1'01" 1' 13" 1'13" 1' 16"
D 95-155 1' 33" 2'04" 1' 44" 1' 58"
8' 156-192 0' 52" 1' 21" 0' 58" 1' 01"
A' 193-237 1'03" 1'21" 1' 12" 1' 20"
Coda 238-280 0' 36" 0' 56" 0' 48" 0' 50"
Total: 6' 10" 8' 04" 7' 07" 7' 40"
Total: 21' 23" 32' 31" 24' 05" 25' 31"
98
The disregard for Hindemith’s tempo and proportional indications may be one
reason Gould’s performance has been called into question:
Such performances exchange the vigor, interest, and lyricism of detail within
a broadly proportioned, readily understandable formal frame for placidness,
aridIty, and a sad predictability that is entirely at odds with Hindemith’s con-
ception of music.17
The meters and durational lengths of the individual movements need to be
addressed as these provide the background upon which the surface rhythms
are placed. Formal divisions in the sonata are clearly defined. In addition to
cadences, other parameters of music such as changes of tempo, texture, and
registration help distinguish formal divisions. The strongest cadences occur at
the ends of the larger sections.
In movements one and four the basic pulse Is 96 quarter notes per minute.
No time signature is given in the score, but there are four strict quarter notes per
measure in the accompaniment pattern of the first four measures of movement
one, clearly establishing 4\4 time (See Figure 11, page 50). In addition, the
performance direction Ruhig bewegte Viertel calls for a strict quarter note
emphasis. The performance direction for movement four calls for a return to the
same tempo and pulse as movement one (Ruhig bewegte Vlertel, wIe im ersten
M). The quarter note pulse Is retained throughout both movements; the alter-
nating 2\4 and 3\4 meter has been previously discussed. Also, 2\4 and 3\4
measures are inserted occasionally during the restatement of the A’ theme. The
coda In movements one and four contains two measures of 1\4 which serve as
anacrusis beats. These measures could easily have been notated as beat four
of the preceding measures or beat one of the subsequent measures. The one-
beat measures help clearly define the three phrases of the coda, however.
The total estimated durations of movements two and five are nearly identical.
17 Neumeyer, The Music of Paul Hindemith 17.
iilltlllll. II t
99
The differences between them arise in the tempos, thematic material, harmonic
organization, and form. Movement two is a slow march in ternary form with con-
trasting wctions in 4\4 and 12\8 meters; movement five is a fast rondo in com-
pound triple and simple triple meters. Both of these movements use changing
meters on a limited basis but always in keeping with the over-riding meter. In
this sonata Hindemith never changes from compound to simple meters within a
section although contrasting sections may do so.
The first theme of movement two, measures 1-26a, retains a quarter note
pulse but the tempo is slowed to 50 beats per minute. The meter is designated
as 4\4. There is one measure of 2\4 inserted about midway through the section.
The B theme from measures 26b to 59 is in 12\8 meter throughout except for
the last measure of the section which is in 6\8. The tempo is marked as dotted-
quarter note = 72 beats per minute. The contrast between the two themes
comes in a change of mood to a livelier melody and compound meter. With 99
pulses for the first theme and 131 pulses for the second theme, the estimated
performance times are 1’ 59" and 1’ 50", a ratio of almost 1:1.
The return of A extends from measure 59b to measure 89 and completes the
ternary form. A’ is 118 pulses instead of 99, an increase in duration of 16%.
The increase is due to the avoidance of the expected cadence in the ninth bar
of the return, the phrase extended by repeating the Opening motive of the theme
with variation. Hindemith also attaches an additional coda to the end of the A’
section.
Movement three is perhaps the most interesting of the five movements. It
has four main thematic sections. Since transitional material relies extensively
on motives from the themes, these have been included as part of the total
duration of the themes. The A theme extends from measure 0 to 59 and con-
sists of 179 pulses of quarter notes at 168 beats per minute. The meter of A is
100
3\4 throughout the section. The estimated time of performance is 1’ 05â€.
The B theme of movement 3 retains the quarter note pulse and tempo as dis-
cussed above and extends from measure 60 to measure 112. The 163 pulses
give an estimated performance time of 0’ 59". There are also several changes
of meter from 3\4 to 4\4 and back again. Once again, the ratio of A to B is almost
exactly 1:1.
The third theme of movement three is more complex than A or B. In the C
theme the dotted-half-note pulse of 72 beats per minute increases the tempo to
216 quarter notes per minute. The pulse, however, should be felt in compound
time. The theme spans measures 113 to 216 and is the longest section of the
movement, being an estimated 1’ 27†in duration. The fast tempo is interrupted
by two short sections with the performance directions of Ein wenig breiter in
measures 146 to 160 and einleiten...Breiter in measures 195 to 197. These
terms “more broadly†and "directly...broader†call for highly subjective changes
of tempo, but in all likelihood could increase the estimated performance time of
this section by as much as 30 seconds or more. This specific section with its
temporal ambiguity lies just beyond the halfway point of the entire sonata in
6 terms of the total estimated time. After a short retransition based upon theme A,
the true reprise of A begins in measure 255 and is shortened considerably to
124 pulses, or 0’ 45â€. A short coda recalls earlier material from sections B and
A.
Movement five is a six-part rondo form. As stated earlier, the total estimated
time for the fifth movement is 6’ 10". Sections A, B, and C each use a com-
pound triple meter of three dotted-half notes per measure at 120 beats per
minute. The estimated performance times for each of these sections are 0’ 40â€,
0’ 25â€, and 1’ 01†respectively. Each section also changes meters from triple
compound to duple compound for only one or two measures at a time, but
1 01
Section C has 11 meter changes.
In contrast, section D of movement five, from measure 95 to 155, is in 3\2
meter. It is also the longest section of the movement with 172 pulses at 112
beats per minute for an estimated time of 1’33â€, or 25% of the movement. 8’
and A’ restate earlier material. In the return, however, each of these is consider-
ably longer in duration.
In reviewing the durations of larger formal units of the sonata, several factors
become apparent. First of all, Hindemith presents a durational arch form with
the outer movements linked to create a 1.5: 1: 1.5 relationship to the middle
movement. This interior movement is also metrically the most interesting as it
presents a certain level of temporal ambiguity just beyond the midpoint of the
sonata. Hindemith’s use of meters in this sonata is somewhat traditional. He
maintains the same type of pulse within sections (as in compound and simple
beat divisions) but does allow limited changes of meter within sections. Some
of the meter changes give actual meter signatures but most rely only on dura-
tional values within the measures to identify the change. Finally, the tempos
given in the score as metronome markings seem to be on the extreme fast or
slow setting. Relationships of durational proportions of each movement and
sections should be maintained throughout for the best interpretation of the
sonata.
Analysis of surface rhythms can explain the differences in durations of each
section and how each section is extended or shortened upon restatement. Ana-
lyzing the surface rhythms also explains the use of motivic material in specific
rhythmic gestures.
The following analysis will present a detailed examination of the surface
rhythms of movement one, which will demonstrate the proposed method of
analysis. Four areas will be addressed in performing the detailed analysis of
102
the first movement. The movement will be segmented using the grouping
parameters discussed in the previous chapter. Longer groups such as sections
and phrases will be identified by length and the tonal centers of each phrase
will be identified. Secondly, note durations used in each phrase unit will be tab-
ulated. The third area will identify how the pitch durations are grouped into
characteristic rhythmic units and how these are used within the longer sections.
Patterns will be described according to Winold's criteria as m (even-note
patterns), lntrametric (confirming patterns), Contrametric (nonconfirming, or
syncopated patterns), or Extrametric (irregular) pattems.18
The Phrase Rhythms of Movement One
Movement one is a binary form. The tempo is 96 quarter notes per minute.
Measures 1-22a constitute Section A, measures 22b-39a Section B, and meas-
ures 39b-51 form a coda. The phrase durations are given in Table 8. It is
noticed that Section A lasts 0’ 49â€, Section B is 0’ 39â€, and the coda is 0’ 21â€,
totaling 1’ 49" in duration. The entire movement with harmonic analysis is
reproduced in Appendix D.
18These are cited previously and explained in more detail in Chapter 3. -
1 03
Table 8
Phrase Durations of Movement 1
NM Measure # Beats Est. Time
Section A
Phrase a 96 1-4a 1 5 0' 9"
b 4b-10 22 0' 14"
a' 11-16a 20 0' 13"
a" 1 6b-22a 21 0' 13"
78 0' 49"
Section B
Phrase d 22b-26a 16 0' 10.2â€
d' 26b-30a 16 0' 10.2â€
d' 30b-39a 29 0' 18"
61 0' 38.4"
Coda (0' 39")
Phrase 9 39b-42 10 0' 6"
9' 43-46 10 0' 6"
a†47-51 1 3 0' 9"
33 0' 21"
Total 1' 49"
Figure 18 plots the tonal plan of Movement 1. The A section begins with a
period structure comprised of two phrases. The antecedent starts on the down-
beat of measure 1 and ends in a cadence on B on the third beat of measure 4.
The total length of the a phrase is 15 beats. Phrase b, the consequent, begins
with the anacrusis to measure 5 and ends in measure 10 with a cadence on 0’,
resulting in 22 beats. Following this cadence, the melodic material from meas-
ure 1 is restated an octave lower in measure 11 beginning phrase a’.
104
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105
This restatement is less conclusive but comes to a partial closure on beat 2 of
measure 16 with the subdominant pitch in the bass voice after 20 beats. The
final phrase, aâ€, extends from the anacrusis to measure 17 to the cadence in the
tonality of A on beat 4 of measure 21.19
Section B consists of a three-phrase group that reiterates the same melodic
material in varied form. The first phrase, labeled d, is 16 beats long, from the
anacrusis to measure 23 to measure 26a. The tonality begins on E minor and
moves to its tritone 85 in measure 26. Phrase d’ moves the same melodic mate-
rial one octave higher in measures 27—30 with a slight change at the end of the
phrase. The ending of the phrase is altered slightly to be less conclusive by
repeating the motive f‘2, e52, a“? three times. The b" from measure 25 is
spelled as A“ in this phrase. The d’ phrase is 16 beats long.
The third phrase of Section B, dâ€, begins with the anacrusis to measure 31.
The third phrase is 29 beats, being extended to measure 31 with a cadential
rhythmic pattern that diverges from the previously established rhythms of the
section. The coda twice repeats a 10-beat phrase in transposition, ending the
movement in measure 51 with a cadence in E major.
The durational values of the upper voice of the two-voice framework of
movement one are listed in Table 9. Hindemith is very consistent in using indi-
vidual durational values. As a general rule In this movement, durations are met-
rically organized; longer note values occur on the first and third beats of the
measure, shorter notes in groups of two on the second and fourth beats.
‘9 A different interpretation of the phrase structure is that the a†phrase
serves as a different consequent to a’, maintaining the basic period structure of
the opening. However, the inconclusive nature of a’ may provide an alternative
interpretation . This second phrase might be heard as an extension of a’ rather
than a new phrase because of the weak conclusion of a’ in measure 16. In my
opinion the grouping of the larger phrases lends itself to calling a†an entirely
new phrase that is based on the same motivic material.
106
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107
Extrametrical note values (triplets, in this case) also occur only in unaccented
metric position, although the longer quarter-note triplets in measure 19 contrib-
ute to the opposition of the metric pulse in phrase a†of Section A.
The eighth note is the predominant note value used in the upper voice of
the two—voice framework of movement one and helps to confirm and maintain
the quadruple simple meter. Of the forty-two eighth notes, 24 occur in pairs.
Three of those pairs are on a metrically strong beat of the measure and six are
in a metrically weak position. Single eighth notes occur as anacrusis beats fol-
lowing a rest 7 times while 10 occur after a dotted quarter note. One occurs with
a quarter note syncopation. The eighth notes of Section B and coda are more
frequent, 66 as opposed to 42. The greater frequency of shorter note values
creates a faster rate of melodic motion in Section B than Section A, even though
the metric tempo remains constant. The eighth notes in Section B always occur
on beats two or four in a similar fashion to Section A. They are always grouped
in pairs or singly follow dotted quarter notes. The coda uses three eighth-note
values following an eighth rest in the three measures of 1\4 meter. These are
construed as anacrusis beats that divide the coda into three distinct phrases.
The second most frequent note value is the quarter note, occurring 52 times
in movement one (23 in A and 29 in B plus the Coda). As mentioned earlier, the
meter is established in the first phrase by the strict quarter notes of the accom-
paniment and the duple division of the beat in the melody of measure one.
The meter is also confirmed by the longer note values that occur on metri-
cally accented beats. Of the longer note values in the movement, in the upper
voice of the two-voice framework 37 quarter notes occur on metrically accented
beats while 11 are on weaker beats. Only one of the half notes occurs in a met-
rically weak position. Fifteen of the dotted quarter note values occur on strong
beats while 6 are metrically unaccented.
1 08
In this movement, Hindemith uses note values in a very regular way. Longer
note values tend to occur on strong beats while shorter note values are on
weaker beats. Analyzing the surface rhythms gives detailed explanations of the
similarities and differences between unified groups and how a composer uses
unified groups to expand and enlarge the rhythmic structure. Hindemith uses
metric dissonance in several ways and for different purposes in the first move-
ment of the piano sonata.
The Unified Groups of Section A
Figure 19 identifies the rhythmic units of measures one through ten and com-
pares them to the metric structure. It is important to note that the rhythmic units
are organized around the barlines. Most of the rhythmic units in this phrase
help establish and confirm the basic metric pulse of the movement by agogic
accents on the downbeats of the measures. The seven confirming units, in
measures one, three, four, five, six, and seven, use dotted quarter notes, quarter
notes and eighth notes. The longer note values occur on the strong beats of the
measures except for the paired eighth notes in measure three. Two predomi-
nantly metric groups occur in measures two and nine.
The most interesting rhythmic group in Section A occurs in measure eight.
Measure eight is identified as a contrametric pattern with extrametric subdivi-
sion at the first subdivision level: the change of meter and the syncopation work
against the established quadruple pulse and the triplet eighth notes alter the
normalized duple subdivision of the pulse. It was discussed previously that the
downbeat of measure eight would be the traditional place for the cadential
chord to end this phrase. The deceptive cadence at this point arises by the
109
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tension of the tritone chord of Group mm on the downbeat of measure eight.
The syncopation of the rhythm and change to triple meter also creates effective
motion away from a cadence. A tonic accent by leap upward to be as well as a
thinning of the texture to a single melodic line continues to emphasize measure
eight. Hindemith then adds a two-bar extension to the regular eight-bar struc-
ture to end the phrase.
Two patterns in the rhythmic groups stand out as more prominent than the
others. These two patterns help to unify the first part of Section A, and also
become motives for expansion in the second part of Section A. The two pat-
terns are given in Figure 20. The opening figure of measure one, identified as
Pattern N, occurs later in measures five and six. Pattern O with the characteris-
tic dotted quarter note, is also present in measures three, four, seven, and nine.
It also occurs in the lower voice of the two-voice framework in measure three.
This pattern is also varied (Pattern O-1) in measure eight to become the contra-
metric rhythmic group of the deceptive cadence.
The rhythmic units are combined to make up the four rhythmic gestures of
measures one through ten. The first rhythmic gesture has a thetic or strong-beat
beginning and closes with an upbeat ending in measure two. The second
rhythmic gesture has an anacrustic beginning and weak-beat ending, although
the ending might also be construed as strong because it ends on the secondary
metric accent, the third beat. The remaining two rhythmic gestures both have
anacrustic beginnings, the first ending on another secondary metric accent in
measure six. The final rhythmic gesture closes with a strong downbeat ending.
The harmonic rhythm of measures one through ten is quite rapid. The har—
monic fluctuation consists mostly of non-tritone chords of Group A, Subgroup III
(refer to the analysis in Appendix C). The roots and quality of the chords
111
ml m. .I M 95,]
Pattern N, Pattern 0, Pattern O-1,
m.1,5,6,11 m.3,4,7,9 m.8
Pattern P, Pattern Q,
m. 12,14 m. 13, 15,16
Figure 20: Unifying Patterns of Section A
change on almost every beat. The few tritone chords of Group B that occur in
this section do so on accented beats: beat one and three of measure two and
beat one of measure eight. The least dissonant chords of Group I are also met-
rically accented as they occur in successions of more dissonant chords.
Measures 11-22 comprise the restatement of the opening and consists of
two phrases: a’ and aâ€. Exact duplication is avoided and development or
extension of the opening phrase is achieved by repetition and variation, two of
the compositional procedures Hindemith identified in Elementam Tra_ining for
Musicians.20 A salient characteristic of Hindemith’s style is the organic nature
of the musical development. Later phrases grow from previously heard material
at least in the first movement of this piano sonata.
Figure 21 identifies the rhythmic units of measures eleven through twenty-
two and compares them to the metric structure. Phrase a’ in measures eleven
2° Hindemith, Elementam Training, 158-159.
112
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1 13
to sixteen consists of six rhythmic units using three different patterns. Pattern N,
the confirming pattern from measure one, begins the restatement of the opening
theme in measure eleven one octave lower than measure one.
Pattern P in measures twelve and fourteen is a varied form of the metric pat-
tern from measure two of the original statement (see Figure 20). It is four beats
long and emphasizes the last note of the pattern. This agogic accent falls on
the third beat of the quadruple meter in measures twelve and fourteen. Pattern
P alternates with a new rhythm pattern (three eighth notes followed by a dotted
quarter or quarter note) identified as pattern Q. In contrast, pattern O is three
beats long and also emphasizes the last note of the pattern which falls on the
second beat of a triple-beat measure each time. The alternation of P and O
combined with the rising pitch of the melody in a sequential repetition of Q in
measures creates momentum which leads to a tentative resting point on the
subdominant on beat two of measure sixteen. The repetition of P and O also
extends the phrase to six bars instead of four.
Measure seventeen begins the final phrase of Section A. In phrase aâ€, pat-
tern N is varied rhythmically while retaining the melodic interval content of
measure one. The motive c132 - e2 - b1 - a1 in measure one is duplicated as d 2 -
f2- 02- bb1 in measures seventeen and eighteen. This final statement of the
motive changes to triple meter which emphasizes the second note of the group
and adds two beats to it. Phrase a†continues with the metric Pattern P followed
by two extrametric patterns of quarter note triplets divided by Pattern Q.
The varied repetition of phrase a as a’ and a†creates the form of Section A.
While the restatements of the thematic material unify the section, especially Pat-
terns N, P and Q, the compositional devices of repetition and variation are used
to extend the phrases and move the musical ideas forward. The strong
1 14
quarter-note pulse of the movement is obscured in the last two phrases of the
section. Metric ambiguity arises in the alternation of quadruple and triple met-
ers in phrase a’ and the change from triple to quadruple meter in phrase aâ€.
The accent is shifted to the second and third beats of the measures as the
rhythm patterns carry through the barlines. Extrametric groups at the first multi-
ple level (the quarter-note triplets in measures nineteen and twenty-one) help to
slow the momentum and blur the metric pulse. Finally, the conclusion of the
section on a weak beat in measure twenty-one further obscures the strength of
the meter.
The Unified Groups of Section B
The unified groups of Section B are built around Pattern N, the same pattern
which organizes the groups of Section A (Figure 19). The primary difference is
the use of an anacrusis beat before each occurrence of Pattern N (See Figure
22). The anacrusis consists of confirming eighth notes in five occurrences and
an extrametric subdivision in the other three; otherwise the pattern confirms the
metric pulse on the first subdivision level, the beat level, and the multiple levels
of the half measure and measure. The addition of the upbeat disguises Pattern
N by dividing the unified groups into two balanced parts aligned so that the
longer notes fall on the strong beats of the measures.
An interesting feature of the first two rhythmic gestures of Section B is the
interaction between the pitch content of the upper voice of the Two-Voice frame-
work and the unified groups. The melody is linked to the second beat of each
measure in an ascending step progression beginning on d1 in measure
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116
twenty-three. (The music is reproduced in Appendix C.) Each repetition of Pat-
tern N sees a rise of a major second at the same point in the rhythmic group:
beat two of measure twenty-four has e1 and beat two of measure twenty-five
has 1131, the melodic high point of phrase d. The rise in the melodic contour is
counterbalanced by static motion created by alternating b and b" on beats one
and three of these measures.21 Phrase (1’ in measures twenty-seven through
thirty repeats these melodic features one octave higher. The pitch a’1 on the
third beat of measure twenty-nine substitutes for the DH on the third beat of
measure twenty-five. Also, a change of rhythm in the last two bars of phrase (1’
increases the momentum by three consecutive statements of the last half of Pat-
tern N (the eighth-eighth-quarter note figure on 1‘2. e132. and a’1 in measures
twenty-nine and thirty).
The key of E is firmly established by the lowest voice with a pedal point E1
on the fourth beat of measures twenty-two, twenty-three, and twenty-four and
measures twenty-six, twenty-seven and twenty-eight. The bass voice also
creates a metric cross rhythm in measures twenty-three through twenty-five and
on octave higher in twenty-seven through twenty-nine. Beat 2 is emphasized
with agogic and pitch accents in an ascending minor second step progression
of c, câ€, and d that is paired with similar motion in the upper voice on d1, e1, and
151. The linear motion of the two-voice framework obscures the chord progres-
sion of the harmonic fluctuation in phrase d and d’, which mostly consists of
chords of groups I and III.
2‘ Neumeyer identifies similarities between measures one through four and
measures twenty-three through twenty-six in a structural analysis. The back-
ground structure reveals a similar neighbor-note motive used in both opening
phrases of Sections A and B of this movement. See Neumeyer, The Music of
flul Hindemith, 200-203.
117
Phrase d†is almost twice as long as the two previous phrases. The length-
ening of the rhythmic gesture is caused by the addition of a new rhythm
pattern—a series of eighth notes in contrary motion in parallel thirds and fifths.
This pattern creates two hypermeasures of 6 beats that begin with the anacrusis
to measure thirty-three?2 Along with this new rhythm pattern the harmonic fluc-
tuation here is the most active in Section B, which causes the greatest area of
harmonic tension in the movement. The phrase closes with a repeated caden-
tial pattern over a pedal point on C.
The Unified Groups of the Coda
The coda consists of three statements of a three-stage sequence of a closing
theme. (Figure 23) Each stage of the sequence begins a diminished fourth or
major third lower than the previous statement which gradually moves the tonal
center to the key of E. The ending of the last phrase is altered to provide two
cadential chords. The last motive is changed to a dotted-half note and an addi-
tional measure added for the cadential chord in measure fifty-one. The total
duration of 33 pulses or approximately 0’ 21†is achieved by exact repetition of
the rhythm. This accounts for the similar durations of each phrase shown in
Table 8 on page 104. Change occurs, however, in the pitch level and harmony.
The harmony consists primarily of chords of Group A subgroup 111, but the final
chord of the first two phrases are tritone chords of Group B subgroup IV and II.
The first phrase of the coda begins in A†and closes in F. The harmonic fluc-
tuation increases through the second phrase with three tritone chords, two of
‘
22 Flood, 77.
118
Figure 23: Movement One, Coda.
(Hindemith, Erste Sonate flJr Klavier, 1936. ©B.
Schott's Soehne, Mainz, 1936. © renewed. All
Rights Reserved. Used by permission of European
American Music Distributors Corporation, sole US.
and Canadian agent for B. Schott's Soehne, Mainz.)
Harmonic Fluctuation [III [Ill 1111 [III IVl
Tonal Funcu'on AI): 4) II (1) VII VI
8: t
x 1 y 1 z————————|
W
1111 II IIIl Ilbl IVl
1111 ® 1 [III nach 11 [Pause
v] anschlieBen
VII 0
Figure 23: Movement One, Coda
120
which occur in metrically strong positions on the downbeats of measures forty-
five and forty-six. The second phrase is also harmonically the most unstable as
it emphasizes the tritone motion 8 to B". This recalls the harmonic relationship
in Phrase d in Section B which begins in E and ends on 8" also. The third
phrase begins in C and closes the movement in E.
The predominating meter of the coda is 3\4, but the question is raised as to
why Hindemith chose to notate measures forty-three and forty-seven in 1\4 met-
er. (The final beat of measure thirty-nine serves the same function as measures
forty-three and forty-seven.) It should also be noted that there is no change of
meter signature to identify the one-beat measures. It was observed previously
that these measures help to separate the three phrases of the coda. A more
conclusive explanation lies in the combination and types of shorter unified
groups. In the coda, the rhythm works against the established pulse by extra-
metric patterns at the first subdivision level (triplet eighth notes) and by dis-
placed accents at the beat level to establish a change of meter from triple to
duple and back.
At the primary beat level, the sequential phrase of the coda is comprised of a
contrametric pattern with extrametric subdivision, a metric pattern, and a con-
firming pattern, labeled as motives x, y, and 2 respectively in Figure 24. Motive
x, the contrametric pattern in measures forty, forty-four, and forty-eight, works
against the established triple meter. The motive can be separated into two dis-
tinct parts: the single eighth-note/triplet pattern serves as an upbeat to the dot-
ted-quarter note, creating the agogic accent on beat two of the notated meas-
ure. This longer note is also harmonically accented. It is the root of the
descending melodic degree progression (A b, E, and C in measures forty, forty-
four, and forty-eight). The off-beat syncopation caused by the displaced accents
121
Rhythmic
(3631111683 (anacrustic) (weak) (anacrustic)
Rhythmic ' ' ' 1
Units: x (contrametric) (meme) 1 z(60111311111118) X 1 y
, 1 2
- l
r r.
established I 1 2
meter l
Figure 24: Unified Groups of Movement 1, Coda
122
creates the feeling of 2 + 2 beats instead of 1 + 3 beats in motive x. The contra-
metric pattern of motive x functions as an upbeat to motive y which reaffirms the
triple meter with its even quarter notes. The additional length of the final phrase
is caused by the cadence, which continues the expected melodic motion at the
beginning of motive 2 but is replaced by a dotted half note value over a chord
from group 1111 moving to a chord from group I.
Chapter 5
Summary of the Research Findings
In this paper, I have presented the necessary information to answer the
research questions and research problem presented in Chapter One. Chapter
Two presented Hindemith’s theories of melody and harmony from The Craft.
Hindemith’s views of rhythm were taken from primary sources written by Hinde-
mith. In Chapter Three, five models of rhythmic analysis were reviewed for com-
parison to Hindemith’s views. A model of rhythmic analysis was demonstrated
in Chapter Four. Finally, in this last chapter I will provide answers to the stated
research questions presented in Chapter One. I will also discuss areas of con-
tinuing research related to the current topic. I will begin with the collateral ques-
tions first, then answer the primary research question: Does Hindemith’s use of
rhythm in his musical compositions reflect his explanations of rhythm in the the-
oretical writings?
Answer to Collateral Question 1
What is the relationship of melody, harmony, and rhythm in The Craft of
Musical Composition and Hindemith's other writings?
Hindemith identified melody, harmony, and rhythm as the three primary ele-
ments of music. According to him, dynamics, tone-color, articulation, timbre,
123
124
and the other parameters of music do not affect the construction of the music,
only the listener’s perception of it.1 As such they are subordinate to melody,
harmony, and rhythm.
In terms of the pitch content, Hindemith’s appeal to a natural order in T_h_e_
Cgft of Mtgical Composition helped him arrive at the basic premise of his theo-
ry: the necessity of tonal centricity.2 This allowed the development of Series
One and Two as the organizational forces of the pitch content and so the Two-
Voice Framework, Degree Progression, Harmonic Fluctuation, and Step Pro-
gression became the tools for planning the harmonic and melodic forces of a
work. Unfortunately, the temporal aspect of music did not lend itself to the same
natural justifications as the tonal aspect. This may be one reason for the lack of
a well-defined theory of rhythm on Hindemith’s part. The pedagogical reasons
for the omission of rhythm in his theory do not excuse his criticism of other theo-
rists for the same fault.3
It was discovered that Hindemith’s interest in developing a theory of rhythm
was in teaching composition.4 While he stressed the interaction of the three
elements, each has a different function in the compositional process. I cite Hin-
demith once again:
Rhythm determines the duration of the chords, and groups them by division
into stressed and unstressed members of the structure. Melody in voice-
leading regulates linear expansion, and in the two-voice framework sets the
pitch limits. In the placing of the harmonic center of gravity and in the regula-
tion of relationships we see harmonic energy at work.5
1 Hindemith, “Methods of Music Theory,†21 .
2 Hindemith, Craft l, 9.
3 Hindemith, Craft I, 110, 179; Elementary Training for Musicians 157; Craft
m, 30, Translation, 21-22.
4 Hindemith, Elementary Training, 157.
5 Hindemith, Craft l,109.
125
The two distinctions of rhythm which Hindemith recognized are duration and
form.6 The primary organizational factor of rhythm itself is meter, which is
grouped in units of two’s and three’s and their compounds. Grouping is deter-
mined by the metric accent. In organizing surface rhythms, meter governs the
placement and duration of individual pitches and harmonies. It also determines
the divisions of melody into motives, phrases, and sections.
It was stated in Chapter Two that Hindemith’s combined references to
rhythm do not constitute a well-developed theory in comparison to his theory of
melody and harmony from The Craft. Hindemith made a rather modest contri-
bution to rhythmic theory.7
Answer to Collateral Question 2
What are the similarities and differences between Hindemith’s concept of
rhythm and other theories of rhythm?
Hindemith was dissatisfied with the general theories of rhythm that he knew.
He understood the analytical emphasis of theories presented by Hauptmann
and Riemann, but he wanted to develop a compositional method of rhythm.8
Even though Hindemith’s theories of harmony and melody provide new insights
into the organization of those elements, his understanding of rhythm is quite
conventional, even orthodox.
The surface rhythms are organized around the meter and metric accent, and
6 Hindemith, Eflementarv Training for Mgsicia_n_s, xii; Craft I, 13-14.
7Neumeyer, “Tonal, Formal, Proportional Design,†93.
8 Hindemith, “Methods of Music Theory,†24.
126
so Hindemith follows Weber, Hauptmann, and Reimann in the organizational
use of the barline.9 The metric background helps organize the durational val-
ues used in the rhythmic groups. Longer note values tend to fall on the
accented beats of the measures. Shorter note values tend to follow longer
notes except when used as upbeats.
Hindemith was able to present some useful ideas concerning musical com-
position and the construction of musical forms. He tried to make a connection
between the surface elements and the longer formal sections. Making composi-
tional decisions and creating a compositional map in the charts and diagrams of
the elements of music was an important step in defining the rhythmic elements
of form. It helps the composer to apply other procedures such as repetition, var-
iation, and change within the context of the larger forms. Also, knowing the
tonal relations in advance allowed him to plot the areas of greatest harmonic
tension and harmonic fluctuation.
As for formal constructs, Hindemith tried to make each composition unique
and individual. He experimented with the use of proportions but did not apply
the principle consistently in his music.10 Balance in his music is achieved
through organic development of melodic and rhythmic material and precise
control of harmonic fluctuation. The insistence upon balanced forms is one rea-
son Hindemith’s music has been labeled neoclassical.
Hindemith’s separation of rhythm from the other elements of music in IE
Cm gives an incomplete description of the importance of rhythm to his theory.
The Cleveland lecture notes presented a more complete idea of the interaction
of all the elements of music. He may have anticipated recent developments in
9 Morgan, 439.
1oNeumeyer, The Mgsic of Pagl Hingemith, 41. Works which Neumeyer
cites as using proportional designs are Angelic Concert from Mathis der Ma_le_r
and the Second Pia_no Sonata 1936).
1 27
rhythmic theory by calling for a comprehensive understanding of music at a time
when music theory was incapable of providing such a solution.
Answer to Collateral Question 3
Can a method of rhythmic analysis be developed which supports Hinde-
mith's concept of rhythm?
The difficulty in developing a method of analysis that Hindemith would
approve lies in assumptions made about what the composer would have done.
Decisions regarding the method of analysis presented here were based on tra-
ditional ideas of rhythm using Hindemith’s theory as a starting point. In the com-
positional planning, Hindemith was concerned with a top-down approach which
identifies large formal structures first, then the elements which combine to
create the larger structures, and finally the individual durational values and how
they are grouped into the patterns of the rhythm. I have attempted to follow this
procedure in the analysis of the First Piano Sonata. At best, one can develop a
model which does not conflict with the presuppositions of Hindemith’s theory.
We cannot know if Hindemith would approve or not approve of the proposed
method, but the success of any analytical method lies in the insights into the
music gained from that method.
128
Answer to Research Question
Does Hindemith’s use of rhythm in his musical compositions reflect his
explanations of rhythm in the theoretical writings?
From the foregoing discussion of the collateral questions, it is evident that
Hindemith’s views on rhythm were quite standard in nature. He did not break
new ground in his thinking. We have addressed Hindemith’s theoretical writ-
ings about rhythm, so in order to answer this question, rhythmic characteristics
of Hindemith’s music need to be identified. Through the analysis of his music
and the survey of his theory, we may conclude that Hindemith’s application of
rhythm does support his explanations of rhythm.
In this movement, meter and the barline are two determinants of rhythmic
grouping at both the primary beat level and subsequent multiple levels. The
quarter note is perceived as the basic pulse, or beat level in Section A, estab-
lished in measure one by the two-voice framework. The majority of opening
rhythmic gestures fall into a metrical organization of equal beats with regular
accentuation at the secondary metrical level. However, as the phrases develop,
ambiguities of accentuation are introduced by changes of meter, syncopation,
and variation of rhythmic elements. Other elements of music such as changes
in texture and harmony reinforce the metric patterns of accentuation.
Table 10 compares Hindemith’s rhythmic style as found in the first piano
sonata with the general characteristics of tonal rhythm identified by \Nll'lOld.11 A
discussion of each comparison is given following the table.
11 Allen Winold, “Rhythm in Twentieth-Century Music,†in Aspecï¬ of Twen-
t_ie;h CentunL M189. ed. by Gary Wlttlich. (Englewood Cliffs, New Jersey: Pre-
ntice-Hall, Inc., 1975): 216-17, 244.
Hindemith's
Style
YES
YES
NO
YES
YES
NO
YES
YES
YES
YES
YES
YES
129
TABLE 10
A Comparison of Rhythmic Styles
Tonal Rhythmic Style
1. Regular pulses are clearly heard or implied and are equal-
timed;
(Table 10, continued)
2. Pulses are grouped into two’s or three’s;
3. Pulse groups are maintained throughout most of the compo-
sition;
4. Pulse groups on various levels coincide with pulse groups
on higher or lower levels;
5. Constant tempo predominates throughout an entire compo-
sition or section;
6. Meter signatures remain constant throughout an entire com-
position or section;
7. One duration predominates as the basic unit of movement,
and very short durations are rare except in special localized
circumstances;
8. Motivic (“rhythmicâ€) units are primarily based on metric pat-
terns; the barline often determines the accent
9. A limited number of rhythmic units combine to create longer
“rhythmic gestures";
10. A strong tendency toward upbeat, or anacrustic, begin
nrngs;
11. Repetition and variation of rhythmic gestures is quite com-
mon as well as rhythmic gestures in ternary patterns;
12. The composite rhythm generally confirms the metric struc-
ture.
130
#1 -3) In movement one, performance directions specify a strong quarter
note pulse which is retained throughout the movement. Pulse is established by
the descending quarter notes of the lower voice. Duple groupings predominate
in movement one, although there is some change to triple meter in repetition of
phrases and extrametrical hypermeasures in Section B and the coda.
Hindemith occasionally uses extrametrical subdivisions. More common are
extrametrical multiples such as triplet quarter notes which obscure the metric
pulse. Shifted accents occur at points of climax and harmonic tension which
sometimes work against the established meter.
#4) The grouping of rhythmic units confirms the pulse on the primary metric
level and the first multiple level. Hypermeasures are created from unequal
parts (4 + 2 in Section B of Movement One and 3 + 2 in Movement Four).
#5) There is only one area of tempo change within a movement or section in
first piano sonata—movement three just after the middle of the piece. Tempos
change between sections but proportional relationships of each section and
movement must be maintained for the most satisfying performance. Interpreta-
tion must be based upon proper reading of the score.
#6) Meter signatures are often not given, so the performer must rely on nota-
tion and grouping. Sections begin with stable metric pulses but often change in
repetitions. (For example, phrase a" of section A changed to 3\4 meter giving
expressive use of meter changes.) In the first piano sonata Hindemith never
changes from compound to simple meters within a section although contrasting
sections sometimes do.
#7) The quarter-note pulse is retained throughout the first movement; other
movements establish one basic duration as the pulse of each section In move-
ment one, the most frequently used note value is the eighth note, followed by
the quarter note and dotted quarter note. Each of these confirms the metric
131
pulse.
#8) The majority of rhythmic units confirm the established meter. Agogic
accents often fall on the first beat of the measure.
#9) Longer rhythmic gestures are built up from shorter rhythmic groups. In
movement one, Pattern N is used in both section A and Section B as an
organizing motive. There is a tendency to use one principle rhythmic pattern
per section, with a new pattern occurring at crucial points in the structure (the
deceptive cadence in Section A, measure 8, and the end of Section B).
#10) Eighteen of twenty rhythmic gestures identified in movement one have
anacrustic beginnings.
#11) Hindemith rarely repeats the same gesture exactly. There is always
some form of variation or change, although rhythm is sometimes the least
changed parameter. Section B uses the same rhythmic gesture twice in a row
with only slight modifications. Shorter rhythmic groups are used to extend
phrase lengths through repetition.
#12) The outer voices of the Two-voice framework are written together and
emphasize similarities of construction.
Some generalizations about Hindemith’s style from his writings also have
been deduced.
1. Meter is the basic organizational factor in Hindemith’s music.
2. Accent groups pulses into basic units of two’s and three’s.
3. Durational patterns make up motivic structures which are the basic units
of form.
4. The process of composition begins with determining the rhythmic form
and rhythmic character of each section, then the tonal relationships and
finally the specific melodic material is written.
5. Shorter rhythmic units undergo repetition, transformation, or change as
they are combined to create longer structures.
6. Rhythm will influence melodic and harmonic structures. .
7. Rhythmic form can be determined by a comparison of the lengths of the
sections of the tonal framework.
132
The basic question to ask regarding this piece is does the sonata “work?â€
Sections, phrases, and musical ideas are easily recognized and traced through
the piece. The aesthetic value of the sonata is based partly on the craftsman-
ship of the composition rather than memorable musical ideas.
Areas of Further Research
Four areas of specific research stand out regarding the topic of rhythm in
Hindemith’s music. First of all, more of his music needs to be analyzed regard-
ing rhythmic structure. The process could be done in stages, finishing the piano
music first, then moving on to other genres. A broader sample of music will give
a better picture of the consistency or changes of Hindemith’s style throughout
his lifetime. His early music from the 1920’s can then be compared to music
from the middle period music and his last works. If his application of rhythm is
consistent throughout his lifetime then it supports the thesis that Hindemith’s
musical application of rhythm and his writings about rhythm are consistent.
A second area is the expansion of the analysis to include other aspects of
music. This study concentrated on specific details of rhythm and only
addressed melody and harmony as needed. There is a need to relate rhythm
more definitely to pitch in both melodic and harmonic analysis, perhaps using
the parameters of contour theory and motion as a starting point. Appending a
rhythmic element to Neumeyer’s method of structural analysis discussed in
Chapter Three might also be beneficial.
A third area that needs further attention is the importance of pillar chords as
unifying and structural elements. Hindemith did not develop criteria for their use
133
or identification. Neumeyer’s five stages are also based upon the identification
of pillar chords but his analytical method also does not define how pillar chords
are identified.12 Hindemith understood the two distinctions of time as the uni-
versal, continuous passage and the chain of events within that passage. ‘3 Hin-
demith’s introduction of pillar chords may have been an attempt to formulate the
boundaries of composition, what the Greeks called the semeia or time-points
that defined structure, but he left no clear method for determining or identifying
specific pillar chords. Some parameters for identifying pillar chords may be
cadential points of repose, important points of harmonic intensity, and the outer
boundaries of the two-voice framework.
The fourth area of continuing research regarding Hindemith’s theory is his
pedagogical method of composition. What are the implications of the rhythmic
concepts for teaching composition? The graphing technique and compositional
planning outlined in Traditional Harmony give practical tools for the beginning
composer. A parallel study would be to look more closely at Hindemith’s
adaptation of the species method. One apparent problem in Hindemith’s use of
species is the brevity of the exercises: he seems to move from first species to a
type of fifth species with little progression of difficulty. So the question is raised,
“How does he make the connection from species to free composition?†Neu-
meyer's proposed restructuring of the sequential method of Hindemith’s primary
texts may provide a better understanding of the pedagogical implications.
A more general area of continuing research is a comparison of recent rhyth-
mic studies. What are the innovations each new analytical methodology pres-
ents? Finding common ground in the terminology, definitions, and concepts
would propel research in rhythm forward extensively.
12 Personal letter from Neumeyer to the writer.
‘3 Hindemith, Elementary Training, 93fn.
APPENDICES
APPENDIX A
APPENDIX A
Cooper and Meyer’s The Rhythmic Structure of Mugjg
The Rhythmic Structure of Music by Grosvenor Cooper and Leonard Meyer
is one of the first modern works to develop a method of rhythmic analysis. The
authors claim a partial influence of Schenker on their work, using limited
melodic and rhythmic reductions to assist in the analysis of longer rhythmic
patterns of phrases, periods, and sections.1 Their reductions are limited to
foreground structures that are dissimilar to Schenkerian structural graphs,
however.2
The basic premise from which Cooper and Meyer work is that rhythm is
essentially hierarchic; it exists simultaneously on many different levels. They
call these levels “architectonic,†which means that individual tones become
grouped into motives, motives into phrases, phrases into periods, etc. The
small rhythmic motives which exist on the “primary rhythmic level†may also
function as integral parts of the larger organization of the piece. Groupings may
change from one level to the next.3
The shortest rhythmic group that is considered complete in itself is said to be
the primary rhythmic level, but these may be subdivided into partial or
incomplete rhythmic motives called inferior rhythmic levels. (These may be
1Grosvenor W. Cooper and Leonard B. Meyer, The Rhythmic Structure of
Music, (Chicago: The University of Chicago Press, 1960), 146.
2 lbid., 70, 84.
3lbid., 2, 60.
134
135
further subdivided into subprimary levels.) When groups of the primary rhythmic
level are combined into longer units they form “superior rhythmic levelsâ€
consisting of phrases, periods, sections, and even whole movements.
Cooper and Meyer identify three types of organization of musical time:
pulse, meter, and rhythm.4 A pulse is defined as “a series of regularly recurring,
precisely equivalent stimuli†which continue in the mind after being established.
Pulses are not considered beats unless they are in a metrical context.
Meter occurs when accents differentiate regular numbers of pulses, or beats.
Cooper and Meyer also believe meter to be architectonic in nature although the
dominant or primary meter tends to remain regular on the primary level.5
Certain principles may work against the regular pattern of the meter, such as a
hemiola (“three groups of two played against two groups of threeâ€) or
asymmetrical accentual patterns (i.e., 3 + 2) within a symmetrical context (i.e.,
common time).
Rhythm is defined by Cooper and Meyer as “the way in which one or more
unaccented beats are grouped in relation to an accented one.â€6 Rhythm is
considered to be independent of meter as it can occur without metric pulse and
any rhythmic organization can occur in any given meter. Furthermore, the bar
lines serve only to indicate metric grouping and do not indicate the rhythmic
organization, although accents of rhythm and meter usually coincide.
Cooper and Meyer identify rhythmic groups according to poetic feet. 7 Of the
seven groups identified, the first five show the relationship of accented to
unaccented beats while the last two are considered unaccented and therefore
4lbid., 3.
5 lbid., 4.
6 lbid., 6.
7 lbid., 6.
136
incomplete rhythms. The accented groups divide themselves into three types,
the iamb and anapest which are end accented, the trochee and dactyl which
are beginning accented, and the amphibrach which is middle accented. The
spondee and pyrrhic are unaccented.
The poetic feet are used to chart the rhythmic relationships in a piece of
music. Cooper and Meyer claim that all levels of the rhythmic structure exhibit
these basic patterns, from the short primary levels to longer structures such as
phrases and periods. Cooper and Meyer identify the dactyl, anapest, and
amphibrach in both duple and triple meters.8 Williams, however, labels the
dactyl and anapest as duple and the trochee and iamb as triple.9
The determination of accent is crucial to the rhythmic interpretation
according to Cooper and Meyer’s scheme. Accent is defined as “a stimulus (in
a series of stimuli) which is marked for consciousness in some way.â€10 This
gives a variety of rhythmic interpretations for each of the poetic feet. Earlier
writers regard note length as well as accent as determining the poetic foot of a
rhythm.11
While a beat can be accented in a number of ways, according to Cooper and
Meyer stress is primarily one of dynamics. Stress can indicate the beginning of
a group and may in some way modify its character. However, the
organizational effect of a stressed group usually lasts no more than two
measures. The effect of a stressed beat or group diminishes on higher levels of
structure.‘ 2
8lbid.,18-26,
9C. F. Abdy Williams, The Rhythm of Modern Mu_sic, (London: Macmillan
and Co., Ltd. 1909), 79-82.
10 Cooper and Meyer, 8.
11 WIlliams, 79.
‘2 Cooper and Meyer, 8, 20, 61, 120.
137
The purpose of The Rhythmic Structure of Music is both prescriptive and
analytic. Cooper and Meyer attempt to formulate a theory of rhythm based on
the principle of accent in poetic meters. They devise a set of symbols for the
analytical procedure. Brackets are placed under the score to identify the
grouping of the rhythm pattern at each level. The primary rhythmic level is
labeled with a numeral 1. Subprimary levels are identified with lower case
Roman numerals while superior levels receive a 2, 3, 4, etc. Other symbols are
placed in the brackets to indicate various characteristics of the rhythmic
structures. (Table 11).13
The Rhythmic Structure of Music has provided the basis for one study of
Hindemith’s Piano Sonatas of 1936.14 In her study, Flood used the higher
rhythmic structures identified by Cooper and Meyer to investigate the harmonic
fluctuation at the phrase, period, section, and movement levels. She concludes
that the choice of harmonic fluctuation as determined by Hindemith’s theory set
forth in gen—l is a planned compositional procedure, and that the musical
climax always immediately follows the point of greatest harmonic tension.
Flood did not study the surface rhythmic patterning of durations nor long-range
harmonic connections. Flood’s study concentrated on pedagogical implications
for the piano teacher.
Several problems hinder the application of Cooper and Meyer’s theory of
rhythm to musical analysis. First of all, there is some divergence from common
definitions in the concepts they discuss. One must learn a variety of new terms
13lbid., 204.
14Dorothy Anne Flood, “The Role of Rhythm in Paul Hindemith’s Concept of
Harmonic Fluctuation as Revealed in the Piano Sonatas of 1936,†(Ed.D. diss.,
Columbia University Teachers College, 1976).
138
in order to apply their principles.15
Table 11
Cooper and Meyer’s List of Analysis Symbols
Imilfu;
l
Accent
Weak beat or group
Felt but unperfonned beats or groups
Initially presumed accent but retrospectively weak
Initially presumed weak beat or group retrospectively
accented
Accent fused to a weak beat or group
Weak beat or group fused to an accent
Fused weak beats or groups
Extended anacrusis
Extended anacrusis at first presumed to be accented
Stress
Grouping, manifest or dominant
Grouping, latent
Grouping without a definite conclusion
Grouping without a definite beginning
Overlapping or pivoted rhythmic groups
Splitting of one rhythmic level into two
15 Ellis B. Kohs, “Review of The Rhythmic Structure of Music, by Grosvenor
W. Cooper and Leonard B. Meyer,†Journal of Mgsic Theom 5/1 (April, 1961):
131-132.
139
Secondly, they admit that no hard and fast rules exist for grouping rhythms
into the various structures, leaving the rhythmic structure open to differing
interpretations.16 Nonetheless, they try to determine some general principles of
grouping, such as similarity and difference, proximity and separation, and
repetition. The other elements of music such as orchestration, dynamics,
texture, etc., may also play a role in determining rhythmic groups.
Lastly, there may be a weakness in the comparative analogy between
language and music.17 Music is considered to be more regular than speech,
therefore the rules of one do not readily apply to the other. The limited choice of
the five basic rhythms (iamb, anapest, trochee, dactyl, and amphibrach)
adapted from poetic feet leads to complications in analyzing difficult rhythm
patterns. They also do not address the nature of musical perception.
16Cooper and Meyer, 9, 11, 23.
17Lewis Rowell, “The Subconscious Language of Musical Time.†Music
TheoLy Spectrum 1 (1979): 105.
APPENDIX B
APPENDIX B
Structural Theories and Rhythmic Analysis
In 1959, Allen Forte published the article “Schenker’s Conception of Musical
Structure." Forte’s purpose was three-fold: to introduce the concepts of
Schenker’s theory of musical structure to American musicians, to encourage
serious musicians to use Schenker’s theory, and to challenge music theorists
with five musical problems which the new theory could help solve.1 The first
problem is stated simply: “Constructing a theory of rhythm for tonal music.â€
In Schenkerian analysis the fundamental organization of tonal music is rep-
resented by the projection of the tonic triad over the temporal span of a musical
composition.2 This “fundamental structure†is known as the Ursatz, or back-
ground. The background consists of two parts in a contrapuntal relationship.
The first is the melodic content of the upper voice which consists of a descend-
ing stepwise descent beginning on a pitch of the tonic triad and ending on the
first scale degree. This Urflie, or fundamental line, works in conjunction with
the bass voice which outlines the tonic and dominant triads represented by the
interval of a perfect fifth. This span of the bass interval is called the
1 Allen Forte, “Schenker’s Conception of Musical Structure,†Journal of
Music Theory 3/1 (April, 1959): 1-30. Reprinted in Yeston, Maury, ed. Readings
in Schenkerian Researgh. (New Haven: Yale University Press, 1977). Refer-
ences are from the original article.
2 A more detailed introduction to Schenker theory can be found in Forte’s
article mentioned above as well as his book lntrod_uction to Schenkerian Analy-
$ (New York: WW. Norton, 1982). See also other articles and books men-
tioned in the bibliography.
140
141
Bassbrechung, or broken bass. The Ursatz represents the total harmonic
conception of the music in the relationship of the tonic and dominant triads.
Structural accents occur at the points where the pitches of the Urlinie coincide
with the pitches of the Bassbrechung.
Two other levels of structure are important to an understanding of Schenker-
ian analysis. The foreground sketch contains the major surface events of the
composition that are most readily tangible in the music itself. The foreground is
not the actual durational values of the notes of the composition but rather repre-
sents the important relationships of the surface events. The foreground con-
tains the closest association to the metrical and rhythmic patterns of the surface
of the music, that which we identify as “the music.â€
The middle-ground sketch is a further reduction of the foreground relation-
ships. It identifies longer-ranging tonal connections and relationships. These
relationships are called prolongations in that the content of the melodic and har-
monic material extends the influence of the tonal area. Each sketch also identi-
fies certain embellishments or diminutions such as neighboring tone and pass-
ing tone motion, and arpeggiations.
Forte asked two basic questions regarding the rhythmic structure of a tonal
composition which he believes Schenkerian analysis could answer.3 The first
addresses the issue of structural levels and the perception of rhythmic events at
each level. How far back do the rhythmic and tonal patterns of the foreground
reach to determine the tonal structure? The second question deals more specif-
ically with the relationship of each level of the structure. How does the rhythm at
each level help to define each subsequent level, either working in conjunction
with or in perceived opposition to each other?
The answer to the first question begins with the identification of foreground
3 Forte, Schenker’s Conception, 21.
142
patterns of motives, phrases, and sections and how they extend into the
background.4 Rhythmic durations tend to give meaning to the tonal relation-
ships, but they do not define them in terms of the structure. The durations of the
pitches at each level are the rhythmic events of the composition.
Pierce claims that the answer to the second of Forte’s questions lies in the
determination of the structural accents at each level.5 Structural accents occur
at the points where the pitches of the Urlinie coincide with the pitches of the
Bassbrechung to form the fundamental structure, or 11%. The structural
accents determine the harmonic rhythm of each level, which becomes gradually
slower with subsequent stages until the M is understood. The durations of
the harmonic rhythm at each level can then be compared to the tonal organiza-
tion of the piece defined by the structural accents. A structural accent may have
a different meaning or emphasis at each stage of the analysis.
Pierce recommends determining how the structural accents divide the com-
position into temporal segments. The metric accents are compared to the struc-
tural accents to see where they coincide or conflict. The structural accents are
characterized by their specific duration and the duration of the group which they
define.6
Morgan claims that Schenkerian analysis is effective in analyzing formal
units. Schenkerian analysis may provide insights into the rhythmic organization
that traditional analysis cannot achieve. He says that,
4 Anne Alexandra Pierce, “The Analysis of Rhythm in Tonal Music,†(Ph.D.
diss., Brandeis Universtiy, 1968): 7, 129-130.
5 Ibid.
6 Ibid, 58.
143
Schenker’s theory, by placing all tonal motion within a structural frame sup-
plied by background pitch elements, supplies a method for locating the
points at which structural motions originate and terminate and thus for defin-
ing the temporal extent of any motion that is complete on some structural lev-
el.7
Morgan goes on to describe three advantages of Schenkerian analysis over
conventional roman numeral analysis. The first is that it identifies a definite
relationship between structural accents and pitch events. Secondly, criteria for
defining accents are precisely defined in reference to the tonal framework.
Finally, accents occur at specific locations at different levels of the rhythmic
structure. The rhythmic structure, then, is dependent on the metric pulse of the
foreground patterns. Morgan establishes criteria to develop a hierarchy of
accents dependent upon their structural weight. The structural weight is deter-
mined by rhythmic goals of the harmony, points of departure and return
(opening and closing accents), and the location of the structural accent in the
_U_r§a_t_z_, among others. Schenkerian analysis claims to identify different organi-
zational structures of tonal music that neither metric nor poetic analysis of the
music can provide.
The question arises as to the usefulness of Schenkerian analysis in the
study of Hindemith’s music. A student of Schenker, Felix-Eberhard von Cube,
analyzed two works of Hindemith according to strict Schenkerian principles: the
song Das Ganze nicht das Einzelne, and Movement One from the First Piano
Sonata of 1936.8 His conclusion regarding both works is that they cannot be
designated as “musical artworks†because they do not adhere to strict tonal the-
ory as set forth by Schenker. In order to use Schenkerian structural analysis to
7 Robert P. Morgan, “The Theory and Analysis of Tonal Rhythm,†The Musi-
cal Quarterly 64:4 (October, 1978): 444-45.
8 von Cube, Felix-Eberhard. The Book gf_Lhe Musical Artwork: An lntegeta-
tion of the Muflal Theories of Heinrich Schenker, translated by Neumeyer,
Boyd, and Harris. (Lewiston, New York: The Edwin Mellin Press, 1988), 353.
144
study non-tonal music, adjustments in the system need to be made as the music
does not conform to the rules of tonal music. Hindemith’s music is based upon
different presuppositions about the organization of tonal forces.
Structural Levels After Schenker
In two articles describing linear reduction techniques, Schachter demon-
strates a method of rhythmic reduction based upon metric organization.9 In
Chopin’s Prelude, Opus 28, No. 3, the rhythm is represented by five graphs.
The quarter-note represents one bar of music. The foreground and third mid-
dleground graphs are very similar to a Schenkerian voice-leading graph. In
fact, Schachter used Schenker’s original analysis of the piece as the basis for
his own study. With each subsequent graph of the middleground, certain
details are stripped away to show the basic underlying elements of composition;
only essentials remain. For example, bar 11 of Schenker’s original analysis is
recognized as an extension of the phrase and is omitted from the lower middle-
ground graphs. Other omissions are the two-bar introduction and the six-bar
coda. What remains in the first middleground graph, which comes closest to the
fundamental structure of the piece, are two eight-bar phrases, indicating that the
“asymmetrical proportions [of the piece] grow out of an underlying symmetry.â€10
Barlines are used to indicate measure groups, or hypermeasures. The
technique of “interruption†of the fundamental structure is also indicated in each
9 Carl Schachter, “Rhythm and Linear Analysis: Durational Reduction,†in
The Mgsic Forum, Vol. V, ed. by Felix Salzer (New York: Columbia University
Press, 1980): 197-232.
10 lbid, 205.
145
graph with the appropriate symbol.
In contrast to the underlying regularity of the Chopin Prelude, Schachter
demonstrates how the Minuet and Trio from Mozart’s Symphony N. 35, K. 385,
is developed from a metrically irregular grouping, especially in the Trio. He
uses a similar analytical technique of the quarter-note representing one bar of
music, each grouped into hypermeasures, but this time the alla breve meter sig-
nature indicates subgroupings of the four-bar hypermeasures into two-bar units.
In the next analysis, instead of a quarter-note, the eighth-note value equals one
bar of Beethoven’s Sonata Op. 14. No. 1, Allegretto. Phrases are grouped into
hypermeasures of eight bars in common time. He uses eighth-notes to more
clearly indicate Beethoven’s use of syncopation. In his last example, Schu-
bert’s Valse Sentimentale, Op. 50, No. 13, the eighth-note is used in the dura-
tional reduction to indicate one measure of triple meter, but a hypermeasure of
two bars is represented by a quarter-note.
There is one difference between Schachter’s use of durational reduction
and Schenker’s middleground reduction: Schachter does not go so far as to
indicate the fundamental structure of the piece. He believes that “the funda-
mental structure does have some rhythmic implications, but these arise out of
tonal function only and have nothing to do with duration.â€11 The meaning of the
basic durations rely upon the grouping of bars and the form of the piece.
Another difference is Schachter’s emphasis on metrical organization. He
defines meter as “a pattern composed of strong and weak impulses in some
kind of regular alternation.†This alternation occurs primarily from bar to bar, but
is also apparent between the downbeats of hypermeasu res. He doubts, howev-
er, that weaker and stronger pulses can be felt over very large divisions of time.
He is also concerned more with the pacing of the musical events in their
I1lbid., 229.
1 46
relationship to the form of the composition.
One problem inherent in Schachter’s attempt at rhythmic reduction is the
deficiency of the notational system. His use of quarter- or half-notes to repre-
sent phrase structure creates ambiguity and confusion in analyzing long sec-
tions of compositions. Also, the voice-leading and structural levels in the rhyth-
mic reductions are less clearly defined than in the true Schenkerian voice-
leading graph. Finally, Schachter’s approach does not concern itself with the
note-to-note durational values, those which we identify as “the rhythm†of the
piece. Instead, he is interested in broader durational values of the measure,
hypermeasure, phrase, and section. Schachter does suggest, however, that the
rhythmic reduction approach be used in conjunction with voice-leading graphs
where they could reveal important features of the piece.
Another contemporary theory of rhythm based loosely on Schenkerian
structural layers is Wallace Berry’s Structural anctions in Music.12 Berry
emphasizes the idea of “structural functions.†Structure is perceived as a slowly
unfolding process defined by motion toward and from the “primary accentâ€
rather than as patterns of melodies and harmonies that are simply repeated. In
his rhythmic theory, accent is the force that defines meter, but musical events
may not coincide with the barlines. He identifies different types or “classes†of
accent, such as metric, dynamic, textural, etc. Metric fluctuation occurs when
accents in the music do not coincide with the accent of the meter signature and
barline. Berry also argues that meter is not only “a stream of marked pulsa-
tions,†but that it also determines organization on several different levels. The
Schenkerian Ursatz does not identify specific elements of duration.
12Wallace Berry, Structural Functions in Music, (Englewood Cliffs, New Jer-
sey: Prentice-Hall, 1976; reprint, New York: Dover Publications, Inc, 1987).
APPENDIX C
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BIBLIOGRAPHY
BIBLIOGRAPHY
Apel, VVIIII. The Harvard Dicticmrv of MLLIC. Cambridge: The Harvard
University Press, 1972.
Bent, Ian. Analysis. New York: WW. Norton and Co., 1987.
. Music Analysis in the Nineteenth Century. Volume I: Fugue, Form
all Style; Volume II: Hermeneutic Approaches. Cambridge: The
Cambridge University Press, 1994.
Berger, Melvin. Guide to Sonatas: Music for One or Two Instyumem. New
York: Anchor Books, 1991.
Bobbitt, Richard. “Hindemith’s Twelve-tone Scale,†The Music Review 26
(1965): 104-117.
Bolitho, Albert George. “The Organ Sonatas of Paul Hindemith.†PhD. diss.,
Michigan State University, 1968.
Briner, Andres. Paul Hindemith. Mainz: B. Schott’s Sbhne, 1971.
Cazden, Norman. “Hindemith and Nature,†The Music Review 15/4 (November,
1954): 288-306.
Cooper, Grosvenor W. and Meyer, Leonard B. The Rhythmic Structure of Music.
Chicago: The University of Chicago Press, 1960.
Creston, Paul. Principles of Rhythm. New York: Franco Colombo, Inc., 1961.
Daehn, Sister R. Christine. “Paul Hindemith: Author, Craftsman, Philosopher.â€
unpublished Master’s thesis, Michigan State University, 1974.
Dunsby, Jonathan, and Whittall, Arnold. Mu_sic Analysis in Theory am! Practice.
London: Faber Music, Ltd., 1988.
Epstein, David. Shaping Time; Musicgthe Brain and Pertormance. New York:
Schirmer Books, 1995.
162
1 63
Evans, Peter. “Hindemith’s Keyboard Music,†The Musical Times 97 (1956):
572-575.
. Review of The Rhflhmic Structure of Music. by Grosvenor W.
Cooper and Leonard B. Meyer. In Tempo 59 (Autumn, 1961): 32.
Flood, Dorothy Anne. “The Role of Rhythm in Paul Hindemith’s Concept of
Harmonic Fluctuation as Revealed in the Piano Sonatas of 1936,†Ed.D.
diss., Columbia University Teachers College, 1976.
Haimo, Ethan. “Rhythmic Theory and Analysis,†In Theory Only 4M: 18-35.
Heiden, Bernhard. “Hindemith’s ‘System’--A New Approach,†Mopern Music 19
(Jan uary-February, 1942): 102-107.
Hindemith, Paul. A Commser’s WorlrLHtLtons and Limitations. New York:
Anchor Books, 1952, rev. 1961.
. A Concentrated Course In Traditional Harmony. Volume I. New
York: Associated Music Publishers, 1943. Volume 2: Exercises for
Advanced Stgdents, translated by Arthur Mendel. New York: Associated
Music Publishers, 1949.
. The Craft of Musical Composition, Volume 1: Theoretical Part,
translated by Arthur Mendel. New York: Associated Music Publishers, 1942,
revised 1945. Volume 2: Exerciees in Twp-Part Writing, translated by Otto
Ortmann. New York: Associated Music Publishers, 1941. Original German
editions Unterweisung im Tonsat_z_. Vol. 1: Theoretischer Teil. Mainz: B.
Schott’s Sbhne, 1937, revised 1940. Ubungsbuch fiir den zweistimmigen
gt; Mainz: B. Schott’s Schne, 1939.
. Elementary Tra_ining for Musicians. New York: Associated Music
Publishers, 1946.
. “Hindemith, The Three Piano Sonatas, Glenn Gould, piano,†Sony
Classical CD SMK 52670.
. “Hindemith, Music for One and Two Pianos,†Bernard Roberts with
David Strong. Nimbus Records, compact disc NI 5459/60, 1996.
. “Methods of Music Theory,†translated by Arthur Mendel. Musical
Quarterly 30 (January, 1944): 20-28.
. “Time, Only, Tells,†Etude 57 (October, 1939): 629-30.
164
Hindemith, Paul. “Old and New Problems of Music Theory," lecture given at the
Cleveland Institute of Music, March 3, 1947. Notes from the Paul Hindemith
Institute, Frankfut am Main.
. “Paul Badura-Skoda Plays Hindemith, Piano Sonatas Nos. 1 and
3,†Sound recording, Westminster XWN 18200, 1956.
. Ubungsbuch fiir den ggeisjmmigen Sag, ed. by Briner, Meier, and
Rubeli. Mainz: Schott and Co., 1970. Typescript translation of Chapters 12-
16, Yale University and Indiana University libraries.
HOIderlein, Friederich. Poems and Fragments, trans. by Michael Hamburger.
Ann Arbor: University of Michigan Press, 1967.
Hoppin, Richard H. Megievel Mtfl, New York: WW. Norton, 1978.
Kemp, Ian. Hindemith. London: Oxford University Press, 1970.
Kidd, James C. “Aspects of Mensuration in Hindemith’s Clarinet Sonata,†The
misic Review 38/3 (August, 1977): 211-222.
Kirby, F. E. A Short History of Keyboard Music. New York: Schirmer Books,
1966.
Kohs, Ellis B. Review of The Rhythmic Structpre of Music, by Grosvenor W.
Cooper and Leonard B. Meyer. In Journal of Mgsic Theory 511 (April, 1961):
129-34.
Landau, motor. “Hindemith the System Builder: A Critique of His Theory of
Harmony,†The Mu_sic Review 22 (1961): 136-151.
. “Paul Hindemith, a Case Study in Theory and Practice,†The Music
Review 21 (1960): 38-54.
Lester, Joel. The Rhythms of Tonal Music. Carbondale, IL: Southern Illinois
University Press, 1986.
Matthews, Denis, ed. Keyboard Music. Great Britain: Penguin Books, 1972.
Morgan, Robert P. “The Theory and Analysis of Tonal Rhythm,†The Musical
Quarterly 64:4 (October, 1978): 435-473.
Neumeyer, David. “Counterpoint and Pitch Structure in the Early Music of
Hindemith.†Ph.D. diss., Yale University, 1980.
1 65
Neumeyer, David. “The Genesis and Structure of Hindemith’s Ludus Tonalis,â€
Hindemitp Jahrbuch/Annales Hingemith 7 [1978](1980): 72-103.
. The Music of Pa_ul Hindemith. New Haven: Yale University Press,
1986.
. “Tonal, Formal, and Proportional Design in Hindemith’s Music,â€
Mtgic Theory Spectrum 9 (Winter 1987-88): 93-116.
Noss, Luther. P_aul Hindemith in the United Statee. Urbana: The University of
Illinois Press, 1989.
Pack, William D. “Paul Hindemith in Turkey: Some Contributions to Music
Education,†Ph.D. diss., Brigham Young University, 1977.
Pierce, Anne Alexandra. “The Analysis of Rhythm in Tonal Music.†Ph.D. diss.,
Brandeis University, 1968.
Randel, Don Michael, ed. The New Harvard Dictionan of Music. Cambridge:
The Belknap Press of Harvard University Press, 1986.
Ratner, Leonard G. “Eighteenth-Century Theories of Musical Period Structure,
The Musical Quarterly 62:4 (October, 1956): 439-454.
Redlich, Hans. “Paul Hindemith: A Re-assessment,†The Mueic Review 25
(1964): 241-53.
Reese, Gustave. msic in uie Mid_dle Agï¬. New York: WW. Norton and Co.,
1940.
Rothfarb, Lee A. Ernst Kurth as Theorist and Analyst. Philadelphia: University
of Pennsylvania Press, 1988.
Rowell, Lewis. “The Subconcious Language of Musical Time,†M_usic Theory
Spectrum 1 (1979): 96-106.
Sachs, Curt. Rhflhm and Temm: A Sttfly in Music Histogy. New York: WW.
Norton, Inc., 1953.
Sadie,StanIey, ed. New Grove’s DictionaLy of Music and Musicians, Volume 8.
New York: MacMillan Publishing Co., Ltd., 1980. S. v. “Paul Hindemith,†by
Ian Kemp.
Schachter, Carl. “Rhythm and Linear Analysis: A Preliminary Study,†in The
Music Forum, Vol. IV, ed. by Felix Salzer, 281-334. New York: Columbia
University Press, 1976.
166
Searle, Humphrey. Twentieth Centuty Counterppint: A Guide for Students.
London: Williams and Norgate, 1954.
Slonimski, Nicholas, ed. Baker’s BipgraphicaI_Dictionary of Musicians, 7th ed.
New York: Schirmer Books, Inc., 1992.
Smith, Charles. “Rhythm Restratified.†Review of The Stratification of Musical
BIMDE. by Maury Yeston. In Perspgtives of New Music 16/1 (1977): 144-
76.
Smither, Howard. “The Rhythmic Analysis of Twentieth Century Music, Journal
of Music Theory 8 (1964): 54-88.
Taylor, Clifford. “The Hindemith Theories: A Revaluation of Premise and
Purpose,†The Music Review 4 n3-4 (August/November, 1983): 246-262.
Thomson, William. “Hindemith’s Contribution to Music Theory,†Journal of
Music Theom 9(1965): 52-71.
Thurmond, James Morgan. Note Grouping: A Method for Achieving Expression
and Style in Musical Pertorma_nce. Camp Hill, PA: JMT Publications, 1982.
Vernazza, Marcelle. “Paul Hindemith—Music Educator,†T_he Americap Mus_ic_
Teacher (June/July, 1984): 30-32.
von Cube, Felix-Eberhard. The Book of the Musica_l Artwork: An Interpretation
of the Musical Theories of Heinrich Schenker, translated by Neumeyer,
Boyd, and Harris. Lewiston, New York: The Edwin Mellin Press, 1988.
Watkins, Glenn. Soundings: Mueic in the TweLtieth Centum. New York:
Schirmer Books, Inc., 1988.
Westergaard, Peter. “Some Problems in Rhythmic Theory and Analysis,â€
Perspectives gn Contemmrary Mï¬ic Theory. ed. by Benjamin Boretz and
Edward Cone. New York: W. W. Norton and Co., Inc., 1972.
White, John D. The Analysis of Music, 2nd ed. Metuchen, New Jersey: The
Scarecrow Press, Inc., 1984.
. Comprehensive Music Analysis. Metuchen, NJ: The Scarecrow
Press, 1994.
Williams, C. F. Abdy. The Rhythms of Modern Music. London: Macmillan and
Co., Ltd., 1909.
167
Williams, C. F. Abdy. The StonLgf Notation. New York: Greenwood Press,
1903, reprint 1969.
Winold, Allen. “Rhythm in Twentieth-Century Music.†In A_specte of Twentieth_
Centum Music, ed. Gary Wlttlich. Englewood Cliffs, New Jersey: Prentice-
Hall, Inc., 1975.
Yeston, Maury. The Stratification of Musical Rhythm. New Haven: Yale
University Press, 1976.
GENERAL REFERENCES
Barry, Barbara R. Musical Time: The Sense of Order. Stuyvesant, New York:
Pendragon Press, 1990.
Berry, Wallace. “Dialogue and Monologue,†_Cellege Mu_sic Symmsium 21/2
(1981): 84-100.
. “Metric and Rhythmic Articulation in Music,†Music The_ory Spectrum
7 (1985): 7-33.
. Structural Functions in Mtï¬ig. Englewood Cliffs, New Jersey:
Prentice-Hall, 1976; reprint, New York: Dover Publications, Inc., 1987.
. Musical Structure and Performance. New Haven: Yale University
Press, 1989.
Beach, David, ed. ï¬pects of Schenkerian Theory. New Haven: Yale
University press, 1983.
Benjamin, William. “A Theory of Musical Meter,†Music Perception 1/4
(Summer, 1984): 355-413.
Brown, Helen. Review of The Time of Music: New Meanings, New
Temporalities. New I__i_stening Stratmies, by Johathan D. Kramer. In Fontes
Artis Musica 37/1 (1990): 70-71.
Butler, David. Review of The Time of Music: New Meanings, New
Temporalittes. New Listening Strategie_s, by Johathan D. Kramer. In Music
Perception 7/4 (1990): 446-50.
1 68
Cadwallader, Allen, ed. Trenue in Schenkeria_n Reseerch. New York:
Schirmer Books. 1990.
Caplin, William. “Tonal Function and Metrical Accent: A Historical Perspective,â€
Music Theory Spectrum 5 (1983): 1-14.
Carpenter, Patricia. “Aspects of Musical Space,†in Explorations in Music. the
Arts, and Ideas: Essays in Honor of Leonard B. Meyer, ed. by Eugene
Narmour and Ruth A. Solie, 341-374. Stuyvesant, New York: Pendragon
Press, 1988.
Childs, Barney. “Poetic and Musical Rhythm: One More Time.†Music Theory
Sgcial Topies. ed. by Richmond Browne. New York: Academic Press,
1981: 33-57.
. “Time and Music: A Composer’s View," Perspectives of New Music
15/2 (Spring/Summer, 1977): 194-219.
Clarke, Eric F. “Levels of Structure in the Organization of Musical Time,â€
Contemporary Music Review 2 (1987): 211-238.
. “Theory, Analysis and the Psychology of Music: A Critical
Evaluation of Lerdahl, F. and Jackendoff, R., A Generalive Theory of Tona_|_
Music,†Psychology of Music and Music Education 14 (1986): 3-16.
Cone, Edward. “Music: A View from Delft,†The Musical Quarterely 4714
(October, 1961): 439-453. Reprinted in Perspectivesï¬on Contemmram
Music Theery, ed. by Benjamin Boretz and Edward Cone. (New York: W. W.
Norton, 1972)): 57-71.
. Msicat Form and Musical Performence. New York: W. W. Norton,
1968.
Cook, Kenneth E. “A New Approach to Musical Meter: Contributions Towards a
Theory of Metric Perception in Music.†Ph.D. diss., Michigan State
University, 1990.
Damschroeder, David, and Williams, David. msic Theory from Zarlino to
Schenker: A Bibliography and Guide. Stuyvesant, New York: Pendragon
Press, 1990.
Edwards, Paul, ed. The Encyclopedia of Philosophy. New York: MacMillan
Publishing Co., Inc., 1967.
Erickson, Robert. “Time-Relations,†Jeurnal ofMusic Theory 7I2 (Winter, 1963):
174-193.
169
Fay, Thomas. “Perceived Hierarchic Structure in Language and Music,†Journal
of Mueic Theom 15/1 and 2 (1971): 112-137.
Flew, Anthony, ed. A Dictionam of Philosophy, 2nd ed. New York: St. Martin’s
Press, 1979.
Forte, Allen, and Gilbert, Stephen. Introduction to Schenkerian Ana_|ysis. New
York: WW. Norton, 1982.
Forte, Allen. “Foreground Rhythm in Early Twentieth-Century Music.†In Models
of Musicel Analysis: Early Twentieth-Centum Music, ed. Jonathan Dunsby,
132-147. Oxford: Basil Blackwell, Ltd., 1993.
.“Pitch-class Set Genera and the Origin of Modern Harmonic
Species,†.Mmal of Music Theory 32/2 (Fall, 1988): 187-270.
. “Schenker’s Conception of Musical Structure,†Journal of Music
Theoty 3/1 (April, 1959): 1-30. Reprinted in Yeston, Maury, ed. Readings in
Schenkerian__Reseerch. New Haven: Yale University Press, 1977.
Fraser, J. T. “The Art of the Audible Now,†Music Theom Spectrum 7 (1985):
181 -184.
Fricke, Jobst Peter. “Hindemiths theoretische Grundlegung der
Kompositionstechnik in seiner ,Unterweisung im Tonsatzâ€," Festschrift
Heinrich Hii_schen gum 65Gepurtsteg. Kassel: Barenretter, 1980.
Friskin, James and Freundlich, Irwin. Music for the Piano: A Handbook for
Concert and Teghinflllateriejs from 1580 to 1952. reprint edition New
York: Dover Publications, 1973.
Gauldin, Robert. A Practical Approach to Sixteenth Century Cpunterpoint.
Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1985.
Gould, Murray. "Species Counterpoint in Tonal Structure." Ph.D. diss., New
York University, 1973.
Graves, William. Twentieth Centum Fugue: A Handbook. Washington, DC.
The Catholic University of America Press, 1962.
Graybill, Roger. Review of Phrase Rhythm in Tonal Music, by William Rothstein.
In Notes 48 (December, 1991): 502-3.
Griffes, Paul. Mofln Music: The Avant Garde Since 1945. New York: George
Braziller, Inc., 1981.
170
Hanson, John. "Pedagogy of Sixteenth Century Counterpoint: Selected
Examples with Commentary." Theoryapd Practiee IV/I (March, 1979): 5-14.
Hasker, William. God, Time, and Knowledge. Ithaca, New York: Cornell
University Press, 1989.
Heath, Louise Robinson. The Concept of Time. Chicago: The University of
Chicago Press, 1936.
Hinton, Stephen. The Idea of Gebrauschmtgk: A Study of Musical Aesthetics
in the Weimar Republic (1919-1933) with Pï¬icutar Reference to the Works
of Paul Hindemith. New York: Garlands Publishing, Inc., 1989.
Hopkins, Robert G. Review of The Stratification of Musical thahm, by Maury
Yeston. In Notes 34/1 (1977): 77-80.
Jeppesen, Knud. Counterppint: me Polyphonic Vocal Style of the Sixteenth
Centug. New York: Prentice-Hall, Inc., 1939.
Jaques, Elliott. The Form of Time. New York: Crane, Russak & Co., Inc., 1982.
Joseph, Charles. Review of Structural Functions in Music by Wallace Berry. In
Notes 33/3 (March, 1977): 600-601.
Jonas, Oswald. Introduction to me Theory of Heinrich Schenker, trans. by John
Rothgeb. New York: Longman, Inc. 1982.
Karkoshka, Erhard. Review of Ubungsbuch fur den dreistimmigen Sa_tz_, by Paul
Hindemith, ed. by Briner, Meier, and Rubeli. In pie Mugkforechung 28/1
(January/March, 1975): 130-1.
Komar, Arthur J. Theom of Suepensions: A Study of Metrical and Pitch
Reletione in Tonal Music. Princeton: Princeton University Press, 1971.
Kostka, Stefan. Review of Structural Functions in Music, by Wallace Berry. In
Americen Music Teacher 27I3 (Janusary, 1978): 42-3.
Kramer, Jonathan D. “Studies of Time and Music: A Bibliography,†Music
Theory Spectrm 7 (1985), 72-106.
. ‘me Time of Music: New Meanings. New Temporalities New
ListeninLStrategies. New York: Schirmer Books, 1988.
V Lacey, A. R., ed. A Dictiongy of Philosophy, 2nd ed. London: Routledge &
Kegan Paul, Ltd., 1986.
171
LaRue, Jan. Guidelines for Style Analysis, 2nd ed. Warren, MI: Harmonie Park
Press, 1992.
Laskowski, Larry. Heinrich Schenker: An Annotated Index to His Anelyses of
Musical Works. New York: Pendragon Press, 1978.
Lederer, Josef Horst. “Zu Hindemiths Idee einer Rhythmen- und Formenlehre.â€
Musikforschung 29 (1976): 21-36.
Lerdahl, Fred, and Jackendoff, Ray. fleneretive Theory of Tonal Music.
Cambridge: The MIT Press, 1983.
Lewin, David. “Some Investigations into Foreground Rhythmic and Metric
Patterning,“ In Music Theory Specia_l Topics, ed. by Richmond Browne.
New York: Academic Press, Inc., 1981.
Mancini, David. “Using Species Counterpoint in the Undergraduate Theory
Curriculum†Journel of Muisic Theory Pedagogy III/2 (Fall 1989): 205-219.
McGraw-Hill Dict_iona_ry of Scientific a_nd Technicatl Terms, 3rd ed. New York:
McGraw-Hill, Inc., 1984.
Morgan, Robert P. Review of The Time of Music: New Meanings, New
Temporalities. New Listening Strategies, by Jonathan D. Kramer. In Music
Theory Spectrgu 12/2 (1990): 247-55.
NestIer, Gerhard. Review of Ubungsbuch flir den dreistimmigen Satz, by Paul
Hindemith, ed. by Briner, Meier, and Rubeli. In Melos 38 (April, 1971): 145-
6.
Palmer, Caroline, and Krumhansl, Carol. “Mental Representations for Musical
Meter,†Journal of Exgrimental Psychology: Huma_n Perception an;
Performafl2e 16/4 (1990): 728-741.
Rexroth, Dietrich, ed. Paul Hindemith Briefe. Frankfurt: Fischer, 1982.
Rochberg, George. “The New World of Sound,†Etude 71 (January, 1953): 19,
50.
Rothstein, William. Phlï¬g Rhythm in Tona_LMusic. New York: Schirmer Books,
1989.
Rowell, Lewis. Review of The Time of Music: New Meanings, New
Temoralities. New Qstenirtq Strategies, by Johathan D. Kramer. In Journal
of Mu_sic Theog 34l2 (1990): 348-59.
172
Salzer, Felix. Structural Hearing: Tona_I Coherence in Music. Volume One gig
Two. New York: Charles Boni, 1952. Reprint edition New York: Dover
Publications, Inc., 1962.
Salzer, Felix, and Schachter, Carl. Counterpoint in Composition: The Study of
Voice Leading. New York: McGraw-Hill, Inc., 1969.
Samson, Jim. Review of Structural Functions in Music by Wallace Berry. In
Temm 122 (September, 1977): 20-24.
Schachter, Carl. “Rhythm and Linear Analysis: Durational Reduction,†in The
Music Forum, Vol. V, ed. by Felix Salzer, 197-232. New York: Columbia
University Press, 1980.
Schubert, Giselher, ed. flul Hingemith. Aufsatze, Vortrage, Reden. Zurich:
Atlantis Musikbuch-Verlag, 1994.
Seidel, Wilhelm. Review of The Stratification of Musicaj Rhyth_m, by Maury
Yeston. In Muzikforschung 33l2 (1980): 222-4
Shackford, Charles R. Review of Ubungsbuch furuen dreistimmigen Satz_, by
Paul Hindemith, ed. by Briner, Meier, and Rubeli. In Journal of Mg Theoty
16/1-2 (1972): 238-65; In Notes 2913 (March, 1973): 451-2.
Shirlaw, Matthew. The Theory of Harmony. 2nd ed. DeKaIb, IL: Dr. Birchard
Coar, 1955. Reprint edition New York: DaCapo Press, 1969.
Siegel, Hedi, ed. Schenker Sttujies. Cambridge: Cambridge University Press,
1990.
Skelton, Geoffrey. Paul Hirï¬emith: The Mag_ Behind the Music. London: Victor
Gallancz, Ltd., 1975.
SIatin, Sonia. Review of Ubungsbuch fur den dreistimmigen Satz_, by Paul
Hindemith, ed. by Briner, Meier, and Rubeli. In The Musical Quarterly 18
(January, 1972): 141-8.
Solie, Roger. Review of The Stratification of Musica_l Rhytlm, by Maury Yeston.
In Journal of Aesthetics and Art Criticism 35/2 (1976): 244-5.
Strobel, Heinrich. Eeul Hingemith: Teetimony in Pictures. Mainz: B. Schott’s
Schne,1961.
Tischler, Hans. Review of T_he Rhythmic Structure of Music, by Grosvenor W.
Cooper and Leonard B. Meyer. In Journal of the America_n Musicological
Society 16/2 (1963): 270-272.
173
Urmson, J. O., and Rée, Jonathan, eds. The Conciseï¬ncyclopedia of Western
PhilosophLem Philosophers, new edition. London: Unwin Hyman, 1989.
Williams, David R. A Biblipgraphy of the History of Music Theory, 2nd ed.
Fairport, New York: Rochester Music Publishers, 1971.
Yeston, Maury, ed. Readings in Schenkeria_n Research. New Haven: Yale
University Press, 1977.
Wlttlich, Gary E. Review of The Stratificauon of Musical Rhythm, by Maury
Yeston. In J_ournal of Music Theory 21/2 (1977): 355-73.
Zinsser, William, with Ruff, Willi. “Our Far-flung Correspondents: Resonance.â€
The New Yorker 60 (April 23, 1984): 84-105.