Smooth Voice-Leading Systems for Atonal Music
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Yeajin Kim, M.M.
Graduate Program in Music
The Ohio State University
2013
Dissertation Committee:
David Clampitt, Advisor
Anna Gawboy
Marc Ainger
Copyright
By
Yeajin Kim
2013
Abstract
In this dissertation, I introduce and develop a methodology to analyze atonal music from the view-point of voice-leading. I organize pitch-class sets having the same cardinality to demonstrate what is defined here as smooth voice-leading.
I first deal with the historical study relating to voice-leading systems for atonal music, a heavily researched field, and I focus on the interval array in pitch-class series.
This dissertation introduces a partition of Schoenberg‘s signature set, the 6–Z44 hexachord, into four systems (orbits), in close analogy with Cohn‘s four systems of maximally smooth cycles of hexatonic triads, and offers analytical examples of pairs of adjacent sets from this perspective. Selected analyses demonstrate its applicability.
Moreover, I suggest relatively smooth voice-leading systems covering Allen Forte‘s all pitch-class sets. The relatively smooth voice-leading systems are organized into groups according to criteria of shared interval-string contents from the pitch-class sets‘ arrays.
In groups of pitch-class sets sharing a cardinality and sharing interval multiplicities in their interval string arrays, I propose smooth voice-leading systems. Voice leading within pitch-class sets in the same group could be considered relatively smooth compared to other voice leadings, either those defined by Cohn to be maximally smooth, or between pitch-class sets in different groups according to the interval-string content criteria, which
ii are not smooth at all. Definitions of these concepts are provided in the dissertation.
iii
Dedication
Dedicate to
My Mother
iv
Acknowledgment
First, I really want to thank earlier scholars who have studied atonal music since their numerous papers encouraged me to consider various ideas concerning analytical methodologies for atonal music, as well as made me have enthusiasm for searching for the latent logic in atonal music.
I also thank my family for their support during the preparation of this dissertation.
Especially, I would like to thank to my husband, Seung-ho and my daughter, Sarah. They waited for me and encouraged me with love throughout my doctoral studies so that I could finish my doctoral program. Special gratitude from deep in my heart goes to my mother in heaven who gave me plenty of love and support.
In addition, I wholeheartedly appreciate my advisor, Professor David Clampitt who offered his encouragement, criticism, comments and suggestions.
Finally, I want to send my highest appreciation to God who makes it all possible.
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Vita
2004……………………………………….………….……..B.M. Music Composition
Yonsei University, Seoul, South Korea
2007……………………………………………….………..……M.M. Music Theory
Yonsei University, Seoul, South Korea
2008–2009………………………………...…………….Graduate Teaching Associate,
School of Music, The Ohio State University
2011 to present………….…………..…………………………………………Lecturer
Yonsei University, Seoul, South Korea
2012 to present....…………………...………………………………………….Lecturer
Korean National University of Arts, Seoul, South Korea
2012 to present……...………………..……………..…………………………Lecturer
University of Seoul, Seoul, South Korea
vi
Presentations
An analytic Method for Atonal Music based on Cognitive Considerations, Society for
Music Perception and Cognition (SMPC), Indiana University-Purdue University, Indiana,
USA, 2009
Voice-Leading Systems of Schoenberg‘s Signature Hexachord, Music Theory Mid-
West (MTMW) (with David Clampitt) , 21st Annual Conference, Miami University,
Ohio, USA, 2010
Fields of Study
Major Field: Music
Area of Specialization: Music Theory
vii
Table of Contents
Abstract………………………………...……………………………………………...….ii
Dedication…………………………………………………………………..…………….iv
Acknowledgments……………………………………………………………………...…v
Vita……………….………………………………………………………………...…….vi
List of Figures…...... ……………………………………………………………..…...….x
List of Tables…………..…………………………………………………………..……xvi
List of Examples……………………..…………………………………….…….…..…xvii
Introduction …………….……………..………………………………………………….1
Chapter 1: Historical Literature Review…………………………………….…………….3
Chapter 2: An Appropriate Analytic Methodology for Atonal Music……….……..…….17
Chapter 2.1: Consideration of motive …………..……………………………………….17
Chapter 2.1.1: Pattern-completion and Prerequisite conditions………………………….19
Chapter 2.2: Latent axis theory…………………………………………………………..29
Chapter 2.2.1: Analysis of Webern‘s op. 16, no. 2 ………………………………………40
Chapter 3: Theoretical Exposition of Smooth Voice-Leading Systems for Atonal Music
……………………………………………………………………………………………61
Chapter 3.1: Voice-leading systems within the Schoenberg hexachord…………………61
Chapter 3.1.1: Examination of 6-Z44 hexachord systems……………………………….74
Chapter 3.2: Relatively Smooth Voice-Leading Systems……………….……………….86
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Conclusion…………………………………………………………………………..…129
Bibliography….……………………………………………………………………..…130
Appendix A: Forte, 1973, Prime Forms and Vectors of Pitch-Class Sets……….……..136
ix
List of Figures
Figure 1.1. Two musical subjects and the interval from s to t ………………………..…..5
Figure 1.2. Hyer‘s tonnetz………….………………………………………………….…..8
Figure 1.3. Richard Cohn‘s Hyper-Hexatonic System..……………...……...………...... 9
Figure 2.1. Bartók, Fourth String Quartet, mvt. I, X-Y-Z progressions in exposition and development…………………………………………….………….…………………….31
Figure 2.2. Alignment of two inversionally complementary semitonal cycles intersecting at a dual axis of symmetry of sum 3………………………………………..……………32
Figure 2.3. George Perle‘s interpretation of diatonic collection………….……….……..35
Figure 2.4. Latin version and English version…………………...………………………40
Figure 2.5. Sum 0 (=12) axis : C (or F#)………………………………...………………43
Figure 2.6. G Axis between clarinet and voice lines in mm.1–2...……………..………..43
Figure 2.7. Webern‘s Op.16, No. 2………………………………………………………45
Figure 2.8. Voice-leading in pitch space………………………….……………………..46
Figure 3.1. Schoenberg‘s Signature Hexachord…………………………………………61
Figure 3.2. Construction of the 6–Z44 Partition………………………………………...63
Figure 3.3. J and K transformation………………………………………...…………....63
Figure 3.4. prime X-forms and inverted y-forms of 6–Z44 hexachord……………….....64
Figure 3.5. J- transformation……………………………………………………………65
x
Figure 3.6. K-transformation…………………………………………………………….66
Figure 3.7. K-of-J-of-X…………………..…...... 66
Figure 3.8. A General Voice-Leading System of the 6-Z44 Hexachord…………………68
Figure 3.9. Western System (a=0) ……………………………………………………….69
Figure 3.10. Partition of 6-Z44 into 4 Orbits……….……………………………………70
Figure 3.11. A General Voice-Leading System of the 6-Z44 Hexachord………….……..71
Figure 3.12. The Schoenberg signature group G and the hexatonic triadic group H…...72
Figure 3.13. The dual commuting groups, for a = 0, 1, 2, 3…………………..…………73
Figure 3.14. Row Structure of Op. 27, II…………………………………….….……….75
Figure 3.15. Transformation T4 of 4-3…………………………………………..……….87
Figure 3.16. I4 relationship……………………………………………………..………..88
Figure 3.17. Chrisman‘s interval array permutation……………………………..………90
Figure 3.18. ………….……………………………………………………….….………91
Figure 3.19. Particular intervallic arrays………………………………………….……..92
Figure 3.20.………………………………………………………………….….………..93
Figure 3.21. Voice leading from pcs 6-22 to 6-Z24…………………………..………….95
Figure 3.22. Hexachord voice-leading system having two semitones, three whole-tones, one interval class four……………………………………………………………………96
Figure 3.23. 7-note voice-leading system having three semitones, three whole-tones, one interval class three…………………...……………………………………..……….97
xi
Figure 3.24. Isolated symmetrical sets……………………………………………...……98
Figure 3.25. Voice-leading system of hexachord having three ic 1, three ic 3…..………98
Figure 3.26. 7-note voice-leading system having four interval class 1, three interval class
2, one interval class 3………………….…………………..…………………..…………99
Figure 3.27. 8-note voice-leading system having four interval class 1, four interval class 2
…………………………………….…………………………………………………….100
Figure 3.28. 9-note voice-leading system having six interval class 1, three interval class 2
……………………………………………………………………….………………….101
Figure 3.29. Three-note Groups………………………..………………….……………103
Figure 3.30. Tetrachord having three ic 1, one ic 9…………………………………….104
Figure 3.31. Tetrachord voice-leading system having two ic 1, one ic 2, one ic 8……..104
Figure 3.32. Tetrachord voice-leading system having two semitones, one ic 3, ic 7……104
Figure 3.33. Tetrachord voice-leading system having two ic 1, one ic 4, one ic 6……..105
Figure 3.34. Tetrachord voice-leading system having two ic 1, two ic 5………………105
Figure 3.35. Tetrachord voice-leading system having one ic 1, two ic 2, one ic 7……..105
Figure 3.36. Tetrachord voice-leading system having one ic 1, one ic 2, one ic 3, one ic 6
…………………………………………………………………………………………..105
Figure 3.37. Tetrachord voice-leading system having one ic 1, one ic 2, one ic 4, one ic 5
……………………………………………………………………………………….….106
Figure 3.38. Tetrachord voice-leading system having one ic 1, two ic 3, one ic 5….....106
xii
Figure 3.39. Tetrachord voice-leading system having one ic 1, two ic 4, one ic 3……..107
Figure 3.40. Tetrachord having three ic 2, one ic 6…………………………………….107
Figure 3.41. Tetrachord voice-leading system having two ic 2, one ic 3, one ic 5……..107
Figure 3.42. Tetrachord voice-leading system having two ic 2, two ic 4………………108
Figure 3.43. Tetrachord voice-leading system having one ic 2, two ic 3, one ic 4……..108
Figure 3.44. Tetrachord having four ic 3…………………………………...…………..108
Figure 3.45. Pentachord having four ic 1, one ic 8……………………………………..109
Figure 3.46. Pentachord voice-leading system having three ic 1, one ic 2, one ic 7...…109
Figure 3.47. Pentachord voice-leading system having three ic 1, one ic 3, ic 6………..109
Figure 3.48. Pentachord voice-leading system having three ic 1, one ic 4, one ic 5...…110
Figure 3.49. Pentachord voice-leading system having two ic 1, two ic 2, one ic 6…….110
Figure 3.50. Pentachord voice-leading system having two ic 1, one ic 2, two ic 4…….111
Figure 3.51. Pentachord voice-leading system having two ic 1, two ic 3, one ic 4…….111
Figure 3.52. Pentachord voice-leading system having one ic 1, three ic 2, one ic 5……112
Figure 3.53. Pentachord voice-leading system having one ic 1, two ic 2, one ic 3, one ic 4
……………………………………………...…………………………………………...112
Figure 3.54. Pentachord voice-leading system having one ic 1, one ic 2, three ic 3…...113
Figure 3.55. Pentachord having four ic 2, one ic 4…………………………………..…113
Figure 3.56. Pentachord voice-leading system having three ic 2, two ic 3……………..113
Figure 3.57. Hexachord having five ic 1, one ic 7…………………………………...…114
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Figure 3.58. Hexachord having four ic 1, one ic 2, one ic 6……………………………114
Figure 3.59. Hexachord voice-leading system four ic 4, one ic 2, one ic 6…………….114
Figure 3.60. Hexachord voice-leading system having four ic 1, one ic 3, one ic 5…….114
Figure 3.61. Hexachord voice-leading system having four ic 1, two ic 4…...…………115
Figure 3.62. Hexachord voice-leading system having three ic 1, two ic 2, one ic 5…...115
Figure 3.63. Hexachord voice-leading system having three ic 1, three ic 3……………116
Figure 3.64. Hexachord system having three ic 1, one ic 2, one ic 3, one ic 4…………116
Figure 3.65. Hexachord voice-leading system having two ic 1, three ic 2, one ic 4...…117
Figure 3.66. Hexachord voice-leading system having one ic 1, four ic 2, one ic 3…….117
Figure 3.67. Hexachord having six ic 2………………………………………………...118
Figure 3.68. Hexachord having four ic 1, one ic 3, one ic 5……………………………118
Figure 3.69. Hexachord voice-leading system having two ic 1, two ic 2, two ic 3…….119
Figure 3.70. Heptachord having six ic 1, one ic 6……………………………………...120
Figure 3.71. Heptachord voice-leading system having five ic 1, one ic 2, one ic 5……120
Figure 3.72. Heptachord having five ic 1, one ic 3, one ic 4……………………….…..120
Figure 3.73. Heptachord having five ic 1, one ic 3, one ic 4……………………….…..120
Figure 3.74. Heptachord voice-leading system having four ic 1, two ic 2, one ic 4……121
Figure 3.75. Heptachord voice-leading system having four ic 1, one ic 2, two ic 3……122
Figure 3.76. Heptachord voice-leading system having three ic 1, three ic 2, one ic 3....123
Figure 3.77. Heptachord voice-leading system having two ic 1, five ic 2……………...123
xiv
Figure 3.78. 8-note set-class having seven ic 1, one ic 5………………………….…....124
Figure 3.79. 8-note voice-leading system having six ic 1, one ic 2, one ic 4…………..124
Figure 3.80. 8-note voice-leading system having six ic 1, two ic 3…………………….124
Figure 3.81. 8-note voice-leading system having five ic 1, two ic 2, one ic 3……...….125
Figure 3.82. 8-note voice-leading system having four ic 1, four ic 2…………………..126
Figure 3.83. 9-note set-class having eight ic 1, one ic 4………………………..………127
Figure 3.84. 9-note voice-leading system having seven ic 1, one ic 2, one ic 3………..127
Figure 3.85. 9-note voice-leading system having six ic 1, three ic 2…………………...128
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List of Tables
Table 1.1. Components of Generalized Interval System……………..……………..……..6
Table 2.1. Straus‘s axis analyses…………………………..………..……………………38
Table 3.1. P- and Q-Cycles of Larger Set-Classes…………….……………………..…..83
Table 3.2. Definition of P-cycle, Q-cycle………………..……………………...…….....84
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List of Examples
Example 1.1. Voice-leading Uniformity in the progression from {F, F#, B} to {G, Bb,
D}………………………………………………………………………………….....…..12
Example 1.2. Voice-leading Balance in the progression from {F, F#, B} to {G, Bb,
D}……………………………………………………...……………………….……..….13
Example 1.3. Voice-leading Smoothness in the progression from (a) {F, F#, B} to {G, Bb,
D} and (b) {D, G} to {A, C, C#}………………………….…………….………………14
Example 1.4. Straus‘s trichordal voice-leading space….………….……………….….....16
Example 2.1. Principal melodic fragment in Stravinsky, Symphonies of Wind Instruments
……………………………………………………………………..…………………….26
Example 2.2. The principal fragment transposed………………………………………..27
Example 2.3. Associational background of Stravinsky, Symphonies of Wind Instruments
………………………………………………………..………………………………….27
Example 2.4. Even-Sum (Axis) and Interval Array……………...………………………33
Example 2.5. Odd-Sum (Axis) and Interval Array………………..……………………..34
Example 2.6. Motive of the first movement of Symphony in C…………………………37
Example 2.7. Straus‘ axis analysis of the first movement of Symphony in C…………...37
Example 2.8. Webern‘s Op.16, No. 2, mm. 1–2…………………………………………42
Example 2.9. Webern‘s Op.16, No. 2, mm. 1–6……………………...………………….47
xvii
Example 2.10. Webern‘s Op.16, No. 2, mm. 7–13……………………………...….……48
Example 2.11. Seven-note collection segmented by the text……………………………49
Example 2.12. Webern‘s Op. 16, No. 2, mm. 1–6………………………………..……..51
Example 2.13. Webern‘s Op. 16, No.2, mm. 7–13……………………………….……..52
Example 2.14. Webern‘s Op. 16, No.2, mm. 1–6……………………………….………55
Example 2.15. Webern‘s Op. 16, No. 2, mm. 7–13………………………………..……56
Example 2.16. Webern‘s Op. 16, No. 2, mm. 1–6……………………………………….57
Example 2.17. Webern‘s Op. 16, No. 2, mm. 7–13…………………………………..….58
Example 2.18. Webern‘s Op. 16, No. 2, mm. 3–6…...…………………………....……..60
Example 3.1. Schoenberg, Op. 23, No. 4, m. 1………………………………………….62
Example 3.2. Webern, Op. 27, mvt. 2, mm. 1–2….……………………………….…….76
Example 3.3. Common-tone Subset and Covering Superset in mm. 1–3/1……...... ……77
Example 3.4. The Invariant Augmented Triad for the Western System…………………78
Example 3.5. Webern, Op. 27, mvt. 2, mm. 5–6……………………………….…..……79
Example 3.6. The Invariant Augmented Triad for the Eastern System…………….……80
Example 3.7. Webern, Op. 27, mvt. 2……………………………………………………81
Example 3.8. Webern, Three Little Pieces, Op. 11, No. 3……………………………….82
xviii
Introduction
There have been numerous attempts to explain logically structural principles latent in atonal music. Theorists have used diverse analytical methodologies, most of which started from attempts to apply analytical logics of tonal music to the domain of atonal music. Among them, the study of voice leading has been an important concern of music theory in atonal music.
The studies of voice leading in atonal music have focused on the issue of similarities and proximities among collections of pitch classes. Neo-Riemannian theory, as developed principally by Richard Cohn, focuses on harmonic successions. Cohn defined maximally smooth cycles, moving between, for example, triads, in which two voices remain fixed and the other voice moves by semitone, to create a parsimonious voice-leading system.1
Douthett and Steinbach introduced more general parsimonious voice-leading criteria, admitting motion by whole tone as well as by semitone.2
1 Richard Cohn, ―Maximally Smooth Cycles, Hexatonic Systems and the Analysis of Late- Romantic Triadic Progressions,‖ Music Analysis, 15, no. 1 (1996): 9–40 and ―Neo-Riemannian Operations, Parsimonious Trichords and Their Representations,‖ Journal of Music Theory 41, no. 1 (1997): 1–66. 2 Jack Douthett and Peter Steinbach, ―Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition,‖ Journal of Music Theory 42, no. 2 (1998): 241–263.
１
John Roeder and Joseph Straus studied voice-leading systems in terms of the similarity of pitch-class sets in succession. Roeder introduced a theory of voice-leading sets for post-tonal music and Straus shows several methodologies for explaining uniformity, balance, smoothness in atonal voice leading.3
Recently, research into voice-leading systems has been studied within the field of geometry of music. Music theorists, principally Clifton Callender, Ian Quinn, and Dmitri
Tymoczko, have explained voice-leading systems within geometric spaces. By considering mappings on tones and on collections of tones, these studies encourage a deeper understanding of voice leading between collections, in relation to their structures.4
In my dissertation, I first review the history of voice-leading systems in atonal music.
Second, I suggest a new appropriate analytic methodology for atonal music. Third, I introduce a special voice-leading system within the so-called Schoenberg hexachord
(Forte‘s set 6-Z44). Finally, I propose more general smooth voice-leading systems covering all pitch-class sets.
3 John Roeder, ―A Theory of Voice Leading for Atonal Music,‖ (Ph.D. Diss., Yale University, 1984) and Joseph Straus, ―Uniformity, Balance, and Smoothness in Atonal Voice Leading,‖ Music Theory Spectrum 25 (2003): 305–352. 4 Clifton Callender, Ian Quinn, and Dmitri Tymoczko ―Generalized Voice-leading Spaces,‖ Science 320 (2008): 346–348, and Dmitri Tymoczko ―The Geometry of Musical Chords,‖ Science 313 (2006): 72–74.
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Chapter 1: Historical Literature Review
Music theorists continually seek new approaches to understanding music by adapting or extending models from the tonal music perspective. One of the most important of these approaches is the viewpoint of voice leading which is the relationship between successive pitches of simultaneously moving parts or voices. Analytical methods from the point of view of voice-leading focus on discovering the principles governing the motion of individual lines. Voice leading may be described as parsimonious when it follows Schoenberg‘s principle of the ―shortest path,‖ moving as few voices as few steps as possible, while retaining as many common tones as possible.5
One of the influential approaches focused on voice-leading principles is transformational theory, developed especially by David Lewin in the 1980s and formally introduced in his work, Generalized Musical Intervals and Transformations (1987). He explained musical transformations, which are operations defined on a mathematical group that represents potential musical events. This approach can be used to analyze both
5 The term ―parsimony‖ has been used variously and loosely in the music theory literature. Theorists consider different intervallic limitations for moving voices in the formalization of parsimonious voice leading. Cohn (1996) limits the motion of voices to no more than a whole tone, Childs (1998) to a semitone, and Douthett and Steinbach (1998) keep track of semitone and whole-tone motions separately. Clampitt (1997) does not use the term ―parsimony,‖ but in the context of smooth voice leading requires that motions do not involve ―jumping over‖ frozen pitch classes, a topological constraint.
３
tonal and atonal music. While traditional musical set theory focuses on the makeup of musical objects and describes operations that can be performed on them, transformational theory focuses on the intervals or types of musical motion that can be described between the musical events in sounding music.6 The resulting analyses are compelling visually and metaphorically because they show through the use of arrows how one musical event is transformed into another as an audible process in a piece of music.
When experiencing music, a listener perceives certain “objects,” such as pitches, or chords, or motives that change. In transformational theory, an analyst collects together all objects that change into one another and then considers the changes themselves, and all possible combinations of them. This leads one to include in the object group that the listener expects not only the objects one actually hears in the piece, but also all other objects that could conceivably result by applying all changes. The resulting group is, metaphorically, a musical “space” which the piece navigates by characteristic “gestures” or “motions.”
Lewin formalized transformations in music, using the mathematics of sets and mappings, and the algebraic structures of groups and semi-groups. As you see in figure
1.1, Lewin explains the concept of transformation by using two musical elements, s and t, and the interval i from s to t.
6 David Lewin, Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987), p. 159. According to Lewin‟s description of this change in emphasis, “The transformational attitude does not ask for some observed measure of extension between reified „points‟; rather it asks: ‘If I am at s and wish to get to t, what characteristic gesture should I perform in order to arrive there?’”
４
Figure 1.1 shows two points s and t in a symbolic musical space. The arrow
marked i symbolizes a characteristic directed measurement, distance, or motion
from s to t. We intuit such situations in many musical spaces, and we are used to
calling i “the interval from s to t” when the symbolic points are pitches or pitch
classes. 7
Figure 1.1. Two musical subjects and the interval from s to t
A Generalized Interval System (GIS) is an ordered triple (S, IVLS, int), where
S, the space of the GIS, is a family of elements, IVLS, the group of intervals
for the GIS, is a mathematical group, and int is a function mapping S x S into
IVLS, all subject to the two conditions (A) and (B) following.8
7 Ibid., xxix–xxx. 8 Ibid., 26.
５
(A): For all r, s, and t in S, int(r,s) int(s,t) = int(r,t)
(B): For every sin S and every I in IVLS, there is a unique t in S
which lies the interval I from s, that is a unique t which satisfies the
equation int (s,t) = i.
S: Space IVLS: the group of intervals Int: A Function mapping SⅹS into IVLS
Table 1.1. Components of Generalized Interval System
Lewin‘s transformational theory was influenced by the work of Hugo Riemann, but
Lewin fundamentally reconceptualized Riemann‘s approach by reordering the relationship between objects (triads) and processes (functions). Function, in Lewin‘s work, has its strict mathematical meaning (transformation, for functions from set S to S), whereas Riemann‘s functions are more akin to categories, although with a hint of dynamism that must have appealed to Lewin.
Neo-Riemannian theory is a methodology for analyzing chromatic music that includes triadic textures and structures but largely excludes tonal characteristics. Musical repertoire that includes such passages is that of Wagner, Liszt, and other highly chromatic music of the late nineteenth and early twentieth centuries. Cohn, building upon Lewin‘s reconsideration of Riemann, presents three contextual operations on triads: Parallel,
６
Relative, and Leittonwechsel. These three formal transformations on the space of harmonic triads are all those (excluding the identity transformation) that are parsimonious in Cohn‘s sense: they keep the maximal number of common tones (i.e., two – only the identity keeps three), while involving pitch-class voice-leading intervals no larger than a whole tone. A consequence of these constraints is that each of these transformations switches the modal character of its input triad, from major to minor, from minor to major.
That is, each of these basic transformations in neo-Riemannian theory is an inversion.
They are called contextual inversions because their action depends upon the object they act upon, rather than the usual twelve-tone inversion operators, which act uniformly on the space of pitch classes, once a mathematical origin is chosen (conventionally, C=0).
Brian Hyer developed Lewin‘s work by substituting a new graphing technique. Hyer made a graph based on Lewin‘s triadic transformations, PAR, REL, and LT and one transposition (DOM).9
Perfect fifths of figure 1.2 generate the horizontal axis, and major and
minor thirds respectively generate the two diagonal axes. This figure
represents a consonant triad and arrows indicate Hyer‘s four
transformations as they act on a C-minor triad. Each of the three
contextual inversions inverts a triangle around one of its edges, mapping it
into an edge-adjacent triangle. P, for Parallel, inverts around a horizontal
(perfect fifth) edge, mapping C minor to C major; R, for Relative, inverts
9 Brian Hyer, ―Tonal Intuitions in Tristan und Isolde,‖ Ph.D. Diss. Yale University, 1989.
７
around a secondary diagonal (major third) edge, mapping C minor to Eb
major; and L, for Leading-tone-exchange, inverts around a main diagonal
(minor third) edge, mapping C minor to Ab major. The fourth
transformation, D (for dominant), transposes a triangle to the vertex-
adjacent triangle to its left, mapping C minor to F minor.10
Figure 1.2. Hyer’s Tonnetz11
Whenever a movement between two triads involves preservation of two common tones, the single ―moving‖ voice proceeds by step. The cases of L and P are by semitone and the case of R is by whole step. This situation allows extended triadic progressions consisting exclusively of single-voice motion by semitone, represented below figure
1.3.12
10 Richard Cohn, ―Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective,‖ Journal of Music Theory 42, no. 2 (1998), 172. 11 Ibid., 172. 12 Richard Cohn, ―Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late- Romantic Triadic Progressions,‖ Music Analysis, vol 15, no. 1 (1996), 24.
８
Figure 1.3. Richard Cohn’s the Hyper-Hexatonic System13
Cohn 1996 designates each such alleyway as a hexatonic system. Figure 1.3 separates four groups and presents them as four circles, whose larger cyclical arrangement reflects the adjacencies of their respective alleys on the Tonnetz.
Along with Lewin‘s work, there were a number of other studies focused on voice leading. Robert Morris generalized rotational arrays in his paper in 1988.14 Drawing
upon examples from Stravinsky‘s music, he defines several functions CINTN. For example, CINT1 defines an ordered list of the directed pitch-class intervals between
13 Richard Cohn, ―Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective,‖ Journal of Music Theory 42, no. 2 (1998), 175. 14 Robert Morris, ―Generalizing Rotational Arrays,‖ Journal of Music Theory 32, no. 1 (1988).
９ adjacent pitch classes (pcs) in a cyclically ordered pitch-class set P=< P0, P1, …, PN-1>, starting with P0.
Clifton Callender discussed an analytical approach involving voice-leading parsimony in the pitch structure of Scriabin in his 1998 paper between sets of possibly different cardinalities (introducing ―split‖ and ―fuse‖ mappings that allowed for the creation or annihilation of pitch classes).15
Recently, Joseph Straus also attempted to generalize voice-leading systems.16 Straus proposes three criteria, related to the traditional transformation of transposition and inversion, while introducing a broadly applicable model for evaluating voice leadings in his paper. His criteria are uniformity, balance, and smoothness in atonal voice leading: (1) uniformity: the extent to which the voices move by the same interval distance and thus approach traditional transposition; (2) balance: the extent to which the voices move by the same index number and thus approach traditional inversion; and (3) smoothness: the extent to which the voices travel the shortest possible distance.
In example 1.1, Straus shows the voice-leading between two pitch-class sets of the same size: {F, F#, B} and {G, Bb, D}, two triads that are related by neither transposition nor inversion. Within voice-leading of these two sets, he provides six distinct ways of leading the notes of the second. Straus explains uniformity concept by using the example below. He marks with an asterisk transformations that are not ―pure‖ and he
15 Clifton Callender, ―Voice-leading Parsimony in the Music of Alexander Scriabin,‖ Journal of Music Theory 42, no. 2 (1998): 219–233. 16 Joseph Straus, ―Uniformity, Balance, and Smoothness in Atonal Voice Leading,‖ Music Theory Spectrum 25 (2003): 305–352.
１０ writes the number of offsets in parenthesis.
Example 1.1 arranges the six voice leadings in order according to
their relative uniformity, measured by the extent to which the voices
move by the same or nearly-the-same interval. The first voice leading
closely approximates a transposition at T3. The bass actually does move
by T3. The soprano, however, moves by T2 (a semitone ―too low‖) and
the alto moves by T4 (a semitone ―too high‖). This first voice leading is
defined as offset by 2 semitones from a transposition at T3 indicated by
the (2) at the bottom of the analytic diagram. The second voice leading
is an instance of ―near-transposition‖: two of the three voices move by
the same interval. It is nonetheless considered less uniform than the first
because it has a higher offset, (3).17
Straus‘s offset number plays a role in measuring degree of uniformity. He provides two other measures for judging voice leading. The first judges voice leadings as relatively uniform depending on the number of voices that move by the same interval, a condition termed consistency. Another way of measuring voice-leading uniformity involves the extent to which the voice-leading intervals diverge.
17 Ibid., 315.
１１
Example 1.1. Voice-leading Uniformity in the progression from {F, F#, B} to
{G, Bb, D}18
Straus‘s concepts of uniformity and balance involve Ian Quinn‘s fuzzy transposition of pitch sets.19 He considers transposition by a measurable distance. Inversion could be generalized into a concept of voice-leading balance, in other words, the extent to which the voices can be understood to flip symmetrically around some common axis.
Uniformity refers to the transposition-like quality of the voice leading; balance refers to its inversion-like quality.
In example 1.2, the first voice leading is the most balanced, the most inversion-like, because two of its voices invert according to index 1 and the index of the remaining voice is offset by only two semitones. The sixth voice leading is the least balanced, the least inversion-like, because its three voices describe axes of inversion that are widely
18 Ibid., 315. 19 Ian Quinn, ―Fuzzy Transposition of Pitch Sets.‖ Paper presented at the Society for Music Theory Conference, Baton Rouge (1996), cited in Straus 2003.
１２ dispersed among the possibilities and would require an adjustment of 6 semitones of repair the deviation.
Example 1.2. Voice-leading Balance in the progression from {F, F#, B} to
{G, Bb, D}20
The smoothness concept that he applies shows the degree of change in the voice leading by taking the sum of the intervallic distances traversed by the voices. (See example 1.3.)
20 Ibid., 319.
１３
Example 1.3. Voice-leading Smoothness in the progression from (a) {F, F#, B} to {G, Bb, D} and (b) {D, G} to {A, C, C#}21
John Roeder suggests a different perspective on voice leading in atonal music. He uses vectors to present individual motions among chord members and he maps registrally corresponding chord members onto one another through these vectors. He analyzes atonal music by focusing on voice leadings in pitch space.22
21 Ibid., 321. 22 ―Voice Leading as Transformation,‖ in Musical Transformation and Musical Intuition: Essays in Honor of David Lewin, ed. Raphael Atlas and Michael Cherlin, Roxbury, MA: Ovenbird Press, 1994.
１４
Julian Hook developed the neo-Riemannian work by Lewin, Cohn, and others by focusing upon the group structure of triadic transformations.23 In his paper, an ordered triple,
More recently, a theory of voice leading has been developed with a geometrical perspective. Callender, Quinn, and Tymoczko represent voice leading in a geometrical space which mathematicians call an orbifold.25 Line segments in higher-dimensional spaces show relationships between the notes of one chord and those of another. In the geometrical space, shorter line segments mean more similar chords.
Joseph Straus, in his 2005 article, presents a somewhat geometrical system of ―set- class‖ voice leading. He shows a graph of set-class connection by showing how set classes of a given cardinality relate via maximal common-tone retention and minimal voice leading motion. In the trichordal space shown in Example 1.4 below, set classes share an edge (i.e., are adjacent on the graph) if two forms share two common tones and otherwise differ by a semitone. This is very different from Cohn‘s triadic maximally voice leading, where all sets belong to Forte set class 3-11 (and share common tones,
23 Julian Hook, ―Uniform Triadic Transformations,‖ Journal of Music Theory 46 (2002): 57–126. 24 For example, in Hook‘s theory, <-,0,0> indicates the neo-Riemannian operation P because the root remains the same in all cases but the mode is changed. 25 Clifton Callender, Ian Quinn, and Dmitri Tymoczko, ―Generalized Voice-leading Spaces,‖ Science 320 (2008): 346–348 Dmitri Tymoczko, ―The Geometry of Musical Chords,‖ Science 313 (2006):72–74.
１５ otherwise differing by a semitone).26
Example 1.4 Straus’s trichordal voice-leading space27
26 Joseph Straus, ―Voice Leading in Set-Class Space,‖ Journal of Music Theory 49 (2005): 45– 108. 27 Ibid., 75.
１６
Chapter 2: An Appropriate Analytic Methodology for Atonal Music
Chapter 2.1: Consideration of Motive
When most people listen to music, they concentrate on the beginning part to catch the music‘s characteristics. When they perceive the features of the music, their cognitive system sets up an expectation of a particular distinctive character to follow. This prevailing cognitive situation also appears in atonal music. The expectations set up by listeners‘ memorable first perceptions drive their subsequent understanding of the piece. I, like most analysts of post-tonal music, therefore focus on motivic connections in this music. Below is Parry‘s thought about the motive, quoted by Dunsby:
It is in fact the shortest complete idea in music; and in subdividing works
into their constituent portions, as separate movements, sections, periods,
phrases, the units are the figures, and any subdivision below them will leave
only expressionless single notes, as unmeaning as the separate letters of a
word.28
28 Jonathan Dunsby, The Cambridge History of Western Music Theory, Thomas Christensen, ed. Cambridge: Cambridge University Press, 2002, p. 907.
１７
The motive could be a compression of the piece as a whole and might provide the first clue not only to connect to a later part, but also to anticipate the next progression.
Music aggregates many tones, presumably ordering them according to latent principles.
By finding these latent logics, we can begin to realize how the music is coherently organized. Thus, numerous theorists, from Schoenberg himself to Rudolph Reti to
Christopher Hasty and many others, have considered the ―‗motive‘ as a critical element of the whole modern music analytical enterprise.‖29 In atonal music, where the language of tonal music to which listeners are accustomed is absent, motivic patterning is very significant for the music‘s comprehensibility.
29 Ibid., 913.
１８
Chapter 2.1.1: Pattern-completion and Prerequisite conditions30
The ―pattern completion‖ idea is an analytic methodology for atonal music analysis proposed by Joseph Straus in his 1982 article entitled ―A Principle of Voice Leading in the Music of Stravinsky‖: it presumes a combination of assumptions from cognitive psychology and Schenkerian analysis, by which the analyst associates notes interspersed far apart from one another. In this chapter, I will examine the theoretical assumptions and subsequent problems of the pattern completion concept, and then propose ways of modifying its drawbacks.
Straus‘s pattern completion concept has three basic assumptions. First, it takes a four-note cell as a complete pattern in the tonal space of unordered pitch-class sets.
According to Straus, the reason for choosing the specific number ―four‖ is based on an assumption borrowed from results in cognitive psychology, whereby a human being memorizes three- or four-digit numbers easily. However, he failed to provide satisfactory information and theoretical foundation to support this assumption.
Cognitive psychologist George A. Miller discussed a close correspondence between
30 I dealt with this content in my thesis. Yeajin Kim, ―A Suggestion of the Analytic Method Reconciling Joseph Straus‘s Associational Model and Pattern Completion : Analyses of YunIsang‘s Selected Works,‖ Yonsei University, 2007.
１９ the limits of one-dimensional absolute judgment and the limits of short-term memory.31
He asserted people‘s maximum performance on one-dimensional absolute judgment can be characterized as an information channel capacity with approximately 2 to 3 bits of information. Moreover, Miller discussed the second cognitive limitation, memory span.
Miller observed that memory span of young adults is approximately 7 items, the idea of a
―magical number 7.‖
Straus‘s second basic assumption has focused upon the expectation that, if a listener hears a specific pitch-class set as a strongly memorable germ motive, he will tend to recognize it without difficulty even though a member of the pitch-class set is missing because he fills up conceptually the missing note to complete the entire germ motive. In other words, by virtue of his strong expectation, the listener is eager to complete the motive set perceptually, and ultimately will have heard it in conception although the motive is incompletely presented.
The last assumption concerns the Schenkerian perspective although he does not state as such clearly. By means of the concept of prolongation, one could have a license to associate notes interspersed in a large-scale tonal space if one feels the process to be meaningful. Straus allowed the analyst to illuminate musical events not only in the foreground, but also in the middleground and background. But the notion of
31 George Miller, ―The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Processing Information,‖ Psychological Review 63 (1956): 81–97.
２０ prolongation in a post-tonal context is inherently problematic, as Straus himself admits, so we are left unsure as to the criteria for the process of meaningful association.32
As suggested above, Straus‘s pattern-completion idea has some problems in terms of its logic. First, it lacks a full explanation of why he has chosen a four-digit number as the limit for human memory capacity. More than that, the most important question is whether the human capacity regarding numeral information could be equally transferred into other domains, here the domain of information. Straus‘s theoretical justifications for his choice are too weak to accept without a question.
Straus assumes that this normative unit becomes so engrained in the listener‘s consciousness that the sounding of part of the pattern creates an expectation for the completion of the pattern through repetition by diverse transformations such as inversion, retrograde, retrograde-inversion so on. When a normative unit of n elements has been established, the appearance of any subset of that unit containing n1 elements will create an expectation for the single missing element, he maintains. Further, he holds that this principle is also valid for structurally related tones even if they are widely separated in time.
Studies related to the perception of atonal music data back to the 1960s.
Numerous psychologists focused on experiments using dodecaphonic series, non- traditional chords, and Tn, TnI operations on pitch-class sets. Among these psychologists, Bruner and Gibson experimented with the intervallic structure of
32 Joseph Straus, ―A Principle of Voice Leading in the Music of Stravinsky,‖ Music Theory Spectrum 4 (1982): 106–124.
２１ nontraditional harmonic sonorities as a hypothesized basis for their aural similarity or dissimilarity.33 However, the results of their experiments do not suggest any particular pattern that would distinguish the responses to a special set from those of non-special sets. The experimental results have led to the conclusion that the perception of similarity among contemporary pitch structures is not an inherent property of the auditory system.
These results may be explained in part by the widespread difference in experience and training for most listeners between that of tonal music vs. post-tonal music.
According to music psychologists, such as L. L. Cuddy, A. J. Cohen, and J. Miller, perception and memory of musical tones are related closely to the tonality of a melody and the harmonic background, in which the melody unfolds, because listeners are accustomed to tonal music, although they differ according to their individual musical experiences.34
Cuddy, Cohen, and Miller see diatonicism and cadence as central factors in the recognition of melodic structure. In turn, diatonicism and cadence could serve as internal criteria for the judgments a listener makes when he or she encounters a tune, which deviates from his or her expectations. The analytic data obtained through multiple experiments by these researchers demonstrates that a listener‘s capacity for recognition
33 Cheryl L. Bruner, ―The Perception of Contemporary Pitch Structures,‖ Music Perception 2, (1984): 25–39. Don B. Gibson, ―The aural perception of similarity in nontraditional chord related by octave equivalence,‖ Journal of Research in Music Education 36 (1998) : 5–17. Gibson, ―The effects of pitch and pitch-class content on the aural perception of dissimilarity in complementary hexachords,‖ Psychomusicology 12 (1993) : 58–72. 34 Rudolf E. Radocy and David J. Boyle, Psychological Foundations of Musical Behavior, (Illinois: Charles C. Thomas Publisher Press, 1997).
２２ deteriorates when he or she is confronted with non-diatonic musical structures. In conclusion, the cognition of a certain pattern is determined by the listener‘s pre-learning
(the sum of their musical experiences), and thus the serialization of aural information differs among listeners, in response to different works, because of differences among the basic units of information that they are capable of apprehending.
Straus‘s first theoretical assumption remains questionable in that he has neglected
the discrepancies in different listeners‘ pre-learning and the differing degrees of
originality, i.e., unfamiliarity, among individual works. Moreover, it is also in doubt
whether his ―four-digit number‖ condition could necessarily be transferred to the
situation of a four-note cell in atonal music, which by definition lacks diatonicism and
the sense of tonality.
Another problem that Straus‘s approach poses lies in his application of a quasi-
Schenkerian perspective to the analysis of post-tonal music without a full explanation of his theoretical assumptions. Some years later, in 1988, he proposed a new analytical model, the ―associational model,‖ which is in contrast to the previous prolongational model. He still failed, however, to present unequivocal criteria as to when the analyst chooses a note as a more structural one than others.35
I will suggest a new analytical model that depends upon a critical examination not only of Straus‘s theoretical assumptions and his actual analyses, but also of those of
35 These criteria are of course supplemented by later theorists like Fred Lerdahl who proposed the ―salience conditions‖ in his 1988 article. I will discuss his salience conditions later in this dissertation. See Fred Lerdahl, ―Atonal Prolongational Structure,‖ Contemporary Music Review 4 (1989): 65–87.
２３ other theorists‘ supplements and modifications to his methodology. For this project, I will take Straus‘s concept of pattern completion and concentrate on a specific normative set, which plays an important role in a given piece. However, I will not limit the choice of the specific motive to a four-element set. Rather, I will allow for the freedom to choose a three, four, five note collection as the significant unit, if the chosen set can be shown to be related organically to the entire piece. The specific pitch-class set must serve as a minimal motive and must repeat itself either literally or in a disguised form
(in transposition or inversion) sufficiently to create in a listener a strongly memorable impression.
In the selection of a special normative set, I will depend upon what I call the
―prerequisite conditions,‖ which are derived from Fred Lerdahl‘s salience conditions. I have adjusted and supplemented Lerdahl‘s conditions for the effective analysis of atonal works. The prerequisite conditions reflect the following circumstances: 1) when notes to be picked out and set in relief are located in the same register; 2) when notes to be picked out have the same dynamics; 3) when notes to be picked out have the same musical articulations or techniques; 4) when notes to be picked out appear in the same metrical position (it does not matter whether these are located on the strong beat or on the weak beat); 5) when notes to be picked out have relatively long durations; and 6) finally when notes to be picked out appear in an important position such as at a beginning or ending of a phrase.
A comparison between my prerequisite conditions and Lerdahl‘s salience conditions follows: First, while Lerdahl stresses the metrically strong beat, I include
２４ weak beats as well as strong beats because the metrical conditions in atonal music deliberately deviate from the norm. Second, while he presents the extreme register as a factor, I propose the condition of the same register: in other words, two notes to be picked out are not necessarily located at the extreme register, but are eligible if they are located at the same register. Lerdahl‘s condition of the same dynamics could be understood and adjusted in the same vein. The reason why I try to apply Lerdahl‘s salience conditions in a more or less loose way is to reflect diverse and expressionistic compositional techniques since 1900 in the analysis. I also add another condition concerning musical articulation and special instrumental techniques because notes highlighted by such musical devices could be audible and discernible and finally be recognized as significant events by a listener. To put it in another way, my prerequisite conditions are more acceptable to atonal music containing diverse experimental techniques than Lerdahl‘s salience conditions.36
Straus‘s pattern completion consists of two fundamental aspects: 1) establishment of a single collection-type or pattern as the normative structural unit for a composition and 2) exploitation of the listener‘s desire for the completion of that unit. Straus considers pattern completion as a key to understanding the musical organization from melodic motives to background structure. He adds that such a cognitive process of pattern completion involves the large-scale tonal space as well as the foreground, in which a web of adjacent notes forms a specific pitch class set.
36 I will apply these prerequisite conditions to selecting members of specific pitch-class sets later in this chapter.
２５
The following example excerpted from Stravinsky‘s Symphonies of Wind
Instruments shows various uses and interactions of the tetrachord [0 1 3 5] in multiple levels. Straus asserts that, once the tetrachord [0 1 3 5] is strongly memorized through a repetition, the missing pitch class 5 is conceptually heard once the remaining elements 0,
1, and 3 have been heard. The bass progression, F-E-D, which forms a [0 1 3] set in its prime form, leads a listener to complete the missing member C, which corresponds to the pitch-class 5 in this case. Moreover, Stravinsky had the missing member appearing at the end of a specific formal unit and completing the germ cell.
Example 2.1. Principal melodic fragment in Stravinsky, Symphonies of Wind Instruments37
37 Joseph Straus, ―The Problem of Prolongation in Post-Tonal Music,‖ Journal of Music Theory 31, no. 1 (1987), 18.
２６
Example 2.2. The principal fragment transposed38
Example 2.3. Associational background of Stravinsky, Symphonies of Wind
Instruments39
Straus implies that there are two possible pitch classes that a listener can expect when he hears three members of a tetrachord. For example, if a listener hears an incomplete tetrachord whose members are C, D, and E (one member of the tetrachord is missing), he will expect either F or B. Whichever pitch class is added to the incomplete
38 Ibid., 18. 39 Ibid.
２７ tetrachord will become a complete tetrachord [0 1 3 5]. The sense of expectation caused by the possibility of the selection arouses tension and relaxation, which serve as important factors in the piece.
２８
Chapter 2.2: Latent axis theory
In atonal music, the concept of an axis is very important with respect to both composition and analysis. In tonal music, an octave system divides tonal materials into unequal parts. For example, one of the primary tonal materials, the perfect fifth, is divided into a major third and a minor third. Although there is a hierarchy within the tones and harmonies in tonal music, in atonal music every tone and each harmonic entity, at least before contextual considerations enter in, is of equal status.
For representing the latent logic within set-classes, one of the useful methodologies is a theory of axis. In atonal music, symmetrical construction in pitch-class sets can lend coherence to music. An axis may be very significant when we understand the interrelationships in atonal music.
The study of the axis theory for comprehending atonal music was first promoted by
Arthur Berger. The axis theory‘s basic concept is adopted from a tonal music perspective.
Berger calls this concept ―centric,‖ that is, ―organized in terms of tone centers but not tonally functional,‖ and he called almost fifty years ago for ―a new branch of theory.‖40
He thought this objective worthwhile since this theory starts from structure in the music itself. Through this perspective, he makes clear Stravinsky‘s latent organization.
40 Arthur Berger, ―Problems of Pitch Organization in Stravinsky,‖ Perspectives of New Music 2, no. 1 (1963): 11–42.
２９
George Perle and Elliott Antokoletz also suggest symmetrical perspective on pitch construction.41 Especially, they analyzed Bela Bartók‘s pieces with respect to symmetry.
The symmetrical segments in Bartók‘s music have a crucial function not only in the local span but also in the larger structure. Antokoletz shows three symmetrical principal cells of Bartók‘s Fourth Quartet.42 He calls these symmetrical sets cell X, cell Y, cell Z and demonstrates axes by the sum of a dyad. The structural priority of cells X, Y, Z in the first-theme group and transition is partly established by the principle of metric departure and return. Of the three basic cells, symmetrical relations may be shown only between X and Z.
41 George Perle, ―Symmetrical Formations in the String Quartets of Bela Bartók,‖ Music Review 16 (November, 1955): 300–312. 42 Elliott Antokoletz, The Music of Bela Bartók, A Study of Tonality and Progression in Twentieth-Century Music, Berkeley: University of California Press, 1984.
３０
Figure 2.1. Bartók, Fourth String Quartet, Mvt. I, X-Y-Z progressions in exposition and development.43
The basic transpositional levels of X (C-C#-D-D#) and Z (G#-C#-D-G) share the same axis of symmetry, C#-D, a principle which forms the basis for a new concept of
43 Elliott Antokoletz, Twentieth-Century Music, Prentice Hall, 1998, p. 128.
３１ tonal centricity in Bartok‘s music. These two symmetrical tetrachords have the same sum
– dyads C#-D (1+2) and C-Eb (0+3) of cell X and dyads C#-D (1+2) and G-G# (7+8) of cell Z that is, those are symmetrically related at sum 3. (Figure 2.2) The axis of symmetry is expressed by the sum of the two pitch-class numbers in a dyad. (See example 2.4 and
2.5)
Figure 2.2. Alignment of two inversionally complementary semitonal cycles intersecting at a dual axis of symmetry of sum 344
44 Ibid., 129.
３２
Example 2.4. Even-Sum (Axis) and Interval Array45
45 Elliott Antokoletz, The Music of Bartók: A Study of Tonality and Progression in Twentieth- Century Music, Berkeley: University of California Press, (1984), p. 74.
３３
Example 2.5. Odd-Sum (Axis) and Interval Array46
In order to demonstrate their functions and interrelations within the music, the properties of symmetrical pitch collections must first be defined. Any collection of two notes is symmetrical, since the two notes are equidistant from an imaginary axis.
46 Ibid., 74.
３４
The concept of Perle‘s special twelve-tone set is introduced by the interval cycle which is a series based on a single recurrent interval and symmetrical pitch formations.47
As one sees in figure 2.3, he suggests that each transposition of the ―white-note‖ diatonic collection be identified as a given seven-note segment of the cycle of fifths, with positive integers representing the number of sharps and negative integers representing the number of flats within the segment. In addition, if we wish to specify the intervallic ordering of the collection, it suffices to identify the tonic or octave-boundary pitch class. The two will not necessarily be identical, and it is often convenient to cite progressions from one diatonic collection to another.
Figure 2.3. George Perle’s interpretation of diatonic collection48
47 George Perle, ―Berg‘s Master Array of the Interval Cycles,‖ The Musical Quarterly 63, no. 1 (Jan., 1977): 1–30. 48 Ibid., 11.
３５
Joseph Straus also proposed the necessity of recognizing the existence of certain poles of attraction in Stravinsky‘s music.49 He focused on the connection between tone centers, the means of progression of tone poles, the prolongation of tone centers and the issue of voice leading. Straus‘s tonal axis could be defined in three characteristics.
1. It must consist of overlapping major and minor triads (for example, EGBD).
In other words, it must have the appearance of a minor or major seventh
chord.
2. It must function in the piece as a referential sonority. It must occur
prominently as a discrete harmony within the piece, particularly in cadential
situations. It must be the essential harmonic generator of the piece; other
harmonies derive from and relate to it.
3. It must embody a conflict or polarity between its two constituent triads. All
axes have the appearance of seventh chords, but not all seventh chords
function as tonal axes. Each of the overlapping triads which constitute the
axis must be shown to have a palpable identity and centricity of its own.50
In Straus‘s article, he considers the motive as the axis at the heart of the work.
Example 2.7 illustrates the relations of the structures in the first movement of Symphony in C, based on the perspective of axis. The normative pattern of the first movement of
49 Joseph Straus, ―Stravinsky‘s ‗Tonal Axis‘,‖ Journal of Music Theory 26, no. 2 (Autumn, 1982): 261−290. 50 Ibid., 265.
３６
Symphony in C, tetrachord 4-11, is very closely related to the other principal structures of the movement, the tonal axis (C-E-G-B) and the pervasive motive (BCG).
Example 2.6. Motive of the first movement of Symphony in C51
Example 2.7. Straus’s axis analysis of the first movement of Symphony in C52
51 Ibid., 285. 52 Ibid.
３７
Table 2.1 presents an overview of Straus‘s axis analyses. This shows the close relation between tonal axis and basic pattern. Oedipus Rex is very much like the first movements of Symphony in C and Symphony of Psalms and the third movement of
Dumbarton Oaks.
Table 2.1. Straus’s axis analyses53
He discusses the large-scale tonal motion from the first chord of the movement to the last chord of the movement, which constitutes motion along the tonal axis from the lower to the upper elements.
In atonal music, the axis perspective is related to inherent fundamentals. Any collection of two notes is symmetrical, since they are equidistant from an imaginary axis.
An actual axis, as opposed to an imaginary one, is represented by the note or pair of notes, either expressed or implied, that form the center of a larger symmetrical construction.
In the next analysis chapter, I will deal with the axis between two pitches by considering pitch space as well as the voice-leading relationships between pitch-class sets.
53 Ibid., 287.
３８
Moreover, I want to present the pivotal role played by this axis in making a coherent construction in music by Webern.
In atonal music, the voice leading of pitch-class sets is often governed by transformational operations of transposition or inversion. In these transformations, there sometimes exists an axis from which the voice leadings depart. For example, consider the two pitch-class sets, {0,1,3} and {1,3,4}. These two sets are related by I4. The inversion transformation implies that there is an axis of symmetry, one half the index of inversion, in this case, 2. The transformation implies a voice leading of pitch class 0 to pitch class 4, about the dyad 1, 3 which remains fixed under the transformation. This axis perspective on voice leading will be dealt with in the chapter 2.2.
３９
Chapter 2.2: Analysis of Webern’s op. 16, no. 2
The Webern‘s op. 16, no. 2 is a canon on Latin Texts for voice and clarinet. The music is very strict in counterpoint, phraseology, and formal structure. This piece‘s form is largely determined by the text, a poem which is taken from Des Knaben Wunderhorn.
Dormi Jesu, mater ridet, Quae tam dulcem somnum videt, Dormi Jesu, blandule! Si non dormis, Mater plorat, Inter fila cantans orat; Blande, veni, somnule.
Sleep, sweet babe! my cares beguiling: Mother sits beside thee smiling; Sleep, my darling, tenderly! If thou sleep not, mother mourneth, Singing as her wheel she turneth: Come, soft slumber, balmily!
Figure 2.4. Latin version and English version54
54 The English version is by Samuel Taylor Coleridge (1772–1834). This was first published in Courier, August 1811.
４０
Webern reinforces the poem‘s structure by marking the principal formal disjunction with a ritardando in measure 7. For this reason, this piece could be separated into two parts. The first part is from m. 1 to m. 7 and the second part is from m. 7 to m. 13.
Because of the canonic style, the two sections overlap in m. 7.
As you see in example 2.8, the clarinet part begins with the melodic gesture consisting of two brief figures: the wide skips and disjunctive contours. The opening melody has a pointillistic character, a fragmented texture that is characteristic of
Webern‘s music. Following the line will become easier once its organization is better understood. Using the concepts of pitch and pitch class, we can begin to understand how the melody is put together. Gradually, with some knowledge of pitch and pitch-class intervals, the sense of each musical fragment and the interrelations among the fragments will come into focus. The two lines, the voice part and clarinet part, proceed in strict inversional canon. The beginning part starts with a tetrachord, {t, 6, 9, 0}, within a very quiet sound reflecting the content of the poem and this melody is inversionally related to the voice‘s melodic line, {4, 8, 5, 2}.
４１
Example 2.8. Webern’s Op. 16, No. 2, mm. 1–2
Since the tetrachord of the clarinet‘s first measure is inversionally transformed to the voice part, these two sets‘ prime forms are same. In pitch-class set theory, we can deal with pitch-class sets in two ways, unordered pitch-class set or ordered pitch-class set.
The axis of symmetry is expressed by one half the sum of the two pitch-class numbers in a dyad. However, there is the possibility to have two axes in ordered pitch classes. For example, the axis between B3 and C#4 is C4 but, the axis between B3 and
C#5 is F#4. In other words, I0 could have dual axes, C or F#. (Figure 2.5.) So, the axis in a dyad depends upon how we use the basic concept of order in pitch classes. In this analysis, I will consider the pitch-class sets with the concept of ordered pitch classes.
４２
Figure 2.5. Sum 0 (=12) axis: C (or F#)
Example 2.6 shows the G axis between the clarinet and voice lines in mm. 1–2.
The two lines cross, intersecting on the absent G axis.
Figure 2.6. G Axis between clarinet and voice lines in mm. 1–2
４３
In the opening figure, four-note sets each maintain their internal content and ordering, while changing their positions relative to each other. This song presents few difficulties in segmentation; the unique timbral and rhythmic identities of coincident gestures are recognized, as are the ―integral‖ gestures. Of course, the strict canon ensures the repetition of most set-classes.
Figure 2.7 shows not only the analysis segmented by a tetrachord in Webern‘s op. 16, no. 2 but also the inversionally symmetrical relations. The reason why I analyze this piece as based on a tetrachord is that the first four notes separated by a quarter rest are played before the voice part and this may make listeners memorize four notes as a crucial motivic pattern. Every detail, not just of pitch but also of rhythm, remains constant between the leading voice (dux) and companion (comes). Apart from instrumentation, only subtle changes in dynamic markings individuate corresponding junctures of the two textural strata.
If we consider mm.1–2, we can easily figure out that Webern did not willingly use common tones to the next progression. So, listeners may feel the appearance of a new collection to be very strange. However, Webern may focus on the internal coherence.
Most of the voice-leading progressions from the motivic pattern, set class 4-12 [0236], pemit the retention of a maximal common subset. In other words, the voice-leading involves minimal offset since the offset number given in parentheses in figure 2.8 is usually 1.55
55 Here, offset number means the number of different notes from the previous pitch-class set.
４４
Figure 2.7. Webern’s Op. 16, No. 2
４５
Figure 2.8. Voice leading in pitch space
４６
Example 2.9. Webern’s Op. 16, No. 2, mm. 1–6
４７
Example 2.10. Webern’s Op. 16, No. 2, mm. 7–13
４８
The next segmentation follows the text‘s phrase. The first phrase‘s ending marked by the circled comma. The voice part in mm. 2–3 consists of seven notes, {4, 8, 5, 2, 1, 2, 6,
7} whose prime form is 7-4 [0123467]. Once we consider the text phrase, we can easily understand that this seven-note collection could be a melodic line. Interestingly, the clarinet part synchronically playing as an accompaniment in mm. 1–3 also contains the same pitch-class set, 7-4 [0123467]. This identical coincidence in the beginning measures not only suggests that the set class 7-4 could play a crucial role as a thematic pattern in this music but also gives us more reasonable provision.
Example 2.11. Seven-note collection segmented by the text
４９
Examples 2.12 and 2.13 are analyzed by segmentations suggested by the text. The beginning set, set-class 7-4 [0123467] proceeds to set-class 8-13 [01234679], 7-1
[0123456], 7-2 [0123457], 6-Z3 [012356], 7-29 [0124679]. As one can see in the analyzed score, there is not any common pitch-class set.
５０
Example 2.12. Webern’s Op. 16, No. 2, mm. 1–6
５１
Example 2.13. Webern’s Op. 16, No. 2, mm. 7–13
５２
Now I want to focus on the aspect of inversional symmetry in Webern‘s op. 16, no. 2, because the two lines of this piece are perfectly related by the I2 transformation.
Symmetrical construction and the consistent inversionally symmetrical relations provide a further means for symbolizing aspects of the poetry, at the same time contributing to the articulation and control of the work‘s formal structure. A special poetic as well as essential musical function is assigned to the consistent inversionally symmetrical organization of the forms around a single axis of symmetry.
The opening section symmetrically progresses to the voice part from the initial tetrachord, {t, 6, 9, 0}. These four-note segmental properties are motivically exploited as a means of defining sectional and overall formal symmetry. As noted above, the axis of symmetry is G, which is registrally determined according to the principle of symmetrical unfolding in the whole piece. The I2 transformation which includes all transformations in this music has the dual axes, G and C#. These axes are the inherent measure of voice leading between voice and clarinet lines. However, these axes are exploited in the structural functions by means of systematic assignments.
One of the characteristics of Webern‘s op. 16, no. 2 is that Webern hardly uses the common tones in adjacent pitch-class sets. Examples 2.16 and 2.17 show the common tones in this music. Interestingly, all common tones are identified with G and C#, the axes of two parts‘ melodic lines. The appearance of G in m. 6 is placed on the repeated word, dormi, in a relatively high register, and the relatively emphasized rhythmic location. In other words, by means of the common tones, the axes of symmetry are displayed in structurally important points. The occurrences of G and C# in the ending part appear at
５３ rhythmically weak points, and the dynamic level falls, following the progression towards sleep in the text (veni somnule). Structural coherence here is primarily produced by the pervasive use of the axis of symmetry.
５４
Example 2.14. Webern’s Op. 16, No. 2, mm. 1–6
５５
Example 2.15. Webern’s Op. 16, No. 2, mm. 7–13
５６
Example 2.16. Webern’s Op. 16, No. 2, mm. 1–6
５７
Example 2.17. Webern’s Op. 16, No. 2, mm. 7–13
５８
Lastly, I want to try to analyze according to twelve-tone theory, because there is just a four-year difference from 1928, the starting year of Webern‘s twelve-tone serial compositions. In this piece, I find two twelve-tone series, although they are not presented in the same part. Example 2.18 represents four pitch-class sets, two 8-13 [01234679] and two 4-13[0136] sets. As their Forte labels show, these sets are in the complement relation.
The connection between the pitch-class set 8-13 in mm. 3–4 of the clarinet part and the pitch-class set 4-13 in m. 6 of the voice part makes a twelve-tone series. The connection between the pitch class set 8-13 in mm. 4–5 of the voice part and the pitch class set 4-13 in m. 5 of the clarinet part makes another twelve-tone series. The proximity of literal complements would seem significant in light of Webern‘s increasing interest in aggregates, which are fundamental from the point of view of twelve-tone composition, which he adopts systematically in his next works.
Webern‘s op. 16, no. 2 represents not only the peak of his canonic technique but also the most prominent early twelve-tone example of his use of symmetry. Moreover, in this piece, one can verify that both local and large-scale structural unity can be achieved thereby in the atonal idiom.
５９
Example 2.18. Webern’s Op. 16, No. 2, mm. 3–6
６０
Chapter 3: Theoretical Exposition of Smooth Voice-Leading Systems for Atonal Music
Chapter 3.1: Voice-leading systems within the Schoenberg hexachord56
Schoenberg‘s signature hexachord is a well-known name cipher, the pitch-class set
{E-flat, C, B, B-flat, E, G}, making use of the German note names Es, H, and B, as well as C, E, and G: SCHBEG. Berg uses this cipher in his Kammerkonzert, and Schoenberg bases the melody of his op. 23, no. 4 on a form of it.
{E-flat, C, B, B-flat, E, G}
(Es-C-H-B-E-G)
6–Z44 hexachord
Figure 3.1. Schoenberg’s Signature Hexachord
56 I presented Voice-Leading Systems of Schoenberg‘s Signature Hexachord with David Clampitt at Music Theory Mid-West (MTMW), 21st Annual Conference, Miami University, Ohio, USA, 2010.
６１
{0, 3, 4, 7, t, e} 6–Z44 hexachord Inverted form {2, 3, 4, 7, t, e}:
Example 3.1. Schoenberg, Op. 23, No. 4, m. 1
An inverted form of the signature set proper with C exchanged for D, appears in the right-hand melody of m. 1. The hexachord is a member of set class 6–Z44. This chapter introduces a partition of 6–Z44 into four systems, or orbits, in close analogy with
Cohn‘s four systems of maximally smooth cycles of hexatonic triads, and offers analytical examples of pairs of adjacent sets from this perspective. The justification for the term system, here as in Cohn‘s neo-Riemannian formulation, is that each of the four subsets that constitute the partition form a Generalized Interval System or GIS, as defined by David Lewin.
６２
X={0, 1, 2, 5, 6, 9}
Two distinct inversions that:
(1) hold fixed a pentachord (that is, the maximum number of common tones)
(2) exchange pitch classes separated by interval class 2
Figure 3.2. Construction of the 6–Z44 Partition
The partition may be naively constructed on the basis of noting that an element of
6–Z44, e.g., X={0, 1, 2, 5, 6, 9}, may be inverted in two distinct ways that (1) hold fixed a pentachord (that is, the maximum number of common tones) (2) exchange pitch classes separated by interval class 2 (rather than interval class 1, as stipulated under Cohn‘s maximally smooth voice leading).
* J - transformation X={0, 1, 2, 5, 6, 9} ⇒ y'={0, 1, 4, 5, 6, 9} * K - transformation X={0, 1, 2, 5, 6, 9} ⇒ y"={0, 1, 2, 5, 8, 9}
Figure 3.3. J and K transformation
６３
For example, sliding pitch class 2 in X to pitch class 4 yields y'={0, 1, 4, 5, 6, 9}, while sliding pc 6 in X to pc 8 yields, y"={0, 1, 2, 5, 8, 9}. By inspection, y' and y" are distinct inverted forms of X.
prime X-forms:
{ a, a+1, a+2, a+5, a+6, a+9 }
<1 1 3 1 3 3>
inverted y-forms:
{a, a+1, a+2, a+5, a+8, a+9}
Figure 3.4. prime X-forms and inverted y-forms of 6–Z44 hexachord
We have implicitly created two contextually-defined inversions, J and K, based upon the configuration of any element in 6–Z44: Let prime X-forms have the configuration {a, a+1, a+2, a+5, a+6, a+9}, up to rotation, and inverted y-forms have the configuration { a, a+1, a+2, a+5, a+9, a+10} up to rotation, with a being a pitch-class number and the additions understood modulo 12.
One could also describe the configuration in terms of the Chrisman/ Regener/
Morris interval string notation, where the prime form configuration is <113133>. That is, construed as a scale, ―step sizes‖ are either 1 or 3 semitones. Any member of set class 6–
Z44 can be colloquially described as containing an ―island‖ of three interval class-1-
６４ related pitch classes, another ―island‖ of two interval class-1-related pitch classes, and an
―isolated‖ singleton pitch class, each island separated from the others by interval class 3.
Figure 3.5. J - transformation
If we choose the 3-pitch-class island as a point of orientation, in prime X-forms it is followed by the 2-pitch-class island, then by the isolated pitch class. In inverted y- forms, the 3-pitch-class island is followed by the isolated pitch class, then by the 2-pitch- class island (always up to rotation). We may contextually define the J-inversion as that inversion that moves by ic 2 one endpoint of the 3-pc island, such that the old 3-pc island becomes the new 2-pc island and the old 2-pc island becomes the new 3-pc island.
６５
Figure 3.6. K - transformation
We may contextually define the K-inversion as that inversion that moves by ic 2 one endpoint of the 2-pc island, such that the old 2-pc island becomes the new isolated pc and the old isolated pc becomes the new 2-pc island.
{a, a+1, a+2} ⇒ {a+4, a+5, a+6} T4 J transposes the 3-pc island of X ; K leaves the 3-pc island fixed
Figure 3.7. K-of-J-of-X
What is the effect of applying J to a prime X-form, followed by applying K to the resulting inverted y-form? A general X-form has the configuration of a set {a, a+1, a+2, a+5, a+6, a+9}, with the additions understood modulo 12. J sends a prime X-form with C
６６ to an inverted y-form, and K sends that inverted y-form to a prime X-form, a transposed form of the original. Which transposition is it? Clearly, it is T4, since if the original 3-pc island consists of the pitch-class set {a, a+1, a+2}, then after applying J, the new 3-pc island is the pc-set {a+4, a+5, a+6}, with the additions understood to be modulo 12. It follows (since K leaves the 3-pitch class island fixed) that K-of-J-of-X yields T4 of X, and repeating this pair of applications 3 times returns X to itself.
Similarly, applying J to an inverted y-form, followed by K, sends the original 3-pc island {a, a+1, a+2} to the new 3-pc island {a+8, a+9, a+10}. It again follows (since K leaves the 3-pc island fixed) that K-of-J-of-y yields T8 of y. Again, repeating this pair of applications 3 times returns y to itself.
By exactly the same analysis, reversing the order of the contextual inversions exchanges the effect upon prime X-forms and inverted y-forms: J-of-K-of-X is T8 of X, while J-of-K-of-y is T4 of y. Taken altogether, we have a cycle of 6 distinct elements of
6–Z44, with adjacent elements inversionally related, subject to conditions (1) and (2) above—that is, only one pitch class moves, with a pentachord held as common tones, and the pitch class moves by interval class 2. Since the voice-leading move is uniformly by interval class 2 rather than by interval class 1, we might call the Schoenberg signature hexachord a relatively smooth cycle. The general voice-leading system of the 6Z-44 hexachord has the following form:
６７
Figure 3.8. A General Voice-Leading System of the 6-Z44 Hexachord
The construction partitions the 24 elements of 6–Z44 into four disjoint subsets, each of 6 elements, each supporting such cycles. The four disjoint subsets of the partition are understood mathematically as orbits of a group, acting on set class 6Z-44. For example, by analogy with Cohn, we may define the Western System.
６８
Figure 3.9. Western System (a=0)
Below we see the four systems as a partition of 6-Z44 hexachord into 4 orbits.
６９
Figure 3.10. Partition of 6-Z44 into 4 Orbits
７０
Each orbit may be characterized by the augmented triad that remains invariant throughout the cycle, the augmented triads of the form {a+1, a+5, a+9}, as a ranges from
0 to 3. Therefore, each orbit may be identified with the parameter a, with a=0, 1, 2, and 3.
Figure 3.11. A General Voice-Leading System of the 6-Z44 Hexachord
Contextual inversions J and K generate a group (completely analogous, indeed isomorphic, to the neo-Riemannian hexatonic group generated by P and L). The
Schoenberg signature transformation group G consists of {Identity, J, K, JK, KJ, and
JKJ}, where the product operation is composition of functions.
What distinguishes the group G from the group of hexatonic triadic transformations H={Identity, P, L, PL, LP, PLP} is not any aspect of mathematical structure, but the fact that H acts on (that is, consists of transformations of) the set-class
７１
3–11 of harmonic triads, while G acts on the set-class 6–Z44. This is a major musical difference, because of the completely different musical contexts in which these sets arise.
G = {Identity, J, K, JK, KJ, JKJ}
H = {Identity, P, L, PL, LP, PLP}
Figure 3.12. The Schoenberg signature group G and the hexatonic triadic group H
Recall that the neo-Riemannian P and L transformations are contextual inversions, that is, they are applied on harmonic triads, and alter the triadic constituents in characteristic ways. In particular, they are inversions that keep two triadic common tones and move the remaining pitch class by the minimal distance, interval class 1. Composing
P with L yields a contextual transposition or Riemannian Schritt, T4 or T8 of the triad acted upon, according to whether it is major or minor, and so arises the by-now familiar hexatonic cycles, with a partition of set class 3–11 into 4 systems, labeled by Cohn according to the cardinal points of the compass.
In the systems of the Schoenberg signature hexachord, the maximal displacement in terms of pitch content is effected by the transformation JKJ, equal to the transformation KJK. This is comparable to the hexatonic pole transformation PLP=LPL.
All of the machinery of Lewin‘s theory of Generalized Interval Systems is available. In particular, recall that since our group is non-commutative, for each system there is a dual
７２ commuting group GIS. In the Schoenberg signature systems, as in the hexatonic systems, these dual commuting groups are particular 6-element subgroups of the usual group of transpositions and inversions. The dual commuting groups are dependent on the particular orbit of hexachords being acted upon, and are of the form G'a = {T0, T4, T8,
I2a+2, I2a+6, I2a+10}.
G’a = {T0, T4, T8, I2a+2, I2a+6, I2a+10}
Figure 3.13. The dual commuting groups, for a = 0, 1, 2, 3
７３
Chapter 3.1.1: Examination of 6-Z44 hexachord systems
The first of two analytical applications involves an aspect of the second movement of Webern‘s Op. 27 that has been overlooked, because it is secondary to the row-forms employed in the piece, while depending upon the registral deployment of these rows about A4 as a pitch-space axis of symmetry. As is well known (Westergaard
1963, Bailey 1991), the row structure of the movement consists of a canon at the eighth note between prograde and inverted forms. The notes in order position 12 serve also in order position 1 for subsequent rows, creating four pairs of rows, all of which either begin or end with the dyad Bb5-G#3.
７４
P0: G# A F G E F# C C# D Bb B Eb
I0: Bb A C# B D C F# F E G# G D# #: 1 2 3 4 5 6 7 8 9 10 11 12 (=1)
P7: D# E C D B C# G G# A F F# Bb
I5: Eb D F# E G F B Bb A C# C G# #: 1 2 3 4 5 6 7 8 9 10 11 12 (=1) :||:
P2: Bb B G A F# G# D D# E C C# F
I10: G# G B A C Bb E Eb D F# F C# #: 1 2 3 4 5 6 7 8 9 10 11 12 (=1)
I5: C# D Bb C A B F F# G D# E G#
P7: F E G# F# A G C# C B Eb D Bb #: 1 2 3 4 5 6 7 8 9 10 11 12 (=1) :||
Figure 3.14. Row Structure of Op. 27, II
The second movement of the Variations for Piano has a pointillistic, fragmented texture that is characteristic of Webern‘s music. The disjunctive contour encourages the listener to adopt strategies of grouping based upon register. The analysis will therefore take into account registrally determined segmentations, in which forms of 6–Z44 emerge both on the local surface and over larger musical spans.
The piece begins with the crucial dyad, Bb5-G#3, followed by A4‘s, struck by both right and left hands. The initial registral disjunction, a pitch-class double-neighbor
７５ figure about A, but registrally deployed 13 semitones above and below the repeated A4, prepares the listener to hear in terms of compound melody, in three registers—high, low, and middle. The movement is subsequently arranged into pairs of elements, distinguished by changes of register and articulation and separated by rests. Forms of the Schoenberg signature hexachord are prominent resultants of the composite melody in the opening measures.
Example 3.2. Webern, op. 27, mvt. 2, mm. 1–2
The grace notes in mm. 2 and 3 are part of the row structure, but in our initial segmentation the grace notes are omitted. The first six salient notes of the movement
(boxed with a dark border) yield 6–Z44 form {12589t}; discarding the final D3 and returning the following E6 yields the inverted form {14589t}, boxed with a light border.
Another way of putting this: an initial pentachord of set class 5–Z37 is completed in two
７６ ways, according to registral extremes, to produce forms of 6–Z44, as shown in example
3.3.
Example 3.3. Common-tone Subset and Covering Superset in mm. 1–3/1
The two signature sets are contained within a 7–22 superset, which is isolated as an introductory musical gesture by virtue of the 2-eighth-note rest that follows, as opposed to the single eighth-note rests separating dyads elsewhere, until the end of the movement with the 4-eighth-note rest preceding the final Bb5-G#3 dyad. The two 6–Z44 forms are inversionally related under K, in this case I6-related, in particular symmetrical in pitch about A4.
That is to say, the two forms are adjacent in the relatively smooth voice-leading cycle of the Western orbit, that is, the orbit determined by parameter a=0. The adjacent
７７ notes A, C#, F in measures 1 & 2 make up the characteristic invariant augmented triad for this orbit.
Example 3.4. The Invariant Augmented Triad for the Western System
The next appearance of the Bb5-G#3 dyad is in m. 5, now with G# in the dux. The segment marked off by this initial dyad and again concluding in m. 6 with the D3-E6 dyad also contains two adjacent forms within the Eastern cycle.
７８
Example 3.5. Webern, op. 27, mvt. 2, mm. 5–6
This time we include the Eb4/D#5 grace notes that precede the concluding D/E dyad, arguing for their salience by the fact of the pitch-class duplication, the fortissimo dynamic, their sonorous middle-register voicing, and their pitch-class upper/lower neighbor status. They are also the overlapping final and initial elements in the row structure: simultaneously final elements of the initial pair of prograde and inverted rows and initial elements of the second pair of inverted and prograde forms. The pair of 6–Z44 sets result from the exchange of the initial G# and B-flat, rather than the final D and E, to form the adjacent Eastern-orbit sets {2, 3, 4, 7, 8, e} and {2, 3, 4, 7, t, e}. The Eastern system is the system with parameter a=2, and it is characterized by invariant augmented triad {3, 7, 11}. Again, this augmented triad is formed by adjacent notes in the segment, enclosed within the framing G#/Bb and D/E dyads.
７９
Example 3.6. The Invariant Augmented Triad for the Eastern System
We argue for the independence from the row structure of the events in mm. 1–3 vs. those in mm. 5–6 by the fact that the order positions of the dyads in the former are 1,
2, 3, and 5, while in the latter they are 10, 11, 12=1, and 2. This is evidence, although not determinative, for intention on the part of the composer.
We have seen local manifestations of pairs of adjacent 6–Z44 sets, both within the opening 6 measures of the piece, which are of course repeated in the binary form. Now we consider the entire movement, using the registral separation of voices that has been promoted by Webern‘s pitch organization, from the opening measure. Taking extremes of register as a measure of salience, with high and low pitches defined as above F#5 and below C4 (8 semitones above and below the central axis A4), we have {2, 3, 4, 7, t, e} and {2, 3, 4, 7, 8, e}, K- and I6-related, symmetrical in pitch about A4, with the exchange dyad being Bb5-G#3, the structural marker dyad for the movement. This reproduces the pairing found at the local level in mm. 5–6.
８０
Example 3.7. Webern, op. 27, mvt. 2
８１
Another example of Webern‘s employment of forms of 6–Z44 sets from the same orbit is in the Little Piece, op. 11, no. 3. The segmentation resulting in these forms is based on registral and timbral grounds. The ‗cello‘s first four pitches together with the following F#3 and G3 in the right hand of the piano (the highest register pitches in the piano line) form the set {0, 3, 6, 7, t, e}, while the entire right hand of the piano part (all piano pitches F#2 and above) forms the T4-related set, {2, 3, 6, 7, 8, e}.
Example 3.8. Webern, Three Little Pieces, op. 11, no. 3
These sets both belong to the Eastern system. This correspondence, albeit weaker in force than the Variations example, is an aspect of coherence for this extremely brief movement.
I conclude with some theoretical observations on the phenomenon of sets that support relatively smooth voice-leading cycles.
８２
Table 3.1. P- and Q- Cycles of Larger Set-Classes57
Only 6–Z44 and the whole-tone-plus-1 heptachord support cycles where the pitch-classes exchanged are uniformly separated by interval class 2. If we further stipulate that the pitch-class motion be smooth in the sense that the pitch classes exchanged be adjacent in the scalar ordering of the sets, which is to say, that the sets be related by a Cohn-flip, in the sense of Lewin 1996, then 6–Z44 is unique. In the case of the whole-tone-plus-1 set, the moving pitch class must jump over, so to speak, a frozen
57 David Clampitt, ―Pairwise Well-Formed Scales: Structural and Transformational Properties,‖ Ph.D. Diss. State University of New York at Buffalo, 1997.
８３ pitch class. For example, {0, 1, 2, 4, 6, 8, t} moves to {0, 2, 3, 4, 6, 8, t}, but pitch class 1 hops to 3 over frozen pitch class 2.
If we loosen the requirement for uniform motion by interval class 2, keeping the other requirements for a relatively smooth cycle, we have a Q-cycle as defined by
Clampitt (1997, 1999).58 It may be shown that 6–Z44 and 9–11 are the only set classes in the usual 12-note universe that support Q-cycles but may not be construed as either well- formed or pairwise well-formed scales.
Table 3.2. Definition of P-cycle, Q-cycle
Finally, the superset 7–22 which is formed by the union of adjacent 6–Z44 sets in a given system is a member of a class of sets which Clampitt defined as minimal according to a graph-theoretical criterion, in the 1999 Intégral paper ―Ramsey Theory,
Unary Transformations, and Webern‘s op. 5, no. 4.‖ One of the hallmarks of these inversionally symmetrical minimal sets is that disturbing their symmetry by removing
58 David Clampitt, ―Ramsey Theory, Unary Transformations, and Webern‘s Op. 5, No. 4,‖ Intégral 13 (1999): 63–93.
８４ one note yields a member of a set class that supports one of these relatively smooth voice-leading cycles. The analytical example there showed Webern employing the minimal superset 8–9, and portions of the voice-leading cycles by maximal proper subsets from set class 7–7 and its complement 5–7. The similarity of the manifestations in these cases is further evidence of a compositional procedure on the part of Webern.
８５
Chapter 3.2: Relatively Smooth Voice-Leading Systems
In 1973, Allen Forte introduced pitch-class set theory as a methodology for atonal music. In his book, The Structure of Atonal Music, he represents pitch-class sets and their relations.59 He thinks that atonal music can be explained by the occurrence of pitches and their combinations. We should consider two cases in pitch-class sets, ordered sets and unordered sets. For example, [0,2,3] is regarded as the same as [2,3,0] because the two sets have the same pitch-class elements, although the order of the pitch-classes is different. If we regard these two sets as distinct, we consider these sets as ordered sets. If we regard these two sets as equivalent sets, we regard these sets as unordered sets. For comparing several sets, it is useful to be able to make a set conform to a basic ordered pattern, most packed to the left, which Forte called normal order. When the first integer of a pitch-class set in normal order is set at 0, he called it the prime form. He also presents similarity relations, pitch-class similarity and intervallic content similarity.60 In
59 Allen Forte, The Structure of Atonal Music, 1973, New Haven and London: Yale University Press. 60 He defines the type of pitch-class similarity Rp to be that which holds for two sets of the same cardinality n whenever they share a common subset of cardinality n−1. This subset may either be comprised of the identical pitch-classes in both sets or it may be includable in one set only as a transposed or inverted form of the other subset. In the former case Rp is said to be ―strongly represented‖; if the shared subset is not literally the same, Rp is ―weakly represented.‖ Intervallic similarity, based on a comparison of interval vectors, can be either minimal or maximal. Minimal
８６ order to indicate interrelationships between pitch-class sets of atonal music, prime form is very helpful to identify a structural construction. For this reason, when we analyze atonal music by pitch-class set theory, we reconstruct the pitch-class sets on prime form for investigating the original source. Through comparing with prime forms of several pitch- class sets represented on the surface of atonal music, we could understand the similarity of pitch-class sets. After noticing those sets‘ equivalent characteristics, we could figure out the transformational principles like transposition and inversion.
Consider the transformations of transposition and inversion. I first investigate transpositional transformation. In the example below, pitch-class set (C, C#, D#, E) and pitch-class set (E, F, G, G#) have the same prime form as the pitch-class set 4-3 and they are in a relationship of T4. As you can see, the contents of the interval string of two pitch- class sets are the same and the order of the interval string arrays of the two sets coincide exactly.
( C, C#, D#, E ) ( E, F, G, G#)
[ 0 1 3 4 ] [ 4 5 7 8 ]
1 2 1 8 1 2 1 8
Figure 3.15. Transformation T4 of 4-3
similarity, or R0, obtains when the vectors for two sets have no corresponding entries that are identical. That is, for sets X, Y, with vectors [x1 x2 x3 … x6] and [y1 y2 y3 … y6], xi ≠ yi for all i = 1,2,3…6. Allen Forte, The Structure of Atonal Music, p. 50.
８７
In the case of inversional transformation, we can penetrate the meaning of the interval string array of pitch-class set relationships. In figure 3.16 shown below pitch- class sets [12367t] and [12369t] are both in 6-Z44. They are in the related by I4. As one can see in fig. 3.16, the two pitch-class sets have the same multiplicities of interval contents, of three interval-class one elements, three interval-class three elements, even though their order is different. The difference in order involves the the exchange of adjacent elements in the interval string, the fourth and fifth elements in the string <1 1 3
13 3>. I this case, we invert the set. In general, when the multiplicities of the interval contents of two pitch-class sets are the same, we could naturally consider the possibility that they may have some relationship.
[1 2 3 6 7 t] [1 2 3 6 9 t]
1 1 3 1 3 3 1 1 3 3 1 3
Figure 3.16. I4 relationship
Richard Chrisman proposes structural aspects of pitch-sets using successive-interval arrays.61 He shows that the intervallic content of any given unordered pitch-class set can be described by representing intervallic arrays between the pitch classes.62
61 Richard Chrisman, ―Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays,‖ Journal of Music Theory 21 (Spring, 1977): 1–28. 62 From a pitch-class set, a collection of pitches whose duplicate elements have been
８８
For example, the set C, D, F-sharp, B has as its interval array the
succession 2-4-5-1 (C to D = 2 semitones ; D to F-sharp = 4 ; F-sharp to
B = 5 ; B to C an octave higher =1 semitone). The array 2-4-5-1 provides
one measure of the intervallic structure of the pitches in the set C, D, F-
sharp, B since the intervallic distances between successive elements will
remain the same (C, D, F-sharp, B), and addition, of any transposition of
C, D, F-sharp, B since the intervallic distances between successive
elements will remain the same (C, D, F-sharp, B transposed up to D, E,
G-sharp, C-sharp, for instance, will still give the array 2-4-5-1).63
A single array, therefore, shows all the transpositions of that set. He also discusses permutations of the pitch-set:
Also, just as any cyclic permutation of the pitch-set, such as the
permutations D, F-sharp, B, C, or F-sharp, B, C, D, or B, C, D, F-sharp,
(ascending order being retained and the pitches kept within a single
eliminated and whose remaining elements are given in ascending order and within one octave, the ―‗successive-interval array‘ (or, more simply, interval-array) can be formed by finding the interval in semitones, written as an integer, between successive pitches in the set.‖ Richard Chrisman, ―Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays,‖ Journal of Music Theory 21, (Spring, 1977), 7.
63 Chrisman presents the example of the collection of tones (C, D, F-sharp, B) in his paper on p. 7.
８９
octave), does not alter the identity of the set in terms of pitch content,
all such cyclic reordering of the elements in interval-arrays are
equivalent: thus, arrays 2-4-5-1, 4-5-1-2, 5-1-2-4, and 1-2-4-5 are all
equivalent and represent all cyclic permutations of the pitch-set C, D, F-
sharp, B and of all the transpositions of that set. Equivalent arrays,
therefore, are those arrays whose elements can be rendered equal – that
is, with the same numerical elements in the same order – by cyclic
permutation. The two equivalent arrays A = 1-2-3-4-2 and B = 2-3-4-2-
1 can be so ―equalized‖ by cyclically permuting elements of wither
array:
A = 1-2-3-4-2 2-3-4-2-1 3-4-2-1-2 = 1st permutation of B nd 4-2-1-2-3 = 2 ‖ ‖ rd 2-1-2-3-4 = 3 ‖ ‖ th A = 1-2-3-4-2 equal to 1-2-3-4-2 = 4 ‖ ‖
Figure 3.17. Chrisman’s interval array permutation64
The elements of one array can be cyclically permuted and in this case, array B is permuted four times as a result, the two arrays can be rendered equal as shown in the last row above. Array A and the 4th permutation of B are equal arrays, 1-2-3-4-2 by cyclic permutation.
64 Ibid., 7.
９０
Chrisman identifies specific pitch levels of interval arrays. He represents the first pitch of a set for identifying the specific pitch level of an array. The intervallic array of collection of tones, C, D, E, F, is 2-2-1-7 and it can be written (C) 2-2-1-7. By using the notational label P0, he represents the initial ordering; by P1 he represents the first cyclic permutation; by P2 the second, and so on.
E,A,D,E-flat = (E)-5-5-1-1 = P0
A,D,E-flat, E = (A)-5-1-1-5 = P1
D,E-flat,E,A = (D)-1-1-5-5 = P2
E-flat,E,A,D = (E-flat)-1-5-5-1 = P3
Figure 3.18.
The following arrays show the special pitch-class sets that result in total pitch-class invariance under some levels of transposition:
９１
Two-element arrays: 6-6 (the ―tritone‖) Three-element arrays: 4-4-4 (the ―augmented tirad‖) Four-element arrays: 1-5-1-5 2-4-2-4 3-3-3-3 (the ―diminished-7th chord‖) (No five-element arrays have this structure) Six-element arrays 1-1-4-1-1-4 1-3-1-3-1-3 (the ―hexatonic scale‖) 1-2-3-1-2-3 2-2-2-2-2-2 (the ―whole-tone scale‘)
Figure 3.19. Particular intervallic arrays65
For two pitch-sets to be equal in pitch content, their respective interval arrays must be equal or be capable of being rendered equal by means of cyclic permutation. For total pitch invariance to occur between some set and one or more transpositions of the inversion of that set, the interval-array for the set must be equivalent to the array of its inversion.
65 Ibid., 15.
９２
P0 = 1-1-10 I0 = 10-1-1 I1 = 1-1-10 = P0
P0 = 2-2-8 I0 = 8-2-2 I1 = 2-2-8 = P0
P0 = 2-5-5 I0= 5-5-2 I2 = 2-5-5 = P0
P0 = 3-3-6 I0= 6-3-3 I1 = 3-3-6 = P0
P0 = 4-4-4 I0= I1 = I2 = 4-4-4 = P0
Figure 3.20.
Robert Morris also focused on the intervallic array of pitch-class sets and he generalized rotational arrays by analyzing Igor Stravinsky‘s music.66 Through the rotational array perspective, he provides Stravinsky‘s array as a systematic way of generating serial, even twelve-tone, material that would concord with his previously established practice and style.
In this chapter, I will propose relatively smooth voice-leading systems of pitch-class sets within a perspective of the intervallic content‘s array previewed above through the theories of Chrisman and Morris.
Forte indicates all possible prime forms and vectors of pitch-class sets (listed in
Appendix A). Previously, many theorists have focused upon voice leadings between and within set classes, and on the consequent common tones and changed tones.
In my own relatively smooth voice-leading systems, I first divide all pitch-class sets into several groups by considering those with similar interval string contents. Each group
66 Robert Morris, ―Generalizing Rotational Arrays,‖ Journal of Music Theory 32, no. 1 (1988): 75–132.
９３ has the same type of intervallic structure, up to permutation of the elements in the interval string. The interval contents of a pitch-class set could be the criteria for judging the degree of similarity in relationships among pitch-class sets. The reason why we regard pitch-class sets transformed by transposition or inversion as the same set is that their intervallic structure inherent materials remains the same. (This could be confirmed by their prime forms.) If a pitch-class set moves to another set within the group sharing the same multiplicities in its interval string through the exchange of adjacent elements in the string, it could be considered as relatively smooth voice-leading. I therefore suggest relatively smooth voice-leading systems in each group. For example, pitch-class sets 6-22,
6-Z24, 6-21, 6-Z23, 6-Z25, 6-Z26 are in the same group because they similarly have two semitones, three whole-tones, and one interval class four in their interval string arrays.
(See figure 3.22.)
In each group sharing the same interval string contents, I connect the members via relatively smooth voice leading. The voice leading is smooth in the sense described in the discussion of relatively smooth cycles above, except that now instead of within a particular set class, it is between different set classes (defined in terms of relationships between prime forms of set classes), and the graphs do not generally form cycles. That is, the relatively smooth voice-leading connection requires that the sets be in Forte‘s Rp relation, with maximal non-empty intersection, and the pitch classes to be exchanged must be adjacent in the scalar orderings of the sets, as in Clampitt‘s Q-relation, but negating the requirement of set-class consistency.
In figure 3.21, pitch-class set 6-22, [0,1,2,4,6,8], goes to 6-Z24, [0,1,3,4,6,8], and
９４ considering their prime forms, five elements, {0,1,4,6,8} remain fixed, while element 2 moves to 3. Although the pitch-class set is transformed to another, the changed set has an interval string with contents—but not order—that is the same as for the previous pitch- class set. In this voice leading, they have similar interval strings and move by a form of parsimonious or smooth voice leading, as defined above.
Figure 3.21. Voice leading from pcs 6-22 to 6-Z24
Figure 3.22 shows that adjacent set classes, either vertically or horizontally aligned, are connected via relatively smooth voice leading. The elements within each boxed group are thus vertices of a connected graph, where the edges represent the relatively smooth voice-leading relation.
９５
Figure 3.22. Hexachord voice-leading system having two semitones, three whole-
tones, one interval class four
As one can see in figure 3.23, the pitch-class sets 7-28, 7-31, 7-32, 7-30, 7-29, 7-27,
7-24, 7-26, 7-23, 7-25 are in the same group because they have the same contents of three ic 1, three ic 2, and one ic 3. In this group, pitch-class set 7-24 can be related to only 7-27 by parsimonious voice leading, so I have placed set 7-24 to the side and inserted a double arrow to indicate the connection with 7-27.
９６
Figure 3.23. 7-note voice-leading system having three semitones, three whole-tones,
one interval class three
Among the relatively smooth voice-leading systems, I found four remarkable pitch- class sets that are isolated, connected with no other sets by such voice leading. In the hexachord system having three ic 1, three ic 3, pitch-class set 6-20 is isolated in the group.
Similarly, pitch-class sets 7-22, 8-28, 9-12 are isolated in their groups. (See Figure 3.25,
Figure 3.26, Figure 3.27, and Figure 3.28.) One may observe from their interval strings that they are all inversionally symmetrical, but otherwise, they are different. The latter two are maximally even sets classes (as defined by Clough and Douthett). The hexatonic
９７ set 6-20 is similar to the maximally even sets, in that it is both transpositionally invariant and invariant under inversion. The outlier is 7-22; as mentioned above, it is minimal as defined in Clampitt 1999, and it is a representative of the singular pairwise well-formed scale pattern, as defined in Clampitt 1997.67 Why these four set classes are isolated in their groups remains a subject for further investigation.
6-20 014589 1-3-1-3-1-3
7-22 0125689 1-1-3-1-2-1-3 8-28 0134679t 1-2-1-2-1-2-1-2 9-12 01245689t 1-1-2-1-1-2-1-1-2
Figure 3.24. Isolated symmetrical sets
Figure 3.25. Voice-leading system of hexachord having three ic 1, three ic 3
67 David Clampitt, ―Pairwise Well-Formed Scales: Structural and Transformational Properties,‖ Ph.D. Diss. State University of New York at Buffalo, 1997.
９８
Figure 3.26. 7-note voice-leading system having four interval class 1, three interval
class 2, one interval class 3
９９
Figure 3.27. 8-note voice-leading system having four interval class 1, four interval
class 2
１００
Figure 3.28. 9-note voice-leading system having six interval class 1, three interval
class 2
In the remaining figures, I enumerate groups according to the criteria that they share a cardinality (from 3 to 9) and share the same multiplicities in their interval strings.
Adjacency within the boxed groups indicates relatively smooth voice leading, yielding the relatively smooth voice-leading systems. These adjacencies represent connections between the prime forms of set-classes which share all but one pitch class, and where the changing pitch class slides between the fixed subset, thereby permuting the interval string by transposing one pair of neighboring elements. The restriction to prime forms, however,
１０１ is a simplification for this initial investigation of the concept. Allowing this type of voice- leading connection between set-classes where all the transposed and inverted forms are considered yields much more complicated, and much richer, systems, represented by graphs with many more connections. This expansion of the theory is another direction for further research.
１０２
Figure 3.29. Three-note Groups
１０３
Figure 3.30. Tetrachord having three ic 1, one ic 9
Figure 3.31. Tetrachord voice-leading system having two ic 1, one ic 2, one ic 8
Figure 3.32. Tetrachord voice-leading system having two semitones, one ic 3, ic, 7
１０４
Figure 3.33. Tetrachord voice-leading system having two ic 1, one ic 4, one ic 6
Figure 3.34. Tetrachord voice-leading system having two ic 1, two ic 5
Figure 3.35. Tetrachord voice-leading system having one ic 1, two ic 2, one ic 7
１０５
Figure 3.36. Tetrachord voice-leading system having one ic 1, one ic 2, one ic 3, one
ic 6
Figure 3.37. Tetrachord voice-leading system having one ic 1, one ic 2, one ic 4, one
ic 5
Figure 3.38. Tetrachord voice-leading system having one ic 1, two ic 3, one ic 5
１０６
Figure 3.39. Tetrachord voice-leading system having one ic 1, two ic 4, one ic 3
Figure 3.40. Tetrachord having three ic 2, one ic 6
Figure 3.41. Tetrachord voice-leading system having two ic 2, one ic 3, one ic 5
１０７
Figure 3.42. Tetrachord voice-leading system having two ic 2, two ic 4
Figure 3.43. Tetrachord voice-leading system having one ic 2, two ic 3, one ic 4
Figure 3.44. Tetrachord having four ic 3
１０８
Figure 3.45. Pentachord having four ic 1, one ic 8
Figure 3.46. Pentachord voice-leading system having three ic 1, one ic 2, one ic 7
Figure 3.47. Pentachord voice-leading system having three ic 1, one ic 3, ic 6
１０９
Figure 3.48. Pentachord voice-leading system having three ic 1, one ic 4, one ic 5
Figure 3.49. Pentachord voice-leading system having two ic 1, two ic 2, one ic 6
１１０
Figure 3.50. Pentachord voice-leading system having two ic 1, one ic 2, two ic 4
Figure 3.51. Pentachord voice-leading system having two ic 1, two ic 3, one ic 4
１１１
Figure 3.52. Pentachord voice-leading system having one ic 1, three ic 2, one ic 5
Figure 3.53. Pentachord voice-leading system having one ic 1, two ic 2, one ic 3, one
ic 4
１１２
Figure 3.54. Pentachord voice-leading system having one ic 1, one ic 2, three ic 3
Figure 3.55. Pentachord having four ic 2, one ic 4
Figure 3.56. Pentachord voice-leading system having three ic 2, two ic 3
１１３
Figure 3.57. Hexachord having five ic 1, one ic 7
Figure 3.58. Hexachord having four ic 1, one ic 2, one ic 6
Figure 3.59. Hexachord voice-leading system four ic 4, one ic 2, one ic 6
Figure 3.60. Hexachord voice-leading system having four ic 1, one ic 3, one ic 5
１１４
Figure 3.61. Hexachord voice-leading system having four ic 1, two ic 4
Figure 3.62. Hexachord voice-leading system having three ic 1, two ic 2, one ic 5
１１５
Figure 3.63. Hexachord voice-leading system having three ic 1, three ic 3
Figure 3.64. Hexachord system having three ic 1, one ic 2, one ic 3, one ic 4
１１６
Figure 3.65. Hexachord voice-leading system having two ic 1, three ic 2, one ic 4
Figure 3.66. Hexachord voice-leading system having one ic 1, four ic 2, one ic 3
１１７
Figure 3.67. Hexachord having six ic 2
Figure 3.68. Hexachord having four ic 1, one ic 3, one ic 5
１１８
Figure 3.69. Hexachord voice-leading system having two ic 1, two ic 2, two ic 3
１１９
Figure 3.70. Heptachord having six ic 1, one ic 6
Figure 3.71. Heptachord voice-leading system having five ic 1, one ic 2, one ic 5
Figure 3.72. Heptachord having five ic 1, one ic 3, one ic 4
Figure 3.73. Heptachord having five ic 1, one ic 3, one ic 4
１２０
Figure 3.74. Heptachord voice-leading system having four ic 1, two ic 2, one ic 4
１２１
Figure 3.75. Heptachord voice-leading system having four ic 1, one ic 2, two ic 3
１２２
Figure 3.76. Heptachord voice-leading system having three ic 1, three ic 2, one ic 3
Figure 3.77. Heptachord voice-leading system having two ic 1, five ic 2
１２３
Figure 3.78. 8-note pitch-class set having seven ic 1, one ic 5
Figure 3.79. 8-note voice-leading system having six ic 1, one ic 2, one ic 4
Figure 3.80. 8-note voice-leading system having six ic 1, two ic 3
１２４
Figure 3.81. 8-note voice-leading system having five ic 1, two ic 2, one ic 3
１２５
Figure 3.82. 8-note voice-leading system having four ic 1, four ic 2
１２６
Figure 3.83. 9-note pitch-class set having eight ic 1, one ic 4
Figure 3.84. 9-note voice-leading system having seven ic 1, one ic 2, one ic 3
１２７
Figure 3.85. 9-note voice-leading system having six ic 1, three ic 2
１２８
Conclusion
This dissertation presents a methodology for studying voice leading in atonal music by considering the content of interval-string arrays in pitch-class sets. Methodologies for understanding voice leading in atonal music have been studied by numerous music theorists, in order to help account for principles of pitch construction. I could not find, however, previous research that classifies pitch-class sets according to the aspect of having the same interval multiplicities in their interval strings. Voice leading between pitch-class sets having the same interval string contents might be considered differently from voice leading between pitch-class sets having different interval string contents.
Especially in atonal music, interval contents play a pivotal role in progressions of pitch- class sets. For this reason I introduce voice-leading systems grouped by pitch-class sets‘ string interval contents, which permits a kind of smooth voice leading. This gives rise to the concept of relatively smooth voice-leading system, which is the primary concept behind this dissertation.
１２９
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１３５
Appendix A: Forte, 1973, Prime Forms and Vectors of Pitch-Class Sets
１３６
１３７
１３８
１３９