FOUR CASE STUDIES VOLUME 1 by SUN-JU SONG, Bmus, BA (Mus

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Music Analysis and the Avant-Garde Compositions of Post- World War II: Four Case Studies

Author Song, Sun-Ju

Published 2011

Thesis Type Thesis (PhD Doctorate)

School Queensland Conservatorium

DOI https://doi.org/10.25904/1912/3234

Downloaded from http://hdl.handle.net/10072/367570

Griffith Research Online https://research-repository.griffith.edu.au

MUSIC ANALYSIS AND THE AVANT-GARDE COMPOSITIONS OF POST-WORLD WAR II: FOUR CASE STUDIES

VOLUME 1

BY SUN-JU SONG, BMus, BA (Mus)(Hons), MMus

Queensland Conservatorium, Griffith University

Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

October, 2008

When analytic thought, the knife, is applied to experience, something is always killed in the process. That is fairly well understood, at least in the arts. – But what is less noticed in the arts – something is always created too. And instead of just dwelling on what is killed it’s important also to see what’s created and to see the process as a kind of death-birth continuity that is neither good nor bad, but just is. (Pirsig, 1974, p. 77)

Abstract

This thesis examines four iconic piano works composed from 1949 to 1952 in relation to the discipline of music analysis: Olivier Messiaen’s Mode de valeurs et d’intensités (1949), Pierre Boulez’s Structures Ia for two pianos (1952), Karlheinz Stockhausen’s Klavierstück III (1952) and John Cage’s Music of Changes (1951). The primary aim is to investigate how music analysis has engaged with these four works over the last fifty years. Investigating the nature and content of earlier analyses reveals various aspects of the discipline: its aims and functions, its governing ideologies and, most importantly, its achievements. This investigation also allows for recent criticism of the discipline of music analysis to be reconsidered. In total, 29 analyses of the four works are closely evaluated. Building on these observations, the author contributes a further perspective on each work by offering her own analytical insights, at times challenging some of the common analytical approaches to them. Rather than applying an analytical system based on a particular theory, concepts from each composer’s own writings and theories are used to provide an appropriate analytical angle. This approach recognises that analysis should be a meaningful dialectic between the conceptual understanding of compositional theory and the perceptual experience of surface musical phenomena. Chapter 1 outlines some of the issues and critical debates surrounding music analysis and avant-garde music, beginning in the mid-twentieth century. Three ideologies (formalism, organicism and positivism) that have underpinned the discipline of music analysis are considered in relation to the four case study works. Chapter 2 discusses how the four composers influenced each other during the years immediately after World War II, and underlines how the significant personal connections between them stimulated the development of their compositional techniques. Chapters 3 to 6 focus in turn on the four works, starting with Messiaen’s Mode de valeurs then moving on to Boulez’s Structures Ia, Stockhausen’s Klavierstück III and Cage’s Music of Changes. Each case study consists of four sub- sections: the historical and biographical context of the work; a review and re- evaluation of selected analytical writings; the author’s own analytical re-examination of the work; and a summary. Although these chapters follow the above sub-sections, each is tailored to accommodate the unique circumstances of each work. For instance,

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the number of analyses selected—and the issues they raise—differs in each case study. As a conclusion, Chapter 7 summarises four aspects: (1) the commonalities between the compositional techniques developed by the four composers across the chosen works; (2) the predominant analytical approaches to the works; (3) further research directions within the discipline of music analysis indicated by this research; and (4) a reassertion that music analysis is essential for an adequate understanding of such music.

Certification

This work has not previously been submitted for a degree or diploma in any university. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made in the thesis itself.

Signed………………………….. (Sun-Ju Song)

Date 14 October 2008

Acknowledgements

First and foremost, I would like to express my deepest gratitude to my principal supervisor, Dr Stephen Emmerson. He has provided guidance to and support for every step of my research and has been a constant source of inspiration and motivation. I am deeply thankful for his generosity of time, his expertise and his input into this research as well as my educational and musical life. I would also like to thank my two associate supervisors: Robin Maconie for his unparalleled expertise that was such a valuable resource in the preliminary stage of the research, and Dr Philip Truman for making many valuable comments. There are several people whose assistance and support have been tremendously important in this research. In reading various drafts of this thesis, Jocelyn Wolfe not only helped me with proof reading but also encouraged me to improve my writing skills. For translating the German literature used in this research, I am grateful to my friend Christina Young for spending countless hours with me even through her pregnancy. I am also thankful to have received further assistance in translating from Barbara Steinhauser, Armin Terzer and Gabriele Ratzke. For producing all the musical scores for this thesis, I am thankful to my friend Josephine Jin who also inspired me through many conversations about composing music. For the mathematical calculations involved in Chapter 4, I thank my friend Anna Coulson whose knowledge of mathematics and skills with the Excel program greatly helped me. To librarian Olga Lipsky, her help and persistence over many years to locate and obtain many different types of literature has been remarkable. Finally, for the final editing and proof reading process, the generosity and friendship of Dr Jenny Butler, together with her expertise in both editorial work and in music, have been invaluable and I am deeply indebted to her. I would like to thank my parents, Jung-Hee Jee and Se-Won Song for their unfailing support and for financially helping me to study in Australia for many years. I would like to also extend my thanks to my cousin Myung-Shin Kim for encouraging me through many phone calls from Korea and to my neighbours Doreen and Gordon Clarke for looking after me like their own daughter. Portions of Chapter 4 of this thesis appears as conference proceeding in “Rethinking structures: A comparative evaluation of analytical approaches to Boulez’s Structures Ia over the last 50 years” [Electronic versions] In R. Hardie (Ed.),

Music and Locality: towards a local discourse in music, Music Books New Zealand, 2004. Permission to use the entire score of Karlheinz Stockhausen’s Klavierstück III was provided by Universal Edition.

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Table of Contents

Volume One

Abstract ...... iii Certification ...... v Acknowledgements ...... vi Table of Contents ...... viii Volume One ...... viii CHAPTER ONE ...... 1 Introduction ...... 1 Aesthetic positions of music analysis ...... 2 Critiques and debates ...... 10 Post-tonal music in the twentieth century and music analysis ...... 17 Central questions and the research design ...... 21 CHAPTER TWO ...... 25 Four Composers and Four Works ...... 25 Interactions ...... 25 Tape music ...... 31 Darmstadt summer courses ...... 33 Die Reihe ...... 35 Conclusion ...... 36 CHAPTER THREE ...... 37 Introduction ...... 37 Review of Previous Analyses ...... 38 Issues ...... 38 Overview ...... 40 The pre-compositionally designed mode ...... 41 Compositional procedure: integral serialism? ...... 44 Formal Structure ...... 45 Texture ...... 46 Conclusion ...... 50 A re-interpretation of Mode de valeurs et d’intensités ...... 51 Analytical premise ...... 51 Messiaen’s rhythmic techniques applied to Mode de valeurs ...... 51 Seven prominent pitch classes ...... 54 Melodic cadence ...... 56

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Conclusion ...... 56 CHAPTER FOUR ...... 58 Introduction ...... 58 Review of the previous analyses ...... 60 Issues ...... 60 Overview ...... 61 Analyses in the 1950s and 1960s ...... 61 Summary ...... 68 Analyses in the 1970s ...... 69 Summary ...... 73 Analyses in the 1980s ...... 74 Summary ...... 79 The analyses in the 1990s ...... 79 Conclusion ...... 89 A re-interpretation of Structures 1a ...... 91 Analytical premises ...... 91 Understanding serial structure ...... 94 Density ...... 94 Density variability and the serial organisation of pitch ...... 96 Mobile and immobile relationship ...... 104 Density variability and the serial organisation of duration ...... 106 Density variability and the serial organisation of dynamics and attack ...... 109 Temporal structure of Structures Ia ...... 115 Conclusion ...... 126 CHAPTER FIVE ...... 128 Introduction ...... 128 Review of previous analyses ...... 129 Issues ...... 129 Formal structure ...... 131 Pitch organisation ...... 135 Arrangement of durational values ...... 148 Arrangement of dynamics ...... 151 Melodic contour and registral distribution ...... 155 Texture and density ...... 160 Perspectives of a listener and performer ...... 162 A re-interpretation of Klavierstück III ...... 167 Analytical premise ...... 167

Symmetry ...... 170 Grouping ...... 171 Conclusion ...... 180 CHAPTER SIX ...... 182 Introduction ...... 182 Review of previous analyses ...... 183 Issues ...... 183 Pritchett’s approach to Cage’s chance music ...... 186 Pritchett’s analysis of Music of Changes ...... 190 Pritchett’s 1993 analysis ...... 203 Schädler’s analysis ...... 205 Uno’s analysis ...... 213 Conclusion ...... 218 A re-interpretation of Music of Changes ...... 220 Analytical premise ...... 220 The influence of European composers ...... 220 Compositional aesthetics reconsidered ...... 227 Performance practice and listening experience ...... 231 Conclusion ...... 232 CHAPTER SEVEN ...... 234 Procedural connections ...... 234 Summary of the review ...... 238 Significance of music analysis ...... 248 Reference List ...... 250

List of Figures

Volume Two

Figure 2.1 Four composers and four piano works Figure 3.1 Three divisions of the mode Figure 3.2 Analyses of Messiaen’s Mode de valeurs et d’intensités reviewed in this chapter Figure 3.3 Comparison of selected analyses of Mode de valeurs Figure 3.4 Schweizer’s illustration of regrouping of notes Figure 3.5 (a) Toop’s rearrangement of the three divisions Figure 3.5 (b) Sherlaw Johnson’s rearrangement of the three divisions Figure 3.5 (c) Schweizer’s rearrangement of the three divisions Figure 3.6 Fuhrmann’s graphic illustration of the note order in Mode de valeurs Figure 3.7 Sherlaw Johnson’s graphic illustration of the note order in Mode de valeurs Figure 3.8 Six planes of sound in Covington’s analysis Figure 3.9 Covington’s analysis of Mode de valeurs Figure 3.10 (a, b, c and d) Reinterpretation of Sherlaw Johnson’s note order into the arrangement of durational values Figure 3.10 (e, f and g) Reinterpretation of Sherlaw Johnson’s note order into the arrangement of durational values Figure 3.10 (h and i) Reinterpretation of Sherlaw Johnson’s note order into the arrangement of durational values Figure 3.11 (a) Messiaen’s Mode de valeurs (Bars 105–115) Figure 3.11 (b) Messiaen’s Mode de valeurs, Metrical reinterpretation (Bars 105–115) Figure 3.12 Motives Figure 3.13 Appearance of G5 (Bars 51–53) Figure 3.14 Construction of seven-note mode (Hidden mode) Figure 3.15 (a) Comparison between the mode of limited transposition and the hidden mode Figure 3.15 (b) Comparison between the mode of limited transposition and the hidden mode Figure 3.15 (c) Comparison between the mode of limited transposition and the hidden mode Figure 3.16 (a) Melodic cadence (Bars 28–29)

Figure 3.16 (b) Melodic cadence (Bar 78) Figure 3.16 (c) Melodic cadence (Bars 107–111) Figure 4.1 Analyses of Boulez’s Structures Ia reviewed in this chapter Figure 4.2 (a) Boulez’s initial series used in Structures Ia according to Ligeti Figure 4.2 (b) Boulez’s two matrices used in Structures Ia Figure 4.3 (a and b) Comparison between Boulez’s and Ligeti’s attack series Figure 4.4 Sectional divisions in Structures Ia according to Ligeti’s analysis Figure 4.5 Ligeti’s examples for the superimposition of threads Figure 4.6 Wennerstrom’s analysis of combined dynamic levels in Structures Ia Figure 4.7 Wennerstrom’s analysis of combined attack levels in Structures Ia Figure 4.8 Griffiths’ graph illustrating the tempo, density and registral distributions of pitches in each formal sections Figure 4.9 Eckart-Bäcker’s analysis of the arrangement of three tempi Figure 4.10 Uno’s analysis of registral distributions of pitch rows in Boulez’s Structures Ia Figure 4.11 Uno’s analysis of segmental and sectional TG boundaries Figure 4.12 Model of serial communication according to Grant Figure 4.13 (a) Density variability of the 14 abstract formal sections of Structures Ia. Figure 4.13 (b) Density variability and the 11 actual formal sections of Structures Ia separated by pauses Figure 4.14 Opening chords of the 14 abstract formal sections in Structures Ia Figure 4.15 Opening chords with the interval of a compound fifth. Figure 4.16 Opening chords without the interval of a compound fifth Figure 4.17 (a) Summary of tritone relationships Figure 4.17 (b) Pitch series paired in tritone relationships Figure 4.18 Repeated pitches shared by all the pitch series used in each abstract formal section Figure 4.19 Density levels and the number of repeated pitches in each abstract formal section Figure 4.20 Relationship between the density variable and the arrangement of dynamics Figure 4. 21 Categorisation of attacks Figure 4. 22 Relationships between the types of attacks and the density level Figure 4. 23 Two overlapping symmetries for the arrangement of tempo xii

Figure 4. 24 Golden Section ratio Figure 4. 25 Iteration table illustrating the calculations application of the GS ratio Figure 4. 26 The application of the GS ratio to the 14 abstract formal sections Figure 4. 27 The cumulative calculation for the real time of the entire piece Figure 4. 28 The range of section VI and GS primary point in relation to the assumed duration values for pauses Figure 4.29 The range of section VI and GS primary point in relation to the pause durations Figure 4.30 Complementary features of sections VI(a) and VI in relation to the GS ration Figure 4.31 A structural connection between the three tempi and the GS proportional system Figure 5.1 Analyses of Stockhausen’s Klavierstück III reviewed in this chapter Figure 5.2 Comparison of selected analyses of Klavierstück III Figure 5.3 Comparison of four author’s interpretations of formal structure of Klavierstück III Figure 5.4 Blumröder’s interpretation of the formal structure of Klavierstück III Figure 5.5 Schnebel’s interpretation of five-part formal structure of Klavierstück III Figure 5.6 Cook’s interpretation of the of formal structure of Klavierstück III Figure 5.7 Analytical approaches for pitch organisation in Klavierstück III Figure 5.8 Cook’s pitch distribution analysis in Klavierstück III Figure 5.9 Blumröder’s three-part durational divisions in Klavierstück III Figure 5.10 Blumröder’s analysis of tetrachords and durational scheme of Klavierstück III Figure 5.11 Blumröder’s analysis of the serial organisation in Klavierstück III Figure 5.12 Harvey’s illustration of pentachords and melodic contour in Kalvierstück III Figure 5.13 Lewin’s rearrangement of pentachords in Klavierstück III for ear-training purposes Figure 5.14 J – Related forms of pentachords retrieved from Lewin’s analysis Figure 5.15 Maconie’s analysis of rhythmic cells used in Kalvierstück III Figure 5.16 Cook’s interpretation of the rhythmic relationship of five-part form using Cooper-Meyer’s rhythmic symbols

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Figure 5.17 Blumröder’s analysis of the serial organisation of durations in Klavierstück III Figure 5.18 Blumröder’s reinterpretation of Bar 8 Figure 5.19 Blumröder’s analysis of the serial organisation of dynamics in Klavierstück III Figure 5.20 Blumröder’s analysis of the permutations of dynamics in Klavierstück III Figure 5.21 Schnebel’s graphic illustration of Klavierstück III Figure 5.22 Blumröder’s analysis of the serial organisation of registers in Klavierstück III Figure 5.23 Blumröder’s serial permutations of registers in Klavierstück III Figure 5.24 Schnebel’s graphic illustration of a dense network of relationships Figure 5.25 Blumröder’s analysis of the serial organisation for density in Klavierstück III Figure 5.26 Blumröder’s analysis of the relationship between articulation and density Figure 5.27 My analysis of the five groups in Klavierstück III Figure 5.28 Comparison between my interpretation and Schnebel’s interpretation of groups Figure 5.29 Comparison between my interpretation and Cook’s interpretation of groups Figure 5.30 Group 1 (Bars 1–2) from Klavierstück III Figure 5.31 Group 2 (Bars 3–7) from Klavierstück III Figure 5.32 Rhythmic cells used in Group 2 Figure 5.33 Symmetrical relationships between dynamics and registers in Group 2 Figure 5.34 Group 3 (Bars 8–10.3) from Klavierstück III Figure 5.35 Symmetrical arrangements of intervals in Group 3 Figure 5.36 Symmetrical arrangements of durational values in Group 3 Figure 5.37 Group 4 (Bars 10.4–13) from Klavierstück III Figure 5.38 Symmetrical relationships in Group 4 Figure 5.39 Group 5 (Bars 14–16) from Klavierstück III Figure 6.1 Analyses of John Cage’s Music of Changes reviewed in this chapter Figure 6.2 Pritchett’s diagram of phrase group structure Figure 6.3 Relationship between phrases and a phrase group Figure 6.4 Relationship between phrase groups and the structure of the entire piece Figure 6.5 Chart structure xiv

Figure 6.6 Sound chart 2 Figure 6.7 Subdivision of sound Chart: 4×4=16 Figure 6.8 Durational chart 2 Figure 6.9 Dynamic chart 8 Figure 6.10 Density chart Figure 6.11 Summary of the compositional process based on Pritchett’s analysis Figure 6.12 Terminology comparisons between Pritchett, Schädler and Cage Figure 6.13 Mobile and immobile relationships in chart system Figure 6.14 Segmented duration in visual measurement Figure 6.15 Comparison between the compositional systems of Cage’s Music of Changes and Boulez’s Structures Ia Figure 6.16 The range for each pre-determined parameter in Music of Changes and Structures Ia Figure 6.17 (a) Cage’s spatial notation in bars 131–133 of Book III Figure 6.17 (b) Recirculation of Cage’s notation in Bars 131–133 of Book III Figure 6.17 (c) Recirculation of Cage’s notation in Bars 131–133 of Book III

CHAPTER ONE

Introduction

Modern twentieth-century music is inconceivable without the mode of aesthetic cognition which finds expression in analysis. (Dahlhaus, 1989, p. 94)

The aim of this thesis is to explore the close connection between the discipline of music analysis and selected compositions for piano within the decade following World War II. It was a period when many composers were rigorously theorising about their compositional techniques in the hope of developing a new musical language divorced from the principles of the past. More than at any other time, compositional theories and music analysis were inseparably associated with each other: the applications of new theories were frequently explained in the form of analysis; the aesthetic values of the works were justified by music analysis that demonstrated their unfolding structural coherence. Griffiths describes this phenomenon as one of the prominent features of post-war modern music: “The half decade after the war was a great age of pure music . . . But in another sense no music was pure; it was all technical demonstration, and incomplete without its manifesto or analysis” (1995, p. 51). Consequently, music analysis has been central to how subsequent generations have understood and valued this repertoire. The musical environment in which many modernist composers developed their works and theories was strongly governed by three particular ideologies: formalism, positivism and organicism. The same ideologies had powerfully influenced the discipline of music analysis since its establishment in the nineteenth century and they persisted throughout much of the twentieth century. As compositional procedures became increasingly technical and aimed to ensure musical logic, music analysis and compositional technique came to share closely a common aesthetic value. It is worth underlining that these very ideologies that were upheld by the avant-garde composers of post-World War II are precisely those for which music analysis has come under widespread criticism in recent decades. For this research, four iconic piano works are chosen for close examination: Olivier Messiaen’s Mode de valeurs et d’intensités (1949); Pierre Boulez’s Structures Ia for two pianos (1952); Karlheinz Stockhausen’s Klavierstück III (1952), and John Cage’s Music of changes (1951). Although there is geographical diversity in their

origin—France, Germany and the U.S.A.—each of these four works demonstrates a revolutionary concept of organising musical materials. As a result, new sonic relationships were rigorously explored. From the first appearance of these works, analysis has been considered as a necessary step for understanding them. Since then, the complex compositional theories and procedures of these works have been hypothesised through various different types of analysis. Some of these hypotheses are continuing; others are not. As the discipline of music analysis itself has come under much criticism in recent times, it is pertinent to re-evaluate these four works with this criticism in mind. This research examines the range of analytical approaches to these four works taken over the last fifty years and then provides a reinterpretation of each work. Aesthetic positions of music analysis

In recent decades, the discipline of music analysis has been criticised for its approaches and underlying aesthetic positions. Widespread and controversial criticisms have targeted the governing ideologies and aesthetic values of music analysis, which sprang from nineteenth-century Romanticism. A number of cultural studies have encouraged the study of music in its sociological context but, in doing so, may have insufficiently recognised what music analysis can offer. This ongoing debate has caused some music analysts to re-evaluate the underlying principles of their discipline, to readjust their methodologies and to relocate their position in musical scholarship. Prior to addressing the sensitive, yet challenging issues that music analysis faces today, it is pertinent to identify the ideological correlation between music analysis and nineteenth-century Romantic ideologies and the way in which the same ideologies influenced modernist composers of the mid-twentieth century. The discipline of music analysis emerged in the nineteenth century. With the three ideologies mentioned above, a set of interrelated beliefs and attitudes were developed and established both in music composition and music scholarship. These became the foundations upon which the institution of music analysis was built. Formalism assumed that musical meaning was embodied in the autonomous, self- sufficient musical work, the essential structure of which could be appreciated without reference to culture and society. With organicism, a demonstrable structural coherence was desired. The structure of musical works was studied, and even

measured against a theory, in order to validate their worth. Often, as the emotional aspects of a musical work were considered non-quantifiable, an objective understanding of the internal relationships of the musical material was pursued and verified as the most reliable basis of artistic evaluation. Thus, analytical methodologies were developed in order to gain intellectual understanding of works. Analysts examined those demonstrable aspects of music such as motivic and thematic developments or the harmonic and formal structure of a composition. Such a tendency is a characteristic of the third ideology, positivism. Music analysis was often used as a tool to demonstrate what Lydia Goehr calls “the so-called logic” (1992, p. 86) in a composition, and came to play an integral role in explaining modern music. Goehr suggests that “Central to formalism is an overriding emphasis on the well-formed, self-sufficient work, whose material and form are united such that even the relations of expression joining composer to product is overridden by the demand that one looks only to the work itself” (1992, pp. 266–267). In accordance with the Romantic aesthetic of formalism, music was valued not because of its ability to represent external things such as emotions or nature but because of its own internal structure. As a result of music’s emancipation from “its service to particularised, extra-musical goals” and from “its dependency on words”, the concept of musical autonomy was established (Goehr, 1992, p. 155). Under the influence of formalism, instrumental music came to be considered superior to vocal music. Carl Dahlhaus identifies this in E.T.A. Hoffmann’s Beethoven review of 1810 and writes that music is “a language beyond language, capable of expressing the inexpressible and opening up profound depths that words cannot reach” (1989, p. 90). How Hoffmann viewed instrumental music in the early nineteenth century would have a profound impact on musical history, Romantic aesthetics and, most of all for our purposes, on music analysis. The belief that music should be understood as an autonomous entity emerged and it is upon this foundation that music analysis has mostly operated. Closely associated with formalism was the similarly influential doctrine of positivism. Although positivism emerged from the seventeenth century with René Descartes, the principle founder of modern philosophy and well-known mathematician, Auguste Comte coined the word “positivism” in the nineteenth century. Since then the word positivism has been widely used and in the twentieth

century, it came to be associated with “logical positivism”1 or “logical empiricism.”2 Robert Scharff explains positivism in the following terms:

In the English-speaking world, of course, positivism is now inherited through its last and most sophisticated version, namely, Logical Empiricism; and one generalization to be made about Logical Empiricism is that it embodies a sort of ultimate crystallization of two of positivism’s core features—namely, the promotion of a rigorously “scientific” epistemology and a supreme self- confidence about its own objective, systematic, ahistorical outlook. (1995, p. 2)

A “musical positivism” did not exist as a recognised compositional style in the nineteenth century (Dahlhaus, 1989, p. 193), but Hanslick’s views on music are a clear indication nevertheless of how positivistic attitudes developed through the century. According to Dahlhaus, a positivistic approach is clearly articulated

when he [Hanslick] argues that beauty is complete in itself, that a musical work of art represents “specific aesthetic structure not conditioned by our feeling”, and that this structure must be “comprehended by scientific observation, set free from any psychological subsidiaries relating to its origin and its effect.” (Dahlhaus, 1982, p. 54)3

Hence, scientific forms and terms of intellectual enquiry were demanded and applied to the understanding of music. The tendency towards scientific investigation of a musical work brought a strong preference for “objectivity” over “subjectivity” in music analysis. Analysts usually discuss a piece of music in terms detached from their personal feelings and experiences: rather they search for musical phenomena that can be objectively arranged in a coherent system. The aim of this process is to reveal the “logic” that governs the organisation of musical elements. David Beard and Kenneth Gloag argue that “the attempt to determine what constitutes a formal structure for an individual work, are reflections of positivistic thought that, in the case of analysis, can often

1 In Philosophies of History: From enlightenment to postmodernity, Robert Burns defines the term positivism in relation to “the ‘logical positivism’ of the Vienna Circle”. The leading figures in the Vienna Circles were Mositz Schlick and, in the post-World War I, Rudolf Carnap (2000, p. 98). 2 In Comte after positivism, Robert Scharff provides a comprehensive comparison of the nineteenth- and twentieth-century positivism and also discusses the intrinsic connection of the ideology to Descartes. Scharff argues that the logical positivists preferred the title “logical empiricist” because they could claim Hume as their forebear instead of Comte (1995, p. 7). Another name for logical positivism and logical empiricism is “neo-positivism” and these labels are often used interchangeably in the literature. 3 Internal quotation mark indicates Hanslick’s words. 4

assume a scientific aura” (2005, p. 136). The positivistic attitude demands that any type of interpretation can only be validated by supporting evidence. Under the influence of the formalistic aesthetic, music analysis became a positivistic examination of an autonomous work. Indeed, as Ian Bent and Anthony Pople (2001, p. 527) state:

The primary impulse of analysis is an empirical one: to get to grips with something on its own terms rather than in terms of other things. Its starting- point is a phenomenon itself rather than external factors (such as biographical facts, political events, social conditions, educational methods and all the other factors that make up the environment of that phenomenon).

A third ideology that was profoundly influential on critics, composers, and music analysts is organicism. It became influential because of the new way in which the relationship between nature and art came to be conceived in the early nineteenth century. “A notion of human spirit was incorporated into the grand concept of Nature. . . . Aesthetic value or beauty was no longer to be sought first in nature and then in art, but vice versa” (Goehr, 1992, p. 160). This reverse of aesthetic direction is also reflected in Hanslick’s statement: “Art should not slavishly copy nature but remodel it” (1957, p. 112). One way of remodelling nature was to achieve organic unity in a musical work since, in the Romantic view, the concept of nature was represented through organicism. Hanslick further explains that: “The logic in music, which produces in us a feeling of satisfaction, rests on certain elementary laws of nature which govern both the human organicism and the phenomena of sound” (1957, p. 53). The desire to uncover principles of nature operating in a musical work became central to the way music critics, composers and music analysts came to appreciate aesthetic value. This value was then dependent on the way a composer arranged musical materials based on universal and natural principles, especially those of organic unity. Hoffmann’s analysis of Beethoven’s Fifth Symphony (1813) clearly postulates an indivisibility of aesthetic value and organic unity. This has been the alluring goal of music analysts ever since, becoming, in many cases, an obsession for analytical investigator. Dahlhaus cites Hoffmann’s writing:

Just as bookish aesthetes have often carped and cavilled at the complete absence of true unity and inner cohesion in Shakespeare; and just as a tree, sprouting buds and leaves, blossoms and fruit from a single kernel, reveals its

beauty only to the inquiring eye: so only a very deep examination of the inner structure of Beethoven’s music will disclose the lofty circumspection of the master, a circumspection at once inseparable from true genius and nurtured by the continuous study of art. (Dahlhaus, 1989, p. 91)

Here, the outwardly metaphorical concept of a single kernel becoming a tree is demonstrated by actual motivic and thematic ideas that develop throughout the piece; the various musical ideas are intrinsically part of a single organism because, as we might now conceive it, they all have the same DNA. Most significantly, Hoffmann was convinced that the beauty of this organic unity, however, is not obvious on the surface but requires the inquiring eye. An organically unified musical work, in Romantic terms, can only be conceived by a great composer, whose genius can only be revealed by “a very deep examination of the inner structure”. Thus, Beethoven is esteemed as a true genius, who could create such work of art. Such beliefs truly set a course for music analysis, particularly its aesthetic positions. Since the nineteenth century, many composers have strived to achieve musical unity founded upon the concept of organicism, and they claimed that there was a close relationships between individual musical elements of a piece and the whole. Leonard Meyer has explained how the method of serialism was one of ultimate consequences of such ideology (1996, p. 195). The ideal of organic derivation is clearly prominent in the writings of Webern, a figure profoundly influential on avant-garde composers of the 1950s. Firstly, Webern describes unity as being of the highest merit:

Unity is the indispensable thing if meaning is to exist. Unity, to be very general, is the establishment of the utmost relatedness between all component parts. So in music, as in all other human utterance, the aim is to make as clear as possible the relationships between the parts of the unity; in short, to show how one thing leads to another. (1975, p. 42)4

Secondly, Webern’s idea of unity is certainly based on organicism, as he refers to Goethe’s primeval plant, saying “the root is in fact no different from the stalk, the stalk no different from the leaf, and the leaf no different from the flower: variation of the same idea.” He continues: “The same law applied to everything living: ‘variation on a theme’—that’s the primeval form, which is at the bottom of everything. Something that seems quite different is really the same. The most comprehensive

4 This statement was given in Webern’s lecture on 15th January, 1932. 6

unity results from this” (1975, p. 53).5 One could see that, in Webern’s view, musical unity is achieved through the maximum degree of relatedness among the various elements that grow out of one another. As Webern’s comments show, over a century after Hoffmann’s writings, hidden natural order remained a prevailing metaphor for composers contemplating musical from and critics evaluating it. Finally, in composing atonal music, Webern also believed that the same or an analogous musical unity could be obtained through the serial system, as it had done in the past with thematic tonal music. He writes: “The twelve-tone row is, as a rule, not a ‘theme’. But I can also work without thematicism, that’s to say much more freely, because of the unity that’s now been achieved in another way; the row ensures the unity” (1975, p. 55).6 Here once again, one can clearly observe that the ideology of organicism is deeply embedded in Webern’s compositional aesthetic and theory. The extent to which such procedures are perceptible to a listener remains debatable. However, many modern twentieth-century composers came consciously to construct musical forms and compositional theories that would ensure organic unity. This desire perhaps reached its climax with the compositional theories developed in the decade following World War II. From the reception of Beethoven’s Fifth Symphony to Webern’s compositional aesthetic of organic unity, the ideology of organicism prevailed as a powerful force. Significantly, how the musical idea develops is often not obvious at first glance, on the surface, but is hidden underneath. In other words, one musical idea grows logically from another while remaining essentially true to the original unifying principle. Dahlhaus (1989, p. 91) explained this Romantic view in the following terms: “To understand music means, according to the romantic view that Hoffmann shared with Friedrich Schlegel, grasping the structure, the harmonic and thematic logic of a work, so as to be able to fathom its aesthetic meaning, a meaning that remains inaccessible to mere mindless enthusiasm.” Concerning analytical method, an attitude similar to Hoffmann’s can be observed in the explanation given by Pierre Boulez. In defining the “the indispensable constituents of an ‘active’ analytical method”, he has suggested that:

[Such an analytical method] must begin with the most minute and exact observation possible of the musical facts confronting us; it is then a question

5 This statement was given in Webern’s lecture on 26th February, 1932 6 This statement was given in Webern’s lecture on 2nd March, 1932. 7

of finding a plan, a law of internal organisation which takes account of these facts with the maximum coherence; finally comes the interpretation of the compositional law deduced from this special application. All these stages are necessary; one’s studies are of merely technical interest if they are not followed through to the highest point—the interpretation of the structure; only at this stage can one be sure that the work has been assimilated and understood. (1975, p. 18)7

Here, Boulez articulates one of the primary aims and methodologies of music analysis: being able to discern the underlying and often concealed principles of the musical structure. From the point of view of Romantic aesthetics, the musical works most highly valued are those that embody organic unity representative of nature-like beauty. Such works can only be created by an artist who is considered a “genius”. This idea of genius developed at the same time as the concept of a musical work became established. As Goehr (1992) has argued, this concept transformed ways of thinking about music and the discipline of musical analysis could never have flourished without this work-concept. As a result of this confluence, the main interests of analysis became the valuing and validating of the “masterwork”. Analytical methodologies were also developed to reveal the “secrets” in the compositional process, which are concealed by the composer. There has also been a tendency for music analysts to cast a value judgement on a piece of music according to whether its organic unity can be demonstrated. Bent and Pople explain the relationship between the idea of genius and the attitude of music analysis:

In reality, the analyst works with the preconceptions of his culture, age and personality. Thus the preoccupation which the 19th century had with the nature of “genius” led to the phrasing of the initial question not as “How does it work?” but as “What makes this great?”, and this remained the initial question for some analytical traditions in the 20th century. Since the “scientific”, comparative method was predominant over evaluation in such traditions, and since only works of genius possessed the quality of structural coherence, it followed that comparison of a work with an idealized model of structure or process produced a measure of its greatness. (2001, p. 528)

7 The introduction of Music analysis: In theory and practice by Jonathan Dunsby and Arnold Whittall begins with this same quotation and the kind of analytical study proposed in their book based on Boulez’s statement above. 8

In relation to the idea of genius, music analysts have also responded to the challenge of classifying genera into an evolutionary tree within which the musical canon of Western music is revealed as the most highly evolved branch. One of the most widely acknowledged analytical methodologies applied to the study of tonal music is based on Heinrich Schenker’s theory, which was also embedded in the aesthetics of formalism, positivism and organicism. Not only was his theory used to validate a set of masterpieces, but it was used by Schenker to dismiss as organically defective works that did not conform to his theory. Goehr points out other essential aspects of the Romantic genius:

Creative artists would have to demonstrate their independence from their creations by interfering with and destroying the impression their works were created by human hands, and in part they could do this by refusing to follow any pre-established set of mundane rules. (1992, p. 161–162)

From the composer’s perspective, the goal of “refusing to follow any pre-established set of mundane rules” could be achieved through enharmonic stretching. Of course, this ultimately led to the dismantling of tonality altogether. In this process, tonal rules were slowly bent: chromaticism increased and harmonic progressions became non- functional, eventually leading to atonality. There are many influences that contributed to the dismantling of tonality, but all of them were in response to the imperative that composers create new musical languages that progress, and perhaps depart significantly from, the tonal practices of their predecessors. Certainly, most avant- garde composers of the 1950s are on record as expressing views consistent with this attitude. Music analysis was, in many cases, a vehicle to explain and demonstrate the construction of musical works based on certain compositional theories that were considered new and progressive. Pre-established rules needed to be anything but mundane and this often resulted in theories of forbidding complexity. Therefore, the concepts of genius and organicism were intricately related to each other, influencing the composer to develop a range of compositional theories that were not only progressive in their nature but that also ensured musical cohesiveness. In the modern era, music analysis has found its meaning and purpose in verifying whether compositions exemplified the above qualities.

Critiques and debates

The ideologies discussed above have been closely identified with the discipline of music analysis since its inception. However, in recent years, the discipline or rather its basis in these ideologies has been under attack from many quarters. Analytical approaches and methodologies have been widely criticised for being both acontextual—that is, lacking a basis in historical and cultural reality—and positivistic—having an objective orientation. Such claims have been made by numerous scholars from various sub-disciplines of musical and cultural studies. Not all of them are well justified but my purpose here is to identify the core issues that have challenged the ideas and assumptions on which the discipline of music analysis has been based. Critical questioning of music analysis began as early as the late 1960s. One of the very first examples is a lecture, “On the problems of musical analysis” (Zum Probleme der musikalischen Analyse), delivered by Theodor Adorno a few months before his death in 1969. In this lecture, he scrutinised the nature and function of music analysis as well as the aesthetic positions on which it had rested. Adorno exposed how the realisation of the value of musical works is co-dependant on music analysis. He claimed that “Works need analysis for their true content [Wahrheitsgehalt] to be revealed” (2002, p. 167). He underlined how this co- dependency deepened—how the relationship between the two became almost inseparable when composition turned to post-tonal techniques, more specifically, to twelve-tone compositions. Adorno’s thesis rests on a fundamental perception of music as evolving from an earlier condition of tonal simplicity to a present-day modernist condition of insuperable complexity. His paper compares Handel’s works, Beethoven’s Diabelli Variations, and Webern’s Bagatelles. The point Adorno attempted to make was that Handel’s work could be appreciated without analysis but Beethoven’s work is harder to understand without an accompanying analysis. Adorno argued, further, that Webern’s Bagatelles makes virtually no sense unless analysed, even when performed according to what is written in the score. On the contrary, when these pieces are analysed and then performed, the works suddenly make more sense. He explained such a phenomenon thus:

If, without analysis, such music cannot be presented in even the simplest sense as being meaningful, then this is as much as to say that analysis is no mere

stopgap, but is an essential element of art itself. As such it will only begin to be able to correspond to the status of art when it takes the demands of its own autonomy upon itself. (2002, pp. 168–169)

Adorno strongly made the case for how particular types of analyses and their worth are in fact themselves dependent on particular types of composition.8 Adorno’s paper (2002, p. 174) discusses the problems of the so-called holistic method of examining music. The danger of this kind of analytical approach is that it often disregards striking individual characteristics in music. He claimed that analytical approaches that are based on the concept of a work’s totality have “disturbingly positivistic implications” (2002, p. 174). As an alternative he suggests:

If one really takes the whole as one’s point of departure than also simultaneously implied here is the obligation to grasp the logic of the individual moments—that is, the concretion of the isolated musical instants. And correspondingly, if one takes the constituent elements as the point of departure one’s task is to understand how these elements in themselves, and frequently in contradiction to each other, and then through this contradiction, also simultaneously generate the whole. (2002, p. 174)

Moreover, Adorno also made a connection between types of compositional processes and analytical approaches. He observed two generalised approaches in composition: the first type of composition is where musical materials are organised, in principle, from totality to detail; and the other type would be where the approach is from detail to totality. Thus, Adorno (2002, p. 176) argues that “according to which of these dominates the structure of the music, the same will correspondingly direct the analysis itself.” Here, he alleged an inseparable correlation between the analytical approach and the way the composer organises musical materials in the creative process. Adorno concluded his thought-provoking lecture with the statement: “The crisis in composition today . . . is also a crisis in analysis” (2002, p. 178). He urged music analysis to go beyond just collecting the musical facts but rather to be capable of critically responding to and influencing composition itself. A remarkably similar view to Adorno’s was also argued in “Analysis, theory, and new music” 9 by Joseph Kerman in the 1980s. Unlike Adorno, Kerman’s

8 This lecture was given at the Hochschule für Musik und Darstellende Kunst, Frankfurt am Main. Adorno’s lecture was then translated by Max Paddison who made his own transcription from a tape recording of the lecture. Paddison first published it in Music Analysis 1 no. 2 (July 1982), pp. 169–187. The revised version of this article appears in Essays on Music (Adorno, 2002, pp. 162–180). 11

arguments prompted a range of responses and reactions across the various disciplines of music studies. Central to Kerman’s arguments concerning music analysis was the formalistic approach to musical autonomy, the analyst’s search for organic unity in music and positivistic approaches found in systematic analysis. According to Kerman, traditional approaches to analysis are flawed because they do not make music more readily understandable:

Their [analysts’] dogged concentration on internal relationships within the single work of art is ultimately subversive as far as any reasonably complete view of music is concerned. Music’s autonomous structure is only one of many elements that contribute to its import. (1985, p. 73)

He was also critical of the role of music analysis in the post-World War II era, when composers were particularly influenced by the ideology of positivism:

In the postwar years, however, a powerful appeal was exerted by analysis— and by exactly those trains of analysis that relied most dogmatically on a single principle, a monism or (as it was sometimes revealingly put) a “secret” of musical form or musical coherence. . . . The appeal of systematic analysis was that it provided for a positivistic approach to art, for a criticism that could draw on precisely defined, seemingly objective operations and shun subjective criteria (and that would usually now even call itself criticism). (1985, p. 73)

Kerman’s dissatisfaction followed two main threads. Firstly, with tonal music, he seriously questioned the pre-occupations with Schenkerian theory and analysis, highlighting its limitations. Ironically, one of the aspects in Schenker’s approach most disturbing to Kerman was its formalistic nature in viewing music as autonomous. “Schenker was ready to strip away not only salient details of individual compositions, but also distinctions between compositions, composers, and periods” (1985, p. 85). Secondly, with the development of post-tonal compositional theories, Kerman recognised the significance of Schoenberg’s compositional theory of the twelve-tone system as, historically, a point of departure. What concerned Kerman about such compositional theories was what he perceived as their limited relevance to the wider musical community. He was especially critical of the information science-based academic culture created by Milton Babbitt and his followers in America.

9 This is chapter in Kerman’s book, Musicology (1985). Kerman proposed the essence of argument in an earlier article titled “How we got into analysis, and how to get out”, (1980) though the widespread impact of these ideas dates from the publication of his book. 12

Kerman’s Musicology (1985) became one of the most often cited criticisms of the discipline of music analysis. It contributed significantly to the paradigm shift in musicological practice from the mid-1980s (Beard and Gloag, 2005, p. 14), which led to a movement recognised as New Musicology. This movement has explored a range of alternative ways of studying music from various social, cultural and gendered perspectives. Ideas and claims which generally define the movement of the New Musicologists, followed Kerman’s suggestions by replacing positivism with a more critical approach and challenging the idea of autonomous musical works in musical scholarship (Beard and Gloag, 2005, p. 122). New Musicology often promotes a cross-disciplinary approach but has come into criticism itself for failing to offer more specific analytical alternatives. In opposition to Kerman and New Musicologists, Pieter van den Toorn has fiercely defended the value and aesthetic positions of music analysis in Music, politics, and the academy (1995). Attempting to find more mutual ground in these controversies, others have started to critically evaluate the nature and general approaches of music analysis. A recent publication, Rethinking music, edited by Nicholas Cook and Mark Everist (1999), is an insightful collection of essays re- evaluating the role and value of music analysis. The common interests and goals of these essays, as Beard and Gloag see, are categorised as Critical Musicology, which is defined as “the idea of continually rethinking music to avoid establishing new orthodoxies or grand narratives, although, in general, it remains concerned with finding some kind of synthesis between analysis and a consideration of social meaning” (2005, p. 38). In line with Kerman, by the late 1980s Leo Treitler also demanded changes in established methodologies of music analysis in his “To worship that celestial sound: motive for analysis”, Music and historical imagination (1989). He particularly addressed the prevailing ideology of organicism in analytical methodologies that attempted to demonstrate the organic unity of a composition. Treitler summarised his criticisms in five main points (1989, p. 52). Firstly, he attacked the predominant attitude and the aim of music analysis that “The work must be explicable in terms of a single principle, and every detail must be derivable from the idea of the whole” (1989, p. 52). Secondly, he criticised theories holding that the focus of analysis is mainly to discuss the pitch structures of a work at the cost of other dimensions of music, pointing out that the reason for such a tendency is that “theorists have most successfully demonstrated the properties of organic coherence” (1989, p. 52). He went 13

on to explain that proving organic unity in the construction of other dimensions of music is not as neat and convenient as dealing with pitch organisation. Thirdly, he expressed dismay at the widespread inclination to perceive music as a synchronic structure. Using Schenker’s theory as an example, he was critical of approaches that aimed to provide a map-like overview of a work that enabled the entire structure to be perceived in a non-temporal way, claiming some justification that “Analysis whose centre of gravity is a synchronic conception of pitch structure is little interested in the phenomenology of music” (1989, p. 52). In other words, the temporal and experiential aspects of music are unfairly neglected. Fourthly, the rationalistic nature of analysis was questioned in the claim that music analysts are more interested in asking “how Music works” than attempting to discover “how musics work” (1989, p. 52). Finally, Treitler pointed out that methods of structural analysis effectively decontextualise music and give preference to a rationalistic approach. In summary, Treitler’s core arguments lie in his desire to see analytical methodologies being “less normative and more phenomenological and historical; that take account of music other than pitch structures, and that concern themselves not with structures alone, but with the relations of structure and meaning” (1989, p. 55). Treitler also identified a deeper issue underlying the dominance of an aesthetic of organicism in music analysis. He claimed that this was due to the relationship between the concepts of genius and organic unity achieved in a composition, as mentioned earlier: “Genius is the natural creative capacity of mind that provides the controlling force in the production of unified works of art” (1989, p. 5). As analysts were preoccupied with this nineteenth-century belief, analytical methodologies were also developed to reveal the secrets in the compositional process and to discover the underlying principle, which had been concealed at the moment of composition. Music analysis became an agent that measured a musical work against the idealised model of structure, a model that had been observed by music theorists and had been based on a set of inter-related ideologies. Furthermore, the analytical approach used in this type of procedure has often been described as quasi-scientific and positivistic. The two most favoured theoretical systems employed by music analysts have been Schenkerian theory and Allen Forte’s Set-theory for tonal and atonal music respectively. However, a common criticism of such analytical systems is that they often validate the worth of a work in terms that are objective but not necessarily audible. Whether or not analyses based on these theories 14

can help listeners has been strongly questioned, especially in terms of perceiving the coherence of the structures that these analyses profess to demonstrate. As the discipline of music analysis has come under serious criticism, the influence of formalism and positivism on the claims of these theories has also come increasingly under attack, thereby challenging the way we think about music and value it. This point was strongly argued a few years later by Nicholas Cook in Music, imagination and culture (1992). In response to major criticisms of music analysis and alternative ways of studying music suggested by the New Musicology, in the mid 1990s Van den Toorn reaffirmed his stance against scholars such as Kerman and Treitler. Concerning their claims against the discipline of analysis, Van den Toorn stated: “The sense here is that much established procedure has been misrepresented by that group, who often confuse the aims of analysis with those of the institutions in which it is taught, practiced, and debated” (1995, p. 7). Van den Toorn provided a detailed critique of what New Musicologists professed to do and also defended Schenkerian analysis by articulating its fundamental rationale. By presenting his own analyses to demonstrate the significance and value of knowing the detail of music, he ultimately targets the shortcomings of New Musicology. The credibility of New Musicologist’s claims against so-called “technical” analysis was undermined by Van den Toorn:

Among historians or “historical musicologists,” only Taruskin has familiarized himself with the mechanics of pitch-class set analysis to an extent permitting a substantial critique of Forte’s methods. George Perle has launched his own evaluation along compositional as well as analytic theoretical lines. But the new humanists have shied away from the specifics of their arguments in this case, those relating to atonal set theory and its methodologies. And it may be that the “technical” aspects of the case have had their inhibiting effect, which at this point is likely to have come as a self-fulfilling prophecy. (1995, p. 227)

Van den Toorn further exposes the contradictions inherent in New Musicology by saying that:

But notwithstanding today’s preoccupation with socio-political matters, its demand that such matters be taken into account, no such consideration of Forte’s methods or of pitch-class analysis in general has been forthcoming from humanists, feminists, or post-modernists such as Joseph Kerman, Leo Treitler, Susan McClary, or Lawrence Kramer. On the contrary, the protest has been general and promotional rather than specific and to the point. And it has

dealt with political, social, and literary scientists rather than with the politics of its own academic backyard. (1995, p. 225)

Van den Toorn firmly believes in the value of music analysis and the way it has traditionally established its role in music scholarship. At the same time, he welcomed a new openness in music scholarship as suggested by other scholars but without rejecting the discipline of music analysis completely. The arguments raised by Van den Toorn are considered as an important and necessary counter-voice that perhaps many are still keen to hear. While New Musicology does not provide more specific analytical alternatives, by the turn of the century Jim Samson had discussed what alternatives are available to analysts today in his “Analysis and context” (Rethinking music, 1999). Firstly, in order for music analysis to advance further, he has proposed that there should be a change in the field of music theory. In his view, music theory should move beyond “the identification of musical structure” and give more attention to “the identification of musical materials, confronting the social nature of those materials and exploring the mechanism involved in their realization and perception” (1999, p. 53). He urged both music theorists and music analysts to engage in issues concerning performance and perception. In other words, the analytical approach should be less autonomous and should engage with the context of the work. As an alternative to the ideology of organicism, Samson suggested that analytical interpretation of works should “embrace disjunction and indeterminacy, as well as, (theories of) chaos and complexity” (1999, p. 53). Although Samson discussed some alternative analytical approaches, he also recognised the value of formalism and modernism in present day scholarship. He argued that the more traditional approaches of analysis remain as a valid possibility, one among many in our pluralistic culture. What has been provided here is a glimpse into some of the major challenges that have been identified in musical scholarship, especially in relation to the discipline of music analysis: questioning and evaluating the tradition of studying music, and searching for alternative ways to engage with a piece of music. In the midst of such changes and challenges, one then asks how modernist compositions, which are deeply rooted in the aesthetics of formalism, should be studied. The following discussion addresses some of the pertinent issues that arise when analysing modernist twentieth- century composition.

Post-tonal music in the twentieth century and music analysis

The development of new compositional theories in the post-World War II period was a continuation of the evolution from atonal works composed earlier in the twentieth century. Leonard Meyer (1996, p. 338) interprets the momentous consequences of this period in the following terms:

Once the absence of a tonal centre was allowed, compositional choices could no longer be thought of as departures, however distant, from the norms of tonal syntax. Conceptualization thus intensified the problems of compositional choice. What was needed was not new strategies, but new rules. One striking manifestation of this need was the deep and abiding concern of composers— especially composers of “advanced” music such as Schoenberg, Babbitt, Stockhausen, Ligeti and Xenakis—with music theory and aesthetics.

In replacing tonality, new modes and artificial scales were sought by composers such as Debussy, through Stravinsky and Bartók to Messiaen. This resulted in new systems of pitch relationships. Of these various new ways of organising pitches, most famous, radical and influential was Schoenberg’s development of 12-tone technique in the early 1920s10. Perhaps its most significant impact occurred after World War II, with the development of some highly complex compositional theories. More than any other, the aesthetics of Webern’s serial compositions were received with great admiration by the younger generation of composers who became prominent figures in the second half of the twentieth century. Not only did Webern’s serial techniques powerfully influence them but they also had a special place in the Darmstadt summer courses. With “extensions and destructions of tonality” (Albright, 2004, p. 7) being perhaps the most prominent characteristic of modernist twentieth-century composition, it was not only composers who faced the challenges of conceptualising these new possibilities. In dealing with music without a tonal centre, analysts were also entering into a new territory, confronting many challenges that were unknown in tonal music.

10 Another figure, who contributed to the development of twelve-tone technique, was Josef Matthias Hauer (1883–1959). Hauer’s devise is often known as ‘trope.’ According to George Perle, “Hauer’s set, or ‘trope,’ as he terms it, is not a unitary structure but a combination of two six-note segments of mutually exclusive content, within each of which only the content, not the order, is specified. Thus the order in which the notes are remaining terms are properly interchanged in order to reflect the reciprocal relations among the set-forms” (1991, pp. 3–4). Although Hauer’s 12-tone composition also used an ordered series as claimed by John Covach (2002), the primary distinction between Hauer and Schoenberg’s system lies with whether employing unordered hexachords or an ordered series. In this thesis, a term “distributive serial technique” is used to describe when a set of pitch classes is only identified as the content without a specific ordering.

As composers were inventing new rules for their works, music analysts had to uncover what these new principles were by examining and applying various theories to compositions. The change that took place in compositional theories can be characterised by an increase in both complexity and diversity. Decoding a new rule for each composition became a time-consuming process for analysts and the lack of consensus on how to do this became apparent. As a result, as Cross (1994, p. 185) mentions, there remain many works, even by major figures in the history of twentieth- century Western music, which are still incompletely treated by music theorists and analysts. Atonality needed to find new ways of conceiving pitch relationships. However, it is easy to overlook the experiments made with the other parameters of music by composers such as Debussy, Stravinsky, Schoenberg, and Messiaen. For example, while Schoenberg was developing a new system of organising pitches, in the works of these composers, the sense of regular metre was broken down and at times demolished. Schoenberg also experimented with the timbral aspect of music with Klangfarbenmelodie in which the transformation of sound colours became a core structural development replacing those of pitch and rhythm. Again, new ways of conceiving timbral and rhythmic structures were needed although, for reasons of conceptual difficulty, these have not been recognised as central by many music analysts. In the analytical literature of modernist twentieth-century music, these other parameters have always been relatively neglected in comparison to the organisation of pitch. The four works chosen for examination in this thesis and the analytical approaches that have been applied to them are to be understood from this broad perspective. They all embody what were, at the time, new compositional theories and techniques and musical languages based on new and unfamiliar rules. As a result new dimensions of the sonic world were opened up. In Mode de valeurs et d’intensités, every single musical element was derived from a mode with three divisions in which every single pitch was assigned its own duration, dynamic and attack. In Structures Ia, twelve-tone technique was expanded to integral serialism, encompassing the organisation of three other musical parameters. In Klavierstück III, the compositional technique moved, in Stockhausen’s terms, from “point” to “group”. In Music of Changes, chance operations were applied to organise various parameters of the music. Thus, in terms of compositional theories and method, these works were certainly 18

“progressive”. As each piece demonstrates revolutionary compositional techniques and methods, this repertoire has been widely regarded as marking an important stage in the development of twentieth-century music. For this reason, they have attracted much discussion concerning how the composer developed his compositional theory and method, and how these theories were implemented in the compositions themselves. A significant corpus of analytical writings has attempted to provide answers to these questions throughout the last half century. It is important to emphasise again that the aesthetic values that fostered the above-mentioned evolution of compositional theories were based on the ideologies identified in the earlier part of this chapter. Composers of the post-World War II period were working with pre-designed formal structure and were more concerned with achieving musical logic than how their compositions would function in society or entertain audiences. Thus, a strongly formalistic approach infused their composition. Musical unity achieved by structural coherence was an essential aesthetic objective of developing a compositional system. Such an approach nevertheless strongly reflected the ideology of organicism. The concern for composers of this time was not about emotion and musical expression; rather, it was about the internal structure of the composition and how to attain optimum integration of musical elements. Their interest in surface sound phenomena was not directed toward emotional expression but to exploring previously unheard acoustic phenomena. Albright (2004, p. 10) explains that “the listener sometimes feels that the music arises not from expressive urgency or from obedience to the rules of good craft, but from autonomous delight in extending the kingdom of sound.” Moreover, composers seemed never to tire of justifying their theoretical and aesthetic stands in the form of systemic analysis, revealing so called logic within the musical work. Again, these characteristics can be interpreted as an indication of how the ideology of positivism in scholarship had influenced the avant-garde composer’s mind. Although music analysis was an important part of the musical culture in this period, modern music has created certain controversies and challenges for the music analyst. These can be identified as follows. Firstly Adorno (2002, p. 172) believes that radical serial and aleatory music cannot be approached by the traditional methods of music analysis. In his view, when confronting such music, analysts are often confused by an approach that merely records the facts about compositional plan or process.

Secondly, as musical unity was an essential concern for both music analysts and composers, a tendency arose for compositions of this period to be tailored to be analysed. As demonstrable musical unity was the primary goal of both, other features of the compositions were often overlooked. Thirdly, as the musical ideas that are embedded in the pre-compositional design become almost impossible for the listener to identify, the musical surface became a more pertinent issue. All too often, music analysts or composers defend musical structures whose particular pitch and interval relationships seem to make no demands on the listener, calling these structures “coherent”. This asks a simple, yet important question: “But can you hear it?” According to Joseph Straus (1986, p. 10), “Whenever a musical analysis pushed beyond the most obvious surface relationships, this question is likely to be asked.” Interestingly, Babbitt’s answer to this question is: “Of course you can hear it, but it’s not a matter of hearing. It’s a matter of the way you think it through conceptually with your musical mind; it’s a matter of how you conceptualize, how you conceive it” (as cited in Straus, 1986, p. 10). Despite Babbitt’s influence on both the development of compositional theories and of music analysis, his belief about conceptualising and hearing musical relationships—mainly pitch and interval—has been strongly questioned.11 Since only a few people have a listening ability that can appreciate what Babbitt claims, Straus distinguishes two very different approaches in music analysis: one is a listener-oriented approach, which focuses on features of musical surface, and the other is a composer-oriented approach, which demonstrates how pre-compositional materials (sets and arrays) are musically concretised.12 The problem of the relation between perception and concepts is never greater than when dealing with compositions of the post-World War II era. Among the outstanding achievements in the development of compositional theories of this time were the systems devised to organise musical parameters other than pitch; the four pieces chosen for this study demonstrate quite different approaches to this area of possibility. As twelve-tone technique had redefined the hierarchical relationship between pitches, the hierarchical relationship between the other parameters now needed to be reconceptualised. No longer were the “new rules” concerned primarily with pitch but equally or as much with the inter-relationship

11 Cook provides a insightful discussion on this matter in the last chapter of his Music, imagination and culture (1992). 12 Straus presents the analysis of the first eighteen bars of the String Quartet No. 2 by Babbitt from the angle of a listener-orientation in “Listening to Babbitt.” 20

between the other parameters such as rhythm, dynamics, texture, register and accentuation, all of which also required explanation within the system. For instance, under the condition of integral serialism and indeterminacy, the way these parameters relate to each other differs profoundly from all previous music. This directly affects the surface sonic phenomenon, thus presenting a different listening experience. Therefore, in composition of the post-World War II period, music analysis faced a major new challenge in investigating essential structural relationships beyond pitch. Central questions and the research design

From the issues identified above—the criticism of music analysis and specific analytical challenges presented by modern twentieth-century composition—the following questions and sub-questions arise:

How has the avant-garde music from around 1950 been approached analytically in the last fifty years? What were the main aspects on which these analytical studies focused? What do these analytical studies reveal about individual pieces?

What are the ultimate benefits of music analysis in relation to this repertoire?

In summary, this thesis reexamines four piano works composed by Messiaen, Boulez, Stockhausen and Cage during 1949–1952 in relation to the discipline of music analysis. The primary aim is to investigate the nature and content of music analysis as it has been applied to these four works over the last fifty years. In addition, building on these observations, my own analyses of these works will contribute a further perspective. Investigating the nature and content of previously written analyses will reveal various aspects of the discipline: its place in musical studies, its aims and functions, its governing ideologies and most importantly its achievements. Based on these types of concrete data, one could then precisely evaluate the validity of current criticism made about music analysis and identify particular approaches in music analysis that might need adjustment. Such a re-evaluation may contribute to a better way of understanding and hearing these works, a way that will hopefully be more meaningful to performers and audiences of our own time. This thesis therefore hopes to challenge some of the commonly held beliefs about these four works from the last fifty years. Each examination for the four individual case study works consists of four progressive stages:

1 Reviewing the composer’s own writings and interviews

2 Selecting a range of the music analyses of each selected work written over the last fifty years

3 Identifying any controversial issues, discrepancies or congruences in the analytical approaches

4 Analytically re-examining each piece in order to address and clarify the issues identified in the previous stage.

The primary motive for studying a composer’s own writings and interviews on their music is to provide a solid foundation for analytical approaches to these works. Evidently, a significant part of the musical culture of the post-World War II era features composers discussing their own work. All four composers examined in the present study have left volumes of writings and interviews regarding their own music as well as critical evaluations of the works of their contemporaries and predecessors. Studying these resources is an indispensable part of research for the following reasons: they provide vital information about the compositional theories and techniques; they paint a picture of the cultural surroundings in which composers engage with their creativities, thus placing the selected works into a historical context; and they are an authoritative source for establishing convincing analytical approaches for each work. The analytical writings selected for each case study vary in several aspects. For example, the publication dates range from 1958 to 2005, encompassing over five decades. Furthermore, the selected analytical writings are extracted from various types of musical literature. Firstly, discussion of individual pieces appears in books that focus on a single composer’s life and his achievements: Robert Sherlaw Johnson’s Messiaen (1975), Paul Griffiths’ Boulez (1978), Dominique Jameux’s Boulez (1989), Robin Maconie’s Other planets: The music of Karlheinz Stockhausen (2005), Jonathan Harvey’s The music of Stockhausen: An introduction (1975), Christoph von Blumröder’s Die Grundlegung der Musik Karlheinz Stockhausen (1993) and James Pritchett’s The music of John Cage (1993). Secondly, there are several journal articles devoted to the analysis of a single piece. Two analyses can be found in Die Reihe Vol. 4 (1960): György Ligeti’s analysis of Structures Ia and Dieter Schnebel’s analysis of Stockhausen’s Klavierstück III. Thirdly, publications that deal with twentieth-century music and more specifically the music of the post-World War

era often include valuable discussion on the selected pieces; among the authors are Griffiths (1981 and 1995), Roderich Fuhrmann (1974), Morag Grant (2001), Rudolf Stephan (1958). Fourthly, Nicolas Cook (1987), and David Lewin (1993), provide analyses of Stockhausen’s Klavierstück III in their publications in books on music theory and analysis. Finally, the relevant parts from three doctoral dissertations are also reviewed: Yayoi Uno (1994), Pritchett (1988) and Mary Wennerstrom (1967). Non-English publications are also included, with the following authors’ analyses only available in German: Stephan (1958), Fuhrmann (1974), Klaus Schweizer (1973), Ursula Eckart-Bäcker (1986), Blumröder (1993), and Stefan Schädler (1990).13 The third stage involves reviewing and re-evaluating the above selected analyses. In this process, the range of analytical approaches and purposes will be closely examined. Comparisons will be made concerning the findings of each analysis: both agreement and disagreement between authors’ opinions and interpretations will be addressed. The strengths and weaknesses of different methodological approaches and analytical systems used in analyses will be discussed. For example, some authors mainly focus on the compositional process and are interested in answering the question of how it works while others consider more the listener’s perspective, and as a result, they approach the piece from the perspective of reception. Any controversial issues, discrepancies or concordances raised will be identified and any paradigm shift occurring in the way analysts have approached the pieces over the last half century will be observed. Through this stage, the validity of criticisms made against the discipline of music analysis can be reassessed. In the final stage, each selected piece will be analytically re-examined. My approach has been to seek an appropriate angle informed by the previous analyses from which to re-examine them. In the process, much attention is allocated to the composer’s own writings on their works and theories, but an application of an analytical system based on a particular theory is avoided. An attempt is made to provide an analysis that is a by-product of the continual dialectic between the conceptual understanding of compositional theory and perceptual experience of surface musical phenomena.

13 These publications were translated by Christina Young and Sun-Ju Song, except Eckart-Bäcker’s article, which was translated by Barbara Steinhauser, Armin Terzer and Sun-Ju Song. Robin Maconie also translated a relevant part from Stephan’s book as a draft version. 23

Chapter 2 discusses how four composers were influencing each other during the years immediately following World War II. The purpose of this chapter is to illustrate the underlying significance of the personal connection between them. Chapters 3 to 6 will focus in turn on the four case studies in this order: Messiaen’s Mode de valeurs, Boulez’s Structures Ia, Stockhausen’s Klavierstück III and Cage’s Music of Changes. Each case study consists of four sub-sections: putting the work in its historical and biographical context; a review and re-evaluation of selected analytical writings; analytical re-examination; and summary. Although each case study is structured in these sub-sections, each of them is individually tailored to accommodate its own peculiarities. For instance, the number of selected analyses differs in each case study, as do the issues raised by them. Chapter 7 summarises the underlying features of the compositional techniques developed by the four composers in these chosen works. This will be followed by an overview of the predominant analytical approaches across the four case studies as well as identifying further research directions in music analysis. In conclusion, the integral role played by music analysis in relation to understanding the repertoire of post-World War II will be reassessed.

CHAPTER TWO

Four Composers and Four Works

It is interesting to observe the fidelity of Messiaen’s pupils to their master, something unusual enough to be emphasized. I have not personally had any great faith in the virtues of teaching above a certain level, and yet I cannot fail to recognize that Messiaen was the determining influence of my student days. (Boulez, 1990, p. 405)

There are close and intriguing connections between the four composers whose works are examined in this research. Their paths crossed on various occasions during the years immediately following World War II. For example, Boulez and Stockhausen both attended Messiaen’s class at the Paris Conservatoire. Important interactions occurred between them when they heard each other’s works performed, including those at the Darmstadt summer courses. Such interactions led to personal meetings and correspondences. Apart from Messiaen, three of them wrote articles for Die Reihe. The four works investigated in this research, moreover, were chosen not randomly but demonstrate some of the underlying connections and influences that occurred among these four composers. Furthermore, each of these pieces marks an important stage in the compositional output of its composer, and these moments were significantly reinforced by their close personal relationships. Indeed, in the post-World War II era, all four composers developed their own system of organising musical material. Their individual solutions to this same challenge can be seen in four selected pieces. Interactions

The purpose of this chapter is to sketch the various layers of interaction between the four composers at the time they composed these remarkable pieces. Figure 2.1 gives an outline of the chronology of these events. Messiaen in fact taught Boulez, who became his pupil in 1944 (Boulez, 1990, p. 418) and later attended his legendary classes at the Paris Conservatoire. Messiaen’s tutelage was a vital influence on his compositional development, as was Messiaen’s belief in the significance of studying modern music as well as non-European music such as music of Africa and Asia in this class. According to Boulez, another influence was Messiaen’s analyses of important contemporary works in his class, in particular compositions of Bartók, Berg and Schoenberg, which were not part of the regular concert repertory at that time.

Messiaen also shared his own evolution in musical thinking and discoveries with his students (Boulez, 1990, p. 405). In fact, these modern compositions were banned or discouraged by the occupying Nazis because modern music was perceived as a token of defiance against German occupation. As a result, these works became known to Boulez and other pupils through Messiaen’s teaching. Boulez’s description of Messiaen’s influence is as follows:

Personally I rejoice at having had the benefit of Messiaen as a teacher at a time when his novelty was fresh, and I can say without fear of seeming banal that this experience at the outset of my career as a composer was not only something that left its mark, but something irreplaceable. (Boulez, 1990, p. 406)

In January 1952, Stockhausen also came to study with Messiaen in Paris, and attended his class in aesthetics and analysis twice a week for the whole year. Stockhausen expressed the experience in the following terms:

In many respects Messiaen did the opposite of what I wanted. He never tried to convince me. That made him a good teacher. He did not give instruction in composition, but showed how he understood music of others and how he worked himself. (as cited in Kurtz, 1992, p. 48)

To Stockhausen, Messiaen’s teaching was different from the doctrinaire style of some of his Cologne teachers. Indeed, Stockhausen’s decision to continue his music study in Paris with Messiaen was most likely inspired by an event in the previous year. At the Darmstadt summer course in 1951, Stockhausen met Karel Goeyvaerts, a young composer and a former pupil of Messiaen. Even though Goeyvaerts was no longer in Messiaen’s class by then, he talked about his teachers and what was taught at the Conservatoire. Goeyvaerts showed Stockhausen his sonata for two pianos, which incorporates the organisation of four musical parameters (Kurtz, 1992, pp. 34–35).14 Stockhausen also shared his compositional ideas with Goeyvaerts in subsequent years, although their correspondence is not available. Importantly, at Darmstadt that year, Stockhausen and Goeyvaerts had an opportunity to hear a recording of Messiaen’s Mode de valeurs in the lecture given by music critic Antoine Goléa. According to Kurtz, Stockhausen pursued Goléa for another opportunity to hear the recording that, anecdotally, he listened to repeatedly. He was fascinated with the piece and sensed its

14 Goeyvaerts made a detailed study of Webern’s Piano Variations (1936) in the winter of 1949–1950 (Toop, 1974, p. 154). 26

relationship to Goeyvaerts’ sonata (Kurtz, 1992, p. 36). Maconie explains how Stockhausen perceived Mode de valeurs:

To Stockhausen it sounded like “star music:” a reference perhaps to the note- constellations of Hermann Hesse (or indeed, to the serial “constellations” of Thomas Mann’s Adrian Leverkuhn). What is clear is that this music was “point music” in the sense implied by the beads of Hesse’s Glass Bead Game as Stockhausen understood it. (2005, p. 42)

While Stockhausen was in Paris, he also became acquainted with Boulez and they immediately discussed their own works. Clearly, over the following years, both composers were well aware of each others’ compositional output and theoretical concepts on which they were working. As Kurtz puts it Boulez was very impressed with the “radicality” of Stockhausen’s Klavierstücke IIV (1992, p. 50). Kurtz notes specific occasions when there were heated debates between them. 15 Boulez also showed the unfinished Structures to Stockhausen in 1952 in Paris (Toop, 1974, p. 143) and later he sent Stockhausen the first sketches of the five formants of his Third Sonata (Kurtz, 1992, p. 87). Moreover, while Stockhausen was staying in Paris, it was Boulez who invited him to visit the Club d’Essai studio for the first time toward the end of March 1952 (Maconie, 2005, p. 99). Kurtz describes their relationship thus: “in spite of all the differences between the two personalities and their work,” their friendship “was based on mutual understanding and respect. For years Boulez was the only person whose musical judgements interested Stockhausen” (Kurtz, 1992, p. 50). The first meeting between Boulez and Cage took place in Paris in 1949; this was Cage’s second trip to Europe. On the recommendation of Virgil Thomson, Cage visited Boulez. The two became friends and corresponded with each other for the next five years (Shultis, 2002, p. 29). The letters they exchanged from 1949 to 1954 are valuable documents revealing the central interests of both composers in developing a compositional system, as well as reflecting the musical climate of the time. On this account Jean-Jacques Nattiez writes in his “Introduction” to The Boulez-Cage correspondence, “What is recreated through these documents is first and foremost the musical climate of the times, a difficult climate for these two illustrious representatives of the avant-garde” (1993, p. 6). As Boulez was working on the

15 One occasion was about Stockhausen’s Studie I and the theoretical concept of the work in the December of 1953, when Boulez and Michel Fano visited Cologne; another was at Darmstadt in 1956, concerning Stockhuasen’s Klavierstück XI, which embraces aleatoric compositional principles. 27

principles of integral serialism, Cage was also searching for matrices that would enable him to organise rhythmic structures. At Cage’s request, Boulez supplied the detailed technical explanation of the compositional system of Polyphonie X, which Boulez was working on. Cage admitted his admiration for what Boulez sent him, writing back to him on 22 May, 1951: “The long letter you sent me with the details about your work was magnificent” (Nattiez, 1993, p. 92). Later the same year, in August 1951, Boulez also sent a letter to Cage with a detailed discussion on the compositional system of Structures (Nattiez, 1993, pp. 98–103). During Cage’s trip to Europe in 1949, Boulez introduced Cage to Messiaen, who invited Cage to play his Sonatas and Interludes to his pupils on 7 June (Griffiths, 1995, p. 29). Later in the month, it was performed again at Suzanne Tézenas’s salon, where Boulez gave a circumspect introduction (Nattiez, 1993, p. 5). Boulez, in the introduction, explained how the four dimensions of a sound, “duration, amplitude, frequency, and timbre,” are modified by inserting objects between the strings of a piano (Nattiez, 1993, p. 30). Boulez was already aware that Cage had achieved his “pulverization” of the musical language intuitively and not by intellectual principles, but he was astute enough to lend his support on aesthetic grounds. A further discussion on Cage’s contribution toward conceptualising a sound with four dimensional properties is found in Boulez’s article “Possibly (Eventuellement) . . .” that was published in 1952. He also comments that:

We owe equally to Cage the idea of sound complexes; he has written works in which, instead of pure sounds, he uses chords without harmonic function but essentially as a kind of sound-amalgam linked to timbre, duration and dynamics in the sense that each of these characteristics can vary with the different components of the amalgam. (1991, p. 135)

Moreover, Griffiths argues that “the definition of those four parameters, which provided the organizational basis for total serialism, came from Cage.” To support this claim, Griffiths refers to a statement from Cage’s essay “Forerunners of modern music”: “sound has four characteristics: pitch, timber, loudness, and duration” (Cage, 1978, p. 63). Cage’s essay, interestingly, first appeared in the journal The Tiger’s Eye in New York, March 1949. It was then translated into French and published in the Paris journal Contrepoints later the same year. One could argue that preparing the piano encouraged Cage to think of sound as consisting of four parameters: by interfering with the strings, you change the nominal pitch, the timbre and the 28

amplitude. You also change the reverberation time, which may be considered equivalent to the duration. It is important to note here that duration is affected by how long the sound can last, not by how long the key is held down. It is, however, questionable that Cage came up with the four parameters himself, as the source from which he got his definition is unidentified. The performance of Sonatas and Interludes in Paris in 1949 had important consequences for both Messiaen’s and Boulez’s compositions. Messiaen, like Boulez, acknowledged Cage’s innovation in modifying the musical parameters through inserting objects between the strings of the piano, writing “Each note possessed its own timbre, its own attack, its own sonority. It was a brilliant idea without far- reaching effects, except indirectly with the appearance of electroacoustic techniques” (Messiaen, 1994, p. 171). In Mode de valeurs, composed at Darmstadt in the summer of 1949, the influence of Sonatas and Interludes can clearly be seen. Following the American’s lead, Messiaen also pre-determined exact qualities of the four musical parameters of pitch, duration, attack and dynamics; but while Cage achieved this by physically preparing the piano, Messiaen did so conceptually, by designing a mode. Cage’s influence on Boulez was no less striking. Having heard Cage’s work, Boulez himself manipulated the four parameters, serially, in Structures 1a. His choice of musical material, the highest division of Messiaen’s mode from Mode de valeurs, further strengthened the connection between the three composers. In addition to influencing his European contemporaries, Cage played an important role in introducing Boulez and his works to the U.S.A. Nattiez says “Cage made himself the proselyte of Boulez in the United States” (1993, p. 7). Cage arranged concerts for Boulez, most significantly the American premiere of Boulez’s Second Piano Sonata. Being so impressed with Boulez’s serial system, Cage translated a large portion of Boulez’s letter16 into English and published it in “Four musicians at work”, Transformations: arts, communication, environments (1952). As the content of Boulez’s letter includes a detailed discussion about serial principles and materials used in Structures, thank to Cage the compositional system of integral serialism developed by Boulez became known in the U.S.A. Both Music of Changes and Structures were composed during this period when Boulez and Cage wrote to each other frequently. Interestingly, both works have been

16 This letter was sent to Cage in August, 1951 (Nattiez, 1993, pp. 98–103). 29

recognised as landmark compositions. However, what has been less understood are the remarkable similarities in the theoretical conceptualisation underlying these works. Boulez explained this in the previously mentioned 1952 article entitled “Possibly . . . ”:

More recently, he [Cage] has been working on setting up structural relations between the different components of sounds, and for this he uses tables which organize each component into parallel but autonomous distributions. The tendency of these experiments by John Cage is too close to my own for me to fail to mention them. (1991, p. 135)

Despite an intense friendship between the two composers, Boulez’s disagreement on Cage’s use of chance operation eventually soured their friendship. This is how Boulez responded to Cage about Music of Changes:

Everything you say about the tables of sounds, durations, amplitudes, used in your Music of Changes is, as you will see, along exactly the same lines as I am working at the moment. The only thing, forgive me, which I am not happy with, is the method of absolute chance (by tossing the coins). On the contrary, I believe that chance must be extremely controlled. (Nattiez, 1993, p. 112)17

Sadly, their friendship never fully recovered. Like Boulez and Stockhausen, Cage also admired Messiaen and his music. He invited Messiaen to the U.S.A. after returning from his second trip from Europe. This occasion was described in the letter sent to Boulez dated 17 January, 1950, and his admiration for Messiaen is seen in the comments: “Messiaen was here—I love him for his ideas about rhythm. Almost everyone was against him because of his half- religious and half-Hollywood spirit. I invited him here . . . and he explained his Turangalila score to some composers” (Nattiez, 1993, p. 48). It seems natural that Cage was impressed with the rhythmic techniques that Messiaen had explored, as Cage himself experimented intensively with various rhythmic structures. The meeting between Cage and Stockhausen took place in 19 October, 1954, when Cage and David Tudor came to Cologne for their first tour in Germany and performed new American piano music. Cage made a strong impression on Stockhausen. According to Maconie (2005, p. 140), Stockhausen viewed Cage as “the

17 Boulez’s published criticism of Cage’s chance operations can be seen in the following articles: “Alea” in Stocktakings from an apprenticeship (1991, pp. 26–38), originally published in French in 1957; “Where are we now?” in Orientations: collected writings (Boulez, 1990, pp. 445–463), originally published in 1978; and “Cage and chance” in Dialogues with Boulez (Di Pietro, 2001, pp. 27–34). 30

craziest spirit of combination I have ever come across; . . . he has that indifference towards everything known and experienced that is necessary for an explorer.” However, Cage’s compositional theory of indeterminacy caused quite a reaction from European composers. His lecture “Indeterminacy” at Darmstadt in 1958 especially generated controversy. The adverse reaction toward Cage is evident in the fact that Cage did not return to Darmstadt until 1990 (Shultis, 2002, pp. 38–39). Despite the controversy over Cage’s compositional concept of indeterminacy, Stockhausen was open minded enough to give a lecture on Cage’s Concert for Piano and Orchestra (1957–1958) at Darmstadt in 1959, having heard it performed the previous year in Cologne. This shows Stockhausen’s interest in Cage’s compositional system. Stockhausen also requested Cage’s lecture, “Indeterminacy”, which was delivered in October 1958 in Brussels, for publication in Die Reihe (Shultis, 2002, p. 34). Kurtz writes that for the New York avant-garde Stockhausen was “their man in Europe, the most friendly to Cage’s music and that of the other Americans” (1992, p. 87). Tape music

Though at first it may seem not directly related to the four case study works, experimentation with tape music was another important intersection between the creative worlds of the four composers. In Europe, this experimentation was begun by sound technician, Pierre Schaeffer, in 1948 at the French National Radio. The informal group of composers led by Schaeffer later constituted the Groupe de Musique Concrète in 1951 (Morgan, 1991, p. 464). Musique Concrète is considered “the first ‘school’ of electronic music” (Russcol, 1972, p. 79). Schaeffer’s studies were centred on the techniques of transforming sounds, initially on acetate disc, and from 1952 on tape, by a limited number of mechanical transformations and editing procedures: playing the sound recording backward, varying the speed of playback (which also transposed the sound in pitch), looping sound samples, adding reverberation, isolating elements from the sound and superimposing different sounds. The manipulations were far from random or guided by intuition. Under Schaeffer, composers were seeking a rigorous methodology to control and manipulate pre- recorded sound samples. However, like Cage's prepared piano, at the earliest stage the emphasis was necessarily on discovering what possibilities existed in the first place. What they did in the studio was to open up what appeared to be unlimited possibilities of manipulating sounds: this ultimately provided a new understanding about the

relationships between various dimensions of sound. These new ways of conceiving music, of course, had a significant bearing on the instrumental works that the four chosen composers were composing at the time. Maconie explains the very essence of these experiments:

Once subject to relativistic alterations of speed, sounds tend to lose their natural identity, becoming indeed “musical objects” identical in their inner structure but more or less differentiated in timbre, tone, and gestural significance. The art of musique concrète was by definition an art of modulating sounds in time. (2005, p. 99)

It is important to emphasise that the experimentation with various methodologies continued until a sound quality that the composers desired was achieved. Thus, composers were constantly adjusting and changing procedures until they were satisfied with the aural outcome.18 Messiaen, Boulez and Stockhausen were attracted to Musique Concrète and composed in the studio, each of them producing a notable work in 1952: Messiaen’s Timbres-Durées, Boulez’s Étude I sur un son and Étude II sur un son and Stockhausen’s Étude (Russcol, 1972, pp. 81–82). Although Messiaen and Boulez soon moved away from this type of composition, in the following years Stockhausen experimented beyond Musique Concrète in the ostensibly more scientific regime of electronic music.19 He left Paris in 1953, and he joined the Cologne Studio, which was established by Werner Meyer-Eppler and Robert Beyer—experts in mathematics, acoustics, physics and electronics—together with the composer Hebert Eimert (Russcol, 1972, p. 87). While Messiaen, Boulez and Stockhausen were exploring new possibilities with tape music, Cage in America also formed a group of musicians and engineers, which in fact used similar procedures of manipulating tape music to those applied in Musique Concrète. From 1951 to 1953 Cage named these experiments “Project of music for magnetic tape”. For this project Cage composed Imaginary Landscape No. 5 (1951–1952) and Williams Mix (1952). The organisation of these two works was based on chance operations (Russcol, 1972, pp. 91–92). Incorporating the chance procedures was one very different aspect from the experimentation in Musique Concrète.

18 An informative and detailed discussion about Musique Concrète can be found in the chapter “Music on Tape” of Other planets: The music of Karlheinz Stockhausen (2005) by Maconie. 19 Boulez moved away from working directly with tape, but pursued the implications of tape composition in instrumental terms. 32

One should not underestimate the connection between the compositional theories developed by Messiaen, Boulez, Stockhausen and Cage for their instrumental works and their experiences with composing tape or electronic music. As Frederick Judd explains: “The composer of electronic music must first create his sounds before he can decide which he will use and how they shall be arranged, but he has a far greater, in fact almost infinite range at his command” (1961, p. 66). Arguably the composer created his systems in order to discover the sounds that would be the product of the systems, which is the reverse procedure to traditional instrumental composition. Thus, the concept and process of pre-organising the essential musical parameters for instrumental music have a close parallel with the four composers’ experimentations in tape or electronic music. It is important to point out that the case study works in this thesis were being conceived in the period when these composers were experimenting with tape music. More significantly, when composing instrumental music, the composers were no longer thinking primarily about organising pitches, but about those dimensions that are to the fore in tape music, such as duration, attack, dynamics, register and density. The influence of Musique Concrète on the instrumental works of these composers is not easy to detect beyond the underlying concept. There are no obvious parallels between them, as they involve two very different approaches to sound creation: instead of controlling operations on pre-existing instruments, the Musique Concrète’s composers operated on systems of classification and notation, or pre- recorded taped sounds. Experimenting with the pre-recorded sound samples certainly allowed composers to expose themselves to new sonic worlds previously unavailable in the sphere of instrumental music. It also encouraged compositional theories and techniques to be driven toward a scientific conception of music more than at any other period of music history. Darmstadt summer courses

The personal interaction of the four composers and the development of their compositional theories were clearly heightened by their participation in the Darmstadt summer courses. The achievements of this crucial period of post-war modern music owe much to these summer courses, which were initiated in 1946 by a cultural affairs adviser to Darmstadt, Wolfgang Steinecke, and a Harvard-trained musicologist, Everett Helm (Maconie, 2005, p. 38). These summer courses provided opportunities

to study works from the inter-war years written by prominent composers, for instance, Bartók, Hindemith, Stravinsky, Varèse, Milhaud and, importantly, the Second Viennese School. They also fostered the composition of new music by presenting performances and lectures on new compositional techniques, including those of electronic music. Since the Festival of 1949, a series of concerts were reserved for works of the younger generation of composers. This pursuit of new music attracted some prominent composers, teachers, music critics and, most of all, the younger generation of composers, who became leaders in the second half of the twentieth century. Maconie discusses the multi-faceted significance of Darmstadt:

First, the new music symbolized a new freedom, a sense that anything was possible; second, the new spirit was international, and aimed to do away with narrow nationalist dogmas that had contributed to the disasters of two wars; third, a new language of music would also arise: objective, scientific, reflecting the universality of human experience and not the inherited prejudices of history. (2005, p. 39)

The four composers’ involvement with Darmstadt can be outlined as follows. Messiaen gave lectures at Darmstadt summer courses alongside René Leibowitz and others and wrote Mode de valeurs at Darmstadt in 1949 (Griffiths, 1995, p. 29). As mentioned earlier, it was at Darmstadt that Stockhausen first heard a recording of Messiaen’s Mode de valeurs, and met Belgian composer Goeyvaerts, who influenced him to see the significance of Webern’s Op. 27 Piano Variations and inspired him to study with Messiaen in Paris. Boulez also participated in these summer courses, both through his compositions being performed and through giving lectures. Both Stockhausen and Boulez became influential figures in Darmstadt, particularly as advocates and exponents of integral serialism. Stockhausen’s Klavierstücke were first performed in Darmstadt in 1954. Cage’s involvement with Darmstadt Summer Courses came later. Cage first gave lectures there in 1958, which were mainly concerning indeterminacy in his compositions. Some of these were reprinted in Die Reihe and generated major controversy in Europe (Shultis, 2002, pp. 33–36). Evidently, the Darmstadt summer courses played an important role in cultivating the progressive musical culture of post-war Europe, which then spread internationally. Therefore, the musical achievements of the four composers and their conceptual developments at this time were strongly influenced by these summer courses.

Die Reihe

If the Darmstadt summer courses provided opportunities for advocating modern music, the journal Die Reihe had a similar purpose. The series, published in German from 1955 to 1962 and in English from 1958 to 1968, came to define for the rest of the world the main characteristics of the musical avant-garde of post-World War II. The editors of the Die Reihe were Herbert Eimert and Stockhausen, the publisher Universal Edition (Theodore Presser in the US).20 The aim of this series of articles is explicit on the title page: “Devoted to developments in contemporary music”.21 Each of the volumes, from one to eight, was given a distinctive focus: I Electronic Music, II Anton Webern, III Musical Craftsmanship, IV Young Composers, V Reports-Analyses, VI Music and Language, VII Form-Space and VIII Retrospective. Griffiths says of Die Reihe, “the pathways of communication were being opened, and the musical achievements of the avant garde were beginning to impinge on a wider world” (1981, p. 89). A number of articles were written by Stockhausen, Boulez and Cage and, importantly, it was here that the first music analyses of Boulez’s Structures Ia and Stockhausen’s Klavierstück III were published. The analysis of Structures Ia, especially, influenced many other writers who wrote about the work in later decades. To this day, Die Reihe serves as an essential source for gaining an understanding of the progressive musical climate of the time, especially in Europe. Although Die Reihe was a reflection of the ideas on music developing at the time, it received severe criticism for its pseudo-scientific stance. It is a common observation that articles in Die Reihe are often long and complex analyses accompanied by many tables, graphs and mathematic formulas. Additionally, they borrow technical terminologies borrowed from fields of science, making them impenetrable to musicians without scientific training. For instance, John Backus, an acoustician, comments: “We are continually baffled by a technical language with which we are unfamiliar. In our frustration we may begin to wonder if perhaps the authors are as confused as their language appears to be, and if the unintelligibility is our fault or theirs” (1962, p. 160). Backus goes further, examining the four volumes (I, IV, III and V) of Die Reihe and pointing out that many of scientific terminologies are

20 The publication was preceded by Hermann Scherchen's Gravesano Review, published by Ars Viva Verlag, and was briefly in competition with the French Domaine Musical series Cahiers de la Compagnie Renaud-Barrault; it was also one factor influencing the formation of Perspectives of New Music in the US. 21 The German edition has “Information on serial music” (Grant, 2001, p. 1). 35

misunderstood and misused. Grant, more recently, puts forward a different perspective. “Die Reihe has become almost a watchword for the discontent people felt with serial music and the way in which its creators discussed it” (2001, p. 2). As a way of justifying Die Reihe, Grant writes:

The very use of “jargon” from the field of physics and communication science is one indication of serialism’s common ground with a number of other aesthetic movements which emerged in central Europe around this time. The journal reflects this wider context, particularly in its later volumes where there are contributions from the realms of architecture, abstract film and literature. (2001, p. 3)

There is no doubt that the publication of Die Reihe has a close association with the development of serialism after World War II, and that it contributed to the perception of serial music as being primarily an intellectual pursuit. Conclusion

The musical languages that the four composers developed during the years immediately following World War II were largely influenced by their common interests. There was a mutual respect among them and they were eager to exchange ideas through personal interactions and correspondence. The experimentation with tape music, especially for Boulez, Stockhausen and Cage, influenced their instrumental music such that musical parameters other than pitch gained structural importance. Also, the Darmstadt summer courses and Die Reihe provided an essential platform for the promotion of their works and ideas.

CHAPTER THREE

Olivier Messiaen’s Mode de valeurs et d’intensités

I am obliged to speak again of the Mode de valeurs et d’intensités, the fame of which is totally unjustified. In it I used a sort of super-series in which pitches of the same name passed through different regions, changing octave, attack, intensity, and duration. I think it was an interesting discovery—but no one noticed it. Everyone talked only of the super-serial aspect! (Messiaen, 1994, p. 80)

Introduction

Mode de valeurs et d’intensités was composed in 1949 at Darmstadt. Messiaen’s pupil, Boulez, specifically referred to Messiaen’s output during the years 1949–1950 as “experimental” and noted how they reveal the composer’s new interests at the time (1990, pp. 412–413). The composer explored these new ideas and techniques primarily in works for piano and organ, instruments with which he had extensive experience in playing and composing for. One of the outstanding works of this period was a set of piano pieces titled Quatre Etudes de Rhythm, which consists of four separate pieces: Mode de valeurs et d’intensités and Neumes rythmiques composed in 1949 and Iles de feu 1 and 2 composed in 1950. “It is significant”, Boulez notes, “that the two works dating from 1949, Mode de valeurs and Neumes rythmiques, were written at Darmstadt and Tanglewood—symbolical names associated with the teaching of composition—at the very time when his work as a teacher was becoming known to the larger public outside its normal setting, the Paris Conservatoire” (1990, pp. 412–413). Boulez further explains what seems to be a move away from Messiaen’s established style in his teacher’s compositions in this period. The most obvious changes, in Boulez’s opinion, are “his use of disjunct[ive] intervals and his relinquishing of his highly melodic, highly conjunct modal writing” (1990 p. 413). Messiaen’s strong attachment to harmony and chordal writing was loosened and he came to write melodic lines without harmonic support: this resembled the approach to serial composition where the horizontal lines have no connection to traditional harmonic laws. It is evident that Messiaen was influenced by the Second Viennese School at this time, with Boulez noting that “the idea of the series was engaging his

[Messiaen’s] maximum attention during these years” (1990, p. 414). This evidently led him to relinquish established aspects of his musical language. The piece received fame due to the pre-compositionally designed mode where four primary musical parameters (pitch, duration, dynamics and attack) were pre- determined. As mentioned in Chapter 2, such a revolutionary concept was immediately taken up by Boulez and Stockhausen, especially in their development of integral serialism. This chapter discusses from various angles how Messiaen’s experiments with various compositional techniques are manifested in Mode de valeurs. The first section of the chapter reviews selected analytical writings and re-evaluates some of their analytical approaches. This is followed by a reinterpretation of the piece based on my own analysis. Review of Previous Analyses

Issues

Almost without fail, the literature discusses Messiaen’s Mode de valeurs in relation to the emergence of integral serialism. Since Messiaen’s division of the mode resembles a Schoenbergian serial row (Figure 3.1), the predominant analytical premise for understanding the organisational scheme has been the 12-tone technique. Some even claimed that the piece is serially organised or at least that a form of serial technique is partially incorporated in the compositional procedure. By way of example, therefore, the comments from three authors are briefly reviewed. Firstly, Roger Sutherland states the work is based on serial permutation:

His Mode de valeurs et d’intensités (1949) was the prototype of the new idiom, being part of the group Quatre etudes de rythme. Although the work is not based upon a single twelve note series but on a mode which comprises three divisions of the chromatic scale, it does predetermine every parameter—pitch, duration, registers, dynamics—by a method of serial permutation. (1994, p. 23)

Unfortunately, the method of serial permutation mentioned by Sutherland is neither proven nor further explained by the author, thus leaving a reader much in doubt. Secondly, Paul Griffiths makes a rather surprising but debatable comment that suggests automatic writing as the basis of operation for the piece:

Indeed, it hardly sounds like Messiaen at all: it sounds, rather, only like itself, which is some measure of the extent to which its composition was automatic, entailing an abstention from creative choice much more severe than in other 38

instances of automatic writing by Messiaen, for composition here becomes simply a matter of rotating the same twelve pitch-duration couples in each of the three voices. The experience was, like Cage’s 4’33’’ (1952), an extreme extrapolation of tendencies in its composer’s music for some years, and it may be that Messiaen’s automaton as much as Cage’s silence was motivated by a wish to annihilate the expressive, ordering ambitions of the conscious mind. (1985, p. 153)

One could ask what Griffiths really means when he uses the term “automatic writing” in this context. The term obviously relates to a compositional technique, as his statement, “simply a matter of rotating the same twelve pitch-duration couples in each of the three voices”, indicates Griffiths suggests here that this type of compositional technique does not require much creative decision-making. Griffiths, after positing such a controversial statement about the compositional technique of automatic writing, concludes that: “the composition of the Mode de valeurs may have been a meditative exercise, producing music that certainly demands an intensive mental activity from the performer but that may leave the listener with the sense that infinity has been contemplated without the intervention of any human agency” (1985, p. 153). Here, Griffiths’ curious claim about automatic writing was perhaps influenced by his aural impression of work, which may have suggested to him the randomness of Cage’s piece. A decade later, however, Griffiths discusses the work from a different angle, thus: “Messiaen’s preface to the Mode de valeurs describes how the piece is composed as a three-part counterpoint, each part using a different set of twelve chromatic pitches and twelve ‘chromatic durations’” (1995, p. 29). Griffiths does not explicitly deny the automatic writing, but replaces it with the concept counterpoint. One thing that he clearly states at this stage is that the piece is not serially constructed (1995, p. 30). Finally, Arnold Whittall’s perspective on this prominent piece of twentieth- century music is surprisingly misleading. He writes:

Messiaen’s short piano composition Mode de valeurs et d’intensités (Mode of Duration and Dynamics), written in 1949, is often loosely described as “totally serial”, but the title itself suggests that the piece presents an extension of Messiaen’s own modal techniques. It does not even follow the Schoenbergian principle of fixing the twelve pitches in sequence; as in all Messiaen’s earlier modal works, the notes of the mode need not appear in any specific order: some may be omitted altogether, other used as often as the composer desires.

More significant in terms of serialism in this piece is the association between certain registers, durations and dynamics. (1999, p. 252)

Whittall recognises that there is no evidence of sequential pitch ordering but he does suggest that serial technique is nonetheless incorporated in the compositional procedure: in this case not with pitch but with register, duration and dynamics. The above quotation suggests confusion between serialism and parameterisation.

Overview

Although this piece is widely known and has been considered in many books and articles, most of these discussions stop at identifying the pre-compositionally designed mode where connections between the four musical parameters are pre- determined. In this case study, five analytical writings on Mode de valeurs have been chosen for a close examination. Though the piece was also well-known and studied by a younger generation of composers in the 1950s, the actual publication of analytical studies came more than 20 years later. Among the five analyses chosen here, the earliest, by Klaus Schweizer (1973), is one of the most detailed. Despite the quality of Schweizer’s analysis, there is virtually no references to his study in the later literature. This is probably because the article was written in German. Another German article, by Roderich Fuhrmann (1974) is comparatively short, but he is the only author who examined the note sequence of the entire piece.22 In the 1970s, literature in English written by Richard Toop (1974) and Robert Sherlaw Johnson (1975) provide valuable insights into the work. As Figure 3.2 shows, Mode de valeurs was a popular analytical subject in the 1970s. Unlike Schweizer’s article, which devotes itself entirely to the piece, Toop’s discussion appears as part of a survey of post-World War II musical development, and Sherlaw Johnson’s appears as part of his discussion of the composer’s experimental period in 1949–1951. In 1980, Kate Covington’s study on Mode de valeurs provided a remarkable turn in terms of its analytical approach, which was based on the author’s repeated listening experiences. Prior to Covington, such an aural perception had not been the primary focus in analysis of Mode de valeurs. In the course of reviewing these five analyses of Mode de valeurs, similarities and differences in the various analytical approaches will be addressed, highlighting many valuable observations and discoveries. These analyses together provide a

22 Both these analyses were translated by Christina Young and Sun-Ju Song. 40

detailed examination of the work from various angles: the pre-compositionally designed mode, the 12-tone technique, formal structure, contrapuntal texture and the listener’s perspective (see Figure 3.3). Since Covington’s analytical approach significantly differs from the others, the majority of her analysis of Mode de Valeurs will be discussed separately while the other four authors’ analytical approaches will be compared.

The pre-compositionally designed mode

The five authors all start their discussions by describing the pre-compositionally designed mode with three divisions from which all the musical materials employed in the piece are derived (Figure 3.1). Although the emphasis of their discussion varies considerably, their observations can be summarised as follows. Firstly, twelve chromatic pitches are arranged in descending order from the highest pitch to the lowest, so in all three divisions, the sequential arrangement is based on the registral placement. Secondly, the durations of the notes in each division are arranged arithmetically proceeding from the first, basic value given in each division. The first division is based on the demisemiquaver, the second division, on the semiquaver and the third division, on the quaver. Thus, in each division the durations are organised from the shortest to the longest. Thirdly, the dynamics and attacks are arranged so that louder dynamics are assigned stronger and forceful attacks while soft dynamics are linked with a softer attack. One of the differences in arranging the dynamics and attack in comparison with durations and pitches is that Messiaen does not employ all twelve types of attacks and seven degrees of dynamics in each division. Moreover, there seems to be no apparent order or sequence in the arrangement of attack and dynamics within each division. Toop, Sherlaw Johnson and Schweizer do not just describe three divisions separately but investigate the inter-relationships between them. For instance, Sherlaw Johnson notices that although the distribution of dynamics within each division is not rigid, there is a tendency for notes appearing in the same register to be given the same or similar dynamics (1975, p. 105). Sherlaw Johnson also comments on the effects which the pre-determined parameters have on each other: “. . . unlike the pitches, durations and intensities, which have an absolute value in their own right, the attacks effectively have the function of modifying the intensity or duration of a note” (1975, p.

106). For example, the absolute durational value between one note to another is fixed, but as the composer assigns staccato or other attacks that can shorten the durations, the actual duration can be varied. Sherlaw Johnson’s analysis shows how Messiaen may have considered the effects these pre-determined parameters have on each other. Toop makes insightful comments about the intervallic structure of each division of mode and about the distinctive features of each voice. He notices that “Messiaen’s ‘triplum’ division ends with a falling tritone” (1974, p. 145), noting that “this fact is significant simply because Messiaen makes great play with these tritone figures” (1974, p. 145). Concerning the pitch repetition, he observes that the “head motive” of the divisions appears more often than other segments of the three divisions and he further suggests that these head motives have a pivotal function (1974, p. 151). It seems that Toop is more interested in how the composer organises pitches in the mode rather than in the duration, dynamics and attack, as he compares the interval content within the three divisions. Toop justifies his approach despite the fact that the organisation of four musical parameters as a mode makes this work unique, saying “But of the two primary parameters (pitch and duration) it is pitch organization that constitutes the major innovation in Messiaen’s work: rhythmic cell organization is a constant characteristic of the works preceding the Quatre Etudes (cf. Cantéodjayâ)” (1974, p. 146). Moreover, the author considers attack and dynamics as secondary parameters since their function is to “emphasis the ‘point’ character of each note, and to minimize association between the same notes in different octaves” (1974, p. 147). He explains that, in the three-division mode, the arrangements of dynamics are conditioned by durations and these two same parameters influence the attack (1974, p. 147). Therefore, Toop’s interpretation implies a hierarchical relationship between four pre- determined parameters, descending in significance from pitch, duration, dynamics to attack. Contrary to Toop’s approach, Schweizer does not only thoroughly examine all four parameters but also draws attention to the distinctive effects created by the arrangement of dynamics and attack. His approach can be understood by the quotation of Antoine Goléa which appears in the early part of his article: “the types of attack, dynamics and the durations or time values have been brought onto a level with pitches, whereby there is even an emphasis on giving preference to duration and dynamics

over sound colours, types of attack and especially pitches” (1973, p. 131).23 Thus, in his discussion Schweizer give these aspects a very different priority than Toop. Schweizer investigates relationships that exist among the pre-determined parameters. For instance, Figure 3.4 shows how he regroups notes according to their similarities in dynamics, attacks and registers and, as a result, demonstrates how notes with common features are categorised within the same registral region (Schweizer, 1975, p. 140). Although this short example seems a minor point, Covington developed the same concept in her analysis about twenty years later when its significance, especially in connection with the aural perception of the piece, became fully realized. The absence of a reference to Schweizer in Covington’s analysis suggests that she was unaware of his investigation. However, Schweizer’s 1975 analysis remains significant because he was the first to propose an understanding of this piece from various dimensions. In addition, Toop, Sherlaw Johnson and Schweizer all present graphs in which all 36 notes of the mode are rearranged according to their registral placement from highest to the lowest. Figure 3.5 (a), (b) and (c) show three graphs from the three analysts. Since these three divisions of the mode differ from the way Messiaen originally arranged the musical parameters in the mode, these graphs encourage us to examine the mode from different angles. By examining the mode in this way, these three analysts share some valuable insights. Toop, for example, rearranges the three divisions into one graph (see Figure 3.5 (a)). He observes that there are “general zones where certain kinds of attack and dynamic predominate, but no systematic organization” (1974, p. 148). Sherlaw Johnson (1975, p. 106) says that “the three groups interlock so that the greatest connection of notes occurs in the middle register. All twelve notes in the octave above and including middle C are used” (see Figure 3.5 (b)). Similarly, Schweizer indicates the area where the three divisions overlap and especially identifies the “chromatic centre” (Chromatischer Kern), where pitches are arranged chromatically (see Figure 3.5 (c)). His graph is very effectively presented and has two advantages: firstly, it illustrates interlocking elements of the three divisions without deforming the original shape of each individual division; secondly, a corresponding table shows the four pre-determined musical parameters. Such

23 die Anschlagsarten, die Intensitäten und die Dauern oder Zeitwerte sind auf eine Ebene mit den Tonhöhen gebracht worden, wobei sogar eher noch eine Bevorzugung der Dauern und Intensitäten gegenüber den Klangfarben, den Anschlagsarten und vor allem gegenüber den Tonhöhen betont wird. Diese Bevorzugung kommt ja schon im Titel zum Ausdruck (Translated by Christina Young and Sun- Ju Song). 43

insights shared by Toop, Sherlaw Johnson and Schweizer on the subject of the pre- determined parameters laid the foundation for further studies of Mode de valeurs. Though different in detail, the three graphs in Figure 3.5 do in fact demonstrate the same approach. There are several advantages in observing the mode in this way. Firstly, it allows us to see the overall plan for all the pre-determined musical parameters. Secondly, it demonstrates the places where the three divisions are interlocked. Thirdly, one can observe vital relationships between the register and the four pre-determined musical parameters, thereby clearly illustrating the important role of register. These relationships can be classified as follows: a chromatic stepwise pitch arrangement occurs in the middle register (C5 to B3); shorter durational values are assigned to the pitches occurring in the higher register and louder dynamics are given to the pitches placed in the lower register. This means that these notes also receive a stronger attack, whereas pitches in the middle register are dominated by softer dynamics. This dispersion is idiomatic for the piano as the lower register has more sustaining power than the upper register. There is a greater range of dynamics assigned to the pitches placed in the upper register of the piano, where the actual contrasting effects of dynamics are not as apparent as those in the middle and lower registers. Finally, after examining all three divisions re-arranged according to their registral placement, it is evident that Messiaen carefully considered the acoustic nature of the piano.

Compositional procedure: integral serialism?

In light of earlier comments, it is necessary to ascertain if the work’s construction demonstrates any degree of serial technique. Since the mode used in this piece resembles three 12-tone rows, it would seem logical to associate Mode de valeurs with 12-tone technique. The five chosen analyses of the piece establish that its organisation is not based on serial technique; this is particularly evident in the analyses of Fuhrmann and Sherlaw Johnson, who examine the sequence of the notes in meticulous detail. Fuhrmann provides a graph where the sequence of every note of the entire piece is traced (1974, p. 181). As seen in Figure 3.6, Fuhrmann also indicates symmetry in the arrangement of notes. Sherlaw Johnson comes to the same conclusion as Fuhrmann, writing:

. . . at eight points in the composition all twelve notes of one of the three groups appear in succession, but in only one part at a time. The notes of each group are permutated in a type of symmetrical order at each appearance, but in three cases the symmetry is slightly disturbed, apparently in order to avoid sounding a note at the same times as another part. (1975, p. 107)

This statement is supported by a graphic illustration where these eight occasions are shown (see Figure 3.7). It is however interesting to note the accompanying description for the graph, which in Sherlaw Johnson’s words “shows the order of the notes in these eight groups, each note being indicated by its duration in the unit appropriate to the group (I = demisemiquavers, II = semiquavers, III = quavers)” (1975, p. 107). Although he uses a duration value to indicate a note, there is no mention in his discussion of durational arrangement, only note orders. This point will be further explored in the later part of this Chapter. These two authors’ investigation of note order clearly demonstrates that there is no application of Schoenbergian 12-tone technique. Neither of them considers other types of serial technique, such as the distributive types of serialism though, given the detail with which they examined the pitch sequences, one would expect that distributive forms of serial technique would have been observed if they existed. One is still left with an important question, if the underlying compositional principle is not serialism, what is it?

Formal Structure

Toop, Schweizer, Sherlaw Johnson and Covington consider the formal structure of the work. While Toop and Schweizer interpret the piece as a tripartite form, Sherlaw Johnson and Covington suggest that sectional division is not discernable. Both Toop and Schweizer see the three appearances of C#1, the last note of the third division in the mode, as an indicator which functions as a closure to each of the three sections. Toop describes the effect and characteristic of C1 as follows:

one is struck by the way the study is divided into three parts, ended by a low C, the “omega” of the three main parameters—longest duration, lowest pitch, loudest dynamic whose almost pitchless, bell-like timbre is the piece’s nearest concession to pictorialism. (1974, p. 144)

Moreover, a remarkably similar observation is made by Schweizer, who says:

The prominent characteristics of the note contra C# (the longest, lowest, loudest and most distinctive of all 36 pitches) enable it to carry out an inter- punctuated and therefore formal function. Its economically handled placing of a mere three times (bars 28–31, 78–80, 112–115) is sufficient to impose a total of a three-part segmentation with the simplified proportions of 3:5:4 (27 ½ :49 ½: 37 ½ bars ) on the Etude.24 (1973, p. 139)

Both authors seem to concentrate on the distinctive features of C1. However, it is doubtful that the three sections suggested by Toop and Schweizer are contrasted enough to be aurally distinguished and therefore one might claim that the tripartite form is more conceptual than perceptual. To this account, Toop adds “paradoxically, despite this triple division, Mode de valeurs’ most important characteristic is to describe the outlines of one enormous constellation” (1975, p. 150). Contrary to Toop and Schweizer, Sherlaw Johnson’s interpretation of the piece largely depends on the perceptual experience; he states that “there is no trace of sectional form, which is so characteristic of Messiaen’s other work and there is no thematic working” (1975, p. 107). Covington, whose main approach is based on aural perception, agrees with Sherlaw Johnson’s interpretation that it is hard to determine sectional form in this piece though she only considers the matter in a single sentence (1980, p. 9).

Texture

Texture is another aspect of the compositional techniques in this piece investigated by Schweizer, Toop and Covington. One of the integral aspects of the compositional procedure is that each of the three divisions of the mode is assigned a separate staff. As mentioned earlier, the entire range of the keyboard is divided into three regions in the pre-compositionally designed mode: division I is assigned to the highest range of the keyboard, division II to the middle register and division III to the middle to lower register. Notes and their assigned characteristics in the first division of the mode are only notated in the top staff, those in the second division for the middle staff and those in the third division for the lowest one. Since there is an inseparable connection

24 Die hervorstechenden Eigenschaften des Tones Kontra-Cis (längster, tiefster, lautester und markantester aller 36 Tonpunkte) befähigen ihn, eine interpunktierende und damit formale Funktion auszuüben. Seine ökonomisch gehandhabte, nur dreimalige Setzung (T. 28–31, 78–80, 112–115) genügt, um der Etude insgesamt eine dreiteilige Gliederung mit den vereinfachten Proportionen 3 : 5 : 4 (27 ½ : 49 ½ : 37 ½ Takte) aufzuerlegen (Translated by Christina Young and Sun-Ju Song). 46

between the pre-designed mode and the distribution and organisation of notes in each voice, understanding the texture of the work is imperative. Similar to their investigation of formal structure, the analytical approaches of Schweizer and Toop share common perspectives in examining the texture while Covington’s approach significantly differs. Again, Schweizer and Toop focus on conceptual matters of composition and their analytical premise is based on the fact that the piece has a three-voice structure. On the contrary, Covington strongly questions whether one could hear the piece as counterpoint between three voices. She argues:

. . . a contrapuntal analysis, considering the three staves or Divisions as three contrapuntal lines, has been done, but there is little justification for this since overlapping ranges and characteristics (such as dynamics and articulation) of the three Divisions prevent the aural perception of the three-part counterpoint. (1980, p. 6).

The theoretical basis for each author’s approach will be examined below. Schweizer interprets the piece as a three-voice structure without exploring a particular type of counterpoint technique. He correlates the texture to an organ piece in which an organist is required to master a work structured in three distinct voices. Knowing that Messiaen himself was an organist and composer of many organ pieces, it is reasonable to suggest that he hoped to translate the organ’s capacity to differentiate the timbre of each layer to a piano piece. According to Schweizer,

In the simultaneous progression of three individual voices [in Mode de valeurs], the composer obviously sees also for the pianist a movement structure corresponding to his conception, dense enough in sound variety of the mode elements, clear enough to just about allow the exact pianistic execution of its value levels.25 (1973, p. 130)

In this work, the longer duration of notes occurring in the lowest voice means they move at quite a different rate than the shorter duration of notes in the upper voices. It is a convincing argument that one could hear division III as the pedal of an organ. Toop, on the other hand, is particularly interested in the contrapuntal techniques used in Mode de valeurs. Prior to presenting his own view on this subject, he critically

25 Im Simultanablauf dreier individuellen Stimmen sieht der Komponist offenbar auch für den Pianisten eine seiner Konzeption entsprechende Satzstruktur, dicht genug in der klanglichen Vielfalt der Moduselemente, überschaubar genug, um die genaue pianistische Ausführung ihrer Stufenwerte gerade noch zu erlauben (Translated by Christina Young and Sun-Ju Song). 47

reviews Goléa’s suggestion that the piece is a three-part canon. Concerning Goléa’s claim, Toop argues that:

Since the pitch order of all 3 divisions is different, successive presentation of all values from 1–12 would hardly qualify as a pitch canon; on the other hand, it would constitute precisely the kind of rhythmic canon to be found in many earlier works. (1974, p. 148)

Toop further explores the effect of how different durational values characterise each of the three voices. He writes:

What is present is a typical feature of the late 13th-century motet, namely the simultaneous presentation of three “speeds” in the three voices, with the duplum written in longer values than the triplum, and the tenor in still longer values (there is a striking analogy with certain pieces by Petrus da Cruce). The other quasi-medieval feature is the separation of the triplum tessitura from those of duplum and tenor, which tend to overlap substantially. (1974, p. 150)

Toop pays significant attention to the temporal aspect of Mode de valeurs. His analytical approach is no doubt influenced by the knowledge that Messiaen was interested in medieval music and also developed sophisticated rhythmic techniques as an essential part of his compositional language. Covington’s analytical methodology and focus significantly differ from the two authors considered above. While the others examine the vertical relationships between the voices in terms of pitch and durational aspects, Covington completes the picture in a way by investigating primarily the other two pre-determined parameters. In developing an analytical methodology, Covington first establishes the premise that “there is little or no relation between the aurally perceived strata and the physical construction of the three Divisions” (1980, p. 6). Such a statement was derived from the author’s listening experience. Instead of accepting the piece as a three-voice structure, she aims to find an analytical solution that is congruent with her aural perception. Covington states that:

Each aurally perceived stratum or plane of sound is composed of notes which coalesce by register and dynamics and also share articulation and duration similarities. Hence, notes which have similar characteristics seem to sound in a line or linear plan of sound. (1980, p. 6)

Thus, in Covington’s analysis, notes are regrouped into six different planes of sound and, as seen in Figure 3.8, each plane is made up by notes with similar characteristics as described above. (She also indicates a note that does not belong to any of planes.) Once all the notes in the pre-compositionally designed mode are regrouped, Covington reinterprets the entire piece in relation to these planes (see Figure 3.9). In this way, the piece is no longer understood as three-layered counterpoint of pitches or duration: instead, the contrapuntal techniques are based on the various arrangements of attacks, dynamics and registral placement. Covington further considers three distinctive features which affect the aural perception of these six planes as a form of counterpoint: “closed placing,” “rhythmic concurrence” and “simultaneous attack” (1980, pp. 9–10) (see Figure 3.9). Firstly, closed placing means that two very similar planes sound simultaneously. For instance, closed placing occurs between planes V and VI as these two planes share similar characteristics. Secondly, rhythmic concurrence refers to note frequency and how it changes over a span of time. In other words, the variance of density has an impact on aural perception. Thirdly, simultaneous attack describes the phenomenon of soft or loud notes simultaneously appearing. For example, the soft dynamic of plane IV produces a linear voice. On the other hand, the continual appearance of notes with loud dynamics and stronger attacks tends to create intensity. These three features are essential to the perceptual experience of the work and the author points out that: “It should be emphasised again that these characteristics are not shaping individual lines, but planes of sound counterpointed against each other” (1980, p. 11). Covington’s analytical approach is remarkable in a sense that it illuminates how registers, dynamics and attacks have an impact on and guide one’s aural perception. Her analysis of Mode de valeurs is pioneering as her analytical methodologies are primarily derived from a perceptual viewpoint instead of the conceptual aspect of composition. She distinguishes these two important and interrelated facets, saying:

Thus, we see a piece in which the visually perceived construction differs notably from the aural interpretation of the structural materials. Both are valid: one shows the composer’s initial methodology; the other shows a listener’s approach toward an understanding of the structure of Mode de valeurs as it moves through an aural range-space and time space. (1980, p. 11)

Conclusion

This seminal work has been analysed from various angles, in relation to the pre- compositionally designed mode, formal structure, texture and serial technique. All five authors chosen have determined that the pre-compositionally designed mode is fundamental to organising the four primary parameters, and have based their analyses on them: Schweizer, Toop and Sherlaw Johnson even provide graphs that illustrate the relationship between the four predetermined parameters and their registral placement. In the course of examining each division of the mode, Toop gives foremost importance to the intervallic content of each division whereas Schweizer explores the interrelationships among dynamics, attacks and register at great length. In the analyses of Schweizer, Toop and Covington, the significance of understanding the texture is revealed and different ways of analysing it are proposed. The analytical approaches of Schweizer and Toop reflect these two authors’ awareness of Messiaen’s compositional output and his interest in rhythmic techniques by focusing on conceptual aspects of compositional techniques. On the other hand, Covington’s interest lies in the aural interpretation of structural materials of the work, which are distinguished from the procedural and conceptual organisation of materials. Her analysis above all illustrates the functions of dynamic, attack and register in a profound way. As mentioned in the earlier part of this chapter, assumptions and speculations have been made as to whether Messiaen incorporated serial techniques as part of his compositional procedure in Mode de valeurs. All five chosen authors state that the serial procedure does not apply to this piece. Fuhrmann and Sherlaw Johnson actually prove the fact by accounting for the sequence of every note, and they conclude that there are regions where the symmetrical arrangement can be observed but that the symmetry is often either incomplete or varied. Although the analyses reviewed here provide valuable insight into the work, analytical approaches seen in this chapter rarely refer to Messiaen’s own compositional language. In the following part of the chapter, Mode de valeurs will be re-examined as the evolution of Messiaen’s own compositional language.

A re-interpretation of Mode de valeurs et d’intensités

Analytical premise

Since the pre-compositionally designed mode and compositional procedure cannot be explained via serialism, this analysis seeks to find an alternative principle underlying the organisation of the work. The analytical premise for re-interpreting Mode de valeurs is to approach the work by directly referring to compositional techniques illustrated in Messiaen’s The technique of my musical language (Technique de mon langage musical)(1956). This book, first published in 1944, is an essential reference point for anyone wanting an understanding of Messiaen’s music, especially that from the 1940s. He provides many musical examples from his music in the accompanying volume and the reference to his future works in the following statement is particularly revealing. “Aside from a few very rare exceptions pointed out in passing, all the examples quoted here will be drawn from my works (past and future!)” (Messiaen, 1956, preface). Therefore, it is surprising how rarely this primary source is referenced in relation to Mode de valeurs. In the course of the following analysis, the application of Messiaen’s various rhythmic techniques will be identified and the influence of the modes of limited transposition will be discussed. The approach taken here reflects the content of the book, in which the composer discusses his techniques from three points of view: rhythm, melody and harmony (Messiaen, 1956, preface).

Messiaen’s rhythmic techniques applied to Mode de valeurs

Mode de valeurs is ametrical; a familiar feature of the composer’s music. Messiaen defined ametrical music as follows:

We shall replace the notions of “measure” and “beat” by the feeling of a short value (the sixteen-note, for example) and its free multiplications, which will lead us toward a music more or less “ametrical”, necessitating precise rhythmic rules (1956, p. 14).

The ameterical quality of this music is directly connected to the sequence of notes in each division in the mode. For instance, the rhythmic techniques of added value and augmentation are used in order to organise durational values in the mode. In each division, the durations of the notes are arranged arithmetically, the first value given in each division being the basic value. The augmentation of rhythmic value can be seen

as the composer augments the basic rhythmic value of division I to II, and II to III; the first division is based on the demisemiquaver, the second division, on the semiquaver and the third division, the quaver (see Figure 3.1). Other rhythmic techniques of Messiaen can be identified. The concept of superimposing a rhythm upon its different forms of augmentation and diminution is employed: this is what Messiaen would call “polyrhythm” (1956, pp. 22–23). As mentioned previously, the three staves strictly correspond to the three divisions of the mode. Since the arithmetical arrangement of each division is based on three different rhythmic values, three rhythmically different lines are superimposed. Moreover, the technique of “nonretrogradable rhythm” is often employed. This technique is unintentionally demonstrated by Fuhrmann and Sherlaw Johnson as both of them identified regions where notes are arranged in the form of symmetry (see Figures 3.6 and 3.7). An ironic aspect of these observations is that both authors set out to investigate the organisational scheme from the perspective of pitch sequential patterns rather than from the durational values. In particular, the purpose of Fuhrmann’s graph is to demonstrate that Schoenbergian serialism was not applied in the piece. Similarly Sherlaw Johnson’s intention to include a table (shown in Figure 3.7) is to illustrate the “order of notes” (1975, p. 107). However, since numeric representations in both authors illustrations refer to duration of notes, various arrangements of durations can be observed; the sequential patterns are made up by either adding or subtracting particular durational values. If those authors had been less concerned with the serial aspect of note ordering, various types of Messiaen’s rhythmic techniques might have been obvious. Sherlaw Johnson’s table in Figure 3.7 categorises the eight examples of note arrangements, but it is revealing to reinterpret these from the perspective of durational values as Figure 3.10 illustrates. The eight examples of note arrangements demonstrate a highly advanced approach to rhythmic construction through the application of mathematical sequences. In relation to the first example (see Figure 3.10 (a)), one can observe that the number of successive pairs of durations add up to thirteen which, being a prime number, was of special significance to Messiaen (1956, p. 4). The pair starts with the outer notes, that is, the shortest and longest durations of the division, and moves towards the middle point of the division. As a result of this regular pattern, the first example demonstrates six such pairings and it is worth noting that this pattern is balanced around a central point, akin to a nonretrogradable rhythm, 52

with three units of thirteen demisemiquavers on each side. The third example is similarly constructed from pairs of successive durations that add up to twelve semiquavers rather than to thirteen demisemiquavers (see Figure 3.10 (b)). The second example uses another pattern for manipulating groups of successive durations, in this case a pattern of three notes. Again the shortest and longest durations are contrasted and the combined durations either side of a central point are equal (see Figure 3.10 (c)). The fourth example bears an interesting relationship to the first: its pairs are arranged in a regular pattern that moves from the outer pairing to the inner pairings shown in Sherlaw Johnson’s first example (see Figure 3.10 (d)). The fifth type is a mathematical sequence produced by successively adding six quavers and then subtracting five quavers (see Figure 3.10 (e)). In this arrangement, the duration of notes increases gradually as the pattern progresses, while the proportion of the increase in durational value remains the same since the distances between the notes are fixed by the alternation of six quavers and five quavers. The sixth example employs another kind of pattern of successive durations, in this case a pattern of three notes (see Figure 3.10 (f)). The length of each group equals 21 demisemiquavers. The seventh example is a mirror formation of the sixth example though the values are augmented and it applies to the second voice rather than the first (see Figure 3.10 (g)). As the bar numbers indicate, the seventh example immediately follows the sixth one. Here, the composer combined two of his major rhythmic techniques: the principles of nonretrogradable rhythm and augmentation (see Figure 3.10 (h)). The eighth example is explicitly based on the technique of added value (see Figure 3.10 (i)). Despite what Sherlaw Johnson attempts to demonstrate, Messiaen clearly was experimenting here with constructions of rhythm rather than pitch. These patterns prove that duration is a higher priority than of pitch in the design of this music. It certainly confirms why this piece is called “mode de valeurs”! But the search for serialism distracted from appreciating this fundamental aspect. The sense of closure created at the end of the piece is reinforced by manipulating metrical arrangement. As stated previously, Mode de valeurs is ametrical music where no perceivable regularity of metre has existed up to bar 106. But, from this point, an appearance of regular pulsation emerges, giving the temporal sense of 3/8 and 6/8 (see Figure 3.11 (a and b)). Thus, Messiaen rather suddenly, yet subtly, allows a sense of pulsation to be established. This is demonstrated in Figure 3.11 (b), in which the music is re-barred to underline this. The sense of regular beats 53

is enhanced primarily by the frequent synchronisations between voices. For instance, bars 106–114 can be reinterpreted as three repeated patterns of two bars of 3/8 and one bar of 6/8. In the first pattern (bars 106–108), every strong beat is indicated by the louder dynamic, creating a stronger accent. When the second pattern appears from bars 109–111, every first beat is emphasised by the vertical synchronisation between two lower voices that automatically have longer durations than the top voice. A sense of closure is achieved by the regular pulsations and frequent uses of the vertical synchronisation of two voices, especially when the lower voices are synchronised. Indeed, the appearance of regular pulsations to prepare the ending interrupts the timelessness evident until bar 106.

Seven prominent pitch classes

Although all the twelve chromatic pitches are employed in each division of the mode, seven pitch classes are given more emphasis than others due to how they recur throughout the piece. I shall refer to these as the “hidden mode”. Firstly, the composer seems to have favoured these pitch-classes over others. The initial pitches of each division of the mode function like a motivic figure: Toop calls them “the head motive” of each division (1974, p. 151). One of the important characteristics of these motivic figures is that they are rarely far from the listener’s attention, since each of them displays distinctive characteristics. As shown in Figure 3.12, Motive I comprises the three highest pitches in the piece; both motives II and III are assigned loud dynamics; motive III does not appear as often as the other two but has the advantage of sharing the same pitch-classes with motive I (Eb and D), which can reinforce the listener’s attention. Motive III is also assigned a forceful attack. Secondly, despite G5 belonging to motive II, it frequently occurs by itself, disassociated from the two other pitches in motive II (see Figure 3.13). The G5 is assigned a strong attack, the dynamic of ff and is placed in a register that is comfortable to listen to. Griffiths also notices this unusual phenomenon and faithfully counts the number of notes used in the second staff. He notes that “of the 185 notes in the line, 34 are Gs” (1985, p. 151). Thirdly, the final significant pitch, C1, is not only the loudest and longest one but also concludes the piece, leaving a lasting impression on the listener. Unlike the motives mentioned above, Messiaen only uses C1 three times throughout the entire piece, in

bars 28, 78 and 112. Both Toop and Schweizer interpret the C1 as an indication for the sectional division of the piece, which, they logically conclude, is tripartite. Given the evident importance of these seven pitch-classes, the next logical step is to assemble and rearrange them starting from C1. There are several reasons for arranging these pitches in the chosen order starting with C1. C1 is the lowest and longest note used in the piece and, most importantly, it is the last note occurring in the piece. By closing the piece, C1 resembles the finalis or final of a church mode according to medieval music theory whereby the modes were built on the last note of the melody. Thus, considering these unique characteristics of C1, it is the best candidate for the starting note of this seven-note mode. Once the seven pitch-classes are arranged in the manner shown in Figure 3.14, immediately an intervallic construction of this mode recalls Messiaen’s modal theory, the foundation of his compositional system.26 As with all of Messiaen’s modes of limited transposition, this hidden mode can be asymmetrically divided by the interval of the augmented fourth, which divides the octave equally. It is not strictly a mode of limited transposition because the two halves do not share the same intervallic order. However, the comparison of the modes of limited transposition is revealing. In Figure 3.15 (a, b and c), comparative analyses of the hidden mode to the modes of limited transposition are made. Firstly, the last four notes of both the second modes of limited transposition and the hidden mode are identical. This means that the second halves of two modes comprise the same intervallic structure: alternating minor-major-minor seconds (see Figure 3.15 (a)). Secondly, the hidden mode can be compared with the fifth mode of limited transposition. The prime intervallic features of the fifth mode of limited transposition, namely the combination of the minor second and the major third, are also found in the hidden mode (see Figure 3.15 (b)). Thirdly, the following intervallic patterns characterise the fourth mode and hidden mode: the first two and the last intervals of both modes are minor seconds (see Figure 3.15 (c)). Based on the numerous similarities that exist between the hidden mode and the modes of limited transposition, the influence of Messiaen’s modal theory on the pre-compositionally designed mode in this piece becomes evident. The hidden mode suggested here is ingeniously integrated into three divisions. As has been frequently noted, each division contains all twelve chromatic notes, resembling 12-tone rows used in

26 Messiaen explains his modes of limited transpositions and their uses in chapters XVI to XIX of The technique of my musical language. 55

Schoenbergian serial techniques. However, the hierarchical relationships that prioritise the seven pitch classes defy the concept of serialism where all the twelve notes are theoretically equal.

Melodic cadence

Messiaen provides an example of a form of melodic cadence in the chapter “Melody and melodic contours” in The technique of my musical language. He writes “this F- sharp is endowed with an attraction toward the C, which becomes its normal resolution” (1956, p. 31). In Mode de valeurs, the interval of augmented fourth G–C# suggests a similar cadential role, with G resolving to C#. For example, in bars 28 and 78, this interval appears in a synchronised manner between the second and third staves as motive II and C1 coincide (see Figure 3.16 (a) and (b)). In this closing part of the piece, Messiaen’s compositional craftsmanship in highlighting this core interval G–C is exceptional. Prior to the final appearance of C1, Messiaen pre-empts it by employing C6 in the top voice as marked in bars 108 and 111. The location of C6 clearly anticipates the final C1 and also draws attention to the augmented interval with the G5 in bars 108–109 as well as in bar 112. The intervals of the augmented fourth formed in bars 108–109 are not as forceful as the ones in bars 28 and 78 but they foreshadow the final appearance of G5 to C1 in bar 112. Therefore, the significance of the interval of the augmented fourth formed by G–C is realised and achieved in two dimensions: at the micro level, where it frames the “hidden mode” employed by the composer in Mode de valeurs and at the macro level, where it articulates the structural divisions. Conclusion

As Mode de valeurs has been mostly understood as a necessary step in the evolution of serialism, appreciation of the piece from the perspective of the composer’s own compositional techniques has been somewhat neglected. Although Messiaen left a significant volume describing his craft, the analyses reviewed in this chapter do not seem to refer to his compositional techniques as a primary source. This case study has revealed that Mode de valeurs demonstrates the compositional craftsmanship that is deeply rooted in Messiaen’s musical language, a language that embraces both past and contemporary music. As the composer states:

We shall not reject the old rules of harmony and form; let us remember them constantly, whether to observe them, or to augment them, or to add to them some other still older (those of plainchant and Hindu rhythmic) or more recent (those suggested by Debussy and all contemporary music). (1956, p. 13)

Interpretations of Mode de valeurs over the past half-century have emphasised its relationship to “more recent” musical developments such as serialism. Undeniably, Messiaen’s piece had a profound impact on the three other composers—and their works—considered in this thesis. The reinterpretation of the piece in this chapter has provided a more holistic picture of how the piece fits within the composer’s evolution, allowing it to be viewed not as an isolated experiment, but as an integral part of the development of Messiaen’s musical language.

CHAPTER FOUR

Pierre Boulez’s Structures Ia

All my attention this past year [1951] has been given to widening the scope of the series and making it homogeneous. With the thought that music has entered into a new form of its activity—serial form—I have tried to generalize the notion of series. (Boulez, 1990, p. 137)

Introduction

Boulez composed Structures for two pianos in 1952. The work consists of three separate pieces called Structures Ia, Ib and Ic that were all written within a relatively short period of time: approximately a month and a half (Boulez, 1975, p. 56). The fame of Structures derives from its embodiment of integral serialism and its significance has been widely recognised as part of both Boulez’s compositional output and broader musical developments in the twentieth century. The application of integral serialism however differs in the three pieces. According to Boulez (1975, p. 56), one can perhaps observe “the process of re-introducing personal invention” although he purposely numbered the pieces in a non-chronological order in order to obscure the process and create “an anti-evolutionary impression of whole”. Composing Structures was an important experiment in which Boulez himself explored a crucial relationship between the role of pre-determined musical material and the role of the composer in the process of composition. This relationship is explained by Boulez in his comment that “I gradually moved from the point where the material suggested itself to me until the situation was reversed: at the end of the second piece it was in fact I who was suggesting to the material that we make something together” (1975, p. 56). Concerning the chronological order of these three pieces, Jameux clarifies that Structures Ia was the first to be written, in the spring of 1951; the others were written later (1991, p. 48). Yvette Grimaud and Yvonne Loriod first performed all three Structures in Cologne on 13 November, 1953 (1991, p. 52). Further, in an article from 1952 Boulez wrote: “Let me throw in the extravagant witticism that there has rarely been such a musically stimulating period in which to live” (1991, p. 113). He was stimulated musically by the innovative musical languages his immediate predecessors had explored and evidently also felt a sense of responsibility to extend those possibilities. He wrote that, “If we are arriving at a

period of stocktaking and organisation, there was previously, starting about 1910, a phase of destructive experiment which abolished, on the one had, tonality and, on the other, regular meter” (1991, p. 114). Here, he attributed the destruction of tonality to serial techniques developed by the three Viennese composers Schoenberg, Berg and Webern, and the destruction of regular metre to the rhythmic techniques developed by Stravinsky. While serialism brought a new structural system to pitch relationships, the rhythmic organisation in serial music had remained conventional and largely unchanged from that of tonal music. While Stravinsky had developed radical techniques for rhythm, his pitch organisation had remained based on tonality, “dealing with a kind of audio-colouring” (Boulez, 1991, p. 114). Thus, in both cases, Boulez perceived a disassociation between the organisation of pitches and that of durations. He interpreted this phenomenon of dissociation as being “a powerful force in the evolution of both structural methods” (1991, p. 114). Since the evolution of pitch and rhythmic structures had already taken place independently from each other, Boulez believed that the next step was to relate them in a coherent manner. The method he chose was “to link rhythmic to serial structures through a common organisation which will also embrace the other characteristics of sound: dynamics, mode of attack, timbre; and then to expand this morphology into an integrated rhetoric” (1991, p. 115). Therefore, Boulez was compelled to write Structures with the intention of expanding twelve-tone techniques into integral serialism. Among the three pieces of Structures, the first (Ia) is chosen for this case study to examine how integral serialism is applied in the construction of the work. One of the benefits of choosing Ia is that there is a sizeable number of analyses of the work in comparison to the other two.27 Since Structures Ia has often been considered the prototype of integral serialism, a review of this literature can provide insight not only into the development of integral serialism but also into the trends of musical analysis over the last half century. As with the other major case study works, this examination is divided into two stages; the first stage involves the review of earlier analytical studies and the second is based on my own analysis.

27 In many cases, a short discussion of Structures Ib and Ic appears immediately following the analysis of Ia, but their content is mostly brief, especially in comparison to Ia. Irina Ivanova in 2000 wrote an article, “Transformations and invariant structure in Structures Ib”, though it only discusses the serial techniques involved in organising pitch. Analyses of Structures Ic can be found in “The abstract system as compositional matrix: An examination of some applications by Nono, Boulez, and Stockhausen” written by Philip Bracanin (1971) and in “P. Boulez: Structures I pour 2 pianos” written by Ursula Eckart-Bäcker (1986). 59

Review of the previous analyses

Issues

Analytical studies of Structures Ia can be reviewed from two points of view: one examining what each individual analysis reveals about the piece, the other discussing the analytical approaches employed. The task of reviewing analyses of this work differs significantly from the other case studies. Firstly, the composer has revealed the fundamental aspects of his compositional technique.28 Perhaps the first person to be aware of Boulez’s pre-compositional serial organisation was John Cage. Boulez, in fact, discussed his serial technique in a letter to Cage in August 1951 (Nattiez, 1993, pp. 98–103). This letter includes the series designed for each of the four musical parameters and two matrices on which Boulez’s entire serial operation is based. Boulez’s comments on serial techniques in this letter are almost like that of a scientist reporting on his own scientific experimentation, faithfully describing all the stages of a particular experiment. In 1952, Cage published a part of this letter in Transformations: arts, communication, environments under the title “Four musicians at work” (Nattiez, 1993, p. 99). Subsequently, the same letter was published in English in Pierre Boulez: Orientations in 1981. In 1952 Boulez also brought Structures to the attention of Stockhausen, who was then in Paris studying with Messiaen (Toop, 1974, p. 143). Thus, at this time both Cage and Stockhausen were closely aware of Boulez’s serial techniques used in Structures. The second unique aspect of this case study is that the appearance of one predominant analysis has coloured significantly all the following writings about the work. This is a highly detailed analysis by György Ligeti of Structures 1a, which was published in Die Reihe in 1958 (in German and then in English two years later).29 This lengthy and comprehensive study discusses various aspects of Boulez’s serial techniques and entire compositional procedure. Cage’s article was already published and therefore the initial series for the four parameters and two matrices were already available to him (Ligeti, 1960, p. 38). Despite the fact that “Ligeti’s notorious analysis” (Grant, 2001, p. 131) has also been criticised for its forbidding complexity,

28 “Possibly . . .” in Stocktakings from an apprenticeship was first published in “Éventuellement . . .” in La Revue musicale in May 1952. 29 According to Grant (2001, p. 131), Stockhausen played a role in encouraging Ligeti to analyse this work of Boulez. As the editor of Die Reihe, Stockhausen was aware of the serial system employed in Structures 1a. 60

subsequent analyses have all tended to rely not on Boulez’s own writings but on Ligeti’s article.30 In re-evaluating subsequent understanding of the work it is essential to determine the extent to which this analysis has set the agenda.

Overview

Here, ten analytical writings on Structures Ia are reviewed in chronological order. (see Figure 4.1). The analyses differ in their length and purposes. A number of them are relatively short—those by Smith-Brindle, Griffiths, DeYoung and Ekart-Bäcker— but six of the authors analyse the work in detail and present significant new insights. Most analyses follow the general direction set by Ligeti. In fact all the subsequent authors faced the challenge of surpassing the quality and depth of his ground-breaking analysis. In the course of this evaluation, the following aspects will be discussed: the similarities and differences between the various analyses in their approaches and observations; the strengths and weaknesses of the various analytical approaches; and how the earlier analyses influenced and shaped the later ones. As one reviews this particular selection of analyses, a gradual change in analytical focus becomes evident. Different aspects of Ligeti’s analysis recur in most writings on this work up to the 1980s and this reflects the prevalence of certain analytical approaches over the preceding decades. However, from the 1990s a clear change in analytical approaches is evident.

Analyses in the 1950s and 1960s

Ligeti’s article is entitled “Pierre Boulez: Decision and automatism in Structures Ia”. The central focus of Ligeti’s discussion is the compositional process. At the beginning of the article, he attempts to justify the value of analysing Structures Ia: “if one is to demonstrate the way constructional principles were used in the early stages of serial music, Structures Ia is a particularly suitable example. Since this composition is very perspicuously worked out, its anatomy is revealed of its own accord, so it can be

30 An exception of this case can be found in York Höller’ book (1994) Fortschriff order Sackgasse? Kritische Betrachtungen zum frühen Serialismus (Progress or dead end? Critical observations on early serialism). The analytical discussion of Structures Ia appears in Chapter 3 under the subheading “Die Theorien von Boulez und Stockhausen” (The theories of Boulez and Stockhausen) and Chapter 5 “Kritik” (Criticism). The explanation of serial technique applied to the piece is primarily a summary of Ligeti’s analysis published in 1958 in German. Höller then discusses the aspect of automatism, referring to another section from Ligeti’s analysis “Automatism” in his Chapter 5. However, unlike other authors reviewed in this chapter, Höller regularly refers to several of Boulez’s writings. 61

analysed as a ‘textbook example’” (1960, p. 36). Ligeti’s discussion on the constructional principles unfolds in three sub-divided sections: “Decision I”, “Automatism”, and “Decision II”. The layout of these divisions is logical and progressive in that it seems to de-code in turn each phase of the actual compositional procedure. In the following paragraphs, the main topics covered in each section of his article are reviewed. In the section “Decision I”, Ligeti details the serial operations of the four musical parameters (pitch, duration, dynamics, and attack) as well as the organisation of the non-serial elements such as formal structure, density and tempo. With the serially organised musical parameters, Ligeti illustrates the initial series used for each as well as the two matrices that produce all the numerical sequences (see Figure 4.2 (a and b)). Following this, he shows the orders selected for each series of the individual parameters. To clarify his explanation of a rather complicated process of serial ordering of the material for two pianos, Ligeti provides a number of tables. Ligeti’s discussion also compares the serial operation for each serialised parameter. For example, he writes about the fundamental difference between the pitch series and durational series as follows:

The chosen basic unit (demisemiquaver) is multiplied by from 1 to 12, and arranged in an increasing arithmetical series . . . Whereas the twelve elements of the note-quality series are predetermined by the customary temperament, the choice of durations, though in itself logical (as an arithmetical series), is all the same arbitrary. The number of duration elements (12) is meant to equal the number of note-qualities present, but the duration series, because of its additive structure, behaves heterogeneously as compared with the note-quality series, whose organization is proportioned. (1960, p. 39)

Ligeti further points out more significant difficulties in equating the serial operation of dynamics and attacks with those of pitches and durations. With dynamics and attacks, the difficulty lies with accurately measuring the individual value of each number in the series. For instance, making a distinction between mp, quasi p, and p can be a challenging and ultimately impractical task. Both parameters of dynamics and attacks greatly depend on a performer’s technique to differentiate between them subtly and consistently. Ligeti stresses the fact that the serial arrangement for attacks is especially unrealistic:

In this piece Boulez makes use of ten modes of attack. It was hard enough to differentiate twelve degrees of intensity, but the distinctions between individual modes of attack are infinitely subtler, since piano touch (unlike string instruments’ very differentiated modes of attack which each produce a quite different formant-spectrum) results from duration and intensity— proportions that are always the typical ones. i.e., touch is not an independent parameter. (1960, p. 42)

These observations of Ligeti’s, certainly, suggest the practical limitations of Boulez’s schematic serial operation. Concerning the other three parameters (duration, dynamics, and attack), he raises the crucial question as to how accurately a specifically assigned value can be performed.31 When it comes to determining the attack series used in this piece, Ligeti suggests that Boulez chose an arrangement of attacks “as he pleased” due to what he referred to as “the double significance of modes of attack (duration + intensity)” (1960, pp. 42–43). Characteristically, Ligeti made considerable effort to determine the attack series used in this work, applying a methodical process that is explained step by step. In summary, he explains: “to find in the serial table the ordinal numbers for individual modes of attack, one has to look for correspondences of certain recurring modes of attack in the piece, and then look for corresponding number successions in the tabular diagonals” (1960, p. 43). However, it is interesting to note that the attack series Ligeti uses in his analysis differs from that shown in Boulez’s letter to Cage in August 1951 (see Figure 4.3 (a and b)). The difference between the attack series shown in Ligeti’s analysis and that in Boulez’s letter is that the order rather than the types of elements varies.32 Prior to discussing the non-serial aspects of the compositional procedure used in Structure 1a, Ligeti introduces the concept of a “thread”, which refers to a horizontal sequence produced when four serialised parameters are all combined (1960, p. 44). Each thread consists of twelve chromatic pitches and durations, but each is assigned only one dynamic level and type of attack. Ligeti demonstrates the way these threads are ordered according to the numeric sequences derived from each matrix, which he

31 It is worth noting that Boulez was also exploring electronic music at this time and so the technology existed that would have overcome the limitations of human performers had they been of deep concern to him. According to Goléa (1958, p. 165) he did play this piece with Messiaen and the technical demands may have been inspired by the transcendental virtuosity of Yvonne Loriod. Messiaen had of course written his Visions de l’Amen in the 1940s. 32 Another, minor difference is that Ligeti’s series consists of only 10 of the attacks that are employed in the work. 63

refers to as the “high-order serial arrangement” (1960, p. 45).33 He then introduces the concept of the density variable. The threads are arranged into fourteen sections that are contrasted by different levels of density. Several of the sections in fact run together—sections II(a), II(b), II(c) and sections IV(a), IV(b)—to form an overall structure of eleven sections (see Figure 4.4). These eleven sections are assigned with one of three tempi and are further articulated by fermatas that create silence between them. Ligeti points out that the eleven sections correspond to the eleven intervals of the series (1960, p. 49). At this point of his article, Ligeti’s focus shifts to the organisation of non-serial elements and he explains the relationships between density variability and formal sections as well as those between the formal sections and the tempo changes. The section following the discussion of serial organisation is called “Automatism”. It begins with the statement: “Once arranged, the elements are woven, in the predetermined way, into a network in which the detailed results cannot be foreseen. Since the sections are separated from each other, the mechanism manifests itself only in their internal structure” (1960, p. 53). Such a statement is based on the observation that the pre-determined features of the individual threads will be imperceptible to a listener in a particular section when more than one thread is compiled or interwoven. His analysis shows how, in such cases, the identity of the individual threads becomes harder to recognise when the density level is increased. Ligeti addresses all four serialised parameters in relation to automatism, arguing that, as the succession of pitches is automatically regulated, both horizontal and vertical dimensions escape the composer’s control. In relation to durations, Ligeti illustrates how shorter values increasingly dominate a section when the number of superimposed threads rises (see Figure 4.5). As a result Ligeti proposes that “the extent to which the individual threads in the web are distinguishable could also be regarded as a mechanical end-product.” (1960, pp. 54–55). He further posits that “three factors are determinant: dynamics, mode of attack and the number of threads present” (1960, pp. 54–55). However, when similar dynamics or attacks are used in a high-density section, attaining the maximum clarity of threads becomes difficult—as for example in sections XI and VI(a) (1960, p. 55).

33 A detailed explanation concerning how Boulez ordered series for the four parameters from the matrices is not necessary here since this aspect of the piece is well known. 64

Ligeti’s argument about automatism is concerned with the surface structure of the music. His primary concerns are twofold: firstly, that the identity of individual threads dissolves as the density levels increase and, secondly, that the degree of unpredictability seemingly increases as the identity of each thread becomes unrecognisable. However, though Ligeti does not underline the point, it must be noted that it was Boulez who determined the number of threads to be superimposed in each section. As will later be argued, Ligeti does not recognise the full implications of the density variable in relation to the surface structure. In the final section of his analysis, “Decisions within the products of automatism”, Ligeti considers two inter-connected subjects in relation to the organisation of pitches. He observes that the serial organisation determined the sequence of pitch-classes but did not dictate their registral distribution: the register for each pitch-class was determined by the composer. Further, he observes that Boulez has followed one rule, to avoid octaves (1960, p. 55). Ligeti investigates the relationship between the registral distribution and its interference with the pitch series, claiming that “this dialectic between the apparently regular and the seemingly accidental is in fact one of the most attractive characteristics of the piece” (1960, p. 56). Ligeti further comments that, due to the registral distributions, note repetitions and certain intervallic patterns recur, and that “the multiple note-repetitions attract our attention, preventing us from perceiving the rest of the interval-relationships” (1960, pp. 60–61). His statement is supported by a number of musical examples. Here, he makes an important connection between this musical phenomenon on the surface structure and a listener’s experience of it. A factor worth drawing attention to is Ligeti’s observation is that these repeated pitches and intervallic patterns are not the consequence of the serial organisation but of the composer’s choice. One of the strengths in Ligeti’s analysis is that it successfully informs two inter- connected aspects of the compositional process: those factors governed by the serial operation, and those that are not. In other words, the non-serial aspects of composition, which were the result of the composer’s choice, are effectively incorporated into the discussion of the serial technique. Ligeti’s analysis illustrates how these two principles are inter-dependent and how they influence each other. Ligeti in fact underlines at the beginning of his article that these two compositional principles, decisions and automatism, are not the opposite ends of a spectrum. Towards the end of the article, he concludes: 65

For the listener, knots, relationships, connections of many kinds emerge; the result is an organism as ramified as it is elastic. We have seen that this organism is the result as much of decisions on the composer’s part as of automatic mechanisms; and that these decisions and automatisms are not opposed principles but two aspects of the same principle. Interacting decisions lead unavoidably to automatism, determination creates the unpredictable; and, vice versa, neither the automatic nor the accidental can be created without decision and determining. (1960, p. 61)

The first analysis of Structures Ia in English was written by Wennerstrom in 1967 as part of her PhD dissertation “Parametric analysis of contemporary musical form”. This dissertation examined ten works written from 1950 to 1965 to reveal the elements of composition that contribute to the articulation of musical form in music from this period. Wennerstrom investigated multiple parameters in relation to three questions: “How do these 10 works deal with the basic problems of putting together musical material? What creates differentiation and continuity? In essence, how are the principles of variety and unity still relevant for all music?” (1967, p. 2). Boulez’s Structures I is analysed in Chapter 4. Unlike many other analyses which mainly or only focus on Structures Ia, Wennerstrom includes analyses of Structures Ib and Ic, applying the same analytical approach equally to all three parts in her attempt to address the above questions. However, the current study focuses primarily on her analysis of Structures Ia. Prior to providing the detailed analysis of Structures Ia, Wennerstrom briefly discusses the historical background of the piece followed by a list of the four initial series on which the three movements of Ia, Ib and Ic are based. She briefly considers the generating process and function of the two matrices, but questions the way the matrices are used in the serial procedure: “Obviously, Boulez has applied an inconsistent technique in deriving material, for while the pitch interval relationships remain the same in transposition, the number series of durations, dynamics, and attacks retain no relationships” (1967, p. 48). However, beyond such general claims, she does not elaborate any further on this topic; instead, she refers readers to Ligeti’s analysis for a more complete description of the serial techniques. The first half of Wennerstrom’s analysis examines, in the following order, how six different parameters of the music contribute to the shape of the piece: pitch, duration, density, tempo, dynamics and attack. It is important to point out that the

author makes a distinction between the organisation of pitches and durations and the other four parameters because density, tempo, dynamics and attack are “much more apparent aurally than are the series of pitch and duration, which serve a sub-formal organizational purpose” (1967, p. 49). Evidently, Wennerstrom’s analytical approach focuses on the listener’s perspective. She grades the remaining four parameters based on how these parameters contribute to the articulation of the overall shapes and her discussion is organised accordingly, starting with the most aurally significant parameter to the least, that is, from density, tempo, dynamics to attack. Wennerstrom’s analysis of the above four parameters can be summarised as follows. She recognises that the density variable is the most audible element that articulates the shape of this piece, categorising the densities into three different levels: thin (1–2 threads), thick (3–5) and thickest (6). Like Ligeti, Wennerstrom expresses concerns about the audibility of each thread in relation to the different levels of density. According to Wennerstrom, “In denser sections the listener receives only an impression of a gradual thickening, reaching a climax in measure 32–39 and 106–115, where six threads are combined” (1967, p. 50). Regarding the arrangement of the three different tempi, the author is convinced that the changes of tempo provide “discernable variations in time” to the listener; in particular, the placement of fermatas further enhances recognition of each tempo change. Although dynamics and attack are organised serially, the author’s interest lies only in the resultant sound effect of these two parameters. Wennerstrom categorises the resultant dynamic levels into five levels: softest, soft, mixed, loud and loudest (see Figure 4.6). She observes that some sections have an overriding dynamic level due to the combination of similar dynamics, while in other sections, the combination of sharply contrasted dynamics results in rapid alternations of different dynamics. Similarly, in relation to attack, the author is again interested primarily in the resultant effect rather than the serial ordering of this musical element (see Figure 4.7). Wennerstrom concludes that the above six parameters operate independently: “This interplay of differential parameters creates a form that, despite its arbitrary symmetries, presents to the listener a fluctuation of continuation and division, depending upon which parameter is most apparent aurally at the moment” (1967, p. 53). However, she notes that at bar 64, the beginning of section VI, “all the paramedic elements coincide” (1967, p. 53) and thereby create the clearest formal division in the piece. 67

In the next section of her analysis, Wennerstrom identifies musical elements that contribute to a sense of musical cohesion both within each section and between the sections. For instance, examples of pitch repetitions are provided as an element that establishes the coherence within a section (1967, p. 55), and the author claims that “even a totally serial composition can have relationships above the level of automatism, if the composer chooses to create such relationships” (1967, p. 57). On the other hand, she contends that the level of unity between sections is weak because sections are juxtaposed without recognisable reference points between them. Although there is the continual repetition of pitches and durations, the organisation of these parameters is not aurally apparent enough to create musical coherence between the sections. The only exceptions are those places where the tritone is emphasised. Therefore, she does not consider the repetition of pitches and durations occurring within every section to be unifying elements. Wennerstrom’s purpose in analysing Structures Ia is summarised in the following statement: “we must learn to understand contemporary music and to perceive it on its own terms—not as a compilation of mathematical operations but as an unfolding of relationships” (1967, p. 18). Wennerstrom first identifies the musical attributes that shape the piece and then discusses how well these attributes can be recognised aurally. As part of her conclusion, she also voices a doubt about Boulez’s serial techniques, saying, “We can question the validity of Boulez’s system. [We can also question the validity] of his series of dynamics, attacks and durations, which as series are surely inaudible, but the results of the system need not be a completely non- differentiated mass” (1967, p. 86).

Summary

Two analyses of Structures Ia were available in English in the 1960s: those written by Ligeti and Wennerstrom. While Ligeti’s focus was to investigate the compositional process as both serial and non-serial, Wennerstrom’s main concern was identifying the way in which different parameters contribute to shaping the overall form in ways that are aurally discernable. It is striking to note that none of the later writers who have examined this work have referred to her analysis. Moreover, few writers on this piece over the following decades shared Wennerstrom’s concern for the resultant sonic shape. In contrast, Ligeti’s analysis and his approach to the piece had a

profound impact upon all the subsequent writers in various ways; without an exception, Ligeti’s analysis has been the continual point of reference when considering the serial operations of this piece.

Analyses in the 1970s

Fuhrmann’s “Pierre Boulez’s Structures Ia” (1974), written in German, is part of a book designed to assist students to understand modern compositions written after World War II. In this chapter, the author organises the discussion into four subsections: (1) “Information (Information)”, (2) “Analysis (Analyse)”, (3) “Material (Material)”, (4) “Consideration of didactic and learning objectives (Überlegungen zur Didatik und Lernziele)”. The second part of this chapter is most relevant to this study since the serial operations are discussed here. In the opening paragraph of “Analysis”, Fuhrmann points out some of the challenges in analysing new music as well as those challenges pertinent to Structures Ia. He notes that it is often the case when approaching such music that one must depend upon the composer’s explanation of compositional techniques or intentions. Fuhrmann (1974, p. 171) also claims that “even Ligeti’s detailed analysis of Boulez’s Structures Ia was based upon a note by the composer; without such a note it would not have been possible.” 34 Prior to discussing the serial techniques in detail, Fuhrmann questions how Boulez’s serial technique affects the resulting surface structure. He says:

There is another problem which is to make a work transparent to the listener through interpretation. Herein, the serial technique has its limits, particularly regarding the sound volume in the rows. Over-organisation founders on the increasing possibility of chance occurrences resulting from the limited concentration ability of performers and audiences. (1974, p. 172)35

Here, Fuhrmann comments on the discrepancy between the intense serial organisation and chance-like surface, a problem acknowledged by many who have studied the work. Though he does not develop the idea, by raising this issue, he counteracts the

34 Auch Ligetis ausführlicher Analyse der Structure I von Boulez (vgl. Anmerkung 6) lag eine Notiz des Komponisten zugrunde; ohne eine solche wäre sie nicht möglich gewesen. (Translated by Christina Young and Sun-Ju Song.) 35 Ein anderes Problem besteht darin, ein Werk durch die Interpretation dem Hörer transparent zu machen. Hier jedoch stösst die serielle Technik, zumal in den Lautstärken-Reihen, an ihre Grenzen. Überorganisation zerbricht an der wachsenden Möglichkeit unkontrolliertbarer Zufälle, die sich aus der begrenzten Konzentrationsfähigkeit der Interpreten und Hörer ergeben. (Translated by Barbara Steinhauser, Armin Terzer and Sun-Ju Song.) 69

common tendency to overemphasise the serial construction at the expense of the listener’s perspective. There are several notable features of Fuhrmann’s “Analysis” section. Firstly, he provides a substantial discussion on Messiaen’s Mode de valeurs as ‘the model of Structures’ (1974, p. 172). Messiaen’s work is studied in detail (as was discussed in the previous chapter) and another of Messiaen’s piano pieces, Ile de feu II, is also briefly examined. Secondly, a short but valuable comparative analysis of Structures Ia, Ib, and Ic is illustrated with an accompanying table. The table shows the following elements: tempo and tempo changes, dynamics and density in each movement. Although this analysis is brief, it is rare in the fact it compares Structures Ia, Ib and Ic in this manner. Finally, the actual analysis of Structures Ia in Fuhrmann’s article is directly derived form the first section, “Decision 1”, of Ligeti’s article. The final section of Fuhrman’s article, titled “The learning objectives” (Lernziele), has a clear pedagogical purpose, being intended for students studying musical analysis, and encouraging intellectual understanding of the structural principles that govern the music. The most relevant issue here is Furhmann’s distinction between “Analytical aspects” and “Cognitive aspects”. In the latter section, he does categorise the “most important rules of serial technique” (1974, p. 185) and “principle of formal design”. However, under the sub-title of “Analytical aspects,” he writes (1974, p. 185):

The student should be able to classify the structures according to the following criteria:

(a) sound pattern according to degree of density,

(b) variable and relatively constant parameters,

(d) Time sections.36

This immediately illustrates the priority Fuhrmann gives to aspects of density, dynamics, attack and the time span of sectional division. Moreover, the above criteria are more strongly associated with aural perception than with intellectual and theoretical principles. The aspects of pitch and duration are notably absent from his

36 Der Lernende sollte die Gliederung der Structures nach folgenden Kriterien vornehmen können: a) Klangewebe nach dem Grad der Dichte, b) variabel und relativ konstante Parameter, c) Klangfelder nach Intensitäts- und Anschlagsgraden („Attaques“), d) Zeit-Sektionen. (Translated by Christina Young and Sun-Ju Song) 70

criteria. Fuhrmann thus differentiates perceptual aspects from conceptual ones and underlines that musical analysis must not neglect issues of aural perception. Although this is little more than a passing suggestion, none of the other analyses from the 1970s drew attention to these aspects so clearly. One year later, Smith Brindle considered Structures 1a in some detail in a widely influential book in English entitled The New Music: The Avant-garde since 1945 (1975). Though some other works such as Mode de valeurs are considered briefly, the analysis of Structures Ia dominates his chapter, “Integral serialism”. Following the same pattern as the analyses of Fuhrmann and Ligeti reviewed above, Smith Brindle underlines the connection between Messiaen’s pre-compositionally designed mode for Mode de valeurs and Structures Ia. Concerning Boulez’s work, his discussion mainly focuses on the serial techniques. Although he acknowledges Ligeti, his analysis is little more than a condensed version of the first part of Ligeti’s article, “Decision I”. Nonetheless, Smith Brindle’s account does offer the practical benefit of simplifying Ligeti’s analysis considerably without compromising the content. Although his analysis may not offer much insight beyond Ligeti’s analysis, his explanation of the basic operational system involved in this piece is considerably easier for a reader to understand. However, the non-serial aspects of compositional procedure are glossed over. Regrettably, he omits Ligeti’s fascinating observations about pitch distribution and register and its effect upon the surface structure. This reflects his overall priority for the serial operations over other musical aspects that would be perceptible to a listener. DeYoung (1978) presents another type of analysis in his “Pitch order and duration order in Boulez’s Structures Ia.” As suggested in the title, DeYoung only examines the relationships between pitch and duration, and attempts to prove a relationship between the serial ordering of these two parameters. By doing so, DeYoung disputes Ligeti’s statement, which is quoted in the early part of his article (DeYoung, 1978, p. 28): “the choice of durations, though in itself logical (as an arithmetical series), is all the same arbitrary.” The analytical methodology applied to this investigation is based on the theory of twelve-tone invariance. This theory is expanded from an analysis of Anton Webern’s Piano Variations written by Peter Westergaard, who posits that a relationship exists between pitch ordering and durational sets (DeYoung, 1978, p. 28). DeYoung argues that, with only a few exceptions, several dyads formed between the pitch series of Piano I and II are 71

accompanied by similar dyads occurring in the accompanying durational series. The current author’s only concern regarding his argument is that many other aspects of compositional procedure do not seem to be taken into account. For example, the registral distribution, another very important compositional procedure that predominately shapes the piece, is not considered in this analysis. Moreover, some significant differences between Structures Ia and Webern’s Piano Variations, unfortunately, are not carefully examined or considered. Evidently, Webern organised the registral placement in relation to invariance even to the extent that it can be recognised through careful listening. The other factor distinguishing Webern’s Piano Variations from Structures 1a is the texture, which is far simpler and far less dense than Boulez’s work for two pianos. Even though the analytical methodology applied by Westergaard to Webern’s Piano Variations is effective, it is not directly applicable to Structures Ia without considerable modification. DeYoung’s analysis proves that invariance controls certain relationships between pitches as well as durations. However, the meaning of such invariance in this case must be seriously questioned. As his example shows, the notes of the dyad themselves occur widely apart (DeYoung, 1978, pp. 29-30). The numerical relationship he sees between pitch orders and duration orders is only apparent on the matrices, pointing to a purely conceptual rather than perceptual approach. 37 The hypothesis in DeYoung’s analysis, though it is rooted in the principle of serial organisation, neglects many other pertinent aspects of Boulez’s serial techniques. Uno’s discussion of Structures Ia demonstrates the serious limitations of DeYoung’s hypothesis (Uno, 1994, pp. 95–96). Unlike the previously reviewed analyses, Griffiths (1978) focuses exclusively on the non-serial aspects of Boulez’s compositional technique, which he refers to as the “secondary decisions” (register, tempo and density) (1978, p. 24). 38 Griffiths courageously dismisses the need to explain the serial operations, justifying this by underlining that “secondary decisions give shape to Structures Ia, that shape being

37 Though Boulez’s work is not considered, Joel Lester’s Analytical approaches to twentieth-century music (1989, pp. 190–191) discusses this specific matter of invariance under “Common interval and subset”. Lester explains that “the difference between possible relationships and those that are used in a given piece is the difference between pre-compositional and compositional factors. Pre-compositional factors are those that exist whether or not they are used.” So Lester proposes specific terms to distinguish between potential invariance and the invariance that has a prominent role and can be discernable. Pre-compositional factors become compositional factors when common intervals are supported by same rhythm and register. 38 Even this choice of terminology suggests that these are of less importance than primary decisions. 72

imposed on a serial structure much more obvious to the analyst’s eye than the listener’s ear” (1978, p. 24). The decision to focus on the non-serial aspects of composition can further be explained by Griffiths’ view on the serial operations and automatism: “the attempt at total serialism had had to be made, and it was made in the full knowledge that it represented an abstention from creative decision-making” (1978, p. 24). In his analytical discussion, Griffiths provides a graph to illustrate how these three elements (register, tempo and density) vary section by section throughout the piece (see Figure 4.8). The changes that occur as a result of these three elements are shown to be significant, and Griffiths states: “Boulez achieves variety and symmetry in what might have been a piece of undifferentiated stasis” (1978, p. 22). Although the graph draws attention to many features that could have been further explored, Griffiths proceeds mainly to summarise Ligeti’s observations regarding these secondary choices. Once again, Griffiths’ discussion is largely derived from Ligeti’s analysis, but his focus, unlike that of Smith Brindle and Fuhrmann, diverges from the technique of serial operation to the function of non-serial aspects.

Summary

In summary, four authors writing on Structures 1a in the 1970s (Fuhrmann, Smith Brindle, DeYoung and Griffiths) all clearly demonstrate the strong influence of Ligeti’s article. However, although Ligeti spends a considerable time in discussing the non-serial aspects of composition as well as the pitch behaviour of the surface, such as repeated notes, these crucial elements are often overlooked or passed over briefly by the later authors. The only one to refer to the work’s surface structure and overall shape is Griffiths, while the others focused on explaining the serial organisation of the four parameters. This is understandable since the compositional technique employed in Structures Ia defies simple explanation but requires several stages of analysis merely to demonstrate Boulez’s serial operation. Evidently, discussing the schematic serial operation, presenting the series with two matrices and tracing the order of series throughout the piece, received higher priority than any other aspects. This preoccupation with explaining serialist composition reflects the general trend in the discipline of music analysis during the 1970s. However, this trend was observed and

seriously questioned even prior to the 1970s. In 1967 Meyer39 was critical of the way music analysis generally approached serial music:

An account of the repertory of materials (pitches, durations, etc.) used in a piece of music and their manipulation cannot serve as an analysis of the work of art itself. To “explain” a piece of serial music by discovering its row structure and detailing its permutations and combinations in the work is almost as pointless as trying to explain a joke by discussing theories of humor . . . One does not understand a piece of serial music by “discovering” the tone rows or rhythmic rows, watching for inversions, retrogrades, and the like any more than one listens to the Eroica by “conceptualizing” the E-flat major scale and the rules of harmony and counterpoint. (1967, p. 268)

Given the historical and theoretical importance of Structures Ia, in which serial organisation is expanded to non-pitch parameters, it is obviously pertinent to discuss the way serial techniques are applied to these parameters. However, knowing the order of series by identifying where the series begins and finishes is not sufficient to understand or appreciate serial compositions, especially a work such as Structures Ia.

Analyses in the 1980s

An analysis of Structures can be found in Jameux’s book Pierre Boulez, which was published in French in 1984 and in English in 1991.40 In the first part of the book, “Trajectories”, Jameux provides a comprehensive discussion of the composer’s life and works from 1925 to 1986. The analysis of Structures appears in the second part of the book titled “Commentaries” among his analyses of eleven other works by Boulez. As such, of all the analyses reviewed here, his is the only one to be presented in the context of a detailed study of both Boulez’s life and other works.41 In Chapter 4, ‘“Eventuellement” (1949–1952)’, Jameux outlines the historical context of Structures. The following two quotations, particularly, inform Jameux’ analytical approach. The first one concerns automatism within Boulez’s serial operations:

39 Although there is a second edition of this book with a new addition of a “Postlude” published in 1994, I have purposely chosen the first edition to represent the time that this idea emerged. 40 Although this was published in English 1991, Jameux’ analysis is reviewed as ideas and analytical approaches representing the 1980s based on its first, French publication. 41 Although the author includes all three parts of Structures I, Structures Ib and Ic are mentioned only briefly and author underlines that it is not his intention to provide a detailed analyses of these two parts (Jameux, 1991, p. 283). 74

It was not a matter of wholly automatic music, partly because the possibilities were not completely exhausted, and also because the choice of register—a fundamental element of the keyboard writing which often transcends the level of the pitches themselves—was left to the composer. (1991, p. 52)

Jameux draws attention to another important element of choice made by Boulez in the process of composing Structures Ia—the density variable:

The organization of these densities has the twofold advantage of involving the composer once more, and of being quite perceptible to the listener, thus providing formal landmarks. My attempt at analysis . . . rests entirely on this notion, and effectively succeeds in demonstrating a very simple antiphony of the verse/response kind. (1991, p. 52)

The author also emphasises choices that Boulez made in the process of composing the work and accounts for the listener’s perspective. His approach therefore is similar to that of Wennerstrom and Griffiths though he does not refer to them. He was most likely not aware of their work, in particular Wennerstrom’s, which was unpublished. Like many analysts of Structures Ia, Jameux provides the series for the four parameters and briefly explains the basic operational system. However, the attack series listed in Jameux’s analysis differs from that of Ligeti’s analysis. It is perhaps the first analysis that refers to Boulez’s attack series that was directly derived from Messiaen’s mode from Mode de valeurs (Jameux, 1991, p. 273). Unlike all of the preceding writers, instead of referring to Ligeti’s analysis Jameux refers to Boulez’s own writing, in this case a letter sent to Cage (Jameux, 1991, p. 273) (Figure 4.3).42 In this part of his discussion, Jameux challenges the common belief that the serial operation is automatic by calculating the possible combinations of pitch and duration series: 2,304 options (48  48 = 2,304). He further explains that “in terms of combinatory analysis, the number of possible permutations of a finite group of twelve objects is about 480 million” (1991, p. 272). Based on these figures, the author argues that choices had to be made from an “almost infinite” number of possibilities in the course of composition (1991, p. 272). By way of emphasising this point, he contrasts the procedure with that of Cage, who advocated that choices be made through chance operations.

42 There is a minor mistake in the attack series presented by Jameux as he has in fact used incorrect symbols for attack 12: normal (1991, p. 273). Neither Messiaen nor Boulez use any symbol to indicate this type of attack. 75

In Jameux’s analysis, various kinds of compositional choice are addressed: (1) the relationship between the 11 formal sections and the overlapping symmetrical arrangement of tempi; (2) the distribution of the 48 pitch series across the 11 formal sections that introduces the concept of an Antiphon and a Response; (3) the relationship between the density variable and the resultant sound effect of dynamics; and finally (4) a durational series that introduces the concept of a rhythmic cantus firmus. Like many other analyses, the content of Jameux’s analysis overlaps with Ligeti’s analysis. Here, only the content of Jameux analysis that goes beyond Ligeti’s analysis is reviewed. Jameux observes that an Antiphon and Response effect is created through Boulez’s distribution of the 48 forms of the pitch series. Based on the serial organisation, Ligeti had identified two Parts within the work and Jameux’ theory underlines these divisions. As the entire original and inversion series are employed in the first part of the piece (sections I–V,) Jameux calls it ‘Antiphon’, while the second part of the piece, where the composer uses the entire retrograde and retrograde- inversion series (sections VI–XI) the author calls the ‘Response’. Though the other parameters are not mentioned in relation to these two terms, Jameux uses these two terms throughout his analysis. Jameux examines the level of dynamics throughout the entire piece in relation to density and formal structure. He claims that there is “a certain dynamic agitation in the Antiphon where, despite the peak represented by section 3, dynamic intensity is less sustained than in the Response” (1991, p. 277). The methodology Jameux uses to gauge the dynamic level is seemingly logical but its effectiveness is somewhat doubtful. He calculates the compound dynamic level of each formal section by, firstly, grading the dynamic spectrum from pppp to ffff in the form of a logarithmic scale of 1 to 12, and then adding the corresponding numbers of the assigned dynamics. As Jameux explains: “Within an acoustic approximation that takes account of the approximate nature of a ‘series’ of dynamics, simple addition of the numbers gives a dynamic index for each section” (1991, p. 227). Although the author acknowledges the “approximation” involved, the result of his measurement remains questionable because, due to the density variable, different sections are assigned a different number of dynamics. As a result, sections with very different dynamic characteristics may turn out to have a similar dynamic index. For instance, the sum of section I is 17 (12+5, that is, ffff and quasi p) while the sum of section XI is also 17 (1+2+3+7+3+1, 76

that is, pppp, ppp, pp and mf). Although the dynamic index of these two sections is the same numerically, the resultant sounds for these two sections, in terms of dynamics, are radically different. Section I would be characterised by the contrast between two dynamic levels but section XI is dominated by soft dynamics.43 When discussing durations, Jameux introduces the concept of a rhythmic cantus firmus in Structures Ia. As he explains it:

[Boulez] adopted a twofold durational world: on the one hand, with certain strands depending on serially organized duration; on the other, with serial pitch statements whose notes are always, and sometimes violently, detached. These latter surround the former with embellishing elements, in the manner of tropes on either side of a cantus firmus. (1991, p. 279)

Therefore, the author distinguishes between those occasions when the duration of a note is to be sustained through its full length and when the note value is shortened by its attack. For example, if a note is to be played staccato, the remainder of the assigned duration is made up by rests. He counts that 26 durational series are used as corresponding series to those of pitch, and interprets these durational series as the cantus firmus in a given section. For instance, in section 10, where only two threads are employed, the note values used in the first piano are always notated short, as either semiquavers or demisemiquavers. Jameux gives this as an example to illustrate the rhythmic cantus firmus in the second piano but claims that “the rhythmic counterpoint of the first piano is not part of a rhythmic series” (1991, p. 279). Jameux concludes his analysis by combining all aspects discussed in one diagram that includes the density variable, formal sections and performance durations based on the 1965 recording of the Kontarsky brothers, tempi changes, dynamic levels and the rhythmic cantus firmus. Jameux claims section VI to be the “expressive centre of the movement as a whole” (1991, p. 282), which can be aurally discerned because of its distinctive dynamics, tempo and density. He describes the latter as “a superb polyphonic quintet” (1991, p. 282) made of five cantus firmus lines. Although his analysis is descriptive, Jameux attempts to define Boulez’s compositional choices in relation to the serial organisation and, in doing so, aims to enhance the listener’s

43 It is interesting to compare this with Wennerstrom, who also attempts to investigate the change of dynamic levels throughout the entire piece. As outlined previously, she categorises the resultant dynamic levels into five areas: softest, soft, mixed, loud and loudest. Her categorisation seems to be more convincing than Jameux’ because she identifies the sections where a wide range of dynamics occur. 77

experience; “my efforts will be justified if they facilitate listening, and reveal the almost involuntary beauty of Boulez’s chosen approach” (1991, p. 283). Eckart-Bäcker’s article, written in German in 1989, includes analyses of both Structures Ia and Ic. It is another rare case where both these pieces are analysed in equivalent depth though, in fact, the analyses are relatively brief and the comparison is not pursued in detail. She explains that her primary purpose for introducing a short analysis of Structures Ic is to show how different aspects of serial technique can be applied to the same materials—the four series for the four parameters derived from the same two matrices. Another unusual feature in this article is that, prior to presenting the analyses, Eckart-Bäcker quotes some of Boulez’ views on how one should approach analysing serial compositions. Though the attempt to include Boulez’s opinions is commendable (especially as most of the other analyses fail to do so), the author does not build these quotations into a clear argument or use them in her later discussion of the work. Concerning the serial operations, Eckart-Bäcker refers to Ligeti’s analysis, but one can clearly notice that the author avoids unnecessary repetitions of Ligeti’s ideas as much as possible. On the other hand, Eckart-Bäcker lists four parameters as “shaping parameters” (Gestaltungselemente)—tempo, density, dynamics and attack— that help a listener to orientate themselves to the work because these parameters do not change as frequently as pitches and durations (1989, p. 395).44 At some point of the piece (section II(a), (b) and (c)), she regards tempo as a unifying parameter as it is the most stable.45 Eckart-Bäcker goes to considerable lengths to examine the arrangement of tempi across the work, suggesting that there are four different, symmetrical units. Figure 4.9 summarises her analysis of tempi. She explains the order of three tempi in this piece in the following ways. Firstly, the tempi arrangement of sections I–V is

44 In this regard Eckart-Bäcker’s approach is similar to Wennerstrom’s, though her dissertation is not cited and was presumably unknown to her. 45 In the same year of Eckart-Bäcker’s analysis (1989), Helga de La Motte-Haber published an article titled “Fundamental factors of music comprehension” in which a monophonic section from Structures Ia was used in an experiment in the field of music psychology. The purpose for this experiment was to determine whether “The ease or difficulty with which subjects conceive music depends on the level of complexity inherent in the structural organization” (La Motte-Haber 1989, p. 31). However, the methodology of this experiment focused primarily on two musical parameters: pitches/register and rhythm. (La Motte-Haber, 1989, pp. 27–28) and so disregarded the crucial roles played by dynamics and attacks. As such, the monophonic section on which de La Motte-Haber chose to focus is not representative of the way the different parameters interact across the work. Her experiment does not aim to reflect a listener’s experience of integral serialism. 78

described as “symmetry from a qualitative perspective” (qualitativem Aspekt symmetrische Strucktur) with the centre of symmetry occurring at section III. Sections VII–XI are described as “qualitatively analogous” to Sections I–V as the same five tempi are again used. Secondly, there are two larger structural units of tempi that are interlocked. Sections I–VIII are a symmetrical unit with Sections VI–V as the middle point. Sections V–XI are also proposed as a symmetrical structure, with its the centre of symmetry being section VIII (1989, p. 393). Eckart-Bäcker concludes that “within the progression of the work, various forms of the component tempo are interwoven” (1989, p. 393). 46 As such, the author has expanded Ligeti’s observations on the symmetrical arrangement of tempo across the work.

Summary

Two analyses of Structures Ia written in the 1980s were discussed here: Jameux’ analysis, which was firstly available in French and then translated into English in 1991, and Eckart-Bäcker’s analysis, which has remained only accessible in German. These two authors are both keen to identify the aspects of the work that are aurally apparent whether or not they are the result of serial organisation. Jameux specifically states that his purpose for analysing the work was to “facilitate listening” (1991, p. 283). Therefore, a change of analytical approach may be seen to be emerging in the 1980s; the focus is no longer confined to the conceptual aspects of compositional techniques but the listener’s perspective is taken into account. It is interesting to note that in the same decade the discipline of phenomenology was being applied to music.47

The analyses in the 1990s

Following the trends identified in the 1980s, two authors, Yayoi Uno (1994) and Morag Grant (2001), provide analyses that explicitly investigate the surface structure of Structures 1a from a listener’s perspective. Previously, several authors had expressed concern that the uncontrollable chance-like sound structure of Boulez’s serialism profoundly challenges aural comprehension (for example, Fuhrman, 1974, p. 172). By the 1990s, the sonic shape of the piece became a central focus of analytical

46 So sind im Ablauf des Stückes verschiedene Gestalten der Komponente Tempo miteinander verflochten. (Translated by Barbara Steinhauser and Armin Terzer.) 47 For example Thomas Clifton’s Music as heard: A study in applied phenomenology (1983). 79

studies. Both Uno and Grant express puzzlement at how, supposedly, logical and serially organised parameters project themselves to a listener as being random and even seemly chaotic. Although their analytical methodologies differ, the issues informing their analytical enquiries are alike. Both authors identify a discrepancy between the compositional theory, which embodies ideas of order, and the aural perception of disorder. Uno’s analysis of Structures Ia appears in Chapter 3 of her PhD dissertation “The roles of compositional aim, syntax, and design in the assessment of musical styles: analyses of piano music by Pierre Boulez, John Cage, Milton Babbitt, and Iannis Xenakis circa 1950” (1994). As Uno puts it, her aims are

first to re-examine the aesthetic and musical contribution of compositional trends in European and American art music circa 1950–1960; second, to develop an analytical method that responds to the specific problems posed by the repertory of this period. (1994, p. 1)

The “specific problems” mentioned refer to the fact that the pre-compositionally organised musical elements are not perceptible on the surface of the musical fabric. Thus, Uno’s analysis attempts to reveal the compositional design responsible for such an outcome, and to empirically assess the musical style of the composition (1994, p. 1). In making an empirical assessment of Structures 1a, Uno uses a technical and systematic analytical methodology. The aim of analysis, she says, is “to define the ‘fit’ among the compositional aim, generative syntax (pre-compositional structure), and design (the gestural and formal designs that are discernable at the musical surface)” (1994, p. iv). Concerning generative syntax, Uno refers to the articles by Ligeti and DeYoung, re-evaluating their analyses and integrating them with her own. She then examines the gestural and formal design of the surface structure. The analytical technique employed here is based on three elements, which are discussed under three subheadings: “Analysis of contour”, “Distributional analysis” and “Analysis of formal segmentation”. According to the author, this is an expansion of James Tenney’s theory of musical perception concerning shape, state and structure. Uno explains the difference between contour and distribution: “contour presents a qualitative analysis of musical gestures or shapes with respect to individual dimensions, while distribution presents a quantitative or statistical analysis of musical elements in determining their

mean values or states” (1994, p. 79). In order to determine the formal segmentation, “the combined input of musical dimension is used to explore the relationship between the relative weightings of musical dimensions and the gestural/formal boundaries” (1994, p. 33). Uno also discusses the procedure of encoding the input value of all musical dimensions.48 The purpose of Uno’s analysis stems from Boulez’s assertion that “the general theme of this piece is really the ambiguity of a surfeit of order being equivalent to disorder” (Boulez, 1975, p. 57), so she aims to provide concrete musical evidence to support this fascinating paradox (Uno, 1994, p. 81). Her discussion comprises three areas: Boulez’s ideological aim, the compositional method, and the surface design, which she examines through a statistical computer program. In Uno’s opinion, the fundamental purpose of integral serialism is to provide a system that can unify various musical parameters. In other words, a greater degree of structural unity can be achieved by organising the four parameters in relation to the same serial principles. However, she recognises that

The apparent “randomness” or lack of thematic or motivic organization in the musical unfolding of this work should not be attributed to the serialization of musical dimensions per se, but to the lack of mathematical and psychoacoustical isomorphisms in the serial derivation of the values for the four musical dimensions, namely, pitch, sustained duration, dynamics, and attacks. Neither do the additive durational series nor the dynamic and attack series provide effective musical analogues to the serialization of the pitch series. (1994, p. 129)

The second part of Uno’s analysis involves the discussion of compositional methods, where she re-evaluates of the analyses of Ligeti and DeYoung and assimilates their ideas into her own analysis. Here she provides detailed explanation of Boulez’s serial techniques. In addition, she investigates the potential of having invariance among the pitch relationships, which had not fully been explored previously.49 According to Uno, the properties of pitch series and the way they are arranged clearly demonstrate the possibilities of invariance. Nonetheless, these possibilities do not eventuate on the surface structure in a form recognisable by a listener. The reason for this is that “the rhythmic alignment governs the extent to

48 The encoding system that Uno applies is similar to Leland Smith’s SCORE code (Uno,1994, p. 36). 49 In order to perform this investigation, Uno converts Boulez’s order-numbered two matrices to pitch- class ones (Uno, 1994, pp. 85–86). 81

which segmental invariances and subset relationships are brought out at the compositional surface. Furthermore, the serialisation of duration restricts the contexts under which the invariance properties between adjacent rows are brought out at the compositional surface” (1994, p. 94). Following the discussion of invariance, Uno critically reviews DeYoung’s analysis, which proposes an important structural relationship formed by the invariance of dyads between pitch and duration. Uno underlines the fundamental differences between the nature of a twelve-chromatic-pitch series and that of a twelve-chromatic- duration series. The main difference is perceptual. “While the dyad pairs for pitch translate into associations between concrete musical objects,” Uno says, “those for duration translate into series of intervals” (1994, p. 96). As demonstrated by Ligeti, when the density level increases, the durational value assigned to each pitch becomes obscure to the point that a high density section is characterised by a mass of short durational values. An ironic aspect of the dyad relationships between the pitch and duration is that the invariance of the pitch dyad is mostly negated by the serialisation of duration. Uno’s investigation of invariance reinforces the discrepancy she claimed to exist between the compositional method and the surface of the composition. Uno expands Ligeti’s analysis in her discussion of the pitch relationships formed as a result of Boulez’s registral placement. Prior to a detailed investigation, Uno provides a graph that shows the registral distributions of all the pitch series. This is very similar to Griffiths’ graph (Figure 4.8) although Griffiths is not cited (see Figure 4.10). Uno then provides a table that shows the relationship between the various degrees of density and the common pitches shared by different series within each section. Ligeti also mentioned that, because one of Boulez’s stated aims was to avoid octaves, when the density increases within a section the likelihood of literal repetition of a pitch occurring in the same register is greater. Here, Uno makes a crucial observation regarding the relationship between the density and the registral placement of pitches:

Boulez, nonetheless, emphasizes the contrast between sections through inducing different degrees of registral fixity. Notice that in sections III and VIII where the polyphonic density is six, the registral placement of pitches within rows is completely fixed, while other sections are more or less fixed with respect to registral distribution of individual row elements. (1994, p. 99)

Uno then takes it a step further and discusses whether these literal repetitions can be aurally recognised. She concludes that “there does not seem to be discernible patterns in which the common pitches are organised” but says, “certain pitches are emphasized more frequently over others” (1994, p. 100). Uno provides a list of these discernable common pitches: for instance, pitch-class Eb appears as a common pitch in ten sections out of the total fourteen sections in the piece. In this part of the discussion, Uno again acknowledges Ligeti’s contribution on this matter, since he was the first author to have recognised the significance of pitches being repeated as a result of registral placement. As Uno demonstrates, DeYoung’s argument that there is a dyad relationship between pitch and duration is not supported by the surface of the composition. In addition, the several pairs of invariance occurring between different pitch series hardly appear as recognisable pitch relationships once the serialisation of duration has been applied. On the contrary, recognisable pitch patterns regularly appear on the surface, not because of invariance but as a result of the registral distribution of pitches. As such, Uno points out that these recurring pitches were not determined by the serial operation but are purely the result of the composer’s choice.

One may conclude, therefore, that the non-serialised dimensions, i.e., registral placement of pitches, polyphonic density, and tempo, in this work serve to create aurally discernable order of events in the compositional unfolding of this work. There are perceivable orders, contrasts, and musical connections that make this piece highly deterministic at the global level of structure, in spite of Boulez’s intention to eliminate subjectivity in this work through the so-called “process of automation”. (1994, p. 130)

While Uno investigates the serial organisation of pitch and duration at length, she gives relatively scant attention to the serial organisation of attack and dynamics. She justifies this by claiming that “these dimensions are attributive as discussed under the preceding chapter, and their psychoacoustical identity is dependent on the roles of other musical dimensions, e.g. pitch register, duration, timbre, and texture” (1994, p.101). The author continues her justification based on Ligeti’s claim of automatism in relation to the serial operations of these two parameters, writing,

Ligeti points out the impracticality of certain dynamics and attack combinations that result from this process of automation. He comments that the derivation of dynamic and attack values from the diagonals of the order-

number matrices amounts to mere numerical abstraction. The series produced from the diagonals bear no apparent musical connections with the pitch and durational series derived from the horizontal and vertical extraction of rows. (1994, p. 102)

In the section titled “Analysis of contour”, Uno demonstrates that there is no syntactical unification among the four serialised parameters as they occur throughout the piece but, on the global level, non-serial elements produce a certain level of contrast between adjacent sections. Her distributional analysis uses statistical techniques to “uncover global distribution attributes of individual musical dimensions and their relationships with one another”. She adds that “these techniques compute the frequency distribution, mean value, standard deviations, variance, correlations and entropy” (1994, pp. 32–33). A considerable knowledge of statistics and information theory is required to comprehend this part of Uno’s analysis and to interpret the various graphs and charts. Although the analytical methodology here is statistical, and the expression is correspondingly technical and dry at times, these investigations reveal valuable insights into different aspects of both the serial and non-serial operations. For instance, she demonstrates a relationship between the variability of pitch and degree of pitch fixation in particular registers. Another relationship occurs between the variability of attack-time duration, dynamics and attacks, and the degree of density. Furthermore, while contour and distributional analyses evidently illustrate a lack of unification among serialised parameters on the surface structures, thus confirming the author’s hypothesis, the analysis of formal segmentations shows a closer ‘fit’ between the compositional design and the surface structure. The change of shape is mostly articulated by the change of density and tempo. However, these changes are not always concurrent with the sectional divisions determined by Boulez. According to Uno (1994, pp. 125–126), eight out of the fourteen structural divisions show distinctive changes that are mainly the result of tempo and density changes (see Figure 4.11). In summary, Uno’s analysis of Structures Ia is the most detailed and comprehensive analysis written since Ligeti’s. Her analytical approaches are pluralistic. She firstly examines the compositional method, addressing various aspects of serial techniques as well as the organisation of non-serial aspects, though she focuses more on the organisation of pitches more than on the other parameters. The statistical analysis is applied effectively to measure the correlation between various

parameters. Backed by this statistical analysis, Uno argues that there is a discrepancy between the compositional plan, specifically the serial operations of the four parameters, and the surface structure of the work. It is important to note that Uno’s application of statistical methods in analysing Structures 1a suggests a breaking away from prevailing analytical approaches. The last analytical study to review in this case study is that written by Grant. It appears in the fifth chapter, “Serial music as an aleatoric process”, in Serial music, serial aesthetic: Compositional theory in post-war Europe (2001). The issues raised by Grant are similar to those addressed in Uno’s analysis. Grant strongly questions the relationship between the rational process underlying the serial operations and its resultant sound, which suggests the irrational process of an aleatoric composition. To understand the serial composition, the author encourages the reader to study the “aesthetic categories” on which serialism and its adaptation are based (2001, p. 131). Grant states that the “seeming paradox of rationality versus irrationality, decision versus automation, holds the key to serialism itself” (2001, p. 131). However, most analyses of Structures Ia, in Grant’s view, “cease at the exact moment where the serial reordering ends, and the piece begins: they deal primarily with the reordered material and only secondarily if at all with those parameters which are not controlled” (2001, p. 131). As has been shown above, this is misleading, as all of the analysts discussed here, from Ligeti onwards, have acknowledged the role played by non- serial aspects in Structures 1a. Unlike many scholars who have written about the piece, Grant aims to address the various issues in interpreting both this work and serialism in general by investigating an aesthetic theory of serialism. Prior to presenting an analytical study of the piece, Grant lays the foundation of several aesthetic theories that are related to information theory. Her investigation starts with a brief review of Meyer-Eppler’s article “Information-theoretical problems of musical communication” (Informationstheoretische Probleme der musikalischen Kommunikation), which Grant considers a starting point for a general criticism of serial music. Meyer-Eppler’s article draws particular attention to the issues of communication between composer (producer) and receiver (listener). Following this, Grant discusses Abraham Moles’ aesthetic theory as well as that of Max Bense, as both explore the connection between information theory and modern aesthetics. According to Grant, Moles’s theory can be particularly useful in dividing information into two types: the semantic and the aesthetic. They are explained thus: 85

Semantic information is translatable, aesthetic is not, but [is] only approximately transferable. Semantic information is symbolic: aesthetic information is statistic. In reality, all messages are a mixture of both, but certain types of message will tend to one or the other. (2001, p. 134)

Grant sees the above distinction between different forms of information as analogous to the fundamental difference between serial music and tonal/thematic music. She applies Bense’s aesthetic theory to make a connection between modern aesthetics and serial aesthetics. The key concept of Bense’s theory is that “all art . . . is a sign of process” (2001, p. 146). He argues that “the difference between abstract and figurative art lies in the nature of these signs: in abstract art the material itself becomes the sign. Thus, modern aesthetics represents a transition from a sign world that functions to a sign world that is” (2001, pp. 146–147). In other words, the presentation of abstract art itself becomes the focal point of that which is presented. If one applies Moles’s terminologies of semantic and aesthetic information, in modern abstract art, the aesthetic information can be as central as the semantic information. Grant is therefore approaching serial music, particularly Structures Ia, from the perspective of a modern aesthetic, where, as a result of fundamental aesthetic changes, interpreting abstract art requires a different attitude from the receiver. Thus, “serial music, unlike thematic music, is not defined by foreseeability; it is surprising, unforeseeable; it is not closed, but open. Aesthetic rather than semantic; statistic” (2001, p. 164). The primary focus of Grant’s analysis of Structures Ia concerns perception of the piece: how the listener experiences and perceives the work as a sound object. Grant points out that, in the main, the fundamental problem of many analytical approaches to atonal or new music is their acceptance of the classical and romantic aesthetics of musical organicism. Such approaches are driven by the search for structural coherence and are often “purporting to re-create the composer’s ‘intention’” (2001, p. 156). From this perspective, she is critical of Ligeti’s analytical approach for what she perceives as its focus on the composer’s perspective rather than that of the listener. Grant further elaborates:

Ligeti’s favourite metaphor in this analysis is the loom: the serial threads are inserted into the machinery and woven into larger structures. Here too the order and indeed the observation of individual thread is much less important than the larger structure they create, the importance of individual elements

being disproportional to the total number of elements—it is in this sense that point and mass structures are both special cases of statistical serial music, and “mobile” or “open” form merely special cases of serial form. (2001, p. 157)

Grant therefore does not provide a detailed discussion of serial operations, though she does mention a number of the essential characteristics of Boulez’s serial techniques, for example the difference in the permutation system between the arrangement of the dynamics and attacks and that of pitches and durations. Grant provides the initial series for dynamics and attack as well as two matrices but excludes those for pitch and duration. The diagonal numerical sequences are marked in the matrices by different shading to show which numerical sequences are chosen for the corresponding dynamics and attacks. Such an approach implies that the dynamics and attack are more discernable from the listener’s perspective. As part of investigating the effect of serial arrangements of dynamics and attacks, Grant provides a graph that illustrates the gradual change occurring throughout the piece in terms of dynamics and density (2001, p. 152). In the graph, the varying degrees of dynamic and attack are indicated by different shadings: louder and stronger dynamics and attacks are represented with darker shadings while the softer and less forceful dynamics and attacks are represented with lighter shadings. The length of each section and the pauses placed between each section are signified proportionally. Grant acknowledges that showing this type of graph could be interpreted as contradicting the perceptual focus of her analysis, though she justifies its use by saying that:

The aim of this portrayal is, however, to give some idea of the manner in which the serial ordering transforms into a serial proportioning: used as a guide for listening, it demonstrates the gradual change in dynamic and textural qualities during the piece. (2001, p. 152)

The graph is followed by a descriptive analysis that identifies the elements that define the overall structure of the piece. These are as follows:

(1) the density of the sections

(2) average attack and dynamic quality of each section

(3) relative speed: defined both by tempo and by the rate of succession of events, thus section II appears to decelerate as the density decreases and there is a greater time lapse between the appearance of the notes

(4) octave placing of each note within a section (2001, p. 153).

Interestingly, the above categories, which are similar to Fuhrmann’s lists, further identify those elements of the music that are more likely to be recognised perceptually and shape the piece meaningfully. As mentioned above, Grant does not give much attention to the serial operation of pitches and durations but discusses issues related to organising these parameters from a different angle. She questions the serial organisation of these two primary parameters and their effect on the listening experience of the work. Since pitch and duration are central to the perception of tonal music, Grant discusses their organisation in Structures Ia in relation to traditional styles. This would then allow a direct comparison of one’s listening experience of traditional tonal music with that of serial music. Grant (2001, p. 154) points out that while tonal music is generally a “foreseeable, forward-motivated form of thematic music”, serial music “often has a static quality.” This impression, she claims, is mainly due to the “microstructure” of the work, which is “formed of the two parameters most strictly controlled, and effectively neutralised; these are exactly the primary parameters of thematic music” (2001, p. 154). She further claims that integral serialism aims to achieve maximum unity among various musical parameters, but, for this to be perceived, the compositional methodology requires a far more systematic approach than that applied in Structures Ia. Therefore, Grant concludes that:

If we take into account that the “rational” procedures of serialism were first employed to create an ‘irrational’, unforeseeable music, i.e. one that is specifically athematic in character, it follows that the method of pitch and duration ordering in Structures Ia is effectively irrational. Rather than a method of ordering, serial technique thus appears as a method of unordering. It was a method of dissolving particular ties, so that others could come to the fore; its constraint was, not so paradoxically, its freedom. This depends on realising that the relationship between working method and audible result is discrete rather than direct, and this is exactly where most analyses of serial music get into difficulties. (2001, pp. 154–155)

In relation to Structures Ia, Grant advocates a model of analysis that is influenced by Nattiez’s semiological theories (see Figure 4.12). In this model, a listener’s role is greatly emphasised as she believes that “analysis deals with the communication of a musical message; analysis is in this sense a particular type of

musical receiver” (2001, p. 155). With thematic tonal music, there is a direct relationship between the working process of musical ideas and the audible result. She describes serial music generally as non-linear, non-progressive, unforeseeable and further comments that in serial music “there is not a logical process of events, rather a field of relations” (2001, p. 158). Therefore, from the perspective of reception, serial music can be interpreted as open form or aleatoric process, even though the unforeseeable effect of the surface results from rigorous organisation.

The single most devastating misunderstanding regarding serialism is exactly the functioning of apparently dialectical opposites—rationality and irrationality, control and freedom—extremes which, in the serial aesthetic, become as intertwined and as interdependent as the actual musical material. The relation composer–listener is just as important, and begs that we reflect on our relationship to these works, and our discussions of them. Until we do, most debates on serial music, based as they are on a mode of musicological thinking still discretely influenced by thematic thinking—the proposition of a direct relationship between the creation of a musical idea and its transmission—will remain fundamentally misjudged. (2001, p. 160)

The way Grant has approached Structures Ia clearly demonstrates a change of ideologies; the author has challenged some mainstream analytical methods that are rooted in the ideology of organicism. The influence of Nattiez’s semiological theories also suggests a move away from positivistic approaches. Furthermore, through the attempt to understand and interpret the piece via studying the aesthetic theories that emerged at the time the piece was composed, Grant even disputes formalistic ways of analysing serial music and exposes their limitations, especially their inadequacy in accounting for the surface structure of the piece.

Conclusion

Reviewing ten analyses of Structures Ia has revealed several noteworthy aspects. As previously pointed out, Ligeti’s analysis evidently overshadowed most subsequent analyses over the following half century. His analysis has been viewed in diverse ways. For example, his rigorous investigation of serial operations of the piece was influential, particularly in the 1970s and the 1980s, when, conversely, the non-serial aspects of his analysis received comparatively little acknowledgement. In the 1990s, Uno not only gives attention to the serial techniques but also expands Ligeti’s

examination of the pitch movements on the surface structure of music. Uno’s efforts on this matter are the corollary of her analytical investigations of the surface structure and its aural perception. While Uno refrains from making any subjective comments on Ligeti’s analysis, Grant is the first to criticise Ligeti’s “notorious” analysis (2001, p. 131) by questioning its relevance to a listener. This chapter has traced the way analytical approaches have developed over the last half century and, in parallel, observed the way scholars have viewed Ligeti’s analysis over that period. Understanding the operation of integral serialism appears to have been of central importance to many analytical studies until the 1980s. Nonetheless, analytical writings that include the examination of the surface structure and the listener’s perspective—Wennerstrom, Griffiths, Jameux, Uno and Grant— have been traced from the 1960s right up to the 1990s. Thus, Structures Ia has been analysed in both conceptual and perceptual aspects over this period. In the 1990s in particular, the issues of perception come most strongly to the fore. The analytical enquiries of both Uno and Grant specifically address the discrepancy between the organisational system and the resultant sound. In both cases, the underlying compositional process and the ‘intention’ of the composer are considered no longer the highest priority; how the surface phenomena will be experienced by a listener becomes the focus. These two authors’ analyses demonstrate their willingness to examine the surface structure of music in ways that do not always fit comfortably with the governing ideologies and methodologies that have underpinned the discipline of music analysis. These idealogies and the methodologies they have given rise to are characterised well by Robert Fink.

It does seem that there is a ‘fear of the surface’ running through much analytical work, from the canonical studies of Schoenberg to the most recent exploration of semiotics and the anxiety of influence. . . The word ‘depth’; the hierarchic relationship between surface and depth is more important than any feature of the surface itself; in most cases, surface or through voice-leading; the complexity or incoherence of the surface before analysis is often noted, and the promise of analysis is that is will be made to disappear. (1999, p.104)

Moreover, by reviewing two analytical studies of Uno and Grant, one can certainly recognise a paradigm shift where the analytical approaches depart from the ideologies of formalism, positivism and organicism. Consequently, they reflect a significant change in the discipline of music analysis.

A re-interpretation of Structures 1a

Analytical premises

It has been widely acknowledged that, in Structures Ia, Boulez expanded the principle of Schoenberg’s twelve-tone technique beyond pitch to other parameters of music such as duration, attack and dynamics. However, this chapter will argue that the primary purpose of writing Structures Ia went far beyond just serialising the four musical parameters. As the title of the work itself makes explicit, Boulez’s aim was to create an appropriate musical structure that could architecturally support the new compositional techniques available to him. These include the twelve-tone techniques formulated by The Second Viennese School and rhythmic techniques developed by his teacher, Messiaen. Boulez’s own writings at this time often reflect his preoccupations with integral serialism, especially when he was discussing his immediate predecessors and their works. However, as noted in the previous section, the predominant influence of Ligeti’s analysis contributed to the relative neglect of composer’s own writings in relation to understanding Structures Ia. Re-interpretation of the work here begins with reviewing aspects of Boulez’s discussion of his predecessors that are pertinent to both his serial techniques in general and to this work in particular. Such an approach has a twofold purpose. Firstly Boulez’s comments provide valuable insight into the way he understood serial techniques at that time. Secondly, they illustrate Boulez’s polemical style of writing not merely as a criticism of his predecessors’ work but as a process of critical evaluation that, I believe, eventually helped him to define his own compositional language. On many occasions Boulez expressed his dissatisfaction with Schoenberg’s serial compositions; most notably in the 1952 article, “Schoenberg is Dead (Schoenberg est mort)”, where he suggested the need for musical structures that could more logically accommodate serial techniques.50

We may recall that the series arose, in Schoenberg, from an ultra- thematicization in which, as we have seen, the intervals of the theme can be regarded as absolute, free of any rhythmic or expressive obligation. . . . It has to be admitted that this ultra-thematicization is the underlying principle of the series, which is no more than its logical outcome. Moreover, the confusion between theme and series in Schoenberg’s serial works is sufficiently

50 This article was published in English in The Score no. 6, (Feb, 1952), 18–22. 91

expressive of this inability to envisage the world of sound brought into being by serialism. . . . But how could the new technique be properly tested if one took no trouble to find specifically serial structures? And by structure I mean everything from the generating of the component materials right up to the global architecture of the work. In a word, Schoenberg never concerned himself with the logical connection between serial forms as such and derived structure. (Boulez, 1991, pp. 211–212)

In this article, written in the same year as Structures, Boulez argued specifically for the design of appropriate serial structures that related the series not just to the generation of “component materials” but, significantly, to the broader designs of the “global architecture”. In essence, Boulez proposed that the entire scope of the musical structure on the broadest scale be reconsidered in relation to the series. While Boulez saw Schoenberg’s series as an “ultra-theme” that merely replaced the function of the theme in tonal music, it was Webern’s serial technique that inspired him to see a vital connection between the series and musical structure. Boulez states, “With Webern, by contrast, the series at once assumes the aspect of an intervallic function, from which the piece itself derives its basic structure; this is the definition which will prevail in future developments” (1991, p. 235). Although Boulez was well aware that rhythmic elements had not been part of the serial operations in Webern’s music (Boulez, 1991, p. 173), he realised that further rhythmic organisation could enhance the development of his own serial techniques. As pitch relationships were no longer governed by tonality in serial composition, he proposed that the other components of music, which had formerly been developed and established in relation to tonal structures, should be reconsidered accordingly. In this analysis, it is suggested that in Structures Ia, one can observe and understand the “future developments” to which Boulez referred. Messiaen, and specifically his work Mode de valeurs, also profoundly influenced Boulez at this time. Boulez envisaged a serial structure emerging out of the organisation of musical materials he observed in Messiaen’s work. He wrote:

Recently, Olivier Messiaen has crystallized these scattered preoccupations of valid contemporary music in his Mode de valeurs et d’intensités, in which the idea of global—in this case modal—organization is applied not only to register, but equally to duration (that is to say, the rhythmic organization of music time), dynamics (that is, the amplitude of the sound) and attack (or the initial profile of the sound). (1991, p. 176)

As was discussed in the previous chapter, Messiaen predetermined the individual modes for pitch, duration, dynamics and attack and Boulez construed Messiaen’s approach as “the notion of chromaticism” being applied to primary musical elements other than pitch (1991, p. 225). He proposed that any single durational value could be treated as a chromatic entity that, when added arithmetically, could generate a corresponding series of durations. A similar principle could also be applied to generate a series of dynamics and attacks where the interval between the elements would be equivalent, even though the arrangement of 12 elements might not be arithmetical. By applying such chromatic entities to parameters other than pitch, Boulez found a solution for extending the twelve-tone series to duration, attack and dynamics. Boulez’s own reaction to his immediate predecessors provides some insight into his intention in developing integral serialism, which went far beyond incorporating the four primary musical parameters into a coherent system of serial organisation. In other words, Boulez realised the necessity to further develop serial operations that could encompass the three other primary musical parameters. This is widely known. However, it is most significant to underline his realisation that every level of composition needed to be reconceptualised in order to create appropriate serial structures. Such a musical structure was perhaps first actualised in the composition of Structures Ia. Therefore, the purpose of analysing this work here is to go beyond understanding it merely through the serialisation of four parameters but to comprehend it from the broader perspective of serial structure. This analytical approach aims to challenge the commonly held, dichotomised view that, on one hand, automatism is the key factor of the serial organisation and, conversely, that the non- serial organisation of musical parameters is merely a reflection of the composer’s intuitive choice. By examining how various musical aspects function and interact in Structures I, this analysis hopes to demonstrate what it means to develop an appropriate serial structure, not only as a significant compositional principle but as one that is also aurally meaningful. This analysis is in two parts, the first of which examines the role of the density variable in relation to four serialised parameters as well as other non-serial aspects. The second part provides the way in which the overall structure is designed based on proportional analysis.

Understanding serial structure

Whereas the majority of detailed analyses of Structures 1a have followed Ligeti’s model of analysis by starting with the series for four parameters, the matrices, serial operation and then non-serial elements, my analysis here will approach the analysis of the serial structure from a different angle by considering firstly the issue of density. As a significant component of the work’s organisation, the density levels were determined through a process separate from the serial operation. The density variable can be seen to have had a deep impact on almost every facet of the composition. Beyond the organisation of the four serialised parameters, the density variable determines the overall shape and, in fact, the most aurally discernable aspects of the piece. In other words, the function of the density variable had far-reaching consequences in the organisation of both serialised and non-serialised parameters as well as on the work’s global architecture. Although not directly referring to Structures Ia, the following quotation explains how Boulez saw the role of density in composition:

Certainly, in a musical world where all notion of symmetry is tending to disappear, and a concept of variable density is assuming a more and more basic role on every level of composition—from the material up to the structure—it is logical to look for a form which is not fixed, an evolving form which rebels against its own repetition; in short, a relative formal virtuality. (1991, p. 29)51

The following analysis demonstrates how the density variable can be seen to play a central role in the serial structure of Structures 1a.

Density

In Structures Ia, the level of density is determined by the number of threads employed in a given temporal span. As outlined above, the term “thread” was first coined by Ligeti and refers to a line produced when all four serialised parameters are combined; therefore each thread consists of twelve pitches and twelve durations with one level of dynamic and attack. Forty-eight threads are used throughout the entire piece and there is no duplication among them in terms of the arrangement of pitch and durational

51 This article was originally published in “Aléa” in La Nouvelle Revue Française, no. 59, (Nov, 1957), 839–857. 94

value. While the order of 12 pitch-classes and the order of 12 durational values differ in each thread, repetitions of dynamics and attacks occur because only ten types of these parameters are used. In each section Boulez created a different density level through superimposing up to six threads (Ligeti, 1960, pp. 49–51). However, the density variable should be understood beyond merely counting the number of threads superimposed within a given temporal span. It influences a range of musical parameters in the compositional process whether they are serially organised or determined by the composer’s personal choices. None of the forty-eight threads used in this piece are identical; each thread has its unique musical features. Therefore, superimposing different threads not only produces various degrees of density levels but also results in different sonic outcomes. My analysis argues that the musical balance and contrast throughout the piece was achieved by deliberately manipulating these sonic possibilities. Ligeti was the first to examine the way Boulez arranged these forty-eight threads to make up 14 different sections and subsequent analyses followed his approach. According to Ligeti, 14 sections are further divided into Part 1 (eight sections) and Part 2 (six sections). These divisions are directly linked to the serial organisation as the series of permutations used in each part are derived either from the Original or Inversion matrices; Boulez allocated 24 threads into each part of the piece. Although, as an abstract structure, 48 threads are distributed across 14 sections, the work is mostly recognised as having 11 formal sections because these are articulated by non-serial elements such as tempo changes and with fermatas between them. As a result, three of the 14 sections are combined as one (section II(a), II(b) and II(c)) and two sections as another (section IV(a) and IV(b)). In this analysis, the “abstract” formal structure of 14 sectional divisions is distinguished from the “actual” formal structure of the piece, with 11 sectional divisions. When not specified, references to the formal structure refer to the “actual” one. Figure 4.13(a) and (b) show the density variable in relation to these two types of formal structure. Over the last half century a number of different analytical approaches have attempted to explain the function of density variability in this piece. For example, how the density level influences the relationship between the pitch-class distribution into registers and the number of repeated notes has been considered, as has how the varying density level can affect the accumulated dynamic levels and the arrangement of attacks. The discussions on these analytical findings were often accompanied with 95

a form of graphic illustration but were weak in drawing convincing analytical conclusions or hypotheses. Despite the fact that the density variable overrides the entire compositional process, in my view, its role and its relationship to other parameters have not been examined in sufficient depth. This analysis will demonstrate how the density variable relates to the serial organisation and, in turn, to the work’s formal structure.

Density variability and the serial organisation of pitch

Boulez made an important decision regarding the articulation of the formal structure in relation to the density variable and the serial organisation of pitch. It can be seen most clearly in the way each formal section and sub-section begins. As Ligeti first noted, “every beginning of a section is marked by a simultaneity of attacks that has the maximum density specific to that section, and this gives a typical physiognomy to the density flow of the entire composition” (1960, p. 53). Thus, “chordal” sonorities are heard regularly throughout the entire piece. As a fermata is placed between each of the 11 formal sections—that is, with the exception of sub-sections II(b), II(c) and IV(b), each chord is preceded by a silence and, thereby, all of them are aurally discernable. Each of these chords consists of the first pitch of the assigned pitch series for that section, with 24 pitch-classes appearing in each of the two parts. Figure 4.14 shows how the composer vertically arranged the first pitch of each assigned series at the beginning of each abstract formal section. Beyond determining a specific level of density (either monad, dyad, trichord, tetrachord, pentachord, or hexachord), the composer chose specific registers and intervals for each of these. Evidently these specific sonorities were neither randomly nor automatically organised but were placed purposefully by the composer to create particular sound characteristics at the start of each section. Therefore, in this part of discussion, several emerging patterns are identified.52 By further examining Figure 4.14, it is apparent that the chords at the beginning of each abstract formal section draw attention to important pitch relationships. Firstly,

52 The semibreves used as note values in Figure 4.14 do not represent the actual durational values. Since there is hardly any timbral distinction occurring between the two pianos, no indication is given in this Figure to identify whether notes are played by piano 1 or piano 2. Though the material is distributed between two pianos, this analysis has made the assumption that they function as one instrument. None of the other analyses have considered the relationship between the two instruments to be an important issue. 96

two pitch-classes (Eb and A) are given particular emphasis. They are isolated by registral placement and by having the two pianos play the same pitch at the same time (sections I, II, VIII and IX). Eb7 is played in unison by both pianos 1 and 2 at the opening of the piece and then, in the following section, A1 is clearly prominent in the lower register as the lowest note of a trichord. When these two pitch-classes reappear in Part 2, their order is reversed. Again they are isolated by the composer’s placement of them at extreme registers. Pitch-class A0 is the lowest note (and the lowest note on the keyboard) in the opening of section VIII and Eb1 is the lowest note of the trichord which opens section IX. Thus, in the second part of the piece these two notes occur within the same low register. Among these two pitch classes, Eb is further emphasised by its structural placement: it is the last note to be heard both at the end of Part 1 and at the end of the piece, thus conveying a strong structural function. Secondly, it is important to note that the intervallic relationship between these two notes is the tritone: the one common interval between two initial pitch series (Original and Inversion). Since the interval of a tritone connects two series, it seems logical to assert its structural importance. Further examples of the tritone emerging as a prominent interval can be observed; the tritone can be clearly heard at the beginning of sections IV(a), VI and XI (see Figure 4.14). The composer used the dyad F#1–C2 to begin section IV(a). At the start of section VI, which is also the beginning of Part 2, the composer purposely exposed the interval of the tritone in the outer voices: the two lower notes are B0 and F1 and two higher ones are C#6 and G6. These intervals are clearly audible due to their placement in these registers. It cannot be accidental that Boulez chose to emphasise the beginning of sections IV(a) and VI with the interval of a tritone; the significance of these sections is discussed later. With section XI, the tritone formed by C#4–G4 is not as exposed as the ones appearing at the beginning of sections IV(a) and VIII, but the interval of a compound tritone (F2–B4) outlines the opening hexachord of XI. Thirdly, beyond that example at the beginning of XI, intervals of a compound tritone are used in several other ways. At the beginning of sections VIII, two highest notes form that interval (F#6–C8). The interval of a compound tritone that opens section VIII may be heard almost as an echo of the beginning of section IV(a) (F#1– C2). Again the composer chose to isolate the interval F#6–C8 by drawing attention to it in the top registers. The contrasting effect in this case is particularly acute since the other notes are placed in an extremely low register. (The trichord in section VIII 97

includes the highest and lowest notes on the keyboard.) Another compound tritone (F#4–C6) is formed within the hexachord that begins section III. Unlike other tritones, the interval here seems to have a specific role in weakening the interval of a perfect fifth (F#4–C#5), which appears in the middle register and is marked fff. The composer perhaps wanted to avoid drawing excessive attention to such a stable interval. Evidently, Boulez utilised these compound tritones in various manners. They are not as forceful aurally as the tritones heard at the beginning of section IV(a) and VI but their presence is still significant. As this interval of the tritone is regularly embedded in the opening chords of the sections, it may be said to resonate throughout the piece. Fourthly, while Boulez emphasised the interval of a tritone (in both pure and compound form) and strategically placed them at the openings of certain sections, other intervals occurring through these chords illustrate patterns that are worth noting. For instance, the interval of a compound perfect fifth appears as frequently as the tritone, at the beginning of sections II, III, IV(b), VI and IX (see Figure 4.15). Interestingly, these compound perfect fifths are formed across the distance of several registers and in their formations they always contain either the highest note in the chord (sections IV(b), VI and IX) or the lowest (section III). In the case of section II, the interval of a compound fifth actually outlines the trichord and it is clearly audible. It seems rather unexpected to find such a consonant and stable interval as the perfect fifth so prominent in such a composition. The consonant sound quality of the interval is often overshadowed by other dissonant intervals (such as the tritones and sevenths and seconds), but with its sense of stability and function as the second overtone of the harmonic series, it provides a resonant sonority that cannot be ignored. Here, one could speculate on two possible functions of these compound perfect fifths in terms of their sound characteristics and their registral placements. These sections (II(a), III, IV(b), VI and IX), where Boulez specifically incorporated the interval of a compound perfect fifth, have high density levels (between four to six threads): their opening chords consist of many dissonant and unstable intervals spreading across the outer registers of the pianos. Instead of sounding as two groups of notes separated by registral placement (with one group of notes in the lower registers and the other in the upper), Boulez may have wanted to make these dissonant intervals sound nonetheless as one chord. For instance, particularly in sections IV(b) and VI, where tritones and compound tritones are exposed in extreme registers, the compound perfect fifth is able in both cases to integrate them into one resonant 98

sonority. Thereby, the composer links seemingly unrelated and dissonant intervals separated by a gulf of registers with the stable interval of the perfect fifth. Since the compound perfect fifth is used instead of the perfect fifth due to the registral distance, the chance of this interval overshadowing other intervals is minimised nonetheless. The proposed function of the compound perfect fifth could be further supported by examining the opening chords that do not contain this interval. Figure 4.16 shows sections II(b), IV(a), VII, VIII, X and XI. With sections IV(a) and X, the dyads are placed in the lower registers. Contrary to sections IV(a) and X, the notes of the trichords opening sections II(b), VII and VIII are spread out across extreme registers. However, each of these trichords is unified by one specific interval: the compound minor second for section II(b), the compound major second for section VII and the compound minor third for IX. Although the registral placement might be extreme, the interval used in each trichord is able to unify the sonority. Lastly, the notes of the hexachord opening the final section (XI) are placed in the middle register thus presenting a uniquely compressed sonority—it is the only one of the chords not to involve ledger lines. In this case, the outer notes of the hexachord outline the interval of a compound tritone and, as the last section of the piece, it reminds the listener of the interval’s structural importance. Finally, Figure 4.14 is another way of illustrating the serial ordering of pitches. As mentioned previously, a number of authors have analysed this aspect. To summarise, in Part 1 of Structures Ia, twenty-four forms of pitch series are employed: the entire twelve forms of the original series for piano 1 are arranged according to the numeric sequence of I1 and the entire twelve forms of the inversion series for piano 2 are arranged according to the numeric sequence of O1. In Part 2 another twenty-four forms of the pitch series are used: the entire twelve forms of the retrograde-inversion series for piano 1 are arranged according to the numeric sequence of RI1 and the entire twelve forms of the retrograde series for piano 2 are arranged according to the numeric sequence of R1. Despite the fact that, seen in this way, the serial ordering of pitches seems mechanical and automated, this author is convinced that, due to the composer’s interference through the density variable, this was not so. The choices made by the composer in this regard resulted in an outcome that was not predetermined by the mere ordering of the rows on the matrices. A pre-compositional design is evident through the arrangement of these opening chords for each formal section, which reveals how Boulez was able to manipulate the serial organisation of 99

pitches to give particular prominence to certain pitches and intervals. While maintaining the numeric sequence derived from the matrices, Boulez has combined certain pitch series by superimposing them. Through the density variable Boulez further exploited the pitch relationship of the tritone in serial organisation. Ligeti is the first one to mention the interval of an tritone in this piece as follows:

The accumulations of tritones are produced automatically, yet they result from choices—choice of the tritone as the series’ peripheral interval, choice of the series’ simultaneity within the sections. Cleary the automatism of the serial loom can be artistically exploited, if elements and operations are well chosen.

Here, the composer’s decision to manipulate this core interval can be seen in the deeper level of the serial organisation of pitches. Figure 4.17(a) shows the relationship between the tritones, discussed above in relation to Figure 4.14, as four pairs of tritones. Eb–A and F#–C occur in both Parts 1 and 2. F–B also appears twice, in the opening and closing sections of Part 2. C#–G occurs once but is also part of the opening pentachord of Part 2. Since Eb–A and F#–C appear in both parts, they may be considered primary. (Though Eb–A never appears as a simultaneity, as argued above, the tritone relationship between these two pitch- classes receives special treatment and is of structural importance.) Moreover, each of the eight pitch-classes mentioned here is the first pitch of the series chosen for that specific section. Consequently, pitch series are regularly paired in a tritone relationship. Figure 4.17(b) categorises each pair of pitch-classes used to form the interval of tritone (in both pure and compound form) in relation to the corresponding pitch series. For instance, pitch series I9 and RO7 both begin with pitch class C while O6 and RI12 both begin with F#. Thus, in Part 1, series I9 and O6 are linked, as are series RO7 and RI12 in Part 2. Here, one could speculate that Boulez purposely planned to link particular pairs of pitch series in a tritone relationship, which are placed in structurally prominent places in the piece. As a common denominator, the interval of tritone is projected on the surface structure of the piece in various ways but, as this analysis has argued, it also fulfils a structural function in the background. The possibility of linking certain pitch series in a tritone relationship is significant in so far as it refutes the assumption of automatism

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being the central in the serial operation. Boulez himself evidently recognised the importance of linking the series, as the following quotation from 1952 suggests:

How to link series is one of the naïvest yet apparently most delicate questions in serial techniques. . . . To be able to connect series through similarity of field is infinitely more satisfying, since it allows a considerable role to harmonic ambiguity; but this presupposes series with remarkable properties, and implies a more or less continuous play on these properties. In any case it would seem to me to give more scope if one were to extend the serial functions to the actual linking of series. This would give functions of function, from the simplest line right up to the most complex derivations. And it would avoid the pitfall of technical automatism. I frankly deplore the ludicrous faith in the absolute power of arithmetic. (1991, p. 119)

The denial of automatism governing the entire compositional process could hardly be expressed more strongly.53 In summary, the framework of Structures Ia shown in Figure 4.14 consists of sonorities of vertically-aligned notes that articulate the beginning of each section and subsections. As discussed above, the opening sonority of each formal section was designed by deliberate selection of registers and interval structures. The entire serial organisation of pitches throughout the work is built on this framework while also signifying the level of density across each section. This process was completely the result of the composer’s choice. It is pertinent to point out that the composer’s decisions here were evidently guided more by musical intuition than predetermined by the matrices or indeed any form of automatism. Boulez determined the register of every pitch-class in each pitch series. As is widely known, Boulez borrowed the musical materials for his 12-pitch series from Messiaen’s Mode de valeurs; Boulez took the exact sequence that appears in the first division of Messiaen’s mode. Messiaen had predetermined the relationship between pitch class and register in Mode de valeurs and so it would have been conceivable for Boulez to organise registers within his serial system similarly. It is important to underline that he did not do this. In fact, as the following part of this analysis will

53 This statement was originally published in French in “Éventuellement . . .” in La Revue musicale in May 1952. In this article Boulez extensively discussed his own serial techniques applied in his earlier serial compositions, including Structures Ia. Though Boulez later (1975) characterised Structures Ia as automatism in a conversation with Célestin Deliège, this assertion is questionable. It is evident that, with the passing of time, he came to view his serial compositions from a different perspective and thereby gave different emphasis to the role of automatism within his serial operation. 101

demonstrate, this element of choice remains a crucial aspect in the compositional process that has been somewhat overlooked and under-appreciated in relation to this work.54 It was Ligeti who first noted that, in terms of determining the register for each of the pitch-classes, the composer ensured that no octaves are formed throughout the piece (1960, p. 56). According to Boulez, octaves create “a weakening, or hole, in a succession of sound relationships” so Boulez completely avoided them “at the risk of structural non-sense” (1971, p. 49). With the sections of lower density, the chances of forming octaves are rare, but when the density level increases in a given section, the chances of having octaves rise accordingly. In other words, since each pitch series is used in its entirety, the number of occurrences of each pitch-class increases according to the density level. In order to prevent octaves, the composer chose certain pitch- classes that would be repeated only in one designated register within a given section. Uno describes this phenomenon as follows: “When the polyphonic density is greater than four rows within a section and the literal repetition of pitches (or common pitches held between the rows within the section) is greater than five, the texture sounds increasingly ‘static’ or homogeneous” (1994, p. 99). Hence, the registral distribution of pitch-classes is inseparably linked to the different density levels.55 Figure 4.18 shows the common pitches that are shared by all the series employed in each section throughout the piece.56 (Sections II(c) and V are exceptional here in having a density level of only one thread.) In other words, these selected notes are literally repeated even when they have been derived from different pitch series within that section, thereby creating an individual sonic frame for each section. Although Figure 4.18 illustrates the aggregation of these pitches in chordal form, these sonic frames do not function as chords but as a supple background structure that produces a unique resonance for each section. The degree of registral fixation may be seen to vary in three levels: from a high number of repeated notes (that is, above 9

54 The relationship between the registral distribution of pitches and repeated notes has been examined by Ligeti, Wennerstrom, Griffiths, Uno and Grant though the extent of their discussions vary in both length and depth. The significance of this topic in relation to the surface structure has also been recognised, in particular concerning the successive repetitions of particular pitches being aurally recognisable. But how the registral distributions of pitches impact the global-structure of Structures Ia has not been investigated in any depth. 55 Uno (1994, p. 130) further recognises that it is these two procedures that largely determine the shape of the resultant surface structure, which is obviously central to a listener’s experience of the music. 56 Figure 4.18 does not include the registral distribution of all 12 pitch-classes of each section, only those that are completely fixed in their register. Griffiths (1978, p. 23) and Uno (1994, p. 98) provide graphs where the entire registral distributions of pitch series are illustrated. 102

repeated pitches in sections I, III, VIII and XI), a medium, (between 5 and 9 of them in II(b), IV(a), IV(b), VI, VII and X), to a low number (less than 5 of them in II(a) and IX). Note particularly that all 12 chromatic notes are repeated in the designated registers in sections III, VIII and XI.57 It is interesting to observe how Boulez placed these sonic frames across the overall form, contributing to a satisfying sense of architectural balance. There is a notable temporal distance between sections with high degree of repeated notes (sections I, III, VIII and XI). A possible reason for this is that the texture of the surface structure of these sections is, as Uno noted, static. Conversely, when only a few pitches are repeated in their designated registers, the sense of animation on the surface level increases. Musical flow and contrast are thereby maintained while excessively static textures of sounds are avoided across the structure. While the composer did not place two static sections immediately following each other, he often juxtaposed two sections that are significantly different in this degree of registral fixation: for example, between sections I–II(a), II(c)–III, IV(b)–V, IV(b)–V and VIII– IX.58 The repeated notes in successive sections are concentrated in different registers thereby creating a striking registral shift from one sonic frame to another. For example, Boulez underlined the contrasts between sections VI, VII and VIII by manipulating such registral shifts: the shift from the upper registers in section VI to lower registers in section VII and then the move back to upper registers in section VIII are clearly audible. Moreover, the composer preferred the middle registers for sections II(a), IX and X while no common note is held in the middle registers in section VII and IV(a). With sections II(b), IV(b) and VII, lower registers are preferred. In contrast, the composer clearly favoured the middle to upper registers in sections VI and VIII. The degree of contrast in this respect is clearly shown in Figure4.18 and suggests a satisfying musical variety and sense of shape. The examples of registral shifts demonstrate how the musical fabric has been woven by interlacing the registral lines in different ways across the entire piece.

57 Despite B4 substituting B6 at the beginning of section XI, this author still considers section XI as having its registers completely fixed. Here, Boulez purposely chose B4 as a part of the opening hexachord, deviating from the fixed register of B6 in order to create a tight and compressed sonority as pointed out earlier. 58 Uno also briefly mentions the compositional technique of juxtaposing sections with different degree of registral fixation (1994, p. 33) but does not emphasise the effect of contrast this creates between sections and across the work. 103

Mobile and immobile relationship

In summary of the above argument, an intrinsic, causal relationship exists between the density levels, organisation of the pitch series and the registral distribution of pitch classes. Through the principle of avoiding octaves, such relationships have resulted in the repeated notes that form the sonic frames illustrated in Figure 4.18. For example, in sections III and XI, six threads are superimposed with six different pitch series (the highest level of density) and the registral distributions of 12-pitch classes are entirely fixed. Beyond the superimposition of multiple series, the frequent pitch repetitions on the surface structure cause the intervallic structure of the individual pitch series to dissolve completely. As previous analyses have pointed out, such dissolution has been one of most disturbing elements of Boulez’s compositional technique because the identity of each pitch series is not recognisable aurally even when following the score. The issue is identified by Ligeti in his section on automatism (1960, p. 53). In contrast, the composer employed only pitch series in sections II(c) and V where the sequential ordering of pitches within each series is preserved in the absence of interference from other pitch series. Having only one thread means no fixation of the register is required as each pitch-class of the series is free to change register throughout the section. Overall, it is evident that, as a part of his compositional procedure, the composer devised a specific relationship between registral distribution and the pitch series. Boulez delineated their relationship as “mobile” and “immobile” in serial composition: an apt description of the interface between serial and non-serial organisation. Boulez explains the concepts of mobility and immobility as follows:

There are many kinds of interference to be set up between the series itself and the register, on the basis that either of these two elements can be mobile or immobile in relation to the other. One has only to imagine the instability arising out of the relation between an unchanging series and a continuously changing register, or between changing series and a completely fixed register: the extreme points in the play of ambiguities of pitch, which may equally combined with ambiguities of rhythms or dynamics. (1991, pp. 119–120)

The degree of mobility and immobility formed between the series and the register constantly changes section by section throughout the entire piece, depending on the degree of pitch repetition. Figure 4.19 summarises the number of common notes held among the pitch series in each section in conjunction with the density levels.

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The terms mobile and immobile used by Boulez can also be found in Cage’s writing. Cage used the identical terminologies to describe an important compositional procedure developed during the time when he was devising chance operation for Music of Changes. In this case, he categorised his charts as either mobile and immobile. Boulez was probably inspired by Cage’s letter written on 22 May, 1951 where Cage outlined the compositional techniques underlying his chance operations in great detail (Nattiez, 1993, p. 94). In reply to this letter on December 1951, Boulez wrote, “The idea that I find most interesting in all that you have explained to me is the opposition between mobility and immobility of the constitutive elements of a table” (Nattiez, 1993, 113). Evidently, Boulez was interested in such compositional techniques and it is revealing to observe how he implemented them in his own serial composition. Although Boulez and Cage both embraced the idea of changeable and unchangeable musical elements co-existing in the compositional process, there was a great difference between them in terms of actualising this idea. For Boulez, this idea was implemented by devising inseparable connections between pitch and its register, thus setting up a delicate interface between serial and non-serial elements. For Cage, it was a way of expanding the range of musical materials available for chance manipulation. With mobile charts, whether they contain materials for pitch, duration, dynamics, or tempi, he permitted himself to change and renew the existing musical materials while the content of immobile charts remained the same. In Chapter 6, the operating system of mobile and immobile charts system in Music of Changes will be discussed at some length. Returning to Structures 1a, the importance of register extends beyond distributing the pitch classes of the chosen series. As pointed out previously, different regions of register are carefully selected to provide a sonic frame in the background of each formal section. The function of register in Boulez’s composition is underlined in the following quotation from the composer:

For me, when I compose, the package is very important, not the inside. The inside of the composition as you approach it more and more, when you are listening, will begin to make sense in its details. But for me, the package is important. For instance, the register: If you have a long segment in a register which is very tight and closed, then this place will be a remembrance, not for thematic material or for dynamics, but because it is in the frame. And then the

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next one will be in another frame—very large, for example. (Di Pietro, 2001, p. 8)

The registral distribution of the pitch series plays a defining role in organising the pitch relationships perceptible on the surface structure. From the composer’s point of view, such a process can be seen as an act of transcending the intervallic arrangement of each pitch series to create what the composer referred to as the “package”.

Density variability and the serial organisation of duration

So far, the relationship between the density variable and the organisation of pitch series has been discussed. The next aspect to be considered here is the serial organisation of durational values: the aspect that is directly influenced by the density variable. The central aim in serialising durations was to ensure that the rhythmic organisation would no longer be subordinate to the pitch organisation but be of equal importance to pitch within the serial structure. In “Possibly . . .” published in 1966, Boulez said “it would doubtless suit us to bring rhythm into line with the serial structure. How might this be achieved?” Here, the composer referred to the practice of isorhythm in the motets of Machaut and Dufay where the rhythmic organisation was detached from the polyphony of pitches (1991, p. 120). As widely recognised, in order to achieve a series of permutations equal and parallel to the permutation principle applied to the pitch series, Boulez constructed a duration series consisting of twelve “chromatic” durations ranging from the demisemiquaver to the dotted crotchet. Like the organisation of the pitch series, Boulez employed 48 forms of durational series, the entire duration of each series lasting 78 demisemiquavers. Direct compatibility between the series of pitch and duration was essential in the serial operation since each pitch within each series is assigned with a particular durational value. As discussed above, the density variable influences the organisation of the pitch series in various ways. It also significantly affects the way each durational series is perceived on the surface structure. As seen in Figure 4.5, Ligeti demonstrates how, when the number of superimposed durational series increases in a given temporal span, the composite sequence of durations becomes increasingly shorter. In sections of high density, the resultant texture thereby is characterised by one of almost regular attacks. In contrast, when just one series is employed, as in sections II(a) and V, the

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identity of the durational series can be recognised as all twelve different durational values have not been affected by the superimposition of other durational series. This means that the composite sequence of durational values of each section changes across the entire piece, and the degree of the attack intervals are perceived as being more or less regular or irregular in accordance with the different density levels. However, Ligeti’s understanding of this phenomenon is explained in relation to automatism: “Variation in horizontal density (frequency of attack)—horizontal alteration of the vertical attack-density: both are now taken out of the composer’s hands” (1960, p. 53). Such a view continues from his explanation of how the individual pitch series becomes unrecognisable and seemingly unpredictable when the density level increases. Ligeti’s claim above may be questioned in so far as the density level was not part of a mechanical process but was clearly determined by the composer. Depending on the number of superimposed durational series, Boulez could, at least, approximately gauge how far the attack intervals could become regular or irregular in each section. If the density level is high, the attack intervals become more regular and if the density level is low, the opposite result can be heard on the surface structure. Thus, it is possible that the composer purposely regulated the degree between irregularity and regularity of attack intervals as a compositional feature of the work. Furthermore, the concept of regular or irregular organisation of durations in a serial piece was introduced by Boulez himself in the letter written to Cage on 30 December, 1950. Boulez explained the necessity of developing various serial techniques that could be particularly applied to the duration series, as distinct from the serial organisation of pitch. The reason, as pointed out by Boulez, is that there is a fundamental differences between the pitch and durational series:

The difference with rhythm is that (1) rhythm is not invertible, so it has 2 fewer dimensions: the inversion and the retrograde inversion; (2) it cannot be transposed homothetically onto any of its values. Thus several transformations valid for the general principle have to be found. (Nattiez, 1993, p. 88)

He included a list of transformations that could be applied to durational series, one of which was the contrast between “regular or irregular” (Nattiez, 1993, p. 88). Therefore, contrasting sections of relative regularity and irregularity may be seen to compensate for these limitations.

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The density variable determines the composite sequence of durational values in each section as it does the organisation of pitch series on the surface structure. In a high-density section, one can observe the correlation between the degree of regularity and the appearance of literal note repetitions. For example, the sound texture of sections III and XI is relatively static because all 12 chromatic notes are fixed in their assigned registers. At the same time, these sections consist of mainly short durations; therefore, the attack intervals are mostly regular. In contrast, sections with one or two threads exhibit an animated surface texture because of the registral distribution of pitches and the irregular attack intervals preserved in the durational series. As such there is some compatibility between the series of pitch and duration. Nonetheless, the order of individual pitch classes and durations within any series becomes impossible to recognise once the density level increases beyond two superimposed threads. Many of the analysts discussed above have been seriously concerned with this aspect, particularly Uno (1994) and Grant (2001), to the point of doubting the effectiveness of compositional techniques within such works of integral serialism. Ligeti was the first one to recognise and discuss the issue. Instead of acknowledging the profound role the composer played in regulating density, Ligeti interprets such an outcome as a result of automatism: mapping the four serialised parameters is achieved through a predetermined mechanism relying on the numerical sequences of matrices. Alternatively, Uno does perceive the importance of density variability and specifies how it affects the surface structure, but she is disappointed with the absence of any thematic or motivic elements that could have been derived from the series. Uno writes:

The fusion of the durational and pitch series serves to randomized the musical effect of the twelve-tone rows; it also dissolves any inherent syntactical relationships, i.e., inversional mapping of dyads, that may have been brought out compositionally, otherwise. In addition, when the polyphonic density is greater than two, the musical effects of the individual duration series are subsumed, to lesser to greater degrees, by the composite attack-time durations. Paradoxically, the attack-time durations project a pattern-oriented, deterministic facet to the rhythmic organization of piece. In short, these problems stem from the aspect of human agency—Boulez’s selection of the material and operations—and not from the mechanical process of “automating” the structure. (1994, p. 129)

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Seven years later, Grant reinforces Uno’s view by saying that “the method of pitch and duration ordering in Structures Ia is effectively irrational” (2001, p. 154). The question arising here is whether the composer ever intended to establish any sense of thematic or motivic pitch relationship on the surface structure that could be recognised as segmental invariance. Searching for invariance on the surface structure of serial compositions has been one of the common analytical approaches, especially when studying the compositions of the Second Viennese School (for example Straus (2005) and Lester (1989)). However, as noted earlier, Boulez did not see the series being an evolution from the theme in tonal music and was critical of what he perceived to be Schoenberg’s confusion between the two. He claimed that Schoenberg had been unable to imagine the worlds of sound that were available through the serial system (1991, pp. 211–212).

Density variability and the serial organisation of dynamics and attack

As the density variable influenced the organisation of pitch and duration series, it also has a bearing upon the operation of the remaining two serialised parameters— dynamics and attacks. Prior to discussing these matters it is useful to summarise the difference between the serial operations of dynamics and attacks from those of pitches and durations. Two main aspects are discernible. The first is that the rate of change for dynamics and attack is considerably slower than that of pitches and durations. For pitches and durations, new values are assigned note by note, but for dynamics and attacks, new values are given to each set of 12 notes. The other aspect is the way Boulez ordered the succession of dynamics and attacks. He no longer followed the numeric sequences of horizontal or vertical lines from the matrices, as he had done in ordering the pitch and duration series but, for ordering dynamics and attacks, he chose numeric sequences diagonally. By doing so, 10 values from each dynamic and attack series are used in this piece instead of 12, and consequently the repetition of values occurs regularly. Such difference in the serial organisation of dynamics and attacks has a combined impact on the surface structure. Grant discusses this as follows:

The significant difference to the horizontal and vertical is precisely that there are not twelve different numbers, and that each consecutive figure is not necessarily different from the previous one. It is just this decision which transforms Structures Ia from a conjunction of eleven independent sections to a piece flowing through time. (2001, p. 152)

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Grant further recognises that the average attack and dynamic level of each section has a role in defining the overall structure (2001, p. 153); sectional divisions are reinforced by changes in the various dynamic and attack levels.59 Apart from Wennerstrom and Grant, there has been relatively little analytical investigation of dynamics and attacks, and especially into how these two serialised parameters impact upon the surface structure. Such a tendency is perhaps largely due to fact that it is difficult to measure the exact quality of these two parameters and further, in practical terms, the execution of each dynamic and attack by the performer can only be approximate. Presenting a further challenge to the performer, the combinations of dynamic and attack seem to be somewhat contradictory in places; for instance, in section III, poco sforzando is coupled with quasi p and in section VII, the staccato accent is coupled with pppp.60 In addition to these challenges, traditionally, these two parameters are considered to be parameters of secondary importance, having decorative rather than structural functions. That the majority of analytical approaches have heavily focused on the pitch aspect suggests that the structural function of these two parameters have not been considered fully. However, it is doubtful that Boulez has developed serial operation to include these two parameters just for a decorative purpose. In Figure 4.20, dynamic values assigned to each abstract formal section are arranged according to the density levels. The purpose of Figure 4.20 is to illustrate the relationship between dynamics and the density levels. Starting with the highest density level, sections III and XI contrast in their predominant dynamic level, section III being dominated by loud dynamics and section XI being almost exclusively soft. The two sections that have a density level of five also display a certain degree of dynamic contrast though it considerably less than that between sections III and XI: section VI is dominated by extremely soft dynamics, more so than section IV(b). There is hardly any distinction in terms of dynamics among the sections with density level four (II(a), VIII and IX), as these sections all combine medium to soft dynamics. Sections with density level three again show clear contrast in their average dynamic level with section II(b) being exclusively loud and section VII having a mixture of

59 Wennerstrom also points out the significant role played by the arrangement of dynamics and attacks in this piece. “There are four other elements in [Structures] Ia that determine sectionalization: simultaneous density, tempo, dynamics and modes of attack. These factors are much more apparent aurally than are the series of pitch and duration” (1967, p. 49). 60 Note that when in the dynamic of quasi piano he adds the descriptor poco to the Sfz in order to match the dynamic level. Poco Sfz does in fact not appear in the initial attack series. 110

less strong dynamics. When only two threads are superimposed, that is in sections I, IV(a) and X, each thread in the section has a distinct dynamic level (while in the sections of higher density more than one thread may share the same dynamic level). Lastly, in V and II(c), a loud dynamic level (quasi p and ffff respectively) is assigned to the one thread of each section. A strong dynamic level was preferred here perhaps to avoid a sudden diminution of sound volume when the density level drops to one. When discussing the compositional technique involved in serialising dynamics, the composer himself described the functions of dynamics as being either “linear” or “complex” (Boulez, 1971, p. 61). In relation to this work, these terms can be applied to describe the relationship between dynamics and the density variable. When two threads are superimposed (sections I, IV(a) and X), the distinction between the two different dynamics is clearly discernable aurally and the function of dynamics thereby may be described as linear. However, as the density level increases, the arrangement of dynamics gradually becomes more complex: the contrast is no longer between threads but between sections. These sections of great dynamic complexity may illustrate what Boulez refers to as “blocks of dynamics” (1971, p. 61). Sections III and XI, where six threads are superimposed, can be seen as being the most extreme cases of such blocks of dynamics. As noted earlier, the registral fixation of 12 chromatic pitches in these sections results in a relatively static texture. Moreover, due to the number of superimposed durational series, the attack intervals of duration are here also most regular. In relation to these serialised aspects, these sections thereby are relatively homogeneous, and as a result, the factor that most distinguishes them is that of dynamics. Those sections with a density level of four may be seen to counter-balance the rest of the piece by only containing similar dynamics. These sections—II(a), VIII and IX—do have some dynamic complexity but there is a little distinction in terms of the dynamics between the sections as well as between the dynamic levels of the threads. In fact in this regard sections VIII and IX are identical. Similar to the relationship between dynamics and the density variable, each abstract formal section displays a different combination of attacks that relate to the number of superimposed threads. However, investigating the combined impact of various attacks on the surface structure of the work can be even more challenging than exploring the dynamics. Although their serial organisation is similar, there is a fundamental difference between two parameters. As a result, the analytical method 111

applied above to demonstrate the structural function of dynamics would not be effective in the case of attacks. For a start, unlike the other series, there is no sense of progressive arrangement in the sequence. An analytical approach to investigate the range of attacks must recognise that many of the forms of attack require that pitches are not always sustained for their full durational value. As a result, the assigned durational values are frequently notated as a semiquaver with a rest (or rests) making up the remainder of the assigned durational value. For example, a durational value of six may be notated as a semiquaver and a quaver rest instead of as a dotted quaver. The primary purpose for such a practice is to accommodate the type of attack assigned to the pitch. Among the 10 attacks, six require non-sustained sounds while four require sustained sounds (see Figure 4.21). The combinations of attacks found in each abstract formal section as a result of superimposing the threads can be categorised into three groups: sustained sounds, non-sustained sounds or a mixture of both. In Figure 4.22, the 14 abstract formal sections are sorted according to these categories, and the density level of each section is also listed to facilitate comparison. As far as attack is concerned, the sections containing similar kinds of attack, whether it required exclusively sustained or non- sustained sounds, can be heard as homogeneous. Thus, in such cases, attacks become a unifying feature. Among the six sections, that use the homogeneous attacks, section VI is most discernable aurally as it has the highest density level, yet all the notes of the section are sustained for their assigned durational value. On the contrary, section IV(a) would display the most pointillistic texture due to the absence of any sustained- notes. The other 8 sections of this work include both sustained and non-sustained attacks. Employing non-homogeneous attacks creates a contrasting effect between the series of sustained notes and that of short notes. In section X, containing two threads, the contrast between these threads is characterised by different forms of attacks, which are further enhanced by the contrasting dynamics (mp and f). With the high- density sections, especially where the entire 12 chromatic pitches are all fixed in their allocated registers, the repeated notes are distinguished by the different attacks assigned to them. For instance, Eb1 in section III is heard differently each time it gets repeated throughout the section because five types of attacks are applied to it. (Staccato is used twice but the staccato notes are coupled with two different dynamics respectively (fff and quasi p) Similarly, different attacks distinguish the remaining 11 pitches that are all repeated in the same register six times in this section. 112

Previous analysts, starting with Ligeti (1960, p. 42), have been quick to conclude that the combination of assigned dynamics and attacks in this piece is not always complementary but, in many cases, is mismatched. However, as seen in this analysis, the different combinations of attacks and dynamics subtly serve to create sonic variety when the remaining two parameters are homogeneously arranged. It is worth noting that, though Boulez borrowed Messiaen’s mode of attacks and expanded Messiaen’s mode of dynamics, Boulez created a variance in organising them. As noted earlier, the poco sforzando is called for in cases where a sforzando would have corresponded to a quasi piano dynamic even though, strictly, it is not one of the options in the original attack series. This certainly suggests that the composer was aware of the issue and willing to adjust where necessary. In summary, both dynamics and attacks are ordered according to the numeric sequence derived from two matrices as part of the entire process of integral serialism. Intrinsic characteristics of each abstract formal section are enhanced through the serial organisation of these two parameters. However, the structural function of these two parameters is only fully realised as one examines their relationship to the density variable. This analysis has illustrated the way superimposing different numbers of threads, which ultimately results in the various combinations of dynamics and attacks, has achieved musical contrast and balance across the work. These various combinations can bring out not only the distinctive feature of each thread or the repeated notes within a section but can also create an audible musical contrast between sections, further defining the shape of the overall structure. This analysis also has identified that, as the higher the density level of a section, the arrangement of pitches and durations becomes more homogeneous on the surface structure. In such cases, using the non-homogeneous combination of dynamics or attacks plays a significant role in obtaining musical balance and contrast between the sections. Boulez’s view from 1952 on serialising dynamics and attacks in Structures Ia is summarised as follows:

As for the attacks and dynamics, we need only separate them from the structures which have previously governed their elaboration and give them their autonomy, for them to participate equally in the global organization of a musical structure. (1991, p. 132)

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There is no doubt that these two parameters contribute to the enhancement of the pointillistic texture, but their contributions go far deeper than decorative purpose: they function strongly in the structural design of the work. Unlike previous approaches to Structures 1a, this analysis has underlined the significant role of the density variable in greater depth, particularly demonstrating its effect upon the four serialised parameters. It has been shown that the concept of variable density influences every level of compositional process, including both serial and non-serial aspects. Importantly, the composer’s decision to superimpose a certain number threads for each abstract formal section was not made arbitrarily but with a specific aim: the aural outcome. Thus, by varying between one to six threads in each section, the composer was able to organise the sonic possibilities to create a desired formal shape. As noted in the beginning of this analysis, my re-interpretation of the work has aimed to clarify the way in which the composer’s concept of serial structure applies to Structures 1a. The four initial series for the four primary parameters were set up in relation to the matrices as the basic musical materials but then evolved through different stages of the compositional process. Previously, analyses focussed on the serial operation and were quick to separate the serial aspects from the non-serial ones. Such approaches led to a belief that there is a discrepancy between the compositional system and the surface structure: the former mainly referring to the serial operation and the latter referring to composer’s decisions relating to the non-serial matters. The significance of the varying density levels across the piece is not evident when it is merely considered as one of the various non-serial aspects unrelated to the four serial parameters. However, as this analysis demonstrated, the composer’s deliberate varying of the density levels—with up to 6 threads being superimposed—is largely responsible for the resultant sound of the piece. It is misleading to attribute this sound structure to processes that were merely automatic. The composer’s own words below sum up the surface structure of Structures 1a described above, which suggests the role of the density variable:

The development of contemporary music shows an increasing dependence on concepts that are inherently variable, and obey evolving hierarchies. That is why we have already seen the series of twelve equal notes replaced by series of sound blocks of unequal density; metre replaced by series of durations and rhythmic blocks (whether rhythmic cells or a number of superimposed

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durations); finally dynamics and timbre no longer content with their decorative or expressive role, but, while preserving these qualities, seeking also a functional importance, which increase their power and importance. 61 (Boulez, 1991, p. 35)

Temporal structure of Structures Ia

The following analysis explores the possibility that Boulez applied an underlying proportional system to govern the overall serial structure of Structures 1a. The formal structure of the work, often described as the juxtaposition of eleven seemly-unrelated sections, can be understood in a more meaningful way that encompasses the operation of both serial and non-serial parameters and, moreover, fundamentally influences the aural experience. Previously, this analysis has demonstrated the way Boulez organised the sonic materials (threads), which are the product of the serial operation, through the density variable. It has also pointed out that this organisation was directly responsible for the resultant sonic shapes. The density level is one of the key variable factors in the serial structure shaping and organising musical materials, and the change of tempo likewise controls the linear unfolding of time. This analysis reveals that creating a sense of large-scale motion in this piece relies on two types of proportional system that counter-balance and articulate clearly defined focal points in the piece. The tempo arrangement in this piece is an important and determining element that distinguishes the actual 11 formal sections from the abstract 14 formal sections (see Figure 4.13 (a) and (b)). The distinction made earlier to describe the formal structure of this piece in two forms may seem of little significance but, as will later be argued, it is directly connected to the way Boulez designed the proportions of the temporal structure.62 This structure is comprised of units of the same durational length, that of 78 demisemiquavers, that is, the length of each thread.63 All the units are performed at one of three tempi and so, in temporal terms, they take different amounts of time. Consequently, the structure of the entire piece is based on 14 temporal units and the total length of the piece can be expressed as 14×78=1092 demisemiquavers. As outlined earlier, not only are the 14 temporal units fundamental to the serial

61 This was originally published in French “Aléa” in La Nouvelle Revue française, no. 59, (Nov. 1957), 839–857. 62 “Temporal structure” here includes the elements that relate to musical time: duration and tempo. 63 Temporal units are the same in terms of notational values but, as will be discussed, not the same in temporal length because of the different tempi that are applied. 115

operation, but also they are also the units that comprise the structure articulated by the changes of the density variable as was shown in Figure 4.13. While the composer serialised rhythm through the units of duration, he purposely avoided serialising tempo because tempo is

ultimately a rate of unfolding of the musical text, and by its nature pragmatic. It will be seen that tempo cannot therefore change every instant . . . the crucial factor in the choice of different tempo is perhaps not a strict observance of the metronome mark . . . but rather a hierarchy of tempo relationships, which comes back to a relation between speeds of musical unfolding. (Boulez, 1991, p. 132)

Boulez used three different tempi in this piece (semiquaver=120, quaver=120, and quaver=144) and they are arranged in the form of two overlapping symmetries. The three tempi used have been frequently referred to as slow, medium and fast, abbreviated respectively in Figure 4.23 as S, M and F. Ligeti and the subsequent analysts have acknowledged this and have recognised that these tempo changes contribute to the aural recognition of the 11 sections of the piece. However, any further investigation of tempo arrangement—how it might relate to other musical parameters, particularly to the overall temporal structure—has not been attempted. For instance, Uno claims that “neither the changes in tempo nor the polyphonic density seems to bear any systematic relationships with the serial plans for pitch and duration. One can assume, therefore, that the designs of tempo and polyphonic density were made on ad hoc bases” (1994, p. 89). However, similar to the relationship between the serial organisation of pitch and registral distribution, it can be seen that the tempo arrangement has been carefully designed to control the rate of musical flow and contrast. Therefore, two forms of structural organisation can be shown to co-exist in this work: one relates to the actual formal division into 11 sections and the other to the abstract divisions into 14 sections. The first of these is represented in Figure 4.23 above in relation to tempi and, as can be observed, it involves two overlapping symmetries. The other reflects the additive nature of durational series and results in a structure of 14 temporal units. In order to explain how these two significantly different temporal systems co-exist and complement each other, it is necessary to understand the overall proportional scheme in relation to the ratio of the Golden Section. Prior to examining the aspect of two temporal systems in further detail, the

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concept of the Golden Section should be briefly outlined. Roy Howat, who has applied the theory to analyse Debussy’s works, defines the concept as follows:

Recognized since ancient times as important in architecture, painting and natural organic growth, the Golden Section (Golden Mean, Golden Ratio— henceforth ‘GS’) is the way of dividing a fixed length in two so that the ratio of the shorter portion to the longer proportion equals the ratio of the longer portion to the entire length. . . . The ratio’s exact value is irrational, its decimal places continuing indefinitely; it approximates to 0.618034. (1983, pp. 1–2)

The author further clarifies an important and unique property of the Golden Section ratio in reference to Figure 4.24. He writes:

C divides the line AB by GS; D is then added to divide AC by GS. But in doing so, D also divides the whole length AB by GS in the other direction, the shorter portion lying to the left. No other ratio has this property. (1983, p. 2)

Here, C is called the “primary” GS point and D is called the “secondary” GS point.64 By continually dividing the length by the GS ratio in both directions, the divisions could be endlessly interlocked while the ratio of each division remains identical.65 The following analysis demonstrates how this mathematical phenomenon of the Golden Section can be applied to Structures Ia. To do this, a minimum standard unit of measurement must be determined: this is a demisemiquaver (DSQ) that is, the smallest value used in the initial series for duration. The total length of the piece (1092 demisemiquavers) can then be divided according to the GS ratio. Figure 4.25 is the iteration table developed to illustrate the calculations clearly. The shaded box is the starting point for the calculation and the GS ratio is applied to value in the “DSQ count” column, which is then carried through from the “Result” column in the previous row. It is interesting to note that while the duration of 1092 DSQ is divided into the 14 abstract formal sections, taking exactly 14 iterations of applying the GS proportion for the values approximates zero, as iteration 14 shows that a figure of 2 is approximately halved (1.236068) and a value of 1 is returned. Therefore, having 14 abstract formal sections is not an arbitrary incident but rather appears to be a calculated decision of the composer. Given Boulez’s background in mathematics

64 Howat differentiates these two points of GS with these terms (1983, p. 22). In relation to this analysis, naming a GS point in two different ways is necessary to clarify the direction that the GS is applied. 65 More detailed discussion on this subject can be found in Chapter 1, “Proportional structure and the Golden Section” in Debussy in proportion: A musical analysis by R. Howat. 117

(Jameux, 1991, p. 7), it seems unlikely that this is coincidental or the result of intuition alone. Figure 4.26 illustrates the application of the GS ratio to the 14 abstract formal sections. There are two key components used to divide the 14 sections or 1092 total demisemiquavers: primary and secondary golden sections. The GS ratio has been applied to the entire length from two directions: from the beginning working forwards and from the end point working backwards. These provide, nominally, primary and secondary GS divisions respectively. Figure 4.26 also shows the names individually assigned to each of the identified golden section lengths. GS.P1 designates the GS primary one length, that is, the first application of the GS ratio to the piece in the direction from the 0DSQ point to 1092DSQ. Similarly, GS.S1 designates the GS secondary one length, that is, the first application of the GS ratio to the piece from the opposite direction, starting from 1092DSQ. It is also important to note that the point at which GS.P2 ends, 417DSQ (417:258) is identical to the point at which GS.S1 ends. As mentioned above, this demonstrates how the golden section ratio creates a network of interlocking relationships. Moreover, Figure 4.26 indicates that the point at which GS.P1 and GS.S1 intersect is the midpoint of the piece, 546DSQ. These calculations suggest that the primary nodal point of GS in this piece is at 675DSQ (675:417) and occurs within section VI. The secondary one is at 417DSQ (417:675) and occurs within section IV(a). It is interesting to observe that these two points do not coincide with the beginning or end of the sections VI or IV(a) respectively but are close to the GS points within the sections: the primary GS of section VI is at 672DSQ and the secondary GS of section IV(a) is at 419DSQ. Since GS is an irrational number, a minor difference in value, such as two or three demisemiquavers, is of minimal significance. Additionally, the minimum standard unit of measurement in this calculation (the demisemiquaver) is very short in musical terms. The variance between the GS divisions of both primary and secondary points of the entire piece and that of the abstract formal sections may be considered negligible. This variance is the difference of two or three demisemiquavers, approximately 2.5% to 3.8% of the total DSQ count for the individual section.66 The following discussion addresses the ways in which these proportions are musically

66 Howat also points out the issue of inaccuracy in the proportional system applied to Debussy’s music; achieving absolute mathematical accuracy in musical terms is in any case not the main purpose. He argues that “the question rather is how much deviation is permissible, or how closely one can approach the exact irrational value in terms of the music’s natural flow” (1983, p. 14). 118

defined so their proportions are not just abstract relationships but also can be experienced aurally. The primary GS for the overall piece (GS.P1) is emphasised in terms of other musical means that are a direct outcome of the serial operation. Wennerstrom, Jameux, Uno, and Grant, whose analytical approaches have focused on the surface structure of the piece, and emphasised the listener’s experience, have all recognised the structural significance of section VI. Wennerstrom argues its structural importance by examining the particular correlation of six parameters: density level, tempo, dynamics, attack, pitch and duration (1967, pp. 53–54). Jameux describes section VI as “the expressive centre of the movement as a whole, and clearly be heard as such” (1991, p. 282). Uno sees the overall form as being a palindrome with section VI—with its particular arrangement of dynamics and attacks—being the palindrome’s centre (1994, p. 124). Grant recognises the distinctive sonic quality of section VI:

Section VI is the apex of the piece, the exact middle point, and there could be no greater contrast to the previous section, or indeed to the entire first half of the piece. It is characterised by quiet and sustained tones, a respite questioned by the tritone heard at the end (a tritone built into the original row, but particularly evident here because of the octave distribution). (2001, p. 153)

Interestingly, the three musical parameters mentioned in this quotation describing this very special and memorable section are all part of the serial operations with “quiet and sustained tones” referring to the arrangement of dynamics and attacks, and with “the tritone” referring to the most prominent pitch relationship. My analysis contends that the remaining parameter of duration also plays a role in underlining the proportional significance of this section. Wennerstrom, Jameux, Uno and Grant all claim that section VI is the “midpoint”—for Grant “the exact middle point” (2001, p. 153)—of the piece. While Uno and Grant do not clarify how this is calculated, Jameux and Wennerstrom offer some justification. For instance, Jameux provides a real-time axis in some of his diagrams based on the recording by the Kontarsky brothers (1991, p. 277) but one must remember that the real time value may vary from one recording to another. Evidently, another recording not by the Kontarsky brothers, to which Wennerstrom refers, suggests a different outcome. She notes in her footnote that

It is interesting that although the first half contains a greater number of thirty- second notes (8 sections to 6 sections in the second half), in performance the 119

first section takes a shorter time (1: 20 to 1:40 of the second section). This disproportion results, of course, from the two Lent sub-sections in the second half. (1967, p. 53)

Such comments however do not acknowledge that these particular pianists are evidently not performing the sections close to the metronome marks—and thus the tempi relationships—that are marked by the composer in the score. Certainly in terms of the durations, independent from tempi, section VI is not at the centre. In such terms, the exact half-way point occurs after 546DSQ, which is in fact the beginning of section V (See Figure 4.26). As the previous calculation has demonstrated, section VI does in fact lie over the primary GS point of the entire piece (GS.P1). Since performances of the piece may not accurately realise the composer’s specified tempo relationships, it is necessary to calculate the temporal proportions independent from recordings. To justify the fact that this primary GS of the piece can be also experienced in real time, the effect of the tempo changes needs to be taken into account. In order to measure an accurate real-time realisation of not only the entire piece but also of the 11 actual formal sections individually, the timing of notational units need to be calculated in terms of split-seconds.67 The basic durational unit of a demisemiquaver (DSQ) can thereby be expressed for each of the three tempi designated by the composer:

1 DSQ for slow tempo = 15.0 split seconds

1 DSQ for medium tempo = 7.5 split seconds

1 DSQ for fast tempo = 6.25 split seconds

Based on the fact that each section has a duration of 78 demisemiquavers, a further calculation can measure a completely accurate timing of sections at each of the three tempi:

Total time for a section at the slow tempo = 19.5 seconds

Total time for a section at the medium tempo = 9.75 seconds

Total time for a section at the fast tempo = 8.13 seconds

The summation of the timings for the 14 abstract sections can now be obtained: 154.38 seconds. (This can also be expressed as 9262.5 split-seconds or 2.57 minutes).

67 A split-second is one 60th of a second. 120

Figure 4.27 is a table that shows in detail the figures that confirm the above calculation. In addition, the cumulative duration is shown in the last column. These calculations enable the GS proportion to be identified in relation to timing. GS.P1 occurs at 5947.5 split-seconds (9263 split-seconds ÷ 0.6180340 = 5947.5 split- seconds), which again occurs within section VI and GS.P2 occurs at 3538 split- seconds (5947.5 split seconds ÷ 0.6180340 = 3538 split-seconds), that is, within section IV(a). Of course these precise calculations would apply only to a perfectly accurate performance and that is probably unachievable and unrealistic. However, as with all proportional relationships, a lack of mathematical exactitude does not invalidate the effectiveness of the proportional scheme (Howat, pp. 14–15). There is one other musical element that needs to be accounted for in relation to this proposed scheme. Boulez notated two types of pauses—short or long—that are used between the 11 actual formal sections. Undoubtedly, these 10 pauses will have a bearing upon the total temporal length of the piece, and thereby affect the aural experience for the listener. Obtaining an exact timing for each pause or the cumulative duration for all the pauses is practically impossible since their length is dependent on the musical judgement of performers. For this reason, it is impossible to include these pauses in the analysis of the GS application in this piece given that the durations of pauses are implicitly infinite variables. However, in the following part of my analysis, a methodology is developed to determine the effect of certain time value for both the short and long pauses upon the overall proportional scheme. The primary aim is to test whether different durations of pauses displace section VI from its position corresponding to the primary GS point of the entire piece (GS.P1). This examination is carried out in the following order:

(1) Identifying the placement of pauses in relation to section VI

(2) Obtaining the cumulative calculation of pauses for the whole piece with the inclusion of a range of assumed values for the two types of pauses

(3) Applying the GS ratio to the above total duration of the piece based on the assumed values for the pauses

(4) Examining whether section VI corresponds to the primary GS (GS.P1) when pauses of different durations are included

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(5) Considering the possible limitations of this methodology in relation to performance

Firstly, in relation to section VI, Boulez has distributed the 10 pauses in a way that is symmetrically balanced with five of them placed either side of it. The only variance is the combination of long and short pauses. The five before section VI are comprised of two long and three short pauses while those after section VI are comprised of three long and two short pauses. Secondly, a range of assumed time values for short and long pauses are applied. Three real time values are necessary to test their effects upon the proportional scheme: the entire duration of the piece, the timing of the beginning of section VI, and the timing of the end of section VI. The range of time values calculated for the short pause varies from one to five seconds and, for the long pause, varies from two to 15 seconds. Such values are deemed to be realistic when the length of the pauses is considered in relation to the durations of the sections themselves. Given that the real- time duration of the actual formal sections range from between 8.125 seconds to 24.375 seconds, it is therefore unlikely that performers would exceed the above suggested durations for two types of pauses in determining the optimum duration for each pause. One might propose that pauses that approach the duration of a complete formal section would seriously disturb the continuity and sense of balance of the work, and thereby be aesthetically undesirable. The results of these calculations are provided in Figure 4.28. Using the given time values for pauses, the table provides the total time durations of the piece, expressed in split-seconds (the resultant figures are shown in the fourth row of the table) and the timings of both the beginning and the end of section VI, similarly given in split-seconds (the resultant figures are shown in the second and third rows respectively). Thirdly, the GS ratio is applied individually to the total duration of the piece as it increases according to the increasing time values assigned for the pauses. The figures shown in last row of the table in Figure 4.28 are the timings of the primary GS (GS.P1) expressed in split-seconds. One can immediately notice that as the duration of the entire piece increases (due to the length of pauses) the timing of the primary GS also changes accordingly. Fourthly, the calculations shown in Figure 4.28 can verify whether section VI lies over the primary GS point of the entire piece when the durations of two types of

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pauses are considered. The table illustrates that, even after applying the time values of less than two seconds for short pauses and three seconds for long pauses, the primary GS point remains within section VI. Applying any value greater than two and three sections for short and long pauses respectively would result in the primary GS point shifting into the long pause immediately following section VI. This finding is also graphically demonstrated in Figure 4.29. In relation to the graph there, the GS relationship is maintained when the dark line falls between the two lighter coloured lines. Finally, the above result has some implications for performance. It is unrealistic to assume that in performance the two types of pauses will be consistent in their length or in their ratios. Figure 4.28 is not intended to suggest that this would be the case. However, the conclusion to be drawn here is that, as long as the performers take less than 2 seconds for the short pauses and less than 3 seconds for the long pauses, the GS primary point will still fall within section VI. If performers do apply pauses within these ranges, then the optimum sense of balance across the work will be achieved. Based on the calculations above, it is not inconceivable that, in arranging the two types of pauses, Boulez deliberately structured the piece to accommodate some variance in the length of pauses. However, as the calculations demonstrate, within a certain range of variance, the overall arrangement of pauses allows for section VI to be heard as part of a proportional design based on the GS ratio and thereby retain an aesthetically pleasing proportional balance. The 10 pauses bear a significant function in this work: they enhance the aural recognition of the 11 actual formal sections, yet do not compromise the effectiveness of the proportional structure. Through the density variable, Boulez arranged the sonic materials (threads) of Structures 1a in a way that the primary GS and the secondary GS of the entire piece are emphasised on multiple levels. The distinctive musical qualities of section VI have been recognised by four authors previously cited in this analysis. The other section of significance in the proportional structure is section IV(a), as the secondary GS falls within it. However, unlike with section VI, previous authors have not noted any distinctive musical characteristics in this section, perhaps because it does not stand out from the other sections as obviously as section VI does. Nonetheless, this analysis contends that the sonic qualities of this section IV(a) are significant in the sense that they were designed to complement the impending musical events in section 123

VI. Figure 4.30 underlines these complementary features of sections VI(a) and VI to illustrate not only how these two sections are positioned proportionally but also how they are contrasted by musical means. As pointed out previously, one of the distinctive characteristics shared by both sections is the combined impact of dynamics and attacks on the surface structure. In section IV(a), the detached tenuto is coupled with ff and sforzando is coupled with quasi f. The attacks used in section IV(a) are exclusively non-sustained and two relatively close dynamic levels are assigned. In section VI, attacks and dynamics are paired as follows: accent with pp, legato with ppp, and detached but slurred attacks with pppp, pp and mf. Contrary to section IV(a), the types of attacks used in section VI are exclusively sustained and the dynamics are relatively homogeneous, being primarily soft. Whether the dynamics are loud or soft and whether the attacks are non-sustained or sustained, the combination of both attacks and dynamics is homogeneous in these two sections. It is worth remembering here that Wennerstrom, Eckart-Bäcker, Uno and Grant all agree that, among the four serialised parameters, the arrangement of dynamics and attacks are more apparent to a listener than that of pitches and durations. It is interesting, then, that the two most significant focal points within the proportional structure are clearly marked by these two parameters. The primary and secondary GS are also emphasised by pitch relationships. The composer intentionally exposed the interval of the tritone in the opening chords of sections IV(a) and VI. As has been argued above, the tritone carries a primary structural role in this piece. This interval is particularly placed and isolated in the lower register at the beginning of IV(a) while no other pitches are played. At the beginning of section VI, the same interval is placed twice, in both the lower and higher registers. To further emphasise the structural importance of this section, Boulez maintained the aural presence of this interval through the registral distribution of pitches (see Figure 4.18). The arrangement of tempi across the work also bears a relationship to the proportional system of GS outlined above. Figure 4.31 shows how a structural connection can therefore be made between the three different tempi and the proportional system of the GS. The medium tempo, which both starts and ends the piece, is placed where two symmetries overlap (section V). This starts at 547DSQ that, in terms of notated durations, is the very centre of the proportional structure. Returning to the argument above regarding the special significance of sections IV(a) 124

and VI, the tempi assigned to them further emphasise their complementary relationship beyond those of dynamics, attacks and pitch noted above. The composer assigned two contrasting tempi: the fast tempo for section IV(a) and the slow tempo for section VI. The positions of the three types of tempi (circled in Figure 4.31) can be interpreted as the connecting points of the overall proportional scheme, although the tempi are arranged in two overlapping symmetries. As such, these tempi may be said to bear a stronger structural significance in underlining the GS relationships, and a hierarchy of tempo relationships exists within the symmetrical arrangement of tempi. In this work, two structural systems—GS and symmetry—are operating and may be seen to counter-balance each other. This recalls Howat’s concept of “structural counterpoint” (1983, p. 13) where different proportional schemes operate in relation to different parameters of the music. In relation to Debussy’s compositions, Howat describes the co-existence of GS and the symmetrical division in musical form as follows:

There is also the duality between GS and symmetrical division. It is widely recognized that GS is more characteristic of organic than of inorganic nature, its presence usually associates with growth or tension, whereas symmetry is more characteristic of inorganic forms (such as snowflakes), associated with stability. (1983, p. 22)

Boulez can be seen to have applied a similar concept in designing the multiple layers of structures that are fundamental to the global architecture of Structures 1a. On the one hand, the proportional division of the GS ratio has been shown to relate to the 14 abstract formal sections, which are inseparably linked to the entire serial operation of the four parameters. For this reason, section VI, where the primary nodal point of GS division of the entire piece falls, stands out for its distinctive sonic qualities produced through the serially organised parameters and the density variable. On the other hand, the piece can also be understood in terms of 11 actual formal sections, which are governed more by the concept of symmetry. These formal sections are underlined by the tempo changes and placement of the pauses between sections of the piece. As the tempo changes are directly responsible for how one hears the speed of musical unfolding, it is interesting to note the relationship between the arrangement of tempi and the registral fixation of repeated pitches. The stability of the symmetrical arrangement is further strengthened through the registral fixation of repeated pitches in sections where a static texture is created as a result of a high number of notes being 125

fixed in their designated registers. Figure 4.32 illustrates the relationship between the tempo structure and the static sound textures across the work. The first and last sections both employ static sound textures. The remaining sections where all 12 chromatic notes are fixed in their designated registers—sections III and VIII—are also noteworthy in this regard. These two sections coincide with the centres of both the first and second symmetrical arrangements of the tempo respectively. However, in section V, at the centre of the piece, the composer determined that only one thread would be played. Unlike the outer sections and particularly sections III or VIII, he purposefully placed an animated texture here that would enhance the arrival of the section VI. It is possible that the composer deliberately manipulated such contrasts, as section V immediately precedes the focal point of the proportional scheme based on the GS ratio. In summary, the overall design of this work is based on a temporal structure that employed two different proportional schemes: 14 temporal units that display GS proportions and 11 formal sections that are built upon an overlapping symmetrical arrangement of three tempi. Although their characteristics differ significantly, the two temporal systems do not function independently but inter-dependently. Together, they govern the compositional process of both serial and non-serial parameters in a cohesive manner. Importantly, the temporal structure of the piece is delicately balanced according to proportions that are recognised to be aesthetically pleasing. Conclusion

As reflected by the title of this piece, Boulez has developed a serial structure in which every level of compositional process was reconceptualised. This included, but also extended significantly beyond, the serial organisation of the four primary parameters. The specific nature of its serial structure is characterised most clearly by two variable factors: density and tempo. Boulez used the first factor, the density variable, to organise the musical materials derived from the serial operations and to provide a clear overall sonic shape. The composer then controlled the rate of musical unfolding through the arrangement of three tempi, which is inter-connected to another proportional scheme based on the GS ratio that governs the entire piece. This analysis has challenged the dichotomised view—automatism verses decision-making—that is commonly portrayed in relation to understanding this work. The serialisation of four parameters resulted in a radical change in sonic relationships and thereby required an

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appropriate and coherent structure to accommodate the new language. The interpretation in this chapter has provided a holistic picture of how Boulez realised this in Structures Ia.

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CHAPTER FIVE

Karlheinz Stockhausen’s Klavierstück III

You have to switch very quickly when you listen to my music and to change with the music from one character to another. . . . from the most quiet to the most excited, from most abstract to the most concrete, from quotes to newly invented moments—yes, that is the best example. (Stockhausen, as cited in Cott, 1974, p. 36)

Introduction

Klavierstück III was composed in 1952 in Paris and published as a part of the volume Klavierstücke I–IV in 1953. Although the piano piece is numbered as the third in the volume, it was the first piece of the four to be composed. According to Blumröder, the exact date of composition cannot be traced for the four pieces. However, through investigating various sources, Blumröder concludes that the composition date of Klavierstücke III and II was before 28 February, 1952, the birthday of the composer’s wife, Doris, to whom the two pieces were given as a gift. The Klavierstücke I and IV were written more or less immediately afterward, at the latest the end of June 1952 (pp. 109–110).68 In November 1953, Wolfgang Steinecke asked Stockhausen for a new work to be performed in the following year at Darmstadt; Stockhausen suggested Klavierstücke I–IV. Thus, in 1954, the first four piano pieces were premiered by Belgian pianist Marcelle Mercenier at Darmstadt (Kurtz, 1992, p. 66). The significance of the four piano pieces lies in the fact that the composer experimented with a new compositional technique with these works. Stockhausen wrote “They [Klavierstücke I–IV] demonstrate the transformation from “pointillist” structures (Piece IV) to complex, higher organised shape (Piece I); with this, so-called ‘Group Composition’” begins (1964, p. 19). 69 Maconie even suggests that Klavierstücke I–IV “in many ways may be regarded as a sketchbook for his later electronic studies” (2005, p. 118). Although the first four piano pieces were written in a relatively short time and use traditional notation, the composer explored a new

68 Blumröder provides a detailed explanation of the order of the four works composed and points out that Jonathan Harvey is in error saying Klavierstücke III and II were composed in 1952 and the others in 1953. 69 . . . zeigt sich bereits ein Übergang von ‘punktuellen’ Strukturen (Stück IV) zu komplexen, höher organisierten Gestalten (Stück I); mit ihnen beginnt die sogenannte ‘Gruppen-Komposition’ (1964, p. 19). (Translated by Christina Young and Sun-Ju Song.) 128

approach to both compositional technique and the instrument. The importance of Stockhausen’s piano pieces in the composer’s output as well as in the compositional development in the avant-garde of 1950s is perhaps best captured by Schnebel:

The Piano Pieces are examples of possibilities realized. The epochal quality of some of the pieces is manifest not only in the terseness of their construction but in just this fact, that such random samples of development encompass more evolution than a whole late work even in the days of Webern. (1960, p. 124)

Klavierstück III is the subject of this chapter. As with the two previous chapters, the first stage involves reviewing previous analyses of the piece and then re-interpreting the piece myself. Since this chapter contains the largest number of analyses to be reviewed, an unusually large proportion is devoted to detailed discussion of the previously written analyses. Review of previous analyses

Issues

Despite the short length of Klavierstück III, the work is remarkably complex. For this reason, it has been analysed from many different angles. Jerome Kohl says: “no doubt because of its small size, this piece has attracted analysts before [Blumröder], but has stubbornly resisted convincing explication” (1999, p. 216). In addition to Kohl, several other authors who have analysed the work willingly admit the challenges they faced with it. Maconie begins his discussion on the work with the following statement: “Piece III, deceptively simple in appearance, is as hard to analyse as it is to perform” (1976, p. 63). Cook also places the analysis of Klavierstück III in the chapter “Some problem pieces” in A guide to musical analysis, saying “the pieces I talk about in this chapter are all difficult to analyze satisfactorily; this is to say, it is difficult to find any unified analytical approach that show them to be coherent” (1987, p. 335). Cook’s comment suggests that the commonly applied analytical tools do not easily reveal the organisational logic of Klavierstück III. The first part of this chapter reviews the endeavours of authors who have attempted to do this.

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Overview

This case study includes a review of ten analytical studies of Klavierstück III written between 1958 and 2005: two by Maconie70, two by Griffiths and six by other authors (see Figure 5.1). Griffiths’ writing of 1995, however, is more like a short review of previous studies of the piece than a separate analysis, while Maconie’s analysis of 2005 is an extension of his earlier one. As seen in the Figure 5.1, the selected analyses are well spread over the six decades since the work’s composition: two analyses written in each decade of the 1950s, the 1960s, and the 1970s, three written in the 1990s and one in 2005. Among the ten analyses selected here, three were published in German—Stephan (1958), Schnebel (1960) and Blumröder (1993)—and only Schnebel’s article in Die Reihe is available in English translation, from 1960. Unfortunately, these analytical studies are not well known outside Germany. In the 1970s, the first two pioneering analytical studies of the work were published in English by Harvey and Maconie. In the 1990s, Lewin and Blumröder wrote their analyses; both are lengthy in discussion and exhaustive in content. Among the eight authors, Schnebel, Maconie and Blumröder had the privilege of having personal contact with Stockhausen. Since Schnebel’s article, which includes detailed study of Klavierstück III, was published in Die Reihe and Stockhausen was one of the editors for the journal, one can surmise that the author had contact with the composer. Maconie and Blumröder both had access to Stockhausen. As clear evidence, Stockhausen wrote a forward for Maconie’s The works of Karlheinz Stockhausen, opening with the following statement: “Robin Maconie leads the reader through my work like a travel guide. He offers an overall and coherent view” (Maconie, 1976, p. v). Blumröder, on the other hand, acknowledges the composer:

And therefore I am grateful to Karlheinz Stockhausen for giving me his continual support by supplying me with photocopies of earlier, unpublished written documents whose scientific evaluation he allowed without any limitations; also by permitting to sight the compositional sketches. Moreover, in our private conversation, he never grew tired of sharpening my sense for the unheard. (1993, pp. vi–vii)71

70 Since the content of the analysis is identical in the first and second editions (1976 and 1990 respectively) of Maconie’s The works of Karlheinz Stockhausen, the first edition of book is considered in this review. 71 Und so gilt mein herzlicher Dank Karlheinz Stockhausen dafür, dass er mir seine stete Unterstützung gewährte durch die Überlassung von Photokopien früher, unpubliziert gebliebener schriftlicher 130

Considering that these three authors were connected in some way to the composer, it would be interesting to observe what their approaches and interpretations have in common and also the extent to which their analyses differ from others. Over the past fifty years, many aspects of Klavierstück III have been analysed, including pitch organisation, durational value organisation, dynamic arrangement, registral distribution, melodic contour, density, tempo, formal structure, the listeners’ perspective and performance practice. Figure 5.2 shows which authors investigated these specific aspects. A central feature of this case study is the diversity of many levels. Unlike the case of Boulez’s Structures Ia, where the first analysis written by Ligeti greatly influenced others for several decades, there is no underlying agreement on how best to analyse the work. Disputes and disagreements exist at various levels, from analytical approaches to conclusions. In the process of reviewing these analyses, an emphasis is given to comparing the different views. I believe that the selected analyses of Klavierstück III in this chapter provide rich materials for one to contemplate both the objectives of the piece itself and music analysis.

Formal structure

Stephan, Schnebel, Cook and Blumröder discuss structural divisions of the piece. Figure 5.3 shows how the individual authors have interpreted musical aspects that they have taken into account in the course of determining their structural divisions. Some similarity can be seen between Stephan’s and Blumröder’s analyses as the placement of the first and last sections are identical, although Stephan interprets the piece as a three-part song form and Blumröder analyses it as a four-part structure. Schnebel and Cook both agree that the piece consists of five structural divisions, but disagree on the proportional relationship between sections. While there are certain similarities in determining the structural divisions, the ways these authors arrive at such decisions vary greatly. This is mainly due to the different musical aspects of the piece they consider. Thus, as one reviews these four authors’ analyses on the topic of formal structure, it is essential to examine the reasoning behind each author’s approach.

Dokumente, deren wissenschaftliche Auswertung er uneingeschränkt gestattete, sowie durch die Erlaubnis, Einsicht in die Kompositionsskizzen zu nehmen. Darüber hinaus wurde er nie müde, mir im privaten Gespräch die Sinne für das Unerkörte zu schärfen. (Translated by Christina Young.) 131

According to Stephan, the form of the piece is close to the design of a three-part song form. He further describes the characteristics of the three parts as follows:

The overall conception of Piano Piece III by Stockhausen is thus no longer a puzzle. In the first section we hear a gradual strengthening of the tonality of E (e); adjacent to this follows a segment centred again on B. A new segment unfolds briefly, referring to E, through to the start of the final section, again B (b) as if launching itself into yet another unknown region. (1958, p. 64)72

Stephan’s interpretation of formal structure, therefore, is based on a tonal structure that is very close to functional tonal analysis. By suggesting a tonal centre of E and B, he seems to imply a traditional tonic to dominant relationship. However, it is questionable whether anyone can really hear different sections of such a piece through their tonal centres. Blumröder is also critical of Stephan’s interpretation, remarking that his analytical approach failed to recognise the fundamentally new compositional technique and aesthetic embodied in Klavierstück III (1993, p. 141). Unlike Stephan, Blumröder does not approach the piece in terms of functional harmony, but focuses on the melodic contour. By using this approach he discerns a four-part formal structure. Blumröder introduces a concept of “the wave-like unfolding of sound-space (wellenförmige Enfaltung des Klangraums)” rooted in the opening interval of the major second ‘A–B’ (major ninth, A4–B5) (1993, p. 114). This interval is named the core-interval (das Kernintervall). According to Blumröder, the fundamental musical idea of the wave-like unfolding of sound-space is introduced in the first four bars; he considers the continuation of this idea throughout the piece as a formal process. The author interprets this phenomenon as “the style of organic growth and evolutionary process of nature” (1993, p. 113). Thus, the wave-like sound-space, whether it is in opened or closed movements (öffnende Bewegung or schlieβende Bewegung), is determined by the development of the melodic contour in relation to the highest and the lowest pitches occurring in each section. (“Open or closed sound- space” can also be explained in terms of “wide or narrow sound-space”). As seen in Figure 5.4 (retrieved from 1993, p. 114), the highest and lowest pitches are indicated in each section with the section number in the box and each is called a “corner-tone”

72 Die Gesamtkonzeption des Klavierstücks III von Stockhausen ist also nicht länger mehr zweifelhaft. Im ersten Abschnitt können wir die allmähliche Festigung der Tonalität E(e) hören; anschlieβend folgt ein um B zentrierter Abschnitt. Rückgreifend auf E entfaltet sich kurz ein neuer Teil, bis dann wieder der Schluβteil bei B(b) anknüpft, um sich endlich in zunächst unbestimmter Region zu verlieren. (The text was translated by Robin Maconie and Christina Young.) 132

(der Eckton). In Figure 5.4, one can also observe the various appearances of core- intervals, which are circled. The author considers them individually. In some cases, the core interval bears a double function, being also a corner-tone: in the third section, B5 is the highest note and in the final section both notes from the core-interval function as corner-tones. Blumröder’s analysis reveals how the registral placement of pitches and the core-interval of the major second relate to the wave-like melodic contours he sees as being central to the development of the formal shape. It is important to note that the musical aspects he considers can be identified on the surface structure of the work and could be perceived by attentive listeners. Schnebel’s analysis proposes that the piece consists of five closely related parts that are governed by the compositional principle of symmetry (see Figure 5.5). Schnebel describes the piece as follows:

In the course of the piece, which uses five different, closely related time- structures, intended symmetry is to become apparent. Thus, on the one hand, the piece is grouped around a centre, on the other it will evince a directed flow. Such dialectic presupposes that the formal course be ambiguous, and in order to achieve this, without deviating from close relationships in the time-structure, the piece receives a homogenous structural course. (1960, p. 126)

Understanding the piece as a five-part structure is central to Schnebel’s analytical approach. The examination of the individual musical elements is conducted according to the structural divisions within each part and then each part is compared. He concludes that each of the five time-structures has its own characteristics, namely “a particular number of notes per group (1–5), the emergence of particular note-intervals (1–5), different degrees of complication in duration-intervals [and] particular combinations of intensities” (1960, p. 130). Through these comparisons, the author reveals the interlocking of symmetry and asymmetry. Cook also looks at the piece as a five-part structure, observing symmetrical relationships; his analytical approach is firmly associated with the formal divisions (see Figure 5.6). Both authors initiate their analyses with discussion of formal structure, which becomes a starting point for further analytical discussions about aspects such as melodic contour, pitch distribution arrangements, durations, register, dynamics and density. Despite these apparent similarities, their structural divisions are not identical, apart from the beginning of part three (bar 8) (see Figure 5.6).

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Additionally, Cook’s argument for a five-part formal structure significantly differs from Schnebel’s. Cook’s begins his discussion of formal structure by outlining some advantages in segmenting the piece into five parts. He says “When you are faced with a piece in a style you know nothing about, the best starting point for analysis is usually to try chopping it into segments” (1987, p. 357). Cook is convinced that the segmentation of a piece can aid performance, can provide analytical convenience and can ultimately convey a meaningful formal structure. His formal structure of Stockhausen’s piece is based on the following durational scheme: 8 + (8 + 8) + 11 + 11 + 11 quavers (1987, p. 357).73 Cook then discusses the characteristics of each part in more detail. Like Blumröder, Cook observes that the melodic contour is often designed in relation to the lowest and highest notes in a section. In some cases, Cook makes further divisions for a section; his part three (bars 8–10:3), for example, consists of three subsidiary groups (1987, p. 360).74 Importantly, Cook supports his way of segmenting and analysing the piece by saying that “each segment constitutes a single, but expanded, moment of time.” (1987, p. 361) His further supports his interpretation of formal structure and his analytical approach in general by considering Stockhausen’s compositional technique developed at the time of writing Klavierstück III, called “groups”. Cook explains that “he [Stockhausen] was reacting against the totally pointillistic style of Punkte, written the year before, and instead organizing his ‘points’ of sound into ‘groups’” (1987, p. 361). Cook’s view of the compositional technique of “grouping” clearly influences the way he examines pitch relationships and arrangement of durational values. For the overall relationship amongst the five parts, Cook has considered two elements: pitch-class distributions and temporal motion. Cook’s analysis of these two aspects will be further reviewed in a later part of this chapter. Reviewing the above four authors’ interpretations of the formal structure of Klavierstück III has already revealed many different facets of the work. Evidently, these authors examined a range of musical features in order to determine the formal structure. In the process of comparing the different interpretations, it is not a matter of

73 There is an inconsistency in Cook’s discussion concerning the basic durational value. Fig. 178 (1987, p. 359) shows that each section is calculated by the number of quavers it contains, while Cook’s explanation refers to counting of semiquavers (1987, p. 357). It is most likely that the “semiquaver” explanation is a mistake. 74 Here,10:3 means the third beat of bar 10. 134

determining whose analysis is correct. Whether they determine the piece is in three-, four- or five-part form, all four authors provide supporting arguments. The aspect of this review relevant to my research is identifying what musical features the individual authors have considered in the process of analysing the piece. The diversity of formal structure interpretations alone suggests the level of ambiguity the piece presents to analysts.

Pitch organisation

Despite efforts to decode the overall pitch organisation of Klavierstück III, the piece has resisted giving one clear, satisfying answer. It has been a highly favoured subject, yet a challenging task. As Figure 5.2 illustrates, all eight authors have attempted to understand the pitch relationships in the piece. As Figure 5.7 shows, these can be divided into those who apply serial techniques and those who search for an alternative. Firstly, I discuss the non-serial methods in the analyses of Stephan, Schnebel, Cook and Griffiths. Stephan argues that the pitch relationships in Klavierstück III are clearly dissociated from Schoenberg’s 12-tone serial technique. He proves this point by demonstrating the frequent pitch repetitions occurring throughout the entire piece. As mentioned above, Stephan explains the pitch relationships in terms of functional harmony. For instance, the author describes bars 5–9 as follows: “One thus hears as the tonal events in this first segment of the middle section, the succession of Bb–F(f)– Bb–Eb, when transferred into the terminology of functional harmony: tonic— dominant—tonic—subdominant” (1958, p. 63).75 Stephan’s final conclusion is that two tonal centres, E and Bb, govern the piece (1958, p. 64). Schnebel examines the pitch organisation, an approach that illustrates how each of the five parts is individually designed and exhibits unique features. Schnebel investigates the number of notes and the series of intervals occurring in each part (see Figure 5.5). Throughout the piece he sees the pitches grouped in collections of five notes, which he terms “note-flocks”. Perhaps the translation of his analysis has not helped to clarify his points, though in its published form, his descriptions seem so brief that his intended meaning is hard to grasp fully. For example, regarding “the

75 Man hört also als tonale Ereignisse in diesem ersten Abschnitt des Mittelteils die Folge B—F(f)— B—Es(es), das heiβt, in die Terminologie der Funktionsharmonik übertragen: Tonika—Dominant— Tonika—Subdominant. (Translated by Robin Maconie and Christina Young.) 135

region of notes—five note-flocks of varying width” (1960, p. 127), he analyses with little detail and does not demonstrate its application beyond the first five notes of the piece. Schnebel only offers discussion about the first five notes of the piece, referring to the number five as a governing structural principle. He writes:

. . . this overall time-structure regulates the work’s material, not only its organisation. Thus the succession of 5 proportions runs subcutaneously through the whole piece. This series is set out in the first five notes of the piece. It consists of four notes in chromatic succession and another note a minor third lower (1960, p. 130).

Despite the detail in which he has evidently explored the piece, repeated reading and great familiarity with the piece is required to understand it. (The translation from German to English may have contributed to this difficulty.) In Cook’s view, an analysis that attempts to reveal serial ordering of pitch relationships seems to be ineffective. For instance, concerning the serial ordering derived from the first five notes of the piece, he says “in order to do this you have to invoke transformations so complicated as to make the music’s serial origins practically unintelligible” (1987, p. 354). Following this statement, Cook reviews Griffiths’ and Maconie’s analyses. He argues that these types of decoding processes are incomplete and as far as they concern the compositional procedure, are highly speculative in nature. However, his underlying objection is that they seem to be removed from more immediate musical perception. As he explains,

. . . it is possible to think that Stockhausen started with a serial scheme in mind, or one based on chromatic wedges, but that he started deviating from the scheme as the work progressed. But whether they are right or wrong in this sense, none of these analytical approaches is going to tell us much about the way the music is experienced (1987, p. 356).

As an alternative to investigating the serial technique, Cook describes the pitch-class distribution in relation to his understanding of the work’s five-part formal structure. This approach is again justified by referring to Stockhausen’s interest at the time in “group” composition. Cook explains:

And if each group, or segment, projects a single musical quality within a moment of time, then is it reasonable for the analyst to ignore temporal distribution within groups and look at the profile each segment makes when the pitches or intervals in it are totalled. (1987, p. 361)

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His analytical methodology to see pitch-class distribution is straightforward: the pitch classes occurring in each section are counted and then compared as shown in Figure 5.8. However, the conclusions he draws from them are less than convincing. He notes that:

You can see that the pitch classes that don’t appear in the first segment have all been used in each of the three middle segments, whereas only two of them

(F and Eb) appeared in the first segment; what is more, neither of these appeared in the first half of the first segment. (1987, p. 361)

As such, and as was mentioned earlier, this pitch-class distribution allows him to perceive the formal structure as symmetrical: similarity exists between segments I and V and segments II and IV, while segment III, being central, contrasts with the other four. Cook also states that a similar profile can be observed when one examines the distribution of interval classes in each segment (1987, p. 361). Examining the distribution of pitch-classes in this way has some limitations. It neither reveals a satisfying answer to why particular pitch classes are predominant in certain parts of the piece and not in others, nor does it help to clarify the underlying compositional purpose. Griffiths suggests the Fibonacci series as an alternative way to examine the pitch organisation. This governing numerical series is applied to interpret the following aspects: the number of notes used in the piece; the number of pitches; the total pitch range between the lowest to the highest pitches, and the intervals which seem to have a particular function. Griffiths’ explanation is as follows:

It is somewhat tempting to see here, alternatively, the operation of the Fibonacci series 3–5–8–13–21–34–55, which is undoubtedly important to the construction of much of Stockhausen’s later music: there are 55 notes, 34 different pitches, and a total pitch range, traversed at a stroke between the last two notes, of 50 (= 3 + 5 + 8 + 13 + 21) semitones. Furthermore, intervals of three, eight or 13 semitones often appear at significant junctures; examples include the chordal minor 3rds in bars 7, 10 and 13.76 (1981b, pp. 85–6)

It is important to note that, in this regard, Griffiths examines more than just the pitch- classes but distinguishes various other aspects of pitch organisation.

76 It is worth noting that, according to Blumröder (1993, p. 128), Stockhausen originally planned for the piece to contain 54 notes. As such, Griffiths’ suggestion that the Fibonacci series was part of the original conception is doubtful. 137

While Stephan, Schnebel, Cook and Griffiths all seek an alternative approach to serial organisation in order to examine the pitch relationships in Klavierstück III, the aim of the other four authors is to understand the pitch organisation based on serial techniques. But even among them, there are important differences. On this account, Maconie says:

Most of the arguing over serial orders arises from trying to apply rules of analysis derived from 12-tone theory, in particular the ideas, one, that the series is an intervallic sequence and two, that the rules are consistent. Neither apply [sic] here. Stockhausen is using a different code. The piece is not based on a theme, rather on a distribution of elements, so to decode it involves an understanding of wartime code-breaking, or its equivalent in J. M. Hauer’s Zwölftonspiel, itself a distributive system. (2005, p. 119)

In the following paragraphs, a further comparison is made on how the remaining four authors examine the pitch relationship of 55 notes in the piece and the central dispute is disclosed. As Figure 5.7 shows, there are two different views of serial organisation. On the one hand, Maconie’s and Blumröder’s analyses show that the pitch arrangement is based on chromatic tetrachords. On the other hand, Harvey and Lewin are convinced that various permutations of the first five pitch-classes (A, B, D, Ab, Bb) govern the pitch relationships of the entire piece. It is worth noting that the dispute over whether the piece is based on the series of chromatic tetrachords or a pentachord (0, 1, 2, 3, 6) emerged around the same time, being initiated by the publications of Harvey in 1975 and Maconie in 1976. These two divergent paths of investigating the pitch relationships still remained in the 1990s as can be seen in the analyses of Lewin and Blumröder; the former analysis shows the organisation and interrelationship of pentachords and the latter illustrates the pitch organisation based on chromatic tetrachords. Both authors provided comprehensive and lengthy analyses that are preceded by brief evaluations of other analyses. Maconie’s analysis suggests that three chromatic tetrachords are the building blocks for organising the pitch relationships in Klavierstück III. He writes:

. . . the pitch organization based on three abutting groups of four adjacent pitches: d–f, f–g sharp, and g sharp–b, an arrangement leaving c and c sharp as “free radicals”—they do not appear until measure 8, giving the sense of a change of key. As the piece progresses, the groups exchange notes and change

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their identities; the final sequence of seven pitches filling the interval of a perfect fifth g–d and, measuring from the c sharp of measure 13, forming the interval series 3 5 6 1 4 2.77 (2005, p. 119)

There are a few points here worth emphasising. Firstly, Maconie indicates a change occurring at bar 8 with the first appearance of the pitch, C#. Interestingly, bar 8 is where both Schnebel and Cook perceive the beginning of the third section. Maconie’s observations about pitch relationships are also reflected in Schnebel’s and Cook’s analyses of formal structure, although neither of these authors considers examining pitch relationships based on chromatic tetrachords. Secondly, one could expect more change as the piece progresses in terms of the three chromatic tetrachords mentioned earlier. Maconie, unfortunately, does not provide more detailed examples of how notes are exchanged but leaves this for the reader to investigate. Finally, Maconie observes that the last seven pitches have a unique intervallic relationship to each other, which are obviously a deviation from the four chromatic tetrachords. On the one hand, suggesting another type of pitch relationships seems to weaken the argument claiming the importance of the chromatic tetrachords. On the other hand, it can be taken as a positive indication that Maconie does not manipulate the music to fit into his analytical logic. In fact, he acknowledges that the last seven pitches do not neatly mould into the rest of the piece, where pitch relationships are governed by three chromatic tetrachords. Almost twenty years after the first publication of Maconie’s analysis (1976), Blumröder wrote an exhaustive analysis in which he interprets the entire pitch organisation in the piece. Prior to discussing this aspect, Blumröder points out the three dimensions of serial order applicable to other parameters of music. For instance, three dimensions even exist with the three time signatures chosen for the piece: 3/8, 4/8 and 5/8 (1993, p. 123). (This will be discussed later, as other parameters of music are examined.) According to Blumröder, “Continuing from Stockhausen’s theory on ‘music as tone-order’, the compositional technical analysis of Klavierstück III must first of all decode the homogeneous, governing principle of serial organisation valid for all dimensions of these tones” (1993, p. 123).78 Since Blumröder’s analysis of

77 Although the citation is from Maconie’s later publication, the content of the analysis is the same as his first published work (1976). 78 Anknüpfend an Stockhausens Theorie der ,Musik als Tonordnung‘, muβ die kompositionstechnische Analyse des KLAVIERSTÜCKS III zuerst das einheitliche, für alle Dimensionen der Töne gültige, 139

pitch organisation is directly related to durational division, he reinterprets the piece in units of 24 + 24 + 6 quavers. This however involves leaving out the last three quavers: he views this as a compositional deviation from the serial plan (1993, p. 124). The argument of compositional deviation is justified by the way serial organisation neatly fits into the durational scheme. Similarly, the numbers of notes are divided into units of 24 + 24 + 6 (see Figure 5.9). Therefore, the arrangement of tetrachords corresponds to the three durational units as well as to the number of notes (1993, pp. 123–124). However, it is important to underline that this pattern is only evident if the last note is not considered part of the calculation. He calls that last note a “one off”, and his justification for this is that it is the final and highest note. However, elsewhere in his discussion, for example, it is viewed as a “corner-note” or “core-interval” of special significance (1993, pp. 113–114)(see Figure 5.4). Therefore, there is an incongruity in Blumröder’s analysis: in the melodic contour and formal structure the last note is seen to carry an important function whereas, in the serial arrangement, it is not even accounted for at all. Like Maconie, Blumröder recognises bar 8, the beginning of the second durational division and the note number division, as a significant point of change in the piece. The operation of chromatic tetrachords up to bar 8 is identical in the analyses of both, but Blumröder explicitly demonstrates the pitch relationships from bar 8, providing an answer to how the chromatic tetrachords vary in the second half of the piece (see Figure 5.10). In Blumröder’s analysis, another three chromatic tetrachords are utilised in bars 8–13: Db–E, F–G#, and A–C. These three tetrachords form a chromatic scale from Db to C. The last six pitches of the piece are derived from two trichords, (G, Ab, A#) and (A, B, C), forming another chromatic scale from G to C. Again, Blumröder’s recognition of change occurring at the end of the piece is similar to Maconie’s analysis concerning the intervallic series of the final seven pitches deviating from the rest of the piece. The serial organisation of pitches varies in relation to the three durational divisions and note-number divisions suggested by Blumröder. Here, one can see that 24 notes match up with the number of notes for six chromatic tetrachords in the first two divisions. As seen in Figure 5.10, the ratios of repetitions for the chromatic tetrachords in the first seven bars are 2: 1: 3 and in bars 8–13 the ratios are 3: 2: 1.

übergeordnete Prinzip der seriellen Organisation aufschlüsseln. (Translated by Christina Young and Sun-Ju Song) 140

Following his discussion of the overall pitch organisation plan, Blumröder demonstrates how pitch organisation is manifested on the surface of the musical fabric. To aid his rather complex discussion, Blumröder gives a very cogent musical example where the pitch distributions of each chromatic tetrachord are clearly shown on the score and illustrated to readers (see Figure 5.11).79 In Figure 5.11, Blumröder also indicates deviations within the serial system: when a pitch is used twice it is shown with dotted lines and when a pitch from one tetrachord is substituted by a pitch from the other tetrachord, it is shown with a bracket. Blumröder offers an explanation with each occasion of deviation in the pitch serial system (1993, p. 128). For instance, between bars 1 and 8, deviations occur due to the substitution that occurs between the notes Bb and D. Among all the deviations marked, Blumröder accounts for the deviation found in bar 12 with pitch B5 as being very significant since it prepares the last note of the piece, which is also B and the highest note of the piece. Here, he admits that at the end, the total number of notes is not 54 but 55 because the pitch B is used twice (1993, p. 128). However, there is potential confusion for the reader, especially when comparing the two Figures (Figure 5.10 and Figure 5.11) in terms of the last durational segment (6 quavers) and last note number divisions (6 notes). Figure 5.10 neatly illustrates the correspondence between two trichords and the number of notes as well as durational divisions, whereas Figure 5.11 is not identical to what has been suggested in Figure 5.10, especially concerning the last six notes. If one is referring to Figure 5.11 the pitch collections of the last two trichords neither fully match the durational divisions nor the number of notes. Having a deviation of note B5 in bar 12 in Figure 5.11 further complicates matters. On the one hand, this is regrettable in what was otherwise a most comprehensive and convincing analysis. On the other hand, perhaps Figure 5.10 is intended to be a compositional theory, whereas the illustration that appears in Figure 5.11 is the reality of the work, including a number of deviations. As Maconie notes, this is a challenge often faced when analysing Stockhausen’s works: “Stockhausen’s music is difficult to characterize as a totality for two main reasons: one, because its terms of reference tend to be momentary, changing from work to work; and two, because the composer himself does not consistently follow his

79 It is worth quoting the entire example for two reasons. The first reason is that Blumröder’s musical illustration is precise and comprehensive. The second reason is that his analysis is not yet translated into English and as a result, it is not well known to non-German speaking readers. 141

own rules” (2005, p. 47). Finally, though these few deviations seem insignificant, in terms of knitting the musical fabric of Klavierstück III, from the composer’s perspective, they were evidently indispensable. This suggests that the desired sound was more important than the compositional theory. Here, one can observe another similarity between Maconie’s interpretation of serial technique applied in the piece and the way Blumröder demonstrates pitch organisation. Both agree that Stockhausen uses a distributive serial system rather than the pitch ordering system of Schoenberg. Maconie comments that this distributive system used in Klavierstück III may be founded on either a wartime code-breaking technique or the serial technique developed by J. M. Hauer (Zwölftonspiel)(2005, p. 119). As seen above, Blumröder’s analysis proves this point; he also explains the distributive system of serial technique used in the piece: “However, first of all the selection process itself needs to be looked at; which is left undetermined within each tetrachord group in so far as the group-tones can be picked out randomly one after the other” (1993, p.126).80 The tetrachords he identifies consist of only minor seconds but a distributive system allows other intervals to be derived from a tetrachord. Thus, three interval-classes (minor, major and minor thirds) are available. The significance of these three intervals, according to Blumröder, is consistent with the overall, serial organisation principle, which is governed by this number three. The author observes that Stockhausen maximises this compositional possibility by continuously choosing different modes (1993, p. 126). As a result, the serial technique employed by Stockhausen in this piece differs greatly from the commonly understood Schoenbergian 12-tone technique. Although Blumröder’s analysis is significantly longer and more detailed than Maconie’s, there are a number of similarities between essential parts of their analyses: outlining the choice of three tetrachords used in the first seven bars, recognising the deviation from the chromatic tetrachords at the end of the piece and identifying the type of serial technique applied to organise pitch. Moreover, both authors are well aware and have analysed many of the composer’s other pieces. As mentioned earlier, both authors had personal contact with Stockhausen. We do not know how much the composer influenced these two authors with regard to their understanding of the pitch

80 Aber zuvor muβ erst einmal das Selektionsverfahren an sich betrachtet werden, das innerhalb jeder Viertongruppe insofern indeterminiert belassen ist, als die Gruppentöne beliebig nacheinander herausgegriffen werden können. (Translated by Christina Young and Sun-Ju Song) 142

organisation of Klavierstück III. Whether or not it is a pure coincidence, the fact is that Maconie and Blumröder both approach the pitch organisation in the piece very similarly. Contrary to these claims of pitch relationships being based on the chromatic tetrachords, Harvey and Lewin suggest that the overall pitch organisation of Klavierstück III can be explained through various permutations of the pentachord (0, 1, 2, 3, 6) formed by the first five notes. However, their objectives and methodological approaches differ greatly. Briefly, perhaps the most obvious dissimilarity is their purpose for presenting an analysis of the pitch organisation. Harvey only shows how the pentachords, most of them being transposed or inverted forms of (0, 1, 2, 3, 6), are formed by certain pitch collections. This is illustrated in reference to the entire score, as seen in Figure 5.12 (1975, p. 26). This illustration appears alongside Webern’s Fünf Geistliche Lieder in order to prove Harvey’s statement: “to show how the pitch structure of this early period tends to work, and how similar it often is to the cellular thinking and contour consciousness of pre-serial Webern” (1975, p. 25). Unlike Harvey, Lewin’s aim in analysing Klavierstück III is to “develop a transformational network analysis that will organize and interrelate all (0, 1, 2, 3, 6) pentachords” in the piece (1993, p. 16). While Harvey’s analysis is very brief and includes no detailed discussion about his selection of pentachords, Lewin’s analysis is exhaustive and is accompanied by justifications of the analytical approach. He also explains how all the transposed and inverted forms of pentachords used in his analysis were selected.81 The systematic way Lewin investigates the pitch relationships in Klavierstück III is extremely abstract. His analytical methodology involves three progressive steps. Firstly, he selects various forms of pentachord (for example, the transposition, the inversion and the transposed inversion) and these pentachords are organised according to the chronological unfolding of the piece. Secondly, he extensively examines all possible relationships existing among the various forms of the pentachord. According to the relationships found among the pentachords, Lewin creates a network, which he describes as follows:

Rather than asserting a network that follows pentachord relations one at a time, according to the chronology of the piece, I shall assert instead a network that

81 Lewin even reviews Harvey’s analysis and discusses meticulously their different selections of pentachords (1993, pp. 19–20). 143

displays all the pentachord forms used and all their potentially functional interrelationship, in a very compactly organized little spatial configuration. This network cannot depict for us how the piece moves through chronological time. But that is not necessarily a methodological disadvantage, for we can view the chronological progress of the piece as a path, or a series of path segments, through the network. And that is interesting, both theoretically and analytically. This piece, in this sense, makes several “passes” through sections of its network; the beginnings and endings of the path-segments thereby acquire special functions. Furthermore, as the path-segments fill or suggest the totality of the network, they constitute one way in which the piece, articulated chronologically into its several “passes,” projects form. (1993, p. 17)

Thirdly, and surprisingly for such an abstract analysis, Lewin suggests an approach to listening that can help to identify the progression of various pentachord forms appearing throughout the piece. For this, the author re-arranges all selected forms of pentachords as “quasi-chorale”, discarding both dynamic markings and durational values (1993, p. 42); this can be seen in Figure 5.13. Griffiths comments critically on Lewin’s method:

Lewin provides a pitch-set abstract of the piece, a kind of chorale, as an exercise in teaching the ear to hear what he hears. By training ourselves in this way, we might think, we are being led to hear Lewin rather than Stockhausen: a particular photograph of a landscape that remains mute. So it may be, but the objection is invalid if it suggests we can ever come directly to the landscape itself, to the piece unfiltered by the commentaries and interpretations of others, and by the prejudices we bring ourselves. (1995, p. 75)

Lewin’s suggested approach to listening is a perfect example of a music analysis trying to dictate what the listener should recognise when he or she experiences music. It is an extreme case where analysis takes a prescriptive role, as explained by Edward T. Cone when he distinguishes two features of analysis:

Analysis, then, exists precariously between description and prescription, and it is reason for concern that the latter two are not always easy to recognize. Description is current today in the form of twelve-tone counting—necessary, no doubt, as preliminary to further investigation, but involving no musical discrimination whatsoever. Prescription, on the other hand, is obvious in the absurd irrelevancies of Werker’s analyses of Bach but is equally inherent in some of Schenker’s more dogmatic pronouncements and in those of his followers. It should be clear at this point that true analysis works through and

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for the ear. The greatest analysts (like Schenker at his best) are those with the keenest ears; their insights reveal how a piece of music should be heard, which in turn implies how it should be played. An analysis is a direction for a performance. In order to explain how a given musical event should be heard, one must show why it occurs. . . . Description, restricted to detailing what happens, fails to explain why. Prescription offers its own explanation, referring to an externally imposed scheme rather than to the actual course of music. (1962, p. 36)

Finally, Lewin makes a connection between his network system and the musical cognitive study of Jeanne Bamberger (1986), who presented an experiment on how pitch organisation can be constructed in both “figural” (narrative, blow-by-blow, temporal) and “formal” (abstract, spatial) processes. In including a rather lengthy discussion of Bamberger’s survey Lewin perhaps wanted to illustrate another case of experimentation in order to strengthen his arguments for a pitch network system (1993, p. 53). However, Bamberger’s experiment was based on tonal music, Twinkle twinkle little star, whereas the pitch relationships in Klavierstück III are remote from tonality. Lewin’s analysis is a good example of the “scientific” approach in musical analysis, and, at one stage, he apologises for using an “unscientific” narrative (1993, p. 20). Lewin strives to find coherent interrelationships among the various forms of the pentachord, which then become the foundation for designing networks that will also be meaningful and helpful to a performer (1993, p. 41). Although the network system can organise and illustrate the interrelationships of all (0, 1, 2, 3, 6) pentachords (1993, p.16) as the author claims, Lewin’s dismissal of other possible forms of pentachords in the network analysis is questionable. It could be argued that the various forms of pentachords used in the network analysis are carefully and conveniently selected for Lewin’s proposed the “transformational network theory”. Furthermore, it is clear that, in many cases, Lewin did not consider register and vertical sonority (for instance, two or three notes sounding simultaneously) when forming the pentachord. The way pitches are selected to form various pentachords is debatable: P6 and p6 are spread across two bars (5–7), P9 and P8 across three bars (8–10), and P1 across three bars (9–11). This is questionable, particularly when Lewin insists on hearing these specific forms of pentachords when these are surrounded by other notes—not to mention dynamics and registral contrasts—that the ear is apparently to disregard. As the

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contour or other surface characteristics of the music have been neglected, one can doubt the validity of the selected forms of pentachords used in Lewin’s analysis. This does reveal a temptation in musical analysis, especially in such a challenging piece, to include those aspects that support the theory, and overlook those that do not. Furthermore, Griffiths exposes the limitation of pitch-class set analysis. He says:

For what any analysis in terms of pitch-class sets must leave out of account is the twinkling of ambiguities between clear intervals (especially thirds and sixths, sevenths and ninths) and uncertain leaps, where the sensation of pitch is attenuated by isolation or extreme register. It may also be that the notes are sometimes to be understood more as programmes for action than as pitches, their placing on the keyboard in relation to one another being important for its effect on timing and attack. (1995, p. 75)

Lewin’s analytical approach raises another question, namely, the extent to which musical analysis can be presented in an abstract form. I am not suggesting here that the abstract form of analysis is not useful. However, how far can or should we dismiss the characteristics exhibited on the surface of the musical fabric and formulate pitch relationships that can only be demonstrated in an abstracted form? In the case of Lewin’s transformational network organisation, “it does not involve dynamics, note values, register, contour, and other such features as they may organize themselves autonomously or in conjunction with P-form structuring” (Lewin, 1993, p. 41). On the one hand, omitting other features in the piece can serve as an advantage in terms of clarifying pitch relationship, as in Schenkerian analysis (Cook, 1987, p. 229), but one could argue that the case of atonal music is quite different from the tonal sub- structures that Schenker was demonstrating. It is a matter of how much omission occurs. Is pitch the only aspect that deserves such attention? On the other hand, the danger of such abstract forms of analysis is that the reader or listener can find it hard to relate to the suggested pitch organisation, especially for the majority of listeners, who lack perfect pitch. In my view, the value of the abstract form in analysis should be judged by whether it can further illuminate the listening or performing experience. Although the abstractness of Lewin’s analytical approach significantly differs from that of the others, there is an aspect of pitch relationships he discusses that coincides with other writers’ views. For instance, Lewin (1993, p. 24) states that the second half of the piece begins at bar 8 and it is initiated by P8 (E, F, F#, G, A#) and p8 (E, F, Gb, G, Db), which do not appear prior to bar 8. As discussed previously,

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both Maconie and Blumröder observe that a major change in terms of pitch organisation occurs at this point. Here, it is important to point out that though Lewin’s analytical approach differs from that taken by these two authors, they all arrive at the same conclusion: that pitch relationships change from bar 8. Moreover, in terms of formal structure, Cook, Blumröder and Schnebel interpret bar 8 as the starting point for another formal section. Therefore, despite the different analytical approaches, there is general agreement on the fact that bar 8 clearly constitutes a place of change in both formal structure and pitch relationships. Another aspect to Lewin’s analysis in fact partially confirms Maconie’s and Blumröder’s understandings of the pitch relationships based on tetrachords. This can be observed in Lewin’s discussion of the “J-related forms of pentachord”. He explains that an important pitch relationship exists between certain inverted forms of the pentachord that are labelled “J”. It involves “the unique form of the pentachord which inverts the given form and leaves invariant the four-note chromatic tetrachordal subset” (1993, p. 26). These J-related forms of the pentachord and their tetrachordal subsets are summarised in Figure 5.14. One can observe that in the J-related forms of pentachord, a particular set of chromatic tetrachords is preserved. In Lewin’s transformational network analysis, the J relationships are considered an important part of the network system (as seen in Example 2.5, p 34). For instance, Lewin comments that “the ‘moves’ J0, T6, and J6 are specially characteristic of complex-formation” (1993, p. 35). What is really fascinating about these J-related pentachords is that two chromatic tetrachordal subsets in Lewin’s analysis (Ab, A, Bb, B and D, Eb, E, F) actually match chromatic tetrachords found in the analyses of Blumröder and Maconie. Therefore, the discovery of J-related forms of the pentachord in Lewin’s analysis reinforces how vital the chromatic tetrachord relationships are in Klavierstück III. In summary, one word to describe the main features for reviewing the analyses of eight authors would be diversity. All eight authors provide different types of analyses varying not only in length, but in their approaches and priorities. The range of approaches found in this part of the review is rather extreme in this case, from Stephan’s functional tonal analysis to Lewin’s pitch-class set analysis. In order to understand the pitch relationships of the piece, four authors examine it in terms of serial organisation and the other four proposed alternative methods. Amongst the four authors, Schnebel and Cook have some similarities in that they both examine the 147

distribution of pitch classes in relation to the five-section formal structure. Those authors who attempt to understand the pitch relationships based on serial technique have a strong tendency towards finding a theory that accounts for everything: the most extreme case being Lewin’s analysis. Amongst the four authors who approach the piece serially, the analyses of Maconie and Blumröder share certain degrees of similarities despite Blumröder’s analysis being longer and more in-depth. Another common feature found with analyses that attempt to discover the serial organisation of pitch is that they tend to examine only the parameter of pitch. One exception is Blumröder’s analysis, where serial organisation of pitch is investigated in relation to the durational divisions. Finding a satisfying analytical approach that is capable of revealing coherent pitch relationships seems to have been almost unachievable for many decades. Perhaps the reason for this lies in the way the piece is composed. One of the causes for such analytical diversity is that the piece itself is profoundly ambiguous. Maconie and Blumröder suggest that, as the piece progresses, Stockhausen may have occasionally diverted, consciously or intuitively, from a pre-compositional plan. It would seem, therefore, that these types of subtle changes have deeply challenged analysts who are neither used to nor expect such inconsistency. In other words, Stockhausen’s concept of musical coherence might not equate with the unity that many analysts have pursued under the influence of the ideology of organicism.

Arrangement of durational values

In comparison with their interest in pitch organisation, the chosen authors show less interest in investigating the organisation of durational values. Only five of them— Schnebel, Maconie, Griffiths, Cook and Blumröder—discuss this aspect and, in most cases, the discussion is considerably shorter than that of pitch organisation. Griffiths in fact merely reiterates Maconie’s analysis in this regard. Again, one can see diversity in analytical approaches. How these authors examine the arrangement of durations in Klavierstück III is now reviewed. Schnebel identifies the range of durational values selected for the entire piece: “0.2 quavers–3 quavers” (1960, p. 126). Since Schnebel’s analytical approach to Klavierstück III is centred on interpreting the “five different time-structures” of the piece, he demonstrates the selection of different durational values for each “time-

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structure” and their distribution (see Figure 5.5). He also examines the duration proportion throughout the entire piece; for example, the proportions of the first four notes are 1: 1: 1: 1. This is correct in so far as the notated individual durations but, by excluding the rests, does not represent the attack intervals between the notes. The purpose of Schnebel’s investigation of the duration is to show how the heterogeneous groups, which Schnebel names as the different time-structures, are juxtaposed. He also notices the symmetry in the way certain durations are grouped: for instance, in the first time-structure (bars 1–2), he says there is “[s]ymmetry emphasised by correspondence at beginning and end” (1960, p. 129). Although Schnebel makes some interesting observations, his analysis is descriptive and generally does not propose why the durational values are organised in these particular ways. Maconie’s analysis of the durations is based on rhythmic cells (see Figure 5.15). He explains:

Essentially the piece is made up of variants and superimpositions of an initial sequence of six values expressed as two groups of three, the first group of three consisting of a longer note followed by two equal shorter values; the second group forming an unbroken succession of which the first is short, the second long, the third of medium duration. (2005, p. 119)

As seen in Figure 5.15 A, each duration of the rhythmic cell is numbered from 1 to 6 with subdivisions (1, 2, 3 / 4, 5, 6). In this analysis, Maconie discusses the various permutations of durational order. For example, he notes that the order of durations is changed in bar 2 from the initial ordering (see Figure 5.15 B). In his later publication, Maconie illustrates the permutations of the rhythmic cell in pictorial form up to bar 8, but does not include the rest of the piece (see Figure 5.15 C). However, the author’s earlier publication, The works of Karlheinz Stockhausen (1976) further explores how rhythmic cells are varied and applied from bar 8 to the end. Here, one can notice that the degree of variation of rhythmic cells increases from bar 8 as the order of durations continually changes from the initial ordering of the rhythmic cell. Though Maconie does not draw attention to it, this phenomenon resembles how the pitch organisation also changes from this point. While Griffiths provides an alternative approach to understanding the pitch organisation from the analyses by Maconie, Schnebel and Harvey by suggesting Fibonacci series as a basis, Griffiths says, “however this may be, the rhythmic structure lies relatively open to investigation” (1981b, p. 86). It seems that Griffiths is 149

reasonably convinced by the fact that the arrangement of durations is based on a rhythmic cell, as he summarises the essence of Maconie’s analysis. Nonetheless, he does suggest that the temporal ordering of the piece may be even more important than the pitch organisation in so far as it is “imparting a new way of feeling time in music” (1981b, p. 86). Cook’s approach to understanding the temporal aspect of the piece differs from that of the previous authors. He does not think it is necessary to examine the relationships between individual durational values because the composer has moved from arranging the sound materials as “points” to arranging them as “groups” (1987, p. 361). He says that

. . . if each group, or segment, projects a single musical quality within a single moment of time, then is reasonable for the analyst to ignore temporal distribution within groups and look at the profile each segment makes when the pitches or intervals in it are totalled. (1987, p. 361)

In order to explain the temporal aspect of the composition, Cook introduces a concept of upbeat and downbeat relationships amongst the five parts with an accompanying diagram that borrows Cooper-Meyer’s rhythmic symbols (see Figure 5.16). “Overall, the piece is directed towards its ending point; segments III and IV are upbeats to the final segment, while at the higher level all these three segments constitute a single downbeat in relation to which the first two segments function as upbeats” (Cook, 1987, p. 361). Cook is convinced that this graph can help to clarify one’s experience of the piece as well as the performer’s experience, and claims that “it demystifies the music” (1987, p. 361). However, Lewin criticises Cook on this matter and entirely dismisses the concept of an upbeat and a downbeat as a means of understanding the piece.82 Blumröder’s discussion regarding the arrangement of durational values seems rather incomplete, especially in comparison with his in-depth study of the pitch

82 Indeed, he writes that “Is it a simple matter to bring a Cooper-Meyer sort of rhythmic theory into the context of Klavierstück III, and then to claim that this usage is descriptive, not speculative? Cook asserts of his breve/macron graph that it ‘clarifies something about the way this piece is experienced’— presumably something beyond his earlier verbal intuitions of expectation or implication. . . . I can see the sense of his asserting that his ideas about expectation can help an intimidated student feel more at ease with the music. But I cannot see how the abrupt incursion of Cooper-Meyer rhythmic symbols into this context demystifies anything. Quite the contrary, the symbols confuse me. Why do we need them? Why is their code better than other codes against which Cook immediately inveighs?” (1993, p. 61). Lewin’s review of Cook’s analysis can be found in “Making and using a pcset network for Stockhausen’s Klavierstück III” (1993, pp. 53–67). 150

organisation. As mentioned earlier, Blumröder believes the musical materials are ordered in three-dimensional arrangements. Thus, firstly he divides the piece into three durational segments that correspond to 24 + 24 + 6 quavers, leaving out the last bar. As discussed previously, these three segments are closely related to the serial organisation of pitch. Additionally, the author carefully examines the way triplets and quintuplets are subdivided, which he calls “separating-proportion” (Unterteilungsproportion)(1993, p. 133). He classifies each triplet and quintuplet into three numbered categories: (1) the separating-proportion being either 1: 2 or 2: 1; (2) the separating-proportion being either 2: 3 or 3: 2; and (3) the separating-proportion being either 1: 4 or 4: 1. Once triplets and quintuplets are categorised, the order of these is observed to find serial permutations used in the piece (see Figure 5.17). However, one can immediately observe in Figure 5.17 that the content of bar 8 is different from the others because it has neither a triplet nor quintuplet. The way Blumröder re-interprets the first two beats of the bar is rather doubtful. He justifies this anomaly by counting up the potential duration of the grace note (see Figure 5.18). Therefore, according to Blumröder, there is a serial organisation of the separating- proportion found among triplets and quintuplets, with the permutations being 1 1 1, 1 1 2, 1 1 3, 1 2 2 and 1 3 3 (1993, p. 134). (As evident in Figure 5.17, these appear in the order 1 1 3, 1 1 1, 1 (2) 1, 1 3 3 and 2 1 2.) Concerning the arrangement of duration, Blumröder’s analysis leaves an impression of incompleteness because the author is highly selective in treating only triplets and quintuplets. Overall, while analysts have strongly tended to understand the relationships of pitch in terms of serial organisation, no one has demonstrated convincingly how the entire organisation of durations might have been organised in such serial terms. One of the most comprehensive analyses of the durational arrangement is Maconie’s, with its theory of rhythmic cells.

Arrangement of dynamics

Four authors discuss the dynamics in Klavierstück III: Schnebel, Maconie, Cook and Blumröder. As with the analyses of other parameters of music, both the analyses of Schnebel and Cook are descriptive; furthermore, their explanation of the arrangement of dynamics in the piece supports the five-part formal structure (as mentioned earlier, Schnebel and Cook do not agree on the analysis of the five-part divisions except at the

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beginning of the third section, bar 8). Although Maconie’s discussion is rather short, he clearly explains the function of dynamics in the piece. Of the four authors, Blumröder provides the more comprehensive analysis. His analysis is abstract in its nature, as he attempts to reveal the series of permutations of dynamics in the piece. Unlike when they examine the relationships of pitch or duration, when authors investigate the arrangement of dynamics, they do so in relation to other parameters of music, such as register, duration, density and melodic contour. The following paragraphs review these authors’ endeavours. Schnebel observes the selection of various dynamics within each section and how they are distributed; he accounts for a number of different dynamic levels used in each section and then describes the features of their arrangements. For instance, he sees that the dynamic arrangements of the first section (bars 1–2) and the last section (bars 14–17) both exhibit a contrast of the soft and louder dynamics, which then rises to the louder dynamic, whereas the middle section (bars 8–11.2) is characterised by the constant alternations of various dynamics (see Figure 5.5). Through Schnebel’s analysis of dynamic arrangement, the symmetrical design of the formal structure (that is, the resemblance between sections I and V and II and IV) is again emphasised. One of the central arguments of Schnebel’s analysis is that the organisational scheme is expressed in five dimensions: the five different time-structures and the first five notes being a structural determination for pitch organisation. Following this logic, Schnebel re-interprets the number of dynamic levels used in the piece as five not four (1960, p. 127). Although Stockhausen only employs four dynamic levels (p, mf, f and ff), the author counts the dynamic level of “space-sound”, meaning silence between the notes (1960, p. 127), as the fifth level, which then conveniently corresponds with his argument. Maconie’s understanding of the role of dynamics relates to the durations. He writes:

The dynamics of Piece III are not intended to delineate melodic or contrapuntal structures, but do have an effect on the apparent durations of individual notes. For instance, in bar 5 the dynamics are graded according to note-duration, whereas in bars 3–4 the reverse in the case, the longest note of the group being the softest, and the shortest, the loudest. (1976, p. 65)

Thus, the author recognises the function of dynamics rather than searching for the governing systematic organisation for the entire piece. If one takes Maconie’s view, 152

the dynamics support the arrangement of durations. The only problem with his argument, however, is that the neat relationship that appears between the length of durations and dynamics in bars 3–5 does not occur in the same manner in the rest of the piece. One can speculate on two possibilities in this case. One is that different types of relationships between dynamics and durations exist in the rest of the piece, and the other is that the unique relationship between dynamics and duration discussed by Maconie only occurs in bars 3–5. The relationship between dynamics and durations needs further investigation and I am convinced that this relationship changes as the piece progresses. This is considered again at a later stage of this chapter. Cook describes the main features of dynamic arrangement as it appeared in each section of the five-part formal structures. For instance, the dynamics and registral distribution of the first section (bars 1–2:4) exhibits an arch-like shape, the middle part being both loudest in dynamics and highest in pitch range. In the second section (bars 2:5–7), however, dynamics are arranged in an inverted arch, as the middle part has predominantly soft dynamics and there is no similar relation between the shapes of the register and of the dynamics. Cook doesn’t mention the dynamics in the third section (bars 8–10:3), and, with the fourth section (bars 10:4–12), he perceives no clear shape in either registral distribution or dynamics. In the final section (bars 13– 16), Cook emphasises the contrast between p and ff, which are assigned to the last two notes of the piece respectively, but he questions how a proper fortissimo can be executed at such a high register. Cook seems most interested in the shape of the surface structure, which is supported by various arrangements of dynamics. This also explains why the author discusses the dynamics in relation to the registral distribution. As he did with the other parameters, Blumröder applies the three-dimensional serial principle to the dynamics. The only problem with this approach is that Stockhausen used four dynamics in the piece and not three. In the serial organisation of the dynamics, the author excludes ff. This is justified by the fact that ff appears only once and is assigned to the last note, which as the reader will recall was not counted in the serial organisation of pitches. The last note is thus considered as a compositional deviation in several respects. Blumröder believes that the last note was freely added to the serial organisation in order to achieve an extreme effect. It is accentuated in every possible aspect (1993, p. 132).

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Blumröder numbers the three dynamic levels: p (1), mf (2) and f (3) and indicates the order of dynamics appearing in the piece on the score (see Figure 5.19). He also transfers the order of dynamics into permutations: six sequences comprising three rows of three dynamic values, which are then further divided into a group of two sequences. The two sequences are grouped under the assumption that the row structure of six permutations of the three dynamics is identical (see Figure 5.20). The author considers this the hypothetical study of serial order. In Figure 5.20, two slightly different sets of permutations are shown: one for the real order and the other for the possible planned order. As with the discussion of serial organisation of the pitch, Blumröder also points out the difference between the possibly planned order and the actual order occurring on the score. He explains the potential reason beneath each deviation (each deviation is circled)(1993, p. 132). The analytical study of dynamic arrangements in Klavierstück III can be summarised as follows. Firstly, both Maconie and Cook observe the dynamics in relation to other musical features. While Maconie points out the interrelationship between dynamics and durational values, Cook constantly discusses the shape of dynamic arrangement in relation to the melodic contour (the registral placement of pitches). Although neither analyst proves these relationships to be the governing principles for the entire piece, they demonstrate clearly the relationship between dynamics and the other two parameters: durational values and registral placement of pitches. This phenomenon resembles the treatment of dynamics in Messiaen’s pre- compositionally designed mode for Mode de valeurs. With Messiaen’s work, the arrangement of dynamics across the mode was inseparably connected to the register. Furthermore, the arrangement of durational values for each division of the mode has a connection to the regions of register assigned to each division. Secondly, Schnebel and Cook investigate the dynamics in relation to the five- part formal structure. Both are interested in the distributional arrangement of dynamics in each structural section, but Schnebel is very keen to prove the dynamic arrangements reinforcing the symmetrical design of the five different time-structures. Finally, it is fascinating to observe the arguments put forward by the Germans Schnebel and Blumröder. The two authors attempt to provide a comprehensive understanding of the dynamic arrangement, whether it is serial or non-serial. They are both convinced that there is a governing numerical scheme that is coherently expressed throughout various dimensions in the compositional plan. However, 154

Schnebel believes the number five governs the compositional plan, whereas Blumröder says the principle of serial organisation is governed by the number three. As a result, in Schnebel’s analysis, an extra dynamic level, “silence”, is added to the organisation and in Blumröder’s analysis, the last dynamic level ff is excluded from the serial organisation. Again, as in the works of his teacher Messiaen, in Stockhausen’s compositions certain numbers can be symbolic. For Stockhausen, these numbers are three and five. Maconie clarifies the connection between the two composers: “The number three is one of Messiaen’s symbolic numbers (five and seven are also important to him, as they are to Stockhausen)” (2005, p. 79). Whether the number three or five is the numerical principle governing the pre-compositional organisation of various parameters, the fact remains that the composer used four dynamic markings. It is another case of analysts trying to account for various musical phenomena in terms of one specific governing principle, an approach that ensures conceptual, but not always musical coherence.

Melodic contour and registral distribution

Since melodic contour and the registral distribution of pitches are closely connected, these two aspects will be discussed side by side. As seen in Figure 5.2, not only are they neglected in comparison to pitch organisation but also they are often treated as subjects unrelated to pitch. This means that in many incidences, pitch organisation is discussed without reference to registral distribution. Some justifications for this approach are given by Stephan and Lewin. Stephan is convinced that there is no system applied to registral distribution (1958, p. 62). Lewin discards the subject matter based on the belief that register and melodic contour are possibly organised themselves “autonomously or in conjunction with P-form structuring” (1993, p. 41). However, these two elements, melodic contour and the registral distribution of pitches, are essential in shaping the piece in the way that performers and listeners encounter it. As seen in the previous case study of Boulez’s Structures Ia, the registral placement of pitches has a significant impact on the surface structure of this piece. Schnebel, Cook and Blumröder discuss both aspects in their analyses of Klavierstück III while Harvey presents only a brief examination of melodic contour. Since their analytical approaches vary widely, the analyses of these authors are first reviewed individually.

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In Schnebel’s analysis, melodic contour and registral distribution of pitches are examined separately as well as concurrently. The author considers these two elements of music in several ways. A graph and figure illustrate the exact value of pitches and durations. The direction of the melody from one pitch to the next is shown with the exact value of intervals, durational proportions, dynamics, and durational lengths. For instance, Schnebel measures the range of registers for each section as seen in Figure 5.5. Elsewhere however, the author concentrates more on examining the general shape of the entire piece and the value of individual parameters is expressed only approximately. Among all the illustrations the author uses, three pictorial graphs show the melodic contour and registral placement of pitch explicitly across the whole piece (see Figure 5.21). Here, the third group resembles a Schenkerian foreground graph. The various types of graphs and figures in Schnebel’s analysis exhibit common features: in every case, they are designed according to the chronological unfolding of the piece, and indicate the five time-structures. Thus, by examining these aspects of melodic contour and registral distribution, Schnebel again demonstrates the way heterogeneously designed time-structures are juxtaposed. Despite the melodic contour of the entire piece being studied and demonstrated in several forms, in each case a substantial explanation supporting the graphic illustrations is lacking. This, among other weaknesses, hinders the clarity of Schnebel’s analysis. Harvey’s short analysis shows how Klavierstück III is similar to the “cellular thinking and contour-consciousness of pre-serial Webern” (1975, p. 25). As seen in Figure 5.12, Harvey indicates the main melodic contours with brackets and, from the same score, extracts the five notes that form various pentachords. The “size” of each melodic contour, which is made up of the succession of notes in a single direction, is measured; the size varies from three to five notes. Although Harvey does not provide a detailed discussion of the melodic contours, one can make a few observations from his musical example. His analysis seems to suggest the primary feature of the melodic contour is rising, rather than falling, since only rising melodic contours are accounted for. It is noteworthy that the author makes no connection between these melodic contours and the various forms of pentachords he identifies, even though both elements are illustrated simultaneously in one example. This emphasises the extent to which the analysis of pitch organisation is divorced from the piece’s surface structure. Cook is more intent on explaining the melodic contour and registral distribution of pitches than on searching for an organisational scheme that would demonstrate 156

coherent pitch relationships. This approach is founded on the author’s conviction that analysis should be able to exemplify what the listener experiences. To this point, Lewin mentions that “My strongest satisfaction with Cook’s exercise comes from it being rich in what I earlier called ‘phenomenological presence’. This makes Cook’s reading of the piece easy to hear without special ear training” (1993, p. 55). Cook’s analytical observations reveal that melodic contour and registral distribution of pitch are interwoven and, again, that these two elements are closely associated with the five-part formal structure. As a result, one can see that each formal section exhibits its own uniquely designed shape. Finally, Cook’s analysis here supports Schnebel’s analytical aims. The essence of Cook’s analysis of these two aspects can be summarised as follows. Firstly, he presents a figure that shows the formal structural divisions and, in this figure (see Figure 5.6), the main melodic contours, both rising and falling, are indicated. Here, it is immediately evident that the indications for the rising melodic contours in Cook’s analysis are almost identical to Harvey’s analysis. Secondly, the author observes that in the first half of the piece, the repetition of notes within the same register is generally avoided, but in the second half of the piece (from bar 8), the repetitions of pitches frequently occur at the same register. Thirdly, the author uses the terms “arch shape” or “inverted arch shape” to describe a specific section. These terminologies imply that various musical parameters are possibly arranged symmetrically, although Cook does not investigate this idea. Finally, he points out that the registral expansion happens gradually from the third section (bar 10: 4) and then reaches its maximum extension at the end of piece. The end of the piece is articulated by the largest leap, employing the lowest and highest pitches that echo the first two. Cook writes “this provides the expected conclusion to the long-term process of registral expansion. If the ending sounds conclusive, then, this is because it fulfils expectations established in the course of the piece as a whole; the music’s form is to this degree organic” (1987, p. 361). Blumröder provides a comprehensive analytical study of the melodic contour and registral distribution of pitch. While he examines the melodic contours in relation to the formal development of the piece, his analytical approach to registral distributions is based on serial organisation. As discussed previously, the melodic contours are described as “wave-like movements” (Wellenbewegung verläuft). He introduces a concept of “sound-space” that is either open or closed depending on the 157

melodic contours. The concept of wave-like development of sound-space is coupled that of a core-interval (das Kernintervall), the major second A–B. Blumröder considers that these two compositional ideas, revealed in the first four bars of the piece, are fundamental in this composition. As shown in Figure 5.4 previously, the first four bars are then followed by a four-bar unit of three subsections. Blumröder discusses the continual progression of wave-like sound-space in each section in detail. For instance, the first four bars set the “open-closed-open movement” whereas the opening of the sound-space movement in the last section (bars 13–16) remains open, as the last two notes are placed more than four octaves apart (1993, p. 114). Since the melodic contour described as wave movements of sound-space is largely influenced by the lowest and highest pitch occurring in each section, named a corner-tone (der Eckton), the author indicates them with boxed numbers. Furthermore, Blumröder points out that these corner tones move further away from each other as the piece progresses, with an exception of the last section. Blumröder notes that his description of the piece here is purposely rather neutral. He uses expressions like “line”, “curve” or progression of either “open” or “closed wave-like development of sound-pace”, which suggests a graphic and not a figurative understanding. He believes that the traditional terminologies, such as melody, voice, period, motives, phrase or chords, are inappropriate for this piece and he deliberately avoids such descriptive terms. One of the reasons for this is perhaps that he considers the perceptual experience of a listener; one hears the piece fleetingly and the movement of such rhythmically differentiated tones gives an overall impression best described in terms of tone colour (1993, pp. 115–116). Blumröder’s analysis here demonstrates a case of searching for more suitable ways of describing and communicating about a composition which, itself, does not fit into the frame of conventional analysis or musical experience. Immediately following his analysis of pitch organisation, Blumröder investigates registral distribution in terms of serial organisation. The number three again governs the permutations he suggests, since the pitches occur predominantly within three octaves. Three different regions of register are numbered: C3–B3 (1), C4– B4 (2) and C5–B5 (3). Those pitches placed outside of these three octaves are indicated with either (3+) or (1-), and are considered a compositional deviation. Blumröder suggests that the deviations occurred because the composer desired more to emphasise certain musical ideas by placing some pitches in a more extreme register 158

than to strictly follow a predetermined serial order of registers (1993, p. 130). Often these compositional deviations are justified through the formation of either open or closed sound-space. In Blumröder’s analysis, the registral distribution of individual pitches is identified and then a corresponding number is written just above each note on the score (Figure 5.22). With the same method he used in determining the serial permutation for the dynamic arrangement, the author transforms the numerical order into a possible permutation: six sequences comprising three rows of three different registers, which are then further grouped into two sequences. The reason for further dividing the six sequences here is that all the possible permutations should appear in a group of two sequences: 1 2 3, 1 3 2, 2 1 3, 2 3 1, 3 1 2 and 3 2 1. However, in determining the order, some ambiguous situations arise whenever two or three notes occurr simultaneously. As with the other serial permutations provided for other parameters, two types of permutation are given: one is the actual order and the other is the possible planned order (Figure 5.23). The differences between the two permutations presented are indicated by circled numbers. Although Blumröder’s analysis is numerical and abstract, it is simply a very rare case where the registral distribution is investigated in terms of serial organisation. In contrast to some other analyses, which are primarily concerned with pitch organisation, clearly Blumröder has considered registral distribution to be of equal significance to the other parameters. In summary, the analytical approaches applied to investigate melodic contours and registral distributions of pitches in Klavierstück III vary widely. It is interesting to note that these subjects are often discussed in relation to other musical features while the organisations of pitch or durational values are mainly analysed as self-contained and separate elements—particularly when the analyses are based on serial organisation—are investigated autonomously. For instance, in Cook’s analysis, the registral distributions of pitches are mentioned in relation to the dynamic arrangements; in Blumröder’s analysis, the development of formal shape is inseparably connected to the melodic contours that are shaped by the highest or lowest pitches of certain regions of the piece; and in Schnebel’s analysis, the note selection and the range of the registers in each time-structure are shown side by side. Finally, it is important to note that Blumröder is the only one who analyses the registral distribution of pitches in terms of serial organisation.

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Texture and density

Like melodic contours and the registral distribution of pitches, texture and density variability greatly contributes to the surface structure of the piece. As shown in the previous chapter, these two aspects were not serialised in Boulez’s Structures Ia but were vital in providing the shapes recognisable to a listener. Interestingly, with the exception of Harvey’s analysis, authors who investigate the melodic contours and registral distribution of pitches in Klavierstück III—Schnebel, Cook and Blumröder— also discuss texture and density variability. While the analyses of these aspects in Schnebel and Cook are rather brief (and again strongly associated with the five-part formal structure), Blumröder’s analysis engages thoroughly with the music and offers valuable insights. Schnebel calculates density for each of the five time-structures. “In this succession of degree of density—1.3, 0.8, 1.14, 1.0, 0.5—the extreme values are at the beginning and end. The straightforward progression from highest to lowest value is interrupted in the middle by a symmetrical formation” (1960, p. 126). Concerning the texture of the work, his description is uncomplicated; the first and last time- structures are monodic and the middle three time-structures vary from one to three parts. Schnebel believes that the significance of each of the musical elements lies in the fact that they do not rely on its pre-determined value, but rather has the power to form relationships, saying “Individual events are important as foci of the relationships. In each piece there arises a dense network of relationships, which is further intensified according to the degree of polyphony” (1960, p. 131). His point here is graphically demonstrated (see Figure 5.24). In Cook’s analysis, texture and density variability do not appear to be central but are subordinate to other features of the work. Despite these two aspects being discussed in relation to the five-part formal structure, it is not easy to grasp the overall picture of texture and density variability in the piece in his analysis. Firstly, his terminology for describing texture seems inappropriate at times. While it is acceptable to describe the beginning of the second section in general terms as “polyphonic” (bars 2:5–7), to propose that the fourth section (bars 10: 4–12) is like “a suggestion of three-part counterpoint” is problematic when the piece is so far removed from any conventional counterpoint technique. Secondly, Cook mentions “note density decline” occurring in the fourth section. Since he does not consistently measure the degree of

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density from the beginning of the piece or from the first section onward, it is hard exactly to know or measure the density decrease. Finally, the author accounts for several musical factors by asserting a sense of cadence in the last section (bars 13–16). One of them is the monodic texture, which he sees as a recapitulation of the first section (bars 1–2: 4). It could be more convincing to interpret the monodic texture both in the beginning and the end of the piece as a symmetrical relationship rather than as a cadential quality. This omission is more striking given the author discusses the symmetrical relationship among his five formal sections elsewhere in the analysis. Overall, the discussion of texture and density seems to be fragmented, a selective rather than systematic approach. Blumröder first mentions density in the formal structure when he notes that the third section (bars 9–12) is denser than other parts of the piece. He investigates density far more thoroughly than the other analysts. Prior to his discussion on this subject, he defines density in Klavierstück III as being where two or three notes sound concurrently (see Figure 5.25). He examines each case and the overall pattern of their appearances in remarkable detail. According to his analysis, there are twelve incidents distributed equally between six cases of two notes sounding together and six cases of three notes occurring simultaneously. The author further categorised these twelve incidents in terms of how two or three notes are played together (the author refers to it as “type of articulation” (Typen der Artikulation)). One is when all the notes are attacked at the same time but released differently as individual notes have different durational values (for example in bar 3) and the other is when the notes are attacked one after the other but they are all released at the same time (for example in bar 5). Again, the author seeks to find possible serial organisation of the degree of density. He transfers the order of the two-note and three-note density degrees from the score seen in Figure 5.25 and then groups these incidents into threes: 2 3 2/ 3 3 2/ 3 3 2/ 3 2 2 (each number here refers to the number of notes). Moreover, Blumröder studies the pattern in which two different types of articulations are arranged. In Figure 5.26 summarises this pattern, differentiating these two articulation types by indicating the number with either a circle or box; the circled numbers signify that the notes are attacked together but released in different times and the boxed numbers mean the individual notes are attacked separately but released simultaneously. However, there is a mistake in the second-last articulation, which should be squared not circled. The

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oversight is significant because it breaks the symmetry of the arrangement that is discussed in the text (1993, p. 135). Blumröder, unsurprisingly, does not interpret these forms of density as chords but as sound colours. He further examines the individual incidents, called “spectra”, measuring the intervals, durations and dynamics, and argues that such an arrangement of density anticipates the process of later electronic music (1993, p. 136). Blumröder points out that in Klavierstück III Stockhausen achieves a balance of consonant and dissonant sound through density grading. Maconie also mentions the relationship to electronic music suggested here, albeit briefly, in his analysis. He refers specifically to bar 5, where three notes occur with three different durations. Maconie observes that the composer’s “cutting” and superimposing techniques and the way he uses time differs from simply ordering durational values, thus foreshadowing the technique applied to organise pre-recorded sounds (1976, p. 65). Significantly, Blumröder and Maconie are the only two authors who link Klavierstück III with compositional aspects of his later electronic music.

Perspectives of a listener and performer

With the exception of Harvey’s analysis, all the analyses reviewed in this chapter mention the perspectives of a listener and/or performer. However, there is no analysis of Klavierstück III where such concerns are the central focus. In comparison with other aspects of composition, such as pitches and durations, aural perspectives and the challenges that a performer faces in playing the piece have received far less attention. This is contrary to the analyses reviewed in the previous chapters where, in some cases, the analytical approach was designed primarily to clarify the listener’s experience. For example, Covington’s analytical approach to Messiaen’s Mode de valeurs is based foremost on the repeated listening experience and Grant’s discussion of Structures Ia is fundamentally from this perspective. The following is a review of the ways analysts of Klavierstück III address the listener’s perspective and issues involved in the performance of the piece. Schnebel recognises that in order to interpret this piece, a performer needs to have a new attitude to be able to recognise aurally the compositional ideas expressed. The author writes:

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He has to play intervals of notes, of durations and of intensities. But in this way the inaudible is made audible—as when one spots proportion in superimposed durations even before their constituent music becomes flexible, and takes on a Chopinesque sensibility. But not only that. Construction becomes audible, and with it the work-idea—and time is compressed to become the moment. (1960, p. 131)

Here, though the performer may articulate such features, the question arises as to what extent a listener could recognise various musical relationships when the piece is directed to be played as fast as possible. This intriguing issue is discussed later. Maconie also advises a pianist especially to be aware of the different lengths of note-values, since they have a clear compositional purpose given to them. He is convinced that the various arrangements of note-values “should be clearly understood and expressed” (1976, pp. 65–66). He even discusses the technical matters involved in performing the piece as follows:

The performer must pay attention to the ends of notes, their liaison, overlapping, and release. For obvious reason no sustaining pedal should be used, but it may help performance to keep in mind the characteristically sustained quality of the organ (though the organ cannot produce the dynamic variation required). (1976, p. 66)

Compared to the compositional techniques he considered, Maconie’s suggestions on performance are brief. However, from this point of view, the pedalling is a pertinent performance element to have identified as it immensely affects the sound colour and the audibility of certain articulations whether it is to do with duration, pitch, or even dynamics. Among all the authors who have published analyses of Klavierstück III, Cook more than any maximises the relevance of his analysis in terms of the listener’s experience as well as considering the issues of performance practice. The underlying principle of his analysis is summed up in the following statement: “We have to think about what the music does to us rather than how it came about. We need to describe it rather than speculate about it” (1987, p. 357). For this reason, he avoids investigating pitch relationships based on serial organisation and he is also convinced that his segmentation of the piece into five parts can aid a performer’s conception of it. At the end of Cook’s analysis, he points out that the way the score is written, the

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“mathematical notation of rhythms” (1987, p. 362) causes more difficulty in analysing the piece than experiencing the music itself. Cook argues that:

. . . what actually happens is that the performer improvises the rhythms more or less in accordance with Stockhausen’s specifications, and the result is a rhythmic fluidity and independence of any fixed beat that probably could not have been easily achieved in any other way. In other words there is a glaring discrepancy between the fearsome mathematical complexities of Klavierstück III’s notation . . . and the way in which the music is actually performed and experienced by the listener. And in general I would say that it is discrepancies between score and experience that, more than anything else, make a lot of contemporary music problematical from the analyst’s point of view. (1987, p. 363)

A similar discussion about measuring the durational values assigned to each pitch in a performance setting is also mentioned by Griffiths. The author recognises the significance of temporal ordering in the piece and writes:

Stockhausen has said that in Piano Pieces I–IV he was concerned with “imparting a new way of feeling time in music, in which the infinitely subtle ‘irrational’ shadings and impulses and fluctuations of a good performer often produce that one wants better than any centimetre gauge”. (1981b, p. 86)

This comment suggests that there is another level of interpretation required when one performs the piece, especially with articulating the subtle differences of durational values assigned to individual pitches. Lewin also discusses the benefit of his analysis to both listener and performer. Firstly, he is convinced of its usefulness despite the fact that his analysis excludes consideration of dynamics, note values, register, contour and other audible features. “Still,” he asserts, “I would find the result of our pentachordal study useful in preparing a performance, that is, useful as more than matters of abstract theory alone” (1993, p. 41). He emphasises that his examples, which spatially illustrate a network of pentachord relationships, can help one to formulate a story that “links meanings of the articulated sections in a coherent and consistent through-line. And that enables a performer, at every moment in the performance, to feel oriented with respect to such a through-line” (1993, p. 41). 83 Secondly, following Lewin’s suggested ear-training

83 The examples mentioned here are example 2.5 appearing on page 34 and example 2.6 on pages 38– 39. 164

exercise, which is designed to help a listener to identify various forms of pentachord and their relationships, he poses himself commonly asked questions, “Do you hear it?” and “Can you hear it?” (1993, p. 43), with regard to his model of analysis (again, referring the same examples 2.5 and 2.6 in his analysis). He suggests here that it is possible to aurally recognise various pentachordal relationships after studying his analysis. Thirdly, and most significantly, he expresses a sense of dissatisfaction about his own analysis of Klavierstück III, relative to what “a good analysis of a good piece by Beethoven” (1993, p. 44) would contribute the listening experience.

If we demand that all music that we examine be on the aesthetic level of the great tonal masterworks, and that all the theoretical equipment we invoke be at the level of sophistication and power that tonal theory has achieved after two and a half centuries of intense development, we will not get very far in coming to terms with the music our recent past. (1993, p. 44)

Although the author does not develop the above point further, he recognises a critical limitation of analysis dealing with post-tonal music by acknowledging the common concern that particular coherent relationships demonstrated in the analysis are not easily heard, or even may be impossible to recognise aurally. Of the analyses of Klavierstück III, those of Cook and Lewin are especially concerned with their usefulness and benefit to a listener and performer, though their analytical approaches do not share any common ground. Though he specifically disputes Cook’s approach, Lewin makes a comparison between them at the end of his lengthy and critical review of Cook’s analysis.

. . . if I were a pianist with little exposure to Stockhausen trying to work my way into Klavierstück III, I would rather use Cook’s analysis as a point of departure than mine. It addresses in a tangible way features of the piece that are much more “phenomenological presences.” And the sorts of dissatisfactions I feel about it are such as to stimulate further thought, by way of response, about the phenomenology of the music. My network analysis might come into play at a later stage of familiarity with the music, should a person develop—as I did—a sense that there is an overall story to be told in the progression of the piece through that field. The differences in segmentation between Cook’s analysis and mine should not be problematic, I think, except for those who believe that form is “a Form,” something a piece has one and only one of in all of its aspects. (Lewin, 1993, p. 76)

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One of the important aspects of Klavierstück III in relation to performance and listening experiences that is often over looked is the effect of tempo. The question one should ask is how to interpret the tempo marking of “as fast as possible” (so schnell, wie möglich), determined by the fastest speed possible for the smallest note value (Maconie, 2005, pp. 122–123).84 A few suggestions are made by three authors. The first author who provides guidance on how one should practice in terms of tempo is Stephan. He advises that a pianist should only attempt to play the piece at the suggested tempo when she or he performs it, but in preparing the piece, it is more appropriate to play with extreme care in a slow tempo (1958, pp. 61–62). The degree to which the tempo marking has an effect on both a performer and listener is a performance issue raised by Stephan. He suggests it is impossible to make any sense of it if one plays as fast as possible immediately, or if one hears such an interpretation unpreparedly (Stockhausen, 1958, p. 62). Secondly, concerning the composer’s choice of tempo description for Klavierstücken I–IV, Maconie suspects that “what is more important is the sense of nervous energy created, rather than the actual tempo” (2005, p. 123). Finally, Blumröder includes a discussion on tempo prior to his analysis. Unlike Stephan and Maconie, Blumröder offers an actual tempo for the piece. He found that the suggested tempo was written on Stockhausen’s original manuscript as a metronome marking of a quaver equalling 120 beats per minute, making the piece 29 seconds in length. Blumröder adjusts this in relation to the composer’s final description (“as fast as possible”) by suggesting an overall tempo of one quaver equal to the metronome marking of 155–145; thus the total duration would then be around between 22 to 24 seconds (1993, p. 111). The tempo description given by the composer has several consequences. Firstly, it opens up a wide range of interpretation for a performer who needs to determine whether to take it literally, as Blumröder says above, or whether a performer should consider it in a more figurative way, depicting the overall sense of the piece. The performer must balance the challenge of playing each note accurately according to its specific duration and dynamic while at the same time playing it at the tempo “as fast as possible”. Such issues have long been of concern to performers. The importance of performing a piece at a “correct” tempo, while illuminating the various detailed musical relationships, is discussed by Leopold Mozart:

84 The following instruction is written by the composer: “Das Tempo jedes Stückes wird vom kleinsten zu spielenden Zeitwert bestimmt: So schnell, wie möglich” (1954, p. 2). 166

Everything taken into account, the end result should be a tempo that allows the piece to be fully played and heard with all its musical details, that allows the phrases to move and the connections and relationships to be perceived, all with a perspective that brings out a sense of the whole. “Who will contradict me if I count this among the chiefest perfections in the art of music?”. (As cited in Rosenblum, 1988, p. 312)

Although Rosenblum here refers to interpreting a classical piece, the principle remains the same in the performance of any form of music. It is, however, regrettable to see that only three authors mention the tempo when this performance aspect has such profound effects on the listening experience. If one has to choose a word to describe the listening experience, it could be “fleeting”, given that the piece lasts barely half a minute. As mentioned previously, Cook and Lewin are conscious of the usefulness of their analysis to a performer as well as a listener, yet have neglected to address the effect of tempo. If one considers seriously the type of challenges faced by anyone wanting to understand this piece, one might question the suitability of the analytical methodologies that have been in place during the last half century. It is important to be cognitively aware of the rich relationships existing among various musical elements, which many analyses attempt to reveal, but analysts must also recognise the musical metamorphosis that occurs when one listens to the piece, especially when played at a fast tempo and within such a compressed time scale. A re-interpretation of Klavierstück III

Analytical premise

The preceding comparative study of the analyses of Klavierstück III shows how different authors have approached the work over the last half century and also the diversity of their approaches and conclusions. Their observations provide a wealth of insight into the piece. Despite their various approaches and purposes, understanding the compositional techniques used in this piece has been one of their primary intentions. This is most evident in the analyses of Schnebel, Maconie, Blumröder and, to a certain degree, Cook. Furthermore, these analyses have revealed two prominent compositional techniques: “grouping” and serialism. Both Schnebel and Cook take account of the transitional phase in Stockhausen’s evolution where, as the composer acknowledged, his compositional technique began to move from pointillism to grouping (Stockhausen, 1964, p. 19). As a result, their

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analyses are centred on understanding the application of the compositional technique. This change in technique also prompted both authors to divide the piece into five segments, aiming to illustrate the way various musical elements are organised in each group. According to their analyses, each group is designed heterogeneously; as a result, each group contains its own musical features. Nonetheless, Schnebel acknowledges that the grouping exists without compromising musical continuity. This musical continuity across the entire piece makes it hard to determine the boundary of each group. It is also worth noting that these authors disagree about the segmental division of the piece, except for the beginning of the third section (bar 8). The differences in their interpretation on grouping the five parts are minimal—for instance, one to three quaver beats—but are significant nonetheless in reflecting the piece’s challenging ambiguities. Finally, their analyses are more descriptive in nature than those of the authors who approached the piece as an example of serial technique. Perhaps due to their focus on the compositional technique of grouping, the Schnebel and Cook tend to explain how each group is distinctive, whether it is due to density variability, texture (monodic vs. polyphonic), melodic contours or types of durational values (longer vs. shorter). On the other hand, Blumröder and Maconie have considered the serial techniques utilised in the piece. As mentioned earlier, they both agree that the technique of distributive serialism is applied, rather than the note-sequencing serial technique developed by Schoenberg. It is perhaps one of the most important aspects of Stockhausen’s serial techniques to have been fully explored. As previously observed, Blumröder provides far more comprehensive analysis on this matter and his analysis illustrates that serial organisation is not only applied to pitch but also to other parameters. Blumröder especially recognises the significance of the serial techniques developed by the composer during 1952 and, prior to analysing the piece, refers to Stockhausen’s own writing, “Situation of the craft (Situation des Handwerks)”.85 In this article, Stockhausen discusses the concept of ordering musical materials and their relationship to the state of what he calls “meditative hearing” (meditatives Hören). As Blumröder notes, Stockhausen explained the concept of “ordering” in music in the following terms: “The assimilation of the individuals into the whole, of the various

85 According to Blumröder, this was written in December 1952 but published in Texte I in 1963. 168

into the uniform. Criteria for order are the richness of relationships and lack of contradiction” (Stockhausen, as cited in Blumröder, 1993, p. 120).86 Stockhausen also mentions that once there is total order in music, all individuals become equal and the sense of an order can be found when the individual accords with the whole. Moreover, the aural recognition of the order presented in the music may then be perceived in a meditative state. Blumröder cites Stockhausen saying that “Music as tone-order concentrates on the human ability to recognise ordering of tones. ‘To recognize’ here means: to exist within and to remain within [this tone-ordering] without intention. . . .” (Stockhausen, as cited in Blumröder, 1993, p. 120).87 The following statement of the composer further clarifies the relationship between tone-order and the listening experience:

Constantly presented totally ordered music representing no “development” can alone cause a state of meditative hearing. One dwells in the music and has no need of what precedes of follows to perceive the individual presence (the individual tone). A prerequisite is that the individual already carries within himself the criteria of order (without any contradiction), which are part of the whole work. (Stockhausen, as cited in Blumröder, 1993, p. 118)88

Here, the composer connects the compositional aesthetics of total ordering of musical materials with experiencing his music in a meditative state; this concept is quite foreign to many musicians in Western culture, especially those under the influence of ideologies such as positivism and formalism. It is important to point out that musical experience as meditation becomes an essential aspect of Stockhausen’s later compositions. This is also reflected in the composer’s statement: “I think identifying with a sound is meditation. A musical meditation is when you completely become the sound” (Cott, 1974, p. 35).

86 Das Aufgehen des Einzelnen im Ganzen, des Verschiedene im Einheitlichen. Kriterien für Ordnung sind Beziehungsreichtum und Widerspruchslosigkeit (Stockhausen 1963, p. 18). (Translated by Christina Young and Sun-Ju Song.) 87 Musik als Tonordnung richtet sich auf die menschliche Fähigkeit, Ordnung von Tönen wahrzunehmen. Wahrnehmen ist hier verstanden als: darin existieren und aushalten ohne Absicht. . . .(Stockhausen, 1963, p. 18). (Translated by Christina Young and Sun-Ju Song.) 88 Ständige Anwesenheit von durchgeordneter Musik, die keine ‘Entwicklung’ darstellt, kann allein den Zustand meditativen Hörens. . . hervorrufen: Man hält sich in der Musik auf, man bedarf nicht des Voraufgegangenen oder Folgenden, um das einzelne Anwesende (den einzelnen Ton) wahrzunehmen. Voraussetzung ist allerdings, daβ das Einzelne bereits alle Ordnungskriterien in sich trägt—und zwar widerspruchslos –, die dem ganzen Werk zu eigen sind (Stockhausen, 1963, p. 21). (Translated by Christina Young and Sun-Ju Song.) 169

Apart from serialism and groupings, there is another important element in the compositional process of Klavierstück III: symmetry. This concept is clearly shown in the beginning of the piece and a number of authors (Schnebel, Cook and Blumröder) have identified its application in various parts of the piece. It is possible that a number of compositional techniques were applied simultaneously as well as in an integrated manner. In support of this view, Stockhausen says:

There are three basic qualities of musical formation: the lyrical, which is the instant, the here and now; the dramatic, which is development, with precise beginning and ending, climaxes, high points and low points; and epic, which is the juxtaposition of different moments, as in a variation form or the traditional form of the suite—that’s an epic form, you can always add a new section, a new chapter so to speak. There’s no strong directionalism as in a dramatic form, but it’s also not static; within a given moment it goes somewhere, it describes some event. And I want all the three. (Stockhausen, as cited in Cott, 1974, p. 35)

The three qualities the composer mentions above could conceivably be related to the three principal techniques of organisation employed in Klavierstück III: serialism for the lyrical, symmetrical arrangement for the dramatic and groupings for epic. However, whereas serial techniques and groupings have been investigated thoroughly by other authors, how symmetrical relationships are applied throughout the entire piece has not been explored autonomously or in full measure.

Symmetry

As is revealed in the following analysis, the idea of symmetry is deeply embedded in the entire organisation of the piece. It can be seen on multiple levels: the arrangement of pitches; registral distribution; durational values and proportions; dynamics; the arrangement of intervals; formal structure, and shape. Symmetry has also allowed the composer to create a musical beginning and ending across the whole piece as well as a climax or anti-climax within individual groups. However, the appearance of symmetry is neither always clear nor completely formed. More significantly, the symmetrical arrangement of various musical elements is inseparable from the compositional technique of groupings. The following analysis and discussion of Klavierstück III reveal why this is so.

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The first stage of my analysis determines the individual groups; here, the groups “entail the bringing together of several notes by means of overall characteristic properties” (Wörner, 1973, p. 91). The second stage of this analysis involves examining each group in the piece; how individually they are designed and how the compositional idea of symmetry is manifested in each group and in the overall structure of the piece. Identifying the different organisational scheme occurring in each group is necessary because the unique organisational scheme in each group creates its own musical identity. Not only does each group have a different sound quality but each has its own distinctive symmetry. The final stage of this analysis considers the compositional aesthetics of Stockhausen in relation to the way this piece is reinterpreted analytically. Similar to the analyses by Schnebel and Cook, my analysis divides the entire piece into five groups (see Figure 5.27). Interestingly, my grouping does not always agree with these two authors (see Figures 5.28 and 5.29). The one place where the three analyses agree is the beginning of group 3 (bar 8) where a change has been noticed by all of the authors. Schnebel and I have the same grouping with groups 2 and 5, while Cook and I agree upon the beginning of groups 3 and 4. It is clear that the piece consists of five groups of notes but the exact boundary of each group is contentious. Such disagreements again show how well the precise groupings have been concealed by the composer. Schnebel points out that “the limits of the group aggregates are ambiguous because of overlapping” (1960, p. 126). Furthermore, as the investigation of the individual groups unfolds, the symmetrical arrangement of musical elements in fact confirms the precise boundaries I am proposing here between each group. The following paragraphs demonstrate the inseparable relationship between symmetry and grouping.

Grouping

As previously observed, Schnebel, Cook and Blumröder agree that Stockhausen presents the idea of symmetry from the very beginning of the piece. The first two bars (Group 1) are monodic in texture and arch-shaped (see Figure 5.30). The arch shape is confirmed in several ways. The centre of the arch is marked by the highest note and the largest leap in the group and it is also reinforced by the dynamic of f. Although the group is metrically made up of one bar of 4/8 and another bar of 5/8, these time

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signatures suggest no strong sense of metre. However, the entire group is most likely to be perceived as 9/8 due to the irregular and rapid movements of the notes; this metric reinterpretation of 9/8 underlines the perception of the group as a single arch- shaped musical unit. The placement of the bar-line perfectly indicates the centre-point of the arch, as the highest note occurs on the fifth beat of the entire group. Thus, the centre of the arch and the centre of durational proportion correspond to each other. In terms of arranging the note values, Schnebel observes the symmetry: “Simple duration intervals … developing from simple to more complex. Symmetry emphasized by correspondence at beginning and end” (1960, p. 129). While the centre of the arch is articulated by the loud dynamic and the melodic contour, the symmetrical arrangement of the individual durational values is rather obscure. As Schnebel writes, durational intervals are only suggested at the beginning of the arch- shape. The second group (see Figure 5.31) in this piece is clearly distinguished from the first group as there are noticeable contrasts between them both analytically and aurally. Firstly, unlike the previous group, the entire group has a time signature of 3/8 and this sense of regular metre is quite audible. Secondly, the second group is dominated by significantly longer durations. Thirdly, one can immediately hear the change of texture from monophonic to polyphonic. The change of texture also affects the melodic contour. Finally, the registral distribution differs from the first group, as the middle register is preferred. The dynamic arrangement of the second group is an inverted arch, as Cook also mentioned (1987, p. 358). It seems that the composer was very clear about making the first two groups contrast in every possible way. These contrasting features are all perceivable aurally since such characteristics of the surface structure are familiar compositional techniques. Again, the second group displays symmetry but differently from the previous group. This can be observed in the arrangement of durational values, dynamics, and registral distribution. With the durational values, the composer further organised durations into three subgroups consisting of three different values (see Figure 5.32). The first subgroup is firstly assigned to the first three pitches (Ab4 G3 Gb5) and then the last three pitches of (A3 F5 G#5). The two remaining subgroups of durations are assigned to pitches (F3 Bb4 E5) and (D4 B3 Eb5) respectively. The three durations used in these subgroups 2 and 3 are not identical; however, in both instances, the durations are ordered in the relative values of long-medium-short, thus providing a common 172

organisational scheme. The way the composer assigns the same subgroup of durations for the first three and the last three pitches suggests an overall symmetry, which incorporates some variation of the three-note pattern in the middle. In the second group, a stronger symmetrical relationship can be found between dynamics and registral distribution. Figure 5.33 rearranges the pitches and dynamics according to register. This abstract arrangement reveals that the composer has placed the louder dynamics in the outer registers and the softer ones in the middle. As the figure shows, the central portion of this arch works almost perfectly, with placement of five dynamics of p, which are then followed by mf and f on both sides of symmetry. The very beginning and the end of the symmetry correspond with the same dynamic of f. This perfect symmetry over the whole group is, however, disturbed by one extra mf placed in the higher register. It seems that the composer has purposely avoided perfect symmetry. A possible reason for this is to ensure the variety in dynamic levels by alternating mf and f for the four highest notes used in this group. As Blumröder has already established, the composer frequently deviated from his own rigid pre- compositional plan in this piece. Although this symmetrical relationship is illustrated on a rather abstract level, it is also reflected on the surface structure as the soft dynamic dominates the middle section of the group, creating an anti-climax or anti- arch. It is exactly opposite to the first group, where the centre of the arch is emphasised by louder dynamics. There are apparent indications for the precise beginning of the third group (see Figure 5.34): the first appearance of pitch Db6 and the change of metre from 3/8 to 4/8. In addition, the overall character of group three contrasts with that of group two but the contrast is less dramatic than between groups one and two. The third group is dominated by medium dynamics and its texture is a mixture of polyphony and monody. Despite these differences between groups two and three, the shape of group three is once more designed as an anti-arch, as especially seen in its melodic contours. As in the previous groups, the third group contains symmetrical relationships in its arrangement of intervals, durational values, registral distributions and dynamics. However, again the way Stockhausen implements the idea of symmetry varies significantly in this group. Firstly, the intervals are arranged in a mirror shape and they are reasonably easy to recognise, being well supported by the dynamics and registral distribution. The following arrangement of intervals provides a sense of mirror structure with this group (see Figure 5.35). The compound minor 3rd, 173

augmented 4th and compound minor 2nd intervals occurring in the first two beats of this group reappear in the last two beats. The compound minor 3rd recurs as a minor 3rd in the upper register. The augmented 4th is transposed an octave higher when it is heard on the opposite side and the interval of the compound minor 2nd is formed between two outer pitches of the final chord. The centre of the mirror is marked by the interval of a major 3rd placed in the lowest register in group three. This major 3rd is approached by a compound minor 3rd in descending motion and followed by an ascending major 6th, which is the interval of a minor 3rd inverted. It seems that that, in arranging intervals, the composer has purposefully used either the compound interval or an inverted one to avoid an obvious pattern of symmetry. From a listener’s point of view, this makes it less apparent but perhaps more musically satisfying. Secondly, the arrangement of durational values is approximately symmetrical (see Figure 5.36). The shortest value is assigned in the centre to create symmetry. Similar to symmetrical relationships discussed previously, there is again some deviation from a precise symmetrical format, as the beginning and end of the group do not exactly match. Interestingly, as the texture becomes polyphonic, the mirror formation of durational values also becomes obscured. Finally, the medium dynamic is preferred in this group; the centre of the mirror is articulated by mf. The soft dynamic, p, is only employed three times and mainly placed in the upper register. In this group, the dynamic arrangement can be further divided into three subgroups: the first four notes are assigned {mf, mf, f, p} and the last four notes {mf, mf, p, p} and the centre is quite clear, with the dynamics {f, mf, f}. Indeed, despite deviations and the lack of exactitude, a symmetrical arrangement is evident. Although group three is in the shape of an anti-arch, its centre is highlighted by various musical features. With registral distribution, the centre note, Eb3, is the lowest one in this group. In terms of duration, it is the shortest. The centre of the anti-arch is further indicated as the density becomes thin and the texture becomes monodic. As has been seen, Stockhausen also gave such musical emphasis to the centre of the first group, but in group three, he has chosen the lower register rather than higher register. The beginning of the fourth group (see Figure 5.37) is clearly marked as the pitch C3 occurs for the first time in this piece with the dynamic of f. However, this group is considered as the most complicated one in the piece because, on the surface, the group does not seem to have any kind of familiar shape. Cook shares this view:

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This is the most complex segment, partly because for the first time there is a suggestion of three-part counterpoint (see Fig. 178), and partly owing to the conflicting shapes outlined by different parameters. There is no clear overall shape either in dynamics or register, though there is some registral expansion; note-density declines, but unevenly (1987, p. 360).

In addition to the musical features Cook observed in the fourth group, several other musical features further contribute to its complexity. This group is the only place where all twelve chromatic pitches are used; in fact it contains the largest number of notes of any group, fifteen. Organising the note values also increases the complexity of the group as various durations are utilised, including three types of quintuplet. The difficulty of analysing this part of the piece can also be seen in that other analyses have been unable to deal with this group convincingly. Despite the complexity portrayed on the surface, a neat relationship is evident between the arrangement of pitches, the durational values and the registral distribution. Moreover, not only are these three parameters related but they are also arranged symmetrically. The following Figure demonstrates this (see Figure 5.38). Pitch E4 is the centre of symmetry in two respects: durational proportion and registral distribution. The entire duration of group four is made up of thirteen quavers and the pitch E4 occurs on the seventh beat. Regarding the registral distribution, seven pitches are placed below the central E4 and seven pitches are placed above it. It is interesting to note that, while the centre of the first group is accentuated by the highest pitch and the centre the third group by the lowest pitch, the centre of symmetry this time is placed at the middle of the registral range. As shown in Figure 5.38, certain note values are arranged to form the mirror, but it seems the composer deviates from an exact mirror formation by altering a few note values and their position. Firstly, the centre of mirror has the longest duration and it only occurs once in this group. Secondly, the very first and last durational values correspond. Thirdly, the placement of quintuplets also forms the mirror although their durational values are not exactly same. Finally, symmetry is found in the number of crotchets placed either side of the centre. One can, therefore, observe that the mirror formation of note values is strongly suggested as the centre and the edges of the mirror strongly imply the presence of symmetry. Since the arrangement of durational values directly connects to the registral distribution of pitches, the composer also had to consider pitch arrangement on the surface structure. Perhaps for

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this reason, the composer was not concerned about those deviations in the symmetrical arrangement of durations. The ambiguity is created by the incongruity between the abstract compositional plan shown in Figure 5.38 and the shape of the surface. Even though symmetry exists as the major part of the compositional plan, it is not manifest on the surface. As Cook mentioned, the fourth group does not suggest a particular shape whereas the surface contour of previous groups has provided some clues about the underlying symmetrical organisation of musical elements. The major incongruity in this fourth group is caused when ten notes appear before the central E3, which appears to be at the exact centre in terms of durational proportion. This E3 stands out due to its sustained duration and the lower density of texture around the pitch. As two thirds of the notes occur prior to the central pitch E3, this creates a contrasting density between the first half of the fourth group and the second half, where the degree of density is low, only having the remaining four notes. However, in the abstract compositional plan, it is clear that there are seven pitches placed below the centre E and the other seven notes placed above. The symmetry has been submerged and concealed but still integrates the pitch, register and durational relationships into a coherent plan. The beginning of group five (see Figure 5.39) is marked by a number of changes: the change of metre from 4/8 to 3/8, the change of texture from polyphonic to monodic, and the change of density from thick to thin. Apart from these features immediately contrasting with the previous group, this group contains the smallest number of notes—five pitches occurring within the range of five octaves—and the sole occurrence of the loudest dynamic level, ff. Here, the shape of the melodic contour suggests neither an arch nor an anti-arch but the durational proportions seem to imply symmetrical arrangement. The five notes appear throughout the duration of the nine quavers but exact symmetrical arrangement is again obscured with the use of the 7:6 proportioning across the penultimate bar (see Figure 5.39).89 Without this, C5 would have clearly been the axis of symmetry. While the symmetrical relationship found in group four is in the abstract form, the symmetry in the last group may be seen to be represented only symbolically, by five notes reflecting the five groups in which the compositional idea of symmetry is deeply embedded.

89 Even though the 9th quaver is not notated as a note, a rest remains part of the music. The composer would have dispensed with indicating the duration of the rest if he did not consider it to be part of the music. 176

Since this group is the last one, the arrangement of musical elements suggests the symmetry over the entire piece, as well as within the group, thus achieving dual purposes. The most obvious way of ensuring the overall symmetrical relationships in the piece is by reiterating the first two notes (A4 and B5) to finish the piece; this occurs here, although these two notes appear in different registers (A2 and B6) with louder dynamics. These two last notes are further accentuated by the biggest leap in the entire piece, leaving an exceptional impression on the listener. Additionally, group five corresponds with group one as both are monodic and are almost of the same durational length (nine quavers). It is possible to perceive these musical characteristics as a listener, once one is aware of them. The overall symmetry in the piece is further emphasised as groups two and four are written in polyphonic texture while group three consists of a mixture of monody and polyphony. Moreover, in both groups two and four, symmetry exits in relation to registral distribution, though only conceptually. In other words, it is only evident abstractly. As this analysis demonstrates, symmetrical arrangement of musical elements is an integral part of Stockhausen’s pre-compositional plan, which provides great richness in relationships. Many of the analyses reviewed here have examined the individual musical parameters autonomously, yet have not shown the interrelationships among various musical elements in relation to one governing concept. Although symmetrical arrangement was not a new compositional idea, Stockhausen explored its possibilities beyond any compositional tradition; the symmetrical relationships and frequent deviations exist in many different forms. Moreover, the symmetrical arrangement in each group changes continuously. The way the principle of symmetry unfolds progressively throughout groups one to five is highly sophisticated and innovative, ranging from obvious to obscure; from familiar to unfamiliar, from surface to depth, and from concrete to abstract. For instance, in the first group the symmetrical relationship can be seen on the surface, although the arrangement of durational values is obscured to a certain degree. The entire group is designed as an arch and no abstract formation of symmetrical relationship can be found. In the second group, there are still some musical elements that suggest symmetry, like the anti-arch shape of the entire group, but the composer first introduces an abstract form of the symmetrical relationships in the arrangement of register and dynamics. In the third group, although a mirror formation can be observed with the intervals, the degree of deviation from perfect symmetry greatly 177

increases. In the fourth group, obscurity can be seen as the prime factor as it becomes almost impossible to discern any shape, let alone symmetry. However, symmetrical relationships occur among register, pitches and durational values in an abstract form that is well concealed underneath the surface. In the final group the symmetry is, in a sense, symbolic. Therefore, all five recognisable groups are governed by symmetry but on their own terms. There are two main compositional aspects in Klavierstück III that make it difficult to recognise the various symmetrical relationships throughout the entire piece. Firstly, as mentioned above, each group has its own unique symmetrical arrangement. Although symmetrical relationships can be found among the arrangement of pitches, durational values, registral distribution, intervals, shapes and melodic contour and durational proportions, throughout the piece the idea of symmetry manifests differently due to various combinations of musical elements. The changes occurring in each group are neither predictable nor transparent. Secondly, since determining the precise boundary of each group allows the symmetry and types of symmetry to be discerned, it is critical to know where the groupings are. However, at first the musical continuity almost forbids drawing the exact boundaries between the groups with certainty. As shown in the analyses of Schnebel and Cook, agreeing on the precise groupings themselves has been challenging. Thirdly, Stockhausen always obscured the symmetry in his works; in other words, he seems to have deliberately deviated from a perfect and complete symmetry. As discussed earlier, the deviations can frequently be seen as to achieve the desired resultant sound but these deviations are never allowed to disfigure the governing frame of symmetry. There are musical elements that suggest symmetrical relationships—sometimes these are obvious, sometimes obscure. Finally, not only do symmetrical relationships exist in abstract forms but also they are abstruse due to the degree of variability. Reinterpreting Klavierstück III would not be fully satisfying if it did not help a listener or a performer. This analysis now presents a journey from the surface to the depths and a journey from groupings to individual elements. Clearly distinguishing what is concrete and what is abstract in the symmetrical relationships can provide a certain level of understanding that can illuminate such an enigmatic piece for performers and listeners. Admittedly, it is almost impossible to recognise aurally all of the symmetrical relationships discussed in this analysis because some are submerged or only exist in conceptual form. However, if the obvious shape, melodic 178

contours, metre changes, intervallic arrangement and dynamics are identified, the listener can be guided to a degree. This distinction between the surface and the conceptual background is an essential aspect of the compositional plan as the composer moves from familiar and more traditional compositional techniques in the beginning of the piece to unfamiliar and innovative methods. In summary, the analytical approach in this analysis involves segmenting the piece into five groups and then providing more detailed analysis of how the musical elements are organised in each group. This analytical methodology is further informed and supported by Stockhausen’s discussion about how a listener should approach his composition. He states:

I go into the deepest possible layer of the individual sound. There’s always this change from the clear gestalt that you can grasp in just one listening to a piece—the very simple subdivisions and blocks. And if you dive into one block you discover multiplicity. In the multiplicity—‘individuality’—you discover individual figures again. Each individual figure has aleatoric components around itself, it becomes a nucleus of a new entity. Deeper and deeper. (Stockhausen, as cited in Cott, 1974, p. 72)

Stockhausen further relates his compositional aesthetics to the phenomena of nature, as one can explore it endlessly. Another important dimension of this piece is that continual and rapid changes occur throughout. Thus, it is necessary for the performer to identify, understand and express these changes appropriately, which will eventually affect the listening experience. Certain changes are more subtle than others. However, some changes make an immediate musical contrast, especially between groups one and two, and between groups four and five. These contrasts can be textural, metrical, or heard in terms of melodic contour. Emphasising musical contrast in this piece is crucial; firstly, since the timbral range of the piano is rather limited, other musical aspects play a larger role in creating contrasts and, secondly, since this piece is performed at a fast tempo and is over so quickly, performers need to know which musical features need to be differentiated as groups unfold, one after the other. Underlining musical changes in the piece is central to a performer’s responsibilities and, as Stockhausen himself explains, an important part of a listener’s experience of his music. He says,

You have to switch very quickly when you listen to my music and to change with the music from one character to another. Like a person who, very

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dramatic at a given moment, then becomes completely quiet, meditative, and then outgoing. . . That’s like flying, flying the globe, or going through all the different aspects of our soul. And from the most quiet to the most excited, from most abstract to the most concrete, from quotes to newly invented moments—yes, that is the best example. (Stockhausen, as cited in Cott, 1974, p. 36)

This analysis is offered as a guide to performing and listening to Klavierstück III. Stephan, Cook, Lewin and Blumröder have suggested other ways of hearing the piece, and the composer himself promoted “meditative hearing”. There are many ways of approaching this piece, as seen in the review of previous analyses, whether it is to do with listening, understanding the pre-compositional plan, performing or theorising the pitch relationships. Central to all of these approaches is the assumption that music analysis should enhance one’s understanding and appreciation of the music. Music analyses explain the internal working of the music: the governing compositional theory that ensures the richness of musical relationships in a coherent plan. Since Stockhausen pre-organised and pre-planned musical elements other than pitch relationships, the compositional procedures became more complicated. Furthermore, three compositional ideas have been identified: serialism, groupings and symmetry. The elusive integration of these principles has ultimately created the great diversity in analytical methodologies, resulting in different conclusions. Conclusion

This case study has illustrated an extreme situation whereby a formalistic approach has dominated music analysis of a piece for over half a century. This tendency in music analysis is undeniably linked to the compositional aesthetics over the same period, which were governed by the ideology of formalism as well as organicism. As demonstrated in the various analyses, including my own, the composer evidently aimed to maximise the richness of the musical relationships as they grow organically throughout the piece. Finally, perceiving the logic and coherence of these musical relationships is challenging, as they often exist conceptually. This difficulty is heightened by the fact that the entire piece is most likely to be heard as little more than a fleeting moment. If one considers seriously the meditative hearing experience that the composer also describes, the rich and coherent musical relationships within the piece take on another meaning. While Stockhausen’s compositional aesthetic was strongly influenced by formalism and organicism, one must be reminded that he

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himself endorsed mysticism in the way he imagined the sound world and wanted it to be heard.

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CHAPTER SIX

John Cage’s Music of Changes

In the Music of Changes, structure, which is the division of the whole into parts; method, which is the note-to-note procedure; form, which is the expressive content, the morphology of the continuity; and materials, the sounds and silences of the composition, are all determined. (Cage, 1978, p. 36)

Introduction

Cage began composing an extensive piano work, Music of Changes, in February 1951 and completed it on 3 December the same year. The first part of the piece was first performed in July 1951 by David Tudor, whom Cage met in 1950 through Morton Feldman. Tudor also gave the premiere of the entire work on New Year’s Day, 1952 in New York (Pritchett, 1988, pp. 107–109). However, Tudor’s involvement with Music of Changes went beyond just performing the work. Cage says:

In all my works since 1952, I have tried to achieve what would seem interesting and vibrant to David Tudor. Whatever succeeds in the works I have done has been determined in relationship to him. . . . When I composed Music of Changes, David Tudor applied himself completely to that music. At that time he was the Music of Changes. (1995, p. 178)

Cage was well aware of the capability and style of Tudor as suggested in the following statement: “When I was composing for David Tudor, he would play what I was writing, and so it suited him” (1995, p. 179). According to Pritchett, Tudor would start to learn the parts of Music of Changes as soon as Cage finished them. Pritchett even asserts that “it was Tudor’s unique abilities that made Music of Changes possible for Cage; without them, such a work would have been a mere compositional exercise” (1993, p. 78). The fame of the piece has largely depended on the fact that chance operations were employed in the composition of the entire work. Although Cage experimented with chance operations in some of his prior works, in this piece, he developed a systematic compositional process in reference to the Chinese book of oracles, I-Ching, which was given to him as a gift by his student Christian Wolff. Cage comments that “On seeing the I Ching table I was immediately struck by its resemblance to the magic square. It was even better!” (1995, p. 43). While incorporating chance

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techniques allowed Cage to be in the state of “purposelessness” which was an influence from the teaching of Zen, he designed the chance procedure to resemble the hexagram of the I-Ching. Cage once explained the relationship between the I-Ching and his attitude toward chance:

I have always accepted everything the I Ching has revealed to me. . . . I never thought of not accepting it! That is precisely the first thing the I Ching teaches us: acceptance. It essentially advances this lesson: if we want to use chance operations, then we must accept the results. We have no right to use it if we are determined to criticise the results and to seek a better answer. In fact, the I Ching promises a completely sad lot to anyone who insists on getting a good answer. If I am unhappy after a chance operation, if the result does not satisfy me, by accepting it I at least have the chance to modify myself, to change myself. But if I insist on changing I Ching, then it changes rather than I, and I have gained nothing, accomplished nothing! (1995, pp. 94–95)

Indeed, Music of Changes represents how Cage embarked on a new and important phase of his creative life from the 1950s, developing chance techniques and grafting oriental philosophies onto his compositional aesthetics. This chapter presents a unique opportunity to review how Music of Changes has been approached analytically and what the content of these analyses are. As with previous case studies, a detailed review of selected analyses is made and my own interpretation of the piece follows. Review of previous analyses

Issues

Although Music of Changes is well known as Cage’s first piece where a chance operation governs the entire compositional process, inquiries into this innovative technique are surprisingly few and in-depth analyses of the work are rare. There are only three authors who have closely examined the work: James Pritchett (1988 and 1993), Stefan Schädler (1990) and Yayoi Uno (1994) (see Figure 6.1). As seen in Figure 6.1, the first analytical study appeared three decades after the piece was published. Not only were the compositional techniques in this piece analytically unexamined, but investigation of Cage’s chance composition in general came much later than one would expect. The long neglect of Music of Changes strongly contrasts

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with the early analytical treatment the other three works examined in this thesis received.90 Prior to reviewing the individual analyses of Music of Changes, it is necessary to address possible causes for the work’s neglect. Firstly, Pritchett argues both in his PhD thesis (1988) as well as in his later publication (1993) that Cage has been treated more as a philosopher than as a composer, although he wrote much music, most of which has been recorded and performed. This is fundamental to why Music of Changes and other chance music were not investigated by music analysts for several decades. This misconception of Cage’s identity as a composer began when chance operation became a central part of his compositions from 1951. Pritchett writes:

The crux of the problem, then, has been a failure to find some way of dealing with Cage-the-composer, his musical compositions, and his chance operations all at the same time. When faced with music composed using chance, critics have drawn a blank. How can one understand a randomly-made composition? What can one say about such a thing? To criticize it would be criticizing a random act; how does one judge the toss of a coin? The way out of this dilemma has traditionally been to ignore the music and dwell upon “the ideas behind it.” (1993, p. 2)

Thus, the way the critic dealt with such problems and challenges was to focus on the compositional aesthetics alone. Emphasis has been given to the compositional aesthetic of chance music at the cost of comprehending both the process of composition and the resultant music. Pritchett suggests that another underlying misconception of Cage’s compositions post-1951 is the generalisation that his individual chance compositions is an “undifferentiated mass of ‘chance music’” (1993, p. 2). In other words, distinctive stylistic features among the indeterminacy compositions are overlooked because of the music’s apparent “randomness”. Pritchett explains that

His adoption of chance techniques is almost always seen as a rejection: a jettisoning of everything traditionally musical. External forces of irrationality (such as Zen Buddhism) are invoked as the cause of this break. Under such influences, it is believed, Cage decided to substitute the throw of dice for his own tastes, so that he could ultimately remove any personality from the composed work. (1993, pp. 1–2)

90 The first analyses of Boulez’s Structures Ia and Stockhausen’s Klavierstück III were both published in 1958 and the first analysis of Messiaen’s Mode de valeurs et d’intensités was published in 1973. 184

The consequence of this has been rather devastating; as Pritchett observes, “histories of his work tend to pass rapidly over the works composed after 1951, with a few brief descriptions and generalizations” (1993, p. 2). Pritchett’s argument explains the unusually small volume of analyses written about Music of Changes even though it is recognised as a significant turning point in Cage’s career. Another possible for the work’s neglect is that the ideologies underlying the discipline of music analysis can be seen to be in contradiction to the aesthetic implication of chance operation. As discussed in earlier chapters, for more than a century a central objective of music analysis has been to reveal organic unity within a work: decoding the way in which various musical materials are interrelated to ensure musical coherence; searching for one musical idea, ‘the seed’, on which the entire work is based, and explaining how one musical idea logically grows and develops throughout the work. The aesthetic value of an autonomous work is often measured against these conditions. It is obvious that embracing the idea of musical material being organised randomly through chance operation defies this ideology of organicism, an ideology that has been upheld by both composers and music analysts since the nineteenth century. Therefore, music analysts faced a dilemma: how can one resolve the ideological conflict between Cage’s compositional aesthetics and the governing ideologies of music analysis? This dilemma seems to have been unrecognised by the discipline of music analysis, since compositional theories and aesthetics share the same ideological ground as the discipline of music analysis. Aesthetic cognition of modern compositions often cannot be separated from music analysis. However, the relationship between music analysis and chance music is different. To face the challenges posed by Cage’s music, analysts need either to find an approach that does not totally refute their own ideological positions, or discard these ideological positions in order to develop new analytical methodologies. As discussed in the introduction of this thesis, analysts have also been preoccupied with the idea of genius, connected as it is to the ideology of organicism. A piece of music is often measured and valued according to its organic unity, which is revealed through music analysis. Therefore, as has been noted by Bent & Pople (2001, p. 528), many music analysts have been concerned with the question ‘What makes this great?’ instead of just answering the question ‘How does it work?’ Another essential quality of genius is that the work of art should no longer be confined by pre- established conventions but embody progressive compositional theories. Based on 185

these two main aspects of the concept of genius, a dilemma with Cage’s chance music can perhaps be identified. On the one hand, analysts could not possibly categorise Cage’s chance composition as the work of genius, since the desired unity cannot be displayed in music that has been organised randomly. On the other hand, chance procedure is a revolutionary approach and can be considered progressive. Thus, Cage’s chance music presents an ideological conflict within traditional musicology as well as within the discipline of music analysis. Finally, there is a practical difficulty for those who wish to understand Music of Changes. Many avant-garde composers of the post-World War II era were willing to discuss their own compositional aesthetics and techniques, not least Cage. As seen in the previous chapters, the composer himself provided details about many of his works and these have commonly been a starting point for analysts; Boulez’s Structures Ia is a perfect example. However, Cage’s writings about chance operation especially as applied in Music of Changes, are not easy to comprehend. He did not discuss his work in any detail, and no musical examples accompany his explanations.91 This is not helpful, since his writings include both unfamiliar theoretical concepts, such as chance operation and the I-Ching, and new terminologies that describe his innovative system. He needed to give clear explanations and illustrations if his procedures are to be fully grasped, but his readers are most likely to be left with more questions than answers about the technical aspects of his composition. Certainly, the difficulty of accessing Cage’s ideas and techniques through his writings is a serious hindrance for music analysts. It is a valuable exercise to review three authors’ analytical studies of Music of Changes, especially in light of the conceptual, ideological and practical challenges mentioned above: how these authors have approached the work; the way they justify their approaches; and most interestingly, the results of their investigations. The review is chronological, starting with Pritchett.

Pritchett’s approach to Cage’s chance music

The first substantial study of Music of Changes was made by Pritchett and appeared in his unpublished 1988 doctoral dissertation, The development of chance techniques in the music John Cage, 1950–1956. His research focuses on the detailed investigation

91 This point is also mentioned by Pritchett (1988, p. 9) and Schädler (1990, p. 191). 186

of Cage’s innovative chance-operational techniques and has been hailed by music theorist David Bernstein as “path-breaking” (2002, p. 208). In the earlier part of his thesis, Pritchett deplores the neglect of Cage’s chance music: “we know next to nothing of how chance actually functions in the music. . . We are faced with a paradox: Cage’s chance music is his most significant work, and at the same time it is probably the least understood” (1988, p. 3). Moreover, the author claims that

We read—perhaps too often—of Cage’s aesthetics, theories of “experimental”, “conceptual”, or “inferential” art; of his influence of other composers, dance, theatre, and the visual arts; of his affinity for the Far East; of Buddhism and Taoism, Indian talas and Indonesian gamelans. Meanwhile, his music—the subject that should stand at the center of our interest in Cage—remains a mystery to us as musicologists; it is ignored, misunderstood, and poorly documented. (1988, p. 2)

As mentioned earlier, Pritchett also speculates on reasons for such insufficient study of Cage’s compositional system of chance music. The function of chance in Music of Changes and the entire compositional procedure are discussed in Chapter III of his dissertation. In his introduction, Pritchett addresses the unique challenges he and other musicologists face when approaching chance music analytically, asserting that “musicologists have been unable, due to conceptual or methodological problems, to handle Cage’s compositions analytically, and dismissed them from our field of critical vision rather than alter our approach” (1988, p. 5). He attempts to clarify the difference between approaching traditional forms of composition and Cage’s chance compositions. The conventional methodology of musical analysis, he posits, uses a musical score as the main source of examining a composer’s musical ideas (1998, pp. 5–6). The score itself can often provide sufficient information for an analyst to investigate various musical relationships and observe the coherence of the structure. However, when chance is integral in the production of the musical score, musical material is organised through random processes. This means that choice in the act of composing is abdicated. Pritchett explains that in order to include the operation of chance, Cage invented a compositional system that enables the incorporation of chance elements; thus, the composer’s ideas can no longer by perceived from the musical score alone. The fundamental difference, according to Pritchett, is that “with chance music the score is arrived at indirectly, via the compositional system. The

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composer designs the system, which, with additional input from some random process, then produces the score” (1988, p. 6). Since Pritchett’s primary aim is to understand the musical thought of Cage in his chance compositions, making a clear distinction between the “final score” and Cage’s invention of a compositional system is central to his analytical approach. In other words, the compositional system is where one can perceive Cage’s thought, more so than in the score: the compositional system is where the composer has control: “We have been so transfixed by the random factors of Cage’s music that we have ignored the domain over which Cage has control: the design of the system” (Pritchett, 1988, p. 8). For this reason, Pritchett’s analysis largely focuses on the compositional system and he attempts to define the nature and parameters of chance operation within Cage’s entire compositional procedure. He proposes that

in approaching this music, one should ask questions “Where does chance enter into the composition?” and “Where has Cage made choices that affect the system and its outcome?” By separating the random from the deliberate, the analytically relevant aspects of the composition are brought into clearer focus, and thus its musical inner working can be discerned. (1988, p. 11)

Therefore, Pritchett’s aim is to distinguish chance elements from non-chance elements in the compositional procedure. One can draw a brief parallel between Ligeti’s analysis of Boulez’s Structures Ia and Pritchett’s analytical approach to Cage’s chance music. Analysing Structures Ia, Ligeti identifies techniques of integral serialism and the non-serial aspects of the pre-compositional plan. As noted in Chapter 4, Ligeti makes a clear distinction between the serial operation of the four main musical parameters and non-serial aspects (such as density, register and tempi), which were determined by the composer. With his analysis of Music of Changes, Pritchett likewise differentiates two paradoxical compositional elements that coexisted in the compositional procedure: chance and choice. The parallels between the compositional system of integral serialism in Boulez’s Structures Ia and Cage’s Music of Changes are further explored later in this chapter. Pritchett examines the compositional system of Cage’s chance music according to his own generic model. He says, “I laid out three basic technical components: a set of fixed, pre-defined elements or ‘givens’; a set of rules to operate on and employ within these givens; and the actual execution of the rules to produce the finished 188

musical score” (1988, p. 12). The significance of this model is that it is not only applicable to Music of Changes but also to other chance compositions of the 1950s, as illustrated in Pritchett’s dissertation. Pritchett has thus developed a generic analytical approach to Cage’s chance composition. Despite the random component of chance music, the randomness exists within the three progressive steps of compositional procedure, which must abide by different types of rules. Therefore Cage’s compositional system in his chance music might generally be understood according to the above three components. Pritchett also examines Cage’s chance composition in the perspective of his compositional evolution. He states that:

Cage’s adoption of chance composition can be seen not as a sudden or even a radical change in outlook, but rather as the logical outcome of his earlier methods. In particular, I will show how Cage’s earlier use of rhythmic structures and sound gamuts was extended to accommodate a more systemic approach to composition. (1998, p. 12)

Throughout Pritchett’s thesis, he attempts to prove this point by carefully investigating the development of the chance system from one composition to the other. The author proves his point by structuring his entire dissertation chronologically, subdividing it into Cage’s pre-chance compositional era (1939–1950), a transitional period (1950–1951) and a chance compositional period (1951–1956).92 In the course of discussing Music of Changes, the author also refers to previous compositions, such as the Concerto for prepared piano and chamber orchestra (1950–1951), and draws similarities. One can observe Cage’s compositional evolution throughout Pritchett’s dissertation. Pritchett had four main sources in understanding and documenting Cage’s chance compositional system (1988, pp. 9–11). Firstly, he had opportunities to interview Cage. Unlike other, previous interviews with Cage, Pritchett’s interviews focused primarily on the composer’s methods. However, Pritchett cautions that one cannot rely solely on these interviews for an understanding of these methods. He writes: “Cage’s memory is not good on matters of details, and he has himself admitted ‘I am not a good historian.’ Therefore, personal statements by Mr. Cage cannot be relied upon by themselves to document his procedures” (1988, pp. 9–10). Secondly,

92 I have briefly described the chapter layout in Pritchett’s dissertation though theses are not the titles of his chapters. 189

Pritchett considers Cage’s own writings about his compositional system, including personal letters to his parents and to Boulez. Again, Pritchett identifies the limitations of Cage’s writings in this respect. As mentioned before, his writing is difficult to comprehend. Thirdly, the author examines various types of musical manuscripts, which

. . . are themselves of various types. There are what I shall call notes, which are early ideas for systems, plans for methods and strategies, or outlines of methods followed. Some works use charts of sounds, durations, dynamics, and so forth, and where these survive, they are of prime importance in understanding the compositional process of which they were a part. There are documents I shall refer to as worksheets: the records of decisions and computation so made in the course of executing the system. . . . There are also a few musical sketches and drafts of the usual sort. (Pritchett, 1988, pp. 10–11)

Pritchett has also organised a numbering system for each page of manuscript in the Cage’s notebook.93 Finally, the published scores are another source but as the author says, again, their usefulness is limited (1988, p. 11). It is evident that Prichett frequently refers to the composer’s notebook when he discusses the chance system of Music of Changes. However, only a few manuscripts are actually cited in the text and, as a result, on a number of occasions it is rather difficult to follow the author’s discussion. Nonetheless, the main strength of Pritchett’s analysis lies in the fact that he was able to examine Cage’s manuscript notes, diagrams and sketches. It seems that without such investigations it would not be possible to understand and document Cage’s compositional system of chance music.

Pritchett’s analysis of Music of Changes

Pritchett provides a comprehensive analysis of the compositional system of Music of Changes. In his analysis every step of compositional procedure is traced: the rhythmic structure, charts system, chance operation and the final production of the musical score. His analysis is descriptive and informative. One of the outstanding features of his endeavour is the way he reveals the complex nature of the compositional process. It is ironic in a sense that randomness, which is the central characteristic of chance music, is part of such a complex and technical compositional system. Pritchett claims

93 A “Catalogue of manuscripts” appears in the Appendix of his dissertation (1988, pp. 311–318). 190

that the process was “not simply a matter of tossing coins and dutifully copying the appropriate sounds from the charts into the score” (1988, p. 151). As he distinguishes chance elements from non-chance elements in the course of his analysis, one can see the detailed pre-compositional plan and many technical and musical decisions made by Cage in the compositional process. In addition, the author attempts to clarify Cage’s terminologies and comments found in the composer’s own writings. Similar to Ligeti’s analysis of Boulez’s Structures Ia, Pritchett’s analysis is a pioneering work in revealing complicated compositional techniques. However, the difference is that Ligeti’s analysis was written three decades earlier, close to the work’s composition, and most of the following analyses have reiterated and built on its findings. This resulted in more widespread awareness of Boulez’s integral serialism than of Cage’s chance procedures. By contrast, the chance system discussed in Pritchett’s analysis has only been available in published form since 1993, although his PhD dissertation was completed in 1988. Due to these circumstances, Cage’s chance system has been far less understood.94 Therefore, this review of Pritchett’s analysis will include a more detailed explanation of the processes and principles than my chapter on Boulez. Pritchett first discusses the rhythmic structure that Cage employs to frame the piece: the proportions of this structure being {3; 5, 6 ¾; 6 ¾; 5, 3⅛}. Although there might be an implication of symmetry,95 he points out that the rationale behind these relationships is not clear. Concerning the rhythmic structure, Pritchett’s explanation is somewhat ambiguous. He says “The overall structure of 29⅝ ×29⅝ is divided into four large parts of one, two, one and two sections respectively, and the whole work lasts well over half an hour” (1988, p. 109). This statement seems to refer to the proportional division of the entire work, meaning the division of the work into four parts. One might assume that what Pritchett refers to as “four large parts” might be the four books that make up the entire piece. Unfortunately, there is no further clarification of the statement and the reader is left in uncertainty about the meaning of “one, two, one and two sections”.

94 Evidently, Boulez’s serial techniques as used in Structures are discussed in several university text books, whereas Cage’s chance operation is rarely considered there and, if so, it is only dealt with in general terms. 95 The imperfect symmetry evident here recalls the way Stockhausen treated symmetry in his compositional design, where minor deviations always upset the balance. 191

Pritchett explains what 29⅝ represents: “One regular, full-sized phrase group consists of 29⅝ measures, expressed as 29 measures of 4/4 metre and one of 5/8” (1988, p. 109). Although the author does not clarify in words, the figure 29⅝ is the result of adding the following numbers: 3, 5, 6 ¾, 6 ¾, 5, 3⅛. According to Figure 6.2, a diagram by Pritchett, the rhythmic structure of the full-sized phrase group is based on the above proportion suggested by the composer. Each number here represents the number of bars. In Figure 6.2, Pritchett also demonstrates two more different sizes of phrase groups; one is a three-quarter-sized phrase group (Figure 6.2 (b)), occurring at the end of the third section, and the other is a one-eighth-sized phrase group (Figure 6.2 (c)), occurring at the end of the fourth section. In order to accommodate the shorter length of phrase group, a change of metre with less common time signatures occurs. Although Pritchett presents a comparison of three different phrase lengths, his explanation does not provide a complete picture of the rhythmic structure in Music of Changes. Since the rhythmic structure and the chance operations are inextricably linked, I now attempt to clarify how the proportions suggested by the composer operate. The suggested proportions (3, 5, 6 ¾, 6 ¾, 5, 3⅛) operate on different hierarchical levels. Firstly, as Pritchett indicates, each number shown in the proportions represents the number of bars, forming a full-sized phrase group (29⅝ bars of 4/4) (see Figure 6.3).96 Secondly, there is a relationship between the given proportion and division of the work into four books (see Figure 6.4). On this level, the proportions are expressed in 3, 11¾, 6¾, 8⅛. Here, each number shown in the sequence corresponds to the number of phrase groups. Therefore, a hierarchical relationship exists in the rhythmic structure of Music of Changes and the number expressed in the sequence means different things depending on which structural level it is referring to. The composer explains the concept of rhythmic structure metaphorically: “In the case of a year, rhythmic structure is a matter of seasons, months, weeks and days” (Cage, 1978, p. 65). Pritchett devotes a large portion of his discussion to the chart systems (1988, pp. 111–129). In reference to Pritchett’s observation, the chart system is directly related to the way Cage creates the polyphonic texture, simply superimposing one layer over

96 In a letter to Boulez, Cage describes the beginning of the phrase group as an “intermediate rhythmic structural point” (Nattiez, 1993, p. 95). However, in the article “Composition”, he uses the terminology “a large unit structural point” (1978, p. 58). 192

another. Throughout the entire piece, the density will vary between one and eight layers.97 Pritchett notes that in order to compose each layer, the composer needed to have three essential musical parameters: sounds, duration and dynamics. 98 These parameters are selected through chance operation. Pritchett explains that “Each set of charts is numbered through consecutively from 1 to 8, the numbers identifying the layer to which they belong” (1988, p. 112). Therefore, three musical parameters used to create the first layer are derived from chart number 1 of sounds, durations and dynamics. Since there are variables of one to eight layers, eight different charts are made for each of these musical parameters (1988, p. 112). Pritchett simply spells out exactly where and how the chance operations occur in the process of composition. Firstly, he explains that the composer designed the chart to correspond to the 64 hexagrams of the I-Ching: “all of the charts contain 64 cells (arranged into eight rows of 8 columns each), so that the cells could be related one-to-one with the 64 hexagrams of the I Ching” (1988, p. 112). Figure 6.5 shows the chart structure given by Pritchett.99 The process of selecting the material from each chart, whether it is sound, duration or dynamics, can be summarised as follows: (1) Cage obtains a hexagram by tossing coins; (2) he finds its number in the 64 hexagrams of the I-Ching; and (3) he chooses a musical element from the corresponding cell in a chart to be used (1988, pp. 112–114). Pritchett discusses the characteristics of charts according to his observations about Cage’s notebook. Firstly, the composer designed sound charts to incorporate sound and silence, as all 32 even-numbered cells represent silence and the 32 odd- numbered cells contain sound materials. Pritchett notices that instead of drawing black cells for silence in the chart, Cage drew eight rows of only four columns in each of the sound charts (1988, pp. 114–115). To illustrate this, Pritchett gives the example of sound chart number two, a scanned excerpt from one of Cage’s notebooks, which Pritchett re-notates since the quality of scanned material is rather poor (see Figure 6.6).100

97 The terminology “layer” used by Prichett here is interchangeable with the terminology “thread” coined by Ligeti in the analysis of Structures Ia. 98 Pritchett clarifies that “The four ‘characteristics of sound’ Cage refers to in his letter to Boulez must be pitch, timbre, duration and amplitude. The first two of these were combined in the charts of sounds, so that only three charts were actually needed” (1988, p. 112). 99 Although Pritchett filled each cell with a number, he noted “The numbers do not actually appear in the charts” (1988, p. 112). 100 Since the quality of this excerpt is too poor to reproduce in this thesis, I have quoted from Pritchett’s later publication, where the same excerpt appears (1993, p. 80). 193

It is evident that for Cage, sound and silence are equally important in a composition. The sound chart is made of single pitches, dyads, aggregates and more complex constellations; “we find various intervallic constructions, the use of grace- notes, flourishes, chords with some notes held and some short” (1988, p. 114). A range of timbral effects are described as well:

There are tones produced by plucking the strings of the piano, by muting the strings with the finger, and by using various sticks or beaters on the strings. In some sounds, notes are depressed silently (notated as diamond-shaped notes) while others are struck sharply, creating resonances by sympathetic vibration. The sound charts also include noises produced on or in the piano, such as by slamming the keyboard lid. In some sounds, the use of pedal is indicated as an integral part of sound. (1988, pp. 114 &117)

Although there are apparent similarities between the Concerto and Music of Changes, Cage introduces a different rule for creating the sound chart for Music of Changes (1988, p. 117). In each sound chart, the 32 odd-numbered cells are further subdivided into two halves (4×4) (the 32 even-numbered cells were omitted in the actual drawing found in Cage’s notebook). Within this sub-chart (4×4) consisting of 16 odd- numbered cells, each row or column (four cells) is required to have all twelve chromatic pitch classes. Pritchett explains that any pitch class is allowed to be repeated across four cells, whether it is a row or column (see Figure 6.7). The benefit of this rule, as the author suggests, is that, firstly, this provides a structure for creating sound materials to be placed in charts and, secondly, it prevents a chart becoming dominated by a certain pitch class or a group of pitch classes (1988, p. 117). Pritchett also notes how the composer abided by this self-imposed rule: “Notations scattered through the sound charts give the pitch-classes necessary for the particular cell to fulfil this plan” (1988, p. 117). 101 It is important to note that, contrary to the composer’s concern in earlier works with percussive and non-pitched sounds, he moved away from the sound world of the prepared piano and in doing so embraced preferences that in fact relate clearly back to Schoenberg’s 12-tone technique and more specifically to those of contemporaries such as Boulez.

101 Unfortunately, the concept of ‘mobility and immobility’ applied to the chart operation is not introduced at this point in Pritchett’s discussion, but the significance of the rule concerning all twelve chromatic pitch-classes here will be further realised once the chart operation of mobility and immobility is understood. 194

The observation made by Pritchett concerning the features of eight charts for duration can be summarised as follows. Like the chart structure for sounds, each of the eight charts is made up of 64 cells. The difference with durational charts is that there are no black cells because duration is applied to both sound and silence. Pritchett comments that Cage desired to achieve a rhythmic variety in Music of Changes, so a large collection of durations are used across the eight charts, creating a “duration gamut” (1988, p. 119). As the author includes an excerpt from duration chart 8 that was scanned from Cage’s notebook, one can see that each cell is often filled by a group of durations instead of one simple durational value (see Figure 6.8). 102 Cage describes this method of adding one note value after another as “segmented”. 103 Some extreme durational values are combined in many different ways to create greater rhythmic diversity, as the author observes: “The constituent parts of these durations consist of values ranging from one thirty-second note to a whole-note, and include sevenths, sixths, fifths, and thirds of beats as well as the common binary division of the beat” (1988, p. 119). Pritchett stresses an important point here that, despite using a traditional rhythmic notation, there is no metrical structure given to the durational scheme. According to Pritchett, in order to facilitate such a situation, Cage has developed a standardised notational technique to communicate, thus:

In Music of Changes, one quarter-note is equal to 2.5 centimeters of length. All other rhythmic values are related to this scale, so that an eighth-note takes up 1.25 centimeters, for example, while a half-note takes up 5 centimeters. Using this system, Cage was able to display easily the ametrical duration within the framework of the metrical rhythmic structure. (1988, p. 122)

This system solved the problem of how to communicate a vast range of segmented durations to a performer. However, it is questionable whether it was the most effective and easy way, as is discussed later in this chapter, when the notational issue is explored. Having discovered lists of durations under the intriguing heading “Generation of Durations” in Cage’s notebook, Pritchett suspects that there was another system that is used to generate durations for Music of Changes. (An excerpt from Generation

102 Since the quality of this excerpt is too poor to reproduce in this thesis, I have quoted from Pritchett’s later publication where the same excerpt appears. (1993, p. 81). 103 This is briefly mentioned by the composer in “Composition”, Silence (1978, p. 59). 195

of Durations is shown in Figure 3-6 of Pritchett’s dissertation.) However, it remains unclear as to how these durations were used or selected. Again, the significance of this will be fully realised as one understands an operational chart regulation called “mobility and immobility”, which is discussed in a later part of Pritchett’s chapter. The following is the observation made by the author concerning Generation of Durations:

The durations of these folios are all clearly in two parts, with each half having either one or two notes. The second halves of these durations change rapidly, going through certain repeated patterns, although the generating principle behind them is unclear. The first half of these durations changes more slowly: in the excerpt of Figure 3-6, all the durations have the same first part of a quarter-note plus a sixteenth-note (this part is not actually written after the first two durations but is implied). All this suggests Cage used some system of permutations or alterations to generate the second portion of a duration, and that when all the permutations had been used, the first part of the duration was simply permuted or altered. (1988, p. 122)

Here, one can see the convoluted process of investigating Cage’s notebook and Pritchett’s attempt to make sense of it all. This one instance reveals Cage’s pre- organisational strategy for creating duration charts. Pritchett explains the features of eight dynamic charts and the rules applied to them (1988, p. 123). He mentions that dynamic charts differ from those of sound and duration in a number of ways. The composer fills dynamics in every fourth cell only, which means that the sequence of numbered cells containing dynamics would be 1, 5, 9 and so on.104 The author further points out that “when the dynamics charts were consulted, it was not just to determine a new dynamic, but also to determine whether a new dynamic was to take over at that point, or whether the previous dynamic was to be continued” (1988, p.123). Similar to the extreme range used for duration and pitch collections, the range Cage employs for dynamics levels is vast, from pppp to ffff.105 As Figure 6.9 shows, a combination of two different dynamics can be seen; Pritchett discusses the meaning and function of it later in the chapter. Moreover, an una corda marking is sometimes indicated, by a dotted line underneath dynamic markings.

104 Pritchett notes that “Cage did not actually draw the black cells, so the charts in the notebook are 2 cells by 8 cells” (1988, p. 123). 105 The dynamic range used here is exactly same as the one Boulez used in Structures. 196

As can be seen, Pritchett’s examination of the composer’s notebook was evidently very thorough. As a result, he was able to explain how the composer dealt with three essential parameters of music (sounds, durations and dynamics). Once these materials are placed in the individual chart, they can be selected randomly through chance operation. It is quite obvious that in determining the range of musical materials, the composer considered the design of each chart in great detail, reflecting his musical preferences. Pritchett discusses the two remaining charts, which are for tempo and density (1988, pp. 123–126). With the tempo chart, Cage puts tempo markings in the odd- numbered cells, as with the sound charts. When an even-numbered cell is chosen, the previously selected tempo is to be repeated. The range of tempi used in the chart is from 40 to 200 per minute. Unlike for sounds, duration and dynamics, though, an example of a chart for tempo is not included. With the density chart, Pritchett demonstrates the organisation in a way that suggests Cage allowed up to eight possible layers: one to eight rows corresponds one to eight layers (see Figure 6.10). An important concept was developed in the operation of the chart system used in Music of Changes: mobility and immobility (1988, pp. 127–129). Pritchett explains that Cage had discovered a deficiency in the chart techniques applied to the Concerto. The problem was that all the charts remained the same. Through composing Sixteen Dances, the composer found a partial solution: replacing eight sounds after each pair of the dances. Pritchett writes: “When setting out to compose Music of Changes, this problem was uppermost in Cage’s mind” (1988, p. 125). The chart technique of mobility and immobility is only implemented in charts for sounds, durations and dynamics. Pritchett observes:

At any given point in Music of Changes, half of the sound, duration, and dynamic charts (either the odd- or even-numbered ones) were designated as “mobile” and the other half as “immobile”. As long as a chart was immobile, its contents did not change. While a chart was mobile, however, any sound, duration, or dynamic in it was replaced as soon it was used. The replacement process, in other words, was random, and dependant on two factors: whether a chart was mobile or immobile at a given point, and which elements in it were used while mobile. (1988, p. 127)

The author later mentions that at the beginning of each phrase group a hexagram number determines charts of mobility and immobility (1988, p. 130). He writes:

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At the outset of Music of Changes, the odd-numbered sound and duration charts as well as the even-numbered dynamic charts were mobile, while the rest were immobile. At the start of each new group in the structure, if the first tempo and density hexagram number were odd, the relationship of mobile and immobile charts was reversed. If the number were even, the prevailing relationship was maintained. (1988, p. 130)

However, he does not explain at which point of composition odd- or even-numbered charts were used as mobile or immobile. In order to operate mobile charts effectively, the composer invented extra regulations to guide the process of renewing the charts (1988, pp. 127–128). According to Pritchett, the composer simply erased the sound material from the cell once it had been selected and used; he then filled the cell with other material. Before Cage removed used material from the sound charts, he copied it to another piece of paper to keep a record of used sound materials. In the process of refilling the cells, Cage followed a self-imposed rule that requires all twelve chromatic pitch classes to appear in each sub-chart (4×4) whether the four cells are in rows or columns. In addition to this, when a particular cell was replaced four times in succession, the composer was obliged to use all twelve chromatic pitch classes by the fourth replacement. Moreover, Pritchett finds four “loose” charts that are numbered 2, 4, 6 and 8 and labelled “static” (1988, p. 128). As a result of examining these charts, Pritchett discovers that these are the initial state of even-numbered charts and he even suspects that Cage’s initial intention, not pursued, might have been to make the even- numbered charts immobile (1988, p. 128). Pritchett discusses the Generation of Durations again in relation to the mobile charts (1988, pp. 128–129). The author speculates that the lists of durations under the title Generation of Durations are not only used to create eight charts for duration but are continually used through the entire compositional process. The prime purpose of this process was to replace durational materials in mobile charts whether they were odd- or even-numbered. Pritchett’s investigation suggests that the Generation of Durations were neither all written at once nor completed prior to the sound charts. Therefore, the Generation of Durations are like a record of how the composer generated the durational materials to refill cells in mobile charts once the used materials were erased. While Pritchett provides a detailed explanation concerning the

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operation of mobile charts for sound and duration, there is no discussion about the operations of mobile charts for dynamics. Following his discussion of the chart system employed to compose Music of Changes, Pritchett explains the actual compositional process, which involved selecting musical materials from charts by chance operations, ordering and combining these selected materials, and finally producing a musical score that could then be performed. One of the fascinating and valuable aspects of Pritchett’s investigation in this part is that it illuminates the several ways the composer intervened, thus emphasising once again the significance of the non-chance elements in the compositional process. Prior to composing each layer, which requires three musical parameters (sound, duration and dynamics), the tempo and density were determined at the beginning of each phrase through tossing coins. In other words, there was a chance of having different tempi and densities for each phrase with an exception that when an even number was chosen, the tempo was to remain the same. Pritchett suspects that the tempo and density for the entire piece were selected before each layer was composed (1988, pp. 131–132). This is based on the observation that if a layer was active in the following phrase—that is, if it crossed the boundaries of a phrase structure—Cage did not abruptly cut it off, but continued to compose it until the chosen duration was filled. A new sound was then used to construct the layer in the new phrase. However, if the layer was inactive in the following phrase, the remaining materials for the layer would stay as silence. During the process of determining the tempo and density for each phrase by the hexagram numbers, the relationship between mobile and immobile charts was also determined at the beginning of each phrase group. As mentioned earlier, the composer used the same hexagram number chosen for the first phase of a phrase group (29¾ bars) to decide which charts became mobile and which were to be immobile (1988, p. 130). Once the tempo and density for a phrase were known to the composer, each layer for this phrase was composed. Pritchett suggests that Cage obtained a hexagram number to select elements from the relevant chart. An element was chosen first from the sound chart, second, from the durational chart, and finally, from the dynamic chart. When silence was selected instead of sound, the composer still proceeded to obtain a hexagram number to select the element from the appropriate durational chart since silence also requires a durational value, but there is no need for a dynamic. The same 199

procedure was repeated until the length of a given layer was completed. Cage then composed the next layer within the phrase, repeating the same process. Every phrase composed in this piece followed the above compositional process (1988, pp. 130– 132). In Figure 6.11, I have provided a summary of the compositional process as revealed through Pritchett’s analysis as reviewed so far. Cage devised another rule to prevent any repetition of sound materials. When the features of the newly selected sound material resembled adjacent sounds, Cage allowed himself to manipulate the situation. He called this “interference” (1978, p. 58). The composer eliminated the similarities by cutting short the previous sound material or by leaving out the like parts of the newly selected sound. In my view, this rule speaks of Cage’s musical taste influencing the compositional process and, more precisely, of his compositional aesthetic. It is hard to know the degree and frequency of these manipulations, and Pritchett does not provide any further discussion on this matter. Nonetheless, he later gives one example of interference where sound material from cell number 45 in chart 2 is modified in bars 98–99 (1988, p. 149). In order to demonstrate the compositional process for Music of Changes, Pritchett dismantles the first, three-bar phrase of the piece (1988, pp. 132–137). In this example, the author says that the number 47 was chosen to determine tempo and density (six layers). He provides the sound material selected for each layer and explains how the six layers were superimposed once these sound materials were assigned durations and dynamics.106 Concerning this example, Pritchett states:

This is but a general outline of the method used to composer Music of Changes, however. The materials used—sound, dynamics, and durations— were complex and diverse, making their coordination far from a mechanical process, but rather one that allowed for a considerable amount of creativity on Cage’s part. (1988, p. 137)

Indeed, as previously mentioned, the materials available in each chart for the three parameters are extremely diverse. In addition to the almost infinite range of possibilities being available to the composer, Cage had no foreknowledge about which sound would be coordinated with which duration and dynamic, as he depended on chance to take over the process of selection. Thus, Pritchett suggests, when sound and duration were put together and when dynamics were applied to a sound that

106 The same example appears in the later publication of the author, The music of John Cage (1993, pp. 84–87). 200

already had a durational value, Cage needed to make a certain level of adjustment. For instance, Pritchett examines cases where the compound dynamic markings ff>mf and f>mp were applied to various types of sound materials such as one note, a group of notes, long trills, tremolo and other figurations (1988, pp. 137-141). On such occasions, the compound dynamics can be reinterpreted as accents, attack and decay and even a crescendo or decrescendo.107 A similar technique was applied when sound and duration were coordinated. According to Pritchett, when a single note was chosen, either the whole duration or the first one or two segments of the whole duration were used. There are times when a note is played as short as possible, particularly when the density is high. It seems that sound material consisting of multiple attacks was treated more diversely: segments of duration are permutated, altered, or even subdivided when the number of attacks in the sound material is greater than the number of segments in the chosen duration; some segments were also turned into silence. Pritchett observes that such frequent manipulations in the process of composition are evidence of the composer intervening in the chance procedures. Pritchett is also convinced that “Every attempt was made, through the manipulation of durations, as well as the use of the sustaining and sostenuto pedals, to make the sound combination produced via the I Ching playable” (1988, p. 151). Pritchett strongly argues that the use of chance operations in this piece was not merely a mechanical process (1988, pp. 137 & 151). He further supports this assertion by closely examining occasions where Cage overrode the chance procedure. Such interventions were no doubt a result of Cage’s creativity and musical taste even though he himself tried to deny these qualities by employing chance operations in the first place. Cage writes in relation to the compositional process of the piece as follows:

It is thus possible to make a musical composition the continuity of which is free of individual taste and memory (psychology) and also of the literature and “traditions” of the art. The sounds enter the time-space centered within themselves, unimpeded by service to any abstraction, their 360 degrees of circumference free for an infinite play of interpenetration. (1978, p. 59)

However, what Pritchett has demonstrated in fact suggests a contradiction between what the composer claimed here and what he actually did through the chance

107 Cage mentioned the various interpretations applied to compound dynamics especially for performance but did not provide any examples to bring clarity to the matter. 201

operations. Furthermore, the musical evidence of Cage’s choice, pointed out by Pritchett, is the chart system itself, where all the possible musical material was predetermined or created. The author writes:

The charts of sounds, durations and dynamics were Cage’s definition of the total range of possibilities. It was at this stage of composition—defining the musical elements—that Cage had the most influence on the ultimate nature of Music of Changes. (1988, p. 153)

Pritchett considers Cage’s innovative way of notating duration, “space-time” notation, as one of the outstanding achievements in the composition of Music of Changes. The author notices that “an eighth-note in Music of Changes is simply an eighth-note, and not a half of a beat needing to be completed”; in other words, the rhythm seems to exist in “the arbitrary succession of durations” (1988, p. 154). As explained by the composer himself on a number of occasions, including in the preface to the piece, in space-measured notation, a crotchet equals the distance of two and half centimetres. Thus, as it is notated in 4/4 time, the length of each bar is ten centimetres. Pritchett sees that such a method of notation was needed to keep vertical alignment. However, he questions the consistent use of bar lines when there is no sense of perceptible metre throughout the entire piece. The only places where the author thinks bar lines are necessary are when the lengths of phrases are shortened due to fractions specified in the predetermined rhythmic structure. Pritchett also observes how the function of the rhythmic structure in Music of Changes differs from his previous works (1988, pp. 155–56). In Cage’s other works, one of the primary reasons for employing a rhythmic structure is to provide a sense of phrasing that relates to the overall structure of a work. Even when tempo changes are incorporated, Cage would make an effort to modify the length of phrases to maintain the general sense of proportion. However, in Music of Changes the rhythmic structure functions differently. Pritchett explains:

By adding the possibility of changing tempi for each phrase, Cage removed the relationship of metrical to clock time; with the removal of a metrical underpinning of the rhythm, there was no longer any apparent sense of phrase structure as well. Instead, structure in Music of Changes served solely to regulate the chance system. . . . [R]hythmic structure no longer dealt in audible proportions, but rather provided Cage with a beginning and an end to the

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process, along with a collection of points in between, which existed only to signal various compositional decisions. (1988, p. 155–156)

Therefore, one can see the function of the rhythmic structure in Music of Changes is subordinated to the chance operations, even though the composer designed the rhythmic structure prior to them. Finally, Pritchett perceives Cage’s primary compositional aim in Music of Changes as achieving greater flexibility among infinite musical possibilities. This aim is reflected through his compositional techniques, which accommodate changes on multiple levels, such as the ametrical arrangement of duration, mobile charts, changing tempi and texture. In my view, these techniques ensure constant changes not only in isolated elements of the piece but in every facet of the musical fabric.

Pritchett’s 1993 analysis

The analysis of Music of Changes reappears in Pritchett’s later publication, the chapter, “Throwing sound into silence (1951–1956)” in The Music of John Cage (1993). While the central focus of Pritchett’s PhD dissertation was Cage’s compositional system, in this book, the author recognises the significant change of Cage’s compositional aesthetic, which eventually led him to explore chance operations. Pritchett identifies that Zen’s doctrine of “no-mindedness” was the primary inspiration in developing chance operations and became a focus of Cage’s experimentation. The author explains that

In Cage’s view, then, any useful compositional method or technique should serve as a means of emptying the mind of thoughts that would exclude possibilities. Chance operations are particularly effective here, since chance effectively blocks the exercise of one’s accumulated knowledge and prejudices. (1993, p. 76)

The author further clarifies Cage’s reinterpretation and application of Zen to his musical world: “For Cage, no-mindedness meant that the mind should be alert to sounds, but empty of musical ideas” (1993, p. 77). Thus, prior to the detailed analysis of Music of Changes, the aesthetic position of the composer is well addressed. Pritchett thus illustrates that Cage developed chance operations to accommodate his newly formed aesthetics.

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The content of Pritchett’s analysis in The Music of John Cage (1993) is derived, with some modification, from his PhD dissertation. It is better organised and is more concise. While the chart system is fully explored and explained as it was in his dissertation, a considerable amount of the discussion concerning Cage’s compositional process is excluded. For instance, Pritchett’s investigation of how the composer permitted himself to intervene or override the chance operations in the process of combining the three primary parameters of sound, duration and attack is omitted. It is unfortunate that such fascinating observations are left out. In my view, the value of this investigation also lies in the fact that the detailed explanation of compositional process was only possible because Pritchett had access to the composer’s notebooks and sketches. Without such an account of these matters, it is hard to fathom the compositional procedure and, more importantly, how the composer employed and interacted with the chance operations. While Pritchett did not consider the listener’s perspective in his dissertation, in the later publication the author briefly mentions the surface structure of the work. He writes:

Changes in density are among the most prominent musical features of Music of Changes. . . . During phrases of low density, the listener attends to the contours of individual events; during periods of high density, the ears are overloaded, the events become unfocused, and the impression is predominantly textural. (1993, p. 88)

In the later publication, Pritchett cleverly accentuates and enhances his analysis by first elaborating on Cage’s aesthetics, then embarking on a complicated discourse on the compositional procedures. His analysis continues the tendency, identified earlier, for analysts to neglect close examination of Cage’s chance operation. It seems that Cage invented the chance system to achieve his aesthetic aim of “no-mindedness” but he was nonetheless unable to fully accomplish this aim.108 Pritchett’s later analysis reveals this clearly by identifying and distinguishing between the roles of chance and non-chance elements. Pritchett writes:

It is the composition of the materials—the charts of sounds, durations, and dynamics—that most strongly determines the effect of Music of Changes and that gives it its unique and unmistakable voice. In this work, perhaps more clearly than in any other, chance appears as a means and not an end in itself. It was necessary for Cage to use the chart technique in order to have his musical

108 This point has been also discussed in Uno’s analysis (1994), which is reviewed later in this chapter. 204

materials—which are completely products of his compositional choice and judgment—speak by themselves, without being forced into a particular sort of continuity. (1933, p. 88)

Through the analytical investigation of chance operations in Music of Changes, especially the critical role played by the charts system, Pritchett argues that one can still trace Cage’s compositional choice and judgement regardless of his aesthetic position.

Schädler’s analysis

Schädler (1990) investigates the compositional process of Music of Changes and provides a comprehensive analysis in the article “Transformation of the time-concept in John Cage’s Music of Changes (Transformationen des Zeitbegriffs in John Cages Music of Changes)”. The detailed account of Cage’s compositional technique, including chance operations, can be found in the subsection “The type of process in Music of Changes (Die Verfahrensweise der Music of Changes)”. This section is followed by three supplements where the author further elaborates on the compositional process and techniques explained in the previous section. As the title of the article suggests, Schädler thoroughly examines Cage’s concept of musical time, which is structured, expressed and communicated in Music of Changes in several ways: through the rhythmic structure, through segmented durations appearing in charts and, most importantly, through the innovative notational system where, as noted above, the value of duration is converted to a measured space-notation, where a crotchet equals 2.5 centimetres. The discussion of these aspects is integrated into Schädler’s analysis of the complex compositional process involved in Music of Changes. Although Schädler’s article was published in 1990, that is, between Pritchett’s PhD dissertation (1988) and his book (1993), these two authors did not seem to be aware of each other’s analysis, since neither of them refer to the other. It is interesting, however, to note the similarities in their methodologies. Like Pritchett, Schädler points out how Cage’s retrospective description of the compositional process used for the piece provides an unsatisfactory and insufficient explanation of the structure and function of the charts and the chance operations. The author thus claims that the function of these charts and the compositional process remain concealed (1990, p.

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191). In order to investigate the compositional process, Schädler examines the composer’s notebook, which includes the entire sketched materials which preceded the writing of the piece.109 Throughout the analysis the author also refers frequently to Cage’s own writing. Since both Pritchett’s and Schädler’s analyses of this piece are based on the composer’s sketches and writings, they both address the same problem, Cage’s rather mystifying explanation of his compositional process. Despite the two authors’ methodologies and the content of their analyses being similar, their emphasis is different. 110 As my review demonstrated, Pritchett’s fundamental aim is to distinguish between how chance and non-chance elements were used; Pritchett spends much time explaining the structure and system of charts and the various self-imposed rules Cage applied to the charts system, which allowed him to intervene in the chance operations. Interestingly, Schädler accounts for the compositional techniques that Pritchett did not fully explore or clarify in his analysis: the rhythmic structure, the relationship between the mobile and immobile charts and Cage’s new innovation of space-notation. These are discussed in Pritchett’s analysis but their significance is further highlighted by Schädler. Although Schädler does not use musical examples to demonstrate how Cage’s musical judgement influenced the final outcome of the piece, he provides extremely useful materials for a reader; for example, twenty-four pages of Cage’s sketches selected from his notebook are scanned and included in the Appendix. Schädler often refers to the relevant pages of Cage’s sketches in his analysis. With the scanned materials provided in his article, a reader can be far better informed and engage more meaningfully with the analytical discourse.111 Schädler’s analysis not only clarifies the rhythmic structure but also connects it to Cage’s concept of the “square root”.112 Schädler suggests the hierarchical nature of the rhythmic structure by using the terms “small” structural point, a “middle” structural point and a “large” structural point (1990, pp. 191–192). A middle structural point occurs at the beginning of each grouping of 29¾ bars. Here, a middle

109 Schädler also provides a clear description of the content of Cage’s notebook (1990, p. 191). 110 Since I have given a detailed description of the compositional process in the preceding review, here I review and discuss only those aspects that are not fully explored by Pritchett. 111 It is unfortunate that an English translation of his analysis is not available. 112 Cage states that “My recent work (Imaginary Landscape No. IV for twelve radios and the Music of Changes for piano) is structurally similar to my earlier work: based on a number of measures having a square root, so that large lengths have the same relation within the whole that the small lengths have within a unit of it” (1978, p. 57). 206

structural point equates to Cage’s ‘intermediate rhythmic structure point’ (Nattiez, 1993, p. 95).113 Each of these units is divided by the almost symmetrical proportions given by Cage, 3, 5, 6¾, 6¾, 5, 3⅛, and Schädler calls each sub-unit, according to these proportions, a small structural point.114 The large structural points mark the division of the overall piece into four books. Schädler explains that subdivisions of the entire piece are derived from the same proportions that govern the middle and small divisions: 3, 11¾ (5+6¾), 6¾, 8⅛ (5+3⅛). It is worth noting that the symmetry evident in the proportions of the lower levels is concealed in this upper level. Thus, Cage’s concept of the “square root” can be perceived in these hierarchical levels, where a large unit is governed by the same proportions as the smaller ones that comprise it. Moreover, the author emphasises the fact that in Music of Change the rhythmic structure is connected with spatial notation where the duration is understood dimensionally, with the duration of a crochet corresponding to 2½ centimetres. Schädler says “From here, the symbolic notation of duration has dissolved, the sound enters on the ‘space-time-point’, which the score shows” (1990, p. 192).115 While Pritchett’s discussion of the rhythmic structure of Music of Changes seems brief, Schädler approaches the subject more systematically. The three hierarchical layers of rhythmic structure discussed by Schädler particularly enhance one’s understanding of how Cage attempted to achieve structural coherence through the element of rhythm. Here, Cage is replacing a hierarchical system dominated by pitch (in tonal music) with a hierarchical system based on rhythm. One might suggest that Cage’s three-layer rhythmic structure almost parallels the way Schenker interpreted the tonal structure of a piece in three layers: foreground, middle ground and background. However, one of the main problems one faces concerns terminologies, since Pritchett, Schädler and Cage all use different ones. This aspect has hindered a clear understanding of Cage’s organising scheme. Figure 6.12 compares all the different terminologies used to describe the rhythmic structure in Music of Changes. As a part of his discussion of the chart system, Schädler gives greater attention to the operation of mobile and immobile charts (1990, p. 194). He contemplates

113 As mentioned earlier, Cage’s term “an intermediate rhythmic structure point” is used interchangeably with another of his terms, “a large unit structural point” (1978, p. 58). This use of two different terminologies to describe the same phenomena is confusing. 114 At the beginning of each small structural point, the tempo and density are determined by chance. 115 Damit löst sich die symbolische Notation von Dauer auf: der Klang tritt ein an der Raum-Zeit-Stelle, die das Papier bezeichnet. (Translated by Christina Young and Sun-Ju Song.) 207

thoroughly the extent to which the action of erasing the used materials in mobile charts has affected the compositional process. Schädler notices numerous signs of an eraser having been used on the charts, which most likely confirms the idea that Cage refilled the mobile charts with new materials. In the Appendix, the author provides the part of Cage’s notebook where the composer copied all sound materials used before they disappeared or, to paraphrase Cage, “passed into history” (1978, p. 58) by the action of erasing. In the process of composing Music of Changes, the function of the eraser is redefined; according to Schädler, it no longer has the function of correcting mistakes but has become an instrument for abolishing musical ideas. The process of erasing musical material separates the past from the future and even creates a vacuum between the sounds (1990, p. 194). In order to refill the sound charts, Cage created new sound materials and then moved the used materials from the mobile charts into the sound archive. Schädler observes that “[t]he whole composition inaugurates a complex process of interlocking moments of insistence and of disappearance” (1990, p. 194).116 Schädler also draws attention to the changes in the relationship between mobile and immobile charts throughout the entire piece, an aspect overlooked by Pritchett. The author observes that in Book 1, only the odd-numbered charts are mobile and the even-numbered ones stay immobile throughout the whole book, while in the other three books, the relationship between mobile and immobile charts changes (see Figure 6.13). At the beginning of the rhythmic structure (a middle structural point), the relationship between mobile and immobile charts is determined by chance. As Figure 6.13 shows, in book 2 the author observes symmetrical division of twelve rhythmic structures with the exception of rhythmic structures 4 and 10, where the relationship remains the same as in the preceding section.117 Book 4 also exhibits the symmetry in the changing relationship between the mobile and immobile charts (the author comments that the last rhythmic structure being so short, ⅛, no decision is made). Although the author does not explore this matter any further, it is interesting to note that, apart from in Book 1, both even- and odd-numbered charts have the chance to be mobile. This means that all the charts have almost equal opportunities to be renewed in terms of their content.

116 Die gesamte Komposition inauguriert einen komplexen Prozeβ von ineinandergreifenden Momenten der Insistenz und des Verschwindens. (Translated by Christina Young and Sun-Ju Song.) 117 Schädler purposely omitted rhythmic structures 4 and 10 in Figure 6.13. 208

Schädler explores various concepts relating to musical time in Music of Changes: spatial notation, segmented durations and their application through chance operations and rhythmic structure and its relationship to tempo changes. Firstly, Schädler says “Setting out from the spatial conception of duration, Cage conceived duration as a division of the space of crotchet equals 2.5 cm and as the addition and combination of spatial division and durational value” (1990, p. 196).118 The author here refers to Cage’s compositional technique of “segmented” duration, where durations are joined together, one after the other, by the addition of up to four components. However, the additive nature of the segmented duration is realised through visual measurement that employs both division and addition119 (see Figure 6.14). Secondly, Schädler further discusses Cage’s method of creating the lists of segmented durations that are recorded in his notebook under the title Generation of Durations. Although Pritchett also investigated these lists of segmented durations, the exact procedure was unknown to him: “Cage clearly had a system for generating the durations of Music of Changes,” he noted, “but this method remains unclear” (Pritchett, 1988, p. 119). According to Schädler, Cage implemented Tarot-games to produce a set of the segmented durations; the composer wrote a different duration on each card and then randomly drew the cards one after another (1990, p. 196). In other words, the process of joining different durational values was entirely dependent on chance. Thirdly, as Pritchett also mentions, once a sound material is chosen, it is assigned a segmented duration which is also selected by chance. Schädler describes such compositional procedures metaphorically as follows: “it needs to be observed that the segmented durations create a ‘refraction’ of the sound constellation due to the unforeseeable oscillation nodes. This is similar to the refraction of light in a prism” (1990, p. 197).120 Schädler tries to relate the compositional procedure to the comment written in Cage’s notebook: “duration, color, speed focus”. The author interprets the

118 Ausgehend von der räumlichen Vorstellung von Dauer, konzipierte Cage Dauern als Unterteilungen der Strecke von [Viertelnote] = 2½ cm sowie als Addition und Kombination von Streckenteilen und Dauernwerten. (Translated by Christina Young and Sun-Ju Song.) 119 This method is similar to the way Messiaen developed his ametrical rhythm. His explanation here clarifies what Cage presents in his letter to Boulez written on May 22, 1951 (Nattiez, 1993, p. 95). 120 Es ist jedoch zu beachten, daβ die segmentierten Dauern durch die unvorhersehbaren Schwingungsknoten eine “Brechung” der Klangkonstellationen herstellen analog der Brechung des Lichtes in einem Prisma). (Translated by Christina Young and Sun-Ju Song.) 209

procedure in which a sound material gets assigned to a segmented duration and then the durational value of sound is realised through spatial notation. He writes:

In the projection of sound into the vibration space, a connection of time and space categories takes place. The concept of duration connects sound with speed, that of segmented duration connects sound with the refraction of light. On the basis of the appearance of sound in time, an optical model—a visual procedure—is established. Time: duration is obviously no longer the absolute difference of spatial expansion, but starts to dissolve in the spectrum which the sound prism projects. In this way musical time is separated from a notion in which the teleological unfolding of an idea, a material, a subject was seen, a notion that still dominated Schoenberg’s row technique.121 (1990, p. 197–198)

Finally, Schädler discusses the relationship between tempo changes and the rhythmic structure. Since the composer allowed tempo changes to occur whenever an odd- number was chosen at the beginning of a small structural unit, Schädler perceives that the tempo in this piece no longer refers to a single musical event but to the length of the rhythmic structure. He says, “tempo is not based on the individual events, but on the space of rhythmic structure: it does not change the events themselves, but rather the music of the space constituting their relation” (1990, p. 198).122 The author, once again, connects the rate of musical motion to the visual realisation of space and points out the compositional procedure Cage chose to interlock tempo changes and the proportional division of the rhythmic structure. While Pritchett’s analysis primarily focuses on the compositional procedure of chance operations in Music of Changes, Schädler’s analysis illustrates several ways in which the composer experimented with musical time in chance music.123 I would like to point out here that Cage’s interest in rhythm considerably predates Music of

121 In der Projektion des Klanges in den Schwingungsraum erfolgt also eine Verknüpfung zeitlicher und räumlicher Kategorien. Der Begriff der Dauer verbindet Klang mit Geschwindigkeit, der der segmentierten Dauer Klang mit der Brechung des Lichts. Auf dem Grunde der Erscheinung der Klanges in der Zeit findet sich so ein optisches Modell, ein visueller Vorgang. Zeit: Dauer ist offenbar nicht mehr das absolut Andere räumlicher Ausdehnung, sondern beginnt sich in dem Spektrum, welches das Klangprisma wirft, aufzulösen. Damit trennt sich musikalische Zeit von einer Vorstellumg, die in ihr die teleologische Entfaltung einer Idee, eines Materials, eines Subjekts gesehen hatte und die selbst noch Schönbergs Reihentechnik bestimmte. (Translated by Christina Young and Sun-Ju Song.) 122Das Tempo aber bezieht sich nicht auf die einzelnen musikalischen Ereignisse, sondern auf den Raum der rhythmischen Strucktur: es verändert nicht die Ereignisse selbst, sondern die Musik des Raumes, der ihre Relation konstituiert. (Translated by Christina Young and Sun-Ju Song.) 123 Pritchett also acknowledges the significance of spatial notation as well as the function of the rhythmic structure being different from Cage’s previous work due to tempo changes and chance operations. However, the focus of his analysis still remains on the compositional system of chance operations. 210

Changes, as his output of percussion music attests. The composer’s ambition to extend and explore the concept of time as a prime compositional technique is revealed in the following statement:

The composer (organizer of sound) will be faced not only with the entire field of sound but also with the entire field of time. The “frame” of fraction of a second, following established film technique, will probably be the basic unit in the measurement of time. No rhythm will be beyond the composer’s reach. New methods will be discovered, bearing a definite relation to Schoenberg’s twelve-tone system. (1978, p. 5)124

Through Schädler’s investigation of Music of Changes, one can see that Cage did not only create a compositional system that was capable of incorporating chance elements, but he also continued to develop a complex rhythmic language. Schädler’s analysis attempts to conceptualise how Cage redefined time, duration and rhythm in Music of Changes, especially through the visual and spatial notation of duration. Schädler often employs the language of metaphor to describe various unfamiliar temporal concepts, and this is one of the prominent features of his analysis. In this respect, Schädler’s analysis greatly differs from the other analyses reviewed in this thesis. In the last part of his article, “Supplement III: repetition and nothingness” (Supplement III: Wiederholung und Leere), Schädler explains “the structural chain” (die Strukturkette)(1990, p. 211) on which the entire compositional process is built. He describes this chain of structure with three words, to which he later adds a fourth. The first three terms are repetition, chance and change: these terms need to be understood as “changing exposed structure” (wechselseitig erhellende Struktur)(1990, p. 209). The fourth element is void. He writes:

Indeed, the musical materials are organised according to the principle of strict non-reversibility in a process vacillating between an insistent and a continuing process. It executes the law of variation as a continuing newness which arises in the face of the few, accidentally defined repetitions. . . . The continuing newness becomes possible through the continuing repetition of action: by the choice of chance.125(1990, p. 209)

124 Cage first delivered the text at a talk in 1937 and the same material reappear as in the brochure accompanying George Avakian’s recording of Cage’s twenty-five-year retrospective concert in 1958, in New York. 125 Zwar ist das musikalische Material organisiert nach dem Prinzip strikter Nicht-Umkehrbarkeit in einem zwischen Insistieren und Fortschreiten wechselnden Prozeβ. Er vollzieht das Gebot derVariation 211

Therefore, Schädler clarifies that the concept of repetition does not refer to recurring musical ideas but to the repetitive action involved in the chance operations. The author explains that the compositional procedure constitutes the continuing repetition of a chance process that produces relentlessly changing musical materials. Cage drew cards randomly to create a list of segmented durations and tossed coins to obtain the hexagram numbers. In order to obtain a hexagram number from which he would select sound, duration and dynamic from the charts, the action of tossing coins had to be repeated many times, producing a constant change of musical materials which appear for a time and pass away (“like history”). The chance-repetition was required to obtain the next new musical material until the determined length of the piece was completed. The result of such action is captured by the author’s following statement signifying the fourth element, void: “Sound lines up next to sound, new next to new and the act of progression repeats itself to: . . . nothing” (1990, p. 210).126 Finally, Schädler connects this aimless procedure of chance repetition to the concept of void, which the composer embraced during the time he developed the compositional system of chance. Cage’s aesthetic aim to achieve void in his compositions is also revealed in his “Lecture on nothing” (1978, pp. 109–127). However, Schädler finds the mention of void in Cage’s autobiographical note and concludes his article with a reference to it. In the last part of the article summing up why Cage was attracted to chance, Schädler quotes:

It was Wednesday. I was in the sixth-grade. I overheard Dad saying to Mother, “Get ready: we’re going to New Zealand Saturday.” I got ready. I read everything I could find in the school library about New Zealand. Saturday came. Nothing happened. The project was not even mentioned, that day or any succeeding day. (1990, p. 212, citing Cage)

It is interesting to note that Schädler believes the appearance of void derived from the composer’s childhood experience whereas Pritchett and others often relate this to Cage’s adoption of Buddhist philosophy. For instance, Pritchett writes: “Cage’s nothing is analogous to what D. T. Suzuki says about the Void of Buddhism: ‘it is zero full of infinite possibilities, it is a void of inexhaustible contents’” (1993, p. 58).

als ein ständing Neues, das sich vor den wenigen zufallsbestimmten Wiederholungen abhebt. . . . Das unablässig Neue wird möglich durch die unablässige Wiederholung einer Handlung: der zufälligen Wahl. (Translated by Christina Young and Sun-Ju Song.) 126 Klang reiht sich an Klang, Neues an Neues, und doch wiederholt sich nur der Akt des Fortschreitens zu: . . . Nichts. (Translated by Christina Young and Sun-Ju Song) 212

But there is no mention of Buddhism in Schädler’s article. Schädler’s approach to understanding the relationship between void and Cage’s compositional aesthetic seems to be based more on Freudian psychoanalytic theory, while Pritchett refers to Cage’s religious belief. In summary, Schädler’s analytical approach to Music of Changes is conceptually oriented. His explanation of the piece is strongly derived from the composer’s point of view rather than that of a listener or performer. It is evident that Cage’s writings about the piece and the operation of the chance system are central to Schädler. Furthermore, he successfully demonstrates that, in the midst of creating the chance procedures for Music of Changes, Cage continued to develop his rhythmic language and explore the dimension of musical time.

Uno’s analysis

Uno examines only the first book of Music of Changes and provides a lengthy and technical analysis of it in her PhD dissertation “The roles of compositional aim, syntax, and design in the assessment of musical styles: Analyses of piano music by Pierre Boulez, John Cage, Milton Babbitt, and Iannis Xenakis circa 1950” (1994). The basis of her investigation is:

I argue, in fact that the repertory circa 1950 poses specific problems with respect to the disparity in the composer’s ideological aim, compositional method, and surface design. . . . My goal in this dissertation is, therefore, to define the “fit” among the compositional aim, generative syntax (precompositional structure), and design (the gesture and formal designs that are discernible at the musical surface) in the assessment of musical style for this repertory. (1994, p. iv)

Since the author applies the same analytical methodology I discussed in detail in while reviewing her analysis of Boulez’s Structures Ia in Chapter 4, this part only focuses on the outcomes of her investigation. Uno relates the aesthetic aim in Music of Changes to the composer’s intention to eliminate “individual taste and memory” from a musical composition. Based on this, the author raises three central questions in her analysis:

Specifically, how are the chance procedures associated with the practice of I Ching—the ancient Chinese “book of changes”—used to this end? Since not all aspects of the composition can be left up to chance, to what extent do the

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predetermined aspects of the musical structure influence the compositional outcome? And lastly, how do the analyses of gestural and formal designs of Music of Changes support Cage’s ideological claim? (1994, p. 131)

Unlike the two previous authors, Uno gives a history and overview of the I-Ching and then links them to Cage’s aesthetic aim. She describes the essence of the I-Ching as follows, by referring to Richard Wilhelm’s explanation:

While the oracles are subject to manifold interpretation, Wiihelm sees the notion of change as the essence of the I Ching, viewing events in transition as more significant than the events themselves. The objective world is seen as being dynamically interconnected with the observer’s subjective state of mind. Cage’s adaptation of the I Ching presents an abstraction of the procedure from the principles and application of the I Ching, e.g., the interpretation of the oracles. His aesthetic aim is to remove his “taste” and “memory” through the coin-tossing method in the selection of sound materials. Nonetheless, the musical realization attests to a dynamic interaction between the “objective” (rational control) and the “subjective” (taste) as will be revealed by the discussion of the generative plan for Book I of Music of Changes. (1994, p. 136)

The author’s statement suggests that she aims to understand how exactly the philosophical aspects of I-Ching influenced both the compositional aesthetics and process. Throughout Uno’s analysis, however, she argues that the composer was only partially successful in abdicating his “taste” and “memory” since the sonic materials and the rhythmic gestures display the composer’s musical preferences. Uno further notes that his preferences shown on the surface structure of the piece even suggest the atonal musical language of the Second Viennese School and Boulez’s piano works from the same period, in particular the Second Piano Sonata (1994, pp. 184–185).127 The only aspect for which Cage utilised the I-Ching was with the coin-tossing method in order to obtain a hexagram number between one and sixty-four (1994, p. 144). Therefore, Uno draws a remarkably similar conclusion to Pritchett’s. Given that the compositional process in Music of Changes involves the chance element, it is interesting to consider what Uno defines as the “generative syntax (pre- compositional structure)”. Her analysis reveals that the pre-compositional structure

127 Schädler also notes Cage’s close familiarity with this piece at the time and suggests that the atonal language in this piece is due to applying a 12-tone technique that, to some extent, resembles that of Schoenberg’s serialism (1990, pp. 188–191). 214

consists of four parts: first, the temporal and rhythmic organisation based on the rhythmic structure; second, the organisation of charts for sound, duration, tempo and dynamic; third, the selection of materials from charts through the method of tossing coins; and finally, the realisation of the score to enable performance (pp. 137–151). The content of her discussion concerning the generative plan was mostly derived from Pritchett’s PhD dissertation, which she acknowledged. One outstanding difference here is that Uno applies pitch-class set analysis to examine the second sound chart. Contrary to Pritchett’s observation that Cage attempted to design a sound chart which “would not be skewed towards any particular pitch class or group of pitch classes” (1988, p. 117), Uno’s pc-set analysis illustrates that related set-classes occur among different cells in the chart, and common chromatic subsets can be found among many of the larger sets: 3-1 {012}, 3-2 {013} and 3-3 {014}. She also notices that many of sound materials contain a leap of a major seventh or minor ninth as linear intervals (1994, pp. 140–141). Uno’s claims concerning the pc-set analysis are insightful, but her case could have been much stronger if she had analysed the remaining sound charts. Uno investigates the pitch structures and rhythmic gestures of Book 1. Her analysis focuses on the surface structure of the piece. She says “While this piece obviously lacks conscious manipulation of thematic or motivic connections, there are different types of pitch and rhythmic gesture that establish patterns of continuity and discontinuity in the course of the piece” (1994, p. 152). It is worth noting that Uno’s approach here differs significantly from Pritchett’s. Pritchett argues that a traditional approach of examining the score as a main source is inadequate in this case since musical materials are randomly organised through chance procedure. 128 Uno subdivides the first book of Music of Changes into three parts: bars 1–29, bars 30–52 and bars 53–88 (i.e. the end of Book 1). As the number of bars indicates, these subdivisions closely correlate to the rhythmic structure (Book 1 consists of three middle structural units, each lasting 29 bars). However, Uno’s reasons for making such divisions are mainly based on the features of surface structure such as types of gestures, predominant pc-sets, textural density and temporal density. Uno

128 Pritchett writes: “In traditional analysis the score functions as a straightforward and direct medium of communication, while in chance composition it is a randomly derived product of a system designed by the composer, it is a result of his work rather than the work itself” (1988, p. 7). 215

demonstrates in her analysis that Music of Changes can be approached in a rather traditional way. Uno expands the pc-set analysis from the second sound chart to the entire Book 1. Although the distribution and permutation of pc-sets in this piece is a result of chance procedure, a unifying feature in the pitch relationships can be found. Uno says, “While not all of the ‘source’ gestures from the sound charts have been examined, one may account for a putative K-complex around 5-2 and 5-18 based on the content of Book 1” (1994, p. 158).129 Since the surface structure of pitch relationships is created randomly, one possible way of explaining such pitch relationships would be that the design of “source” gestures (sound charts) influenced the resulting pitch-class sets (Uno, 1994, p. 159). This is a rather fascinating analytical observation, although the audibility of the above pc-sets is not measured or mentioned. Uno’s final observation of the surface structure of the piece is the co-existence of randomisation and determination in Book 1 of Music of Changes. Uno writes:

At the macroscopic level, the structure of the piece is tripartite, articulated by the overall changes is temporal density, tempo, and gesture-types. At the local level, the chance procedures tend to create a succession of gestures that defy any consistent or predictable patterns in establishing musical continuity. Even if surface gestures imply musical continuity by the sequential repetition of motives or gesture . . . this effect is nullified by the lack of syntactical connections with the preceding and following gestures. (1994, pp. 160–161)

Furthermore, according to Uno, the rhythmic gestures and sonorities of Book 1 illustrate compositional preferences that are quite different from Cage’s previous works, particularly, “the simple rhythmic structure and the modal sonorities” of String Quartet in Four Parts (1949–1950) (1994, pp. 159–160). Uno believes that the sound materials used in Cage’s sound charts resemble those in Boulez’s piano works from the same period. It is a convincing argument when one considers how these two composers corresponded about the compositional ideas of Structures Ia and Music of Changes. For instance, Uno notes that pianistic idioms like “extended trills, arpeggiations in contrary motion, grace-note figures preceding a simultaneity” were not common in Cage’s previous works (1994, p. 160). Evidently, Cage was well aware of Boulez’ Second Piano Sonata and Uno notes that he liked the piece at the

129 5-2 = {01235}and 5-18 = {01457} 216

time and also asked David Tudor to perform it several times to promote Boulez’s work (1994, p. 160).130 By applying statistical techniques through computer programs, Uno measures several dimensions of surface structure. Firstly, she examines the contours of individual parameters of music—pitch, temporal density, duration and dynamics. Her analysis ultimately suggests that definitive patterns can be seen and these enhance the tripartite division of Book 1. More importantly, these patterns are not random. However, since each parameter in this piece is organised independently through chance operations, the contour of each parameter seems to create its own shape and has little relation to any other. The only exception is in relation to attack and pitch, as they were derived in conjunction with each other (Uno, 1994, pp. 162–169). Later Uno suggests a possible reason for tripartite divisions on the surface structure. She posits that this surface phenomenon relates to the pre-compositionally designed rhythmic structure and especially to how density and tempo change throughout the piece according to the rhythmic structural division (1994, p. 189). Secondly, Uno aims to define the “distribution characteristic of musical dimensions” in this piece. Concerning pitch, she notes that while the middle register of the piano is preferred, all twelve chromatic notes are evenly distributed throughout Book 1. This trend is due to Cage’s self-imposed rule of employing all twelve tones in the sound charts. With duration, he mainly used small durational values. Uno explains that this is also a result of Cage’s rhythmic technique of “segmented” duration. A sustained dynamic level can be seen on the musical surface because of the way Cage designed the dynamic charts. As mentioned previously, Cage only uses every fourth cell in the dynamic chart for a new dynamic to occur and this means that the charts operate on different probabilities from that of duration: the rate of durational change is four times greater than for dynamics (1994, pp. 169–175). Through Uno’s analytical investigation, one can see how the pre-compositionally designed chart system actually affects the surface structures of the piece. Uno further proves Pritchett’s point that Cage’s thoughts and selections for designing various charts are reflected on the surface regardless of chance procedures. Uno precisely defines exactly how in her statistical analysis.

130 As Uno notes, this is documented in their correspondence dating April to December 1950 (Nattiez, 1993, pp. 93–122). 217

Finally, Uno examines the degrees of correlation and regression between parameters. The lowest correlation coefficient can be seen between the dimensions of pitch and duration, which means their relationships are essentially random. However, the dimensions of dynamic and attack display an opposite trend; stronger attacks have the tendency to appear with louder dynamics. Uno speculates on this phenomenon as follows: “The fact that dynamic and attack values belong to the same precompositional charts reflects degrees of conscious or subconscious effort on the part of Cage to coordinate these dimensions” (1994, p. 176). Uno’s analysis demonstrates how the musical parameters were randomly organised through chance procedures. The randomness is also manifested on the surface structure as no concrete relationships exist among the parameters and no sense of musical continuity is perceptible at the micro-level. In other words, Cage’s compositional system does seem to achieve an organisation of musical elements that is essentially random. In spite of the randomness of the surface structure, in the overall shape of the surface structure, Uno argues, it is still possible for one to perceive formal junctures which mostly are indicated by textural density and the “composite attack-time duration”. However, Uno concludes that Cage’s aesthetic aim to free himself from individual taste and memory in the musical composition has not fully succeeded.

Conclusion

Reviewing the three authors’ analyses above, it is clear that Pritchett, Schädler and Uno have each analysed Music of Changes in depth. Although their approaches differ in several aspects, they all contain accounts of the compositional process and in particular define the role of chance procedures precisely. It is now evident that: firstly, Cage constructed complex and systematic procedures to incorporate chance elements in the musical composition; secondly, the chance operations only influenced the ordering of musical parameters such as pitch, duration, dynamic, tempo and density; thirdly, Cage played a primary role in creating the “source” materials that are seen in the pre-compositionally designed charts; and finally, Cage’s aesthetic aim to free musical composition from personal taste and memory was only partially achieved since his compositional preferences are recognisable. Moreover, these preferences were strongly in line with those of the European avant-garde.

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The analytical approaches of Pritchett and Schädler have many similarities. They both thoroughly investigate the composer’s notebook in order to understand the chance procedures and are deeply concerned with the intention of the composer and his compositional system. Since their analytical findings and interpretations of the work are validated through evidential proof, their methodologies are positivistic in nature. Central to their analysis is the investigation of the entire compositional process and the revolutionary composition techniques used in Music of Changes; thus, they both concentrate on the conceptual matter of the composition far more than considering the performance point of view or how it would be perceived by a listener. Despite such similar analytical methodologies, Schädler focuses more on how Cage developed his rhythmic language whereas Pritchett’s main interest is to understand the interdependency between non-chance and chance elements in the compositional process. Interestingly, while their analytical focuses differ slightly, the two analyses complement each other well. Uno’s analysis, however, demonstrates the connection between the compositional system and the surface structure of the piece as it would be perceptible to a listener. Like Pritchett, Uno precisely points out the non-chance elements in the compositional process but takes this further to illustrate what effects these non-chance elements have upon the surface structure. In Uno’s analysis, a statistical measurement is used to clarify the various dimensions of musical relationships on the surface structure. Unlike Pritchett and Schädler, Uno briefly comments on the similarity between Boulez’s Structures Ia and Cage’s Music of Changes noting that, even though one employs serial procedure and the other chance operations, the compositional systems of both works display rigorous planning and, moreover, their methods of construction share some common principles. Although the analytical studies of Music of Changes were published as recently as the early 1990s (Pritchett’s doctoral thesis (1988) is an unpublished work), all three authors’ analyses reviewed in this chapter demonstrate the fact that it is possible to analyse Cage’s chance music; Uno even employs a formalistic approach of analysing pitch relationships and rhythmic gestures. Like many traditional analytical approaches, the compositional procedure and composer’s intention are discussed in these analyses. As a result, one can argue that, despite Cage’s attempt to embrace Eastern philosophies and to employ chance operations in his composition, to a certain extent Music of Changes remains governed by the ideology of formalism. The following 219

section will further explore a paradox that exists between what Cage claimed to achieve in Music of Changes and what his compositional techniques reveal. A re-interpretation of Music of Changes

Analytical premise

A common belief is that chance operations and the compositional aesthetic based on Buddhist philosophy are significant keys to understanding Cage’s Music of Changes. However, in the course of reviewing the previously written analyses of Music of Changes, it is revealed that such matters are not as central to comprehension of the work as is widely perceived. All three authors (Pritchett, Schädler and Uno) define the role of chance in relation to the non-chance elements in the compositional process, thus one is now able to observe Cage’s musical preferences demonstrated throughout the charts system and the overall compositional process. Cage’s choice of musical materials used in this piece strongly suggested European influences rather than those of Eastern music. It is highly unlikely anyone listening to this work would be able to identify any sonic resemblance to either Chinese or Indian music. To fully understand the work in its true context, it is therefore necessary to investigate the extent to which Cage’s compositional techniques had been influenced by European composers; the compositional techniques and aesthetics of European composers such as those of Schoenberg, Messiaen and Erik Satie will be discussed in comparison with Cage. In particular, there are striking parallels between Cage’s compositional system for Music of Changes and the integral serialism in Boulez’s Structures Ia. A comparative analysis is made below.

The influence of European composers

All three authors reviewed in this chapter note that Cage designed the sound charts where all 12 chromatic pitches are required within the sub-chart (4×4). This recalls Schoenberg’s 12-tone technique more strongly than any of Cage’s previous work. As a result, the sound world of Music of Changes is undoubtedly atonal, something unique for Cage. The influence of Schoenberg on Cage can be seen in a different light, as Albright points out:

Cage was a pupil of Schoenberg and evidently replaced Schoenberg’s rigor with an equally extreme surrender of will; but it might also be argued that he

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carried out the Schoenbergian revolution through different means, by trading a serialist’s chart of retrograde and inversions for the charts of the I Ching. Chance itself can be a stern taskmaster. (2004, p. 189)

Cage’s encounters with Schoenberg in the 1930s perhaps prepared him to develop a compositional system that was innovative and rigorous but rarely resembled his teacher’s methods. Another prominent figure whom Cage had an opportunity to meet was Messiaen. As noted in Chapter 2, Cage met Messiaen when he visited Paris in 1949 and then invited him to America, as noted in the letter sent to Boulez in January 1950. Cage evidently admired Messiaen, as he wrote to Boulez: “I love him for his ideas about rhythm” (Nattiez, 1991, p. 48). A trace of Messiaen’s influence can be seen in the way Cage developed a rhythmic technique called the ‘segmented’ duration, where several durational values are added one after the other, creating a set of durations. Cage explains his technique as follows: “Because addition is the generation means employed, the duration may be said to be ‘segmented.’ These segments may be permutated and/or divided by two or three (simple nodes)” (1978, p. 59). The underlying concept of the ‘segmented’ duration suggests the influence of Messiaen’s rhythmic techniques such as added value, augmentation and diminution. Music of Changes is ametrical and the way Cage achieves this recalls how Messiaen explains rhythmical techniques in his ametrical music. Cage also admired Satie and perhaps identified with Satie’s music more than that of any other European composer. As Cage comments, he not only was attracted to the French composer’s compositional aesthetics but he also loved his writings and sense of humour (1995, pp. 183–184). But one can observe Satie’s influence most of all in Cage’s concept of rhythmic structure. Cage states “A zero musical structure must be just an empty time. Satie made at least three kinds of empty time structure” (1978, p. 80). Cage identifies all three types of rhythmic structure in Satie’s music in his article “Erik Satie” in Silence (1978, p. 81).131 The common element of these structures is symmetry, which Cage interpreted as ‘zero’. Another of Cage’s statements implies that, in Satie’s music, Cage seemed to identify the ideal way to structure a piece. In his conversation with Daniel Charles, he says,

131 The article originally appeared in 1958. 221

Every time there is, as in the works you’re describing, a time structure, you can divide that time and introduce into it, as material, silence. I attempted to do as Satie or Webern had done: to clarify structure, either with sounds or with silence. (1995, p. 39)

As discussed previously, symmetrical rhythmic structure is an essential part of the compositional system in Music of Changes, and provided the framework on which chance could operate. The connection between the development of chance procedures in Cage’s music and of integral serialism in Boulez’s is often mentioned in the literature: their compositional techniques were developed in the same period; they exchanged ideas in correspondence while the works were composed, and most importantly, the connection has often been supported by the fact that, although both composers’ aesthetic aims differ greatly, their works produce a similar sonic result. Griffiths notes the parallels between Cage’s chance techniques with Boulez’s serial techniques as follows:

The deceptive nature of the link between Cage and Boulez is nowhere more striking than in a comparison of the former’s Concerto for prepared piano and chamber orchestra (1950–51) with the latter’s first book of Structures (1951– 52). Both composers made extensive use of number charts, but where Boulez saw them as an aid in total serial regulation, Cage took them as a means to attain non-intention. Both found themselves confronted by what Cage might have described as the complete dominion of ‘Law’ over ‘Freedom’, but where Boulez immediately attempted a new reconciliation, Cage was delighted by the possibility of removing personal creative wishes. (1981a, pp. 66–67)

Despite the use of the chart system in the Concerto for Prepared Piano and Chamber Orchestra, Music of Changes is Cage’s very first piece in which he applied chance operations uniformly. Below, the compositional system of Cage’s Music of Changes and Boulez’s Structures Ia are compared. This comparative analysis attempts to demonstrate how profoundly these two composers influenced each other and aims to present a more comprehensive view on Music of Changes than has so far been attempted. I hope that it is also more accurate. The compositional system employed in Music of Changes and Structures Ia can be reduced to five successive components (Figure 6.15): (1) pre-determined structure; (2) pre-determined musical parameters; (3) pre-designed series or charts; (4) a

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selection process through serial ordering or chance operations; and (5) process of combining all the necessary musical elements. Clearly, both composers’ methods of construction not only display complex and rigorous planning and organisation but also share this same five-step process. The following comparative analysis illustrates the parallels between each step of this process. As the first stage of composition, Cage and Boulez both designed the structure of their pieces. The function of formal structure in both pieces is similar in that it not only frames the work but indirectly regulates the compositional process. For instance, at the beginning of each formal section in both works, a change of density and tempo occurs. As Uno argues, change of density is the most perceivable musical phenomenon and therefore shapes the listener’s experience of both pieces. Pre-determining the primary musical parameters was a core concept in composing Music of Changes and Structures Ia. A set of parameters was determined in both compositions: pitch (“sound” in Cage’s terminology since he incorporated some non-pitched sound), duration, dynamic, attack, range of density and tempo. There are two essential differences between Cage’s and Boulez’s treatments of these parameters. The first is that, while Cage applied chance operations to govern all the above parameters, Boulez took the four parameters to be serialised and determined density and tempo himself. The other difference is that Cage combined dynamic and attack whereas Boulez treated these two parameters as separate entities. The range determined for each pre-determined parameter also shows a profound resemblance, as illustrated in Figure 6.16. For pitch, both composers used the twelve chromatic pitches. For duration, both composers employed a wide range of note lengths and manipulated them in an additive manner, where individual durations are accumulated to create an ametrical effect. For dynamics, the extreme range from pppp to ffff is used in both pieces. For the levels of density, both composers allowed movement between thin and thick textures. For the tempo, the range from fast to slow tempi is used. Overall, one can notice that both composers preferred using extreme ranges for the individual parameters. The above pre-determined parameters were individually arranged. Cage arranged the individual parameter through the charts system, which contained 64 cells. As mentioned earlier, Cage generated eight charts for sound, duration and dynamics and one chart for density and tempo and wrote “these charts were subjected to a

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rational control” (1978, p. 26).132 Boulez built a series made up of twelve elements for each individual parameter, a technique resembling Schoenberg’s 12-tone series. Boulez then designed two matrices based on the transposition and inversion of the pitch series. Thus, Boulez’s matrices could function similarly to Cage’s charts: the design of Cage’s chart is 8×8, compatible to I-Ching’s hexagram and the design of Boulez’ matrix is 12×12, matching the series. In this part of both compositional processes, an important link is established. No longer do the musical parameters exist and function in their own right but now they were associated and encoded with numbers. In Cage’s chart system, each cell was numbered from 1 to 64 and each number correlated to given musical material whether it was to do with sound, duration, dynamic and attack, density and tempo. In Boulez’s matrix, the numbers from 1 to 12 correspond to the order of the series whether it was related to pitch, duration, dynamic or attack. Therefore, the pre-determined musical parameters in both compositions are inseparably related to numbers. No element could function unless it was encoded to be then interpreted through the charts or matrices. Once the relationship between pre-determined musical parameters and numbers was set up, the compositional process could move to the next stage, which was selection. In order to choose, the composers needed to consider two questions: how to select the already determined musical parameters and in what order they should be arranged. Here, Cage and Boulez arrived at very different solutions: the former chose chance operations while the latter employed integral serialism. Cage’s choice of method symbolises freedom from the personal tastes of the composer; Boulez’s choice, on the other hand, symbolises the laws of logic and order. Despite their compositional process being seemingly contrary to each other, the principles that underpin the two composers’ methodological solutions are astonishingly similar. Firstly, the selection method hardly permitted the composer’s input. Cage had no control over which musical materials would be selected next, as selections were made by chance. The ordering of musical parameters in Boulez’s system largely relied on the numerical sequences occurring in his two matrices. Secondly, the primary parameters (pitch, duration, dynamics and attack) were chosen in a non-related manner. In other words, while Cage was involved in selecting the process of sound through tossing coins, he could not foresee which duration or

132 The statement was made in the lecture delivered by Cage at Darmstadt in September, 1958. 224

dynamic and attack would be assigned to the randomly chosen sound material. Similarly, the ordering of four parameters was made independently and the coordination between the four serialised parameters was influenced by the numeric sequences in the matrices. 133 Thirdly, these characteristics found in the selection process directly resulted in the formation of fundamentally new field relationships among the primary musical parameters. The relationships among parameters dissolved to equilibrium, moving away from the pitch-dominated compositional system; thus, both compositions totally broke away from the thematic orientation of Western music. Both compositions opened up a new sonic world. Similarities to each other include a sense of randomness and a static absence in direction. Further aspects of compositional technique need to be considered. One can also observe similarities in how Cage and Boulez decided on the other matters of composition. Firstly, for Music of Changes, determining the register for a chosen pitch or group of pitches was left to the composer. Cage preferred to use the middle register, as proven in Uno’s statistical analysis, and Boulez carefully selected register in the process of distributing the pitch-classes, avoiding any octaves. Secondly, both composers generally chose “polyphonic” textures by superimposing several layers. Thirdly, in both works, the rate of change occurs for tempo and density concurrently at the beginning of each structural point. Fourthly, both composers saw a greater importance for silence in these compositions. For Cage, silence is as prominent as sound. It is reflected in the design of the sound charts, where silence is treated as equal with sound. For Boulez, the boundary of formal structure is defined by silence, which is a result of using fermata markings at the end of each formal section. Finally, both composers incorporated the dichotomised concept of mobility and immobility in their compositional systems.134 Cage applied the concept to regulate the chart system, in which half of the charts become mobile while the other half stay immobile. Musical materials in immobile charts remain the same while the content of mobile charts gets refreshed with new materials as soon as the musical material is used once. This procedure was invented out of Cage’s concern about “how to become mobile in my thought rather than immobile always.” He concluded that “there [is] no

133 As discussed in Chapter 4, in relation to Structures Ia, Boulez chose which numeric sequences were to be used for ordering the specific parameters and he further organised the serially ordered four parameters through the means of the density variable. 134 As mentioned in Chapter 4, in Boulez’s letter to Cage on December 1951, one can see how Boulez was fascinated by the concept of mobility and immobility used in Cage’s chart system (Nattiez, 1993, p. 113). This confirms Cage’s influence on Boulez in this regard. 225

incompatibility between mobility and immobility and life contains both” (Nattiez, 1993, p. 94). However, Boulez saw the concepts of mobility and immobility in composition from a different angle. As discussed previously, the relation between mobility and immobility is reflected in the intrinsic relation between pitch series and the distribution of registers (1991, pp. 119–120). The compositional systems of Cage’s Music of Changes and Boulez’s Structures Ia are not only remarkably alike but are also based on the same constructional principle. Both systems consist of a balance between the composer’s control and the involuntary and mechanical element: one being chance operations and the other being integral serialism. The similarity of their compositional systems was recognised by Boulez although he disapproved of Cage’s chance-influenced approach. Boulez wrote about Cage’s compositional system in 1952 in the article “Possibly . . .” as follows:

More recently, he [Cage] has been working on setting up structural relations between the different components of sound, and for this he uses tables which organize each component into parallel but autonomous distributions. The tendency of these experiments by John Cage is too close to my own for me to fail to mention them. (Boulez, 1991, p. 135)

Indeed, the comparative analysis between Cage and Boulez strongly reveals how close they were in compositional techniques, although it is hard to tell accurately which composer had the greater influence on the other. Nattiez emphasises Cage’s admiration of Boulez and how Boulez responded to Cage’s request to provide a detailed explanation of his development of integral serialism (1993, pp. 6–7) Nattiez quotes Cage’s comments in his letter written in January, 1950: “The great trouble with our life here is the absence of an intellectual life” (1993, p. 6), which reflects Cage’s desire to engage with an approach from outside America. In another comment from Cage in the same letter, one can also see how much Cage was occupied with Boulez’s musical world: “Since knowing you, our music sounds feeble to me. In truth, it is only you who interests me” (Nattiez, 1991, p. 48). Since the similarities of the two composers’ compositional systems are so evident in many respects, the influence is highly likely to have gone both ways, though Nattiez seems to suggest that Cage was the more strongly influenced.

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Compositional aesthetics reconsidered

The above comparative analyses illustrate the extent to which Cage’s compositional techniques and system were influenced by European composers and by Boulez in particular. These influences go beyond just compositional techniques. They represent an undoubtedly strong formalistic approach to composition. (This may seem at odds with Cage’s avowed fondness for Satie’s compositional aesthetics, which promoted an essentially anti-formalistic approach.) Boulez’s Structures was clearly composed under the governing ideology of formalism and the evident parallel with the construction principles of Music of Changes puts Cage’s chance music under the same ideological umbrella. As the title of his piece clearly suggests, Boulez’s main concern was how to structure the various dimensions of sound and to create a musical structure that can architecturally support serially organised parameters. Boulez chose to write Structures without text; this is one indication, but not the only, that it confirms to formalist ideology. It is also impossible to impose a narrative or program on the work: it is acontextual. Doesn’t Music of Changes conform to the same ideology? Both composers were deeply interested in how to structure sound materials and it was this mutual interest that brought two composers together for a period of correspondence. The only difference between the compositional systems of Cage and Boulez lies in the mechanism by which musical materials were selected and organised. Furthermore, the very fact that Cage was compelled to develop chance techniques conforms to the broader musical culture post-World War II. In other words, Music of Changes was a composition within the culture and ideology of the era. Meyer suggests that scientific and mathematical models may have influenced post- World War II composers, leading them to believe that

. . . compositional constraints should be both comprehensive and consistent. As a result, using secondary parameters to shape script-based forms, and script-based forms as the basis for motivic development, came to seem not merely problematic, but somehow illicit—even immoral. Yet the need for high-level constraints remained. And to meet their need a host of precompositional constraints were devised. (1996, p. 342)

Meyer relates Cage’s chance operations to other compositional pursuits in twentieth- century music as follows:

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What needs to be noticed, first of all, is that Stravinsky’s use of convention is a response to the same need for stylistic constraints as Schoenberg’s use of the twelve-tone method, Xenakis’s use of probability theory, and Cage’s use of tables of random numbers. (1996, p. 347)

As noted earlier, one finds another contradiction between Cage’s professed aim in his chance music and what he really did. Cage states: “It is . . . possible to make musical composition the continuity of which is free of individual taste and memory (psychology) and also of the literature and ‘tradition’ of the art” (1978, p. 59). However, as Pritchett and Uno have argued in their analyses, Cage’s compositional system reflects the composer’s taste and thoughtful decision-making process. Cage’s compositional system reveals that his aesthetic position was indeed formalistic, like many European avant-garde composers of the time. Despite trying to move away from traditional Western compositional aesthetics by embracing Eastern philosophy and chance, Cage only partially achieved his aim and evidently still endorsed formalistic approaches in this work. Concerning Cage’s chance music, particularly Music of Changes, Goehr argues that Cage attempted to challenge the traditional role of the composer in his relationship to his compositions, yet his chance compositions have not been emancipated from the Romantic tradition of “work-concept” that is inseparable from the ideology of formalism. Goehr begins her argument by noting that Music of Changes is a fully composed work. According to Goehr,

That seems to be an endorsement of the work-concept, and so does [Cage’s] explanation. “In other words,” he writes, “though the sounds and then successions were to some extent dictated by chance, the notation is complete and must be observed by the performer.” Were one to replace “chance” with “inspiration”, little would distinguish this statement from one made by the most committed of romantic work-oriented composers. (1992, p. 262)

Goehr further questions the radicalism of Cage’s approaches:

Contrary to the spirit of an anti-survivalist aesthetic, in which music is performed for the present and is not composed to last, Cage’s ‘works’ have survived and have come to be representative, no less, of the avant-garde repertoire of musical works. Performers and listeners receive his music with the same respect many of Beethoven’s contemporaries had for his innovative works. In both cases, such respect involves feelings both of awe and

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bewilderment. Cage’s ‘real’ and ‘random’ sounds have not stayed real or random. (1992, p. 265)

Although Cage claimed that employing chance operations in his composition would allow his music to break away from the past and make his music radically different from that of his contemporaries, the fact is that many of his chance compositions, particularly Music of Changes, still conformed to the tradition of formalism. Therefore, the common assumption that Cage’s chance music relates primarily to Eastern philosophy and aesthetics needs to be challenged. In 1952, Cage himself declared that his development of chance techniques in this piece was influenced by the Chinese book of oracles, the I-Ching (1978, p. 57). However, this Eastern influence is perhaps absorbed by the governing ideology of formalism. If Cage’s chance music did not undermine formalism and could not threaten the Romantic tradition of “work-concept”, what did the chance procedure in Music of Changes really achieve in terms of challenging traditional Western compositional aesthetics? What it did do was to dismantle another long-standing compositional aesthetic, namely organicism. Organicism is one of the powerful nineteenth-century ideologies, and persisted into twentieth-century modernist compositions. Again, as was outlined in an earlier chapter, under the influence of organicism, musical unity is pursued through maximising the relatedness between musical ideas, thus achieving musical coherence within an autonomous work. As a result, motivic and thematic development in a piece was desired as it could represent a metaphor of a single “seed” germinating, growing and becoming a “tree”. However, the significance of chance (as opposed to integral serialism in Boulez’s Structures) defies any development of a musical idea. As Schädler observed, musical materials are laid one after the other as they were chosen, according to the random act of tossing coins. Cage perhaps was not aware of it but his chosen compositional method of chance operation directly opposed the prevailing ideology of organicism. He even wanted to ensure that similar musical material did not sound in close succession. This can be seen in the self-imposed rule applied in the process of composing Music of Changes: “interference.” Cage explains that “In the case of ‘interference’ (the appearance of a sound having characteristics in common with the characteristics of the previously sounded situation) the characteristics that produce the interference are omitted from the newly appearing sound or cut short in the situation that has

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previously sounded” (Nattiez, 1993, p. 106). In other words, Cage was eliminating any chance of relatedness between musical materials to the point of over-riding chance. Traditionally, achieving structural coherence in a piece of music largely depends on the presence and degree of musical relatedness. This became fundamental to many composers as well as to musical analysis. However, through chance operations Cage’s primary aim was to free himself from a composer’s right or privilege to control and manipulate musical materials. Therefore, chance makes the logical development of musical ideas impossible. As discussed previously, since the nineteenth century a relationship has been built between conceptual musical unity and the idea of genius. Treitler defines genius as “the natural creative capacity of mind that provides the controlling force in the production of unified works of art” (1989, p. 51). The ability of a composer who can create such a work of art was recognised and respected by critics and especially by music analysts. In support of this view, Goehr says, “Early romantic theories argued that it was on the transcendent, universal level of the ‘free’ genius that artists gave ‘fine’ content to their works” (1992, p. 162). Here, it is often understood that the “fine” content of a piece of music is organic. The significance of Cage employing chance operation in his compositions is that it challenges the intriguing relationship established between the idea of genius and the aesthetic value of musical unity within a single work. The adoption of chance in the compositional process thus totally opposes the traditional attitude of how one values a work of art. According to Pritchett, one of the main problems faced by critics dealing with Cage’s chance music was how one could criticise or judge a random act such as the tossing of coins. The solution for this dilemma has been, Pritchett says, “to ignore the music and dwell upon ‘the ideas behind it’” (1993, p. 2). Although Pritchett does not mention the concept of genius and how it has been related to creating a unifying musical work, it is now clear that music critics could find it difficult to classify Cage’s chance music as a work of genius because it does not and could not display musical unity. How Cage himself rejected the set of traditional values toward works of art is summed up in the following:

Value judgements are not in the nature of this work as regards either composition, performance, or listening. The idea of relation (the idea: 2) being absent, anything (the idea: 1) may happen. A “mistake” is beside the point, for once anything happens it authentically is. (1978, p. 59. Italics mine)

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Finally, how do chance operations affect the experiences of performing and listening? As the composer states above, making a value judgement is not required. However, chance elements in this composition actually do not have much effect on how one should prepare the piece or perform it since the piece is fully composed and notated. Cage explains the role of performer:

Though chance operations brought about the determinations of the compositions, these operations are not available in its performance. The function of the performer in the case of the Music of Changes is that of a contractor who, following an architect’s blueprint, constructs a building. 135 (1978, p. 36)

Like many avant-garde compositions of post-World War II, the kinds of challenge which a performer faces in this piece are more to do with technical issues: continually changing tempi, rhythmic complexity and making subtle differences in dynamics, attacks and pedalling. Cage was obviously aware of the piece’s extreme complexity and wrote in the preface to the score, “It will be found in many places that the notation is irrational; in such instances the performer is to employ his own discretion” (1961, preface).

Performance practice and listening experience

For a pianist, one of the greatest difficulties of learning the work is to become familiar with Cage’s innovative spatial notation. As noted previously, this notation has been considered revolutionary, as Cage was able to convert the concept of duration into a visual and spatial dimension: a crotchet equals 2.5 centimetres. From the performer’s perspective, however, this new system of notation can be challenging as he or she has to re-orient himself or herself to Cage’s notational system or recalculate Cage’s notation to the conventional method in order to play it accurately. Figure 6.17 demonstrates the difficulties involved. This short passage was chosen as an example as it contains only one layer of sound, but one can but imagine the inherent complexities of executing several superimposed layers. The notation is ambiguous, so I have included two different calculations. Figure 6.17 (b) is the result of counting all four dotted demisemiquavers as they are. Figure 6.17 (c) is the result of respecting the pitch A5 which has the duration of a dotted semibreve occurring right after the bar line.

135 This statement was made in a lecture delivered at Darmstadt in September, 1958. 231

This appearance of A5 suggests that the four dotted demisemiquavers should fit into the previous bar. It is possible that at some point Cage perhaps miscalculated the durational values. Interestingly, I have also found a printing error by Henmar Press (now an imprint of C.F. Peters): the published score does not match Cage’s instruction since throughout it a crotchet is shorter than 2.5 centimetres. It is most likely that the publisher has reduced the actual size of the original score. On the one hand, as noted earlier, Cage’s new development of a notational system has been considered revolutionary. On the other hand, this might not have been the best practical solution for either the performer or publisher and may have contributed to the relative neglect of the work. The experience of listening to Music of Changes can be similar to that of listening to Boulez’s Structures Ia in so far as the sounds are atonal; no recognisable patterns are evident, and one would be aware of perpetual changes in every dimension of the music. One of the main characteristics of Music of Changes a listener notices is its apparent unceasing randomness; nothing seems concrete and there is no discernible development of musical ideas. Such characteristics are mainly a result of chance operations as Uno’s statistical analysis proves. Therefore, chance operations determine the listener’s experience of the work. In such a case, listeners should also be encouraged to accept the result of chance whether they like it or not in so far as, as mentioned in the beginning of this chapter, the attitude of acceptance was central to the composer’s aesthetic philosophical pursuit. Conclusion

Through the reviews of previously written analyses, chance operations appear as part of a complex and systematic compositional procedure that was only articulated during the late 1980s and the early 1990s. These analyses are examples of how Music of Changes can be approached and, surprisingly, they are mainly formalistic in nature. Despite the recent criticism of music analysis as being formalistic, this case study, on the contrary, demonstrates the strength and necessity of such an approach. In fact, non-formalistic approaches to understanding Cage’s chance music, such as emphasising philosophical aspects, cannot fully satisfy one’s curiosity about his chance operations in this piece but can only provide general and somewhat shallow perspectives.

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As proven through my review of analyses in this chapter, Cage’s chance procedures demonstrate common features of avant-garde compositions of the post- World War II era: complexity in pre-compositionally designed plans, atonal quality of sound and high demands on the performer. The commonality has been further argued through the comparative analyses made between Cage’s compositional techniques in Music of Changes and those of other twentieth-century European composers, particularly Boulez’s employment of integral serialism. Even though Cage’s interests in Eastern philosophies have been more widely acknowledged, there is no doubt that Music of Changes was governed by the ideology of formalism. But despite the influence of formalism, Cage’s chance music challenges a long-held ideological position of music analysis: organicism. Up to Cage’s chance music, these two ideologies—formalism and organicism—co-existed and in fact were inseparable within compositional theories and music analysis. Since its inception, Cage’s incorporation of chance has been widely recognised as revolutionary but it is perhaps in this regard—being deeply formalistic while denying the ideology of organicism— that it now appears most revolutionary, even ironic.

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CHAPTER SEVEN

Conclusion

Analysis not only reinforces what is already intuitively known but can also challenge the security that lies in existing knowledge, disturbing the comfort of the familiar, inviting us to reconstitute our perception. (Swanwick, 2002, p. 13)

The four case studies in this thesis provide a platform from which to observe and ponder how analysts since the 1950s have understood and valued four iconic mid- twentieth-century pieces. Particular facets of music analysis have been discussed: the strengths and weaknesses of different analytical methodologies, how music analysis can enhance an understanding of a compositional process, and how music analysis can inform the experience of listeners and performers. As addressed in the Introduction, music analysis has been facing widespread criticism concerning its idealism, its approaches and its position in musical scholarship, and it is necessary to closely examine what music analysis has really achieved over the past half century. In this final chapter, I summarise firstly the commonalities between the compositional theories of the four composers, based both on my and others’ analyses of the four works. In the second half of this chapter, I briefly summarise my review of other analyses of the four case-study works—29 analyses in total—and identify further research areas within the discipline of music analysis. Procedural connections

The four works studied in this thesis have strong historical and procedural connections. Chapter 2 discussed the interaction and correspondence between the four composers during the years immediately following World War II. These moments of personal interaction profoundly reinforced the particular compositional approach of each piece. Although analysts until now have treated these four works as more or less unrelated items, this thesis has investigated the works side by side, allowing one to see similar features in their compositional techniques and ideologies. There were shared interests among the four chosen composers and these were reflected in the way they developed their compositional theories. In this part of the conclusion, I summarise the compositional features that underpin the four selected pieces.

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The principal feature of all four works is that their composers organised musical parameters beyond pitch extensively and systematically, an approach considered a progression from Schoenbergian 12-tone technique, which only pre-determines the sequence of pitches. Pre-determination of four parameters (pitch, duration, attack and dynamics) was first attempted by Messiaen, in Mode de valeurs, although the piece is based on a modal system. Boulez and Cage, with Structures Ia and Music of Changes, not only determined primary musical parameters but also developed a rigorous compositional procedure that could organise them systematically. In Structures Ia, four essential parameters (again pitch, duration, attack and dynamics) are independently and serially organised according to a numeric sequence derived from two matrices, while the composer determined the remaining, non-serial parameters. In Music of Changes, the same musical parameters, as well as density and tempo, are organised through the mechanism of chance. Moreover, in Stockhausen’s Klavierstück III, primary parameters are neatly pre-organised and inter-connections among various parameters are carefully designed through the principle of symmetry. Although Stockhausen employed serial technique, it is distinguished in that the elements of series are organised in a distributive manner rather than a sequential one. As a central part of pre-organising his musical materials, Stockhausen developed the concept of grouping the notes, in contrast to the pointillism that is associated with Boulez’s Structures Ia. Messiaen’s evolution of rhythmic techniques has contributed to the development of techniques in which the rhythmic organisation becomes independent from the pitch structures. Similar in significance to Schoenberg’s break away from tonality in his twelve-tone composition was Messiaen’s composition of ametrical music through applying a number of techniques such as added value, nonretrogradable rhythm and polyrhythm. These techniques, clearly evident in Mode de valeurs, were fundamental to both designing the mode as well as structuring the entire work. As mentioned in the preceding case studies, Messiaen influenced the other three composers, particularly with his rhythmic techniques. Boulez, for instance, designed the prime series for duration where the 12 elements were arranged arithmetically by adding successively the first value of the series, the demisemiquaver. In addition, Boulez created a cohesive musical structure no longer governed by pitch relationships but by the arrangements of duration and tempo. Cage too implemented the technique of added value, and the resulting “segmented duration” enabled him to 235

organise durational values through chance operations. Stockhausen, while not taking up the concept of added value in his piece, applied the technique of nonretrogradable rhythm in organising various durational values to reflect the overall symmetry of the piece. Through employing Messiaen’s rhythmic techniques, then, Boulez, Cage and Stockhausen all composed ametrical pieces in which the organisation of duration is no longer subordinate to the pitch structure. Alongside Messiaen’s rhythmic techniques, the influence of the 12-tone technique persists into the musical language of these composers. Messiaen, for example, did not employ the 12-tone technique in Mode de valeurs, but the “hidden mode” discussed in Chapter 4 was disguised in the three-division mode, which features three arrangements of the 12 chromatic notes. This strongly suggests a connection to Schoenberg’s 12-tone rows and led many analysts initially to approach the work as an example of serial composition in which pitches are organised sequentially. Boulez’s link to Schoenberg’s serial technique was more explicit: he wrote Structures Ia with the overt intention of extending Schoenberg’s technique into integral serialism. The note-sequencing method of serial technique was essential to the serial operations involved in this work. Stockhausen also employed a serial method, but departed further from Schoenberg’s serial technique. Similar to Boulez, Stockhausen, in Klavierstück III, also expanded serial technique to parameters beyond pitch, but organised elements distributively rather than sequentially. Importantly, Stockhausen neither hesitated to divert from his chosen permutations nor obligated himself to use all 12 chromatic pitches with the same frequency, developing a technique that combined elements of Schoenberg’s 12-tone technique with Messiaen’s modal one. The influence of Schoenberg’s serial composition was also found, perhaps surprisingly, in Cage’s compositional plan for Music of Changes. Cage designed the sound charts to be subdivided so that all 12 chromatic pitches had to be used within each subdivision. It is rather ironic to observe such a seemingly rigid rule co-existing with the randomness of chance operations. Though their methods differed, all four composers were occupied with pre- determining and pre-organising primary musical parameters. Prior to World War II, a conceptual shift in composition was becoming evident, with a number of composers starting to treat the organisation of rhythmic elements as equal in importance to pitches. Messiaen developed his rhythmic techniques so that rhythm could function independently from other parameters. Both Boulez and Cage designed the overall 236

structures of their works to be governed by the organisation of duration rather than of pitch: Boulez devised a proportional system based on the Golden Section ratio which, together with the symmetrical arrangement of tempi, controlled the rate of musical unfolding, and Cage developed a rhythmic structure that was inseparable from the entire chance operation. Stockhausen also incorporated various durational values in forming a symmetrical structure. In the post-World War II period, it is evident that parameters other than pitch received almost equal prominence in the pre- compositional plan, as seen in the four case-study works. Such changes in compositional technique had consequences in several dimensions. Most significantly, pitch was no longer the dominant element in the compositional design, and, unlike thematic tonal music from previous centuries, other musical parameters were no longer subordinate to the pitch structure. This brought a dramatic change in sonic relationships and, as a result, a totally different experience for the listener. When parameters other than pitch gain more structural importance, the surface structure of the music changes. In Mode de valeurs, the pitch relationships are not the only source of linear musical motions: dynamics, attacks and registers are, too (as Covington’s analysis demonstrates). As my analysis showed, the seven prominent pitches that make up the “hidden mode” were assigned particular durations, dynamics, attacks and registers. Importantly, the three-layer contrapuntal formation used in this piece is not based primarily on pitch structure but on rhythmic structure: the technique of polyrhythm. In Structures Ia, the density variable becomes a key factor in creating the sonic shape of the piece, which is further articulated by the arrangement of attacks, dynamics, registers, tempi and pauses. By basing the underlying proportional system on the Golden Section ratio, Boulez has organised these parameters not only in a cohesive manner but also in a way that is both aurally discernable and aesthetically gratifying. In Klavierstück III, Stockhausen was conscious of creating various musical gestures within a single piece. The desired musical gestures were a by-product of a compositional technique called grouping. As discussed in Chapter 5, although Stockhausen rigorously organised musical parameters, each group formed in the piece is distinguished not by the pitch structure alone but more predominantly by the change of density, texture, shape and the organisation of durational values. In Chapter 6 it was observed how changes of density and tempo help shape Music of Changes, although the predetermined musical parameters were put together through chance operations. 237

Though a fundamental conceptual shift away from the pitch-dominant compositional techniques can be observed in the post-World War II period, the ways composers organised the predetermined musical parameters differed considerably. Consequently, a listener faces unconventional and perplexing sonic phenomena on the surface structure of these pieces. In the cases of these four composers, such sonic shapes may have been influenced by their experimentation with tape music, which forced them to think about concepts such as sound, timbre and duration in a new way. In these iconic works, all four composers employed a pre-designed formal structure, within which various musical parameters were pre-determined in, systematic detail . In line with the dominant aesthetic mode of modernism in earlier twentieth-century compositions, the four chosen composers’ works embraced the principles of formalism. Summary of the review

Through the four case studies, 29 analyses were examined. These analyses varied in a number of respects: the types of literature, the length of analysis and the analytical approaches taken. Over the last 50 years, the analytical writings of the four chosen pieces appeared in the various types of literature as mentioned in the Introduction. Firstly, analytical discussions of each piece can often be found in books that address a single composer’s life and works. Secondly, historical overviews of twentieth-century music often incorporate analytical comments on the chosen works, even if briefly. Thirdly, there are publications, mainly articles that focus on music theory and analysis, and present analyses of these works as illustrative examples. Fourthly, there are several journal articles that provide more in-depth, autonomous analyses of these works. Finally, three doctoral dissertations contain extensive analyses. The range of literature here strongly suggests that analyses of these works are not limited to specialised theory and analysis texts. The most common goal shared by the authors of analyses reviewed here was to understand the new and individual compositional theories developed by each of the four composers. Compositional theories, techniques and the pre-compositional plans are integral parts of the four chosen compositions, and almost every author attempted to deal with these conceptual matters. The tendency of analysts to focus on the conceptual underpinning of a work is particularly obvious when the composer himself partially revealed a plan or made his techniques transparent on the score; the example

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of the former would be Messiaen’s Mode de valeurs, where the mode is given in the preface, and of the latter, Boulez’s Structures Ia, where the prime forms of the series is revealed explicitly at the start of the work. Accordingly, all the analyses of Messiaen’s Mode de valeurs reviewed in Chapter 3 discussed the pre-compositionally designed mode. With the exception of Covington’s analysis, the pre-compositionally designed mode becomes either the focus or a starting point for the analyses. As pointed out in Chapter 4, the integral serial technique used in Structures Ia was the most favoured topic for several decades. On the other hand, Stockhausen and Cage concealed their compositional techniques at the time of writing Klavierstück III and Music of Changes and only provided brief comments about these works. For this reason, analysts were very keen to discover the composers’ compositional techniques and pre-compositional plans. With Stockhausen’s Klavierstück III, a particular effort had been made to decode the pitch relationships. All the analysts mentioned in Chapter 5 were eager to identify one governing compositional idea that can account for the relationships of every single note in the piece. However, from this perspective, the logic of the piece seems almost impenetrable. There was little agreement on how the composer organised musical materials and, as a result, studies of this piece were characterised by disputes and disagreements among different authors. Extensive analytical studies were also made of Cage’s Music of Changes although they came nearly four decades later than those which examined the three other pieces. A desire to understand Cage’s compositional theory of chance operations was based on inquiries into how chance elements had been incorporated, and these analyses revealed that a chance operation existed within a complex and logical procedure. As discussed in Chapter 6, it was necessary to clarify Cage’s compositional techniques of chance and, as a result, some commonly held views of Cage’s chance music were challenged. As asserted above, when analysts discussed compositional theories and techniques, they sought to find a governing principle that underpins the construction of the entire piece. A “holistic method” (Adorno, 2002, p. 174) of examining music could be seen on a number of occasions. In the process of illustrating the governing principle, analysts have shown a tendency to systematically reduce surface musical events to a more or less abstract uniformity. Such an approach to music analysis closely relates to an influential ideology of the discipline: organicism. As discussed throughout this thesis, one of the central aims of music analysis has been to 239

demonstrate structural coherence by illuminating a single underlying principle. The aesthetic concept of organicism was not, however, a twentieth-century invention: it has profoundly influenced the development of compositional theories since the nineteenth century. It seems logical to assert that the usefulness of an analysis depends on the composer and analyst sharing common aesthetic ideals. It follows, then, that despite the significance of organicism, analyses that seek, above all, to demonstrate the organicism of a work are often more ambitious than successful. This is most evident in some of the analyses of post-World War II compositions reviewed in this thesis. For example, the most common analytical approach to Mode de valeurs was to investigate whether the piece was serially organised. A disconcerting aspect of such an attempt is that the fundamentals of Schoenbergian serial compositions are about a pre-designed and controlled sequence of pitches governing the entire piece; yet, Messiaen experimented far away from serial techniques and far beyond merely the pitch relationships. As discussed in Chapter 3, most analysts were not able to explain how Messiaen explored his unique musical language, but were only able to discuss the work as it related to the characteristics of a pre-compositionally designed mode. Stockhausen’s Klavierstück III was analysed even more “holistically” than any of the other three case-study works. Griffiths’ attempt to explain the organisational scheme based on the Fibonacci series (1981) was one example, while other authors tried to understand the pitch relationships based on either pentachords or chromatic tetrachords. In particular, Lewin’s endeavour (1993) to systematically prove that all the pitch relationships in the work were based on a pentachord provides a case of music analysis being also influenced by positivism. He and others sought to reveal a perfect musical logic—a perfect organicism. More than any of the other analysts of this piece, Lewin overlooked the details that did not seem to fit his systematic approach. In a holistic and logical way, his analytical methodology illustrated the pitch relationships that matched the theory he had proposed. Maconie (1976, 2005) and Blümroder (1993) also examined pitch relationships derived from a series of chromatic pentachords but they both acknowledged compositional deviations occurring on the surface structures and accepted that deviations are part of the compositional features in Stockhausen’s work. In most analyses of Boulez’s Structures Ia, emphasis was given to understanding the integral serial operations which ensured a maximum relatedness 240

between four primary parameters, thereby demonstrating overt organicism in the compositional process. Such a tendency is not unexpected, as the compositional aesthetic itself was rooted in the ideology of organicism. However, it is also interesting to observe that as early as the 1960s (Wennerstrom), the surface structure and the listener’s perspective became a part of analytical studies of Structures Ia though this isolated analysis was rarely cited subsequently. In the 1990s, the seemingly capricious surface phenomena arising from the serially organised musical parameters is the central focus of the analyses written by Uno and Grant. Despite the fact that Music of Changes incorporated the random action of chance, analyses investigated Cage’s compositional procedure in a rigorously positivistic manner. In order to understand the procedure, Pritchett (1988, 1993) and Schädler (1990) had to examine Cage’s notebooks and systematically trace the steps involved in composing the work. Although the governing rule of chance organised musical materials randomly, the analyses of Pritchett and Schädler revealed a systematic and logical compositional procedure designed by Cage. Adorno’s comment that the value of a work is often realised through music analyses may be confirmed through these case studies. Being based on formalistic approaches, many analyses focus on the musical content of works as autonomous entities and their content was often discussed in relation to compositional theories and techniques. The central questions of these analyses were: How did the composer organise the musical materials in this piece? and how can one illustrate the underlying musical logic within this piece? As mentioned throughout the four cases studies, some limitations are evident in the way authors addressed these questions. Asking such questions, and thereby focusing on compositional techniques alone to understand these repertoires, results in analyses that are less than fully satisfactory. For instance, this is evident when analysts discussed the organisational scheme of a work primarily in relation to its pitch structures. With Cage’s Music of Changes, the organisational scheme is not the central issue due to chance operations, but Uno (1994) provided a pc-set analysis for the second sound chart and Book 1. Interestingly, she found predominant set relations in the sound chart as well as a unifying feature in the pitch relationships, even while noting that these relationships were a product of random action. The case for Messiaen and Stockhausen was different. In search of the organisational scheme, Fuhrmann (1974) and Sherlaw Johnson (1975) examined the pitch sequence throughout the entire piece. Their graphs proved that there is no set 241

sequential pattern as far as pitch is concerned but, as discussed in Chapter 3, their graphs unintentionally illustrated Messiaen’s techniques of nonretrogradable rhythm and rhythmic augmentation and diminution. Every single analysis of Stockhausen’s Klavierstück III reviewed in Chapter 5 discussed pitch relationships, while other parameters were not examined with such consistency and rigour. Not only did the approach vary from the application of tonal theory to the use of pc-set theory, but the level of complexity involved in such investigations varied. Lewin’s analysis (1993) was an example where the author totally focused on investigating the pitch relationships. Lewin even suggested how to listen to the piece based on the pitch relationships discussed in his analysis. In the listening exercise, all other dimensions of music—those that actually shape the piece—were excluded altogether. It was almost alarming to see how many analyses were preoccupied primarily with pitch relationships. As noted in the Introduction, such a tendency across the discipline has been criticised by Treitler (1989). It is no longer acceptable to approach the case-study works through pitch relationships alone. Analyses that emphasise the pitch structure of piece at the cost of neglecting other dimensions of music often run the risk of representing the music as merely an illustration of certain theories, rather than engaging with the work itself. Clearly, all four composers were experimenting with organising musical parameters beyond pitches in these works so it is strange that analysts took no time to reflect on the paradigm shift that had occurred in the compositions in the analyses reviewed here. For the composers, pitch was no longer the only dominant dimension of music. For example, Messiaen developed rhythmic techniques that were sophisticated enough to be organised independently from the pitch structure. Cage was also very keen to replace the harmonic structure of tonal music with a rhythmic structure. Boulez and Stockhausen certainly developed compositional languages that were capable of organising various musical dimensions as thoroughly as they did pitch relationships. But analysts have taken time to respond to such a development in compositional techniques. It would be unfair not to acknowldege that some authors did nonetheless perceive the significance of dimensions of music other than pitch in these works. In analysing Messiaen’s Mode de valeurs, Schweizer (1973) and Covington (1980) realised how the composer carefully organised attacks, dynamics and registers and, especially, Covington suggested these parameters had a more prominent effect on the listening experience than pitches. In analysing Boulez’s Structures Ia, Wennerstrom 242

(1967), Griffiths (1978), Jameux (1984), Uno (1994) and Grant (2001) all discussed the non-serial aspects of composition in depth, as they believed non-serial elements contribute largely to the shape of the work perceivable to a listener. Concerning the serialised parameters of music in this piece, Grant took the radical step of investigating the effect of the serial arrangement of dynamics and attacks rather than pitches and duration, arguing that this type of approach is more appropriate to guide a listener than an orientation towards pitch and duration. In analysing Stockhausen’s Klavierstück III, Schnebel (1975), Maconie (1976, 2005) and Blümroder (1993) seriously considered various dimensions of music other than pitch. For example, Schnebel, Harvey, Cook and Blümroder examined the overall contour of the piece; such a descriptive approach can be useful to listeners. Finally, in relation to Cage’s Music of Changes, the composer’s notes clearly indicated the important roles of non- pitch parameters in his compositions, and this is of course reflected in the approach of most writers on this piece. In particular, Schädler investigated the way Cage treated durations and time in detail alongside his discussion of chance operations. One pertinent question to ask is how much a listener can relate to analyses demonstrating the various aspects of musical structure. Many of the reviewed analyses in this thesis approached the works from the point of view of the composer, not that of a listener. These types of analyses often seem irrelevant to listeners, performers and other fields of musical scholarship because of their over-emphasis of the conceptual matter of the compositions, their use of technical terminologies and, sometimes, their application of forbiddingly complex methodologies. As has been noted in the Introduction, such music analysis has been challenged in recent years. It should be noted, however, that authors such as Wennerstrom (1967), Covington (1980), Jameux (1984), Cook (1987), Uno (1994) and Grant (2001) either entirely focused on the surface structure or at least examined it seriously. Here, I will summarise how the above authors were largely concerned with the failure of music analysis to provide explanations of surface musical phenomena. Firstly, Covington (1980) argued that the three-voice structure of Mode de valeurs has little or no relation to how a listener perceives the piece but that listeners would recognise linear motions created by a group of notes that share similar characteristics in their registral placements, attacks, dynamics and durations. Covington’s primary analytical methodology was based on repeated listening and, thereby, attempted to produce an analysis that is aurally meaningful. 243

Secondly, Uno (1994) examined the fact that pre-compositionally organised musical elements are not always reflected on the surface of the musical fabric. Her PhD dissertation includes analyses of Structures Ia and Music of Changes. Uno investigated the compositional design to identify the compositional elements that are responsible for the sonic outcomes of these pieces. As a central part of her investigation, Uno applied a systematic analytical methodology that empirically assessed the musical style of such compositions. In the case of Structures Ia, Uno explored a fascinating paradox existing between the order in the compositional design and disorder on the surface, concluding that the discrepancy between the compositional design and the surface structure was extreme. She took a similar approach when examining the relationship between Cage’s chance operations—and his intention to remove all his personal musical tastes from the compositional process—and the sound product of the piece. In fact, Uno proved that Cage’s compositional aim in this regard was not fully achieved. Thirdly, in her analysis of Structures 1a, Grant (2001) raised a similar concern to Uno, questioning the unusual relationship between the seemingly rational compositional process of integral serialism and the sound product that more closely resembles the irrational process of aleatoric composition. Although Grant and Uno shared similar concerns and conclusions, Grant tried to understand and explain various issues that arose when interpreting Structures Ia and integral serialism by investigating an aesthetic theory of serialism. Finally, Cook (1987) questioned how the analyses of Klavierstück III based on pitch relationships derived from complex series of permutations would enhance one’s experience of the work. Thus, he approached the piece in a descriptive way by focusing on the musical gestures and what elements of music contribute to them. His aim was to produce an analysis that can aid a listener or performer. Although Griffith did not explore the relationship between the conceptual organisation and surface structure in depth like Uno (1994) and Grant (2001), Cook did identify discrepancies between what one sees in the score and what one actually experiences through listening or performing, and pointed out the challenges that these types of discrepancies bring to music analysts dealing with contemporary repertoire. How to conjugate both the conceptual and perceptual aspects seems crucial if an analysis is to be convincing and have a wider appeal. Although only a relatively few authors seriously considered the surface structure of the works they analysed, their 244

attempts are remarkable in the sense that their analytical approaches start to depart from the traditional methods as well as, significantly, the governing ideologies. Both Covington (1980) and Cook (1987) had the courage to move away from the types of analysis that are committed to demonstrating the synchronic structure of pitch relationships. Wennerstrom (1967), Jameux (1984), Uno (1994) and Grant (2001) found analytical approaches no longer confined by the ideologies of organicism and positivism. Moreover, the changing focus from finding a meaning in the internal organisation of the work to producing a meaningful analysis for the listener is a sign of the displacement of formalism. The case studies in this thesis have revealed that recent music analysis is more likely to distinguish the conceptual aspects involved in the pre-compositional plan from the surface musical phenomena that can be perceived aurally. The awareness of such matters needs to be increased and further study is required to develop adequate methodologies for the detailed understanding of twentieth century compositions. Despite the complexity involved in many analyses of the selected works, a small number of analyses included comments that could enhance performance, such as articulation of detail, pedalling, texture, instrumentation and tempo. This is perhaps the least-developed aspect of such analyses and has never been their central focus. One could speculate on several reasons for such a tendency. Firstly, there is an underlying assumption that gaining an understanding of the structure of the piece or having a deeper knowledge of the pre-compositional plan alone can enhance performance. However, as argued previously, with this repertoire the relationship between the pre-compositional plan and the surface structure is not as direct as in tonal music: it is far more discrete and intricate. Secondly, the analyses of these works tend to incorporate graphs and charts, thus relying on visual communication. Yet, musical experience is largely dependent on aural communication. The visual presentations are almost unavoidable due to the complexity of the compositional theories and pre-compositional plans, and they can aid conceptual understanding of works. Unfortunately, some of the graphic representations are so complex or abstract that a performer is likely to find them irrelevant. The works examined here are complex and technically demanding enough without analyses adding another layer of difficulty. Thirdly, moving away from the conceptual matter of structure and theories, there is another challenge in describing these types of music. The traditional terminologies, such as melody, harmony and rhythmic motive, seem inadequate. 245

Often music analysts are left with a choice between using overly general terminology or employing technical terminology from the highly specialised field of music theory. Finally, for various reasons, these repertoires have obviously never been popular in the concert hall, for either performers or listeners. One might suggest this lack of wide appeal could reflect a failure of music analysis to explain these types of composition to performers in meaningful terms. Alternatively one could see that the clear lack of demand means that such an attempt is not urgent. Nonetheless, whatever challenges music analysts may face in developing analytical approaches more relevant to performance practice, it remains a much-needed area of research within the discipline of music analysis, especially when it tackles post-tonal compositions. In the process of examining 29 analyses throughout the four case studies, a wide range of approaches and diverse perspectives in music analysis became evident. Finding a particular angle to approach a work is vital to providing a satisfying analysis. The question is how this angle can be found. The case studies showed that authors have a tendency to produce more convincing analyses when they had access to the composer or the composer’s notebooks. For instance, Blumrőder (1993) and Maconie (1976, 2005) both spoke with Stockhausen, and Pritchett and Schädler had access to Cage’s notebooks. There are also a number of authors who have drawn on closely related fields of music studies such as history, compositional aesthetics, as well as the compositional theories prevalent at the time when the works were composed. To analyse Boulez’s Structures Ia, Ligeti (1960) already knew the serial material used in this piece. In analysing Stockhausen’s Klavierstück III, the way Schnebel (1960) and Cook (1987) divided the piece into groups clearly shows that the influence of Stockhausen’s compositional theory of groups. In analysing Cage’s work, none of the three authors reviewed here—Pritchett (1988), Uno (1994) or Schädler (1990)—approached Music of Changes as an autonomous work, but considered the composer’s aesthetic aim. Uno and Schädler even pointed out Boulez’s influence on the work. On the other hand, authors who analysed Messiaen’s Mode de valeurs did not show an interest in understanding the piece from the perspective of Messiaen’s own musical language. Apart from Covington’s approach (1980), analysts approached the piece from the perspective of serial technique, only to find that the piece scarcely demonstrated the application of note sequencing techniques. However, Chapter 3

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showed how knowing Messiaen’s musical language could be a key element to finding a suitable analytical direction. Similarly, most authors who analysed Boulez’s Structures Ia did not refer to Boulez’s own writings, instead citing Ligeti’s analysis. Moreover, there are two analyses that exhibit an extremely autonomous approach: DeYoung’s analysis (1978) of Structures Ia and Lewin’s analysis (1993) of Klavierstück III. Both authors concentrated exclusively on the internal relationships: DeYoung, on the relationship between pitch and duration, and Lewin, on a network of pitch relationships. The intention of such analyses mainly lies in demonstrating particular theories that the authors have proposed and treating each work as an autonomous entity divorced from its context. These analyses do not consider that unusual amount of writings and interviews the composers have left concerning their own compositional techniques and aesthetics. From my review of analyses throughout this thesis, I can conclude that whether analyses are focused on the conceptual or perceptual aspects of work, whether authors take a single narrative approach illustrating the unfolding of one governing idea or whether they apply interlocking approaches informing a heterogeneous organisational scheme, non-autonomous approaches are preferable to providing more comprehensive and well-rounded understanding of the music. Therefore, in order to find an appropriate angle to illuminate a work, music analysis should engage with compositional aesthetics and underlying ideologies, and should reflect an awareness of how a single work relates both to the composer’s individual development as well as to the broader historical/cultural context. Through reviewing 29 analyses written over the half century, certain broad developments within the discipline of music analysis may be observed. From the 1980s, a number of authors began to break away from the formalistic approach that had dominated music analysis (Wennerstrom’s analysis can also fit into this category, but was published in 1967). In other words, authors’ interests and analytical inclinations did not only focus exclusively on the internal workings of musical materials within a work but they started to enquire into analytical approaches that informed the listener’s experience. Covington’s analysis of Mode de valeurs (1980) was the first analysis in this case study where the focus was shifted from the compositional plan and techniques to the listener’s experience. Wennerstrom (1967), Jameux (1984) Uno (1994) and Grant (2001) all examined the surface structure of 247

Structures Ia and considered the listener’s perspective in depth. Cook (1987) also wanted to move away from a conceptual approach restricted to examining pitch relationships. His analytical focus was how music analysis can enhance a listener’s musical experience. Moreover, Pritchett’s 1988 attempt to investigate Cage’s chance music is notable in a sense that the aesthetic concept of chance operations defies the aesthetic of organicism that has been another powerful governing ideology of music analysis. It is interesting to note that in the very same decade, Kerman’s Musicology (1985) was published and from the 1980s, analytical enquiries by the above authors challenged the longstanding aesthetic position of music analysis and, as a result, a paradigm shift within the discipline started to occur. Indeed, this detailed review of analyses has illustrated how analytical enquiries into the four chosen piano works have gradually moved away from the prevailing ideologies. To this end, Jim Samson writes:

Formalism and modernism were closely tied together, their fortune controlled by the project of aesthetic autonomy. We might expect, then, that any displacement of modernism by postmodernism would demand a comparable displacement of formalism. The “opening up” of analysis can in part be understood in these terms. (1999, p. 52)

This study, therefore, not only provides a wealth of insight into these works but also provides a case for reflecting on how various profound ideological changes within our culture have influenced the analytical study of such seminal works. Significance of music analysis

Despite recent criticism made against the discipline of music analysis, it is still necessary to examine rigorously the musical materials within a piece if one desires to gain a satisfactory understanding of it. For instance, with Messiaen’s Mode de valeurs, although mainstream literature commonly associates the work with integral serialism, a close examination of the work clearly proves how far the piece is removed from Schoenbergian serial technique. With Boulez’s Structures Ia, through detailed investigation a delicate interface between the serial and non-serial elements of compositional process was revealed. At the same time, the significance of density variability and the proportional systems, with which Boulez experimented in creating his serial structure, was realised. With Stockhausen’s Klavierstück III, the composer’s heterogeneous organisational scheme not only attracted many different analytical

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approaches but also fostered valuable and thought-provoking debates among various authors. This was noticeably a case where music analysis enhanced our understanding, illuminating how Stockhausen developed his musical language when composing this piece. With Cage’s Music of Changes, only by closely analysing the compositional process could the extent to which chance operations replaced the composer’s decision-making process be measured. Through these types of meticulous examination, the aesthetic concepts underpinning the compositional processes could be clarified and thus commonly held views of the works could be challenged. In order to fully understand and appreciate the significance of music such as the four compositions selected in this thesis, analysis must continue to reside and be valued within musical scholarship. Although in recent decades many have challenged the precepts of music analysis as a discipline, the post-World War II repertoire, in particular, requires analysis to reveal its inner workings. Without a rigorous analysis that examines and accounts for the details of music, one runs the risk of misrepresenting the piece and of misunderstanding the composer’s intention. Such omissions were pointed out in each case study. It is important to address this issue here because the insights gained from music analysis can influence other disciplines of music such as history, performance, criticism, compositional theory and pedagogy. As Dahlhaus acknowledged, “Modern twentieth-century music is inconceivable without the mode of aesthetic cognition which finds expression in analysis” (1989, p. 94).

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Reference List

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257

MUSIC ANALYSIS AND THE AVANT-GARDE COMPOSITIONS OF POST-WORLD WAR II: FOUR CASE STUDIES

VOLUME 2

BY SUN-JU SONG, BMus, BA (Mus)(Hons), MMus

Queensland Conservatorium, Griffith University

Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

October, 2008

List of Figures

Volume Two

Figure 3.15 (c) Comparison between the mode of limited transposition and the hidden mode Figure 3.16 (a) Melodic cadence (Bars 28–29) Figure 3.16 (b) Melodic cadence (Bar 78) Figure 3.16 (c) Melodic cadence (Bars 107–111) Figure 4.1 Analyses of Boulez’s Structures Ia reviewed in this chapter Figure 4.2 (a) Boulez’s initial series used in Structures Ia according to Ligeti Figure 4.2 (b) Boulez’s two matrices used in Structures Ia Figure 4.3 (a and b) Comparison between Boulez’s and Ligeti’s attack series Figure 4.4 Sectional divisions in Structures Ia according to Ligeti’s analysis Figure 4.5 Ligeti’s examples for the superimposition of threads Figure 4.6 Wennerstrom’s analysis of combined dynamic levels in Structures Ia Figure 4.7 Wennerstrom’s analysis of combined attack levels in Structures Ia Figure 4.8 Griffiths’ graph illustrating the tempo, density and registral distributions of pitches in each formal sections Figure 4.9 Eckart-Bäcker’s analysis of the arrangement of three tempi Figure 4.10 Uno’s analysis of registral distributions of pitch rows in Boulez’s Structures Ia Figure 4.11 Uno’s analysis of segmental and sectional TG boundaries Figure 4.12 Model of serial communication according to Grant Figure 4.13 (a) Density variability of the 14 abstract formal sections of Structures Ia. Figure 4.13 (b) Density variability and the 11 actual formal sections of Structures Ia separated by pauses Figure 4.14 Opening chords of the 14 abstract formal sections in Structures Ia Figure 4.15 Opening chords with the interval of a compound fifth Figure 4.16 Opening chords without the interval of a compound fifth Figure 4.17 (a) Summary of tritone relationships Figure 4.17 (b) Pitch series paired in tritone relationships Figure 4. 18 Repeated pitches shared by all the pitch series used in each abstract formal section Figure 4.19 Density levels and the number of repeated pitches in each abstract formal section Figure 4.20 Relationship between the density variable and the arrangement of dynamics

iii

Figure 4.21 Categorisation of attacks Figure 4.22 Relationships between the types of attacks and the density level Figure 4.23 Two overlapping symmetries for the arrangement of tempo Figure 4.24 Golden Section ratio Figure 4.25 Iteration table illustrating the calculations application of the GS ratio Figure 4.26 The application of the GS ratio to the 14 abstract formal sections Figure 2.27 The cumulative calculation for the real time of the entire piece Figure 4.28 The range of section VI and GS primary point in relation to the assumed duration values for pauses Figure 4.29 The range of section VI and GS primary point in relation to the pause durations Figure 4.30 Complementary features of sections VI(a) and VI in relation to the GS ration Figure 4.31 A structural connection between the three tempi and the GS proportional system Figure 5.1 Analyses of Stockhausen’s Klavierstück III reviewed in this chapter Figure 5.2 Comparison of selected analyses of Klavierstück III Figure 5.3 Comparison of four author’s interpretations of formal structure of Klavierstück III Figure 5.4 Blumröder’s interpretation of the formal structure of Klavierstück III Figure 5.5 Schnebel’s interpretation of five-part formal structure of Klavierstück III Figure 5.6 Cook’s interpretation of the of formal structure of Klavierstück III Figure 5.7 Analytical approaches for pitch organisation in Klavierstück III Figure 5.8 Cook’s pitch distribution analysis in Klavierstück III Figure 5.9 Blumröder’s three-part durational divisions in Klavierstück III Figure 5.10 Blumröder’s analysis of tetrachords and durational scheme of Klavierstück III Figure 5.11 Blumröder’s analysis of the serial organisation in Klavierstück III Figure 5.12 Harvey’s illustration of pentachords and melodic contour in Kalvierstück III Figure 5.13 Lewin’s rearrangement of pentachords in Klavierstück III for ear-training purposes Figure 5.14 J – Related forms of pentachords retrieved from Lewin’s analysis Figure 5.15 Maconie’s analysis of rhythmic cells used in Kalvierstück III

Figure 5.16 Cook’s interpretation of the rhythmic relationship of five-part form using Cooper-Meyer’s rhythmic symbols Figure 5.17 Blumröder’s analysis of the serial organisation of durations in Klavierstück III Figure 5.18 Blumröder’s reinterpretation of Bar 8 Figure 5.19 Blumröder’s analysis of the serial organisation of dynamics in Klavierstück III Figure 5.20 Blumröder’s analysis of the permutations of dynamics in Klavierstück III Figure 5.21 Schnebel’s graphic illustration of Klavierstück III Figure 5.22 Blumröder’s analysis of the serial organisation of registers in Klavierstück III Figure 5.23 Blumröder’s serial permutations of registers in Klavierstück III Figure 5.24 Schnebel’s graphic illustration of a dense network of relationships Figure 5.25 Blumröder’s analysis of the serial organisation for density in Klavierstück III Figure 5.26 Blumröder’s analysis of the relationship between articulation and density Figure 5.27 My analysis of the five groups in Klavierstück III Figure 5.28 Comparison between my interpretation and Schnebel’s interpretation of groups Figure 5.29 Comparison between my interpretation and Cook’s interpretation of groups Figure 5.30 Group 1 (Bars 1–2) from Klavierstück III Figure 5.31 Group 2 (Bars 3–7) from Klavierstück III Figure 5.32 Rhythmic cells used in Group 2 Figure 5.33 Symmetrical relationships between dynamics and registers in Group 2 Figure 5.34 Group 3 (Bars 8–10.3) from Klavierstück III Figure 5.35 Symmetrical arrangements of intervals in Group 3 Figure 5.36 Symmetrical arrangements of durational values in Group 3 Figure 5.37 Group 4 (Bars 10.4–13) from Klavierstück III Figure 5.38 Symmetrical relationships in Group 4 Figure 5.39 Group 5 (Bars 14–16) from Klavierstück III Figure 6.1 Analyses of John Cage’s Music of Changes reviewed in this chapter Figure 6.2 Pritchett’s diagram of phrase group structure Figure 6.3 Relationship between phrases and a phrase group

Figure 6.4 Relationship between phrase groups and the structure of the entire piece Figure 6.5 Chart structure Figure 6.6 Sound chart 2 Figure 6.7 Subdivision of sound Chart: 4×4=16 Figure 6.8 Durational chart 2 Figure 6.9 Dynamic chart 8 Figure 6.10 Density chart Figure 6.11 Summary of the compositional process based on Pritchett’s analysis Figure 6.12 Terminology comparisons between Pritchett, Schädler and Cage Figure 6.13 Mobile and immobile relationships in chart system Figure 6.14 Segmented duration in visual measurement Figure 6.15 Comparison between the compositional systems of Cage’s Music of Changes and Boulez’s Structures Ia Figure 6.16 The range for each pre-determined parameter in Music of Changes and Structures Ia Figure 6.17 (a) Cage’s spatial notation in bars 131–133 of Book III Figure 6.17 (b) Recirculation of Cage’s notation in Bars 131–133 of Book III Figure 6.17 (c) Recirculation of Cage’s notation in Bars 131–133 of Book III

vi Figure 2.1 Four composers and four piano works

Years Four composers and four piano works Other events 1944 Boulez became Messiaen’s student

1946 Darmstadt summer courses commenced. 1948 Schaeffer began experimenting with tape music at the French National Radio 1949 Cage met Messiaen and Boulez in Paris. Cage’s Sonatas and Interludes for prepared piano was Messiaen composed Mode de valeurs. performed in Paris. The correspondence between Boulez and Cage began. 1951 Stockhausen first heard Mode de valeurs at Darmstadt. The electronic studio was established in Cologne. Cage composed Music of Changes. Cage began his experimentation with tape music. 1952 Stockhausen went to Paris to study with Messiaen. Stockhausen and Boulez became friends. Stockhausen composed Klavierstücken I–IV. Boulez composed Structures. 1953 Stockhausen joined the electronic studio in Cologne.

1954 Cage met Stockhausen in Cologne.

Figure 3.1 Three divisions of the mode1

1 From Mode de valeurs et d’intensités, by O. Messiaen, 1950, p. 2.

Figure 3.2 Analyses of Messiaen’s Mode de valeurs et d’intensités reviewed in this chapter

Authors Year of Titles Pages publication Klaus Schweizer 1973 Olivier Messiaen’s Klavieretude “Mode de valeurs et d’intensités” 8–146 Roderich Fuhrmann 1974 “Pierre Boulez (1925), Structures I (1952)” in Perspektiven Neuer Musik: 170–187 Material und didaktische Information Richard Toop 1974 “Messiaen/Goeyvaerts, Fano/Stockhausen, Boulez” in Perspectives of New 141–169 Music Robert Sherlaw Johnson 1975 “The experimental period 1949-1951” in Messiaen 101–115 Kate Covington 1980 Visual perception vs. aural perception: A look at Mode de valeurs et 4–11 d’intensités

Figure 3.3 Comparison of selected analyses of Mode de valeurs

Schweizer Fuhrmann Toop Sherlaw Johnson Covington (1973) (1974) (1974) (1975) (1980)

Pre-compositionally designed mode

Note order Formal structure Texture

Listener’s perspective

Figure 3.4 Schweizer’s illustration of regrouping of notes2

2 From “Oliver Messiaen Klavieretude ‘Mode de valeurs et d’intensités’”, by K. Schweizer,1973, Archiv für Musikwissenschaft, 30(2), p. 140.

Figure 3.5 (a) Toop’s rearrangement of three divisions3

3 From “Messiaen/Goeyvaerts, Fano/Stockhausen, Boulez”, by R. Toop, 1974, Perspectives of New Music, 13(1), p. 149.

Figure 3.5 (b)Sherlaw Johnson’s rearrangement of the three divisions4

4 From Messiaen, by R. Sherlaw Johnson, 1975, p. 106.

Figure 3.5 (c) Schweizer’s rearrangement of the three divisions5

5 From “Oliver Messiaen Klavieretude Mode de valeurs et d’intensités,” by K. Schweizer, 1973, Archiv für Musikwissenschaft 30(2), p.132.

Figure 3.6 Fuhrmann’s graphic illustration of the note order in Mode de valeurs6

6 From “Pierre Boulez, Structures”, by R. Furhmann, 1974, Perspektiven neuer Musik: Material und didaktische Information, p. 175.

Figure 3.7 Sherlaw Johnson’s graphic illustration of the note order in Mode de valeurs7

7 From Messiaen, by R. Sherlaw Johnson, 1975, p.108.

Figure 3.8 Six planes of sound in Covington’s analysis8

Sound plane 1

Sound plane 2

Sound plane 3

Sound plane 4

Sound plane 5

Sound plane 6

Note not belong in any sound plane

8 From “Visual perception vs. aural perception: A look at Mode de valeurs et d’intensités,” by K. Covington, 1980, Indiana Theory Review 3(2), pp. 7 & 9.

Figure 3.9 Covington’s analysis of Mode de valeurs9

9 From “Visual perception vs. aural perception: A look at Mode de valeurs et d’intensités,” by K. Covington, 1980, Indiana Theory Review 3(2), p. 10.

Figure 3.10 (a, b, c and d) Reinterpretation of Sherlaw Johnson’s note order into the arrangement of durational values10

(a) Example 1 (Bars 24–28, Group 1)

1 2 3 4 5 6 12 11 10 9 8 7

Centre 1 + 12 = 13 4 + 9 = 13 2 + 11 = 13 5 + 8 = 13 3 + 10 = 13 6 + 7 = 13

(b) Example 3 (Bars 39–49, Group II)

1 3 5 (4) (2) 11 9 7 8 10 12

Centre

1 2 3 (5) (4) 6 12 11 10 9 8 7

Centre

(d) Example 4 (Bars 53–57, Group I)

1 2 3 6 5 4 7 8 9 12 11 10

Centre 6 + 7 = 13 11 + 2 = 13 12 + 1 = 13 4 + 9 = 13 5 + 8 = 13 10 + 3 = 13

10 From Messiaen, by R. Sherlaw Johnson, 1975, p. 108.

Figure 3.10 (e, f and g) Reinterpretation of Sherlaw Johnson’s note order into the arrangement of durational values

(e) Example 5 (Bars 61–80, Group III)

1 - 5 2 3 4 5 6 +6 7 8 9 10 11 12

(f) Example 6 (Bars 81–86, Group I)

1 2 3 4 8 7 6 5 12 11 10 9

(g) Example 7 (Bars 86–96, Group II)

4 3 2 1 5 6 7 8 9 10 11 12

Figure 3.10 (h and i) Reinterpretation of Sherlaw Johnson’s note order into the arrangement of durational values

(h) Symmetrical arrangement of durations in Bars 81-96

Example 6 (Bars 81–86, Group I) Example 7 (Bars 86–96, Group II)

1 2 3 4 4 3 2 1 8 7 6 5 5 6 7 8 12 11 10 9 9 10 11 12

Centre of symmetry

(i) Example 8 (Bars 103–107, Group I)

1 2 3 4 5 6 7 8 9 10 (12) (11)

Figure 3.11 (a) Messiaen’s Mode de valeurs (Bars 105–115)11

11 From Mode de valeurs et d’intensités, by O. Messiaen, 1950, p. 11.

Figure 3.11 (b) Messiaen’s Mode de valeurs, Metrical reinterpretation (Bars 105– 115)

Figure 3.12 Motives

Motive I

Motive II

Motive III

Figure 3.13 Appearance of G512(Bars 51–53)

Figure 3.14 Construction of seven-note mode (Hidden mode)

12 From Mode de valeurs et d’intensités,), by O. Messiaen, 1950, p. 7.

Figure 3.15 (a) Comparison between the mode of limited transposition and the hidden mode

Mode 2

Hidden mode

Figure 3.15 (b) Comparison between the mode of limited transposition and the hidden mode

Mode 5

Hidden mode

Figure 3.15 (c) Comparison between the mode of limited transposition and the hidden mode

Mode 4

Hidden Mode

Figure 3.16 (a) Melodic cadence13 (Bars 28–29)

Figure 3.16 (b) Melodic cadence14 (Bars 78)

13 From Mode de valeurs et d’intensités, by O. Messiaen, 1950, p. 5. 14 From Mode de valeurs et d’intensités, by O. Messiaen, 1950, p. 9.

Figure 3.16 (c) Melodic cadence15 (Bars 107–111)

15 From Mode de valeurs et d’intensités, by O. Messiaen, 1950, p. 11.

Figure 4.1 Analyses of Boulez’s Structures Ia reviewed in this chapter

Authors Year of Titles Pages publications György Ligeti 1958* “Pierre Boulez: decision and automatism in Structures Ia” in Die Reihe Vol. 4 36-62 Mary Wennerstrom 1967 Parametric analysis of contemporary musical form 46-87 Roderich Fuhrmann 1974 “Pierre Boulez (1925), Structures Ia (1952)” in Perspektiven Neuer Musik: Material und 170-187 didaktische Information Reginald Smith Brindle 1975 The new music: The avant-garde since 1945 25-33 Lynden Deyoung 1978 “Pitch order and duration order in Boulez’ Structures Ia” in Perspectives of New Music 27-34 Paul Griffiths 1978 Boulez 19-27 Dominique Jameux 1984** Pierre Boulez 269-284 Ursula Eckart-Bäcker 1986 “P Boulez: Structures I pour 2 Pianos” In Werkanalyse in Beispielen 390-99 Yayoi Uno 1994 The roles of compositional aim, syntax, and design in the assessment of musical styles: 83-130 analyses of piano music by Pierre Boulez, John Cage, Milton Babbitt, and Iannis Xenakis circa 1950 Morag Grant 2001 Serial music, serial aesthetics: compositional theory in post-war Europe 150-154

* The English translation of this article was published in 1960. ** The English translation of this book was published in 1991.

Figure 4.2 (a) Boulez’s initial series used in Structures Ia according to Ligeti1

Initial series for pitches

Initial series for durations

1 2 3 4 5 6 7 8 9 10 11 12

1 From “Pierre Boulez,” by G. Ligeti, 1960, Die Reihe, 4, pp. 38-39.

Figure 4.2 (a) Boulez’s initial series used in Structures Ia according to Ligeti2

Initial series for dynamics

1 2 3 4 5 6 7 8 9 10 11 12

Initial series for attacks

2 From “Pierre Boulez,” by G. Ligeti, 1960, Die Reihe, 4, pp. 38-39.

Figure 4.2 (b) Boulez’s two matrices used in Structures Ia3 Figure 4.2 (b) Boulez’s two matrices used in Structures Ia17

Original Inversion

1 2 3 4 5 6 7 8 9 10 11 12 1 7 3 10 12 9 2 11 6 4 8 5 2 8 4 5 6 11 1 9 12 3 7 10 7 11 10 12 9 8 1 6 5 3 2 4 3 4 1 2 8 9 10 5 6 7 12 11 3 10 1 7 11 6 4 12 9 2 5 8 4 5 2 8 9 12 3 6 11 1 10 7 10 12 7 11 6 5 3 9 8 1 4 2 5 6 8 9 12 10 4 11 7 2 3 1 12 9 11 6 5 4 10 8 2 7 3 1 6 11 9 12 10 3 5 7 1 8 4 2 9 8 6 5 4 3 12 2 1 11 10 7 7 1 10 3 4 5 11 2 8 12 6 9 2 1 4 3 10 12 8 7 11 5 9 6 8 9 5 6 11 7 2 12 10 4 1 3 11 6 12 9 8 2 7 5 4 10 1 3 9 12 6 11 7 1 8 10 3 5 2 4 6 5 9 8 2 1 11 4 3 12 7 10 10 3 7 1 2 8 12 4 5 11 9 6 4 3 2 1 7 11 5 10 12 8 6 9 11 7 12 10 3 4 6 1 2 9 5 8 8 2 5 4 3 10 9 1 7 6 12 11 12 10 11 7 1 2 9 3 4 6 8 5 5 4 8 2 1 7 6 3 10 9 11 12

3 From “Pierre Boulez,” by G. Ligeti, 1960, Die Reihe, 4, p. 38. 17 From “Pierre Boulez,” by G. Ligeti,1975, Die Reihe, 4, p. 38

Figure 4.3 (a and b) Comparison between Boulez’s and Ligeti’s attack series

(a) Boulez4

1 2 3 4 5 6 7 8 9 10 11 12

(b) Ligeti5

4 From The Boulez-Cage correspondence, by J. Nattiez (Ed.), 1993, p. 101. 5 From “Pierre Boulez,” by G. Ligeti, 1960, Die Reihe, 4, p. 43.

28 Figure 4.4 Sectional divisions in Structures Ia according to Ligeti’s analysis6

Figure 4.4 Sectional divisions in Structures Ia according to Ligeti’s analysis 19

PART A PART B

Section I IIa IIb IIc III IVa IVb V VI VII VIII IX X XI

Number 2 4 3 1 6 2 5 1 5 3 4 4 2 6 of threads Bars 1-7 8-15 16-23 24-31 32-39 40-47 48-56 57-64 65-72 73-81 82-89 90-97 98-105 106- 115

6 From “Pierre Boulez,” by G. Ligeti, 1960, Die Reihe, 4, p. 49.

19 From “Pierre Boulez,” by G. Ligeti,1975, Die Reihe, 4, pp. 49

Figure 4.5 Ligeti’s examples for the superimposition of threads7

Two threads superimposed in section I

Four threads superimposed in section IVa

7 From “Pierre Boulez,” by G. Ligeti, 1960, Die Reihe, 4, pp. 53-54.

Figure 4.6 Wennerstrom’s analysis of combined dynamic levels in Structures Ia8

Measure 1–7 8–15 16–47 48–56 57–64 Dynamic level 12, 5 7(2), 2(2) 11(2), 8 7(2), 2(2), 5 12 (labeled by 8 assigned 5(2), 11(3,) 8 numbers) Resultant effect mixed soft loud soft Loudest

Measure 65–72 73–015 106–115 Dynamic level 2, 1, 3 (2), 7 6, 9, 1 1(2), 2, 3(2), 7 7, 9, 2, 6 7, 9, 2, 6 6, 9 Resultant effect soft mixed (medium) softest

8 From Parametric analysis of contemporary musical form, by M. Wennerstrom, 1967, p. 52.

Figure 4.7 Wennerstrom’s analysis of combined attack levels in Structures Ia9

Measure 1–15 16–23 24–31 32–47 48–64 Mode of attack 12, 5 3(2), 5 12 8, 3, (2), 11, 12, 1(3), 11, 8, 1 (labeled by 12, 5, 8, 11 8 5, 11, 8 assigned 5(2), 11(3,) 8 numbers) Resultant effect legato staccato legato staccato and sfz accented

Measure 65–89 90–115 Mode of attack 6(3), 12, 1 9(3), 1 12, 2, (2) 7, 5 6(3), 1 9 (3), 5, 7, 1

Resultant effect legato accented and

9 From Parametric analysis of contemporary musical form, by M. Wennerstrom, 1967, p. 52.

Figure 4.8 Griffiths’ graph illustrating the tempo, density and registral distributions of pitches in each formal sections10

10 From Boulez, by P. Griffith, 1978, p. 23.

Figure 4.9 Eckart-Bäcker’s analysis of the arrangement of three tempi11

11 From “P. Boulez: Structures I pour 2 pianos,” by U. Eckart-Bäcker, 1986, Werkanalyse in Beispeilen, p. 393.

Figure 4.10 Uno’s analysis of registral distributions of pitch rows in Boulez’s Structures Ia12 (Semibreves represent common notes shared by rows within a section)

12 From The roles of compositional aim, syntax, and design in the assessment of musical styles: Analyses of piano music by Pierre Boulez, John Cage, Milton Babbitt, and Iannis Xenakis circa 1950, by Y. Uno, 1994, p. 98.

Figure 4.11 Uno’s analysis of segmental and sectional TG boundaries13

13 From The roles of compositional aim, syntax, and design in the assessment of musical styles: Analyses of piano music by Pierre Boulez, John Cage, Milton Babbitt, and Iannis Xenakis circa 1950, by Y. Uno, 1994, p. 126.

37 32

Figure 4.8 Model of serial communication according to Grant 23

Figure 4.12 Model of serial communication according to Grant14

Composer Serial Piece Listener Choice of ‘rational’ ‘irrational’ sequences procedures to give of musical events ‘irrational’ results

24 Figure 4.9 Boulez’s pitch series used in Structures Ia

23 From Serial music, serial aesthetics: Compositional theory in post-war Europe, (p. 157), by M. Grant, 2001, Cambridge: Cambridge University Press. 24 From The Boulez-Cage correspondence, (p. 100), by J. Nattiez (Ed.), 1993, Cambridge: Cambridge Unive r s i t y P r e s s . 14 From Serial music, serial aesthetics: Compositional theory in post-war Europe, by M. Grant, 2001, p. 157.

Figure 4.13 (a) Density variability of the 14 abstract formal sections of Structures Ia. (Each square represents a thread)

I IIa IIb IIc III VIa VIb V VI VII VIII IX X XI Sections

Figure 4.13 (b) Density variability and the 11 actual formal sections of Structures Ia separated by pauses. (Again each square represents a thread).

I IIa IIb IIc III IVa IVb V VI VII VIII IX X XI

Figure 4.14 Opening chords of the 14 abstract formal sections in Structures Ia15

Sections: I IIa IIb IIc III IVa IVb V VI VII VIII IX X XI

15 Eb4 and Eb1 are also shown in section V and XI respectively as the last note played in the section.

Figure 4.15 Opening chords with the interval of a compound fifth

Sections: IIa III IVb VI IX

Figure 4.16 Opening chords without the interval of a compound fifth

Sections: IIb IVa VII VIII X XI

Figure 4.17 (a) Summary of tritone relationships

Figure 4.17(b) Pitch series paired in tritone relationships

Tritone Eb–A C–F# F–B G–C# Relationship Series paired O1–O3 I9–O6

in Part 1 I1–I3 Series paired R5–R8 R7–RI2 RI8–RI5 R12–R11

in Part 2 RI12–RI11 RI5–R1

Figure 4. 18 Repeated pitches shared by all the pitch series used in each abstract formal section

Section: I IIa IIb IIc III IVa IVb V VI VII VIII IX X XI

Figure 4.19 Density levels and the number of repeated pitches in each abstract formal section

Sections I II a II b II c III IV a IV b V VI VII VIII IX X XI

Density 2 4 3 1 6 2 5 1 5 3 4 4 2 6

Repeated 9 3 5 0 12 5 6 0 5 5 12 2 5 12 pitches

Figure 4.20 Relationship between the density variable and the arrangement of dynamics

Density Sections Assigned dynamics levels 6 III quasi p, quasi p, quasi f, fff, fff XI pppp, ppp, ppp, pp, pp, mf 5 IV(b) ppp, ppp, quasi p, mf, mf VI pppp, ppp, pp, pp, mf 4 II(a) ppp, ppp, mf, mf VIII ppp, mp, mf, f IX ppp, mp, mf, f 3 II(b) quasi f, fff, fff VII pppp, mf, f 2 I quasi p, ffff IV(a) quasi f, ff X mp, f 1 II (c) quasi f V ffff

Figure 4.21 Categorisation of attacks

Types of attacks Symbols Sustained sounds none

Non-sustained sounds

Figure 4.22 Relationships between the types of attacks and the density level

Types of attacks Abstract formal sections Density level Sustained sounds II(b) 1 V 1 I 2 VIII 4 VI 5 Non-sustained sounds IV(a) 2 Mixture of sustained and X 2 non-sustained sounds II(b) 3 VII 3 II(a) 4 IX 4 IV(b) 5 III 6 XI 6

Figure 4.23 Two overlapping symmetries for the arrangement of tempo (The three tempi are abbreviated as follows: S for Slow: semiquaver=144, M for Medium: quaver=120, and F for Fast: quaver=144.)

M F S F M M S F M F S M

Figure 4.24 Golden Section ratio16

A C B

0.618034 . . .

16 From Debussy in proportion: A musical analysis, by R. Howat, 1983, p. 2.

Figure 4.25 Iteration table illustrating the calculations application of the GS ratio

Iteration No. X DSQ count Result Remainder 1 0.6180340 1092 675 417

2 0.6180340 675 417 258

3 0.6180340 417 258 159 4 0.6180340 258 159 98 5 0.6180340 159 98 61 6 0.6180340 98 61 38 7 0.6180340 61 38 23

8 0.6180340 38 23 14

9 0.6180340 23 14 9 10 0.6180340 14 9 5 11 0.6180340 9 5 3 12 0.6180340 5 3 2 13 0.6180340 3 2 1

14 0.6180340 2 1 1

Figure 4. 26 The application of the GS ratio to the 14 abstract formal sections

Figure 4.27 The cumulative calculation for the real time of the entire piece

Sections Duration Duration Duration DSQ (Split Sec) (Sec) cumulative Cumulative (Split Sec) I 585 9.75 585 78 IIa 487.5 8.125 1072.5 156 IIb 487.5 8.125 1560 234 IIc 487.5 8.125 2047.5 312 III 1170 19.5 3217.5 390 IVa 487.5 8.125 3705 468 IVb 487.5 8.125 4192.5 546 V 585 9.75 4777.5 624 VI 1170 19.5 5947.5 702 VII 487.5 8.125 6435 780 VIII 585 9.75 7020 858 IX 487.5 8.125 7507.5 936 X 1170 19.5 8677.5 1014 XI 585 9.75 9262.5 1092

Total 9262.5 (Split Sec) Total 154.375 (Sec) Total 2.57 (min)

Figure 4. 28 The range of section VI and GS primary point in relation to the assumed duration values for pauses

Range of section VI and GS primary point (within VI)

S|L Pauses (sec) 0|0 1|2 1|3 2|3

Starting point 4777.5 5197.5 5317.5 5497.5 Ending point 5947.5 6367.5 6487.5 6667.5 Total Duration 9262.5 10162.5 10462.5 10762.5 GS Primary 5725 6281 6466 6652

Range of section VI and GS primary point (out of range)

S|L Pauses (sec) 1|4 2|4 2|5 3|6 4|8 4|10 5|10 5|15

Starting point 5437.5 5617.5 5737.5 6037.5 6457.5 6697.5 6877.5 7477.5 Ending Point 6607.5 6787.5 6907.5 7207.5 7627.5 7867.5 8047.5 8647.5 Total Duration 10762.5 11062.5 11362.5 11962.5 12862.5 13462.5 13762.5 15262.5 GS Primary 6652 6837 7022 7393 7949 8320 8506 9433

Figure 4.29 The range of section VI and GS primary point in relation to the pause durations

Figure 4.30 Complementary features of sections VI(a) and VI in relation to the GS ration

Figure 4.31 A structural connection between the three tempi and the GS proportional system

Figure 4.32 The relationship between the tempo structure and the static sound textures

Tempi: M F S F M S F M F S M

Figure 5.1 Analyses of Stockhausen’s Klavierstück III reviewed in this chapter

Authors Year of Titles Pages Publication Rudolf Stephan 1958 Neue Musik [New music] 60–64 Dieter Schnebel 1958* “Karlheinz Stockhausen” in Die Reihe Vol. 4 121–135 Jonathan Harvey 1975 The music of Stockhausen: an introduction 22–27 Robin Maconie 1976 The works of Karlheinz Stockhausen 62–66 Paul Griffiths 1981b Modern music: The avant-garde since 1945 85–87 Nicolas Cook 1987 A guide to musical analysis 354–363 David Lewin 1993 “Making and using a pcset network for Stockhausen’s Kalvierstück III” in 16–67 Musical form and transformation: 4 analytic essays Christoph von Blumröder 1993 “Kriterien der seriellen Komposition (1952)” in Die Grundlegung Der 109–154 Musik Karlheinz Stockhausen [“Criteria of serial composition (1952)” in Laying the foundation of Karlheinz Stockhausen’s music] Paul Griffiths 1995 Modern music and after: Direction since 1945 72–75 Robin Maconie 2005 Other planets: the music of Karlheinz Stockhausen 118–120

* The English translation of this article was published in 1960.

Figure 5.2 Comparison of selected analyses of Klavierstück III

Stephan Schnebel Harvey Maconie Griffiths Cook Lewin Blumröder (1958) (1958) (1975) (1976, 2005) (1981, 1995) (1987) (1993) (1993) Formal ● ● ● ● Structure Pitch ● ● ● ● ● ● ● ● Organisation Durational ● ● ● ● ● Organisation Dynamic ● ● ● ● Arrangement Registral ● ● ● Distribution Contour ● ● ● ●

Texture and ● ● ● Density Perspective of a listener ● ● ● ● ● ● ● and performer

Figure 5.3 Comparison of four authors’ interpretations of formal structure of Klavierstück III

Authors Number of divisions Proportion relationship Considerations for determining the formal structure

Stephan 3 4–8–4 (bars) Harmonic implication

Schnebel 5 9–15–16–10–9 (quavers) Closely related time-structures in symmetry

Cook 5 8–(8 + 8)–11–11–11 (quavers) Aid for performer, convenient basis for more detailed analysis, a neat durational scheme and “groupings”

Blumröder 4 4–4–4–4 (bars) Melodic contour, corner tone (the lowest and highest notes in each section) and core interval (major second A–B).

Figure 5.4 Blumröder’s interpretation of the formal structure of Klavierstück III 28

28 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 114..

Figure 5.5 Schnebel’s interpretation of five-part formal structure of Klavierstück III29 Group Notes Durations Intensities Selection Distribution Selection Distribution Sel. Distr. I Notes within a Fairly even distribution of the Durations within a region Mainly high durations, a few 4f Contrasts of p-mf regions of 3.5 notes—density decreases toward the from the “middle” upwards. low ‘vertices’ 4mf f-mf rising to f octaves top (elucidation). 4p

II 2.25 octaves Notes arranged in three register- Durations within the whole Mainly durations in the 4f Three complexes: mf bands—at the borders and in the available region. middle lower region. Few 3mf mf→p→mf middle. Elucidation toward the high vertices. 5p f bottom.

III 3.1 octaves Notes distributed fairly evenly, Durations within the whole Mainly medium durations 6f Contrast alteration and grouped in high and low complexes, available region. 6mf mingling high density. 4p

IV 2.75 octaves Notes arranged in four register- Durations within the entire Mainly durations within the 4f Complexes bands. Elucidation toward the top. available region. middle and upper region. Few 3mf p-mf low vertices. 3p p-f mf-f

V 4.1 octaves Average even distribution of notes. Durations within a low Medium durations, one low 1ff f Elucidation toward the bottom. Low region. vertex. 1mf Contrast of mf-p and f- density. 2p ff (rising to ff)

29 From “Karlheinz Stockhausen,” by D. Schnebel, 1975, Die Reihe, 4, pp.126–127.

Figure5.6 Cook’s interpretation of the formal structure of Klavierstück III30

30 From A guide to musical analysis, by N. Cook,1987, p. 359.

Figure 5.7 Analytical approaches for pitch organisation in Klavierstück III

Figure 5.8 Cook’s pitch distribution analysis in Klavierstück III31

31 From A guide to musical analysis, by N. Cook, 1987, p. 362.

Figure 5.9 Blumröder’s three-part durational divisions in Klavierstück III 32

32 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 124.

Figure 5.10 Blumröder’s analysis of tetrachords and durational scheme of Klavierstück III33

33 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 125.

Figure 5.11 Blumröder’s analysis of the serial organisation in Klavierstück III34

34 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 127.

Figure 5.12 Harvey’s illustration of pentachords and melodic contour in Klavierstück III35

35 From The music of Stockhausen, by J. Harvey, 1975, p. 26.

Figure 5.13 Lewin’s rearrangement of pentachords in Klavierstück III for ear- training purposes36

36 From Musical from and transformation: four analytical essays, by D. Lewin, 1993, p. 42.

Figure 5.14 J-Related forms of pentachords retrieved from Lewin’s analysis37

J relationships Conversion to Pitches Invariant chromatic tetrachordal subsets

P0 and p0 (Ab, A, Bb, B, D) and (F, G#, A, Bb, B) Ab, A, Bb, B

P6 and p6 (D, Eb, E, F, Ab) and (B, D, Eb, E, F) D, Eb, E, F

P8 and p8 (E, F, F#, G, A#) and (Db, E, F, Gb, G) E, F, F#, G

P2 and p2 (A#, B, C, C#, E) and (G, A#, B, C, C#) A#, B, C, C#

37 Note. “P” for pentachord and “p” (lower-case) for inverted form of P and the content of the above figure is derived from Lewin’s analysis. From Musical from and transformation: Four analytical essays, D. Lewin, 1993, pp. 26–32.

Figure 5.15 Maconie’s analysis of rhythmic cells used in Klavierstück III38

A. Original form of rhythmic cell in bar 1

B. Variation of rhythmic cell in bar 2

C. Pictorial illustration of rhythmic cells to bar 8

Figure 5.16 Cook’s interpretation of the rhythmic relationship of five-part form using Cooper-Meyer’s rhythmic symbols39

38 From Other planets: The music of Karlheinz Stockhausen, by R. Maconie, 2005, pp. 119–120. 39 From A guide to musical analysis, by N. Cook, 1987, p. 362.

Figure 5.17 Blumröder’s analysis of the serial organisation of durations in Klavierstück III40

Figure 5.18 Blumröder’s reinterpretation of bar 841

40 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 133. 41 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 134.

Figure 5.19 Blumröder’s analysis of the serial organisation of dynamics in Klavierstück III42

Figure 5.20 Blumröder’s analysis of permutations of dynamics in Klavierstück III43

42 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 131. 43 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 131.

Figure 5.21 Schnebel’s graphic illustration of Klavierstück III44

44 From “Karlheinz Stockhausen,” by D. Schnebel, Die Reihe, 4, page number is not provided.

Figure 5.22 Blumröder’s analysis of the serial organisation of registers in Klavierstück III45

Figure 5.23 Blumröder’s analysis of permutations of registers in Klavierstück III46

45 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, pp. 128–129. 46 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 130.

Figure 5.24 Schnebel’s graphic illustration of a dense network of relationships47

47 From “Karlheinz Stockhausen,” by D. Schnebel, Die Reihe, 4, p. 131.

Figure 5.25 Blumröder’s analysis of the serial organisation for density in Klavierstück III48

Figure 5.26 Blumröder’s analysis of the relationship between articulation and density49

48 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 134–135. 49 From Die Grundlegung Der Musik Karlheinz Stockhausen, by C. Blumröder, 1993, p. 135.

Figure 5.27 My analysis of the five groups in Klavierstück III50

Figure 5.28 Comparison between my interpretation and Schnebel’s interpretation of groups (My interpretation is indicated by squares and Schnebel’s one is indicated by curved lines.)51

Figure 5.29 Comparison between my interpretation and Cook’s interpretation of groups (My interpretation is indicated by squares and Cook’s one is indicated by curved lines.)52

Figure 5.30 Group 1 (bars 1–2) from Klavierstück III53

Figure5.31 Group 2 (bars 3–7) from Klavierstück III54

53 Karlheinz Stockhausen, Klavierstücke 1–4|für Klavier|Nr. 2 (c) Copyright 1954 by Universal Edition London (Ltd.), London/UE 1225. 54 Karlheinz Stockhausen, Klavierstücke 1–4|für Klavier|Nr. 2 (c) Copyright 1954 by Universal Edition London (Ltd.), London/UE 1225.

Figure 5.32 Subgroups of durations used in Group 2

A. Subgroup 1

B. Subgroup 2

C. Subgroup 3

Figure 5.33 Symmetrical relationships between dynamics and registers in Group 2

Figure 5.34 Group 3 (bars 8–10.3) from Klavierstück III55

Figure 5.35 Symmetrical arrangements of intervals in Group 3

Com. Min.3rd Min.3rd

Maj. 6rh

Com. Min.2nd Aug. 4th Com. Min.3rd

Aug. 4th Maj. 3rd

Figure 5.36 Symmetrical arrangements of durational values in Group 3

Centre

Figure 5.37 Group 4 (bars 10:4–13) from Klavierstück III56

Figure 5.38 Symmetrical relationships in Group 4

Figure 5.39 Group 5 (bars 14–16) from Klavierstück III57

Centre of metrical division

Figure 6.1 Analyses of John Cage’s Music of Changes reviewed in this chapter

Authors Year of Titles Pages publication James Pritchett 1988 The development of chance techniques in the music of John Cage, 1950- 107–156 1958 Stefan Schädler 1990 Transformationen des Zeitbegriffs in John Cages Music of Changes 185–236 [Transformation of time-concept in John Cage’s Music of changes] in Musick–Konzepte James Pritchett 1993 The music of John Cage 78–88 Yayoi Uno 1994 The roles of compositional aim, syntax, and design in the assessment of 131–186 musical styles: analyses of piano music by Pierre Boulez, John Cage, Milton Babbitt, and Iannis Xenakis circa 1950

Figure 6.2 Pritchett’s diagram of phrase group structure58

58 From The development of chance techniques in the music of John Cage, 1950–1956, by J. Pritchett, 1988, p. 110.

Figure 6.3 Relationship between phrases and a phrase group (Cage refers to them as small and large unit structures)

A phrase group (Large unit structure)

3 5 6¾ 6¾ 5 3⅛

Phrases (Small unit structures)

Figure6.4 Relationship between phrase groups and the structure of the entire piece

Book 1 (Three phrase groups)

Book 2 (11¾ phrase groups) 11¾ ×

Book 3 (6¾ phrase groups)

6 ¾ ×

Book 4 (8⅛ phrase groups)

8⅛ ×

Figure 6.5 Chart structure59

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16

17 18 19 20 21 22 23 24

25 26 27 28 29 30 31 32

33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48

49 50 51 52 53 54 55 56

57 59 60 61 62 63 64 65

59 From The development of chance techniques in the music of John Cage, 1950–1956, by J. Pritchett, 1988 p. 113.

Figure 6.6 Sound chart 260

60 From The music of John Cage, by J. Pritchett, 1993, p. 80.

Figure 6.7 Subdivision of sound Chart: 4×4=16 61

1 2 3 4 2 3 4 12 chromatic pitches need to 1 2 3 4 be used within subdivisions 2 3 4

* elements coloured with grey are silences

61 The diagram is produced based on Pritchett’s explanation from The development of chance techniques in the music of John Cage, 1950–1956, by J. Pritchett, 1988, pp. 117–118.

100

Figure 6.8 Durational chart 262

62 From The music of John Cage, by J. Pritchett, 1993, p. 8.

101

Figure 6.9 Dynamic chart 863

ffff>f ff>p f>ppp f

p>ppp f>ppp

ffff f>pp

ffff>f ffff>ff

mf>pp fff>mf

p>ppp p>pp

ffff>ff mf>mp

63 From The music of John Cage, by J. Pritchett, 1993, p. 81.

102

Figure 6.10 Density chart64

Hexagrams Active layers 1 – 8 1 9 – 16 1 + 2 17 – 24 1 + 2 + 3 25 – 32 1 + 2 + 3 + 4 33 – 40 1 + 2 + 3 + 4 + 5 41 – 48 1 + 2 + 3 + 4 + 5 + 6 49 – 56 1 + 2 + 3 + 4 + 5 + 6 + 7 57 – 64 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

64 From The development of chance techniques in the music of John Cage, 1950–1956, by J. Pritchett, 1988, p. 124.

103

Figure 6.11 Summary of the compositional process based on Pritchett’s analysis

Composing each layer within a Using the same Determining the phrase hexagram number tempo and density 1. Hexagram number for Repeating the chosen for the first for each phrase sound or silence entire process phrase, determining the according to the 2. Hexagram number for until the end relationship between hexagram number duration of the piece mobile and immobile selected by chance 3. Hexagram number for chart for the given dynamic when sound is phrase group selected.

104

Figure 6.12 Terminology comparisons between Pritchett, Schädler and Cage

Rhythmic Structure Pritchett’s terminologies Schädler’s terminologies Cage’s terminologies (3, 5, 6¾, 6¾, 5, 3⅛ ) 3 , 5 , 6¾ , 6¾ , 5, and 3⅛ Phrases Small structural units Small unit structural point (each number refers to the number of bars) 29⅝ bars A phrase group Middle structural unit Large unit structural point Intermediate structural point 4 books (3, 11¾, 6¾, 8⅛) 4 parts 4 Large structural units 4 Books (each number refers to the number of phase groups and middle structural units)

105

Figure 6.13 Mobile and immobile relationships in chart system65

Book 2 Section (Abschnitt) 1 2 3 5 6 7 8 9 11 12 Mobile (beweglich) 1357 2468 1357 2468 1357 2468 1357 2468 1357 2468 Immobile (unbeweglich) 2468 1357 2468 1357 2468 1357 2468 1357 2468 1357

Book 3 Section (Abschnitt) 1 2 3 4 5 6 7 Mobile (beweglich) 2468 2468 2468 2468 1357 1357 1357 Immobile (unbeweglich) 1357 1357 1357 1357 2468 2468 2468

Book 4 Section (Abschnitt) 1 2 3 4 5 6 7 8 9 Mobile (beweglich) 2468 1357 2468 1357 1357 2468 1357 2468 Immobile (unbeweglich) 1357 2468 1357 2468 2468 1357 2468 1357

65 From “Transformatione des Zeitbegriffs in John Cages Music of Changes [Transformation of the time-concept in John Cage’s Music of Changes],” by S. Schädler, 1990, Musik–Konzepte, 90, pp. 195–196.

106

Figure 6.14 Segmented duration in visual measurement66

66 From The Boulez-Cage correspondence, by J. Nattiez, 1993, p. 95.

107

Figure 6.15 Comparison between the compositional systems of Cage’s Music of Changes and Boulez’s Structures Ia

Five components Music of Changes Structures Ia Pre-determined structure Rhythmic Structure 11-section formal structure Pre-determined musical parameters Sound, duration, dynamics, density and tempo Pitch, duration, dynamic, attack, density and tempo Pre-designed series or charts 8 charts each for sound, duration and dynamics Series for pitch, duration, dynamic and attack 1 chart for density and 1 chart for tempo Selection process Chance operations through tossing of coins Serial ordering according to matrices Process of combining the all necessary Creating a layer and superimposing layers Creating a layer and superimposing layers musical elements according to the given density level according to the given density level

108

Figure 6.16 The range for each pre-determined parameter in Music of Changes and Structures Ia

Pre-determined parameters Music of Changes Structures Ia Pitch (Sound) 12 chromatic pitches and percussive sounds 12 chromatic pitches Duration Demisemiquaver – Semibreve Demisemiquaver – Dotted-crotchet Dynamic pppp – ffff pppp – ffff Attack Combined with dynamics 12 types of attack Density 1 to 8 1 to 6 Tempo Mixture of slow, medium and fast tempi Mixture of slow, medium and fast tempi

109

Figure 6.17 (a) Cage’s spatial notation in Bars 131–133 of Book III67 (The length of one bar is 10 cm according to Cage’s instruction, although the published score is reduced to 85% of this size.)

67 From Music of Changes, by J. Cage, 1961, p. 58.

110

Figure 6.17 (b) Recirculation of Cage’s notation in bars 131-133 of Book III

Figure 6.17 (c) Recirculation of Cage’s notation in bars 131-133 of Book III

111