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Some interparametric correlations in Edgard Varèse’s Déserts Morrison, Kenneth John 1992

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Some Interparametric Correlations in Edgard Varèse's Déserts by K E N N E T H J O H N M O R R I S O N B.Mus., University of British Columbia, 1989 B .Sc , University of Manitoba, 1976 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF Master of Arts in THE F A C U L T Y OF G R A D U A T E STUDIES (School of Music, Music Theory) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A June, 1992 © Kenneth John Morrison, 1992 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Music The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT Some Interparametric Correlations in Edgard Varèse's Déserts Varèse's later works can be described as an integrated network of parallel processes. As the composer's last completed work for instrumental ensemble, Déserts represents the culmination of Varèse's compositional style. At both the microstructural and macrostructural levels, there are correlations among a myriad of musical parameters. At the grossest formal level in Déserts and other works, the numbers of measures are often a whole number multiple of a primary motivic number. The number of beats and the number of rhythmic attacks are often related to the total number of measures in a piece such that there is an overall attack density of 1 attack/beat. There is also a regular metric background that belies the apparent rhythmic irregularities of the musical foreground. Similar correlations also control events at the level of microstructure: among pitch interval in semitones, the number of attacked time-points, and the total number of attacks in the entire instrumental texture in a given segment. These interparametric correlations enable the analyst to explicate some of the structural processes operating in this music. T A B L E OF CONTENTS Abstract ii List of Figures iv Acknowledgements vi Chapter 1 - Introduction 1 Chapter 2 - Microstructure in Déserts 10 Instrumental Section I (mm. 1-82) 10 Instrumental Section II (mm. 83-224) 25 Instrumental Section III (mm. 225-63) 40 Instrumental Section IV (mm. 264-325) 44 Conclusion 50 Chapter 3 - Macrostructure in Déserts 51 Conclusion 74 Bibliography of Works Cited 77 LIST OF FIGURES Figure 1.1 - Page 1 of Intégrales 4 Figure 1.2 - Interval/Attack Correlations in the First-Half of m. 21 of Déserts.... 6 Figure 2.1 - Interval Symmetry in mm. 1-20 of Déserts 12 Figure 2.2 - Symmetrical Interval Collection in m. 21 13 Figure 2.3 - Symmetrical Discrete Attack Complex in mm. 21-29 14 Figure 2.4 - Symmetrical Time-Point Attack Complex in mm. 21-29 15 Figure 2.5 - Projection of Attack/Interval Complex from m 29 to m. 30 16 Figure 2.6 - Interparametric Correlation in Upper Subsegment of mm. 34-38.. 17 Figure 2.7 - Interval Symmetry in mm. 41-45 18 Figure 2.8 - Multiparametric Symmetries in mm. 63-64 21 Figure 2.9 - Time-Point Attacks in mm. 54-78 22 Figure 2.10 - Rotation of Tetrachords in mm. 85-93 26 Figure 2.11 - Symmetrical Time-Point Attack Complex in mm. 85-94 27 Figure 2.12 - Symmetrical Discrete Attack Complex in mm. 83-94 28 Figure 2.13- (a) Timpani Attacks in mm. 93-117 29 (b) Timpani Attacks in mm. 216-38 Figure 2.14 - Horizontal Rotation in mm. 127-30 32 Figure 2.15 - Symmetrical Pitch Collection in mm. 157 34 Figure 2.16 - Interparametric Correlations in mm. 210-11 39 Figure 2.17 - Imperfectly Symmetrical Pitch Collection in m. 270 45 Figure 2.18 - Groups of 13 Discrete Attacks in mm. 271-75 46 Figure 2.19 - Imperfectly Symmetrical "Framing" of m. 278 47 Figure 2.20 - Imperfect Interval Symmetry in mm. 304-10 49 Figure 3.1 - M m . 38-44 of Ionisation 54-Figure 3.2 - Sectional Divisions in Ionisation and Section I of Déserts 58 Figure 3.3 - Establishment of Similar Attack Points in m. 1 and m. 54 of Déserts. 59 Figure 3.4 - Table of Durations in Déserts 61-62 Figure 3.5 - Numbers of Beats at Consolidated Tempi 63 Figure 3.6 - Comparison of the Two "Endings" in Déserts 66 Figure 3.7 - Regular Attacks of Eb4 in mm. 312-318 67 Figure 3.8 - Parallel Macrostructure Segmentations 70 Figure 3.9 - Latent Interval Symmetries 73 A C K N O W L E D G E M E N T S I would like to thank the faculty of the School of Music at the University of British Columbia for their constant support and rigorous instruction—especially Dr. William Benjamin, Dr. Wallace Berry, Dr. Keith Hamel, and Dr. Eugene Wilson. Special thanks are due my thesis advisor. Dr. John Roeder, whose careful editing and insightful questions were invaluable. I would also like to thank Dr. Jonathan Bernard for kindly and constructively criticizing my original thesis proposal. I must also acknowledge Dr. Bernard's work on the music of Edgard Varèse. Much of the work in this thesis owes a debt to it, especially to the comprehensive bibliography—the primary reference source for all work on Varèse. Finally, I would like to thank CPP/Belwyn for kindly granting me permission to include musical examples from die published scores by Varèse to which they own the copyright. Chapter 1 - Introduction Varèse insisted that "form is a result ~ the result of a process. Each of my works discovers its own form."l Although this description suggests that Varèse's conception of musical form is open to rational inquiry, most analysts have had littie success at describing processes in Varèse's music. The most successful analysis to date has been the seminal work of Jonathan Bernard. His theory convincingly describes the organization and transformation of some of the pitch materials in Varèse's works. However, he does not consider in detail how other musical parameters are coordinated with pitch. Rather, he asserts that "of all the domains of musical sound, pitch and register—or, as they are called here, pitch/register— are primary in the music of Varèse."^ I am convinced that Varèse's music is better described as a network of interparametric associations witiiout insisting on any one parameter's hegemony. This thesis will treat some aspects of rhythmic organization that have hitherto gone unnoticed, aspects that facihtate a more complete description of the formal processes operative in Varèse's music. When describing his conception of rhythm, Varèse stated: Rhythm is the element in music that gives life to the work and holds it together. It is the element of stability, the generator of form. In my own works, for instance, rhythm derives from the simultaneous interplay of unrelated elements that intervene at calculated, but not regular time lapses [emphasis added].3 Referring to Déserts y srtst claimed that: The work progresses in opposing planes and volumes. Movement is created by the exactly calculated intensities and tensions which function in opposition to one another; the term 'intensity' referring to the desired acoustical result, the word 'tension' to the size of the interval employed.'''' ^Edgard Varèse, "The Liberation of Sound," Perspectives of New Music 5/1 (Fall/Winter 1966), 16. J^onathan Bernard, The Music of Edgard Varèse, (New Haven: Yale University Press, 1987), 128. ^Edgard Varèse, "The Liberation of Sound," Perspectives of New Music 5/1 (Fall/Winter 1966), 15. '^ Edgard Varèse, quoted in Femand Ouellette, Edgard Varèse, translated by Derek Coltman, (New York: Orion, 1968),183. Of course, the composer did not elaborate on what he meant by "the desired acoustical result." However, given his explicit statement about interval sizes as "tension," causes one to speculate as to the nature of the "intensity" that functions "in opposition" to the interval. In the program notes, the composer further stated that although the intervals between the pitches determine the ever-changing and contrasted volumes and planes, they are not based on any fixed set of intervals such as a scale, or series, or any existing principle of musical measurement. They are decided by the exigencies of this particular work.^  I am convinced that the interparametric correlations I have found better represent the nature of Varèse's compositional aesthetic. The rhythmic processes treated in this thesis seem to be operative in most of Varèse's matiu-e output. However, I will concentrate on Déserts — a work for large chamber ensemble and tape composed in 1954. I will also discuss briefly, aspects of three other works: Intégrales for large chamber ensemble (1925), Ionisation, his percussion ensemble masterpiece (1931), and Density 215 for solo flute (1936). These pieces are, of course, quite different, confirming what the composer asserted: the "exigencies" of individual works give rise to different processes in each. This does not mean, however, that Varèse's compositional aesthetic and indeed compositional method were not both highly developed enough to be consistent across all his mature output. That is, even though the composer has insisted on the uniqueness of each piece, there are still many elements that are common to all of his mature works. Most of my observations seem to involve pre-compositional planning. My interest in analysis is manyfold, but one of my primary interests in analysis is the attempt to glean some aspects of a composer's working method. Therefore, much of what I have to say will be conjecture on the composer's working method and compositional aesthetic rather %dgard Varèse, quoted in Henry Cowell, "Review of Performance of Déserts," Musical Quarterly 51 (1955), 372. than the role of the perceiver. That is, I will concentrate on what Jean-Jacques Nattiez calls the poietic level rather than the esthesic.^ M y fundamental assertion is that many parameters of Varèse's compositions are controlled by numerical relationships. That is, the quantities of many parameters in a given composition correspond to a few different integers and simple arithmetic relations among them. These parameters include: pitch interval in semitones, time-points at both the local and global level, attack numbers and attack density. Even the chosen size and instrumentation of each ensemble reflects a primary numerical relationship. Déserts opens with the reiteration of the interval of 14 semitones and not coincidentally the ensemble of wind instruments consists of 14 players. Similarly, Density 215 for unaccompanied flute begins with the interval of 1 semitone. Ionisation, which is essentially devoid of intervallically caUibrated pitch materials stUl maintains operative numerical relationships: an ensemble of 13 players, a score marked by 13 rehearsal numbers, and 13 X 7 = 91 measures. As we shall see, this correlation with measure numbers is very important in the formal plans of these works. One common type of correlation is between the pitch interval and number of discrete attacks. A clear example of this is the first page of the score of Intégrales (reproduced as Figure 1.1). The opening figure in the Eb-clarinet (sounding D5-Ab5-Bb5) contains intervals of 6 and 2 semitones, making a total span of 8.^  The attacks of Bb5 are clearly grouped into 2 and 6, again for a total of 8. In m. 4, the operative quantity of 8 is maintained, this time involving the 6 attacked time-points in the Eb-clarinet plus the 2 attacked time-points in the tam-tam and gong. ^Nattiez would probably describe my approach as "inductive poietics"—an attempt to discover a composer's working method by relatively "neutral" observations—see Jean-Jacques Nattiez, Music and Discourse: Toward a Semiology of Music (Princeton, NJ.: Princeton University Press, 1990). 7ln this thesis, I shall conform to the American Acoustical Society's convention of denoting "middle C" as C4. Figure 1.1 - Page 1 of Intégrales INTÉGRALES AnoUzntixio. Edgard Varèse Petites Flûtes. Hautbois. en mit Clarinettes. en sil» Cor en fa. en ré Trompettes. ' en ut Trombone-ténor. | Trombone-basse. < Trombone-contrebasse.' Cymbale suspendue. 1 Caisse claire. ) Caisse roulante./ Tambour à curde. Castagnettes. 2 Cymbales. Blocs chinois." \ Grelots. _ Chaînes. 3 Tambour basque." Gong. Tam-tam-Triangle. 4Cymbale chinoise. Verges et Fouet. Grosse caisse. - m Andantina-These interparametric relationships are not idle numerological fantasies, but demonstrable consistencies by which the composer is able to create a highly integrated texture. They appear in most, i f not all, of Varèse's mature output, suggesting that = 9 5 6 7 3 ? -? \ 7 a 1 ^ -i 1 i^^o^ o >^J^'^^ 3 5 6 Varèse's constant analogies to crystallization and other geometric phenomena really did guide the composer's compositional aesthetic. It is also important in assessing Varèse's place in twentieth-century music to realize that Intégrales was premiered in 1925, long before Messiaen and his students experimented with interparametric consistencies. Central to my theory is the observation that there is often a correspondence between the number of attacks and the prominent pitch intervals measured in semitones. I will refer to this aspect of the music's organization as its microstructure. For example, in many sections of the piece, the number of discrete attacks equals the number of semitones between the pitches at either registral extreme. By discrete attack, I mean each attack in a given section, including attacks of pitches that double other pitches at the unison or in other octaves. There are two types of event which I do not consider to possess a discrete attack. First, in the rare instance when the exact same pitch is attacked at the same timepoint with the same articulation and intensity by two or more identical instruments, the simultaneous attacks are counted as one. Figure 1.2 shows a reduction of the first half of m. 21. The first two pitches, C#4 and G#3 are attacked in exactiy the same rhythm, and with the same articulation and intensity by trombones I and 11 (I is muted here and II has been muted since m. 17). Thus we count only 2 discrete attacks. However, the subsequent pitches played by trombone n in the same measure differ from the simultaneously attacked events in trombone I, and therefore each counts as a discrete attack. The A4 that is sounded on the third quarter-beat of the bar simultaneously by the flute, third trumpet, first trombone, and piano all count as discrete attacks since they are played by different instruments. This reflects Varèse's insistence that orchestration is a fundamental parameter of music and differentiates between exact orchestral doubling and even the most subtle of orchestration changes. The other kind of event that we will consider not to possess a discrete attack is the grace note. Grace notes generally are used as articulation factors, and so do not have the event status of discrete attacks. Figure 1.2-6 Discrete Attacks in the first half of m. 21 of Déserts a b c d e 5 time-point attacks (a-e) Although the number of discrete attacks is often significant, as I will show below, significant segments are sometimes characterized simply by the number of attacked time-points they contain. In Figure 1.2, for instance, 5 time-points are articulated by attacks. In contrast, the last G#3 and D4 shown, which the three trombones all attack at once, articulate together only one time-point. These two types of attack parameters are not contradictory, rather they represent two different ways of measuring attack density. This is consistent with Varèse's aesthetic.^ The number of attacked time-points and the number of discrete attacks can be thought of as related, and yet still distinct. One aspect of the orchestration sometimes makes the number of discrete attacks unclear. Often, the piano exacdy doubles attacks in the wind instruments or the percussion instruments. At the beginning of the piece, the piano attacks contribute to the numerical correlations between the pitch interval and attack parameters. Later, however, there is sometimes a numerical correlation between interval and the number of discrete attacks in the ^The discrete attack parameter is a useful concept in terms of microstructure only since it reflects more the vertical dimension of sonic space—the entire texture over a short span of time. The time-point attack parameter is operative at the level of macrostructure as well as microstructure. wind instruments only. This variable role of the piano thus creates some difficulty in determining how many events have discrete attack status. In his description of Déserts, Varèse stated that the "instrumental ensemble [is] composed of 14 wind instruments, a variety of percussion instruments played by 5 musicians, and a piano as an element of resonance. "9 If the composer rigorously adhered to this conception of the instrumental ensemble throughout the piece, then my discrete attack counts that include the piano may be an inaccurate reflection of the piece. However, in some passages, the piano plays a more structural role than merely contributing to "resonance." In the description of the microstructure of Déserts in Chapter 2,1 will provide altemate readings of discrete attack totals to acknowledge this ambiguity. Accounting for both the parameters of discrete attack and time-point attack, a multi-dimensional network can be described that relates them to pitch-intervals. In Figure 1.2, the first interval we hear is 5 semitones, presented as the melodic interval between C#4 and G#3. 5 is also the number of attacked time-points in this segment. The next interval we hear, 6 semitones, is presented as a "vertical" interval between trombones I and II (although it is subsequently presented melodically in trombone I). 6 is also the number of discrete attacks in the segment. So we see that the two values of attack quantities correspond to the size of the two melodic and harmonic intervals in this section. Such analysis of the microstructure of Déserts will constitute Chapter 2 of this thesis. Another important aspect of microstructural organization is the arrangement of repeated notes into groups of motivic numbers of attacks. The clearest example of this is the 13 septuplets of reiterated pitch C7 in the xylophone in mm. 157-64 of Déserts. As we shall see, both the numbers 13 and 7 are prominent throughout this piece. In Chapter 3,1 shall discuss other parameters in regard to events over larger temporal spans. This study of the macrostructure of Déserts is based on hearing the piece ^Edgard Varèse, quoted in Henry Cowell, "Review of the first American performance of Déserts" Musical Quarterly ^ \(\955), 371-72. as comprised of overiapping, simultaneously moving "sound masses." The resulting complex textures are heard as organized into sections of motivic durations measured in numbers of measures or beats. These durational consistencies are also evident in Ionisation. As I have already pointed out, the number 13 governs the large-scale durational planning of the piece and, as we shall see, the piece is divided into large sections that are symmetrical like the first instrumental section of Déserts. As part of the following discussion engages pitch-interval processes, I would like to give a brief overview of Jonathan Bernard's elegant theory of pitch/register for the music of Varèse. This theory demonstrates that there is a high degree of consistency in the ways that pitch materials are transformed throughout each of Varèse's mature works. Bernard convincingly asserts that the essential syntactic units of Varèse's music are the intervals between pitches in a specific register, not pitch-class-intervals nor pitch-classes themselves. He believes the trichord is the fundamental collection in Varèse's music, but my analysis does not depend on the validity of this assertion. He has discovered that many of the pitch collections in Varèse's music exhibit pitch-interval symmetry, and describes three basic processes by which collections of pitches can be continuously transformed. One or more pitches of a segment may be "rotated" around one pitch which serves as an axis. "Rotation" is related to the famiUar process of inversion. Often in symmetrical collections, there is symmetrical "expansion" at both ends of the span. The opposite transformation, "contraction," is also operative. An interval collection may also be "projected" in register, which is essentially the same as transposition. An understanding of these processes is necessary to understanding part of the following analysis. That is, the existence of a numerical correlation between the realms of pitch and rhythm suggests there may be processes operating in the rhythmic domain that parallel Bernard's processes in the pitch domain. This possibility will be explored below. ^^'or an introduction to these processes, see Chapter 2 of Jonathan Bernard, The Music of Edgard Varèse (New Haven: Yale University Press, 1987). Bernard's theory demonstrates that complete octave equivalence is not a valid concept in Varèse's music: A spatial environment would collapse upon the introduction of inversional equivalence.... Octave equivalence is different, if it is consistentiy employed with reference only to interval sizes and not to the positions of individual pitches. That is, octave equivalence does not mean that a pitch in one octave means the same thing as it would in any other octave, but diat an interval enlarged or shrunk by an octave (or compound octave) has,/or some purposes only, a meaning equivalent to its unaltered form.^l This question as to the role of octaves is especially important in the present study. As the piece progresses there is an increase in the statements of single pitch-classes simultaneously sounded in two or more octaves. The possibility of a corresponding, limited type of modulo 12 equivalence in the rhythmic domain will also be considered in Chapters 2 and 3. However, this does not necessarily mean that inversional equivalence is operative in this music. At the end of Chapter 3, where I outline the overall formal process of Déserts, I will briefly taUc about referential pitches and global pitch design. Bernard prefers not to speculate on this possibility, but I consider it to be an important organizational device. In a 1930 interview, Varèse characterized his music as "naturally atonal [such that] certain tones [serve as] axes around which the sound masses seem to converge."^2 Much of the analysis of the pitch/register parameter of Déserts has ah-eady been thoroughly described by Bernard in his three publications that contain analytical commentary on this work—^particularly, in his 60-page analysis of Déserts.The reader may assume that virtually all of the analysis of the pitch/register parameter of Déserts in the present study is based on Bernard's theory. ^hbid., 102. ^2josé André, "Edgard Varèse y la mûsica de Vanguardia," La Naciôn, Paris, March 1930; trans. David R. Bloch in "The Music of Edgard Varèse," (Ph.D. dissertation, University of Washington, 1973), 260; quoted in Jonathan Bernard, The Music of Edgard Varèse, (New Haven: Yale University Press, 1987), xx. ^^This complete analysis of Déserts can be found in Jonathan Bernard, "A Theory of Pitch and Register for the Music of Edgard Varèse," (Ph.D. dissertation, Yale University, 1977), 235-94. Chapter 2 - Microstructure of Varèse's Déserts The instrumental score of Déserts is readily divided into four sections by the three interpolations of "organized sound." In this chapter, I will describe the instrumental sections of the piece as a series of "moments" that display interparametric correlations at the local or microstructural level. As we shall see, however, the piece is not simply a linear series of discrete, contiguous segments; rather, the microstructural components of the form often overlap with each other, or are interrupted by interpolations of other materials (on tape or otherwise). To facilitate a description of the microstructure, I will segment the piece in the following manner: the tape interpolations—at m. 82,224, and 263—divide the instrumental material into 4 sections; each of the 4 sections is parsed into a few subsections—determined by a strong sense of closure or contrast; which, in turn, are parsed into segments; which can be composed of component subsegments. The very complex issues of alternate segmentations, overlap, and interruption wiU be addressed in Chapter 3, where I describe the possibility of parallel processes in the piece. Instrumental Section I (mm. 1-82) Measures 1-20 constitute a static subsection which opens the piece by establishing many of the motivic quantities and interparametric relationships that are operative throughout the entire composition. These quantities and relationships are quite apparent in this first section, compared to later in the piece. That is, the opening has a relatively expository nature, in that the transformational processes occur more directiy than in many of the later sections of the piece. The opening F4-G5 dyad, an interval of 14 semitones, is reiterated in slighdy varying instrumental combinations throughout the first twenty measures. While this dyad is sounded by itself (to the third quarter of m. 6), it is sounded in pairs, each of which consists of a total of 14 discrete attacks. That is, the first six measures divide into three segments— m. 1 to the first beat of m. 3, second beat of m. 3 to the second beat of m. 5, and third beat of m. 5 to the third beat of m. 6— each of which consists of 2 time-points at which the F-G dyad is attacked, and 14 discrete attacks. Also, of the 14 discrete attacks in each segment in the opening six measures, 7 of the discrete attacks are on F4 and 7 are on G5.1 This division of 14 attacks into 7 plus 7 is subsequently (at m. 7) reflected in the domain of pitch by the division of the F4-G5 dyad by C5 into two vertical intervals of 7 semitones each. Prior to this division, the texture gets more complex in m. 6 with the sounding of the D2-E3 dyad. Not only is this another interval of 14 semitones, but it establishes other intervals with the existing F4-G5 dyad. Of primary importance is the interval of 13 semitones separating the two established vertical constructs: E3 to F4. Later in this opening subsection (mm. 14-20), the interval of 13 semitones is sounded by the Bb l -B2 dyad. This quantity, 13, becomes motivic, not just in the domain of pitch, but also with respect to discrete attacks. For example, after the opening 7 measures, the F-G dyad is heard in alternating groups of 14 and 13 attacks throughout the remainder of the opening section: m. 7 contains 14 discrete attacks of the F-G dyad; mm. 8-11 feature 13 discrete attacks of F and G (again, assuming the cymbal attacks are associated with the F-G dyad). Metaphorically, one might understand mm. 1-11 as the clear articulation of a sound mass, (the F-G dyad) moving through space. As it approaches another body, (the D-A dyad) the attractive and repulsive forces between the two bodies start to change the sound masses themselves. In mm. 14-20, the new trichord, {Bbl,B2,C#4}, is heard as related to the opening dyads (see Figure 2.1). The interval span of 27 semitones becomes very significant to the pitch parameter of the macrostructure (see Chapter 3, p. 72). ^ Actually, some of the attacks associated with the F-G dyad are on "unpitched" cymbals. However, the cymbal attacks are always simultaneous with the attacks of the F-G dyad and the registral difference suggests that the composer may have intended the low cymbal to correspond to the F4 and the high cymbal to the G5. Similarly, in the subsequent passage (specifically, mm. 10-11) there are 3 relative-pitched gongs associated with the D2-A2-E3 trichord. Figure 2.1 - Interval Symmetry in mm. 1-20 Other correspondences between attack quantities and pitch-interval sizes are also apparent in this section. During the first 13 measures, there are 28 attacked time-points. 28 is twice 14, the span of die two sonorities: F4-G5 and D2-E3. The subsequent passage of this introductory section, mm. 14-20, contains 16 attacked time-points. Thus, there is a total of 44 time-point attacks in the opening 20 measures. The interval span in the opening is 45 semitones (Bbl-G5). Thus, i f there is a correlation between time-point attacks and interval over the entire opening 20 measures, it is inexact. We shall see the significance of the number 44 in the next subsection, mm. 21-29. To summarize, in the opening subsection of Deserts, the same quantities— 7, 13, and 14— seem to govern the parameters of pitch-interval, attacked time-points, and discrete attacks. These are all inextricably enmeshed, creating an overall texture that is consistent in all parameters. Another manifestation of interparametric relationships is apparent in the second subsection of the opening of the piece, mm. 21-29. The first two quarters of m. 21 consist of 6 discrete attacks. As we have seen in connection witii Figure 1.2, the numbers 6 and 5 govern both intervals and attacks here (see p. 7). The G#3 and D4 that conclude the 6-attack gesture are absorbed at the end of m. 21 into an exacdy symmetrical pitch structure— G#3-D4-A4-Eb5-Bb5-E6 (see Figure 2.2). Figure 2.2 - Interval Symmetry in mm. 21-22 1 } 6st. )7st. } 6st. Considering only the pitch domain, we may understand the rest of the passage as part of the sound-mass breaking off, thus changing the entire spatial configuration. That is, the G#3-D4 dyad, which is first heard as part of the exactiy symmetrical vertical collection m m. 21, is "projected" down an octave to G#2-D3 in the timpani and bass clarinet, which are the lowest pitches in this subsection. A larger-scale symmetry is also apparent in mm. 17-20, where there is an interval of 27 semitones between the lowest pitch B b l and the C#4. In mm. 21-29, the highest pitch is the piccolo's E6—27 semitones above the C#4 that begins the section.2 We observe a corresponding symmetry in the attack design: from the third quarter of m. 21 to the downbeat of m. 22, there are 16 discrete attacks; and in the remainder of the passage, there are also 6 discrete attacks in the winds (Bb clarinets in mm. 24-27). ^  This creates an arrangement of 16 discrete attacks "framed" by segments of 6 discrete attacks each. 6+16+6 makes a total of 28 discrete attacks in the wind instruments and piano throughout this passage. We have already seen the number 28, a multiple of the motivic 14, in mm. 1-13, and the number 16 in mm.14-20. It is also possible to hear in this ^This intervallic consistency is observed by Bernard on p. 18 of his 1981 paper, and is in turn indebted to: Chou Wen-Chung, "Varèse: A Sketch of the Man and his Music," Musical Quarterly 52 (1966), 160-61. ^The total of 16 discrete attacks is arrived at by assuming that the simultaneous attacks on Eb5 by trumpets I and III count as one discrete attack. passage a segment of 22 discrete attacks composed of the central 16-attack segment and either one of the 6-attack segments: a) (the first two quarters of m. 21 as 6 attacks) + (the remainder of m. 21, plus the 3 attacks on the downbeat of m. 22 as 16 attacks) = 22; or b) (the same 16) + (6 in the clarinets in mm. 24-27) = 22. Figure 2.3 represents these hearings. Figure 2.3 - Symmetrical Discrete Attack Complex in mm. 21-29 First half of Second half of Bb clarinets in m. 21 m. 21 + m. 22 mm. 24-27 Discrete Attacks: 6 + 16 I I L . 22 (winds, + 22 (all ^ 44 discrete piano) percussion) ^ ui5,ncLC attacks In this section there are also 22 discrete attacks in the percussion (the grace notes in the field-drum figures are considered to be articulation subdeties, not actual discrete attacks). Thus, this passage contains two segments of 22 discrete attacks each, for a total of 44 discrete attacks. Recall that the first subsection (mm. 1-20) contained exactiy 44 time-point attacks. 44 is also the interval in semitones between the extreme pitches, G#2 and E6 of this second subsection. The number of attacked time-points in this section is also significant. The time-point attacks in the piano and wind ensemble total 14, which, of course, corresponds to the familiar motivic interval. The 14 attacks are composed symmetrically— 4 attacks in the second half of m. 21 to m. 22, framed by 5 attacks on either side: 5 in the first half of m. 21 and 5 in the Bb clarinets in mm. 24-27. This symmetrical structure is itself part of a larger symmetrical structure: 14 attacked time-points in the wind instruments and piano, plus 8 attacked time-points in the timpani, plus 14 attacked time-points in the unpitched percussion. Thus, the number of attacked time-points in the percussion instruments throughout this section is 22 (same as the number of discrete attacks). Figure 2.4 is a graphic representation of the symmetrical features of the time-point attacks in this section. Figure 2.4 - Time-Point Attack Complex in mm. 21-29 Indefinite-Pitched Percussion Timpani Winds and Piano Time-Point 14 + g + 14 Attacks: I II I 22 • 22 The timpani are appropriately afforded the special status of belonging to both the percussion ensemble and the ensemble of pitched-instruments (wind instruments and piano). Thus, there are two possible segments of 22 attacked time-points involving 14 time-points in both the pitched and non-pitched ensembles, and 8 time-points in the timpani. This is similar in design to the two possible segments of 22 discrete attacks shown in Figure 2.3. So, we see that in this short passage, the parameters of pitch/register, discrete attacks, time-point attacks, and even orchestration are carefully controlled, and integrated into a complex and rigorously calculated texture. This passage also shows tiiat the different parameters do not necessarily articulate sectional divisions together. The downbeat of m. 21 does initiate a new segment. This is partly due to the fortissimo dynamic. However, in terms of the pitch/register parameter, the measure begins as a continuation of the previous segment. The B b l in the tuba and B2 in horn II from m. 20 are sustained throughout m. 21. Togetiier with the C#4 tiiat is attacked by trombones n and HI on the downbeat of m. 21, the Bbl-B2-C#4 trichord of the previous segment is reiterated. The C#4 acts as a pivot between the adjacent sections. Measures 30-53 comprise the next subsection, which occupies the centre of section I. Following Bernard's 1977 analysis, we may hear it parsed into three smaller segments— mm. 30-40, mm. 41-45, and mm. 46-53.4 The beginning of the subsection is marked by a sudden change in timbre, tempo, and texture in m. 30. There is, however, intervallic continuity with the preceding section. As Bernard notes, the first trichord (Bb3-E4-B4) in m. 30 can be heard as a transposition of the timpani figure that ends the previous section. The transposition relation between the two trichords in the adjacent sections is the motivic 14 semitones (see Figure 2.5). The similarity between the trichords is reinforced by an attack number correlation: the timpani in mm. 21-29 sound the trichord G#2-D3-A3 with a total of 8 attacks, and the transposed trichord at the first quarter-beat of m. 30 is also sounded with 8 discrete attacks in the brass. Figure 2.5 - "Projection" of trichord from m. 29 to m. 30 m.29 30 14 St. m o ' , te PXJ brass (8 discrete attacks) timpani (8 discrete attacks) Much of the rest of m. 30 also consists of stacks of alternating 6- and 7-semitone intervals resulting in the interval of 13 semitones between the extremes of any contiguous '^ Jonathan Bernard, "A Theory of Pitch and Register for the Music of Edgard Varèse," (Ph.D. dissertation, Yale University, 1977), 235-43. trichord. There is a total of 4 times 13, or 52 discrete attacks in m. 30. The number 13, which first appeared as the pitch-interval between the two original dyads in m. 6, becomes increasingly important in interparametric correlations in this central subsection of the opening instrumental section of the piece. As at the end of the first subsection, there is a correlation here between the number of attacks and the total interval span. There are 78 discrete attacks in mm. 32-40, and the interval between the extreme pitches, C#l-G7, is 78 semitones. At least one of the fragments also demonstrates a similar correlation: Figure 2.6 shows that the upper three woodwinds in mm. 34-38 sound 16 discrete attacks and the interval of 16 semitones (Eb6-G7). The number 16 recalls the discrete attack structure of mm. 21-22. Figure 2.6 - Discrete Attack/Interval Correlation in Upper 3 Woodwinds in mm. 34-38 *Note: double stems signify 2 discrete attacks In the next self-contained segment, mm. 41-45, we find another example of a similar yet slighdy modified relationship between intervals and the number of discrete attacks. As Figure 2.7 shows, the pitches played in this passage form vertical interval structures that are mirror-images around an axis that bisects the interval between Eb4 and Bb4. Similar to measure 30, the adjacent 6- and 7-semitone intervals accrue such that each side of the axis spans 26 semitones by the end of the passage. The correlation to attacks is this: each side of the axis also contains 13 discrete attacks. That is, there are 13 attacks of pitches in the span from C#2 to Eb4 and there are 13 attacks of pitches in the span from Bb4 to C7. Dynamic markings also contribute to the articulation of the motivic quantity: G#2 and Bb4—which also form the interval of 26 semitones—are the loudest pitches at the end the passage (third quarter of m. 45). There is another interesting feature of this 26-semitone vertical interval: while it is clearly the addition of two adjacent intervals of 13 semitones (octave + minor-second), the resulting 26 semitones is, like the interval of 14 setnitones, a compound interval of the major-second. It suggests tiiat, despite Bernard's objections, a limited sort of pitch "octave equivalence" may be heard to be operative. Figure 2.7 - Interval Symmetry in mm. 41-45 5s.t. G 6 7s.t. F 6 13 t.p. attacks 6s.t.V. B5 26 s.t. 7 s t v E5 6s.t.N. Bb4 7 s.t. 6St./. Eb4 6 SX/. D3 G*2 26 s.t. 13 t.p. attacks The next short passage, from the last two triplet-eighths of m. 45 to the end of m. 53 concludes the second subsection. Its duration is exactiy 61 triplet quarters and there are exacdy the same number (61) of time-point attacks in this section. This passage begins at the end of m. 45, where horn n and Bb-clarinet n sound horizontal groups of 7- and 6-semitones respectively. The beginning of this passage, up to the end of m. 47, spans 36 semitones (F#l-F#4), and there are half this number (18) of discrete attacks. The timpani and tubas also sound 18 time-point attacks from m. 48 through m. 51. To summarize: in the second subsection, we have seen a striking number of correlations between prominent pitch intervals and attack complexes in relatively clearly defined segments. In addition, in the last segment discussed—mm. 45-53 there is a simple numerical relationship between the duration of the segment and its time-point-attack total. The prominence of 14 in the beginning of the first subsection has been superseded by 13 and its multiples. This is particularly evident in the central segment of the central subsection (mm. 41-45). Mm. 54-82 comprise the third major subsection in the first instrumental section. In this third subsection of the opening section of the piece, the segmentations begin to get more ambiguous—a trend that increases throughout much of die rest of the piece. There is much compelling evidence for hearing mm. 54-78 as a discrete subsection, rather than hearing the subsection continuing to the first interpolation of "organized sound" at the end of m. 82. These multiple segmentations will be discussed in more detail in Chapter 3. Mm. 54-58 constitute a segment consisting of two registrally discrete subsegments: the reiteration of a Al-C#2-Bb2 trichord and the unusual sounding in different registers of the same pitch-classes E and F. There are 24 discrete attacks of pitches that comprise the trichordal subsegment. On the downbeats of mm. 54 and 56, there is also an attack of the low gong. The low register of the gong and its sustained quality result in us hearing these 2 attacks as associated with the trichordal subsegment. This results in a total of 24 -t- 2, or 26 discrete attacks of the trichordal subsegment, which is twice the interval span of the 13 semitones of the subsegment. The other subsegment in mm. 54-58 spans 25 semitones (E5-F7) which is a mod 12 equivalent of the trichordal subsegment. There are 71 discrete attacks in this latter subsegment, resulting in a total of 97 discrete attacks in the entire segment. 97 is also a mod 12 equivalent of the interval spans of the two subsegments in this passage.^ A second segment begins at the last quarter beat of m. 58, overlapping slightly with the end of the first segment, and can be heard as continuing to the end of m. 62. Each of these segments contains 22 time-point attacks each (counting the E6-F6 in the piano on the last quarter beat of m. 58 as the end of the first segment, and the C-B harmonic dyad and the C-G melodic interval as 2 time-point attacks for the beginning of the second segment). This is similar to mm. 21-29, where 22 was also an important number. In the next segment— m. 63 to the end of the third quarter of m. 65—are stated two interesting symmetrical sound masses that contain 6 time-point attacks each. M . 63 contains a hexachord that is exactiy bilaterally symmetrical both rhythmically and intervallically. Each trichordal half of this hexachord spans 13 semitones; however, the 13-semitone span is not comprised of alternating 6- and 7-semitone spans, as many of the collections in the piece are(cf. mm. 41-45). Rather, the span is comprised of 9- and 4-semitone intervals. The overall span of the entire hexachord in m. 63 is 27 semitones. This can be heard as being related to the intervallic content in mm. 1-20, where 27 semitones was the span firom the extreme pitch of one of the dyads to the closest pitch of the other dyad. Many other features of the hexachord in m. 63 will be discussed in terms of a possible overall global pitch design in the piece, at the end of Chapter 3. M . 64 presents a hexachord that is pitch-class-complementary to the hexachord in m. 63, again with its intervals symmetrically arranged. However, although the hexachord ^There are 22 time-point attacks in this passage. However, if we hear the two component subsegments as discrete, then the 4 time-point attacks of the trichordal subsegment may be added to the 22 time-point attacks of the E-F subsegment, resulting in a total of 26 time-point attacks for the passage, which equals the span of the E-F subsegment is divided by register into two trichiords, as before, the rhythms are not perfectly symmetrical. In the latter hexachord, the intervals of 9 and 4 semitones again contribute to a span of 13 semitones for each trichordal half. However, this time the interval between the two trichords is (C4-A4) 9 semitones, resulting in a perfectly symmetrical alternation of intervals: 9+4+9+4+9 for a total span of 35 semitones.^ Figure 2.8 compares the symmmetries between both hexachords. Both hexachords are inversionally symmetrical around a pitch axis between E4 and F4. Again, at the end of Chapter 3,1 will discuss the significance of this pitch axis (see p. 72). Figure 2.8 - Pitch Collections Symmetrical about E4/F4 Axis in mm. 63-64 m. 63 woodwinds brass ) L > — â k 4 H-^ 1 ^ • m. ^—I ^ f- " m -3±^t=^ «s 3 0 F4 E4 N.B.: Horizontal symmetry about vertical axis in m. 63 A new segment begins at the last eighth note of m. 65 and extends through the attacks on the downbeat of m. 71. The total of time-point attacks in this segment is the motivic number 28. The collection of pitches in tiiis segment are intervallically symmetrical aroimd the pitch E4, which also bisects the interval between C#4-G4 that begins the segment. It is also significant that the one repeated pitch in horn I ( G4) is attacked 13 ^These symmetrical interval collections that contain alternating intervals which result in overlapping pairs of like intervals are analogous to a similar phenomenon in the attack parameter—for example, the overlapping adjacent groups of 22 attacks in mm. 21-29 (see Figures 2.2 and 2.3 ). times in this passage. Many subsequent occurrences of repeated attacks of single pitches will also contain exactly 13 attacks— one of the primary motivic numbers throughout the piece. The change of timbre and texture with the sounding of the timpani and brass from the last beat of m. 71 to the penultimate sixteenth-note of m. 78 causes us to hear this passage as another segment. This segment is comprised of several discrete textural elements sounding together, not unlike the texture in mm. 32-40. This passage also contains correlations between discrete attacks and pitch-intervals. The most obvious is the motivic 13 attacks in the timpani in mm. 71-74 that sound the interval of 13 semitones— D#2-E3. The piano doubles some attacks of the timpani and some attacks of the other instruments, but it nevertherless also sounds a total of 13 discrete attacks. In the trumpets and horns, from mm. 72-76, pitch-classes G# and D are sounded in different octaves. There are exactly 25 discrete attacks (using my doubling rules), a number which nearly matches the 24 semitones of registral space covered by tritones— G#3-D4-G#4-D5-G#5. Comparing this to the nearly exact correlation between die interval of 45 semitones and the total of 44 time-point attacks in mm. 1-20 may cause us to wonder whetiier such a correlation was intended. Figure 2.9 is a table of time-point attacks in die subsection that can be heard ending in m. 78. Figure 2.9 - Time-Point Attacks in mm. 54-78 mm. 54-58: mm. 58-62: mm. 63: 22 t.p. attacks 22 t.p. attacks 6 t.p. attacks 6 t.p. attacks 28 t.p. attacks 46 t.p. attack mm. 64-65: mm. 65-71: mm. 71-78: Total 130 t.p. attacks I regard the number, 130, as significant because it is an even multiple of 13, the number that is featured in many parameters of this subsection and others. This, by itself, supports the segmentation that ends the subsection at m. 78. Now, there is another way of parsing this long subsection that yields attack-interval correlations. It can be heard as two contiguous segments: mm. 54-65 and m. 66-78. Then, the first segment, mm. 54-65, sounds 56 time-point attacks and spans the interval of 56 semitones (A1-F6). However, die question of the nature octave doublings again arises: the maximum interval sounded is actually 68 semitones, A1-F7, which is a mod 12 equivalent to the number (56) of time-point attacks."^ So, the correlation may seem questionable. However, the second segment in this alternative segmentation, mm. 66-78, lasts 13 measures, and is unified partly by being intervallically symmetrical around the pitch E4, and by the fact dtat it contains a correlation between the maximum pitch interval, E)#l-F7 (74 semitones) and 74 time-point attacks. In terms of microstructure, the final 4 measures of the first instrumental section (mm. 79-82) contain 14 time-point attacks and 31 discrete attacks. If the 3 attacks in the piano in m. 81 are not counted as discrete (functioning here merely as "resonance"), then the total of discrete attacks in these measures is 28, or double the number of time-point attacks. The number 14 and its multiple 28 were both prevalent at the beginning of the piece and the reappearance of these numbers at the end of the first instrumental section is significant.^ Jonathan Bernard's analysis connects the last four measures of die first instrumental section to the first two measures of the second instrumental section, mm. 83-84 on the 7However, the occurrence of octave doubling and the sounding of a pitch class in more than one octave in a passage is uncommon in Varèse's music, generally, but increasingly prevalent in Déserts (particularly, see my discussion of the 2 "endings" of the piece in Chapter 3). ^The horns in each of mm. 79-80 and the trumpets in m. 82 sound 3 dyads. In each of these measures, the vertical intervals of the dyads add up to 14 semitones. Such motivic interval sums do not seem to occur elsewhere in the piece. basis of a network of pitch/register associations.^ We can also observe some pitch-interval/time-point-attack correlations that further strengthen the sense of continuity of these measures. If we entertain considering the last sixteenth of m. 81 to the end of m. 84 as a segment, then a new network of pitch-interval/time-point-attack correlations emerges. Firstiy, the interval range is expanded to 44 semitones (B1-G5) which corresponds to twice the number of quarter notes of duration in tiiese six measures (4 ms. of 4/4 + 2 ms. of 3/4 = 22). This numerical relationship is reminiscent of that in mm. 21-29. Chapter 3 will present some further striking evidence for this multipUcity of segmentation with regard to the macrostructure. J^onathan Bernard, "A Theory of Pitch and Register for the Music of Edgard Varèse," (Ph.D. dissertation, Yale University, 1977), 249-50. Section II (mm. 83-224) Section n is extremely difficult to segment. It seems as if the relatively clear textures or sound masses of Section I are here, for the most part, transformed by various collisions, reflections, and penetrations. Consequently, dividing the section into discrete subsections may be inappropriate in some passages, although some sections are definitely self-contained. Bernard subdivides the second instrumental section into roughly five sub-sections: mm. 83-117, 118-45, 146-74, 175-99, 200-24.10 I mostly agree with Bernard's segmentation of this instrumental section, but many other segmentations seem to be operative. That is, I hear in some places overlapping, not discrete segments. Many of the smallest segments, which I will call gestures, demonstrate attack/ittterval correlations. Therefore, in the following section of my analysis, I shall concentrate on these microstructures. The organization of tiiese microstructures into larger structural processes will be dealt with in the next chapter. The opening subsection of Section II— mm. 83-117— contains many interparametric correlations. Mm. 83-84 open this subsection with a quiet sustained texture recalling the last measures of Section I. The opening trichord, Bl-C3-C#4, is a pair of 13-semitone intervals spanning a total of 26 semitones. From mm. 85-92, a 4-note collection— D3, C5, F#6, G#7— is reiterated, spanning 54 semitones. From the last eighth-note of m. 91 to the third quarter of m. 93, this same interval collection is presented, transposed and inverted—Fl, G2, C#4, B5. Figure 2.10 shows the interval content of these tetrachords. The intervals of 14,18, and 22 semitones remind us of some of the primary motivic intervals of Section I. lOjonathan Bernard, "A Theory of Pitch and Register for the Music of Edgard Varèse," (Ph.D. dissertation, Yale University, 1977), 251-73. Figure 2.10 - Rotation of Tetrachords in mm. 85-93 At the fourth quarter of m. 93, the timpani initiates a change in pitch materials and texture. From the last quarter of m. 94 to the end of m. 109 the tetrachord {C#2, G#2, D3, A5} is reiterated. This collection is similar to the akeady familiar alternating 6- and 7-semitone collections, except that the expected A3 is displaced upward by two octaves. This displacement yields a total span for this collection of 44 semitones, which again is reminiscent of mm. 21-29. In mm. 97-99, 102-04, and 108, the pitch-class A is projected up a further two octaves to sound as A7. In measures 85-93, the number 35 seems to be important. The segment is a total of 35 quarter-beats long (the last quarter beat of m. 93 can be heard as the beginning of a new subsection by virtue of the tempo change and the new material presented zi fortissimo in the timpani), and there are 35 time-point attacks each in both the winds and the indefinite-pitched percussion. ^ 1 As well, there are 9 attacks in the timpani in mm. 93-94. The number of time-point attacks in all percussion (that is, including timpani) is 35 + 9, or 44. 11 Also note that the entire first subsection of Section II—mm. 83-117— is 35 measures long and that there are 35 discrete attacks of pitches of the tetrachord in mm. 85-92: D2,C5 J^6,G#7. The number of time-point attacks in all instruments of definite pitch, including the timpani in m. 94, is also 44. Figure 2.11 shows that this establishes a network of relationships similar to that in mm. 21-29. The timpani, when considered as both part of the percussion ensemble and the pitched instruments ensemble, creates totals of 44 time-point attacks. Moreover, the pitch interval in the immediately adjacent subsegment (of which the timpani is a bridge), mm. 94-96, is C#2-A5 or 44 semitones. It is also interesting to note that the pitch-classes involved are also the same as tiiose in m. 21—the "transition" measure in the parallel section in the first part of the piece: altiiough die vertical arrangement is slightly different, m. 21 sounds the same pitch classes, C#,G#, D, and A . Figure 2.11 - Symmetrical Time-Point Attack Complex in mm. 85-94* Winds and Piano Timpani Indefinite-Pitched Percussion T.P. Attacks: 3 5 - 1 - 9 + 3 5 I II I 44 44 •includes timpani only in m. 94 Another clear discrete attack/interval correlation emerges from a slightly different segmentation. It is possible to hear the beginning of the timpani gesture at the end of m. 93 as the end of the previous segment as well as the beginning of the next segment. The former association results in another attack complex that associates the first 4 attacks of die timpani (last beat of m. 93 and the first beat of m. 94) with both die pitched ensemble and the indefinite-pitched ensemble. Figure 2.12 shows the timpani in its "dual role" again creating attack complexes of 54 discrete attacks—thus corresponding to the span of the tetrachords in mm. 85-93. Figure 2.12 - Symmetrical Discrete Attack Complex in mm. 83-94* Indefinite-Pitched Percussion Timpani Winds and Piano Discrete Attacks: 50 + 4 + 50 J L 54 54 •includes first eighth only in m. 94 Many individual gestures demonstrate some interparametric numerical correlations. The timpani features the interval of 13 semitones—C#2-D3 in mm. 94-98 and in mm. 115-17, E2-F3, a transposition up of 3 semitones. The timpani sounds the C#2-G#2-D3-G#3 collection for a total of 26 attacks. 26 is a significant number throughout the piece, being a compoimd interval of the major second (as is 14) and an even multiple of 13—the featured interval in the timpani (C#2-D3). The pitch G#3 is also sounded by the timpani, thus the maximum span in the timpani is 19 semitones. If we hear the timpani attacks on the pitches, E2-B2-F3, in mm. 115-17 as part of the previous segment, the total of timpani attacks throughout the entire segment, mm. 85-117, is then 31. This is a mod 12 equivalent of 19, the maximum interval. Figure 2.13 (a) diagrams some other potential correlations of the entire timpani part in this subsection. Other aspects of the timpani in this passage are discussed in relation to a similar passage in m. 216 on p. 41. Figure 2.13(a) - Timpani (complete) in mm. 93-117 m.93 featured interval = 13 st. = f 106 maximum interval = 19 st. - mod 12 equivalent to 31 attacks 3 ^ r j j -te 3 ^ <ir^ ir V V V I 5 Figure 2.13(b) - Timpani (complete)* in mm. 216-38 featured interval = 13st. m.216 ^3 1 ^ — - 4 - — Î — - J — max. interval = 19 st. 225 p i - J --< ? ? J 1—« 0--ta f f » - J - ^ -J * N.B.: attack in m. 221 is on slapstick, not the timpani themselves The next subsection, mm. 118-145, is divided fairly clearly into several smaller segments that demonstrate large-scale attack/interval correlations. The most convincing large-scale segmentation, however, occurs after the fifth beat of m. 132 where the indefinite-pitched percussion begins. This segmentation does not produce any obvious attack/interval correlations, but die occurrences of the time-point attacks are significant. Each of the segments contains attack complexes of whole-number multiples of 13: mm. 118-32 contains 13 X 7, or 91 time-point attacks, and the pitched instruments in mm. 132-45 contain 13 X 4, or 52 time-point attacks. The indefinite-pitched percussion in mm. 132-45 contain a total of 150 discrete attacks or 142 time-point attacks which is very close to the time-point attack total of 143 in the pitched instruments throughout the entire subsection. 12 Throughout this entire subsection, then, there is a nearly exact time-point-attack correlation between the pitched and indefinite-pitched instruments. While the segmentation at the fifth beat of m. 132 is convincing, there are other possible concurrent segmentations that result in attack/interval correlations. The rest-delimited segment—mm. 118-27—comprises the first such segment where the interval from the lowest to highest pitch (G1-A7) is 74 semitones and the number of discrete attacks is also 74.13 This segmentation is convincing since the segment is rest-delimited. However, diis leaves us with mm. 128-132 which sound very similar and involve similar pitch collections. Mm. 132-36 demonstrate another inexact correlation between discrete attacks in the percussion, 64, and the maximum interval D1-F6, 63 semitones. Mm. 137-45 is another passage that does not easily demonstrate clear interparametric correlations between attack numbers and pitch intervals. However, this 9-measure segment is exactiy 50 quarters in duration and the maximum registral space is C l -D5, or 50 semitones. Other than those already mentioned, there are no other obvious pitch interval/attack number correlations within any convenient self-contained segments. For example, the second and third indefinite-pitched percussion passages (mm. 139-42 and 144-45) contain 54 and 32 discrete attacks, respectively. These numbers do not reflect any important this instance, I have not included within the indefinite-pitched percussion time-point attack total, the 3 attacks in the gongs from mm. 132-37 that are coincident with and akeady included as attacked time-points in the pitched instmments. The other attacks of the gongs—for example, mm. 138,39,44—are not coincident with time-point attacks in the pitched instruments and therefore do contribute to the attack complex of the indfinite-pitched percussion. l^If the C5 sounded by the horns in m. 124 is counted as two attacks due to the different articulations employed, then the number of discrete attacks in this passage is 75—a near-exact correlation with the pitch-span. intervals in this passage, however, the number 54 was well-represented, in the previous subsection (mm. 85-93). There are also some striking attack numbers for certain components of the overall texture. For example, the pitch-class Bb and indeed the specific pitch Bb3 seem to be referential at the beginning and end of this subsection. There are 13 attacks of pitch-class Bb in the opening (mm. 118-21) and also exactiy 13 near the end of the passage (mm. 138-44). These short segments can be heard as fairly discrete from the intervening material, primarily because of the abrupt change of pitch-class content. The repeated attacks on the pitch, A3, in trombone I and tuba I in mm. 127-30 are arranged into a rhythmic pattern demonstrating consistent attack density as well as a motivic number of attacks. From the downbeat of m. 127 to the end of the fifth quarter of m. 130, is 19 quarter beats of duration. It is a coherent segment because the attacks group into arrangements of 13. From the downbeat of m. 127 to the end of the fourth quarter beat of m. 129, trombone I sounds 6 + 1 + 6=13 attacks of A3 in 13 quarter beats of duration. This recalls the reiterated G4 in hom I in mm. 65-69, which also contains 13 attacks. Àn overlapping segment, mm. 128-30, also contains 13 attacks of pitch A3 in trombone I and tuba I (this time within 11.5 quarter beats of duration). It is conceivable that this second segment demonstrates some kind of rotational process in the rhythmic dimension, not unlike that demonstrated by Bernard in the pitch/register dimension. Figure 2.14 rearranges mm. 128-30 such that m. 128 becomes the middle measure of the second segment, mm. 129-30. This "rotation" creates a second arrangement of 6 + 1+ 6 = 13 attacks also in exacdy 13 quarter beats of duration. This process may also be related to the arrangement of adjacent overlapping segments that was isolated in mm. 21-29. At any rate, it is apparent from these observations that 13 governs the number of attacks in many segments. Figure 2.14 - Horizontal Rotation in mm. 127-30 m. 127(trbl) 128 129 130 (tuba D • • • • •0—p-Attacks: L 13 m. 129 •>• Routed from before m. 129 m. 128 m. 130 1»—0-13 Thus far in Section H, we have seen, generally, a less consistent integration of interparametric correlations than in most of Section I. That is, the set of correlates is becoming increasingly large, compared to the interparametric occurrences of 14 in die opening measures of the piece. We have also seen an increase in the complexity of the texture. Increasingly, there is less direct interval/attack correlation for an entire subsection or segment, but a relatively small set of numbers still governs many parameters. The number 13 appears throughout, particularly in deciding the groups of reiterated attacks of single pitches (for example, the reiterated G4 in hom I in mm. 65-69 and die groupings of the reiterated A3 in die low brass in mm. 127-30). Whole-number multiples of 13 also seem to determine attack densities in entire segments, for example, in mm. 54-78, where there are 13 X 10 or 130 time-point attacks. The description of the time-point attacks in both the pitched and indefinite-pitched ensembles in mm. 118-45 also feature whole-number multiples of 13. Such attack stmctures do not add to the pitch structure of the piece, but certainly do contribute to the attack density consistencies that are distinctive features of Varèse's music. Mm. 146-74 is the next major subsection. These 29 measures are comprised of a complex network of segments that would not be well represented by a linear segmentation. Therefore, the following discussion of this subsection will describe the interparametric correlations observable at the level of microstructure in concurrent segmentations. Such apparendy contradictory readings of a passage of music are a manifestation of Varèse's aesthetic. From m. 146 to the downbeat of m. 149 is a rest-delimited segment. The maximum interval span is Eb3-Bb5, or 31 semitones. There are also 31 discrete attacks in the wind instruments. The duration of this segment is 15.5 quarters or 31 eighths (downbeat of m. 145 to the first eighth of m. 149). There are also the familiar 13 discrete attacks in the piano in these 3 measures. Mm. 149-52 also contain an interparametric correlation. There are 23 discrete attacks in the winds, and 16 in the piano, making a total of 39 discrete attacks, which is exactly one-half the framing interval of 78 semitones in these measures, Db 1-07.1"* Mm. 156-64 begin with material reminiscent of m. 118: A3 surrounded by its chromatic neighbour notes, Bb3 and G#3 in the bass clarinet. This interval expands in the next measure to the registral extremes of A0-C7, or 75 semitones. There are two interesting correlations in odier parameters in the previous subsegments to this large intervallic span. The pitch collection is yet another symmetrical interval collection. Figure 2.15 shows this feature. The two tetrachords that comprise this collection each span 31 semitones and are separated from each other by the familiar interval of 13 semitones. 31 was prevalent in many attack and durational parameters in mm. 146-49. l^This is a different criterion for counting than in the previous segment, since we are now counting the piano. This demonstrates just how difficult it is to judge which attack totals are operative. Figure 2.15 - Symmetrical Pitch Collection in m. 157 XT J An alternate segmentation of the previous passage may help account for the cmious empty measure of 1/8 (m. 153). From the beginning of this subsection to the end of this measure (mm. 146-53) is 37.5 quarter beats or 75 eighth-notes of duration, corresponding to the interval span of m. 157We have already seen several passages that contain correlations between an important interval and the duration of a segment in eighth-notes. Most notable is the correlation in die alternate segmentation of this same passage—31 eighth-notes of duration, 31 discrete attacks in the winds, and the interval of 31 semitones. There are also 63 discrete attacks in the winds and piano in mm. 156-64 which is a mod 12 equivalent to the number of semitones in the interval span (75). These measures also feature an interesting arrangement of attacks in the xylophone. From the last quarter beat of m. 157 to the end of the first quarter of m. 163, 13 quarter beats of C7 are played in the xylophone. Each of the quarter beats is a septuple!. From mm. 157-64, there are also 7 attacks of C7 l^ It may be difficult to hear m. 154 as starting a new sub-segment. However, this does not preclude the possibility that the composer planned and wrote the piece using these correlations. Such concurrent segmentations and inter-segment correlations convincingly account for many of the otherwise unaccountable rhythmic/durational structures throughout the piece. The tempo change at the downbeat of m. 154 does support an alternate segmentation there. in the piano. These 7 time-point attacks of C7 in the piano can be heard as being closely related to the septuplet figures of C7 in the xylophone. Another possible numerical relationship is that between the total number of attacks and the overall duration of diis passage. From the first to the last of these septuplets is exacdy 46 quarters of duration. This is almost exactiy half the number of attacks in the xylophone ( 1 3 X 7 = 91). The indefinite-pitched percussion may be heard as organized into groups of motivic attack numbers. To count their attacks, we consider rods on cymbals and snare drum as a sonority with a single attack, distinct from die reiterated sharp attacks of the Chinese blocks, lathes, timbales (tom-toms), wood drum, etc. Using such a discrimination yields the data that m. 152, 154, 157-58, and 160 contain 13 attacks each of non-sustained percussion. Also, the total number of attacks in all indefinite-pitched percussion (all percussion except xylophone) are nearly equal in each of the large segments, mm. 149-55 and mm. 157-64:74 and 73 discrete attacks, respectively. There are other possible groups of attacks that can be heard as contributing to die overall texture: die sounding of percussion attacks in groups of 13 within each of the instruments. That is, percussion 3 from mm. 158-64 sounds 13 X 2 = 26 attacks, and percussion 2 sounds 13 attacks in mm. 159-63.17 This segment demonstrates die difficulty encountered in trying to parse this music into contiguous but discrete segments. In fact, many of the segmentations are not at all l^If all discrete attacks of pitch-class C—except xylophone—are counted in mm. 157-64, there are 13 discrete attacks: on the fourth beat of m. 157, there is a single attack of C7 in the piccolo, and 5 of the 7 attacks of C7 in the piano start with a grace-note attack of C6. This is one of the rare instances where grace-note attacks do contribute to a motivic number of attacks. Therefore, I still prefer to consider grace-notes as not having discrete attack status. In fact, there is only 1 attack of C7 after m. 157 that does not have an associated C6 grace-note (last beat of m. 161) suggesting perhaps a copying error in the published score. 17The 7 attacks by percussion 2 in m. 158 are associated with the similar figure played with the hands in m. 155. At m. 159, player 2 is directed to use timpani mallets thus distinguishing these and the subsequent attacks from the preceding ones. Such articulation, timbrai, and registral subUeties are significant in Varèse's percussion writing. For an extensive treatment of these elements in Ionisation, see Chou Wen-chung, "Ionisation: The Function of Timbre in Its Formal and Temporal Organization," in The New Worlds of Edgard Varèse, edited by Sherman Van Solkema, (Brooklyn, N.Y.: Institute for Studies in American Music, 1979), 26-74; and Jean-Charles François, "Organization of Scattered Timbrai Qualities: A Look at Edgard Varèse's Ionisation," Perspectives of New Music 29/1 (FallAVinter, 1991), 48-79. discrete. This entire subsegment was described as being comprised of 2 large segments— mm. 146-64 and mm. 165-74, but the internal parsing of the segments has been shown to be much less clear in this passage. The emphasis of pitch C7 from mm. 157-64 supports hearing these measures as a subsegment. However, the figures played by Percussion 2 and 3 in m. 158 have ah-eady been associated with the previous subsegment rather than with the repeated C7 figures. Thus, often these networks of associations preclude the parsing of the music into discrete segments. Mm. 165-74 constitute a clear rest-delimited segment of 10 measures that contains many interparametric correlations. Like much of the rest of the entire piece, parameters in this segment are represented by numbers that are whole number multiples of 13. The 10 measures of duration contain 13 X 4 or 52 quarter beats. The maximum interval in this segment is (Ebl-Ab6) 13 X 5 = 65 semitones. Throughout this entire passage, there are 65 discrete attacks in the wind instruments and 65 time-point attacks. In addition, many of the subsegments-or gestures have 13 attacks. For example, there are 13 discrete attacks in the winds in each of m. 165 and m. 166.^^ The segment from the last quarter of m. 167 through m. 168 contains 13 time-point attacks and sounds the interval (B2-C4) of 13 semitones. Mm. 169-170 also sound 13 time-point attacks. There are 7 time-point attacks in each of the following two measures ~ mm. 171-172, and 6 time-point attacks in m. 174. The 15 reiterated G#2's in the timpani in mm. 170-74 can be heard to articulate die number 13, as well, because the 2 attacks in m. 172 may be heard as simply doubling the piano and bass clarinet. Bernard describes mm. 175-199 as a discrete subsection but I hear it extended into mm. 200-203; primarily because this latter percussion passage remains relatively quiet whereas the passage beginning in m. 204 sounds like a subsectional break. 19 This subsection may be l^A tie is missing between the F#'s in Hom I in m. 166. l^Bemard states, "Four measures solo percussion cause a hiatus in pitched material at this point. This suggests a sectional articulation; and in fact the overall spatial connections bridging the percussion passage are not very strong." Jonathan Bernard, "A Theory of Pitch and Register for the Music of Edgard Varèse," (Ph.D. dissertation, Yale University, 1977), 270. parsed into the following segments: mm. 175-77, mm. 178-86, mm. 187-93, mm. 194-99, and mm. 200-03. The fermata on the last beat of m. 193 is the most obvious point of intemal articulation within the subsection. Thus, mm. 175-93 may be heard as a fairly continuous segment. The first three measures of this segment (mm. 175-77) sound a rhythmic figure that contains 7 pairs of time-point attacks for a total of 14. This number of attacks has been prominent throughout the piece, particularly at the beginning. With the addition of the D6-Eb7 dyad in m. 177, the interval span is expanded to F#4-Eb7, or 33 semitones. This correlates to the number of beats in this first subsegment (mm. 175-77), where 11 quarters are divided into triplet-eighths. The texture in mm. 177-88 also shows some correlation between attack numbers and intervals within one of its component subsegments: in die upper register, the pitches Bb5, Db6, D6, B6, C7, D7, Eb7—spanning a range of 17 semitones are attacked 17 times (discrete attacks). There have been other instances where the overall texture and one segment of the same texture have exhibited attack/interval correlations (see the disussion of mm. 34-38 on p. 17). The next segment—from the last eighth of m. 186 to m. 193—contains interparametric correlations involving the number 33 that governed the previous segment in this subsection.20 Mm. 187-93 are 33 quarters in duration (actually, from the last eighth of m. 186 to the end of m. 193 is 34 quarters) and the winds present 33 discrete attacks in diis passage. The maximum interval span in this passage is Bb5-D7, or 16 semitones, which is as close as possible to one-half of the number 33. There are also 32 time-point attacks in the indefinite-pitched percussion in mm. 190-91. Mm. 194-199 are the penultimate segment in this subsection. There are 56 discrete attacks in the pitched instruments and 25 discrete attacks in the indefinite-pitched percussion, for a total of 81 attacks in this section. This total is very close to the total of 82 20while the 3 attacks at the last eighth of m. 186 have ah-eady been included in the attack total of the preceding passage, they are clearly also the beginning of the next segment that continues to the end of m. 193. This "overlap" or association of single events with both the previous and subsequent segments is similar to the concurrent attack complexes described in detail with regard to the passage in mm. 21-29. discrete attacks that occurred in the indefinite-pitched percussion just previously, in mm. 187-91. The maximum interval span in mm. 194-199 is Bl-Bb6, or 59 semitones. However, i f the following 4 measures of unaccompanied percussion (mm. 200-204) are considered to be part of die segment beginning at m. 194, the total duration of the passage is 59 quarters, corresponding to the maximum pitch interval in this subsection. Mm. 204-224 constitute the last subsection of the second instrumental section. There are not any definite segmentations in this subsection. In fact, the following correlation between the duration in quarter beats of die entire subsection and the interval span supports hearing no Unear segmentations at all. The chromatic aggregate is sounded in m. 204, spanning 75 semitones (Gl-Bb7).2l The duration of the entire last subsection (mm. 204-24) is 75.5 quarter beats. Since the first attack in m. 204 does not occur until the second eighth-note, the actual duration is exactly 75 quarter beats. (It is true that the lowest sounding pitch in tiiis subsection is D l in m. 216; however, it will be argued that this is part of another segment entirely.) Even though it is difficult to parse this subsection into discrete, linearly contiguous segments, many individual components or gestures do demonstrate interparametric correlations. One segment in the overall texture does stand out—the percussion in mm. 206-209. There are 7 time-point attacks in die timpani and 6 time-point attacks in the rest of the percussion, for a total of 13.22 This percussion segment—from the second quarter of m. 206 to the end of m. 209—is also 13 quarters in duration. In the segment of mm. 210-11 there are 11 time-point attacks in 11 quarters of duration. The uppermost interval, Eb6-D7 in the piccolos, is 11 semitones and the sounding duration of each of these pitches in the piccolos is 11 eighths, for a total of 11 quarter beats (see Figure 2.16). 2lThe notated G3 in the piano in m. 204 in the published score is undoubtedly an E3—unison with trombone II). 22The single attack of the gong on the last quarter of m. 205 coincides with the attack of a different part of the texture—the brass chord. Figure 2.16 - Interparametric Correlations in mm. 210-11 m. 210 picc. 7 4 11 St. 4 4 11 11 N.B.: there are also 11 t.p. attacks in these 2 measures In the segment of mm. 212-17—as in the previous segment— t^here are some striking correlations when the texture is separated into some of its component subsegments. The interval of 13 semitones is again prominent, this time subdividing two 26-semitone spans, in m. 212 (F#2-G3-G#4) and m. 213 (Eb3-E4-F5). The segment of low brass, piano, timpani, and percussion in mm. 216-17 also features 13-semitone intervals: E2-F3 in the timpani; and Dl-Eb2 in the tubas and piano. There are also 13 time-point attacks in this segment. The maximum interval circumscribed by this low-register segment is 33 (Dl -B3). Recall that this interval was prominent in m. 177. Interestingly, the highest pitches of die 33-seniitone interval in m. 177 were D6-Eb7, while at m. 217, die lowest pitches of the 33-seniitone interval are the same pitch-classes: Dl-Eb2. This instrumental section ends with 7 measures of indefinite-pitched percussion. The attack and duration data for this segment wdl be discussed together with related segments in die following section. Section III (mm. 225-263) Although m. 225 follows a tape interpolation and thus begins the third instrumental section, it is not the beginning of new material. The organized sound interpolation at the end of m. 224 must be heard rather as an interruption of an otherwise contiguous segment. This is very different from the effect of the next interpolation at m. 263, which is preceded by clear closure. The beginning of the tiiird instrumental section then, mm. 225-238, is essentially an extension of the end of instrumental section U. Elements of continuity include the Dl-Eb2 dyad in die tuba and piano with the F2-Bb2-B2 in the trombones and piano, and the E2-B2-F3 trichord in the timpani (m. 216), recurring (at m. 225 and 238). Between the three statements of this material are two similar passages of indefinite-pitched percussion, mm. 218-224 and mm. 232-37. Specific metronome markings are consistently associated with specific segments throughout this passage: all of the low brass/piano segments are marked quarter =132 and both of the percussion passages are marked quarter =100. The attack designs of the indefinite-pitched percussion passages are also very similar to each other. Mm. 218-24 contain 62 discrete attacks. The next occurrence of similar material at mm. 232-37 contains 59 discrete attacks. Each passage is an arrangement of attacks on either side of a single iteration of the slapstick. In mm. 218-24, the discrete attack arrangement is 29 + 1 + 32, for a total of 62; in mm. 232-37, the airangement of discrete attacks is exacdy symmetrical: 29 + 1 + 29, for a total of 59. Also contributing to the essential unity of mm. 216-38 are the durations of the related segments. The first percussion segment (mm. 218-24) is 7 measures long and has a duration of 13.5 quarters (the first attack in m. 218 is on the second quarter beat). The second percussion segment (mm. 232-37) is 6 measures long and has a duration of 12.5 quarters. Adding these two related segments together results in a total of 13 measures containing exactly double the number of quarters of duration—26. The timpani figure in this reiterated passage recalls some odier material as well: it is a transposition up 3 semitones of the timpani figure in mm. 94-108. Even more striking is die correlation between number of attacks and the pitch interval in these parallel passages. In mm. 93-117 tiiere are 26 attacks of pitches C#2-G#2-D3-G#3. The number 26 reflects the 13-semitone interval that is featured in this passage. However, as mentioned in the previous discussion of this passage (see p. 28), the timpani sounds pitches that actually circumscribe 19 semitones of registral space: C#2-G#3. This segment continues to the end of m. 117. In mm. 115-17, three other pitches sound in the timpani: E2-B2-F3. Figiu-e 2.13b (see p. 29) shows that the extreme pitches in this subsequent trichordal segment that again is 13 semitones, are equidistant from the extreme pitches in the C#2-G#3 pitch collection. The five attacks in the timpani in mm. 115-17 bring the total number of timpani attacks to 31 throughout this entire passage. The total of 31 attacks is exactly 12 more than die number of semitones between the extreme pitches in the timpani throughout this same section. This mod 12 relation could be considered simply a coincidence, but for the occurrence of similar material in mm. 216-38. In the latter passage, there is also a total of 31 attacks in the timpani on pitches, E2-B2-F3-B3, which again circumscribe the interval of 19 semitones. Many relationships between the pitched material throughout the beginning of the third instrumental section can be demonstrated, as Bernard has done.23 The D4 in the homs in m. 225, the reiteration of pitch D4 in the brass in m. 226, and the subsequent material may be heard as discrete from die low brass/timpani segment—aldiough still derived from it, as Bernard insists. This results in hearing two other segments that are discrete from each other: a segment starting in m. 228 involving C2-Ab3-Bb3-B3—a span of 23 semitones; and a temporally displaced segment beginning with the D4 (first heard in m. 226), continuing with die D5-Eb5-Eb4 in m. 227, and finally die C#6 that occurs in 23jonathan Bernard, "A Theory of Pitch and Register for the Music of Edgard Varèse," (Ph.D. dissertation, Yale University, 1977), 276. mm. 228-36. This pitch collection, D4-C#6, also spans 23 semitones. This segmentation is supported by the fact that there are 13 discrete attacks of the D-Eb pitch-classes, which is the interval from D4-Eb5. Adding to this total the 10 discrete attacks of pitch C#6 results in a discrete attack total of 23—same as the pitch interval from D4-C#6. There is also a correlation between the pitch interval and the number of discrete attacks in the overall texture. The maximum interval span from mm. 225-38 is Dl-C#6— 59 semitones. This is reflected in the second indefinite-pitched percussion segment in mm. 232-37. As already described, attacks in that segment are arranged symmetrically: 29 + 1 + 29, or 59 in total. As we shall see in Chapter 3, the interval of 59 semitones recurs in structures throughout the rest of the piece.24 The segment from m. 239 to the end of the first beat of m. 247 forms the middle of the three subsections of the third instrumental section. It features a low-register dyad (E l -F2) and a high-register dyad (Db6-C7) that are sustained from m. 242 through to the first quarter beat of m. 247, overlapping with the beginning of the last subsection which begins at the first sounding of the lone F#4 in m. 244. The interpolation of m. 242 with the metronome marking of quarter = 176 between material that is marked at quarter = 80 may suggest that this is not a continuous segment. However, considering mm. 239-47 as a subsection that overlaps witii the final subsection of section i n at m. 244 is supportable. M . 239 to the end of the first quarter of m. 247 form a segment that is 39 quarters in duration. 39, or 13 X 3, is represented in components of the complete texture. In m. 242, die D5-A5-Eb6 trichord (an interval of 13 semitones) is sounded by exacdy 13 discrete attacks.25 The remaining instruments sounding these pitch classes (in either octave) from the second to fourth quarters of m. 242 also total 13 discrete attacks (3 in the piccolo, 3 in the trumpets, 4 in the piano, and 3 in the xylophone). The maximum registral space in 24Actually, the passage in mm. 41-45 is the first symmetrical collection in the piece that spans 59 semitones. See Figure 2.7 on p. 18. 25This includes—from the second quarter to the fourth quarter of m. 242—all woodwinds except the piccolo, which sounds this trichord an octave higher. mm. 239-41 is El-Eb7, or 71 semitones. This is an expansion by mod 12 of the operative interval throughout the end of the third instrumental section, 59 semitones. The octave reinforcement in the upper register emphasizes the descent of 3 semitones in parallel octaves: Eb-D-Db-C. This descent of three semitones also becomes increasingly important for the rest of the piece. From the third quarter of m. 244 to the end of the third instrumental section in m. 263, the single pitch F#4 is reiterated, as is the trichord G2-Bb2-A3. The interval span from the G2 at the bottom of the trichord to the F#4 in diis subsection is 23 semitones. In this passage there are 26 discrete attacks of the pitch, F#4, 32 discrete attacks of die trichord pitches, and 1 gong attack in unison with a sounding of the trichord in m. 251, for a total of 59 discrete attacks—a number that was prominent in the previous sub-section and a mod 12 equivalent of the interval span in the final subsection of Section IH. Section IV (mm. 264-325) The final instrumental section is not readily parsed into discrete subsections. It is better described as a network of segments that are sometimes contiguous, sometimes overlapping, and sometimes rest-delimited. The following description of interparametric correlations in its microstructure wiU therefore reflect diis generally continuous character. The opening segment of this section consists of 6 measures of indefinite-pitched percussion (mm. 264-69) and a sounding of the chromatic aggregate in the wind instruments (doubled in the piano) in m. 270. The opening 6 measures of indefinite pitched percussion, mm. 264-69, contain a total of 51 discrete attacks. In m. 270, there are 22 discrete attacks in the wind instruments. Thus the total number of discrete attacks in this segment is 73, which is also the interval in semitones (Fl-Gb7). Here the piano attacks are heard doubling the wind instruments.^^ The imperfecdy symmetrical collection in m. 270 (shown in Figure 2.17) is similar to that in m. 204 which is also an imperfectly symmetrical collection of all 12 pitch-classes containing 22 discrete attacks in the wind ensemble, doubled by 12 attacks in the piano. A significant difference between these similar passages is that in the earlier passage (m. 204) there is not a correlation between interval span and the number of attacks—suggesting that it is less stable than the passage in mm. 264-70, where there is a clear interval/attack correlation. 26An "8va" sign may be missing above the third to fifth beats of the piano part. Figure 2.17 - Imperfectly Symmetrical Pitch Collection in m. 270 The next segment—from the downbeat of m. 271 to the second beat of m. 277— contains several attack/interval correlations. The first 5 measures of this segment (mm. 271-75) can be heard in overlapping groups of 13 time-point and discrete attacks. There are two groups of 3 attacks each of rolls on drums with snares (field drum in mm. 271-73 and snare drum, field drum, and side drum in m. 274). Beside these drum rolls, we hear 10 discrete attacks in m. 274 and again 10 discrete attacks in m. 275. Figure 2.18 shows how these adjacent groups can be heard as attack complexes of 13 discrete attacks. Another correlation is evident if we consider time-point attacks. The 10 discrete attacks in m. 274 occur at 7 attacked time-points. Thus we can also hear overlapping segments of 13 time-point attacks.27 The corresponding interval of 13 semitones—Eb2-E3, B3-C5, A2-Bb3— is prominent in m. 276, reflecting the groupings of 13 attacks throughout this passage. This segment may be heard continuing to the fourth beat of m. 277: the second 27This simultaneous grouping of overlapping time-point and discrete attacks first occurred in mm. 21-29-seepp. 14-15. sixteenth of which begins the next segment. Thus the duration of the segment from m. 271 to die end of die diird beat of m. 277 is 13 X 3 = 39 beats long. Figure 2.18 - Overlapping Groups of 13 Attacks in mm. 271-75 time-point attacks •13 113 { 3 } 7 3 10 m.271 272 273 274 . 275 } 10 3 10 •13 •13 discrete attacks The numbers 7 and 6 carry the music from m. 274 to 277. There is a correlation in m. 274 between the 7 time-point attacks and die interval of 7 semitones (A2-E3) in the timpani and trombone. M . 276 contains 7 time-point attacks, including the attack of the snare drum, which is differentiated from its earUer attacks by the direction of "no snare." The 6 clearly articulated time-point attacks of pitched instruments in m. 276 correspond to the melodic interval of 6 semitones (Eb2-A2).28 From the fourth beat of m. 277 to the end of the fourth beat of m. 280 may be heard as a short rest-delimited segment that contains famihar motivic intervals that correlate widi the attack structures. In m. 278, the interval of 10 semitones (G5-F6) is sounded. The fourth beat expands diis interval one octave, to 22 semitones (Bb4-G#6). At the end of m. melodic interval of 6 semitones is also heard clearly in the upper voice of the timpani (E3-Bb3). 278, we get a leap of 13 semitones in the piccolos (G#6-A7) over a B5 in trumpet I, thus forming another 22-semitone interval. The maximum interval span here is Bb4-A7, 35 semitones. The pitch material in this measure may be divided into two collections diat are symmetrical within themselves. G5-G#6-A7 form two superposed intervals of 13 semitones each and the remaining pitches, Bb4-E5-B5-F6, form a chain of alternating 6-and 7-semitone intervals. Since the piano attack on the last eighth of m. 279 allows E l , 2, 4, and 5 to vibrate, die registral space circumscribed by the sustained tones in the piano arid vibraphone to m. 280 is El-Eb6, or 59 semitones—already heard in mm. 216-38 and figuring prominently in the penultimate segment of the piece (mm. 306-307).29 In m. 278, there are 10 time-point attacks, recalling the 10 time-point attacks in each of mm. 274-75. M . 278 may also be heard as being framed by 4 attacks on either side that occur in a very similar rhythmic pattern. The two groups of 4 attacks each are imperfecdy symmetrical mirror-images that may be anodier horizontal rotation—analogous to the rotation of imperfecdy symmetrical interval collections (see Figure 2.19). Figure 2.19 - Imperfectly Symmetrical "Framing" of m. 278 Attack-complexes are imperfect mirror-images 2^The piano left hand in m. 279 should have a bass clef, denoting that E2 and E3 vibrate. It is possible to hear a different segmentation of this passage. The 3 attacks on the wood drum from after beat 4 of m. 277 to the end of the second beat of m. 279 sound as a contiguous segment Even though the attacks from beats 3 to 5 of m. 279 are rhythmically related to the attacks in m. 277, the new slower tempo (quarter = 80) and the sustained quality of the vibraphone and piano harmonics differentiate this from the previous material. Such a segmentation results in some segments that exhibit interparametric correlations. The vibraphone/piano segment from the third beat of m. 279 to the end of m. 282 is 16 beats long, as is the next segment—mm. 283-86—which is of a contrasting staccato character. The former segment (mm. 279-82) has 16 time-point attacks—the same as its duration. The next segment (mm. 283-86) appears to contain 45 time-point attacks. However, there is sharp contrast between the staccato attacks in the timpani and xylophone and the slurred articulation in the bass clarinet. So, it is possible to hear each slurred group of notes in the bass clarinet as a single attack resulting in 35 time-point attacks in this segment—corresponding to the interval span (F#2-F5) of 35 semitones. From mm. 287 to the end of the first beat of m. 294 may be heard as a segment that ends with the reiteration of pitch D5 in the brass. The pitch-class D has already been isolated in a similar passage in m. 226. In the earlier passage, pitch-class D was sounded as D4. In the passage being discussed (283-93), the mid-point or axis of bisection of the registral space is D4 (that is, D4 bisects the interval from Bbl-F#6, or 56 semitones). Therefore, this reiteration of D5 may be heard as an octave projection of the mid-point. There are 28 time-point attacks throughout this entire passage (mm. 287-93)—one-half of the interval span. Mm. 294-307 comprise a subsection of 63 quarters of duration corresponding to the registral span of 63 semitones (Gl-Bb6). There are attack complexes within this subsection that contain familiar numbers of attacks. The passage begins in m. 294 widi 13 attacks in the claves. Mm. 296-303 dien may be heard as a contiguous segment since the xylophone and timpani play static dyads (G2-F#3 in die timpani and E5-F6 in die xylophone). The xylophone plays 27 attacks of E5-F6. There are exactly double 27, that is 54 time-point attacks in the timpani, die last 13 of which are doubled by the piano in octaves. In mm. 304-307, a large verticality accumulates, spanning 59 semitones (Bbl-A6). In Chapter 3, the significance of this interval will be discussed (see. p. 71). Figure 2.20 diagrams this nearly symmetrical vertical collection and shows that the next pitches at the registral extremes of the chain would have been E l and Eb6. It seems significant that E and Eb are the extreme pitch-classes in the ending of die piece, and also that at the parallel large vertical collection near the end of the third instrumental section (m. 242) E l and Eb6(7) actually are the extreme pitches. Figure 2.20 - Imperfect Interval Symmetry in mm. 304-10 > ± } 6st. # = does not occur There is a final chain of 13-semitone intervals from mm. 311-14, as well as an octave-displaced B4. There are not any obvious interval/attack correlations in this segment. Mm. 310-325 see a return of the same texture and tempo as was in the end of instrumental section HI, mm. 243-63. Moreover, the pitch materials are an exact transposition of 3 semitones of those at the end of section HI. There are no direct interparametric correlations in this final segment. However, all parameters are mod 12 equivalents. In this final segment of die piece, there are 15 discrete attacks of Eb4 and 3 attacks of the cymbal coincident with 3 of the attacks of Eb—a total of 18 discrete attacks of the Eb4 and its associated percussion. There are 15 attacks of the pitches of the E2-G2-Ab3 trichord and 2 attacks of the gong associated with the trichord—a total of 17 attacks. Thus, there are 35 discrete attacks in this segment. There are 8 time-point attacks of the pitch, Eb4 and 3 time-point attacks of the trichord—a total of 11 time-point attacks in this final segment of the piece. Thus in the final 14 measures of the piece, the interval span (23 semitones), the number of discrete attacks (35), and the number of time-point attacks (11) are all mod 12 equivalents. The significance of these mod 12 equivalents is discussed in Chapter 3 (p. 66). Summary of Microstructure in Déserts In the preceding explication of the microstructure of Déserts I have shown that there are a variety of interparametric correlations operating throughout the piece. Many segments—particularly in die opening of the piece—demonstrate a clear correlation between pitch interval and the number of discrete attacks. We have even seen evidence of transformations in attack-complexes—such as the rotation in mm. 127-30—that are analogous to the pitch transformations described by Bernard. The duration of segments is also governed by interparametric correlations, such as in mm. 210-11 where duration and virtually every other parameter is a manifestation of the number 11 (see Figure 2.16 and the discussion on p. 39). Sometimes, attack-complexes are governed not by direct correlation with an occurring pitch interval, but by correlation with motivic numbers. For example, many complexes of repeated notes contain the motivic 13 attacks. Perhaps the most unexpected correlations are those that are not direct but rather mod 12 equivalents. The vaUdity of such observations is likely to remain open to question, but mod 12 interparametric correlations in certain passages—such as the related timpani figures in mm. 93-117 and mm. 216-38 —are very striking indeed and for understanding the macrostructure, even vital. Chapter 3 - Macrostructure in Varèse's Déserts In the previous chapter, I have shown that Déserts is comprised of an often overlapping series of short segments that exhibit numerical correlations between motivic pitch intervals and parameters that can be categorized under the broad rubric of rhythm. I have demonstrated that many of these segments exhibit an exact correlation between pitch and attack density, as in the equality of pitch interval in semitones with the number of discrete or time-point attacks. There are also correlations between these parameters and the duration of segments, measured both in note values and in numbers of measures. Since there is no necessary correlation between duration and attack density, this durational planning represents an organizational process distinct from that involving attacks. In this chapter, I will describe this organization of the microstructiu-al components into the larger, predominandy sectional form, which I will call the macrostructure. I shall begin with a discussion of the time-point and durational planning and then speculate as to some global pitch and pitch-interval associations. I wdl conclude with a brief description of the formal processes operating, with reference to certain of Varèse's aesthetic descriptions and analogies. Many of the processes involved in Déserts also occur in Ionisation. So, before continuing with the analysis of the macrostructure of Déserts, I will summarize some aspects of structiu^ process in Ionisation that are germane to the macrostructure of Déserts. Attack structures and durational structures articulate the form in both works, but in different ways that are captured nicely by Elliott Antokoletz's description of two parallel processes occurring in Ionisation: "unity in Ionisation is determined by the overlapping and inteirelation of two basic concepts of construction: sectional form and dynamic form.."^ That is, Antokoletz is postulating that at least two parallel processes are occurring. He ^Elliott Antokoletz, Twentieth-Century Music, (Englewood Cliffs, N.J.: Prentice-Hall, 1992), 344. Antokoletz's description, "dynamic form," seems to be loosely based on a term by the Futurist, Umberto Boccioni. For a discussion of possible aesthetic relationships between the Italian Futurists and Varèse, see Jonathan Bernard, The Music of Edgard Varèse, (New Haven: Yale University Press, 1987), 23-31. does not go into detail about the sectional form, but merely mentions that it may be related to that of sonata form.^ That is, he doesn't notice the symmetry of Ionisations sectional form: a central section of 3 X 13 = 39 measures framed by 2 X 13 = 26 measures on either side (see Figure 3.2 on p. 58). Antokoletz's analysis is also indebted to that of Chou Wen-chung's 1979 analysis.^ In this analysis. Chou identifies 7 different timbrai classes throughout the piece. I am suggesting that the 7 timbrai categories and the myriad interparametric occurrences of 13 account for the number of measures in Ionisation—\2> X 7 or 91."* These sectional divisions in Ionisation are not unequivocal. For instance, while the work can be parsed into three large sections that each have numbers of measures that are multiples of 13, the intra-sectional durations seem to be governed by other processes. A more detailed look at one of the internal sections of Ionisation demonstrates that attack-complexes are governed by motivic numbers in diis piece, as well. Measures 38-50 are a fairly clearly delimited contrasting section of (again) 13 measures. One of the most striking features of the first part of this subsection is an accented rhythmic unison among a mixed timbrai group: primarily players 1,2, 10, and 12. One of the sounds that dominates this passage is that of the accented triplet beats. There are 13 such accents to the end of m. 42 where the texture changes (see Figure 3.1). As with many of the segmentations in Déserts, the adjacent sections are not always discrete—there is often overlap at sectional boundaries. Moreover, this sectional analysis seems to account for only part of the process or processes at work in diis music. Again, if there is still any doubt that Varèse was interested in using rigorous pseudoscientific methods in composition, here is his ^This is based on the "sonata form" descripdon by Nicolas Slonimsky, "Analysis," preface to the published score of Ionisation (New York: Colfranc, 1966). ^Chou Wen-chung, "Ionisation: The Funcdon of Timbre in Its Formal and Temporal Organization," in The New Worlds of Edgard Varèse, edited by Sherman Van Solkema, (Brooklyn, N. Y.: Institute for Studies in American Music, 1979), 26-74. '^ Recall ihailonisation is for an ensemble of 13 players, there are 13 rehearsal numbers in the score, and 13 is manifested in many attack and durational parameters. description of Ionisation: "Ionisation has turned out well—cryptic, synthesized, powerful, and terse. And, as for the structure: stunning mechanics. Antokoletz's conception of concurrent formal processes in Ionisation is relevant to the present analysis of Déserts. The "dynamic" form of Déserts is characterized by a continuous transformation of materials, maintaining numerical correlations among the parameters of pitch interval in semitones and numbers of discrete or time-point attacks. Our perception of these correlations has increased our ability to parse die complex texture into its component segments, thus allowing the explication and description of the interacting "sound masses" at the microstructural level. The "sectional" form of Déserts is characterized by the organization of these interacting sound masses into longer sections that exhibit durational and time-point consistencies based on numerical values that are often consistent with those operative in the microstructure correlations. ^Chou, "Ionisation: The Function of Timbre," 74. It is also interesting that the first published edition of the score bears the date November 13,1931. The numerical significance of this date—13/11/31—may be not only its palindromic possibilities, but also the fact that its components are all prime numbers. Varèse has shown an interest in prime numbers elsewhere—for example in his notes for a project that was to have been entitled "l'Astronome:" "Unexpected reception of signals—prime numbers 1, 3, 5, 7. The governments decide that they must answer 11,12 [?]. The answer to that is 17,19." [Edgard Varèse, quoted in Femand Ouellette, Edgard Varèse, translated by Derek Collman, (New York: Orion, 1968), 115.] Figure 3.1-13 Accented Time-Point Attacks in mm. 38-44 oiIonisation 1 2 3 4 5 6 7 8 16 m. 38 m ^ 9 Cencerro GrosM CaiMc ( t m gr;rvei Ccneerro Tun*taffl clAir T u n •tarn grave » «•"•î- i;: Clair g I Sirène c U i r * iTWnbour m corde I 6 Fuuet Guiro 7 3 BlocaChiiKHi cljiir 8.... S Tarole 10. Grelola Cymbale» Giiiro 12 I Enclume*. 13 . Piano T r i a u l e 1 J 7 j 7 J17 , 1 7 J, 7 ,t , a 1 pn •'}•' Fii ,1 / ,1 , ,K, râ"" ' - 3 - ' f - T - . n ^ 1 ' -q=ha^ • J J 1 » 1 . , * ' • 7 - ' A r f -7^ r . r i 7 mj j ] ^ r h h J ran I]/ J J J J J J —J—J—j , — j - j ^» 1 1 nn SID TS}i i ? " " q 7 j i 7 r i 7 1 J"^  7 H 7 — 7 jê ;J3y r f ? 7 - 1 njv^ £J J J i i . ..J — J — k J — ] S î / f t ffl J 7 •) i^Tffl 1—4 1 1 ^ .r > ^ I J its - f^}? ^ ) i f l j . - n < J , J , V ff" > i 1 i . A 5»^  n J 10 11 12 m. 41 C«BC«rro Tam-fam clair Tani'Uni grjve T il fjl 7 i 2_JL Z Rofigoa.. .clair -I C i i a c Rouluin fra< a rTambour mitttairv tCaÏMe roulante i 5. 6. Sirène cUire Tambour à oordv Siràm grave Fouet G d i r 7..< I Bloc» Chinoii clair moyen I C U v e a l Triangle 8.... 9 . Giiaae .claire jC, jGrave Tarole Caiaae claire Cymbale wapendui i. J. ïm. j — n t 10 p"''*' 'Gtiiro Caatagnellea ] 12. Tambour d * Bwiue |f " ^ Bnclunc*. 1= 3-4-m 13 J3îj ài miAi •M. * J 1—J-X a JL (S 4 _ A 5 8 ^ I J U i -) U •4-5 5 4-S: 5 7 4 i-a_) U 15 5 On the grossest level, the overall large sectional form of Déserts can be seen as expressing an important aspect of the microstructure. There are 4 large instrumental sections that altemate with 3 interpolations of "organized sound," residting in a total of 7 separate sections. The number 7 is well-represented in the pitch domain by the interval of 7 semitones and by its multiple 14, which is of primary motivic significance in the piece. There is a direction in the score that entertains the possibUity of the piece being performed without the tape interpolations. However, the composer stated ihai Déserts was composed with these 7 sections in mind: Déserts was conceived for two different media: instrumental sounds and real sounds (recorded and processed) that musical instruments are unable to produce. After planning the work as a whole, I wrote the instrumental score, always keeping in mind its relation to the organized sound sequences on tape to be interpolated at three different points in the score.6 While the tape interpolations therefore do represent an integral part of the piece as a whole, I wdl concentrate on the macrostructure of the instrumental sections in this analysis. As in Ionisation, the number of measures in the instrumental sections of Déserts may reflect an aspect of the precompositional planning of the piece. Altogether the 4 instrumental sections total 325 measures. The number 325 is divided evenly by 13 and 25, both of which have been demonstrated as being significant in many other parameters. Another numerical derivation is relevant here. If we multiply die opening interval of the piece—14 and the final interval—23, we get 322. This is, of course, 3 less than the actual length of 325 measures. I cannot help but speculate that this difference of 3 may refer to the transpositional relation of 3 between the endings of the last 2 sections of the piece. The importance of die factor 13 in the total number of measures leads us to consider the relation of the macrostructure of Déserts to that of Ionisation; the latter is also a multiple of 13 (91) measures in duration. ^Edgard Varèse, quoted in Henry Cowell, "Review of Performance oiDéserts," Musical Quarterly 51 (1955). 371. As in Ionisation, the first instrumental section of Déserts (mm. 1-82) consists of a central subsection framed by two subsections of equal numbers of measures: mm. 1-29, 30-53, 54-82. In both pieces, the segmentation between the first and second subsections is not as clear as that between the second and third subsections. Another relationship between the first 82 measures oï Déserts and Ionisation is the segmentation of the middle section. In Ionisation, the 39-measure central section (mm. 27-65) can be parsed into 4 subsections: mm. 27-37, mm. 38-42, mm. 43-50, and mm. 51-65. The 24-measure central subsection of Déserts can be parsed into 3 segments: mm. 30-40, mm. 41-45, and mm. 46-53. Figure 3.2 shows that the organizations of these segments of seemingly random lengths are actually very simdar. The first three subsections of the central section of Ionisation and die 3 segments of the central subsection of the first part of Déserts are the same numbers of measures: 11,5, and 8. The additional 15 measure segment (mm. 51-65) in Ionisation simply makes up the 39 (13 X 3) measures, which is a motivic number in Ionisation. This correlation with numbers of measures in Déserts may simply be a coincidence. However, the similar airangement of measures in the two pieces is very striking indeed. Figure 3.2 - Sectional Divisions in Ionisation and Section I of Déserts Section I of Déserts m. 1 29 30 41 46 53 54 82 II I l I [11 5 8]* 29 ms. 24 ms. 29 ms. Ionisation m. 1 26 27 38 43 51 65 66 91 I H I I IL [ 11 ' 5 ' 8 ]'* 15 13X2=26 ms. 13X3=39 ms. 13X2=26 ms. Returning to the macrostructure of Déserts itself, there are other aspects of rhythm besides duration that reinforce the formal similarity of mm. 1-29 and mm. 54-82 in the first instrumental section. Figure 3.3 shows the regular metric pulse that opens both of these sections. In the opening subsection, a dotted-quarter-note pulse is clearly established. In the first three measures of the final subsection, (mm. 54-56) a regular quarter-note-triplet pulse is established. Upon closer inspection, it is discovered that in fact, the same time-points are attacked in each respective meter. That is, in both passages, metric-points 1, 2, 7, 8, and 13 are attacked (a similar articulation of regular time-points occurs at the beginning of each of the 2 "endings" of the piece, mm. 243-63 and mm. 310-25). The numbers of discrete attacks in these two sections are also related. Each pair of attacked time-points in mm. 1-6 contains 14 discrete attacks. In mm. 54-56, the first and third pairs of attacked time-points contain 14 discrete attacks each, while the second pair of attacked time-points (m. 55) contains 13 discrete attacks. Figure 3.3 - The Establishment of Similar Attack-Points in m. 1 and m. 54 = 92 / =69 m I 2 I I 14 altacks r3-, J=50 (_J_='5<' 54 =: 2 T 7 8 I I 14 allacks 55 13 IS 14 attacks 56 L a 17 T T 13 14 < 3 -I I 14 altacks I 3 1 etc. J 13 attacks 14 attacks Another striking consistency between these outer subsections is not just that they have the same number of measures, but also that their durations in quarter notes are the same. This correspondence of subsectional duration suggests that a simdar principle of formal organisation might be operative in the rest of die piece. To investigate diis possibility, the table in Figure 3.4 catalogues the quarter-beats of duration of all of segments at each of the tempo markings in Déserts. This table facilitates an accurate count of the durational parameters of measure numbers and notated durations. From this table, we see that the first 29 measures last 116.5 quarter notes. The final 29 measures of the same section, mm. 54-82, last 117.5 notated quarters of duration. Taking into account a note on the first page of the published score that the "first interpolation of organized sound enters on die fourth beat of m. 82" makes the durations of the first and last sections exactiy the same—116.5 quarters."^ 116.5 is almost exacdy the number of quarter beats in 29 "^ Edgard Warèse, Déserts, (New York: G. Ricordi, 1959), ii. measures of 4/4 time (29 X 4= 116). This same average number of beats per measure occiu-s in Density 21.5, where there are 244 (61X4) quarters of diu^ation in 61 measures. These observations by themselves might suggest a kind of consistent metric background upon which the constantly changing rhythms of the piece are superimposed. However, what is even more striking is that the number of time-point attacks virtually equals the number of quarter notes in the metric background in each piece in question. In Density, there are 245 time-point attacks in 244 quarters of notated duration, and in die first 82 measures of Déserts, there are 342 time-point attacks throughout the 341 quarter beats of notated duration. This consistent attack density may reflect the composer's attempt to find a musical analog of projecting changing relationships of volumes onto planar surfaces. ^  Instrumental sections 2, 3, and 4 do not have an attack density approximately equal to 1 attack per quarter note of duration. This is consistent with the contrast between their character and that of the first instrumental section in that, the first instrumental section has a much simpler texture and displays remarkable interparametric correlations, whereas the continuous transformation of materials in the following sections precludes simple numerical relationships. From Figure 3.4 we see that in all 4 instrumental sections, there are 1873 time-point attacks in 1426 quarters of duration—an inconclusive datum.9 However, it is not aways possible to be certain what constitutes a beat. ^This is a paraphrase of the composer's description of the process involved in his 1925 composition, Intégrales: "The form of the projection at any given instant is determined by the relative orientation of the figure and the surface at that instant." Edgard Varèse, quoted in Femand Ouellette, Edgard Varèse, translated by Derek Coltman, (New York: Orion, 1968), 83. 9ln Figure 3.4 the numbers of beats refer to quarter note values. That is, in places where the half note or eighth note are used as the beat, the corresponding number of quarter beats is recorded. Tempo q c g 3 Q «4-1 0 1 Ti-en ,J=50 Section 1 - 3 4 1 J mm 1-29 29 ms 60 63 66 72 80 84 88 90 92 116.5 100 108 112-116 120 132 144 160 176 200 30-40 11 ms 47 41-53 13 ms 60 54-82 29 ms SfiClifllL2 —658.5 J 117.5 Sectio Il tot il mm 83-93 11 ms 41 94-96 3 ms 13 97-105 9 ms 40 106 1ms 4 107 1 ms 5 108-118 11ms 49 119-123 5 ms 21 124-132 9 ms 47 133-145 13 ms 74 146-149 4 ms 20 150-153 4 ms 17.5 154-164 11 ms 70 165-166 2 ms 12 167-173 8 ms 33 174 1 ms 7 175-177 3 ms 11 178-186 9 ms 26 187-192 6 ms 26.5 193 1 ms 7 194-195 2 ms 12 J=50 60 63 66 72 80 84 88 90 196 1ms 7 116 197-198 2 ms 10 199 1 ms 11 200-203 4 ms 19 204-210 7 ms 26 (temp Chang i on be It 4) 210-215 6 ms 27 216-217 2 ms 8 218-224 7 ms S Ê C l i f l I L 3 - 1 4 0 . 5 j 14.5 Sei lion 2 lota! 225 1ms 7 226-228 3 ms 14 229-237 9 ms 18.5 238 1ms 7 239-241 3 ms 13 242 1 ms 7 243-246 4 ms 18 247-263 17 ms Section 4 —286 J 56 Se clion : tola! 264-270 7 ms 32 271-275 5 ms 29 276-278 3 ms 21 279-286 8 ms 34 287 1 ms 7 288-295 8 ms 41 296-303 8 ms 36 304-309 6 ms 24 310-325 16 ms 62 S( clion I total 22 92 100 108 112- 120 132 144 160 176 200 1191 14 20 39 16 16 9 6 11 101 58 23 21 61 5 40 80 15 Total 32 18 70 47 26 327.5 32 14 56 169 274.5 60 21 133 34 45 36 24 Many of the tempi recorded in Figure 3.4 are related by a factor of 2. The operative range of tempi is considerably less if these related tempi are consolidated. Figure 3.5 tabulates the numbers of beats after consolidating all tempi that are related by a factor of 2. For example, the 12 quarter beats at the tempo of 132 are equivalent to 6 quarter beats at the tempo of 66 in the subsequent 8 measures. For this table, the tempi of 100 and 200 were both consolidated under the tempo of quarter = 50. The resulting total of 1343.75 beats is close to the number of beats expected if the piece was planned with an average of 4 beats per each of the 325 measures (4 X 325 = 1300—a total with obvious motivic associations). Of course, this determination is not unequivocal. Two other alternate readings result in compellingly motivic numbers of beats. If the tempi of 50 and 200 are consolidated under the tempo of quarter = 100, then diere are exacdy 130 (10 X 13) quarter beats in Section EI (mm. 225-63). Also, consolidating the beats at quarter = 60 under the tempo of 120 results in 169 (13^) beats at that tempo throughout the entire piece.i^ Figure 3.5 - Number of Beats at Consolidated Tempi Tempo 50 60 63 66 72 80 84 88 90 92 108 112-116 Total No. of Beats 175.75 284 70 64 48.5 345.5 32 175 56 169 60 21 1343.25 ^^In Figure 3.4 we see that there are also 13^  or 169 beats at quarter = 92. On the largest scale, it is possible to discern some numerical design in the number of measures given to each section. To do so, however, entails hearing points of sectional articulation that are different from the tape interpolations, because the sectional proportions based on those interpolations—82:142:39:62—do not seem signficant. It has already been suggested that only the division between sections HI and IV sounds like a convincing sectional break. That is, the interpolations at m. 82 and m. 224 may be considered to interrupt otherwise contiguous instrumental material. Perhaps the composer organized the large sections into numbers of measures according to motivic numbers. If the first tape interpolation is heard as an interruption of contiguous instrumental material or if the piece is performed without the tape interpolations, a distinct sectional division may be heard as occurring at either the end of m. 78 or the end of m. 84 (see p. 23 a b o v e ) . 7 8 is factored evenly by 13 and 6, and 84 is factored evenly by 14 and 6, so both segmentations involve numbers that are very motivic to the piece. Of course, these altemate segmentations disturb die bilateral symmetry of the first instrumental section described in Chapter 2 (29 + 24 +29 = 82 measures) and the correlation between number beats and time-point attacks. However, such multiple segmentations may better reflect the composer's description of the formal processes involved. For example, perhaps the first instrumental section is not simply a discrete section of 82 measures, but a fluid association of interacting sound masses or "bodi geometrical figure and plane surface moving in space, but each at its own changing and varying speeds of lateral movement and rotation. "^ 2 fhjs spatial imagery, used by the composer, may be analogous to a multiplex sectional segmentation of die music. The piece does not segment into discrete segments arranged in a serial fashion, but into overlapping simultaneous segmentations that sometimes exhibit notated durations and numbers of measures consistent with motivic numerical values in the microstructure ^ ^There is another correlation that supports hearing a sectional division at m. 78—the maximum interval span in the entire fu-st instrumental section is in mm. 37-39 (C#l-G7)—78 semitones. ^^Edgard Varèse, quoted in Femand Ouellette, Edgard Varèse, translated by Derek Colunan, (New York: Orion, 1968), 83. parameters. The sectional form of the first instrumental section (mm. 1-82) is symmetrical with respect to the durational parameters of numbers of measures and beats. The concurrent dynamic form, which allows altemate segmentations at m. 78 and m. 84, reflects some of the important microstructiu-al interparametric correlations. Whde the durational symmetry supports the 82-measure discrete section, there are other correlations that strongly support hearing a 78-measure opening section. Firstiy, the maximum interval span in the entire first instrumental section is 78 semitones (mm. 37-39—C#l-G7)— corresponding to the number of measures. Secondly, while the duration of the first 82 measures (341 quarters) is virtually the same as die number of time-point attacks (342), die duration in quarters of mm. 1-78 is (13 X 25) 325—corresponding exacdy to the number of measures in die entire piece. The fact that altemate segmentations are possible for any given section of the piece does not diminish the validity of these observations, but rather demonstrates a strong correlation with the relationships of events at the microstructure level. At any point in the score, there is often more than one sound-mass "moving," so no moment exists as a clear point of articulation. This is especially true after the first section, as more diverse interactions of subsections and segments occur. Another parameter that may have been subject to a rigorous "calculation" by the composer is that of notated durations; recall that many of segments described in Chapter 2 have a duration diat reflects motivic numbers. The following discussion elucidates some other aspects of durational organization tiiat help articulate die macrostructure. One aspect of the overall stmcture of Déserts that I find unusual is the apparent sounding of two endings. That is, the third and fourth instrumental sections each end with a very similar texture featuring the same intervals. Since these two cadential segments feature the same intervals, we might expect them to feature the same attack parameters. However, this is not quite the case. Figure 3.6 compares the parameters of these two parallel segments. It shows that aldiough the rhythmic quantities do not match, diey are equivalent at mod 12. The total of 25 time-point attacks at the end of the third instrumental section is the only attack total that does not exhibit mod 12 correlation with the pitch interval in both of these parallel sections. Figure 3.6 - Comparison of the Two "Endings" in Déserts Pitch Interval Span Tempo Number of Measures Duration in Quarter Notes Number of Discrete Altacks Number of Time Point Attacks End of Section III mm. 243-63 • = 80 21 74 (Mod 12 = 2) 59 (Mod 12= 11) 25 (Mod 12=1)??? G2-F#4 23 St. Mod 12 = 11 EndofSecUonlV mm. 310-25 = 80 16 62 (Mod 12 = 2) 35 (Mod 12= 11) 11 (Mod 12= 11) E2-Eb4 23 St. Mod 12 = 11 The final segment of the fourth instrumental section sounds like a cadential reprise, like the series of regular time-point attacks in mm. 1-6 and mm. 54-56. We have already noted that at m. 310, the material first presented at the end of the third instrumental section (mm. 243-63) returns, transposed and rhythmically altered. Figure 3.7 shows that, beginning on the second quarter of m. 312, each of the first 4 attacks of pitch Eb4 occur at the durational interval of 13 eighth notes. ^^Bemard does notice the regular occurrence of the first 4 attacks of the Eb4—see Jonathan Bernard, The Music cf Edgard Varèse, (New Haven: Yale University Press, 1987), 158. Figure 3.7 - Regular Attacks of E M in mm. 312-18 This return to a regular metric pulse is consistent with the return to completely static pitch materials for this final segment, and the significance of the number 13 is obvious. This importance of 13 eighths in governing the durational parameter of static pitch segments may help elucidate the macrostructure of the first subsection of the piece. In the discussion of the microstructure of the opening 29 measures, a convincing segmentation was described at m. 21 (see p. 16). While m. 21 was considered to be the beginning of a new segment, the F4-G5 dyad sustains through the measure—Percussionist 3 is directed to let die chimes vibrate until the end of m. 21. Therefore, the F4-G5 dyad is sounding almost continuously from m. 1 through to the end of m. 21.14 -phese 21 measures are exactiy 169 (13 X 13) eighth-notes in duration. 169, or 13^ would seem to be a likely candidate for a motivicaUy important number—an hypothesis that is confirmed by a striking recurrence at the end of the piece. A l l notated durations of the Eb4 and its associated cymbal attacks in mm. 312-25 total exactiy 169 eighths. This is further confirmation that the static material at the end of die piece is strongly related to the static material diat opens the piece. Remember, too that there are 169 quarter beats throughout the piece at the tempi of 92 and 120. The tempo of 92 is important—the piece opens at quarter = 92 which becomes essentially the mean tempo for the entire piece. The actual duration of all notated segments at their respective ^^In fact, on the third and fourth beats of m. 15 there is no F-G dyad sounding. metronome markings is 944 seconds, wiiicli is very close to the result of simply using 92 as a mean tempo for the 1426 beats—^930 seconds. If we consider attack points in the two parallel endings (mm. 244-63 and mm. 312-25), another interesting numerical correlation emerges. Counting the first attack of F#4 at the third quarter beat of m. 244 as time-point 1, the last attack of F#4 at the downbeat of m. 263, occurs at time-point 65 (that is, the sixty-fifth quarter). At the parallel place in the fourth instrumental section, the first attack of Eb4 occurs at the second quarter of m. 312. The last sounding of Eb4 occurs at the downbeat of m. 323—time-point 39 (that is, the thirty-ninth quarter after the first attack of Eb4). Both 65 and 39 are whole number multiples of 13, which is, of course one of die primary motivic numbers in the piece. A simdar relationship exists between die first and last attack points of die trichords in diese respective sections. The trichord {G2-Bb2-B3} is first sounded at the sixth eighth note of m. 246. Considering this as time-point 1, the last sounding of the trichord occurs at time-point 46 (the fifth eighth note of m. 250). In die parallel passage in the fourth instmmental section, the first attack of the trichord {E2-G2-Ab3} is at the downbeat of m. 317. Again, considering this as time-point 1, the last sounding of the trichord occurs at time-point 23 (the second quarter note of m. 323). 46 is simply twice the number 23, which is also the size of the outside pitch interval in semitones in both of these passages. Together widi die correlations of mod 12=11 (discussed on p. 66 with respect to Figure 3.6) between these two related passages, there are many striking simdarities in non-pitch parameters. There is yet another possible segmentation that convincingly demonstrates a correlation between the duration of the last segment and the very first segment of die piece. From the second quarter beat of m. 312 (die first attack of Eb4) to the end of the last measure is 53 quarter notes. The opening 13 measures of the piece (before the first true interaction of sound-masses) are also 53 quarter notes in duration. Many other aspects of durational symmetry or near-symmetry influence the overall formal process in Déserts. Some observations about durational parameters may also support hearing a climax at approximately m. 242. From mm. 242 to the end of the piece is 84 measures. This is the same as the number of measures in one of the altemate segmentations of the first instrumental section of the piece. If we decide that the first articulation of pitch F#4 in m. 244 is the beginning of post-climax section of the piece, then the numbers of measures are essentially die same in the first section and last section of the piece—82. This might establish an approximate durational symmetry over the entire piece: 82 -I- 161 + 82 = 325 measures. By using such descriptive words as turning point and climax, I am not suggesting that this piece reflects a classical conception of form relating to a narrative curve of some sort. I agree entirely with Bernard and other analysts that this music instead reflects a decidedly non-classical formal process. However, this durationally symmetrical form reflects many of the symmetries in all of the parameters at the micro-structural level. Figure 3.8 is a schematic diagram that summarizes some of these altemate macrostructure sementations. Figure 3.8 - Parallel Macrostructure Segmentations I 1—84.5 H J-53H .325 m. 1 13 21 m. 1 l _ 29 30 53 54 78 82 83 Il 117.118 145. 146 3^ 204 21^: 174^ 175 20 ,20  215216 243 244 263.264 ^84.5JJ 1 312 325 312 325 1-29 ms. HI—24 29 •35 •28 •29 Il—29 •28 h 13H There are correlations within the pitch realm that also support hearing the passage around mm. 239-44 as a turning-point. In diese measures, there are imperfectly symmetrical vertical pitch collections that, for the most part, feature alternating 6- and 7-semitone intervals. As pointed out in the discussion of the microstructure of this passage (see p. 42), the extreme pitches in these collections are E l and the pitch class Eb occurring in several different octaves as Eb5, Eb6, and Eb7. Thus, the interval span in this collection is 59 semitones with octave reinforcement (71 semitones). This interval is a mod 12 equivalent of the outside interval of the end of both die third and fourth instrumental sections (G2-F#4 and E2-Eb4)—23 semitones. Moreover, the extreme pitches at m. 242 (El-Eb5/Eb6/Eb7) are a compound octave expansion of the extreme pitches at the end of die piece (E2-Eb4). As pointed out in Chapter 2, die final large vertical coUection in the penultimate subsection of the fourth instrumental section (mm. 304-09) is also a mod 12 equivalent of that in mm. 242-43 (Bbl-A5/A6, 59 semitones). In the discussion of these parallel passages in Chapter 2, mod 12 numerical correlations between this pitch interval and several other parameters were described. Remember, too, that the symmetrical collection of rhythmic attacks in the indefinite pitched percussion (mm. 232-37) is organized into 59 attacks. The interval of transposition (3) between the parallel static pitch collections in die 2 endings is manifested at other points in the piece. This "difference of 3" controls some other transformations in the piece. For example, the interval of transposition between the two simdar timpani passages—mm. 93-117 and mm. 216-38— is 3 semitones.^^ Immediately prior to the static ending of the third section, there is a chromatic descent of 3 semitones doubled in octaves in m. 242—Eb-D-Db-C. Note that this descent is anticipated in the clarinets in the previous measure (m. 241). The difference of 3 is also operative in 15The lowest pitches of these collections are C#2 and E2 which is a rotation (inversion) of the relationship between the lowest pitches in the 2 endings— G2 and E2. the attack realm: as described on p. 40, there are two related passages of indefinite-pitched percussion—mm. 218-24 and mm. 232-37. The former segment contained 62 discrete attacks and the latter segment contained 59—3 less than the former. In many ways, then, the entire form and process of the piece can be related to the first interaction of sound masses in the piece. The opening features multi-parametric occurrences of the number 14. The first interaction of sound-masses (between the F4-G5 and D2-E3 dyads) establishes the motivic importance of the number 13 and die superposition of the first two interval numbers—14 +13 = 27—establishes a relationship of mod 12 = 3 diat is ultimately "worked out" in the last quarter of the piece. This "difference of 3" is manifested in other parameters, as well. Another potential significance of the retiterated pitches in the two endings of the piece is a symmetry widi respect to a hypodietical pitch-axis. In Figure 3.9, we see that the Bbl /B2 dyad in m. 14 can be heard as a projection of the axis of symmetry (Bb3/B3) around which the 2 symmetrical opening dyads are placed. The E3 and F4 which are symmetricaUy placed about this original axis become the pitch-axis (E4/F4) for the end of the piece—the reiterated single pitches in the two endings of the piece (Eb4, F#4) are symmetrical about this axis.^^ It is also significant that the projection of this axis (from Bb3/B3 to E4/F4) is 6 semitones—a motivic interval and an interval of projection used frequendy by Varèse. are other examples of pitch coUecUons that are symmetrical about the E4/F4 axis, including mm. 63-64. ^^One of the other rare occurrences of repeated material in Varèse's music is the reprise of the opening oboe solo, transposed 6 semitones, at the end of the fu-st movement of Octandre. Figure 3.9 - Latent Interval Symmetries Beginning of piece End of piece (N.B. - axis of symmetry is projected 6 semitones) Axes of Symmetry o Sounding Pitches "Projection" # Hypothetical "projections" of pitches producing latent interval symmetry I also consider it significant that two of the pitch-classes conspicuously absent from the opening of the piece are the reiterated single pitches in the two endings of the piece (F# and Eb). This conjecture about pitch-class aggregate completion is not meant to contradict Bernard's assertion diat pitch/register is die primary organizational parameter in the pitch domain in Déserts. However, these apparent octave displacements and the mod 12 numerical correlations in many of the rhythm/durational parameters suggest that some global pitch-class scheme is valid. Conclusion This is the first attempt that I am aware of to describe the music of Varèse in terms of an integrated networlc of parameters. It aims to demonstrate that Varèse's music is best represented as simultaneously unfolding formal processes, even though a complete elucidation of these parallel processes is stdl somewhat elusive. Some detads of the present analysis may seem contrived or explained too vaguely. However, I am convinced that many of the numerical correlations described do exist and were part of the composer's precompositional planning. I have demonstrated that in Déserts these parallel processes generally fall under two categories that I have called micostructure and macrostructure. The foUowing is a summary of the interparametric correlations that define these processes: Microstructure: 1. Attack Complexes: (a) correlation of discrete attack and interval—for example, the 14 discrete-attack-complexes of the F4-G5 dyad in mm. 1-6; (b) correlation of time-point attack and interval—for example, the 44 time-point attacks in the opening 20 measures and the 45-semitone interval span in this opening subsection are nearly equal. The exact correlations in the subsequent passage (mm. 21-29) confirm that 44 is a significant number. (c) motivic numbers of attacks in groups of reiterated pitches—many groups of repeated attacks of single pitches contain die motivic 13 attacks. Other discrete segments contain correlations between the number of attacks and die pitch interval (for example, die timpani attacks in mm. 93-117 and mm. 216-38). There is also evidence that some of the transformations in the attack-complexes are analogous to processes in the pitch domain such as the "rotation" in mm. 127-30 and the bdateral symmetry of the attack-complexes in mm. 21-29. Macrostructure: 1. Regular metric background—Density 215 and Instrumental Section I of Déserts demonstrate an attack density of 1 time-point attack per quarter beat of duration. 2. Controlled Durational Parameters: (a) Many segments and subsections are of motivic numbers of measures, sometimes corresponding to important pitch intervals. (b) Many segments and subsections are of motivic numbers of beats (quarter- and eighth-note beats are both operative). There seem to be two related processes: (i) where the overall duration of a subsection is of a motivic number of beats—for example, mm. 1-21 are 13^, or 169 eighths long; and (ii) where the sounding duration of all elements of the segment total a motivic number of beats—for example, in mm. 312-25 die Eb4 sounds for exactiy 169 eighdis. 3. Overall interparametric consistency: die altemate segmentation of Section I that places a sectional division at the end of m. 78 results in interparametric correlations between the number of measures and the maximum interval span (78), and a relationship between die number of beats in the section (325 quarters) and the number of measures in the entire piece. 4. Bdateral symmetry: there is a simdarity between die bUaterally symmetrical or mirror-image pitch collections and the large-scale durational planning which also exhibits such symmetry—^for example, the 29 + 24 + 29 measures of the opening section and overall symmetrical design of 82 + 161 + 82 measures for the whole piece. In the microstructure, the arrangement of attacks in the indefinite-pitched percussion in mm. 232-37 (29 -i-1 -f- 29) also demonstrates a simdar symmetry. 5. Overall number of measures related to motivic pitch intervals: the total number of measures (325) is related to die product of the beginning and ending interval numbers—14 X 23 -i- 3 = 325 (remember die 13 X 7 = 91 measures in Ionisation). In both micro- and macrostructures, many of the parameters in Déserts are governed by the numbers 13 and 14. In the pitch realm, these numbers correspond to the minor-ninth and major-ninth—intervals that are prevalent in all of Varèse's works. Several aspects of process in Déserts seem to be directiy derivative of the same processes operative in Ionisation—^particular the importance of the number 13. Future analysis with these principles should include the complete analysis of die rest of Varèse's œuvre. Many of the processes shown to be operating in Déserts are related to processes in other works by Varèse—^particularly Intégrales, Ionisation, and Density 215. Such analysis may be able to determine whether these compositional processes occur in all of Varèse's mature output or whether there is a development in die composer's compositional method, culminating in his last completed work for an instrumental ensemble—Déserts. It is my hope that other analysts wUl pursue these avenues of inquiry toward a more complete description of this music. Varèse was known to show a great deal of disdain for theory and analysis: "By its very definition analysis is sterile. To explain by means of it is to decompose, to mutilate the spirit of a work."18 My purpose in the present study is not to "decompose" or to "mutilate," but simply to try to gain a better understanding and appreciation of the work of a composer whom I hold in the highest regard. ^^Edgard Varèse, "Jérôm' s'en va-t'en guerre," translated by Louise Varèse, The Sackbut 4 (1923): 147, quoted in Bernard (1987), xix. BIBLIOGRAPHY OF WORKS CITED André, José. "Edgard Varèse y la mùsica de Vanguardia." La Naciôn. Paris (March, 1930). Translated by David R. Bloch in "The Music of Edgard Varèse." Ph.D. dissertation. University of Washington, 1973. Quoted in Bernard, Jonathan. The Music of Edgard Varèse. New Haven: Yale University Press, 1987. Antokoletz,Elliott. Twentieth-Century Music. Englewood Cliffs, N.J. : Prentice-Hall, 1992. Bernard, Jonathan. The Music of Edgard Varèse. New Haven: Yale University Press, 1987. . "Pitch/Register in the Music of Edgard Varèse." Music Theory Spectrum 3(1981): 1-25. . "A Theory of Pitch and Register for the Music of Edgard Varèse." Ph.D. dissertation, Yale University, 1977. Chou Wen-chung. "Ionisation: The Function of Timbre in Its Formal and Temporal Organization." In The New Worlds of Edgard Varèse, edited by Sherman Van Solkema, 26-74. Brooklyn, N .Y . : Institute for Studies in American Music, 1979. . "Varèse: A Sketch of the Man and his Music," Musical Quarterly 52, no. 2 (April, 1966): 151-70. Cowed, Henry. "Review of the first American performance of Déserts," Musical Quarterly A\ (1955): 370-73. François, Jean-Charles. "Organization of Scattered Timbrai Qualities: A Look at Edgard Varèse's Ionisation," Perspectives of New Music 29/1 (Fall/Winter, 1991): 48-79. Nattiez, Jean-Jacques. Music and Discourse: Toward a Semiology of Music. Translated by Carolyn Abbate. Princeton, N.J.: Princeton University Press, 1990. Ouellette, Femand. Edgard Varèse. Translated by Derek Coltman. New York: Orion, 1968. Slonimsky, Nicolas. "Analysis." Preface to Varèse, Edgard. Ionisation. New York: Colfranc, 1966. Varèse, Edgard. "The Liberation of Sound." Perspectives of New Music 5/1 (Fall/Winter, 1966): 11-19. . Density 21.5. New York: New Music Quarterly 19/4, 1946; reprint. New York: G. Ricordi, 1956. . Déserts. New York: G. Ricordi, 1959. . Intégrales. New York: Curwen, 1926; reprint. New York: G. Ricordi, 1959. . Ionisation. Paris: Max Eschig, 1934; reprint. New York: G. Ricordi, 1958. . "Jérôm' s'en va-t'en guerre." Translated by Louise Varèse. The Sackbut 4 (1923): 147. Quoted in Bernard, Jonadian. The Music of Edgard Varèse. New Haven: Yale University Press, 1987. 

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