Varèse - Generative Cell and Symmetrical Operations

Contemporary Music Review Vol. 23, No. 1, March 2004, pp. 17–44

Interaction Between the Generative Cell and Symmetrical Operations in Varèse’s Octandre Eli-Eri Moura

The first part of this paper presents a comparative study of three authors (Chou Wenchung, J. D. Anderson and J. W. Bernard), who based their works, to different degrees, on Varèse’s words about his own music. Despite their common basic reference, the authors arrived at different conclusions concerning the guiding principles pr inciples in Varèse’ Varèse’ss compositions. This was partly due to the open quality of the composer’s comments. Without attempting to utilise Varèse’s terminology, and employing a rather pragmatic approach, the second part discusses a passage of Octandre , where two compositional compositional devices, pointed out by the above mentioned authors, interact determining the pitch organisation of the piece: the generative cell and symmetry symmetr y. Keywords: Edgard Varèse; Octandre ; Generative Cell; Symmetrical Operations in Music To reveal a new world is the function of creation in all the arts, but the act of creation defies analysis. (Edgard Varèse, 1967b, p. 204)

Part 1

If examined from f rom a strictly strictl y technical point po int of view, v iew, Varèse’s comments on his work tend to be metaphorical and analogical rather than analytical, analy tical, as he preferred describing and comparing, rather than analysing and explaining. Direct references to or descriptions of the technical aspects of his musical language are not found in any of his writings. Instead, he referred to them using terms such as sound masses, masses, expanding planes, planes, collision,, penetration collision penetration,, repulsion repulsion,, transmutation transmutation,, sound projection, projection, and crystallisation crystallisation,, among others, which do not belong in traditional musical vocabulary. The following quotations are some examples of his remarks containing such terms. (1) When new instruments will allow me to write music as I conceive it, the movement of sound masses, of shifting planes, will be clearly perceived in my works, taking the

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E.-E. Moura place of the linear counterpoint. When these sound masses collide, the phenomena of penetration or repulsion will seem to occur. Certain transmutations taking place on certain planes will seem to be projected onto other planes, moving at different speeds and at different angles. In the moving masses you would be conscious of their transmutations when they pass over different layers, when they penetrate certain opacities, or are dilated in certain rarefactions. (E. Varèse, 1967a, p. 197)

(2) There will no longer be the old conception of melody or interplay of melodies. The entire work will be a melodic totality. (E. Varèse, 1967a, p. 197)

(3) We have actually three dimensions in music: horizontal, vertical, and dynamic swelling or decreasing. I shall add a fourth, sound projection—that feeling that sound is leaving us with no hope of being reflected back, a feeling akin to that aroused by beams of light sent forth by a powerful searchlight—for the ear as for the eye, that sense of projection, of a journey into space. (E. Varèse, 1967a, p. 197)

(4) Conceiving musical form as a resultant—the result Conceiving result of a process—I was struck by what seemed to me an analogy between the formation of my compositions and the phenomenon of crystallization. (E. Varèse, 1967b, p. 203)

(5) There is an idea, the basis of an internal structure, expanded and split into different shapes or groups of sound constantly changing in shape, direction, and speed, attracted and repulsed by various forces. The form of the work is the consequence of this interaction. Possible musical forms are as limitless as the exterior forms of crystals. (E. Varèse, 1967b, p. 203)

(6) 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. (quoted in Cowell, 1955, p. 372)

This unique terminology was the common basic reference in the following three analytical approaches to Varèse’s music, briefly reviewed and compared here. Chou Wen-chung: ‘Varèse: A Sketch of the Man and His Music’ In this article, published in Musical Quarterly in 1966, Chou Wen-chung cites some passages of Intégrales and Déserts to exemplify typical Varesian procedures. The composer himself commented on the procedures in texts such as those quoted under (1) and (5) above. Chou examines the pieces in light of Varèse’s terminology. According to his interpretation, the ‘idea’—‘basis of the internal structure’ (see quotation (5) above)—in Intégrales is present in the passage stated by the C trumpet in bar 10. Isolating the pitch-classes of this idea, the author forms the set {D,A  ,B  ,G, F }. Then, he creates five subsets of three notes each, in which ‘the intervallic content

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as defined by the pitches given, is statable in any vertical permutation or linear ordering’ (Chou, 1966a, p. 158): I = {A  ,B  ,D}, II = {F ,G,B  }, III ={F ,G,A  }, IV = {D,G,A  }, V = {G,A  ,B  }1

The formation of ‘sound masses’ is based on these groups. For example, groups I and II originate two distinct sound masses heard throughout the first section of the piece: the vertical sonorities played by piccolos and B  clarinet (A5,E  6,B6), and trombones (C3,E3,C 4) respectively. Besides derivation from specific subsets, other factors are cited as defining and characterising the distinct layers of sound masses: spatial placement (register), independent dynamic organisation and timbre. The idea in bar 10 is also referred to as the ‘expanding plane’ of the first section of Intégrales. In the article ‘Open rather than bounded’, Chou (1966b, p. 3) suggests that when an idea is expanded, sound masses are originated: ‘Sound masses seem to emerge out of the expansion of an idea—the basis of an internal structure—into the sonic space.’ From this, one may observe that by ‘expansion’ the author means basically the process of verticalisation, or registral distribution of the idea. ‘Transmutation’ is related by Chou to a kind of variation or transformation that changes the attributes of the sound masses. He cites an inversion of group II, at bar 25, as an example of such a process (which can be seen in Figure 12): ‘[it] suddenly activates

Figure 1 Transmutation in Intégrales, bb. 25–27, According to Chou.

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Figure 2 Collisions in Intégrales, b. 36, According to Chou.

three versions of the third group . . . which emerge and collide with each other in quick succession, in mm. 26–29’ (Chou, 1966a, p. 159). 3 From this passage, one may infer transmutation also affects in some way the surrounding context provoking the appearance of new sound masses and/or their displacement. Not explicitly stated, the phenomenon of ‘collision’ seems to be registral and/or temporal encounter of sound masses. Figure 2 is an example of how Chou illustrates a series of collisions of sound masses shaped out of the third, fourth and fifth groups. ‘Penetration’ and ‘repulsion’—cited as the possible result of collision—are not explained. Chou (1966b, p. 3) states that ‘when . . . sound-masses collide, the interaction tends to bring about penetration, during which certain attributes of one sound mass are transferred to another’. However, these attributes are not specified.


Symmetry is cited once by Chou. It occurs in a passage of Déserts (bb. 41–45) in which two sound masses are merged through ‘symmetrical expansion’. However, this subject is not discussed further; the mention of it being limited to this sole example. According to Chou (1966a, p. 161), the succession of the processes defined above ‘accounts for the immense sense of growing organism in the entire score [Déserts] and illustrates Varèse’s concepts of “sound as living matter”’. John Davis Anderson: ‘The influence of scientific concepts on the music and thought of Edgard Varèse’ In his doctoral dissertation, Anderson (1984) attempts to demonstrate a philosophical and practical relationship between some scientific concepts and Varèse’s music. First, he examines writings by and about Varèse. Then, two analyses of Octandre are presented: the first, aiming to ‘discover both the actual processes involved and the degree to which they are consistent with Varèse’s aesthetic principals [ sic ]’ (Anderson, 1984, p. 7); the second, applying Varèse’s terminology. Finally, the author relates some Varesian procedures to scientific fundamentals. Based on the several points of view reviewed in the beginning of the thesis, Anderson proposes a working ‘aesthetic model’ of how Varèse’s music functions. The main premises are: (a) it is founded upon the conceptions that sound has inherent intelligence and will, and that music itself is capable of functioning in three-dimensional space; (b) vertical and horizontal material is derived from the same sources—‘germinal cells’ which are in constant process of variation; (c) timbre works as an agent of delimitation (Anderson, 1984, p. 40–41). The author applies these ideas to the analyses of Octandre. In the first, where standard analytical technique and terminology is used, he mentions the descending chromatic tetrachord in bar 1 as the basis for the entire compositon—the germinal cell, which resembles Chou’s ‘idea’. Following Babbitt (1966, pp. 15–17), he extracts the two possible different trichords from this figure: {Gb,F,D  }, presented in the piece by spatial proximity; and {G  ,F,E}, presented by temporal proximity (Anderson, 1984, p. 46). According to Anderson, these trichords were used by Varèse without regard to temporal or vertical ordering and were freely transposed. Their intervallic content functions as a generating device, originating chords and linear material. Concerning the analysis itself, it is mostly dedicated to locating and classifying such cells throughout the piece. In addition, emphasis is given to the sectional aspect of the work. In the second analysis of Octandre, Anderson defines the terms ‘projection’, ‘sound mass’, ‘expanding plane’, ‘transmutation’, ‘collision’, ‘penetration’, ‘attraction’ and ‘repulsion’, and points out particular parts of the piece that exemplify them. Based on quotation (3), he associates the concept of ‘projection’—a term not defined by Chou—with the enormous crescendi of some single chords which occur in passages of periodic rhythm and at endings of sections of Octandre (as in I, b. 15; II, bb. 80–81; III, bb. 33–38). Such configurations represent ‘Varèse’s illusion of sound “departing on a journey”’ (Anderson, 1984, p. 77).

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Three chief criteria are cited by Anderson to identify ‘sound masses’ (as cited in quotation (1)): timbre, articulative-rhythmic-dynamic similarity, and interior intervallic consistency. Chou used roughly the same criteria. The distinction between Anderson’s and Chou’s arguments lies solely in the different degree of emphasis put on each characteristic of sound masses. While Chou emphasises the interval content, Anderson finds the timbre their most important feature. Apart from this, their interpretations are the same. Figure 3 illustrates two contrasting sound masses, and Figure 4 shows sound masses separated in time as pointed out by Anderson. Concerning the term ‘expanding plane’, Anderson applies two different meanings to

Figure 3 Contrasting Sound Masses in Octandre I, b. 18, According to Anderson.

Figure 4 Sound Mass Separated in Time in Octandre I, bb. 19–21, According to Anderson.


it. In the first part of his dissertation, he cites quotation (2) to draw the following conclusion: Most sound-masses are constructed by the ‘verticalisation’ of previous material. Pitch-classes or intervals which constitute a melody or ‘plane’ are often reorchestrated and sounded simultaneously as a chord or ‘sound-mass’. . . . Planes are capable of evolving into masses and vice versa. . . . Varèse terms this process the ‘expanding plane’. (Anderson, 1984, pp. 33–34)

This conclusion is basically the same as Chou’s. However, in analysing Octandre, Anderson defines expanding plane by taking the literal sense of the word: range expansion, or widening of melodic lines, apparently without direct connection to the germinal cell. The example given by the author is the second part of the initial oboe solo in the first movement (bb. 6–10), where the range is extended to G6. Yet a third sense is given to the term ‘plane’—without the adjective ‘expanding’. He applies the word to certain one-note lines found throughout the piece which, according to him, help the music’s continuity, as they link different sound masses or rhythmic events (an example cited is C6, played first by the B  clarinet, and then by the oboe, in Octandre I, bb. 26–32). Concerning ‘transmutation’ (from quotation (1)), Anderson and Chou also express basically the same opinion. The former states: ‘Sound masses do not remain constantly stable, but are subject to variation. Varèse terms his process “transmutation”’ (Anderson, 1984, p. 81). However, he does not specify what kinds of variations can possibly occur. Regarding the other terms of quotation (1), the two authors have different views. In contrast to Chou, Anderson ‘flattens out’ the phenomena of ‘penetration’ and ‘collision’ resulting in the invasion of one sound mass on the registral boundaries of another. However, a distinction is made between the two concepts. In penetration, a single pitch breaks into the registral limits of a sound mass, while in collision several do so. Chou interprets Varèse’s (1967a, p. 197) words—‘When these sound masses collide the phenomena of penetration or repulsion will seem to occur’—more strictly and attributes to the specific term ‘penetration’ a more general sense: the transference of ‘certain attributes’ of one sound mass to another. According to Anderson, as a result of penetration or collision, ‘attraction’ 4 or ‘repulsion’ may occur, depending on whether the pitches of the sound masses subsequently move toward a common or a distinct register. Two passages cited by the author are illustrated in Figure 5 (penetration resulting in attraction) and Figure 6 (penetration resulting in repulsion). Anderson concludes his dissertation by establishing analogies between some of the Varesian processes described above, and some theories in physics which were formulated in the first decades of the twentieth century (e.g. relativity, quantum).

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Figure 5 Penetration Resulting in Attraction in Octandre I, b. 18, According to Anderson.

Figure 6 Penetration Resulting in Repulsion in Octandre II, bb. 37–40, According to Anderson.

Jonathan Walter Bernard: The music of Edgard Varèse Bernard’s The music of Edgard Varèse (1987) was the first full-length book on Varèse’s music. Its first part is devoted to exploring the influences of three aesthetic trends


(Cubism, Futurism and Expressionism) on Varèse’s work, while the main body deals with Bernard’s theory concerning Varèse’s compositional method. 5 Like Chou, Bernard begins his approach by citing quotation (1). The author suggests that the composer actually considered space in a physical sense when he described events such as ‘movement of sound masses’, ‘collision’, ‘penetration’, ‘repulsion’, etc. In order to explain how this physical space is translated into musical space, Bernard first calls attention to Varèse’s concerns on the vertical dimension of music. These are not only reflected in the composer’s own music, but also in his remarks on a kind of melodic line, which is not horizontal but vertical (L. Varèse, 1972, p. 211). The author concludes: ‘If the vertical dimension has primary status in Varèse’s musical frame of reference, then the partitioning of vertically defined space takes on crucial significance’ (Bernard, 1987, p. 41). Next, Bernard takes quotation (6) to refer to an apparently self-generating method of composition that allows order to emerge from within. This method is related to Varèse’s analogy to the process of crystallisation, where the ‘growth through orderly expansion of a minimum of an idea, cell-like in nature [is emphasised]’ (Bernard, 1987, pp. 42–43). While this part of Bernard’s analysis resembles Chou’s and Anderson’s concepts of ‘idea’ and ‘germinal cell’, his conclusion differs essentially from their approaches. He cites quotations (4) and (5) to suggest that ‘the initial idea is not necessarily anything more than a point of departure. . . . Having appeared once, the initial event may never be heard again’ (Bernard, 1987, p. 43). Based mainly on these two premises—the partitioning of vertically defined space as a fundamental factor in Varèse’s music, and the existence of a mechanism originating continual change throughout this space—Bernard creates his method to approaching Varèse’s works. Fundamental is the view of the tempered pitch system as a neutral arrangement of semitones, where criteria of absolute size and distance are pertinent. In this context, two maxims of set theory are discarded: octave and inversional equivalence. Finally, he presents the principle behind the manipulation of events (as described in quotation (1)) that ultimately provides the continuum of change in Varèse’s music: symmetry. Bernard classifies symmetry into two principal kinds, which can work either in a simultaneous, or in a partial way: (a) ‘mirror’, where two segments mirroring each other have one or two pitches as an equidistant centre; (b) ‘parallel’, which displays two or more segments with the same adjacent intervallic ordering from the lowest pitch to the highest. In Bernard’s point of view, while symmetrical procedures describe on the simplest level individual formations, an extension of their principles can describe larger contexts as instances of process. In his theory, this process is carried out by four basic operations, namely ‘projection’, ‘rotation’, ‘expansion’ and ‘contraction’. They are designed to be applied both singly and in combination—the first is related to parallel symmetry, and the others to mirror symmetry. ‘Projection’ is basically spatial transposition: the transference of a pitch structure from one spatial location to another over time (as in Déserts bb. 29–30, between the

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pitches of timpani and brass). 6 Despite the fact that both Bernard and Anderson cite quotation (3) as a reference, their conclusions about this term are completely different. As mentioned above, Anderson related projection to the crescendi of some chords. By ‘rotation’ Bernard means inversion in a literal sense (as in Intégrales b. 151, between the chord played by the trumpets and horn, and the chord played by the trombones). In ‘expansion’, the outer pitches of a structure migrate in contrary motion, by the same number of semitones, to a new location in the chromatic spectrum. Bernard does not relate this term to Varèse’s ‘expanding plane’, which is interpreted by Chou as the verticalisation of previous materials, and by Anderson as the widening of the range of a melodic line. However, Bernard’s expansion resembles very closely the example of Déserts given by Chou, where two sound masses are merged through ‘symmetrical expansion’. ‘Contraction’ is defined as the opposite of expansion. According to Bernard (1987, p. 57), the confluence of processes suggests the occurrence of the ‘phenomenon referred to by Varèse as “penetration” of one sound mass by another’. In another source, Bernard (1981, p. 18) states: Analytical complications often result when the musical fabric splits and the process of transformation proceeds in two or more different directions. When the disparate structures produced by such fragmentation reunite in a single structure, penetration may be said to occur.

A divergence in interpretation is again apparent. In Bernard’s view, penetration is defined at a structural level—‘confluence of processes’—whereas in Chou’s and Anderson’s views, it is defined at a more superficial level: transference of ‘certain attributes’ of one sound mass into another, and invasion of a single pitch from one sound mass into another. In order to account for internal details, Bernard applies trichords that are connected by certain operations, and grouped under what he calls a ‘constellation’. In coherence with the rest of his approach toward external dimensions of structures, these ‘internal’ operations are undertaken because of spatial considerations. Accordingly, from a basic form, say, [3][8] (to use his own example and labelling), present in a trichord such as {G 4,B4,G5}, derivative forms called ‘unfolding’ are created by replacing the inner note in an outer position, symmetrically related to the lower or higher note: [3][11]:{E 4,G 4,G5}, and [8][11]:{G 4, G5,D 6}. In a third derivative form, ‘infolding’, the inner note serves as symmetrical centre to an outer note which moves into the chord: [3][5]:{B4,D5,G5}. In actual application, the symmetrical aspects of these operations remain only on an abstract level. This is due to the fact that only the absolute intervallic content is considered in the relationship between trichords. Trichords of the same ‘constellation’ may be found in different spatial locations (transpositions), cancelling the realisation of their inherent symmetrical properties. In any case, this is not a contradiction in Bernard’s approach, as his major concern in using these trichords is related to the measurement of intervallic consistency throughout Varèse’s works (Bernard, 1987, p. 77).


Leaving aside the question of influences on Varèse’s thought (science versus art), one can observe that the approaches of Chou and Anderson coincide in many ways, while Bernard’s approach seems to move in a different direction. In fact, the divergence begins in how the words of the composer are taken into account. The former attempt to explain Varèse’s terminology in a rather literal way, and to identify instances in the scores which illustrate the terms. Bernard, using the same sources, does not take Varèse’s terms literally, and limits himself to drawing two basic conclusions: (a) the relevance of the vertical dimension and its partitioning; (b) the existence of a process of continual change in the sonic space. Regarding terms such as ‘expanding plane’ ‘transmutation’ ‘projection’, ‘collision’, ‘penetration’, ‘repulsion’ and ‘attraction’, it seems that Chou and Anderson take Varèse’s words as the very descriptions of compositional processes and generating mechanisms. Accordingly, they set forth to find events that could be depicted in those terms, and cited isolated passages as evidence. They did not identify the underlying principles. For example, in their conception of the phenomena of transmutation, collision, penetration, repulsion and attraction, these are responsible for the movement and transformation of sound masses—and this movement and transformation can also be regarded as taking part in the generating processes of the music. However, the definitions they present for the phenomena lack theoretical support. One may ask what, besides the composer’s arbitrary will, makes the sound masses behave as described. Varèse suggests that there are ‘various forces’ that are responsible for their movement (quotation (5)). However, Varèse did not explain what he meant by ‘forces’, nor did Chou and Anderson attempt to define them. Bernard did not specify or explain these terms, except for ‘projection’. 7 Rather, he seemed to consider Varèse’s terminology as a metaphorical way of referring to the musical results, or to the more superficial levels of the music. Accordingly, he undertook the task of finding the generating mechanisms that led to the production of those results. The compositional devices and procedures suggested by him are to some degree all related to the principle of symmetry.

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Part 2

Below are some observations on the compositional procedures and their practical application in the first 15 bars of Octandre. There is no attempt to apply Varèse’s terminology. The goal is to show how two of the principles described above interact in order to function as generating devices, and subsequently to determine the pitch organisation of the piece. These principles are the ‘idea’ or the ‘germinal cell’ (mentioned by Chou and Anderson), and symmetry (studied by Bernard).8 Observations on Compositional Procedures The descendent tetrachord at bar 1 of Octandre will be called the ‘generative cell’ and the trichords derived from it (as pointed out by Babbitt and Anderson), ‘cell 1’ and ‘cell 2’ (as shown in Figure 7). 9 The way the pitches of these cells are spatially (registrally) realized is of fundamental relevance. Considering cell 1, one can extract the following dispositions from the first four bars (see Figure 8). Dispositions a, b and c represent the three possible registral combinations of the pitches of cell 1, if one arranges two pitches adjacently in the higher register and displaces the remaining an octave in the lower register (as shown in Figure 9). With the inverse operation, i.e. arranging two pitches adjacently in the lower register and

Figure 7 Cells 1 and 2: Octandre I, b. 1.

Figure 8 Octandre I, bb. 1–4.

Figure 9 Dispositions a, b and c.


Figure 10 Registral Combinations of Cell 1 (Type 1).

Figure 11 Registral Combinations in Octandre I, b. 10.

displacing one an octave in the higher register, one will obtain a complete set of six such registral combinations (see Figure 10). To make this clearer, this kind of spatial organisation of cell 1 will be called ‘type 1’. It can be noted that all of these registral combinations of cell 1 appear systematically, without regard to the pitch ordering, throughout Octandre. For instance, at bar 10, 10 the initial statement of the oboe (bb. 1–4) is ‘recapitulated’ by the flute, the B  clarinet and the trumpet in a contracted and overlapped form. In this ‘recapitulation’ a different combination (chosen among the possible six) is used for each cell of the oboe line (as seen in Figure 11). Cell {B  4,A  5,A5} (shown in black note heads in Figure 11) has also an equivalent cell in the opening passage of the piece, if one takes into account the pitches G 4 (b. 4)

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Figure 12 Cell Bb4-Ab5-A5 in Octandre I, bb. 4–6.

Figure 13 Registral Combinations of Type 1 in bb. 1–4 (a) and m. 10 (b) of Octandre I.

Figure 14 Spatial Organizations of Cell 1: ‘Type 2’ (a), ‘Type 3’ (b) and ‘Type 4’ (c).

and A4 (b. 5) in the oboe, and B  3 (b. 6) in the B  clarinet (see Figure 12). Exceptionally, in this case both cells display the same registral combination: b1. Figure 13 shows the registral combinations in the two passages. 11 All the combinations of type 1 are present, except for c2, which appears later in the first movement. Other spatial organisations of cell 1 occur in Octandre. Types 2, 3 and 4 are shown in Figure 14. The spatial organisation type 2 is also present in the initial statement of the oboe (x in Figure 8). It appears less frequently than type 1 in the first movement, but its appearance is noticeable in other parts of the work as in the opening of the second


Figure 15 Spatial Organization Type 2 in Octandre II, bb. 2–4 (Piccolo).

Figure 16 Cells 2 in Octandre I, bb. 26–27.

movement (see Figure 15). Types 3 and 4 play a structural role in the first fifteen measures of the piece, as will be shown below. Concerning cell 2, its most frequent spatial organisations are the same as its basic referential sets (see note 9). An example of their appearance is found at bars 26–27 of the first movement (Figure 16). The reason for emphasising these types of spatial organisations is that they reflect the organising principle that underlies all the work: symmetry. Symmetry is defined here as reflection—that is, the result of turning a figure around a linear axis, which is called the axis of symmetry. Because of the characteristics of the tempered pitch system, an axis of symmetry can comprise one single note or an m2-dyad. Taking F4 and E4–F4 as centres of symmetry, for example, the dyads in Figure 17 represent the simplest forms of reflection. More complex symmetrical formations are possible through the superimposition of these dyads. From cell 1, types 2, 3 and 4 are visibly symmetrical, and type 1 exhibits a property that will be called ‘potential symmetry’. Type 1 is not symmetrical by itself, but depends on the sum of two registral combinations (a1 and a2, b1 and b2, . . .) to achieve symmetry. The two spatial organisations of cell 2 are symmetrical in relation to each other.

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Figure 17 Symmetrical Dyads Around the Axes F4 and E4–F4.

One can say that Octandre is made of a sequence of symmetrical formations whose pitches may be simultaneous (forming single chords) or spread out in time (forming diverse configurations). Some of these symmetrical formations determine large-sale structures, which serve as a frame for the pitch organisation of sections. Outer notes may also determine the registral and/or temporal boundaries of these sections. Among the use of other devices, spatial organisations type 3 or 4 of cell 1 play these structural roles in Octandre. Other symmetrical formations, such as cell 2 and types 1 and 2 of cell 1, determine the small-scale structures of these sections or areas. They establish the ‘immediate’ the ‘next’ step, fulfilling the spatial and temporal vacuum within the large-scale structures. The confluence of the large and small-scale structures implies or results in the following features: (a) various centres of symmetry (axes) may be found in a determined section or area of the piece (sometimes, even in a single chord), with each axis bearing a different degree of importance in the section where it appears; (b) the pitches originated by the processes of symmetry may serve as an axis for new symmetrical formations; (c) the missing pitches of certain apparently incomplete symmetrical formations may be found either before or after the location of their axes. Some other features can be observed in this ‘symmetrical system’. One of them is what can be called ‘implicit symmetry’, in which the pitch-classes above the axis are identical to the pitch-classes under it. The simplest formation exhibiting this property is an octave divided by two tritones. Some other instances involving a major number of pitches are shown in Figure 18. Implicit symmetry is basically used by Varèse in order to change registral activity and displace the range of influence of some axes within the sections. Because of its implicit symmetry, the tritone may be considered the most important articulative agent in Octandre. In each area of the piece, emphasis is given to one of the cells. The most frequent one is cell 1, and the most repeated kind of spatial organisation is type 1. Emphasis is also given to certain transpositions of the cells, which makes possible the focus on


Figure 18 Implicit Symmetry.

certain regions of the chromatic spectrum in each area of the piece, according to the ‘tonal’ plan of the composer. In general, two cell 1’s whose pitch-classes are separated by tritons work together. They are, in some way, related to each other within the sections or at the articulative points of the piece. These related cells represent an extension of the idea of the tritones as articulative agents in Octandre. Figure 19 (both a and b) show two instances where such related cells appear. Spatial organisation type 1 was said to exhibit the property of potential symmetry. As an extension of this idea, one can establish a ‘potential axis’ for each pair of the registral combinations of type 1. The aforementioned related cells result from the symmetrical expansion of these potential axes (Figure 20). These procedures account for the building up of a compositional system in which the total chromatic is flexibly cycled. It is important to mention that they do not function as an end to themselves, but as a means to realise musical ideas that emerge in the superficial structure of the work. An example of such superficial structure is the passage at bar 15, clearly designed to be one of the pivotal points of the first movement (in this passage, all the instruments participate and the total chromatic is reached for the first time; a large amount of tension is subtly released as the first most dense sonic mass of the piece is followed by a single descending line played by the bassoon). Application The pivotal point at bar 15 in Octandre delineates the end of the first important section of the piece. This section can be divided into two subsections, which, in turn, can be separated into smaller areas. The first subsection (A) extends from bar 1 to the end of bar 8 where B  4 appears. The second subsection (B) goes from B  4 of bar 8 to the point of bar 15 where the note A3 is presented by the bassoon (with this note the pivotal point reaches the total chromatic).12 In this way, one can say that B  4 in bar 8 represents the articulative point of the section and belongs to the two adjacent subsections.

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Figure 19 Related Cells: (a) in Octandre I, bb. 8–9; (b) in Octandre I, b. 20.

Figure 20 Potential Axis/Related Cells.

Each subsectiion is characterised by the presence of a specific symmetrical configuration, which determines its large-scale structure. Figure 21 shows the large-scale structure of subsection A.13 It is built up by cell 1s spatially organised according to type 3. The axis of this symmetrical configuration is A4. It is noticeable that the highest pitch of this configuration (G6) crosses the


Figure 21 Large-scale Structure of Sub-section A: Octandre I, bb. 5–9.

Figure 22 Large-scale Structure of Sub-section B: Octandre I, bb. 10–15.

the piece. The crossing pitch may or may not take part in the symmetries of the neighbouring section. In this case, the presence of G6 is important to complete the symmetry of the first chord of bar 11 (which will be discussed later). But after this point, it is suppressed and does not appear again. The symmetrical configuration that determines the large-scale structure of subsection B is more complex (Figure 22). It is built up by two superimposed cell 1s spatially realised according to type 4. The adjacent outer notes of these cells (C4 in the horn and B4 in the bassoon at bb. 11–13) are stated simultaneously and form an M7dyad, which can be regarded as a provisional axis of the configuration. With the two next notes of the bassoon at bar 13 (F  3 and F3), the actual centre of the symmetrical configuration is reached. As in the former case, one of the outer notes of the configuration crosses the boundaries of its section, this time the lowest note (B  1). The pitch-classes of this axis (F and F) also represent the potential axes of the related cells of the two superimposed cell 1s that frame subsection B’s large-scale structure

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Figure 23 Related Cells of Large-scale Structure (Sub-section B).

Figure 24 Articulative Point/Related Cells: Octandre I, bb. 8–9 (Oboe).

Some transpositions of cell 1 play a more prominent role than others within the section. For example, the cell whose pitch-classes are B  , B and C, is emphasised in subsection B (it is present in both internal and large-scale structures), while it is hardly found in subsection A. Its presence in the articulative point at bars 8–9 signals the beginning of subsection B. It is also noticeable that there is a confluence of two related cell 1s in this articulative point of the section (Figure 24). Another feature that characterises the two subsections and signals the referred articulative point as their border is the use of the spatial organisation of the cells. At the close of subsection A, the cell 1 from the beginning of the piece appears again, now presenting the symmetrical pair of combination a1, which is combination a2 (Figure 25). This procedure reflects what happens from this articulative point to as far as the pivotal point at bar 15; that is, all the cells formerly presented are repeated with their spatial organisation changed (examples of these procedures will be presented in the discussion of the small-scale structures). The study of the rhythmic and proportional aspects of the piece is outside the scope of this paper. However, it is interesting to note the existence of a possible temporal symmetry working together with the spatial symmetries in this section of the piece.


Figure 25 Registral Combinations a1 and a2: Octandre I, bb. 1 and 8.

Figure 26 Temporal Symmetry in Octandre I, bb. 1–15.

The middle point of subsection A coincides with the attack of the axis A4 by the oboe at bar 5. In subsection B, the provisional axis—C4 in the horn and B4 in the bassoon— at bars 11–13 represent the centre of the temporal symmetry in this section. The point where this dyad is formed (attack in bar 11), and the point where it is undone (bar 13 where the bassoon abandons B4 to reach F  4 and F4), are equidistant to the two boundaries of subsection B. B  4 at bar 8 (oboe), marked by a contrasting dynamic, functions as a common (neutral) point where the two subsections intercept each other. Figure 26 shows some aspects of this temporal symmetry in more detail. The lacunas left by the described large-scale structures are filled by small-scale structures. These are also determined by symmetrical formations projecting cells 1 and 2 of the generative cell. Comments on the most important of these symmetrical formations are below in a summarised form. Bars 1–4: taking into account the potential axes of the cells 1 of the initial generative cell (bb. 1–3) and its transposition (b. 4), the complete chromatic range can be said to be implied in this area. The first five potential axes represent the five pitch-classes of the large-scale configuration of subsection A (Figure 27). The last potential axis (F ) is the point of departure to reach A4 (the principal axis of the subsection) at bar 5. The one before the last (G) serves as an axis for the local symmetry. Three cells 2, as well as smaller symmetries, appear in the context (Figure 28). Bars 6–9: at bar 6, E5 in the oboe is a result of implicit symmetry, if one takes B  3 in the B  clarinet (to which it is related by tritones) as the local axis (Figure 29). Not taking into account the notes of the large-scale structure, only four pitch-classes appear in this area (b. 6 to B  in b. 8): E, D  , D and F. Together with the pitch-classes of the large-scale structure (excluding G ), they form two pairs of related cells (Figure 30). At bars 8–9, two related cells determine the articulative point of the section:

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Figure 27 Potential Axes Projected in the Large-scale Structure.

Figure 28 Local Symmetries in Octandre I, bb. 4–6.

Figure 29 Implicit Symmetry in Octandre I, bb. 5–6.


Figure 30 Related Cells: Octandre I, bb. 6–8.

Figure 31 Simultaneous Symmetries in Octandre I, bb. 8–11.

{F 4,F5,E4} and {B  4,C5,B5}. As already mentioned, the first presents the symmetrical pair of the registral combination presented by its equivalent cell in bar 1. However, its pitches also participate in the local symmetries of the area. There are two simultaneous symmetries working in this passage. The axis of one is E5, and the axis of the other is F5–F 5. E5, which also works as an important axis for formations of subsection B, is already present in the context of the passage. F5–F  5, whose pitch-classes coincide with the axis of the large-scale structure of subsection B, is located ahead at the second chord of bar 11 (see Figure 31). Bars 10–15: the two formations at bar 11 (beat 1/beat 4) have two different sources. The second (beat 4) is the result of implicit symmetry, used in this case to change the register of its pitches. The dyad F–F  does not belong to the symmetrical formation. It is the axis of symmetry in bars 7–8 as mentioned above (see Figure 32). The first chord of bar 11 is an apparently incomplete formation. Having A5–A  5 as axis, its other ‘side’ is found earlier in the articulative point (shown in Figure 33). The registral combinations (type 1) of the cells in bar 10 have already been discussed. Most of the pitches of this passage are also related to the first symmetrical formation of bar 11 (Figure 34). At bar 13, the pitches F 4 and F4 in the bassoon represent the axis of the large scale structure of subsection B, as already mentioned (see Figure 22). B  5 in the B  clarinet is related to the second symmetrical formation of bar 11 in the way shown in Figure 35. G5 in the trumpet can be considered as an axis of the two preceding axes, and E5

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Figure 32 Implicit Symmetry in Octandre I, bb. 8–11.

Figure 33 Apparently Incomplete Symmetrical Formation in Octandre I, bb. 9–11.

Figure 34 Pitches Related to the First Symmetrical Formation of b. 11: Octandre I, bb. 10–11.




Figure 37 Local Symmetry at End of Sub-section B: Octandre I, bb. 13–15.

(also in the trumpet) as the reappearance of the axis of the second chord in bar 11, as illustrated in Figure 36. The superimposition of F4 (bassoon) and E5 (trumpet) at the end of bar 13 implies a new axis (B4–B  4), which will serve as the centre of the last symmetry of subsection B (shown in Figure 37). It is important to note that the

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Figure 38 Related Cells in Sub-section B.

Figure 39 Registral Combinations of Type 1: Octandre I, bb. 11–15.

pitch-classes of the main axes of subsection B (E,F,F  , B  , B, C) form a set which corresponds to two related cells 1 (see Figure 38). Similar to what happens at bar 10, all described symmetries (from b. 11 to b. 15) create cell 1s, spatially organised according to type 1. The six registral combinations of type 1 are distributed among the cells (as shown in Figure 39). The devices and procedures studied here are found throughout Octandre, determining its pitch organisation. In interacting, the generative cell and the symmetrical operations not only function as generating mechanisms, but also serve to establish the basis of a system to unify and organise the composition. Notes [1] It is noticeable that among the nine possible subsets of cardinality 3 from the ‘idea’ {D,A  ,B  ,G,F }:[0,1,2,4,8], Chou applies only five in his analysis: [0,2,6], [0,1,4], [0,1,2], [0,1,6] and [0,1,3]. [2] This figure, as well as numbers 2–6, present musical reductions I made myself. They intend to either clarify some of the arguments exposed by the authors or exemplify passages cited by them. [3] At first, Chou seems to regard each of his ‘groups’ as a generic interval collection (like a set type in set theory), to which any set related by inversion is identical in terms of interval content. Accordingly, sound masses should be drawn from the groups without reference to any qualification. However, at this point, he qualifies inversion as ‘transmutation’.

Contemporary Music Review 43 [4] This term is probably taken from quotation (5). [5] Diametrically opposite to Anderson’s points of view (science as the major influence), Bernard sees in the visual arts of around the time of the First World War the main sources of influence on Varèse’s aesthetic views. [6] ‘Partial projection’ is also possible: the outer boundaries of the formations are preserved while the internal details are not. [7] The term ‘penetration’ is also used by Beranrd. However, it is not referred to as a process, but as a confluence or result of processes. [8] Only the principles are taken, not their developments by these authors. With respect to symmetry, Bernard’s approach is not used here. [9] The basic referential sets for these trichords are shown in Figure 40. They represent the four possible combinations of the four initial pitches of the piece (the ‘generative cell’), taken three at a time, and accomplish two conditions: scalewise ordering; and registral organisation resulting in the two smallest possible interval contents between the adjacent pitches. Any given three-note configuration is said to be ‘cell 1’ or ‘cell 2’ if it coincides with one of the basic sets shown in Figure 40 or with one of their transpositions, independently of the pitch ordering. [10] Bars given without allusion to the movement they belong, refer to movement I. [11] The cell E5,D  5,D5 is the only one that does not belong to this kind of registral organisation. Its specific type is discussed later. [12] It is important to mention that, while the pivotal point is present in the ‘superficial structure’ of the work (thus clearly perceived by the listener), the other points of division that are mentioned here belong to the ‘deep structure’ of the piece (thus not necessarily perceived by the listener). The presented segmentation (section, subsections, areas) is offered here in an attempt to clarify this deep structure. [13] In the following examples the notation of rhythm, dynamics and articulation are omitted. Some indications are given to clarify the context of the passages in Figure 41.

Figure 40 Basic Referential Sets for Cell 1 and Cell 2.

Figure 41 Signs Used in the Examples.

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References Anderson, J. D. (1984). The influence of scientific concepts on the music and thought of Edgard Varèse (Doctoral dissertation, University of Northern Colorado). Ann Arbor, MI: University Microfilms, (UMI No, 8418115). Babbitt, M. (1966). Edgard Varèse: A few observations of his music. Perspectives of New Music , 4(2), 14–22. Bernard, J. W. (1981). Pitch/register in the music of Edgard Varèse. Music Theory Spectrum, 3, 1–25. Bernard, J. W. (1987). The music of Edgard Varèse. New Haven, CT, and London: Yale University Press. Chou, W. (1966a). Varèse: A sketch of the man and his music. Musical Quarterly , 52(2), 151–170. Chou, W. (1966b). Open rather than bounded. Perspectives of New Music , 5(1), 1–10. Cowell, H. (1955). Current chronicle. Musical Quarterly , 41(3), 370–373. Varèse, E. (1967a). New instruments and new music. In B. Childs & E. Schwartz (Eds.), Contemporary composers on contemporary music (pp. 196–198). New York: Holt Rinehart. Varèse, E. (1967b). Rhythm, form and content. In B. Childs & E. Schwartz (Eds.), Contemporary composers on contemporary music (pp. 201–204). New York: Holt Rinehart. Varèse, L. (1972). Varèse: a looking-glass diary (Vol. 1). New York: W. W. Norton.

Varèse - Generative Cell and Symmetrical Operations

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