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COPYRIGHT 2003 University of Washington
INTRODUCTION
BETWEEN THE EARLY 1970s and the late 1990s Iannis Xenakis composed several short works for solo instruments and for small ensembles. Because of their limited duration and minimal instrumentation, these works may be thought of as miniatures in comparison to his lengthier and more numerous chamber and orchestral works. (1) Xenakis's miniatures are attractive subjects for analytical study because, in spite of their brevity and relative simplicity, they nevertheless demonstrate a masterly coordination of texture, pitch materials and temporal structure comparable to that found in the larger works. The miniature that has been chosen for analysis here is a r. (Hommage a Ravel) for piano, which was commissioned by the Radio-France International Festival at Montpellier in 1987 as part of its commemoration of the fiftieth anniversary of Ravel's death. At twenty-one measures and just over two minutes in duration, it is one of the most concentrated of Xenakis's works. While the subtitle of a r. indicates that it is a tribute to Ravel, the work contains little stylistic resemblance to the older composer's work except for a general virtuosity and brilliance of harmonic color. Stylistically it is much more closely related to Xenakis's own chamber works from the same period, works such as Jalons (1986), XAS (1987), and Waarg (1988), which are particularly notable for their exuberance and intensity of expression.
This analysis will examine the structure of a r. from three perspectives: first, texture, segmentation and formal design; second, sieve theory and pitch-class sets; and finally temporal structure. Though these three perspectives inevitably intertwine somewhat through the course of the article, each will be the principal focus of a separate section below.
TEXTURE, SEGMENTATION AND FORMAL DESIGN
In Xenakis's instrumental works, large numbers of individual sounds tend to coalesce into textures that are homogeneous with respect to rhythm, density, dynamics, articulation, and pitch collection. Relative homogeneity is maintained even when the temporal intervals between the sounds are randomized, as in the process of stochastic composition, for in such cases the average density of the resultant texture remains constant. (2) Sudden changes in any of these characteristics have the effect of setting off a given texture from other, surrounding textures, thus dividing the musical surface into clearly perceptible segments. These segments are the containers for the sound masses for which Xenakis is well known, and they serve as the primary structural units upon which the form of a work is based. As such, they are subject to hierarchical groupings that form the basis for larger structural units. (3) Segments group together to form sections and sections, in turn, group together to form parts. A small work will typically divide into no more than two or three parts.
In Xenakis's music from the 1970s and 1980s, two or three distinct types of textures are generally featured in a given work. In a r. two types of textures are used: simultaneities and random walks. Simultaneities are common to several different styles of music, and therefore do not require any particular technical explanation. Their role in Xenakis's stylistic development, however, is worth noting. As Mihu Iliescu has pointed out, the simultaneities in Xenakis's late music appear to be a transformation of the idea of sound masses--formerly manifested as collections of disconnected, individual sonic elements--into large chords, resulting in "vertical blocks" of "harmonic/timbral color." (4)
In contrast to the transformation of sound masses into simultaneities, the stylistic evolution of the random walk in Xenakis's music is not so simple, and therefore requires a more detailed explanation. A general definition of a random walk is "the movement of something in successive steps, the direction, length, or other property of each step being governed by chance independently of preceding steps." (5) Random walks are used as models of physical processes such as Brownian motion, which is "the ceaseless erratic motion of fine particles in suspension." (6) Xenakis has used both terms interchangeably to describe a type of sinuous melodic contour introduced into his instrumental music in the early 1970s. These melodies have their origin in Xenakis's first attempts to extend the principles of stochastic composition into the domain of sound synthesis through the generation of stochastic waveforms. (7) Graphic representations of stochastic waveforms consist of continuous lines that feature irregular jumps in amplitude with respect to time. When the vertical dimension of the graphs was interpreted as pitch instead of amplitude, the irregular shapes contained in the graphs resulted in a distinctive type of instrumental melody. In order to facilitate the performance of the melodies by instrumentalists, pitch-time graphs of random walks were transcribed into standard musical notation. Xenakis's first instrumental work to employ random walks is Mikka for violin (1971), in which a continuous glissando moves through pitch space with unpredictable changes of speed and direction. (8) When the notion of random walks was subsequently transferred from music for strings to music for keyboard instruments, the continuous motions that had formerly been represented by glissandi were replaced by rapid linear motions through large pitch collections, the latter lending a distinctive harmonic flavor to the random walks.
Xenakis explains the connection between random walks and pitch collections in the preface to the score of Mists for piano (1980), where he states that one of the main ideas in that work is "the exploration of pitch series (scales) and their cyclic transpositions.... Their exploitation in sound is made either in a contiguous manner (melodically) or by means of stochastic distributions ... in order to produce sound clouds of defined density; in short, with the aid of random walks (Brownian movements)." (9) This statement suggests that the sound "clouds" (i.e., masses) and the melodic presentations of the scales both may bc referred to as random walks. While it is indeed true that the individual elements in stochastic sound masses are connected by randomly occurring intervals in pitch and time (like the irregular jumps in a random walk), the distinguishing feature of the random walks that Xenakis introduced into his instrumental music in the 1970s is their perceptible continuity. It is in this specific sense that random walks are referred to in the present analysis, for those that appear in a r. are of the melodic variety only, the stochastic sound masses having been replaced in this work by simultaneities. (10)
The random walks in a r. are invariably presented in pairs, thereby establishing a two-voice texture as a norm within the work. Segment 1, shown in Example 1, contains the work's first pair of random walks. Both walks are rhythmically identical, consisting entirely of triplet thirty-second notes, but each one changes direction independently of the other. The walk in the lower staff changes pitch on every note, but the one in the upper staff passes initially from A4 to G#4, where it remains for much of the first beat of the measure. (11) Similarly, once the upper walk reaches G6 near the end of beat 2, it remains there for three triplet thirty-second notes. The presence of repeated pitches in these two places in the upper walk effectively changes the rate of speed at which it moves through its scale. (The pitch repetitions also prefigure the more extensive use of repeated articulations later on in the work, the structural importance of which will be discussed shortly.) Because of the pitch repetitions in the upper walk the two walks differ in the rate of speed at which they move through the pitch material, despite their identical surface rhythms. The walks are coordinated harmonically, however, since both of them move through the same scale. Another coordinating factor is dynamics, for the markings in the score indicate simultaneous changes in both walks. The first pair of walks thus demonstrates a balance between independent and related characteristics, a balance that is maintained within the other pairs of random walks in a r.
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Segment 2 introduces the work's first simultaneity, an eight-note structure on the first beat of measure 2. The juxtaposition of the random walks in segment i with the simultaneity in segment 2 initiates an alternation between contrasting textural types that continues through segment 10, as shown in the form chart in Example 2. Textures are represented in the center of the chart by a strip of black and white regions, the white regions symbolizing the random walks and the black ones the simultaneities, respectively. The accumulation of black regions in segments 10 through 13 represents a succession of simultaneities that breaks up the previous pattern of alternating textures. These simultaneities are represented in the chart individually because of the precedent established in segments 2, 4, 6, and 8 that single simultaneities may stand alone as discrete segments.
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The break in the pattern of alternating textural types that occurs following segment 10 is the basis for the work's first sectional division. This is shown in Example 2 where the first section, A1, includes segments 1 through 9 and the second section, S1, begins with segment 10. (The "A" in the section designation A1 signifies "alternating textures," while the "S" in S1 signifies "similar textures.") Although it may at first appear that segment 10 should be included in section A1 in order to complete the alternation of random walks and simultaneities that was established by segments 1 and 2, its placement at the beginning of S1 is intended to reflect the particular sense of cohesiveness resulting from the direct succession of similar textures in segments 10-3. While it is true that the break in the pattern of alternating textures does not become apparent until the arrival of segment 11, the force of the direct succession of two simultaneities in segments 10 and 11 effectively overwhelms whatever links segment 10 may have had to the random walks in segment 9. The impression of radical structural change is reinforced by the succession of two additional simultaneities in segments 12 and 13. The inclusion of segment 10 in this group is further supported by the pitch structure of the simultaneities. (See Example 3.) The exact repetition of the pitch contents of segment 10 in segment 12 effectively divides the succession of four simultaneities into pairs according to their pitch contents: the differing pitch contents of the odd-numbered segments contrast with the identical pitch contents of the even-numbered ones. In this way the alternation of contrasting structural elements persists into S1, but now it is temporarily transferred from the characteristic of texture to that of pitch.
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The identifies of the textures have been very clear up to this point in the work. With...
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