ECMST ~ MASA Empirics, Computation, Mathematics, Science and Technology in Music and Acoustical Signal Analysis. Staatliches Institut für Musikforschung PK Berlin.

Sonority and Proportionality in Xenakis’s Dikhthas

Dimitris ExarchosResearch Fellow, Staatliches Institut für Musikforschung, Berlin;

Contemporary Music Research Unit, Goldsmiths, University of London

IntroductionThe notion of randomness is frequently met in the music of Xenakis. As he would deny improvisation (see Varga, 1996: 55), randomness was for him related to the princi-ple of causality and determinism in the compositional procedure. ‘The fact that now we have discovered how to separate, for example, chance in different degrees, provides us with a valuable tool in order to organize and manipulate it [chance, randomness], if not to define it (see Xenakis, 1992: 4). The question of systematizing the properties of sound – after dividing the orchestra to the extreme – in order to create mass events was explored in his second major orchestral work Pithoprakta (1955-6), where he mainly used Maxwell and Boltzmann’s kinetic theory of gases (see Xenakis 1992: 15). In gases molecules move rapidly in all directions free from the influence of other molecules. According to the kinetic theory the in-ternal energy of a body is equal to the energy of the mo-tions and positions of the molecules that it consists of. Maxwell’s law of the distribution of velocities, based on the basic concepts of classical statistics, gives the dis-tribution of speed among the molecules in equilibrium state. Phenomena that provide evidence of molecular agitation are diffusion and Brownian movement.

Brownian MovementsBrownian movement has a central role in Dikhthas (for piano and violin, 1979). In physics this concept refers to the increasingly irregular (chaotic) motion of molecules when held in suspension in a liquid. The idea of using Brownian movement in composition coincides with Xenakis‘ first experiments of sound synthesis with com-puters at Indiana University in Bloomington (USA) in the late 1960s (see Solomos, 2001: 245). He rejected Fou-rier analysis that expresses sound as summation of si-nusoidal components of frequencies that are multiples of the fundamental frequency of the sound. Instead he proposed the inverse: ‘we wish to construct sounds with continuous variations that are not made out of unit ele-ments. This method would use stochastic variations of the sound pressure directly’ (Xenakis 1992: 246). By unit elements he means the sine functions that are piled up to compose the sound. He would substitute them by var-iations of the pressure curve directly, which is now our starting point. In order to do so, he used the metaphor of a particle moving capriciously according to Brownian movement (random walk) theory. However, at that time Xenakis did not compose any electroacoustic works us-ing this method of sound synthesis probably because the technologic means available to him were not powerful enough (see Solomos 2001: 246-7). What he did was to transfer this experience with electronic means to instru-mental composition. This was done by simply transcrib-ing graphs of microsound synthesis, on a Cartesian sys-tem of coordinates where the x-axis represents time and the y-axis pitches. The last step would only be to convert this into traditional notation. The first work he composed this way is Mikka (1971, for solo violin), which consists of continuous glissandi without vibrato – an austere meta-phor of a random walk.

ArborescencesFor a great part of the 1970s Xenakis would use mate-rial derived from Brownian movement, which eventually evolved and appeared in various forms in his works. He was interested in achieving continuity – continuous vari-ation and continuous transformation. In 1973 this prac-tice led the composer to start composing by using sev-eral Brownian-movement lines simultaneously, starting from a common source resulting to a tree-like expansion that he called ‘arborescence’. He firstly used this meth-od in Evryali, his second solo piano piece composed in 1973. The challenge here was ‘how to achieve continuity on an instrument which has an opposite nature’ (Varga 1996: 90). ‘We start out of a point in space. This can be pitch versus time space or any other. In order for it to ex-ist the point has continually to repeat itself. In this way a line is formed which can have any shape. Any point on the line can also reproduce itself and bring about an ar-borescence’ (Varga, 1996: 88). Xenakis was deeply inter-ested in physical forms. He considered the idea of shape essential in nature, logic and the arts. He would speak of several natural shapes, such as the veins and nerves

of the human organism, lightning or the tributaries of a river and would point out that what is lacking today is a theory of shapes and he would call for a science of gen-eral morphology, which would deal with the forms of all the scientific areas and the laws of their evolution and transformation.

AnalysisThe task of representing time in a literal way is an in-tricate one in Dikhthas. The composer indicates several metronome markings in the piece in a manner that it is hardly feasible to count in a single time unit. There are 13 tempi in the work and most of them are short resting points within the range of an accelerando or a rallentan-do. We have chosen to express the duration of sections in seconds and not only in measures. This inevitably has to be approximate (in accordance with Xenakis’s indication on the score that “the metronome markings are approxi-mate, indicating the minimum tempo”).

Xenakis conceived arborescences as shapes that he can manipulate and transform in various ways using altera-tions such as stretching, rotation, etc. Figure 1 shows the first arborescence of the work (where the y-axis is pitch and the x-axis is time). Another one, with the same con-tour and direction, is heard in mm. 15-23, so that the lat-ter can be thought as a stretching of the former.

The way Xenakis arranged these materials in Dikhthas reveals a close relationships to the proportions of the Golden Section and the Fibonacci sequence. The golden section refers to the division of a given line in such a way that the ratio of the larger segment to the smallest one is equal to the ratio of the whole segment to the larger one. The Fibonacci sequence is a sequence of numbers formed by adding successive members to find the next member: 0, 1, 1, 2, 3, 5, 8, 13,… The values of the ratios 1, 2/1, 3/2, 5/3, 8/5, 13/8,... approach the value of the Golden Section as the sequence proceeds.

The length of the two arborescences mentioned above result in the proportion 8/5, which are two successive numbers in the Fibonacci sequence. Consequently the ratio of their duration is strongly related to the Golden Section.

There are 5 sections in Dikhthas. Table 1 shows the dis-tribution of the sonorities in section 3 and how the in-teraction between them relates to the formal structure of the section. Apart from arborescences and Brownian movement, other types of sonorities appear, such as sc-ochastic (‘clouds’ of piano sounds), or a long held note. The total duration of the section is 234’’. In the table we see how three types of Brownian movement have been

distributed until m. 109 where all the three forms ap-pear superimposed. The third type is shown in Fig. 2. This Brownian movement has grace notes at the end of short and consequent glissandi, shown with the thin lines at every turn of the main glissando line. If we focus on the temporal proportions we can clearly see the relations be-tween the three sub-types which present interesting ra-tios. The first sub-type (mm. 71-80) lasts for 49’’ and leads to a subsection dominated by a long-held C#, which has duration 29’’; a third sub-section (mm. 85-95) lasts for 51’’. The fourth subsection (mm. 95-108) is divided in two parts of the third form of Brownian movement (Fig. 2) with duration 30’’ and 20’’ respectively. We can see that the quest for formal clarity has been attempted through subdivisions, based on numbers from the Fibonacci se-quence. In Fibonacci numbers – in approximation – this would result in the numerical expression 5, 3, 5, 5. Taking into account the further division of the subsection ex-pressed by the last number (5) this expression will be 5, 3, 5, 3, 2.

Table 2 represents the form of Dikhthas. In it we can see, in synoptic view, how the work is structured according to sections of certain sonorities. In terms of temporal pro-portions, section 3 with the Brownian movements is the most prominent one. We have already seen how signifi-cant the role of this sonority is to the work. This section is positioned not in the middle of the work, so as to re-sult in a symmetric division, but wee see that the same schema for the temporal proportions of the sonorities in the sections, also holds for the structure of the work as a whole, namely the golden ratio: the point of the golden section of the ‘central’ section 3 coincides with the one of the composition as a whole. This point is found at the point where the third sub-type of the Brownian move-ment (m. 98). Furthermore and as a logical consequence, the temporal proportion of the music before and after section 3 is the golden ratio as well (311/190). So, if the duration of the whole work is AB, the duration of section 3 CD and their common golden section point P, the tem-poral proportions in Dikhthas can be expressed by the following expression:

The question arises whether this arrangement was in-tended by the composer or not. In an interview said that such a result is not necessarily intentional: ‘Musicologists may analyse scores and come up with their conclusions – and they may be perfectly right – but their findings need not indicate anything conscious on my part’ (Varga 1996: 204).

ReferencesSolomos, Makis. 2001. “The Unity of Xenakis’s Instru-mental and Electroacoustic Music: The case of Brownian Movements”, Perspectives of New Music 39/1: 244-54.Varga, Balint Andras. 1996. Conversations with Iannis Xenakis (London: Faber and Faber).Xenakis, Iannis, 1992. Formalized Music: Thought and Mathematics in Composition (NY: Pendragon).

Fig. 1: Arborescences in mm.1-6 and 15-23 of Dikhthas

Figure 2. Brownian Movement in mm. 95-108.

Tab. 1: Chart of Selection 3

Tab. 2: From Chart of Dikhthas