Analysis of S.709 by Iannis Xenakis
Xenakis was particularly remembered as a pioneer in electronic and computer music, and for the stochastic mathematical theories in his compositions, including probability, aleatory, Markov chains, Brownian motion. He characterized his music as “a form of composition which is not the object in itself, but an idea in itself, that is to say, the beginnings of a family of compositions.”1 His intention of the use of stochastic methods was that, the things that are happening according to determinate rules can be changed without changing the overall meaning. The use of mathematical processes in the music of Xenakis creates the most radical forms and structures, which mostly lefts the listener not to be aware of. Mathematical formalization in music was not a novelty at the time of Xenakis, but unlike serialists who were using simple mathematical operations to compose, or John Cage who introduced the probability in composition but without strong mathematical base (the IChing probabilistic methods), Xenakis was aware of the indeterminism, strongly covered by the mathematical probabilistic theories. He was also aided by the psychoacoustics, which is another element of which the serialists or John Cage were not so aware of. Xenakis also differed from John Cage or the serialists from the fact that he used the mathematics to formalize the everyday sound phenomena as the rain or the tumult of the protesting riots. Xenakis' electroacoustic works are quantitatively inferior to the instrumental works, but they leave a revolutionary quality, both technically and musically. The early experimentations with musique concrète in the 1950s, resulted with the creation of pieces like Diamorphoses, Concret PH, Analogique B, Orient-Occident and Bohor and their realization in GRM (from 1951-1958 called Groupe de recherches de musique concrète-GRMC, lately renamed Groupe de Recherches Musicales-GRM), which as a concept will last until the development of the UPIC and GENDYN. However, in 1972 he created computer-generated sounds in a multimedia piece called Polytope de Cluny. It is in the late 1970s that Xenakis initiated to experiment with the synthetically created sound. In the 1978, together with the engineers in the CEMAMu (Centre d’Études de mathématique et automatique musicales), he realized a computer music system that enabled sound synthesis, using a graphic interface, called UPIC (Unité Polygogique Informatique de CEMAMu). With this system, one was able to draw the elements of the sound; the waveform, the dynamic envelope and the notes2. He composed 5 pieces using this system. This system derived from the idea of sketching the music. This interest derived from the multimedia experiences, and also from the fact that he was always opposing the Fourier Analysis as the basis of sound synthesis3. The sketching was the chance to experiment freely with the sound.
GENDYN program and non-standard synthesis method In the same period of the 1970s, Xenakis introduced a complete new synthesis technique. It was considering the discrete time-domain presentation of the waveform (time vs. amplitude). Xenakis developed it for the implementation of the stochastic functions in the digital sound, called nonstandard sound synthesis. This method creates an unpredictable timbre results, which was only supporting the Xenakis' idea that learning the sound, could not be necessarily related to the traditional timbre knowledge and instrumentation.
1 Xenakis, I. 1962. Formalized Music. Thought and Mathematics in Composition. Indiana University Press 2 This was a consequence of the projects he was doing in that period. Xenakis in the 1970s was very involved in the multimedia. 3 Harley, J. 2002. “The Electroacoustic Music of Iannis Xenakis” Computer Music Journal 26(1)
In electroacoustic music, the stochastic data was more related to the organization of the sounds, both on a macro and micro level, but never directly to the technical properties of the digital sound. In the late 1950s and early 1960s he speculated that his stochastic algorithm could create new sonic waveforms. Although, he continued to research and to experiment on this field, he finally got the opportunity to develop it seriously, when the microcomputers of Hewlett-Packard appeared. Even having as an idea from his beginnings in composing, he could not realize it, considering the technical disabilities. The last three electroacoustic pieces Gendy301, Gendy3 and S.709 were considered as the last phase of his electronic/electroacoustic creation. The stochastic synthesis consists of generating waveforms that could vary continuously according to pre-formalized stochastic function. Instead of 'curving' the waveform, he interpolated the breakpoints (samples) in a linear way (fig. 1). The horizontal and vertical proceeding of the points in the successive cycles is calculated on the basis of a probability formula, causing kind of stochastic amplitude modulation (vertically) and frequency modulation (horizontally). To control the timbres, Xenakis had to determine the range of the variation of points of a cycle, so that more radical is the variation, noisier is the timbre, and vice versa. The audible stability of the stochastic process is relative.
Figure 1. Linearly-segmented wave-form In Concret PH, Xenakis showed the interest of the idea of the grains (since he was considered as one of the first composers to see the potential of granular synthesis), as the essence of the sound. Leaded by Gabor's theories, Xenakis was the first to experiment musically with it (more concentrated in the instrumental music4). The invention of the “dynamic stochastic synthesis” is the result of this interest, which in this case (technically) is the sample. Xenakis was always pushing the idea of “Automated Art”, where the algorithmic procedure is used for the stochastic generation without having a score. By default, the probabilistic methods are generating rich variety of musical structures, which at the same time, are controlled by an algorithm used. Instrumental works, composed stochastically by a computer, were already composed by Xenakis in the 1960s (ex. Achoriripsis). However, according to Xenakis, the computer was involved only in a few steps of composing5. He considered eight phases of a musical creation: 1. Initial conceptions (intuitions, provisional or definitive data) 2. Definitions of the sonic entities 3. Definition of the transformations (setting up of relation between sonic entities and (re)ordering them in time) 4. Micro-composition (examining the sonic entities in details)
4 Xenakis, I. 1962. Formalized Music. Thought and Mathematics in Composition. Indiana University Press 5 Xenakis, I. 1962. Formalized Music. Thought and Mathematics in Composition. (p.22) Indiana University Press
5. Sequential programming of 3. and 4. (the macro-schema) 6. Implementation of the calculations (verification of the mathematical formulas) 7. Final symbolic result (transcribing the computer data into a traditional notation for an instrumental music) 8. Sonic realization of the program (orchestral performing, manipulation of the type of electroacoustic music, computerized construction and transformation of the sonic entities) This schema could undergo variations. Xenakis used the computer only for the phases 6., 7. and eventually 8.. In 1991 Xenakis completed a GENDYN computer program that introduces a stochastic algorithm in dynamic stochastic synthesis. The synthesis was initialized at the CEMAMu in Paris. Gendy301 and Gendy3, even if both are related on the basis of their principle, they sound different. Gendy301 was lately canceled from his official work-catalog, so two products remained: Gendy3 and S.709. Xenakis did not continue to compose “a family” of “Gendy” works, as he did in the 1960s with the “ST” compositions. The term “non-standard synthesis” is referred to the fact that it is not related to any of the known physical behaviors of the acoustical sounds, and is formalized in more abstract way. However, dynamic stochastic synthesis is describing the method more concretely, meaning that the values for the breakpoints of the waveform are randomly generated elements (created by the strict mathematical processes), and it is “dynamic” because it is continuously varying (fig. 2)6. Three years after Gendy3, in 1994, Xenakis composed S.709 using the same synthesis method. Compared to the previous two, this was more oriented to the creation of the unstable sonorities. For a creation of music using this method, we could distinguish 5 phases of elaboration starting from the piece as a whole, and going into a detailed level or in this case the main and the most elaborate phase, the sample. Hierarchically, the structural “zoom” seems like this: 1. Musical Piece, 2. Track, 3. Wave, 4. Segment, 5. Sample. The waveform is divided into segments which are represented by the succession of their sample values. Linearly interpolated breakpoints is raising the question of spectrally rich sound. Thus, the aleatory variations should be kept in a certain range, otherwise the saturation will be produced.
6
Methods of non-standard synthesis were also invented by Gottfried Koenig, Herbert Brün, Agostino di Scipio and others.
Figure 2. Time vs. amplitude breakpoint variations GENDYN is controlling the following parameters: 1. number of segments for the waveform 2. duration of the synthesis process 3. type of stochastic functions for axis-x and abscissa-y (Xenakis used various probability density functions) 4. the range ('elastic barriers') in which the variations are obtained The only input data is the choice of the probabilistic functions used to generate random numbers, and their distribution is formalized using the GENDYN algorithm and then applied to the coordinates, sonification of the abstract algorithmic action. This results with a sound, produced completely out of the already known synthesis methods, and at the same time aesthetically very rudimentary.
S.709 S.709 was the last electroacoustic piece written by Xenakis7. With the invention of the dynamic stochastic synthesis and the application in this piece, at the same time Xenakis returned us to the roots of the electroacoustic music. With its fairly simple structure and form and the lack of features like the spatialization, S.709 is a study of the microstrucure and an example of a musical creation from the micro properties of the sound. The complexity of the piece is concentrated on the microlevel, i.e. on the waveform. The piece gave radically new sonoristic results. It was important that GENDYN is a system that accomplished the principle idea of Xenakis to create music out of nothing. This means that the composer is not precisely conscious about the final
7 In 1997 wrote a piece called Erod commissioned from the Bath Festival in England. He re-used the UPIC system for this composition (even if he was occupied with the stochastic synthesis program), thus using sounds derived from his earlier recordings.
result, which was also one of Xenakis' “anti-ego” ideals8. However, the expressivity in the music is determined by the indeterministic calculations. If we try to confront the way how the stochastic methods were implemented in the instrumental compositions and here, we will find them very related to each other. In order to formalize the disorder, three elements are of crucial importance in all Xenakis' musical creation: 1. the determination of the mean value of the stochastic laws; 2. its deviation, that is shaping the music; 3. and the “invention” of the elastic barriers. However, this piece differs from the rest of the music by Xenakis. Taking a waveform and using its apparently simple one-dimensional structure (modulating the breakpoints of the waveform in two directions) is enough powerful idea for a musical creation. The first artistic action by the composer is to choose the musically interested and their later formalization that creates the form. The macro-form of the piece is achieved by mixing the created sequences of sounds. Mentioned before, in the music of Xenakis, the form is excluding all previous models of formal development. In this case, it seems like a mixture of different blocks of music. The division of the blocks is logically made by their diversity. Even if the stochastic synthesis resulted with very alienated timbres, the “timbre opus” is quite small. Confronting the two GENDYN pieces, they are complementary one to another. While, Gendy3 is tending to stability, S.709 is examining the unstable sonoristic properties of the stochastic synthesis (fig. 3 and 4).
Figure 3. The pitched stability in Gendy3
Figure 4. Randomized character of S.709
Even of the formal vagueness, I tried (intuitively) to categorize 3 sections. In Gendy3, the formal partition is far more obvious and technically related. There, it almost seems like a technical exposition of various mathematically-formalized audible stabilities. Here instead, we have shorter, and blender sequences, which are the permutations of the same sound-material. I considered them in a more traditional way. So that, the first part is an exposition of all sound-events, thus varied and layered in a various ways and is streaming from 0'00” to 2'20”. The exposition is divided in 4 subparts, which gives an impression of 4 iterations. The second is partly contrasting the first. It achieves more stable character with the wider use of the pitched sounds (2'20”-6'10”). With its beginning, it is suddenly stabilizing with the low pitched sounds, that follows with augmenting the tension and going forward to the third and last section. The attitude of 5 different sub-parts could be recognized. From 2'20” to 3'08” a clear use of the low-pitched sounds. The next sub-part (3'08”3'24”) consists of two wide glissando-sounds, which structurally are very related to the previous sounds. In the next two sub-sections (3'25”-4'07” and 4'07”-4'52”) we have a re-development of the sound-material characteristic for the second part. With the last sub-section (4'52”-6'12”) the tension is increasing, forced also with the appearance of the rhythmic sound. The third part is reassuming 8 Pape, G. 2002. “Iannis Xenakis and the 'Real' of Musical Composition” Computer Music Journal vol.26(1)
the properties of the first part, finishing with an interrupted end (6'10”-7'00”) (fig. 5)
Fig. 5. The macro divisions and sub-divisions in S.709 Compared to Gendy3, S.709 has more homogeneous form and gives an impression of continuity and infinite interplay between the sound-figures or the spectro-morphologies. It is intriguing the fact that Xenakis did not develop the formal structure of this piece, as much as for the main microidea. Nonetheless, the complexity on micro-level is complementing the “relative” simplicity on macro-level. To examine this argument, we should also take a look of how the GENDYN functioned and to see its technical limits. In the original GENDYN program, the composer had on disposal a separate tool which served as sequencer. It was kind of “patchwork” tool that offered the user to switch on and off up to 16 tracks of streaming synthesis sound material. Maybe it was the technical difficulty to compute complex synthesis data with the macro-structure. The “collage” technique is one of the characteristics of Xenakis' electroacoustic music. In S.709, we almost could say that the whole piece is constructed as one sequence or one sound streaming. The “sequence” formalization is hardly implementable here (aware of the important role of the term “sequence” in Gendy3). Examining the S.709 gives an impression of “discrete” continuity, both on macro as well as micro-level. However, we could distinguish timbres of almost stable character (almost pitched) and those noisier, which here are even enhancing the instability, with the rhythmical character. Considering the fact that for this kind of synthesis there is no score input, means that the only limit are the elastic barriers, that are restricting a “bandwidth” of the formalized random walks. The elastic barriers were applied to the following elements: 1. determines the margins for variations of the abscissa x 2. determines the margins for variations of the ordinate y 3. the range of samples per segment 4. the amplitude range (from -32768 to +32768, since the signal is presented by 2's complement 16-bit integers) The possibility to vary the x and y was accomplished by the choice among various probabilistic density functions (exponential, Cauchy, Lehmer, logistic).
Figure 6. Discreteness on micro-level
Sound-events in S.709 The sound events are actually forming the composition. The pitch behavior was one of the tools to distinguish the sound-events. Obviously, we have two sound structures interfering with each other. In the both cases, there are both methods implemented, but it is the range of the elastic barriers that determine the width of the modulation. However, I distinguished 4 types of sound-events, concerning their timbral diversity, but grouped in 2 categories. I
Timbre modulation
1. Rhythmical sequence (sound example 1) II Pitch modulation 2. Glissando sounds (sound example 2) 3. Scales – ascending, discending or in both manners (sound example 3) 4. “Ground-sounds”9 (sound example 4) The selected timbres were managed as patterns (or configurations). The first rhythmical sound is a result of greater modulation of the ordinate. We could describe this as:
yi,j+1 = yi,j + fy(z) (variation in amplitude) xi,j+1 = xi,j+ fx(z) (variation in frequency) where i is the segment, j is the waveform and z is the value determined by the stochastic laws. It is the first great sound characteristic of the piece. The rhythm of sound is caused by the modulation of the ordinate y. And the pitch ascendants are obtained by a rapid compression of the period. But, greater is the band of the elastic barriers, bigger is the timbrical change. Listening to the piece, the iteration of the same sound-events is obvious. If we try to confront to each other, we could see that they are fairly equal, but not exactly. It seems that it is the question of editing that Xenakis chose to select different parts of the pre-recorded sound data. Or, as we said before, it is only the question of switching on and off the streaming tracks in the GENDYN program. The glissando in this 9 I called them “ground” because of the “brace” impression of the sound polyphony.
morphology is having a range from cc. 1000 Hz to cc. 3000 Hz, which means that the elastic barriers are shifting from cc. 40 samples per period to cc. 15. It is obtained by compressing the period, which is responsible for the fundamental frequency determination in the piece. The fundamental frequency, thus in the entire piece is not surpassing the 5000 Hz, resulting with the minimal number of 8 samples per period. The second type of categorized sounds (the glissando) is conceptually very common and very important from the earliest composing period of Xenakis. However, it has not much to do with his earlier conceptions of glissando (ex. like in Metastasis). Here we have a sound-product of the algorithm. If the rate of the FM is not too fast, it results with perceptive scales. Though they are not continuous. The values that create the scales are triggered by less-randomized variations withing the “elastic barriers”. The third type are the micro-tonal scales and melodies, which derive from deviation of the mean value. Their dynamic depends on the probabilistic functions used, that could give a variation from stable pitch to more vast frequency deviation. The stability is rare, but is present. It is determined by the “attractors” of the probabilistic functions. Technically, the “attractors” are the result of a particular combination (caused stochastically) of the synthesis parameters, such as small variance of the distribution functions. However, the discreteness of the note-jumps in the scales (considering even those of the glissando) are the result of the characteristic of the distribution function used (for ex.: the steepness of the Cauchy, Gaussian or Laplace distribution or the smoothness of the Lévy or Hyperbolic distribution), so that steeper is the curve of the distribution, more discontinued is the note jump10 (fig.6). In fourth category I set the sounds that I called “ground sounds”. As mentioned before, the decision to divide the sound-objects in two groups was brought because all the sounds from the second group have harmonic spectrum, no matter of their sonoristic role in the piece. From 2'18” to 3'24” there is a section where there is an increased presence of this sound-event. The timbre is the “trademark” of both GENDYN compositions. With all due respect of the invention of the dynamic stochastic synthesis and its musical application, it seems that the harmonic timbre is almost inevitable (fig. 7). It is raised from the fact that, the “steeper” waveform is a result of a spectrally harmonic sound. Considering as a lack or not, the phenomena of harmonic timbre will stay present as long as the interpolation of breakpoints in the waveform is linear. Just to compare, this kind of interpolation reminds of a waveform analogue to the sawtooth waveform.
Fig. 7. One of the pitched sounds in S.709, confirming the harmonic spectrum (55.5”-56.5”)
10
Hoffmann, Peter. 2001.“Analysis through Resynthesis. Gendy3 by Iannis Xenakis” Presences of Iannis Xenakis Paris: CDMC: 185-194
Conclusion S.709 is not reminding of the previous compositions of Xenakis. Even if this was the last electroacoustic piece by Xenakis, it gives an impression of very youth-like work. The musical “shock” is the same as listening to Concrete PH in the late 1950s. In fact, the structural homogeneity of S.709 reminds of Concrete PH. The sudden end of the composition is philosophically related. On one hand, we have again the “anti-ego” ideal by Xenakis, confirming that “grande finale” is not needed for a conclusion of one piece11; on the other hand we could relate it to the indeterminate “flowing” of the generated sound material and not having time to produce also a complex macro-structure (concerning his age and his healthy conditions in the 1990s). This piece is rare example of pure “computable music”, without post-processing elaborations (except for the mapping of the events). It seems that Xenakis achieved the idea he had since the beginnings of his compositions; to control the physical properties of the sound, stochastically. In 1995-96, he visited CEMAMu and developed a program of almost real-time Stochastic Synthesis12. It did not function well, because of the great latency for the computation of the sound. Even if he stopped composing electroacoustic music after this work, he left a powerful synthesis technique to be explored in the future.
11 Pape, G. 2002. “Iannis Xenakis and the 'Real' of Musical Composition” Computer Music Journal vol.26(1) 12 Hoffmann, P. 2001. “Analysis through Resynthesis. Gendy3 by Iannis Xenakis” Presences of Iannis Xenakis
References and bibliography Di Scipio, Agostino. 2001. “Clarification on Xenakis: The Cybernetics of Stochastic Music” Presences of Iannis Xenakis. Paris: CDMC: 71-84. Harley, James. 2001. “The Electroacoustic Music of Iannis Xenakis” Presences of Iannis Xenakis Paris: CDMC: 33-57 Hoffmann, Peter. 2001.“Analysis through Resynthesis. Gendy3 by Iannis Xenakis” Presences of Iannis Xenakis Paris: CDMC: 185-194 Hoffmann, Peter. 1996. Implementing the Dynamic Stochastic Synthesis Pape, Gerard. 2002. “Iannis Xenakis and the 'Real' of Musical Composition” Computer Music Journal vol.26(1) Xenakis, Iannis. 1962. Formalized Music. Thought and Mathematics in Composition. Indiana University Press Xenakis, Iannis. 1963. Musique Formelles. La Revue Musicale
Ivan Penov 2008
