##### Peter Hoffmann

This technique refers to a method of computer sound synthesis. The original conception dates back to 1962, when Xenakis was working with his ST algorithm, resulting in the stochastic compositions *ST/x* (see Stochastic Music). Stochastic synthesis rejects Fourier analysis or pure electronic sounds and favors applying stochastic processes directly to the sound pressure curve, in order to control the level of disorder (see Luque 2009). While the first experiments in stochastic synthesis were used in *Polytope de Cluny* (1972), in 1977 Xenakis developed this idea and introduced *dynamic stochastic synthesis*, used in *La Légende d’Eer*. In the late 1980s Xenakis constructed the GENDYN algorithm (*génération dynamique stochastique*). In the narrow sense, this is an algorithmic procedure to generate musical sound (and if music is organized sound, as Edgard Varèse once put it, to generate music). Stochastic synthesis is an extension of stochastic composition from the domain of the musical score (to be performed by instrumentalists) to that of digital sound (to be performed by a digital-to-analog-converter). In other words, while stochastic composition calculates notes (whose musical parameters are, among others, temporal onset and duration), stochastic synthesis calculates digital samples. A musical score of stochastic composition does not necessarily need to be generated by a digital computer, but stochastic synthesis is dependent on computer execution, if only for the sheer amount of sample data (44100 samples per second per track).

In the case of Xenakis’s reference works, *Gendy 3* (1991) and *S.709* (1994), stochastic synthesis is a combination of sound synthesis as such (calculating digital samples) and aspects of the stochastic composition procedure that Xenakis used for instruments before, i.e. calculating an “architecture” of sound patches (onsets and durations) in multiple layers of sound. In other words, the algorithm generating GENDYN’s multitrack sound patch architecture is the same that generated the architecture of Xenakis’s stochastic compositions *ST/x* in 1962: one stochastic law (density formula) is used for the partitioning of a track into patches of duration and another (coin-tossing) to decide for each one of these patches if they will sound or not.

Stochastic synthesis conceives the sound waveform (the succession of digital samples) as a polygonal line (breakpoints linked by linear interpolation). Each of these breakpoints performs a *random walk *simultaneously in time (the spacing between successive breakpoints ranging from zero to hundreds of interpolated sample ticks) and in amplitude (between -32767 to +32767 when using 16-bit quantization). See video example below.

A random walk is a mathematical concept of a stochastic process also known as the walking path of a “drunken sailor”. In stochastic synthesis, random walks are one-dimensional (only back- and forward); however, applying these to both time and amplitude, effectively results in a two-dimensional random walk of a breakpoint in time-amplitude space. An integral part of the mathematical concept of a random walk is the notion of a (reflecting or absorbing) barrier which confines the space of the walk (i.e. the drunken sailor being locked up in a sobering cell). In stochastic synthesis, only reflecting barriers are used (the drunken sailor bounces off the walls of his sobering cell instead of sticking at them).

The numbers of breakpoints per waveform is a parameter of stochastic synthesis to be chosen by the composer: the greater the number of breakpoints, the lower the fundamental frequency of the resultant sound, since each additional breakpoint makes the waveform polygon longer. Sound generation first starts with a “flattened” waveform polygon whose spacings and amplitude values are all zero (i.e. perfect silence; it could of course be of any shape). The breakpoint positions of the succeeding waveforms are then derived by means of the aforementioned pair of random walks on each of the breakpoints. Therefore, for a waveform that consists of *n* breakpoints there are 2×*n* random walks acting in parallel for its deformation over time. Since the movements of the breakpoints are independent, the entire initial shape of the waveform is quickly lost over time. But GENDYN’s sound characteristics are not governed by waveform shape *per se*, but by the dynamics of that shape’s stochastic change.

In *S.709*, the random walk steps for the waveform’s breakpoints are random numbers distributed according to a set of predefined stochastic functions. In *Gendy3*, the same random numbers govern not the breakpoint random walks directly, but another intermediary set of random walks whose positions are then taken as the steps of those random walks that govern the breakpoint positions. In other words, in *Gendy3*, for each breakpoint, the random numbers drive a pair of primary random walks which in turn drive a pair of secondary random walks whose current position is taken as the sample spacing and amplitude of the corresponding breakpoint. This second-order random walk feature is the reason for the relative stability of *Gendy3*’s sound-world (in comparison to *S.709*), acting as attractors of maxima and minima of sample spacing and amplitude value. Another reason for the stability of *Gendy3*’s sounds are the barriers that are set by Xenakis in a way so as to reduce the “freedom” of the random walk, sometimes even to zero, resulting in frozen pitch or frozen spectra. In general though, the random walks are free to move and create various phenomena of (stochastic) frequency modulation, from slow glissando movement to buzzing or even noisy sounds, as well as (stochastic) amplitude modulation, with transient spectra similar to flanging or phasing effects, although not periodic but random.

When observing stochastic synthesis in action, the choice of specific random distribution functions which were so dear to Xenakis (such as Cauchy, Logistic, Hypercosine, Arcsine, or uniform distribution) proves to be of limited effect onto the sonic features of the resulting sound. It is true that random numbers wildly distributed according to the Logistic function account for rougher timbre and more nervous buzzing, in contrast to the Cauchy distribution, whose sparse distribution allows for delicate pitch movement and slow spectra transients. However, the position of random walk barriers is much more important, as is the chosen number of breakpoints per waveform. The various combinations of breakpoints and random walk barrier positions open up a composition space for stochastic synthesis within which a composer can navigate. This brings up Xenakis’s own picture of a composition space vessel:

With the aid of electronic computers the composer becomes a sort of pilot: he presses the buttons, introduces coordinates, and supervises the controls of a cosmic vessel sailing in the space of sound, across sonic constellations and galaxies that he could formerly glimpse only as a distant dream.

Xenakis 1971, 144

It is striking how far Xenakis stretched out within that space, given the limited means he had at his disposal (blind flight, as it were, in a lame vessel). To an analyst like the author of these lines, the composition space would later be found to be governed by combinatorial laws (see Hoffmann 2009). It is almost certain that these were unbeknownst to Xenakis. Therefore, his masterpiece *Gendy3* must be considered as the fruit of the intuition of a genius.

After Xenakis, many composers have elaborated upon stochastic synthesis and used it in numerous compositions. There is a stochastic oscillator in the widespread CSound library and other music computing languages, and stochastic oscillators have even been built into synthesizers. However, only few composers have followed the pure approach of Xenakis to create an entire composition “out of nothing”, direct-to-disk, by a single algorithm and with no post-processing.

#### References

Luque, S. 2009. “The Stochastic Synthesis of Iannis Xenakis.” *Leonardo Music Journal *19: 77–84.

Hoffmann, P. 2009. “Music Out of Nothing? The Dynamic Stochastic Synthesis: a Rigorous Approach to Algorithmic Composition by Iannis Xenakis.” Doctoral dissertation, Technische Universität Berlin.

Xenakis, Iannis. 1971. *Formalized Music: Thought and Mathematics in Composition*, translated by Christopher Butchers, G. W. Hopkins, and Mr. and Mrs. John Challifour. Bloomington and London: Indiana University Press.