Most synthesis methods start with something periodic and then destroy it a little. A sawtooth wave with a filter and LFO. A sine cluster with amplitude envelopes. Even granular synthesis chops up recorded audio into controlled clouds.
Xenakis started from something different: probability. Not as a compositional metaphor, but as the literal mechanism that determines each audio sample. His Dynamic Stochastic Synthesis (DSS) generates waveforms by displacing a set of breakpoints each cycle using random walks drawn from probability distributions. No oscillator equation. No Fourier components. Just constrained randomness walking through amplitude-time space.
The result is a synthesis method that sits on a continuous spectrum between a pitched tone and broadband noise, controllable by adjusting how far the random walks are allowed to wander. It sounds unlike anything else, and it remains underused.
Stochastic Music Before Synthesis
To understand DSS you need to understand where Xenakis came from. Born May 29, 1922, in Braila, Romania, to Greek parents, he moved to Greece as a child and later fled to Paris after the Greek Civil War. He joined Le Corbusier's studio and worked as an architect for twelve years, collaborating on Sainte Marie de La Tourette and the Philips Pavilion at the 1958 Brussels World's Fair.
He was simultaneously composing. His 1955-56 work Metastasis applied probability theory to the mass behavior of orchestral lines, treating sound clouds statistically rather than melodically. The individual lines were less important than the aggregate density and direction of motion.
By 1957, he had formalized this into a computational procedure. Working with programmers M. F. Genuys and M. J. Barraud, he implemented the "Free Stochastic Music" program in FORTRAN on an IBM 7090. The program generated entire orchestral works parametrically: pitches drawn from a distribution, inter-event times from an exponential law, glissando speeds from a Gaussian law.
The most rigorous example was Achorripsis. Xenakis constructed a 7 by 28 matrix of 196 cells, mapping timbral classes against time units, and populated each cell with event counts drawn from a Poisson distribution with mean density lambda = 0.6. The number 0.6 was not arbitrary: at this density, cells average less than one event each, producing the sparse, isolated-event texture the work requires. These calculations appeared in his 1963 treatise Musiques formelles (published in English as Formalized Music in 1971).
All of this was composition-level stochasticism. The randomness controlled note parameters: when, at what pitch, how fast. The sound itself was still produced by traditional instruments.
Pushing Randomness Down to the Sample
Thirty years later, Xenakis asked the obvious follow-up question: what if you applied stochastic processes not to the score but to the waveform itself?
The answer was GENDYN, a program he wrote in BASIC with programming assistance, producing his final electronic works GENDY3 (1991) and S.709 (1992). GENDYN operates at the sample level, directly computing each audio sample as the output of a breakpoint interpolation process.
Here is the core algorithm:
A waveform consists of N breakpoints distributed across one period, each with an amplitude position and a time position. On every cycle, both positions of every breakpoint are perturbed by independent random values drawn from a chosen probability distribution, scaled by a perturbation parameter. The waveform for the next cycle is constructed by linear interpolation between the updated breakpoints. This is repeated for every subsequent cycle.
The period length is not fixed. Because the time positions of breakpoints change, the total cycle length changes, which means the fundamental frequency wanders. The "pitch" of the result is determined by minimum and maximum frequency constraints that bound the random walk.
Xenakis used second-order random walks rather than first-order. In a first-order walk, the position at time t+1 equals position at t plus a random step. In a second-order walk, the position at t+1 equals position at t plus a velocity, where the velocity itself is incremented by a random step. This produces smoother trajectories and richer timbral motion.
Boundary conditions use reflection rather than hard clipping. When a random walk would carry a breakpoint amplitude above the maximum or below the minimum, the excess is reflected back into range: a step of +0.3 beyond the ceiling returns 0.3 below it. This keeps the signal bounded without introducing the spectral discontinuities that hard limiting would create.
The Pitch-Noise Continuum
The key parameter in DSS is the perturbation scale: how large the random steps are relative to the available amplitude range. At small perturbation scales (close to 0.0), the breakpoints move slowly and the waveform changes gradually between cycles. The output sounds quasi-periodic, a buzzy, evolving tone. At large perturbation scales (close to 1.0), the breakpoints jump wildly each cycle and the output approaches broadband noise.
The fundamental creative insight is that this is not a binary. Any value between 0.0 and 1.0 produces a sound that is simultaneously pitched and noisy, with a character that is continuously adjustable. There is no clean separation between tone and texture: at mid-range perturbation values you get something that resembles a filtered noise source modulated by a distorted oscillator, but with an organic instability that neither can replicate on its own.
The number of breakpoints also shapes the result. A waveform with 4 breakpoints per cycle produces simple, angular shapes. Sixteen breakpoints allow more spectral complexity. More breakpoints also mean more independent random walks happening simultaneously, which increases the richness of the modulation.
Probability Distributions and Timbre
The choice of distribution determines the character of the steps. In Xenakis's implementation and in modern software recreations, the available distributions include:
LINEAR: uniform distribution. Steps are equally likely across the perturbation range. Produces relatively flat spectral results.
CAUCHY: heavy-tailed distribution. Most steps are small, but occasional large jumps occur. Produces more sudden discontinuities and a rougher timbre.
LOGISTIC: sigmoid-shaped probability, clustering steps near the center. Produces smoother, more continuous timbral change.
ARCSINE: probability concentrated near the extremes. Steps cluster at maximum and minimum perturbation. Produces highly nonlinear, discontinuous waveforms.
EXPONENTIAL: probability decays from zero. Steps are usually small and occasionally very large, with a different distribution shape than Cauchy.
Each distribution produces a distinct spectral character. The practical implication is that you can navigate timbral space not just by changing perturbation scale but by changing the statistical personality of how the waveform deforms.
Implementation in SuperCollider
Nick Collins implemented DSS as three UGens for SuperCollider in 2003. They are part of the core distribution.
Gendy1 uses first-order random walks. Gendy2 uses second-order walks (closer to Xenakis's GENDYN). Gendy3 adds a hard frequency constraint, allowing you to lock the fundamental to a specific pitch while the timbre still evolves stochastically.
// Second-order random walk, logistic distribution, 12 breakpoints
{
var sig = Gendy2.ar(
ampdist: 2, // 2 = LOGIST
durdist: 2,
adparam: 1.0, // distribution parameter
ddparam: 1.0,
minfreq: 150, // frequency wanders between these bounds
maxfreq: 350,
ampscale: 0.4, // small steps = pitched character
durscale: 0.4,
initCPs: 12,
knum: 12
);
(sig * 0.3).dup
}.play;
// Raise ampscale and durscale toward 1.0 to push toward noise
{
var sig = Gendy2.ar(
ampdist: 1, // 1 = CAUCHY: rougher, more discontinuous
durdist: 1,
adparam: 1.0,
ddparam: 1.0,
minfreq: 200,
maxfreq: 400,
ampscale: 0.85, // near noise
durscale: 0.85,
initCPs: 8,
knum: 8
);
(sig * 0.3).dup
}.play;Csound also ships a gendy opcode with equivalent functionality, documented in the Csound manual.
Why This Still Matters
DSS produces textures that oscillator-based synthesis cannot approximate. The tonal-noise continuum it operates on is distinct from filtered noise (which stays noise regardless of cutoff) or FM synthesis (which produces defined harmonic relationships). The random walks introduce a kind of deep structural instability: not a modulated signal but a signal whose fundamental shape is different every cycle.
For sound designers working on organic textures, this is directly useful. Wind, breath, bowed noise, electronic interference, and biological rhythms all share the quality of being almost-periodic. DSS generates exactly that, with the probability distribution acting as a precise dial between regularity and chaos.
The more interesting implication is compositional. Xenakis did not use randomness to add unpredictability to a structured work. He used it as the generative mechanism itself: structure emerging from the aggregate behavior of constrained random processes. That distinction is worth sitting with.