He was not. In 1955–56 he composed Pithoprakta — “actions through probability” — for 46 strings, each of which is, in his own account, a molecule of an ideal gas. This post works through the mathematics he used and asks what it means when a composer takes statistical mechanics seriously as a compositional tool.

The Problem with Post-War Serialism

To understand why Xenakis did what he did, it helps to know what everyone else was doing. By the early 1950s, the dominant tendency in European new music was total serialism: the systematic extension of Schoenberg’s twelve-tone technique to rhythm, dynamics, articulation, and register. Every parameter of every note was determined by a series. Messiaen had sketched this direction in Mode de valeurs et d’intensités (1949); Boulez and Stockhausen had taken it to its logical extreme.

The result, as Xenakis observed with characteristic bluntness in Formalized Music (1963/1992), was a kind of sonic indistinguishability: because every parameter varied according to independent deterministic series, the textures produced by total serialism sounded essentially like random noise. The maximum of local determinism had produced the appearance of global chaos.

His diagnosis was precise and, I think, correct: if the perceptual result of maximum determinism and maximum randomness is the same, then the path forward is not to find a better deterministic scheme but to embrace randomness explicitly, at the level that governs the macroscopic structure. Control the distribution; let the individual events vary within it. This is exactly what statistical mechanics does for a gas: it does not track every molecule, but it knows with great precision what the distribution of velocities will be.

Statistical Mechanics in Brief

In a classical ideal gas of $N$ molecules at thermal equilibrium with temperature $T$, the molecules move in all directions with speeds distributed according to the Maxwell-Boltzmann speed distribution:

where $m$ is the molecular mass and $k_B$ is Boltzmann’s constant. The parameter $a$ sets the characteristic speed scale: it grows with temperature (hotter gas means faster molecules) and shrinks with molecular mass (heavier molecules move more slowly at the same temperature).

The distribution has a characteristic shape: it rises as $v^2$ for small speeds (few molecules are nearly stationary), peaks at the most probable speed $v_p = a\sqrt{2}$, and falls off as $e^{-v^2/2a^2}$ for large speeds (very fast molecules are exponentially rare). The three characteristic speeds are:

No individual molecule is tracked. The distribution is everything: once you know $f(v)$, you know all macroscopic properties of the gas — pressure, mean kinetic energy, thermal conductivity — without knowing the trajectory of a single molecule. The individual is sacrificed to the ensemble.

Pithoprakta and the Orchestra as Gas

In Pithoprakta (1955–56), Xenakis assigns each of the 46 string instruments to a molecule of a gas. The musical analogue of molecular speed is the velocity of a glissando: the rate at which a glissando moves through pitch, measured in semitones per second. Slow glissandi are cold molecules; fast glissandi are hot ones.

For a given passage with a specified musical “temperature” (an intensity-and-density parameter he could set as a compositional choice), the 46 glissando speeds are drawn from the Maxwell-Boltzmann distribution for that temperature. No two strings play the same glissando at the same speed. The effect, to a listener, is a dense sound-mass — a shimmer or a roar — whose internal texture varies but whose overall character (the temperature, the density) is under the composer’s control at exactly the level that matters perceptually.

Xenakis worked out the velocities numerically by hand. The score of Pithoprakta was among the first in which the individual parts were derived from a statistical distribution rather than from a melody, a row, or an improvisation instruction. The calculation is tedious but not difficult: for each time window, choose a temperature, compute $f(v)$ for the 46 values of $v$ that tile the distribution, and assign one speed to each instrument.

The connection between macroscopic structure and microscopic liberty is deliberately preserved. The shape of the sound-mass — its brightness, its turbulence, its rate of change — is controlled. Each individual line is unpredictable. This is, structurally, the same trade-off that makes thermodynamics work: you give up on the individual trajectory and gain exact knowledge of the aggregate.

Musical Temperature as a Compositional Parameter

The analogy is worth making precise. In the physical gas, raising the temperature $T$ increases $a = \sqrt{k_B T / m}$, which shifts the peak of $f(v)$ to the right and widens the distribution. More molecules have high speeds; the variance of speeds increases.

In Pithoprakta, raising the musical “temperature” has the same effect: more instruments perform rapid glissandi; the pitch-space trajectories are more varied; the texture becomes more active and more turbulent. Lowering the temperature concentrates the glissando speeds near zero — slow motion, near-stasis, long sustained tones that change pitch only gradually. The orchestra cools.

This mapping is not metaphorical. Xenakis computed it. The score contains numerically derived glissando speeds; the connection between the perceptual temperature of the texture and the statistical parameter $T$ is quantitative. When musicians speak of a passage “heating up,” they are usually using a figure of speech. In Pithoprakta, they are describing a thermodynamic fact.

The Poisson Distribution and Event Density

Pithoprakta uses a second physical model alongside the Maxwell-Boltzmann distribution: the Poisson process, which governs the density of independent, randomly occurring events.

If musical events (pizzicato attacks, bow changes, individual note entries) occur at a mean rate of $\lambda$ events per second, the probability of exactly $k$ events occurring in a time window of length $T$ is:

The Poisson distribution has a single parameter $\lambda$ that controls both the mean and the variance (they are equal: $\langle N \rangle = \mathrm{Var}(N) = \lambda T$). A high $\lambda$ produces a dense cluster of events; a low $\lambda$ produces sparse, widely spaced events.

Xenakis used this to control the density of pizzicato attacks independently of the glissando texture. A passage can be cool (slow glissandi) and dense (many pizzicati), or hot and sparse, or any combination. The two distributions operate on independent musical parameters — pitch motion and event density — giving the composer a two-dimensional thermodynamic control space over the texture.

Markov Chains: Analogique A and Analogique B

In Analogique A (for string orchestra, 1958–59) and its companion Analogique B (for sinusoidal tones, same year), Xenakis moved to a different stochastic framework: Markov chains.

A Markov chain is a sequence of states where the probability of transitioning to the next state depends only on the current state. The chain is specified by a transition matrix $P$, where $P_{ij}$ is the probability of moving from state $i$ to state $j$:

Under mild conditions (irreducibility and aperiodicity), the chain converges to a unique stationary distribution $\pi$ satisfying:

The convergence is geometric: if $\lambda_2$ is the second-largest eigenvalue of $P$ in absolute value, then after $n$ steps the distribution $\pi^{(n)}$ satisfies $\|\pi^{(n)} - \pi\| \leq C |\lambda_2|^n$ for some constant $C$. The gap $1 - |\lambda_2|$ — the spectral gap — controls how quickly the chain forgets its initial state. A transition matrix with a large spectral gap produces rapid convergence; one with $|\lambda_2| \approx 1$ produces long-memory dependence between distant states. This is a compositional choice: the spectral gap determines how quickly a piece’s texture changes character.

In Analogique A, Xenakis divided the sonic space into a grid of cells defined by pitch register (high/middle/low), density (sparse/medium/dense), and dynamic (soft/loud). Each “screen” — a brief time window — occupies one cell in this grid. The progression of screens through the piece is governed by transition probabilities: from a high/dense/loud screen, there is some probability of moving to each adjacent cell, specified by Xenakis’s chosen transition matrix.

This is a Markov chain on a discrete state space of sonic textures. The macroscopic trajectory of the piece — its overall movement through sound- quality space — is determined by the transition matrix, which the composer sets. The details of each screen are filled in stochastically, within the parameters of the current state. Again, the individual is sacrificed to the aggregate; control is exercised at the level of the distribution rather than the event.

Game Theory: Duel and Stratégie

The most extreme and, to my mind, most interesting of Xenakis’s formalisations is the use of game theory in Duel (1959) and Stratégie (1962).

A two-player zero-sum game is specified by a payoff matrix $A \in \mathbb{R}^{m \times n}$. Player 1 (the “maximiser”) chooses a row $i$; Player 2 (the “minimiser”) chooses a column $j$; Player 1 receives payoff $A_{ij}$ and Player 2 receives $-A_{ij}$. In a pure-strategy game, each player selects a single action. In a mixed-strategy game, each player chooses a probability distribution over their actions: Player 1 uses $\mathbf{x} \in \Delta_m$ and Player 2 uses $\mathbf{y} \in \Delta_n$, where $\Delta_k$ denotes the standard $(k-1)$-simplex.

The expected payoff to Player 1 under mixed strategies is:

Von Neumann’s minimax theorem (1928) guarantees that:

where $v^*$ is the value of the game. The pair $(\mathbf{x}^*, \mathbf{y}^*)$ that achieves this saddle point is the Nash equilibrium: neither player can improve their expected payoff by unilaterally deviating from their equilibrium strategy.

In Stratégie, each conductor leads one orchestra. Each has nineteen “tactics” — six basic musical textures (e.g., sustained chords, staccato pizzicati, glissandi masses, silence) plus thirteen combinatorial tactics that combine two or three of the basics. The payoff matrix is a $19 \times 19$ integer matrix, also defined by Xenakis, specifying how many points Conductor 1 scores when their orchestra plays tactic $i$ against Conductor 2’s tactic $j$. A referee tracks the score.

The conductors make decisions in real time during the performance, choosing tactics based on what the other conductor is doing and on the evolving score. The piece ends when one conductor reaches a predetermined score threshold.

The Nash equilibrium of the payoff matrix tells each conductor, in principle, the optimal distribution over tactics to play: if both play optimally, the expected score trajectory is determined. In practice, conductors are not expected to compute mixed strategies on the podium; Xenakis’s point is structural. The game-theoretic formalism is used to design the payoff matrix so that no tactic dominates — every choice has consequences that depend on the opponent’s choice — guaranteeing that the piece will always contain genuine strategic tension regardless of who is conducting.

Duel (1959) is the earlier, simpler version for two chamber orchestras. Stratégie (1962) was premiered in April 1963 at the Venice Biennale with two conductors competing live. The audience was aware of the game, of the score, and of the payoff matrix. The premiere was by most accounts a success, though the practical complications of running a zero-sum game in a concert hall (including the question of whether conductors were actually computing Nash equilibria or just following intuition) were never fully resolved.

Formalized Music

Xenakis assembled his theoretical framework in Musiques formelles (1963), translated and expanded as Formalized Music (1971; revised edition 1992). The book is one of the strangest documents in twentieth-century music theory: part treatise, part manifesto, part mathematical appendix. It covers stochastic composition, Markov chains, game theory, set theory, group theory, and symbolic logic — all presented with the confidence of someone who is equally at home in the engineering faculty and the concert hall, and with the occasional obscurity of someone writing simultaneously for two audiences who share almost no vocabulary.

The core argument is that musical composition can and should be treated as the application of mathematical structures to sonic material, not because mathematics makes music “better” but because mathematical structures are the most powerful available tools for controlling relationships between sounds at multiple scales simultaneously. The statistical distributions control the macroscopic; the individual values vary within them. The game- theoretic payoff matrix controls the strategic interaction; the individual tactics fill in the details. Mathematics operates at the structural level and leaves the acoustic surface free.

This is a different relationship between mathematics and music from the ones in my earlier posts on group theory and Messiaen or the Euclidean algorithm and world rhythms. In those cases, mathematics describes structure that already exists in the music — structure the composers arrived at by ear. In Xenakis, mathematics is the generative tool: the score is derived from the calculation.

What the Analogy Does and Does Not Do

The Maxwell-Boltzmann analogy in Pithoprakta is exact in one direction and approximate in another.

It is exact in the following sense: the glissando speeds Xenakis computed for his 46 strings genuinely follow the Maxwell-Boltzmann distribution with the parameters he chose. The score is a realisation of that distribution. If you collect the glissando speeds from the score and plot their histogram, you will find the characteristic $v^2 e^{-v^2/2a^2}$ shape.

It is approximate — or rather, it is analogical — in the sense that strings in an orchestra are not molecules of a gas. They do not collide. They have mass and inertia in a physical sense that has no direct mapping to musical parameters. The temperature $T$ is not a temperature in any thermodynamic sense; it is a compositional variable that Xenakis chose to parameterise with the same symbol because the formal relationship is the same. The analogy is structural, not ontological.

This is worth saying plainly because it is easy to be misled in both directions: either to over-claim (the orchestra is a gas) or to dismiss (the orchestra is merely labelled with physical vocabulary). The actual claim is more modest and more interesting: the mathematical structure of the Maxwell-Boltzmann distribution is the right tool for specifying a certain kind of orchestral texture, namely one where individual elements vary stochastically around a controlled macroscopic envelope. The physics provides the formalism; the music provides the application. This is how mathematics works in engineering, too.

The Centenary and What Remains

Xenakis died in 2001, by then partially deaf and with dementia. His centenary in 2022 produced a wave of new performances, recordings, and scholarship — including the Meta-Xenakis volume (Open Book Publishers, 2022), which collects analyses of his compositional mathematics, his architectural work (he designed the Philips Pavilion for Le Corbusier’s Expo 58 in Brussels using the same ruled-surface geometry he was using in Metastaseis), and his political biography.

What remains resonant about his project is not the specific distributions he chose — the Maxwell-Boltzmann is not the only or even necessarily the best distribution for many musical applications — but the epistemological position it represents. Xenakis insisted that the right question to ask about a musical texture is not “what is the note at beat 3 of bar 47?” but “what is the distribution from which the events in this section are drawn?” This shift from individual determination to statistical control is precisely the shift that makes thermodynamics possible as a science, and Xenakis was the first composer to apply it deliberately and systematically.

When a composer writes “let the orchestra be a gas at temperature $T$” and then actually computes the consequences with Boltzmann’s constant in front of him, I do not feel that physics has been appropriated. I feel that it has been recognised — seen, from a different direction, as the same thing it always was: a set of tools for thinking about ensembles of interacting elements whose individual behaviour is too complex to track but whose collective behaviour is not.

The orchestra is not a gas. But the Maxwell-Boltzmann distribution describes it anyway.

References

• Ames, C. (1989). The Markov process as a compositional model: A survey and tutorial. Leonardo, 22(2), 175–187. https://doi.org/10.2307/1575226

• Jedrzejewski, F. (2006). Mathematical Theory of Music. Delatour France / IRCAM.

• Nash, J. F. (1950). Equilibrium points in $n$-person games. Proceedings of the National Academy of Sciences, 36(1), 48–49. https://doi.org/10.1073/pnas.36.1.48

• Nierhaus, G. (2009). Algorithmic Composition: Paradigms of Automated Music Generation. Springer.

• Matossian, N. (2005). Xenakis (revised ed.). Moufflon Publications.

• Solomos, M. (Ed.). (2022). Meta-Xenakis. Open Book Publishers. https://doi.org/10.11647/OBP.0313

• von Neumann, J. (1928). Zur Theorie der Gesellschaftsspiele. Mathematische Annalen, 100(1), 295–320. https://doi.org/10.1007/BF01448847

• von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.

• Xenakis, I. (1992). Formalized Music: Thought and Mathematics in Composition (revised ed.). Pendragon Press. (Originally published as Musiques formelles, La Revue Musicale, 1963.)

Changelog

• 2026-01-14: Corrected the description of Stratégie (1962): each conductor has nineteen tactics (six basic plus thirteen combinatorial), with a 19 x 19 payoff matrix — not six tactics and a 6 x 6 matrix. The six-tactic, 6 x 6 description applies to the earlier Duel (1959).
• 2026-01-14: Added “in April 1963” to the Stratégie premiere sentence. The composition date is 1962; the premiere took place on 25 April 1963 at the Venice Biennale.
• 2026-01-14: Changed “studying briefly with Honegger” to “approaching Honegger (who was too ill to teach).” Xenakis sought instruction from Honegger circa 1949, but Honegger was in declining health and did not take him as a student.