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Théorie des jeux

James Harley

In Chapter 4 of Formalized Music (1992), “Musical Strategy—Strategy, linear programming, and musical composition,” Xenakis explores musical applications of game theory. First developed by computer science pioneer John von Neumann (1903-57), game theory was presented in pure mathematics first, then applied to economic behavior theory, among other fields. In essence, game theory is “the study of mathematical models of strategic interactions among rational agents” (Wikipeda, “Game theory”).

Xenakis adopted the straightforward zero-sum game in which the gains or losses of each participant are exactly balanced by those of the other players. In his exegesis, he describes what he calls “heteronomous music” in contrast to “autonomous music,” the latter being traditional notated music, where any “game” or “conflict” are fully notated in the musical score (Xenakis 1992, 110-1). In heteronomous music, one could “introduce a concept of external conflict between, for instance, two opposing orchestras or instrumentalists. One party’s move would influence and condition that of the other” (111). Xenakis worked out the mathematical guidelines for two pieces, each for two orchestras and two conductors: Duel (1959), and Stratégie (1962).

In short, the fundamental interest […] lies in the mutual conditioning of the two parties, a condition which respects the greater diversity of the musical discourse and a certain liberty for the players, but which involves a strong influence by a single composer.

Xenakis 1992, 113

Interestingly, Xenakis saw musical precedence in musical competition:

in certain traditional folk music in Europe and other continents there exist competitive forms of music in which two instrumentalists strive to confound one another. One takes the initiative and attempts either rhythmically or melodically to uncouple their tandem arrangement, all the while remaining within the musical context of the tradition which permits this special kind of improvisation.

Xenakis 1992, 112

In Duel, the composer defined six types of sonic event: 1) a cluster of sonic grains such as pizzicati, blows with the wooden part of the bow, and very brief arco sounds distributed stochastically; 2) parallel sustained strings with fluctuations; 3) networks of intertwined string glissandi; 4) stochastic percussion sounds; 5) stochastic wind instrument sounds; 6) silence (113-4).

Each of these events is written in the score in a very precise manner and with sufficient length, so that at any moment, following his instantaneous choice, the conductor is able to cut out a slice without destroying the identity of the event. We therefore imply an overall homogeneity in the writing of each event, at the same time maintaining local fluctuations.

Xenakis 1992, 114

Xenakis then proceeded to create a list of possible combinations of the two orchestras, assigning an evaluation grade to each coupling, from “passable” to “good” to “very good,” with gradations within each of those. From there, he was able to produce a matrix for each conductor, containing a list of events, and the evaluation grade for each. This way, each conductor could make choices about what to do next based on this matrix in conjunction with what the other conductor is choosing to do at the same time.

In performance, the conductors are back-to-back, so they do not see signals that the other makes to their musicians. An automated light system could be used to track the choices and the scores, or a referee can track the game, and display the scores for the audience. The creative involvement of the audience would work better if there were multiple performances of the music in one presentation. Be that as it may, Duel was not performed until 1971, and has been rarely heard since. There does not seem to be a recording available. In the foreword to the published score, Xenakis took pains to separate value judgment from the winner-loser aspect of the game:

the losing conductor most absolutely not be considered less good than the winner… The winner has won simply because he has better followed the rules imposed by the composer, who, by consequence, claims all responsibility for the “beauty” or “ugliness” of the music.

Xenakis 1959

Stratégie, also for two orchestras and conductors, was premiered more quickly, at the Venice Biennale in 1963, then performed again in Paris in 1965. The basic six “tactics”—1) winds; 2) percussion; 3) strings’ sound-box struck with the hand; 4) string pointillistic effects; 5) string glissandi; 6) sustained string harmonics—are expanded with thirteen additional tactics, combinations of two to three of the basic tactics. Note that “silence” was dropped as a tactic in Stratégie; the possibility that the response of silence in orchestra B to silence in orchestra A (a possibility in Duel) must have been reconsidered. Xenakis generated all of the materials for Stratégie by computer using his ST algorithm in 1962. With a 19x19 matrix (the 6 basic tactics plus the 13 combinatorial tactics), a value for each cell in the grid was filled in, then a few simplified matrices were created (3x3) that group sets of tactics, the aim being to provide more accessible material for the conductors, who are making choices about what to do next on-the-fly.

In both of these pieces for two orchestras, the composed materials are the same for each ensemble, but those materials are heard in different orders, overlapping much of the time. The chance elements are very restricted, so both works will sound like Xenakis. The composer put a great deal of effort into thinking through how concepts and techniques adapted from game theory could be turned into music. The practical difficulties of presenting the music has dampened their impact, although the theory behind them has been influential for other composers, primarily in the domain of computer music, where the game theory elements can be integrated into the programming.

Around the same time as Duel was premiered in 1971, Xenakis composed a chamber work based on game theory, Linaïa-Agon, premiered in 1972 at the London Bach Festival. In this score, the game is set in the context of an ancient contest between Linos and Apollo. In the forward to the score, the stage is set: “according to the legend, Linos, the celebrated musician, provokes Apollo, who strikes him down. Here the legend is incarnated by a musical game between two adversaries—Linos = trombone, Apollo = French horn or tuba. Contrary to the legend, this game gives a chance to Linos to extricate himself. His actual chance is mathematically provided by decision matrices. “To throw the gauntlet to the gods is not blasphemy but is to surpass them by surpassing oneself” (Xenakis 1972). The score opens with a fully composed section, the “Commencement.” Then there follow three “Battles,” in any order, each guided by a 3×3 matrix conveying pitches, dynamics, and playing mode. Each section also has a choice of composed material the players choose to perform. The matrices are set for Linos (trombone) and Apollo (tuba), but the horn is otherwise treated as part of the Apollo side. These battles include “in between” passages, then a “Destiny Suspended” that leads to a “Victory Chant and Requiem” to finish. Because Linaïa-Agon is scored for solo players, there are more liberties built-in to the structure than the earlier orchestral works. By now, there have been many performances, and recordings to compare. One musician, Benny Sluchin, a trombonist who had performed this music many times, developed an interactive version of the game matrices guiding this work (Sluchin 2015).

References

Sluchin, Benny. 2015. Xenakis: Linaia-Agon (DVD), New York, Mode Records.

Wikipedia, The Free Encyclopedia s.v. “Game theory,” last modified 8 October 2023 15:59 UTC, https://en.wikipedia.org/wiki/Game_theory

Xenakis, Iannis. 1959. Duel, jeu pour 56 musiciens divisés en 2 orchestres avec 2 chefs, Édition Salabert, MC 554.

Xenakis, Iannis. 1972. Linaïa-Agon, jeu musical pour cor en fa, trombone ténor et tuba, Salabert, EAS 17055.

Xenakis, Iannis. 1992. Formalized Music, Hillsdale: Pendragon Press.

How to cite

HARLEY, James. 2023. “Game Theory.” In A Xenakis Dictionary, edited by Dimitris Exarchos. Association Les Amis de Xenakis. https://www.iannis-xenakis.org/en/game-theory