##### Dimitris Exarchos & Peter Hoffmann

*Metamusic*, a term coined by Iannis Xenakis, is the idea of a music of a higher order which would contain any existing music of any tradition as a special case. In other words, Xenakis’s postulated metamusic is universal in the sense that it encompasses all musics of all time and all places in the world. In order to construct such a music, Xenakis tried to find fundamental structures that would allow him to “compose something which could include any form of expression” (Varga 1996, 50). Technically, metamusic involved a set of methodologies that would help lay down the foundations of music composition. This set could include virtually any basic musical concept, such as scale, harmony, rhythm, etc., and would thus constitute a metalanguage, that is, a technical (second-order) language that would describe the natural (first-order) language of music. The term first appeared in “Towards a Metamusic” (Xenakis 1992, 180–200); the essay offers no definition, however a related such concept is what Xenakis speculatively called “general harmony” (Xenakis 1992, 182): a theory, or a set thereof, that would subsume existing theories of harmony, scales, etc.

Although the first mention of a “metamusic” appeared in the mid-60’s, Xenakis’s earliest allusion to such concept dates back in 1960, in the first text of *Musiques formelles*, “Musiques Stochastique (générales, libres)” (Xenakis 1963, 11-56; reprinted as “Free Stochastic Music” in Xenakis 1992, 1-42). The opening lines of this texts read much like a manifesto:

Art, and above all, music has a fundamental function, which is to catalyze the sublimation that it can bring about through all means of expression. It must aim through fixations which are landmarks to draw towards a total exaltation in which the individual mingles, losing his consciousness in a truth immediate, rare, enormous, and perfect. If a work of art succeeds in this undertaking even for a single moment, it attains its goal. This tremendous truth is not made of objects, emotions, or sensations; it is beyond these, as Beethoven’s Seventh Symphony is beyond music. This is why art can lead to realms that religion still occupies for some people.

Xenakis 1992, 1

Such “transmutation of every-day artistic material”, is what Xenakis calls “meta-art” (1992, 1). As often, Xenakis’s vision is also critical of his contemporaneous artists, who fail to account for the universal “substratum” that can enable “a formalization which will unify the ancient past, the present, and the future”, exceeding localities of time and space, reaching a “planetary scale” (Xenakis 1992, 182).

In “Towards a Metamusic”, Xenakis offers two possible instances of such a metamusic: his idea of a *stochastic music* on the one hand and his *sieve theory* on the other. As for the first, it is easy to see, at least theoretically, that composing with sound clouds, as is the case in stochastic music, could statistically produce serial music or any other way of voice leading as an extremely rare but possible case. The sounds of the stochastically generated cloud just would have to fall by chance into positions corresponding to the rules or habits of that music. (Think of the popular metaphor of infinitely many chimpanzees hacking away on a typewriter; one of them is ultimately to type Shakespeare’s collected works.) Clearly, Xenakis was not interested in creating, by chance, instances of any existing music. Instead, his interest was to create, with the help of stochastics, a music that would transcend any known musical style by abstraction of its fundamental structure which is in this case the notion of a cloud of sonic events.

With sieve theory on the other hand, Xenakis claims to have found a universal mechanism for creating any musical interval structure (pitch, meter, intensities, etc.) with any degree of complexity. Conversely, according to him, any intervallic structure of any music can be reduced to (a combination of) sieves. Therefore, with sieves, at least theoretically, Xenakis would be able to construct all musics of the world, Indian Ragas, Javanese Gamelan, Byzantine chant, Western well-temperated music, etc. For instance, he offers the following mathematical formula for “one of the mixed Byzantine scales, a disjunct system consisting of a chromatic tetrachord and a diatonic tetrachord […], separated by a major tone, […] notated in Aristoxenean segments” (Xenakis 1992, 197):

*8*_{0}* * (9*_{0}* + 9*_{6}*) + 9*_{6}* * (8*_{2}* + 8*_{4}*) + 8*_{5}* * (9*_{5}* + 9*_{8}*) + (8*_{6}* * 9*_{3}*)*

Each pair of numbers stands for a residue class *M*_{I} where *M* is the modulo and *I* is the residue; symbols * and + refer to the intersection and union of residue classes respectively. Again, Xenakis was not interested, with sieves, in reproducing existing styles of music but in composing a form of generalised harmony: dense, irregular orchestra clusters or non-octaviating scales played out of phase, to name but two examples.

In “Towards a Metamusic”, Xenakis also points out that his two formalisms, stochastic music on the one hand and sieves on the other, by their axiomatization and formalization, lend themselves to execution by computer. This is possibly the deeper meaning of metamusic, in analogy to *metamathematics*, which is a term coined around 1920 by mathematician David Hilbert, itself in analogy to metaphysics. Metamathematics means deducing the whole of mathematics from a set of axioms (obvious facts taken for granted) with the help of formal deduction rules. In other words, Hilbert postulated a formal procedure that all mathematicians in the world could agree upon, in the sense of Leibniz’s dictum “Let us calculate!” (“If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other [. . .]: Let us calculate.” [in Russel 1900, 170])

We would today say that Hilbert in effect strived to make mathematics computable and, in this way, universal. (The notion of computability was formed in response to Hilbert by Turing, Church and Post around 1936.) The idea is that if mathematical truth can be computed by a machine, then there is no more need for debate among mathematicians about mathematical questions. In analogy, metamusic means making music computable and thus universal (Hoffmann 2019), leading a way out of the “innumerable possibilities in a state of chaos, [the] abundance of theories, of […] styles, of […] ‘schools’ […]” (Xenakis 1992, 182).

Xenakis created two metamusic computer programs. The first was for computing music using stochastics (for his 1962 series of “ST” pieces, later rewritten as GENDYN to include sound synthesis, for *Gendy3*, 1991 and *S.709*, 1994). The second was for creating scales from any given sieve formula as well as creating a sieve formula for any given scale (Xenakis 1992, 277-288). Unfortunately, unlike GENDYN, Xenakis did not extend his sieve program to algorithmic composition and sound synthesis. Had he done so, he would have created a work of reference in the realm of sieved sound. However, in his compositional practice, Xenakis did use stochastics, sieves and even the combination of both, bending one to the other, as has been demonstrated by Benoît Gibson (2022).

Xenakis’s claims as laid out at the beginning of this article, may lead one to infer that metamusic can help discover structures that exist in a time-less realm. However, the benefit of a historical perspective allows us to assess such claims. Makis Solomos (2004, 127) argues that Xenakis’s search for universal structures “was very closely related to his time, being somehow equivalent […] with the mathematical structures defended by the Bourbaki mathematicians” (the latter referring to a group of French mathematicians preoccupied with pure mathematics who indeed managed to formalize important fields of mathematics and therefore met Hilbert’s challenge). However, he goes on to clarify, this is not meant to suggest that Xenakis was in search of an invariable structure of all music; rather, Xenakis had a tendency towards abstraction as an expression of “solidified” and “materialized” human intelligence. Tapping into the history of mathematics, we can identify a parallel debate that was taking place in the twentieth century, between mathematical Platonism and formalism. The former attitude assumes that mathematical “truths” exist, in some form or other, independently of our potential discovery; whereas the latter refers to mathematics as a symbolic meaningless game. A leading figure of the Broubaki group, Jean Dieudonné, accounted for this debate in 1968, as follows:

we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say: “Mathematics is just a combination of meaningless symbols,” and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. This sensation is probably an illusion, but is very convenient. That is Bourbaki’s attitude towards foundations.

Dieudonné 1970, 145

We may assume that to Xenakis, a music-mathematical Platonism was a necessary working assumption, but not necessarily a deep-rooted belief system. Like the mathematicians of his time, in order to keep working with abstract ideas, one tends to assume one is dealing with objective reality, not so much to defend its existence, but in order to simply be able to keep going. As mathematician Rueben Hersh eloquently put it: “The working mathematician is a Platonist on weekdays, a formalist on weekends” (1997, 39).

Xenakis used stochastics and sieves most of the time, but he did not always do so in a strict way. Formal applications are found in his early and middle periods, and are relatively few on the ground, mainly found in works that immediately follow the introduction of these concepts in the 50s and 60s. In his late years, it seems that he departed from such strict formalization, himself stating that he did not use mathematics anymore as he was “afraid that it interfere[d] with the freedom of the music itself” (Harley 2002, 17). To him, mathematical applications were a “natural tendency” given his training as an engineer, or a way for him to “escape from tradition” (Harley 2022, 14).

#### References

Dieudonné, Jean A. 1970. “The Work of Nicholas Bourbaki.” *The American Mathematical Monthly* 77 (2): 134–45. https://doi.org/10.1080/00029890.1970.11992437.

Gibson, Benoît, 2022. “The Use of Stochastic Distributions in the Instrumental Works of Iannis Xenakis: Between Chance and Intuition” in: Makis Solomos, Anastasia Georgaki (eds.), Proceedings of XENAKIS 22, Centenary International Symposium, Athens and Nafplio, pp. 1-14.

Harley, James. 2002. “Iannis Xenakis in Conversation: 30 May 1993.” *Contemporary Music Review* 21 (2–3): 11–20. https://doi.org/10.1080/07494460216659.

Hoffmann, Peter, 2019. “’My music makes no révolution’. Thoughts on the role of mathematics in the work of Iannis Xenakis”, in: Roberto Illiano (ed.), *Twentieth Century Music and Mathematics*, Turnhout, Brepols, 2019 (= Music, Science and Technology I, publications of the Centro Studi Opera Omnia Luigi Boccherini, Lucca), pp. 41-60.

Hersh, Reuben. 1997. *What Is Mathematics, Really?* New York: Oxford University Press.

Russell, Bertrand. 1900 . *A Critical Exposition of the Philosophy of Leibniz, with an Appendix of Leading Passages*. Cambridge, [Eng.] The University Press. http://archive.org/details/cu31924052172271.

Varga, Bálint András. 1996. *Conversations with Iannis Xenakis*. London: Faber.

Xenakis, Iannis, 1963. *Musiques formelles*. Special issue of *La revue musicale* 253/254.

Xenakis, Iannis. 1992. *Formalized Music: Thought and Mathematics in Composition / Iannis Xenakis.* Rev. ed. Stuyvesant, NY: Pendragon Press.

##### How to cite

EXARCHOS, Dimitris, and Peter HOFFMANN. 2023. “Metamusic.” In *A Xenakis Dictionary*, edited by Dimitris Exarchos. Association Les Amis de Xenakis. https://www.iannis-xenakis.org/en/metamusic