Implementing Xenakis’ theoretical concepts in OpenMusic visual programming language : probability, sieve, set and group theory

Moreno Andreatta and Carlos Agon (IRCAM, France)

(Presented by Moreno Andreatta )

We present our recent implementations in OpenMusic visual programming language of some concepts proposed by Iannis Xenakis in his theoretical writings. We focused on four families of theoretical tools belonging to the two main categories of Xenakis' compositional universe: stochastic and symbolic music. After introducing some implementations of general stochastic methods used by Xenakis (Poisson law, Markov processes, etc.) we firstly focus on a computer-aided model of Iannis Xenakis piece Achorripsis (1956/57) for 21 instruments. This implementation, which was originally realized in real time in Patchwork, has been recently rewritten in OpenMusic by also using Max/MSP as the real-time engine. We will present this model by focusing on the concept of maquette that enables to describe in a dynamic way the probability distributions used by the composer in this piece. Concerning symbolic music, we focus on three classes of theoretical concepts that have become standard tools in computational musicology: the sieve-theoretical construction of musical scales and rhythmic patterns, the set-theoretical organization of pitch-spaces and the group-theoretical organization of the musical form. After discussing the implementation of sieve theory in OpenMusic libray RepMus, we show how we combined these ordered structures with some more general algebraic tools (circular representation, group actions, orbits …) that we recently integrated in the last version 5.0 of OpenMusic. These tools enable to analyze structural properties of some musical scales and rhythmic patterns that Xenakis used in pieces belonging to the category of Symbolic Music(Akrata, Nomos Alpha, Psappha, Pléïades…). Through the implementation we will also discuss the link between sieve theory and set-theoretical methods in music theory, analysis and composition. A good example of piece which is based on set-theoretical operations is provided by Herma (1960-61) for piano solo. For building the computer-aided model of the piece, a maquette object was used, that is, a container that displays information concerning different levels of temporal organization. This maquette reproduces the temporal flow chart provided by the composer in his theoretical writings. By describing the temporal blocks objects in OpenMusic, we show how the computer-aided model deals with the two fundamental categories: the outside-of-time and the in-time categories. More abstract tools are provided by group theory that has been used by the composers in pieces like Akrata, Nomos Alpha and Nomos Gamma. After presenting some implementations of group-theoretical methods dealing with the rotations and the reflections of regular polyhedra in the space, we will discuss the computer-aided model of two pieces based on such a groups: Akrata (1964-65) for 8 winds and 8 brass and Nomos Alpha (1966) for solo cello. These two implementations clearly shows the music-theoretical properties of the Fibonacci process according to which the different sound objectsare organized and the underlying formal model based on the group of rotations of the tetrahedron and the cube in the space.

The implementation of all these theoretical tools originally proposed by Iannis Xenakis is an important aspect of the composer's legacy in the music-theoretical community. Computational musicology as a well established academic discipline can still find some interesting inspirational ideas by trying to generalize these tools and apply them in different analytical and compositional contexts.  

Moreno Andreatta graduated in mathematics at the University of Pavia in 1996 and in piano performance at the Conservatory of Novara in Italy in 1998.  He also studied composition, music analysis and conducting with Francesco Valdambrini.  As the recipient of a fellowship of the University of Sussex at Brighton, he studied aesthetics and sociology of music with David Osmond-Smith and specialized in group theory with Roger Fenn.  In 2001 he was awarded the European scholarship of the Marcel-Bleustein Foundation for his researches in the relationships between music and mathematics.  He graduated in computational musicology in 2003, with a thesis on algebraic methods in XXth century music and musicology (EHESS/IRCAM).  He is presently associate researcher at IRCAM/CNRS (UMR 9912) where he also coordinates the math/music activities.

Carlos Agon first studied computer science at the Universidad de los Andes, Bogota (Columbie).  After his PhD degree in computer science at the University of Paris VI in 1998, he recently obtained the HDR (Habilitation à diriger des recherches) with a thesis on “Programming Languages for computer-aided composition.”  Together with Gérard Assayag, he conceived the OpenMusic visual programming language that, with more than five hundred users, has been used in the composition of several musical works. He is currently preparing the edition of the “OM composer’s Book” (together with Gérard Assayag and Jean Bresson), which will present a panorama of several compositional processes analyzed by composers who utilized OpenMusic for the main conception of their pieces.