Institut de Recherche et Coordination Acoustique/Musique (IRCAM), Paris, France, 3-4 December 1999

Reviewed by Marc Chemillier (Caen, France)

The cycle of Diderot Mathematical Forums, held under the authority of the European Mathematical Society (EMS), features one conference a year taking place simultaneously in three European cities linked by telecommunications. Each cycle addresses three different aspects of the discipline: fundamental mathematics, mathematical applications, and the relation of both to society. The subject of the Fourth Diderot Forum was "Mathematics and Music." It was held in parallel in Lisbon (organizer: Jose-Francisco Rodriguez), Vienna (organizer: Hans Feichtinger), and Paris, on 3-4 December, 1999. This review deals with the Paris forum, which took place at IRCAM and was organized by Gérard Assayag (IRCAM) and Laurent Mazliak (Université de Paris VI).

The specific theme of this conference was "Mathematical Logic and Musical Logic in the Twentieth Century." The scientific and musical committee included, in addition to the organizers, Marc Chemillier, Laurent Fichet, François Nicolas and André Riotte. Four sessions were scheduled: 1) historical perspectives about both formalization of logic and formalization of music in the twentieth century, 2) implicit computation and unconscious computation in relation to ethnomusicology and cognitive science, 3) formal systems and their application to computer-assisted composition software, and 4) the limits of formalization.

The conference was introduced by Jean-Pierre Bourguignon (President of the EMS) and Laurent Bayle (Director of IRCAM). Mr. Bayle noted that while mathematics is not a subject studied in itself at IRCAM, it appears as a transversal discipline involved in many different domains from sound synthesis to computer-assisted composition. Mr. Bourguignon pointed out that this edition of the Diderot Forum has focused on the relation between mathematics and music in a much deeper way than preceding ones had focused on the relations between mathematics and other domains (finance - 1996, environment - 1997, culture - 1998).

At the end of the first day, a videoconference (sponsored by France Télécom) provided a meeting point for the three forum locations: Paris, Lisbon (where the theme was historical aspects of the relation between mathematics and music), and Vienna (where the theme was sound processing). Paris participants in the round table included Daniel Andler (cognitive science), Simha Arom (ethnomusicology), and Guerino Mazzola (mathematics and music), with Mr. Bourguignon as moderator. The Lisbon participants were unfortunately lost soon after the exchange began, due to technical problems typical of this kind of experience.

The session on "Historical Perspectives" began with a presentation by Marie-Josée Durand-Richard on the formalization of logic since the work of George Boole. This was followed by Laurent Fichet, who, speaking from a critical perspective, gave examples of formal musical analysis of pieces inspired by set theory (such as Piano Sonata No. 2 by Pierre Boulez).

The second session, "Implicit Computation, Unconscious Computation," addressed the question of whether mathematical structures exist only in formalized music or in all types of music. The speakers were Daniel Andler, who gave a brilliant lecture on the cognitive framework of this question, Marc Chemillier, who presented examples taken from sand drawings of the Vanuatu, harp music from the Central African Republic, and polyrhythmic music from the Aka pygmies, where "ethnomathematical" structures can be found, and Marc Leman, who explained how formal logic can be used as a meta-level description system for understanding what he calls "musical images," a model of music perception.

The session on "formal systems" presented a collection of mathematical models underlying computer-assisted composition systems. Mikhail Malt gave an overview of mathematical models used by Iannis Xenakis in his compositions, and demonstrated a recomposition of the piano piece Herma using OpenMusic, software developed at IRCAM. Yann Orlarey presented Elody, an impressive composition software developed at GRAME (Centre National de Création Musicale, Lyon), which is entirely based on the lambda calculus. Guerino Mazzola gave a lecture on the sophisticated mathematical tools, including Grothendieck's factorial algebraic geometry and Lawvere's topos theory of logic, that he has used in the implementation of music software such as Rubato. Shlomo Dubnov described the use of information theory for discovering hidden structures and redundancies, and presented research done in collaboration with Mr. Assayag on statistical learning methods applied to music generation, illustrated by an amazing computer-generated pastiche of J. S. Bach's Musical Offering.

At the final session, "The Limits of Formalization," François Nicolas gave a lecture in the form of a sequence of variations on the theme "what is musical logic?" He asserted that musical logic is dialectic (built from writing, perception, and variations). Mathematician Jean-Paul Allouche described his collaboration with composers Marcel Frémiot and Tom Johnson, who use mathematical tools in a creative way. André Riotte, whose works on formalized musical analysis has been a rich source of inspiration for many of the participants, concluded the conference with some remarks on the dialectic of rigor and freedom in the creative process.

As an addition to the concluding session, Benny Sluchin, trombonist with the Ensemble InterContemporain, gave a lecture-recital on the piece Solo, for melodic instrument with electronic feedback, by Karlheinz Stockhausen. Before the performance, his talk, entitled "Musical Logic and the Open Work," provided technical explanations about the way the electronic part is based on recording, transforming, and playing back in real time the music performed by the soloist.

Altogether, the forum was well-attended, with approximately 120 participants. The proceedings are to be published by Springer Verlag (for further information on the Diderot Forum, see www.ircam.fr/equipes/repmus/diderot99/EnglishPage.html).