Motivic rhythms

Abstract

In this article on mathematics and music, we explain how one can “listen to motives” as rhythmic interpreters. In the simplest instance which is the one we shall consider, the motive is simply the $H^1$ of the reduction modulo a prime p of an hyperelliptic curve (defined over $\mathbb{Q}$). The corresponding time onsets are given by the arguments of the complex eigenvalues of the Frobenius. We find a surprising relation between mathematical properties of the motives and the ideas on rhythms developed by the composer Olivier Messiaen.

Keywords:

• motives
• hyperelliptic curve
• primes
• Eratosthenes sieve

2010 Mathematics Subject Classification:

• 00A65

Acknowledgments

The author is grateful to the referees for their useful comments, to Fernando Rodriguez Villegas, Moreno Andreatta, Karim Haddad, Carlos Agon and Ugo Moschella for their help in the realization of these ideas, and also to our IHES group of discussions on mathematics and music with, besides Moreno, Charles Alunni, Olivia Caramello and Pierre Cartier. The principle of the above developments was obtained during the collaboration with Danye Chéreau and Jacques Dixmier which appears in the novel “Le Spectre d’Atacama” published by O. JACOB Paris, 2018.

Disclosure statement

No potential conflict of interest was reported by the author(s).