Abstract The subject of rhythmic canons has been revivified by new concepts, coming from the field of music composition. The present article is a survey of the mathematical knowledge in this field, from the 1950’s to state of the art results.

1 What Is this All About ? Paving the Way

I was introduced to the fascinating subject of rhythmic canons by people working at the Ircam, especially Moreno Andreatta and later on Tom Johnson. On closer investigation, this is the meeting point of numerous musical and mathematical issues, from spectral theory (Fourier transforms and Hilbert spaces) (Lagarias and Wang, 1996) to factorisations of abelian groups (de Bruijn, 1955), from mosaics and tilings (e.g. the azulejos in the Alhambra as mirrored in Debussy’s La Puerta del Vino (Amiot, 1991)) to Galois theory and Galois groups on finite fields.

1.1 Rhythmic Canons and Tiling

There are already many different and conflicting definitions for rhythmic canons (see for instance Mazzola (2002, p 380-382), Vuza (1990-91), Fripertinger (2001)). Our definition of a rhythmic canon will stress the regularity of the overall beat, allowing to work with integers.

Definition A rhythmic canon is a tile (a purely rhythmic motif) repeated in several voices (for instance with several different instruments) with different offbeats, so that two distinct notes never fall on the same beat.

This is the musically intuitive definition. To be more precise, let us modelise it with two sets of integers: let be the the set of beats of the rhythmic motif, and the sets of the offbeats (meaning the beats where a new voice starts).