Why rhythmic canons are interesting

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 1.1 Rhythmic canons and tiling I was introduced to the fascinating subject of rhythmic canons by people working at the Ircam, especially M. ANDREATTA and later on T. 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 (like the azulejos in the Alhambra mirrored in DEBUSSY’s La Puerta del Vino (Amiot, 1991)) to Galois theory and Galois groups on finite fields. There are already many different and conflicting definitions for rhythmic canons (see (Mazzola, 2002, p 380-382), (Vuza, 1990-91), (Fripertinger, 2001) for instance). 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 A be the the set of beats of the rhythmic motif, and B the sets of the offbeats (meaning the beats where a new voice starts). Definition 1 If two subsets of the integers,A,B ⊂ Z have the property that the map A×B 3 (a, b) 7→ a+ b ∈ A+B is one to one, then we shall write A+B = A⊕B and call this sum direct. Definition 2 We have a rhythmic canon with motif (or »inner rhythm«) A and set of entries (or »outer rhythm«) B when A,B are subsets of Z such that A is finite and the sum A+B is direct. Perspectives in Mathematical Music Theory 2 Figure 1: a rhythmic canon with five voices Thus A is tiling A⊕B. Without stress on B, we will simply say that A tiles. This definition is already quite restrictive, as it forbids two different voices to play on the same beat (which happens often in musical canons such as the Musical Offering). The restriction of a common pulsation however, which allows to rescale to integers, is not as stringent as it seems as proved in (Lagarias and Wang, 1996). An example of a rhythmic canon is given by A = {0, 1, 4, 5}, the inner rhythm, and B = {0, 6, 8, 14, 16}, the outer rhythm of the canon. We also use a second description writing the beats of one voice from left to right in one line, where 1’s stand for beats and 0’s stand for silences, and different voices in consecutive lines vertically aligned. 11001100