New perspectives on rhythmic canons and the spectral conjecture

Abstract

1.1. Purpose. After some impressive developments in 2003-2005, the study of rhythmic canons had slowed down, or almost stalled. Discussion of our results and ideas with specialists of more general tiling problems, also featured in this issue of the journal, opened new alleys for exploration. In particular, the quest for all Vuza canons, which is interesting both practically and theoretically, has started anew. Fresh results also suggest future directions for exploration and renew hope for solution of long standing conjectures. 1.2. Some definitions. This might be redundant with other papers in this issue, but it seemed desirable for clarity’s sake to state the basic definitions. Definition 1. For the purposes of the present paper, a rhythmic canon (RC) is a tiling of the cyclic group Zn by translations, i.e. Zn is the disjoint union of translates of some subset A: Zn = A ∪ (A+ b1) + (A+ b2) + · · · = A ⊕ {0, b1,...} = A⊕B A is called the inner rhythm and B the outer rhythm. Musically this can be rendered as a canon (say with percussion instruments) repeating with period n, playing any motif (modelized as a collection of integers corresponding to the beats played) that reduces to A modulo n, beginnning each instance of this motif on beats congruents modulo n to the bi’s. The algebraic notion closest to the idea of a tile in Zn would be the orbit of A under the action of Zn by translation, i.e. the set {A+ k, k ∈ Zn}. If we visualize Zn as a circle, it means A is viewed up to rotation – for a periodic event, the question of a ‘starting beat’ is irrelevant, though musically of course this is different. In practice, we will assume for starters that 0 ∈ A, and the results will usually be stated up to such a rotation. Let it be said once and for all that it may be convenient to interpret any tile A ⊂ Zn as a subset of the non negative integers, beginning with 0 and with minimal largest element: