Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

NEW PERSPECTIVES ON RHYTHMIC CANONS AND THE SPECTRAL CONJECTURE
EMMANUEL AMIOT
Abstract. The musical notion of rhythmic canons has proved to be relevant to some non trivial mathematical problems.
After a survey of the main concepts of tiling rhythmic canons, we discuss recent developments that enable to make, or
expect, definite progress on several open mathematical conjectures.
1. Introduction
1.1. Purpose. The story of rhythmic canons is long and fullsome, as this notion is in fact equivalent to the problem of
tiling the integers. Musical ideas arrived with Dan Tudor Vuza [26] in 1991, and gave rise to interesting developments,
especially after this theoretical model was implemented in the it OpenMusic visual programming language [10]. One
hard remaining problem is the computation of Vuza canons, which are fundamental bricks out of which all canons can
be constructed, and all their features derived. But as there is no known polynomial algorithm for this task, it had been
impossible until recently to enhance Fripertinger’s results, verifying that all Vuza canons up to a period of 108 were
classified.
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, including a complete list of all Vuza canons for the two next possible periods, n = 120
and n = 144. 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 is called the inner rhythm and B the outer rhythm.
Zn = A ∪ (A + b1) + (A + b2) + · · · = A ⊕ {0, b1, . . . } = A ⊕ B
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, beginning each
instance of this motif on beats congruents modulo n to the bi’s. Already this model loses something of the actual musical
object, as motif (A) and offset list (B) are reduced modulo n. This makes sense, perceptually, for periodic rhythms with
enough repetitions for the periodicity to be perceived. Also, though the reference to rhythm is traditional, originating
in [26], the concept is really adequate to any discrete, periodic structure, such as pitch classes for instance. Perhaps one
might talk about ‘musical canons’ instead, but this is evocative of a fuzzier, contrapunctual, concept.
Figure 1. A tile and one of its rotated forms
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} – think of the bass instrument in a classic tango, with A = {0, 3, 4, 6} and n = 8 for instance.
1