Rhythmic canons and Galois theory

The study of rhythmic canons by mathematicians with musical interest has given rise to unexpected connections to non trivial mathematical problems. The author tried a new approach based on a Galoisian interpretation of the novel results and concepts of Coven and Meyerowitz [4]. Extending this in a second part to finite fields, he concludes with unexpected results. 1 Rhythmic canons, tiling the line, polynomials 1.1 Canons 1.1.1 Basic definition As an abstraction of the musical notion, a rhythmic canon is a canon debarred of all pitches, intensities and other musical parameters, keeping only rhythm. There are several voices, each playing the same motif, or rhythmic pattern, but beginning at different onsets. A motif (meaning ‘rhythmic motif’ in the context of this paper) can be modelised by the set A of the onsets of its different notes. For musical ([17]) and mathematical ([13]) reasons, it is desirable to modelise all these onsets by integers, measuring multiples of the unit beat. Usually a musical canon is periodic, and some musicians have added the constraint that on each beat one and only one note is played. Translates of A (i.e. the different voices) will have the form bi + A; putting all these offsets bi in one set B, one gets a first possible definition including the periodicity condition (the rhythmic motif A is usually called 〈〈 inner rhythm 〉〉, the set of onsets B is called 〈〈outer rhythm 〉〉): Definition 1. A canon with inner rhythm A, outer rhythm B and period n is given by two finite subsets A,B ⊂ N and an integer n satisfying A⊕B ⊕ nZ = Z Mathematics Subject Classification 2000: Primary 05E20, Secondary 33C80.