Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

For instance the tile $A = {0,1,4,5}$ becomes ${0,3,4,7}$ when multiplied by $3 (mod 8)$ . It still tiles with the same $B$ , taken to be ${0,2,8,10,16,18}$ in figure 6.

Musically it means it is possible to change the rhythmic motif without changing the table of entries of the different voices - or the reverse, as $A$ and $B$ play symmetric roles.

Figure 6:

A canon, and one of its affine transforms.

This kind of transform applies readily to the category of rhythmic canons. Indeed Dan Tudor Vuza (Vuza, 1990-91, part 3) and (Tijdeman, 1995) have independently proved what amounts to the following result :

Theorem 4 If $p$ is coprime to $n$ and $A o+ B o+ nZ = Z$ then also

(pA) o+ B o+ nZ = Z A o+ (pB) o+ nZ = Z

This means that any affine transform (with the affine group modulo $n$ ) of a tile also tiles, with the same period (indeed with the same outer rhythm). Thus we have a classification of rhythmic canons with fewer classes, up to independent affine transform of the inner or of the outer rhythm. Finally, this opens intriguing possibilities of a generative canon theory.

2.4 Hajós groups and Vuza canons

The question of rhythmic canons with aperiodic inner and outer rhythms boils down to a factorisation of $Z/nZ$ with non periodic factors. As it happens, it was noticed by György Hajós and some others (in several steps and over several years in the 1950’s) that most cyclic groups are Hajós groups in the following sense:

Definition 4 $Z/nZ$ is a Hajós group if, for any decomposition $A o+ B = Z/nZ$ there exists $p / (- nZ$ with $A + p = A (mod n)$ or $B + p = B (mod n)$ .

Non-Hajós groups are sometimes called »bad groups« and admit to several interesting generalizations, irrelevant here; see (Sands, 1962) or (Tijdeman, 1995) for a general discussion of factorisations of finite abelian groups..

The following theorem was rediscovered independently by (Vuza, 1990-91):

Theorem 5 $Z/nZ$ is a bad group, i.e. a non-Hajós group (i.e. there exists an aperiodic decomposition of $Z/nZ$ ) except in the following cases, and in these cases only: when $a n = p$ or $a 22 a n = p q,n = p q,n = p qr,n = pqrs$ , $p,q,r,s$ being primes.

Vuza called the aperiodic canons corresponding to such decompositions »regular canons of maximal category«, and he established an algorithm for producing independently some inner and outer aperiodic rhythms for a »bad« $Z/nZ$ .