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

Let $G$ be an abelian group. A subset $S$ of $G$ is called $g$ -periodic for $g (- G$ if $S = g+ S$ , and it is called periodic if it is $g$ -periodic for some $g (- G$ . Otherwise $S$ is called aperiodic. (Subsets of $Zn$ are aperiodic if and only if they are acyclic. In other words, the $Cn$ -orbit of an aperiodic subset of $Zn$ is of size $n$ , or there is no non-trivial cyclic operation which stabilizes an aperiodic subset of $Zn$ .) The group $G$ is called a Hajós group, or has the $2$ -Hajós property, if in each factorization of $G$ as $S1 o+ S2$ at least one factor is periodic. In Sands (1962) all finite abelian groups which are Hajós groups are classified. This classification yields the following list of cyclic Hajós groups:

Theorem 9. The group $Zn$ is a Hajós group if and only if $n$ is of the form

k k 22 k p for k > 0, p q for k > 1, p q, pqr for k (- {1,2}, pqrs

for distinct primes $p$ , $q$ , $r$ and $s$ .

Corollary 10. $Zn$ is a non-Hajós group if and only if $n$ can be expressed in the form $p1p2n1n2n3$ with $p1$ , $p2$ primes, $ni > 2$ for $1 < i < 3$ , and $gcd(n1p1,n2p2) = 1$ .

Proofs of Theorem 9 and Corollary 10 can be found for instance in Vuza’s papers Vuza (1991, 1992b,a, 1993). The smallest $n$ for which $Z n$ is not a Hajós group is $n = 72$ which is still much further than the scope of our computations.

The pair $(L,A ) 0$ is a regular complementary canon of maximal category if and only if it is a tiling canon and both $L$ and $A 0$ are aperiodic. Consequently, regular complementary canons of maximal category occur only for $Z n$ which are non-Hajós groups (cf. Vuza, 1991; Andreatta, 1997). Hence, we deduce that for all $n$ such that $Z n$ is a Hajós group the following is true:

Lemma 11. If a pair $(L,A ) 0$ describes a regular tiling canon in a Hajós group $Z n$ , then $A 0$ is cyclic.

This reduces dramatically the number of pairs which must be tested for being a tiling canon, since for Hajós groups $Zn$ we only have to test pairs $(L,A0)$ where $A0$ is periodic, whence $A0$ is not a Lyndon word. By an application of Theorem 8 we computed $Kn$ , the numbers of non-isomorphic canons of in $Zn$ , given in the third column of table 2.