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

The first rhythmic pattern is also called inner rhythm, whereas the pattern of coming in of voices is called outer rhythm (Andreatta, 2001, 2002). Inner and outer rhythms replace Vuza’s original ground and metric classes (Vuza, 1991-93), a terminology that could give rise to some confusions for what concerns the characterisation of rhythmic and metric properties of such global musical structures. This tiling condition implies that time axis is provided with a minimal division which holds as well for the inner and for the outer rhythm. Rhythmic canons verifying the tiling condition are also called, in Vuza’s terminology, »Regular Complementary Canons«. In fact, voices are all complementary (there is no intersection between them) and once the last voice has come in, one hears only a regular pulsation (there are no holes in the time axis). It is now easy to show that the construction of the so-called »regular complementary of maximal category« (shortly RCMC-Canons) is equivalent to the problem of providing a pair of non periodic supplementary sets of some cyclic group $Z/n$ . Note that Vuza’s Theorem 2.2 (Vuza, 1991-93, p. 33) is formally equivalent to Hajós’ theorem. Its special case, Theorem 0.1 provides a connection between the theory of supplementary sets and that of limited transpositional structures (i.e. translation classes with transpositional symmetry). It affirms that »given any pair of supplementary sets (in $Z/12$ ) at least one set in the pair has transpositional symmetry«. As previously noted, this result remains true when $Z/12$ is replaced by any cyclic Hajós group $Z/n$ . But the theory developed by Vuza suggests something further: given a cyclic group which does not have the Hajós property, it provides a method of constructing some non periodic supplementary sets of such a group.¹⁶

¹⁶: It has been shown that there are supplementary sets which are apparently not a direct solution of Vuza’s algorithm. This question was originary raised by the composer George Bloch who observed that a RCMC-Canon of period $p$ could be »interpreted« as a RCMC-Canon of period $2p$ simply by replacing each minimal division with two attacks holding a minimal division each. For example this process enables to construct a partitioning set of period $n= 144$ as a simple transformation of a partitioning set of period $n= 72$ . This new set does not belong to the catalogue of solutions provided by Vuza’s algorhythm. In fact it is, in some sens, »redundant«, since it is not maximally compacted. H. Fripertinger did the same discovery independently from the intuitions of the French composer and implemented the algorhythm which includes all those Vuza-Canons which do not have the maximal category property.

Besides the musical relevance it seems to me that this theory could also be interesting from a purely mathematical perspective, for it concerns some structures which are far from being completely classified.¹⁷

¹⁷: In presenting his generalised Hajós property in the Hungarian Colloquium on Abelian Groups (September 1963), Sands admitted that »the problem of obtaining the factorisations of those groups which do not possess this Hajós property remains«. A recent paper on the $k$ -factorisation of abelian groups (Amin, 1999) seems to suggest that the problem is a still interesting problem in mathematics.

In a previous study (Andreatta, 1996) we considered the case $Z/n$ , where $n = 108$ , which Vuza seemed to forget to consider as belonging to the family of groups which do not satisfy the Hajós property. By following a slightly different method for the construction of a pair of non-periodic supplementary sets of $/Z108$ (see Andreatta, 1996, pp. 25-27) we found the following factorisation:

M = {39,51,63,66,78,90} N = {16,18,20,22,26,27,36,52,56,58,62,72,81,88,90,92,94,98}