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

$1 *1 = -1-.1 A B #G G$ where $1 A$ (equivalently $1 B$ and $1 G$ ) means the characteristic function of $A$ and the convolution product is defined in the following way:

1 sum 1A *1B(x) = #G- 1A(x - y)1B(y). y (- G

For a proof of the previous theorem we refer to the quoted works. The following figure shows an axample of supplementary sets. They are respectively $A = {0,8,10}$ and $B = {0,5,6,11}.$

Figure 4:

Factorization of the cyclic group of order n in two subsets.

Partitioning and supplementary translation classes are related to each other by the fact that a given translation class $M (- T (G) p$ (i.e. is a partitioning class) iff a class $N (- T (G)$ such that $M$ and $N$ are supplementary exists. This leads immediately to the ”canonic” interpretation of the partitioning concept and supplementary classes (See Figure 5)

Figure 5:

A tiling rhythmic canon.

It is a canon in 4 voices obtained by the time translation of the pattern $R = [2,8,2]$ in the onset-times 0, 5, 6, 11. The previous intervallic notation means that, given a minimal division $u$ , two successive beats $ri,ri+1$ of $R = [r1,r2,...,rk]$ are separeted by $ri+1 - ri,$ $u$ -temporal units.