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

with arbitrary $p$ each time.

Also it enables to make greater canons from small ones, which has interesting abstract as well as practical consequences: computerized tools for such transformations of canons are currently under development.

Figure 7:

Zooming out, dividing the tempo and number of voices by 1/4.

2.6 Equirepartition

’From de Bruijn (1955) to Tijdeman (1995) there was some concern about equirepartition, that is to say equiprobability of all residues of a set (or sum of sets) modulo some $n$ .’ Now in the case of a canon, if $Z$ can be written $A o+ B o+ nZ$ it means that $A o+ B$ is a complete set of residues modulo $n$ , or in other words there is equirepartition modulo $n$ ; in polynomials this is expressed by:

A(x).B(x) =_ 1+ x+ x2 + ...+ xn-1 (mod xn - 1)

Indeed this equirepartition result can be reversed to build up trivial (but still musically interesting) canons:

Start from $2 n-1 A(x) = Dn(x) := 1 + x+ x + ...+ x$ , a trivial tile, and randomly add multiples of $n$ to each the exponents above, e.g. get

' 1+nr1 2+nr2 n-1+nrn- 1 A (x) = 1+ x + x + ...+ x

which tiles trivially as $A'o + nZ = Z$ . Indeed a reverse is true :

Figure 8:

Example of equirepartition mod 9.

Theorem 7 A finite subset $' A$ of $N$ , beginning with 0, tiles $Z$ trivially with period $n$ - meaning $' A o+ nZ = Z$ , if and only if $' 1+nr 2+nr n-1+nr A (x) = 1+ x 1 + x 2 + ...+ x n-1$ .