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

Theorem 8. The number of isomorphism classes of canons in $Zn$ is

sum Kn = m(d)c(n/d)a(n/d), d|n

where $m$ is the Möbius function, $c(1) = 1$ ,

sum c(r) = 1 m(s)2r/s for r > 1, r s|r

and

1 sum a(r) = - f(s)2r/s - 1 for r > 1, r s|r

where $f$ is the Euler totient function.

Here the description of a canon as a pair $(L,A)$ with certain properties was used in oder to classify all canons in $Z n$ by methods of step 1 or step 2 of the general classification scheme.

4 Enumeration of Rhythmic Tiling Canons

There exist more complicated definitions of canons. A canon described by the pair $(L,A)$ of inner and outer rhythm defines a rhythmic tiling canon in $Zn$ with voices $Va = a+ L$ for $a (- A$ if

the voices $Va$ cover entirely the cyclic group $Zn$ ,
the voices $Va$ are pairwise disjoint.

Rhythmic tiling canons with the additional property

both $L$ and $A$ are acyclic (i.e. invariant only the trivial translation),

are called regular complementary canons of maximal category.

In other words, the voices of a rhythmic tiling canon form a partition of $Zn$ . Hence, rhythmic tiling canons are canons which are also mosaics. More precisely, if $|A|= t$ then they are mosaics consisting of $t$ blocks of size $n/t$ , whence they are of block-type $c$ where

{ ci = t if i = n/t 0 otherwise.

This block-type will be also indicated as $c = ((n/t)t) = (| L ||A| )$ .

So far the author did not find a characterization of those mosaics of block-type $c$ describing canons, which could be used in order to apply methods from step 1 or step 2 of the general classification scheme.

Applying Theorem 6, the numbers of $Cn$ -isomorphism classes of mosaics presented in table 1 were computed. Among these there are also the isomorphism classes of canons, but many mosaics of these block types are not canons!