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

such that for all $i,j (- {1,...,t}$

the set $Vi$ can be obtained from $Vj$ by a translation of $Zn$ ,
there is only the identity translation which maps $Vi$ to $Vi$ ,
the set of differences in $K$ generates $Zn$ , i.e. $<K - K > := <k -l |k,l (- K> = Zn.$

We prefer to write a canon $K$ as a set of its subsets $Vi$ . Two canons $K = {V1,...,Vt}$ and $L = {W1,...,Ws}$ are called isomorphic if $s = t$ and if there exists a translation $Tj$ of $Zn$ and a permutation $s$ in the symmetric group $St$ such that $T j(Vi) = Ws(i)$ for $1 < i < t$ . Then obviously $Tj$ applied to $K$ yields $L$ .

Here we present some results from Fripertinger (2002). The cyclic group $Cn$ acts on the set of all functions from $Zn$ to ${0,1}$ by

Cn × {0,1}Zn --> {0,1}Zn Tj * f := f o T- j.

When writing the elements $Z f (- {0,1} n$ as vectors $(f(0),...,f(n- 1))$ , using the natural order of the elements of $Zn$ , the set ${0,1}Zn$ is totally ordered by the lexicographical order. As the canonical representative of the orbit ${ } Cn(f ) = f o Tj |0 < j < n$ we choose the function $f0 (- Cn(f )$ such that $f0 < h$ for all $h (- Cn(f )$ .

A function $f (- {0,1}Zn$ (or the corresponding vector $(f (0),...,f(n - 1))$ ) is called acyclic if $Cn(f )$ consists of $n$ different objects. The canonical representative of the orbit of an acyclic function is called a Lyndon word.

As usual, we identify each subset $A$ of $Zn$ with its characteristic function $xA : Zn --> {0,1}$ given by

{ 1 if i (- A xA(i) = 0 otherwise.

Following the ideas of Fripertinger (2002) and the notion of Andreatta et al. (2001), a canon can be described as a pair $(L,A)$ , where $L$ is the inner and $A$ the outer rhythm of the canon. In other words, the rhythm of one voice is described by $L$ and the distribution of the different voices is described by $A$ , i.e. the onsets of all the voices of the canon determined by $(L, A)$ are $a+ L$ for $a (- A$ . In the present situation, $L /= 0$ is a Lyndon word of length $n$ over the alphabet ${0,1}$ , and $A$ is a $t$ -subset of $Z n$ . But not each pair $(L,A)$ describes a canon. More precisely we have:

Lemma 7. The pair $(L,A)$ does not describe a canon in $Z n$ if and only if there exists a divisor $d > 1$ of $n$ such that $L(i) = 1$ implies $i =_ d- 1 mod d$ , and $x (i) = 1 A0$ implies $i =_ d- 1 mod d$ , where $x A0$ is the canonical representative of $C (x ) n A$ .

An application of the principle of inclusion and exclusion allows to determine the number of non-isomorphic canons.