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

cardinalities of the sets $A o+ B,A, B$ respectively. Indeed in those conditions we compute easily the cardinality of the set $A$ by plugging 1 as the value of $x$ in $A(x)$ :

Lemma 1 $|A| = A(1)$

In the example above,

sum 21 A(x) = 1+x+x4 +x5 B(x) = 1+x6 +x8 +x14+x16 A(x).B(x) = 1+x+ xk k=4

In this paper, we will usually try to get a set $A o+ B$ without gaps. More precisely, essentially two cases arise:

1.2 Loops and Lines

In the last example, such as in a fugue, there are some gaps in the beginning, but after a time several voices play together without gaps nor double beats, and this could be carried on for as long as wished. Most canons in the musical tradition similarly »tile a Loop ¹

¹: as coined by Johnson.

«, meaning that it takes a while to get all voices singing together. Thus we venture to introduce a generalization in our mathematical model, extending this tiling to the whole set $Z$ of integers.

The above example can be rearranged to something simpler, called »tiling a line« by Johnson ²

²: We will avoid this phrasing because the generally accepted mathematical meaning is different. See below for definition.

, namely

{0,1,4,5} o+ {0,2}= {0,1,2,3,4,5,6,7}

So here is a more sophisticated example: ${0,1,5} o+ {0,6,9,15}$ may be extended to ${0,1,5} o+ {0,6,9,15}o + 12Z$ .

As we will see in Theorem 3, if a finite rhythmic motif $A$ tiles the set of all integers then

Z = (A o+ B) o+ nZ

for some finite $B$ , where $n$ is a period of the tiling. This means that $A o+ B$ gives a complete set of residues modulo $n$ , hence reduction modulo $n$ yields $pn(A) o+ pn(B) = Z/nZ$ where we denote by $pn(A)$ the elements of $A$ taken modulo $n$ . The smallest such $n > 1$ will be called the period of the canon.

This also shows that the reduction to finite $A$ and $B$ yields no loss of generality and entails our most refined definition of a rhythmic canon:

Proposition 2 The set $A < Z$ tiles an $n-$ loop iff there exists a set $B$ with $pn(A) o+ pn(B) = Z/nZ$ .

When the context is clear, we will drop the map $p n$ . The periodic character of such a tiling allows a cyclic representation (figure 2). Indeed it may even be best to set the representation of the canon on a torus (figure 3), though perhaps this is a bit far from musical perception.