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

Definition 2 A rhythmic canon consits of a motif (or »inner rhythm«) $A$ and a set of entries (or »outer rhythm«) $B$ where $A,B$ are subsets of $Z$ such that $A$ is finite and the sum $A+ B$ is direct.

Thus $A$ is tiling $A o+ B$ , i.e. $A + B$ is a disjoint union of translates $A + b,b (- B$ of $A$ . Without stress on $B$ , we will simply say that $A$ tiles. This definition is already quite restrictive, as it forbids two different voices to play on the same beat (which happens often in musical canons such as the Musical Offering). The restriction of a common pulsation however, which allows to rescale to integers, is not as stringent as it seems as proved in (Lagarias and Wang, 1996).

An example of a rhythmic canon is given by $A = {0,1,4,5}$ , the inner rhythm, and $B = {0,6,8,14,16}$ , the outer rhythm of the canon. We also use a second description writing the beats of one voice from left to right in one line, where 1’s stand for beats and 0’s stand for silences, and different voices in consecutive lines vertically aligned.

110011000
      110011000
        110011000
              110011000
                110011000

Figure 1 shows a graphic representation of the same canon, with a black square at every beat and time flowing from left to right.

Figure 1:

A rhythmic canon with five voices.

A further equivalent definition for finite $A,B < N$ , very useful as we will see, uses polynomials. It is customary to introduce a generating function $A(x)$ as follows: Let $sum k A(x) = x k (- A$ and similar for $B$ . As a translate of rhythmic motif $A$ is obtained by multiplying the associated polynomial $A(x)$ by a $xl$ , it is easily seen that

Proposition 1 For $A,B < N$ both finite, we have a canon with inner and outer rhythm $A,B$ iff $A(x).B(x)$ is a polynomial with only 0’s or 1’s for coefficients.

Up to a change of the time origin, it can be assumed that both sets $A$ and $B$ begin with 0 (i.e. $A(0) = B(0) = 1$ for the associated polynomials). This we will assume throughout the paper. Such polynomials are called 0-1 polynomials and may be viewed as elements of several rings (see section 4.2). The limitation on the finiteness of $A$ and $B$ will be discussed below (see Theorem 2 and Theorem 3).

The number of non zero coefficients in $A(x).B(x)$ will be exactly the product of the number of non zero coefficients in $A(x)$ and $B(x)$ . These numbers are the