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

Figure 2:

A tiling mod 8.

Figure 3:

12 voices mod 72, on a torus.

Tiling a line in the sense used by (Johnson, 2001) means tiling a range of consecutive numbers. It is a rhythmic canon with a beginning and an end, without holes or double beats:

Definition 3 The set $A < Z$ tiles a line iff there exists a set $B$ and $* n (- N$ with $A o+ B = {0,1,2,...,n -1}$ .

In our example, this would be the case with $B'= {0,2,8,10}$ for instance. Now obviously

Theorem 1 A tiling of a line gives a tiling of a loop, i.e. tiling a range ${0,1,2,...,n -1}$ enables to tile both $N$ and $Z$ .

It is just a matter of repeating the tiling ad infinitum. As the tiling of a line is a special case of tiling a loop, the same Theorem 3 will show that this is exactly the general problem of tiling $N$ (with a finite tile). But Theorem 6 will show that tilings of a line are much more repetitive than tilings of a loop.

An important remark: from now on, we have a duality because in

(A o+ B) o+ nZ = Z, (A o+ B) = (B o+ A)

both $A$ and $B$ are finite and play symmetric roles. We will call $B o+ A$ the dual canon of $A o+ B$ .