1.3 Aperiodic Canons

It is very easy to make very repetitive canons, by playing a periodic voice:

1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

and stuffing it with copies of itself, like the following:

1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1       1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1         1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1           1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

Indeed this happens quite frequently in »real« music. But we would like to know if less regular canons are possible.

So we will give a precise meaning to »avoid too much regularity in a rhythmic canons«:

• avoiding too regular entries of voices and
• avoiding a rhythmic pattern with a shorter period

As it happens, this connects directly to an old problem of decomposing and more generally abelian groups into a direct sum : see Theorem 5 below. As mentioned above, from Theorem 3, the most general rhythmic canon tiling with a finite motif is equivalent to a decomposition in direct sum of the cyclic group of order :

The condition of non regularity means that there does not exist with and

The existence of such decompositions is non trivial, the first historically found had a period of 108 and the smallest possible period is 72 (cf. (Andreatta, 1997)), so any examples will have to be pretty long to write down.

We will now state a number of mathematical results. It was surprisingly hard to unravel some of these, in old papers ridden with errors and missing proofs. I will not reproduce here all proofs (see bibliography for these, especially (Sands, 1962)) but just the main ideas, especially insofar as musical concepts occur. A good historical review can be found in (Tijdeman, 1995), with extensive bibliography.

2 Older Results

In this section we discuss a number of results known before the present investigation was started.