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

technical but it is the most convincing. I will not reproduce here any of these proofs but only give their gist, which is threefold:

Let $m = max A- min A + 1$ be the width of motif $A$ , and consider time spans of width $m$ . There is only a finite number of possible sequences of entries in such a period of time (at each beat, either a new voice begins, or it does not), at most $2m$ certainly. So in a large enough collection of time spans of width $m$ beginning at different beats, there must be at least two identical sequences of entries by the pigeonhole principle.
In other words, there is but a finite number of possible combinations.
The next point is to prove that such a large sequence determines the whole tiling of $Z$ . The fact that the sequence is wider than the whole motif can be used to prove that there is only one way to plug in the next voices.
In terms of orchestration, an idle musician is always waiting for the next gap to get in.
Now if the construction is the same starting at two different places, and wholy determined by what occurs in such a sequence, it means that the whole phenomenon is periodic (up to the difference of time between the two sequences).
Again in musical terms, when you are recognizing something you have heard before, it means that a repetition is taking place.

We must stress this is particular to tilings of $Z$ with one tile. So it is useless to look for ’random’ repetitions of one single motif, though it is possible with one motif and its retrogradation: (Lagarias and Wang, 1996) mentions ${0,1,5}$ which, together with ${0,4,5}$ , allows any arrangement of the two 9-tiles on figure 5, which had been found also by Johnson by systematic search of short tilings with several 3-tiles.

Figure 5:

Random concatenations of these two tiles give aperiodic tilings.

To this day, little is known about tilings under a group of translations and central symmetries, even in the one dimensional case.

2.3 Affine Transformation within a Fixed Period

It is possible to expand the rhythmic intervals in a motif without changing the period: the process has often been applied by composers, be it in the time or the pitch space (for instance Alban Berg’s multiplies the basic Lulu’s serie by 5 to get Scholl’s), consisting of an affine transform with a ratio relatively prime to the period.