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

This could be called a trivial canon, for classification’s sake: the period of the canon is really the number of notes of the motif. Indeed, it is easily seen that $Z = |_| (A'+ n k) k (- Z$ So $A'$ tiles, as $A' o+ nZ = Z$ .

It must be stressed that this not the general case, indeed this is what Hajós (or Vuza)’s theorems are about. But

Theorem 8 (de Bruijn) When a tile is of prime size, $|A |= p$ , then $A$ is of the above form, i.e.

A(x) =_ 1+ x + x2 + ...+ xp- 1 (mod xp - 1)

The proof is not too difficult (using Lemma 2 below, $A$ must have a $P a p$ as a factor and reduction $(mod xp- 1)$ leaves no choice but $a = 1$ ). See also Theorem 9 (Newman, 1977), extending this to $n = pa$ with some complications.

These questions of equirepartition bring to mind a number of fascinating related issues, among which Fourier analysis.

Indeed several tiling problems in finite dimension have led to the so-called Fuglede-spectral conjecture well explained in Laba (2002), where regularity in a tiling is equivalent to exhibiting a Hilbert base of a function space on a tile. To state it more precisely:

Spectral Set Conjecture 1 A region $T$ tiles $n R$ (by translations) if and only if there is a set of exponentials

S = {ec |c (- /\} ec : x '--> exp(2ipc.x)

whose restrictions to $T$ are a Hilbert base of $L2(T)$ .

To give the simplest example, $T = [0,1[$ tiles $Z$ and the $en : t '--> e2ipnt,n (- Z$ are a Hilbert base of 1-periodic functions. Apparently very slow progress has been made in that direction of late.

3 Recent Results

Now we turn to recent results in connection with our subject.

3.1 Around Cyclotomic Polynomials

3.1.1 A useful tool: the $Pn$

From now on, we will make heavy use of the polynomials associated with finite subsets of $N$ . For the sake of clarity, I recall the definition of a polynomial associated with a subset of $N$ :

sum A(x) = xi i (- A