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

The smallest $n$ for which $Z/nZ$ is bad is $n = p2q3,p = 3,q = 2$ i.e. $n = 72$ . An example is given below. A good historic outline of this is (Andreatta, 1997). Strangely, the first historical example of a bad group was $Z/108Z$ , though this group was forgotten by Vuza (and rightfully restored by Andreatta).

It is worthy of note that from the 36 canons given by Vuza’s algorithm for $n = 72$ , define only two orbits under the full group of affine transformations (as also pointed out by Thomas Noll, Harald Fripertinger).

2.5 Reduction

In 2002, at the MaMuX seminar of IRCAM (Amiot, 2002) I wondered if something alike to non-Hajós groups did exist in the monoid $N$ , that is to say aperiodic tilings of a line, which would have been an even rarer material than aperiodic tilings of a loop. The answer stems from a rather hidden (though often alluded to) result:

Definition 5 The $m$ -zoom of the rhythmic canon with inner rhythm $A$ and outer rhythm $B$ is the canon with inner and outer rhythms $' ' A ,B$ where in terms of the generating polynomials

' 2 m-1 m ' m A (x) = (1 +x + x + ...x ).A(x ) B (x) = B(x )

Meaning musically that:

to get $A'$ , each note (resp. each silence) in $A$ is replaced by $m$ consecutive notes (resp. silences)
to get $B'$ , the metronomic tempo is multiplicated by $m$ : for instance for $m = 2$ , a beat in quarter notes should be replaced by a beat in eighth notes.

Theorem 6 (de Bruijn) Every tiling (by a finite tile of more than one element) of $N$ is reducible to a smaller tiling, i.e. is an $m$ -zoom of a smaller tiling for some $m$ .

The demonstration (by induction) is a lemma (de Bruijn, 1955) in a paper on British number systems (sic!). Several years elapsed before the relevance of this lemma to factorisation of semi-groups was noticed. Hence, as any tiling of $N$ is periodic (a combination of Theorem 3 and Theorem 1), the above Theorem reads

Every tiling of a finite range ${0,2,...,n- 1}$ is reducible to a smaller tiling.

This means that a canon »tiling a line« in Johnson’s sense (with only translations of only one tile) must have--mathematically-- a very simple structure. Indeed it could be built from scratch (that is to say from one note, played once) and recursively replacing

one note by a succession of $p$ notes in the same voice, or
one voice by a succession of $p$ voices (using the duality between inner and outer rhythm)