TILING THE (MUSICAL) LINE WITH POLYNOMIALS: SOME
THEORETICAL AND IMPLEMENTATIONAL ASPECTS
Emmanuel Amiot
CPGE Perpignan
1 rue du Centre
F-66570 St Nazaire,
France
Moreno Andreatta
Ircam-CNRS
1, place Stravinsky
75004 Paris, France
Carlos Agon
Ircam
1, place Stravinsky
75004 Paris, France
ABSTRACT
This paper aims at discussing the polynomial approach to
the problem of tiling the (musical) time axis with translates of one tile. This mathematical construction naturally
leads to a new family of rhythmic tiling canons having the
property of being generated by cyclotomic polynomials
(tiling cyclotomic canons).
1. INTRODUCTION
Tiling problems in music theory, analysis and composition have a relatively old history in mathematical music
theory. Surprisingly, despite the well-known canonical
equivalence (isomorphism) between a well-tempered division of the octave and the cyclic character of any periodic rhythm [19], the study of some tiling properties of
the time-line by means of translates of a given rhythmic
tile (or some usual transformations of it) is a relatively
new research area inside mathematical music theory. Dan
Tudor Vuza's algebraic model of tiling canon construction
by the factorization of a cyclic group into a direct sum of
two subsets [20] gave a strong impulse to the implementation of algebraic methods in music composition. 1 In this
paper we focus on cyclotomic polynomials. Some preliminary definitions about cyclotomic polynomials, tiling
of the line process and rhythmic tiling canon construction will be provided in Section 2. In Section 3 we show
how this approach has been implemented in OpenMusic
visual programming language and discuss some difficulties in directly applying the cyclotomic factorization to the
canon construction. In Section 4 we discuss some interesting connections between Vuza's original model of Regular Complementary Canons of Maximal Category [20]
and the polynomial approach by also showing how both
approaches are intimately related to some mathematical
conjectures.
1 For a detailed presentation of the group factorization approach to
the construction of tiling canons, together with the OpenMusic implementation, see [5]. For a combinatorial discussion of Vuza's model, also
see [9] and [12]. For a different compositional approach to the problem
tiling the line, see [13].
2. SOME PRELIMINARY DEFINITIONS
This section provides some definitions on cyclotomic polynomials and some general factorization theorems.
2.1. 0-1 polynomials and rhythmic tiling canons
A rhythmic tiling canon is a decomposition of a cyclic
group Z, into a direct sum of two subsets:
Z,= AeB
An enhancement of the ambient structure originates to
[16]: put A(x) = CaEA xa, then the above equation becomes a relation between 0-1 polynomials, that is to say
polynomials with coefficients being either 0 or 1:
A(x)xB(x) - 1+x+x2 + +Xn-1 (mod xn-1)
Factors of the polynomial A, (x) = 1+x+x2 +...n-1
are thus of paramount importance, especially those with 0 -1 coefficients. We find a number of them by considering
the cyclotomic approach.
2.2. Cyclotomic polynomials
Definition 1 The nth cyclotomic polynomial is
n(x) = ([ (x-e2i k )
gcd(k,n)=l
This the monic polynomial whose roots are the primitive
units of order n, that is to say the ( E C for which z" = 1
though zXr 1 for 1 < r < n.
A classical result states that these polynomials have integer coefficients, i.e. n,(x) E Z[x].
Another classical result states that they are irreducible
in the euclidean ring Q[x], and hence in Z[x]. Another
way to put it is that any polynomial in Z[x] having a primitive unit root of order n has ~, as a factor.
Directly relevant to rhythmic canons is the fact that the
product of a selection of cyclotomic polynomials can be
expressed by the following equations:
- 1 = d d()
An (-) = dLn,d__1 d(x) (1)
0
