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

(Halsey and Hewitt, 1978; Bazelow and Brickle, 1976). For example Halsey and Hewitt’s algebraic study on enumeration only concerns the interpretation of the cyclic group in the pitch domain. The 11th paragraph is dedicated to the discussion of »Parkettierung« (Tessellation or factorisation) of finite abelian groups. The underlying philosophy consists of considering such groups »die auch nur die geringste Chance haben, jemals in der Theorie der musikalischen Komposition eine Rolle zu spielen«. ⁷

⁷: i.e. that have the chance of playing a role in musical composition (Halsey and Hewitt, 1978, p. 190)

First of all, the family of finite abelian groups is restricted to that of cyclic groups. Non cyclic abelian groups have, in fact, »keinerlei Beziehung zur Musiktheorie im derzeit üblichen Sinne.«⁸

⁸: i.e. no relationship with music theory, as it appears today (Halsey and Hewitt, 1978, p. 200)

The problem is that music theory, as discussed in Halsey and Hewitt’s article, is concerned with chords inside a $n$ -tempered System and the restriction $n < 24$ »schliesst alle Fälle ein, die in absehbarer Zukunft für das Komponieren von Musik in Frage kommen zu können scheinen«. ⁹

⁹: i.e. closes all the possibilities that appeare to be important for music composition in a near future (Halsey and Hewitt, 1978, p. 200). This prediction has been largely refuted, as we’ll see in the following, by Vuza’s model of periodic rhythm, which does not impose any limitation to the order of the cyclic group. This shows that the problem of »tension« between a mathematical construction and a possible musical application is, sometimes, very difficult to define and to predict. For a different example consider Lewin’s GIS construction (Lewin, 1987) which offers many examples of musically relevant non commutative groups.

From this perspective, non Hajós cyclic groups play no role in the discussion since the smallest group which does not have the Hajós property has order equal to 72. Again, there are many reasons for trying to describe Vuza’s results on Canons by means of a more generalised algebraic theory, as we started in (Andreatta, 1996, 1999). For a technical presentation of the problem of classification of rational rhythms and canons by means of Mazzola’s mathematical music theory see the section 16.2.3 of Mazzola (2003).¹⁰

¹⁰: See Fripertinger (2001) for a different approach on the enumeration problem of non isomorphic classes of canons.

With regard to the benefits which music and mathematics could gain from each other, one seems to have to agree with Olivier Revault d’Allonnes that »the sciences can bring infinitely more services, more illuminations, more fecundations to the arts and particularly to music than music can bring to scientific knowledge« (Xenakis, 1985, p. 15). And that not only »musical thinking has not yet sufficiently utilized all the mathematical resources it could« but also that »given the relatively elementary level of mathematics [in the concepts employed] I would say that the interest is null for mathematics« (Xenakis, 1985, p. 15).

This study is an attempt to describe some advanced algebraic appoaches to music compositions which are particularly interesting from a mathematical perspective.

2 The Minkowski-Hajós Problem

As mentioned in the introduction, the family of Hajós groups originated by an old problem of number theory which Hermann Minkowski raised in 1896