Contributions to module-theoretic classification of musical motifs

In this work we present a mathematical description method for music developed by G. Mazzola ([1], [2]) which is especially suitable for the analysis of one-voiced melodies. Operations used in classical music such as pitch shift, time inversion, pitch inversion etc. induce a general classification of musical motifs based on symmetry operations. The main focus was the algorithmical solution of the isomorphy problem consisting in finding methods to determine isomorphy of motifs (to find out whether there exists a geometric transformation transforming one given motif into another given motif) with a computer.

We first put together some basic results of algebra and mathematical module theory and derive various procedures to determine the isomorphyclass of musical motifs. One possibility is to find invariants, i.e. functions whose value is the same for isomorphic motifs.

The classification depends on the measure a motif is noted in. We treat only one case in detail (division of a measure into 12 units, hence 3/4 or 6/8 with 16th as smallest unit). Given this restriction, there are 5 isomorphy classes of motifs with 2 elements, 26 of motifs with 3 elements and 216 of motifs with 4 elements. The latter number was not exactly known at the beginning; part of my work was to verify a list of representants by E. Köhler ([3]). Motifs with more elements can be classified easily as well, but there is a "combinatorial explosion" in the number of classes.

I implemented some of the algorithms on a sun workstation in the programming language C; they allow the complete isomorphy classification of musical motifs with up to 4 elements.

Possible applications could be analysis of the compositional structure of a musical piece (such as to see where the first theme reappears), statistical analysis of the compositional style of a piece, composer or period (Exist there likings or dislikings for certain motif classes in the melodies? Does one composer use exceptionally many or exceptionally few different classes?). In turn the same method could be used into the opposite direction for algorithmic composition. I have analyzed several melodies by J.S. Bach, Mozart and Jon Lord dividing a melody into motifs and calculating the frequency distribution of the isomorphy classes. I also have made some experiments in algorithmic composition generating melodies with a given sequence of isomorphy classes. The application examples are not really serious, but they serve very well to illustrate the possibilities - and the limits! - of the underlying theory.

Literature

• [1] G. Mazzola, Gruppen und Kategorien in der Musik, Berlin Heldermann, 1985.
• [2] G. Mazzola, Geometrie der Töne. Elemente der mathematischen Musiktheorie, Birkhäuser, Basel 1990.
• [3] E. Köhler, Letter to G. Mazzola, Hamburg 1988.