Contributions to module-theoretic classification of musical motifs

First we have to define the mathematical objects we want to talk about.

How do we connnect music and mathematics? In a way much similar to physics: we "translate" musical data in a certain way into mathematical language and then see what assertions can be made with that mathematics.

Looking at a musical score, we observe that the tones depend on many different parameters. We recognize a horizontal orientation (time), a vertical orientation (pitch), different note forms (tone length), various symbols concerning volume, instrumentation, phrasing etc. From this complex sign structure we want to consider only those parameters that are essential for melody recognition ("neglecting" data that are irrelevant for a certain purpose is another common procedure borrowed from physics.) Choosing pitch and entry time (time of the "note on" event) of a tone, we get a representation of music as a set of points in a plane.

In classical western music there is most often a finite set of pitches (12 semitones per octave), and the time coordinates are integer multiples of a smallest unit: both pitch and entry time can be described with integer coordinates. Thus we describe a piece of music as a finite, non-empty point set in Z x Z or Z^2 (Z denoting the ring of whole numbers).

Even in this strong simplification the structures we encounter are still complex.

A piece of music is usually not perceived as a whole but sequentially from begin to end. Since in this work we are mainly concerned with motifs - i.e. local structures, appearing inside a short section in time - we simplify the model a little more by restricting ourselves to structures occurring inside one bar. Let one bar be divided equidistantly into n units. Then, if the time coordinate of a tone is x, its value with respect to the beginning of the bar is x mod n ( x % n in C syntax). Looking at the pitch parameter we observe that the usual denotation of pitches with letters ( c, c#, d etc.) describes an analogous process: pitches differing by one or more octaves are identified. If y is the pitch coordinate of a tone (the absolute number of semitones from a given center) the position inside its octave is y mod 12. Hence as the final base for our studies we get Z12 x Zn (where Zk denotes the quotient ring Z / k*Z). A musical motif is represented as a subset.

The theme of this work is motif classification. When listening to a piece of music, we observe that certain motifs keep reappearing throughout the piece, but often in slight variations: in another key, at other locations inside a bar, expanded in time, with inverted intervals or played backwards... (This is part of common practice compositional technique). Those motif variations can all be translated into mathematical language: Transposition of a motif into another key will appear as a translation (of the set representing the motif) by a vector parallel to the y (pitch) axis, time stretching is a multiplication of all x (time) coordinates with a value >1; interval and time inversion are reflections with axes parallel to the y and x axis. Those are all affine transformations mapping straight lines to straight lines; other affine transformations, such as rotations by 90 degrees, are also used in contemporary music. We conclude that affine transformations are meaningful in music and define:

If a motif is mapped to another motif by an invertible affine transformation, we call the two motifs isomorphic.

Notes

1. It has to be pointed out that the above definition (of course) does not include all "possible" transformations - such as slight "rubber band"-like deformations preserving the shape of a motif or pitch shifts, expansions and inversions based on a given scale instead of semitones. However, if we ask the question what is "possible", the answer will probably be "anything goes". Another point is the obvious fact that no composer (not even the strictest serial one) uses all "possible" transformations; the choice of the transformation group is one of the characteristics of musical style. A deeper investigation would be interesting but is beyond the scope of this work. The group of affine transformations includes surely some of the important transformations, is well-defined and, as we shall see, computer-friendly.
2. The space Z12 x Zn has the advantage that the number of possible motifs as well as the number of possible transformations is finite. On the other hand, motifs lying in different measures cannot be compared directly - a serious restriction if we want to do wide range analysis. What can be done is to refine the subdivision of one bar - e.g. a motif in Z12 x Z8 and one in Z12 x Z12 could be compared in Z12 x Z24. This raises at once the question whether or not the classification depends on the actual subdivision; and unfortunately the answer is: in some cases it does. For the most important measures, however, the dependencies are of a simple nature: depending on beginning subdivision and refinement degree, the classification gets either "coarser" or "finer". The measure-independent classification allows direct comparisons of motifs in arbitrary measures - but the question of the choice of the smallest time unit raises a similar problem.in ZxZ.