Contributions to module-theoretic classification of musical motifs

Some mathematical foundations

To get a closer look at the case Z12 x Zn we will now slightly raise the abstraction level. As we said in one of the first chapters, Z12 x Zn is a Z-module. However, for most of what we want to say in this chapter, we do not need the concept of modules: it is enough to see Z12 x Zn as an abelian group (addition being the group operation).

Observe that there is a natural way to define multiplication of a group element x with an integer n (this, in fact, turns every abelian group into a Z-module) via:

where 0 denotes the unit element of the group and -x the group inverse of x.

Musical motifs appear now as subsets of an abelian group, and classification is done with respect to translations and group automorphisms.

Some basic properties of finite abelian groups and their automorphisms

The following results are merely stated here; they should be found in any basic theory book dealing with groups.

Definition:

If x is an element of an abelian group G, then the order of x (written o(x)) is defined as the smallest non-negative integer n with n*x = 0 (multiplication with integers defined as above).

If the group G is finite, then every element in it has finite order.

For any prime number p, the group elements whose order is a power of p form a subgroup of G; this subgroup is called the p-component of G (denoted as Gp). Gp exists for every prime number p, but in most of the cases it is trivial (i.e. consists only of 0.)

If Gp is non-trivial for only one prime number p (all elements of the group have an order of a power of p, i.e. G = Gp) then G is called a p-group.

The group G can be written as the direct sum of its non-trivial p-components for all prime numbers p.

If f is an automorphism of G, then o(f(x)) = o(x) for all x in G. From this follows that every p-component is mapped onto itself (p-components are characteristical subgroups of a group). Hence every automorphism of G splits up into an automorphism of each of the p-components.

Main theorem on finite abelian groups:

Every finite abelian group can be written as (is isomorphic to) a direct sum of cyclical groups.

Hence, every finite abelian p-group can be written as a direct sum of cyclical groups of the form (Z(p^k))^t. The numbers k and t are unique, but the actual decomposition is not.

If f is an automorphism of a finite abelian p-group, then f defines another decomposition of the group via the summands f( (Z(p^k))^t ) which are isomorphic.

Example: Z7 x Zn

An alternative way to define the pitch coordinate is not to count the number of semitones (from a given center) but the number of scale tones, i.e. if c is 0, then d is 1, e is 2 and so on. With this approach we could cope with pitch shifts, expansions and inversions based on the scale instead of semitones - and there would also be a closer coincidence with classical music notation since the pitch value would correspond directly with the vertical position of the notes in a score. A little complication, of course, would be notes not belonging to the given scale. Again in analogy to classical music notation with its # and b accidentials, we could save the deviation from scale value in a separate coordinate - or, as we will now, we could simply ignore them.

Examining the group Z7 x Zn, we observe that its p-component for the prime number 7 in most of the cases is Z7 - exactly the pitch coordinate. (Except, of course, the rather exotic cases where the rhythm is a multiple of 7.) Now, every automorphism mapping each p-component to itself means in this case that there is NO automorphism that mixes pitch and time coordinate in any way - i.e. no transvections, no rotations, no parameter shifts. The only invertible linear transformations are pitch inversions, time inversions, pitch expansions and time expansions - precisely the ones that are most widely used (at least in the so-called common-practice period). One might say that this is another closer coincidence with common-practice music than Z12 x Zn and therefore prefer this approach; I prefer to see the two as alternatives that both have their justification - varying, of course, with style, composer or even piece.