Contributions to module-theoretic classification of musical motifs
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Classification of motifs with 2 elements: Invertible expansion transformations

There is one step still missing for the completion of the classification of motifs with 2 elements: two elements (0,n) and (0,m) can possibly be transformed into each other by an invertible time expansion. While this is trivial in Z^2 (it just means that we need not bother about the sign), this is not the case in (Z12)^2 where there exist the following non-isomorphic values: 1 (equal to 5, 7, 11), 2 (equal to 10), 3 (equal to 9), 4 (equal to 8) and 6. Hence we have 5 isomorphy classes.

The multiplication with 7, by the way, is called the "cycle-of-fifth-isomorphism". Applied to the pitch parameter, a set of 7 directly succeeding elements {y, y+1, y+2, ..., y+6} is transformed into a diatonic major scale (and, the transformation being its own inverse, a major scale is transformed into a set of directly succeeding elements). As we see, it has a musical meaning as well (for "common-practice" classical music usually not for motif variation, though).

Non-invertible transformations

What does the classification look like if we consider non-invertible transformations as well? If we take two motifs in the standard form derived in the last chapter:

m1 = {(0,0), (n,0)}, m2 = {(0,0), (m,0)}

then there exists a time expansion mapping m1 to m2 if and only if m is a multiple of n. So we can at once draw the Hasse diagram that shows the hierarchy of the isomorphy classes in (Z12)^2:

    1
   / \
  2  3
 / \/
4  6

We see that all motifs hang together in some way; in particular, a motif of class 1 can be transformed into every motif.

Distribution of the classes

When I started to analyze melodies counting the occurring classes, I was first surprised that the distributions all seemed to look alike: there would be certain classes always very frequent and others always very sparse. The reason for this becomes clear if we look at how many elements are contained in the different classes. Below is a table with the classes of all motifs in (Z12)^2 of the form { (0,0), (x,y) } - the class is written on the position (x,y).

Of totally 12^2-1 = 143 motifs, 96 belong to class 1, 24 to class 2, 12 to class 3, 8 to class 4 and 3 to class 6. The differences are huge! Hence, the absolute majority of all motifs being isomorphic, any analysis based on motifs with 2 elements alone is not likely to say a lot.

Invariants of the isomorphy class of a motif

We are now looking for functions (especially easy-computable ones) whose value are the same for isomorphic motifs. Every such function induces a subdivision of motifs into "classes" (one "class" containing all motifs with the same value of the invariant). These subdivisions are generally coarser than isomorphy, but with a combination of two invariants we will be able to classify all motifs in (Z12)^2 with 3 elements

Invariants (1): Recursive classification

Let k1, k2 be arbitrary motifs. Now, if k1 is isomorphic to k2, then, clearly, every subset of k1 is isomorphic to some subset of k2. Hence, the set of the isomorphy classes of all subsets of a motif is an invariant of the isomorphy class of this motif! Now we see how valuable the complete classification of the motifs with 2 elements is, for it provides us with a whole family of recursive invariants - one for each number of elements. Moreover, the subdivision into classes obtained from ANY invariant can in turn be used for the classification of motifs with more elements. Note that the ordering of the subsets does not matter.

Invariants (2): Volume

Let now k be a motif with 3 elements. The 3 elements form a triangle (or a straight line, in special cases). Any invertible affine transformation transforms this triangle into another triangle (and any straight line into another straight line). The area of the new traingle is the area of the original one multiplied with |det m| where m is the matrix belonging to the transformation. Hence we have:

How is the area of the triangle computed? Let the motif - via translation - be brought into the form { (0,0), (x1,y1), (x2,y2) }. Then the area of the triangle is |det m|/2, where m is the matrix

The case of a straight line is the degenerate case where the area of the "triangle" is 0 (the two vectors are linearly dependent).

Explicit formula: |(x1*y2 - x2*y1)/2|.

DEFINITION:

For R=Z or Z12, the VOLUME of a 3-element motif {v1, v2, v3} where v1, v2 and v3 are vectors in R^2 is defined as

det (v2-v1, v3-v1)

For a motif with more than 3 elements, we define the volume to be the greatest common divisor of the volumes of all 3-element subsets.

The volume is an invariant of the isomorphy class of a motif, up to nvertible elements in R.

Notes:

1. The important property - that the volume is invariant - is independent of the division by 2 that we skip for the definition. (Instead of a triangle, we could speak of the parallelogram spanned by the vectors.)
2. We also skip the absolute value in our formula since this is contained in the mention of invertible elements (-1 is invertible).

And now we are ready for the

Classification of motifs in (Z12)^2 with 3 elements

THEOREM:

The isomorphy class of a motif in (Z12)^2 with 3 elements is uniquely determined by its volume and the isomorphy classes of its subsets.

Proof: by trying it out on all motifs. Note: This does not hold for the measure-independent case Z^2.