1.3 Counterpoint

The mathematical model of counterpoint (Mazzola, 1990a) was first used in the context of neurophysiological investigations via Depth-EEG (Mazzola, 1995b), where we tested the perception of consonances and dissonances in limbic and auditory structures of the human brain. In that research project, classical European theories--following Johann Joseph Fux (Fux, 1742) as a typical reference--were our objectives. However, the model later, with the thesis of Jens Hichert (Hichert, 1993), turned out to have a similar extension to other interval dichotomies, and again, it turned out that the European choice was an exemplification of a »anthropic principle«.

We shall only sketch the core structures here to illustrate the modeling methodology. Some more technical details are given in section 5 below. This counterpoint model starts from a specific 6-by-6-element dichotomy of the twelve interval quantities modulo octave which are modeled as elements of , i.e., prime = 0, minor second = 1, etc., major seventh = 11. So the classical contrapuntal dichotomy is . This dichotomy has a unique autocomplementarity symmetry , i.e., . In this theory, such dichotomies are called strong dichotomies. There are six types (i.e., affine orbits) of strong dichotomies. If we draw the dichotomies as partitions of the discrete torus given by the Sylow decomposition of (in fact the torus of minor and major thirds!), then it turns out that the classical dichotomy has a maximal separation of its parts on the torus among the six strong dichotomy types. It has a remarkable antipode dichotomy which has its parts mixed up more than any other strong type, this is the major dichotomy the first part of which consists of the proper intervals of the major scale when measured from the tonic!

For each strong dichotomy, the results of Hichert enable a new and historically fictitious counterpoint rule set. These six >worlds of counterpoint< are quite fascinating for several reasons, one of which we shall now make more explicit. It deals with the seven-element scale in which the counterpoint rules are realized²

. If one looks for the diatonic scales (those having only semi-tone and whole-tone intervals for successive notes) where the freedom of choice of a successor interval to a given interval is maximal in Fux counterpoint (dichotomy ), then the major scale is best, and it has no cul-de-sac, i.e., it is always possible to proceed from one consonant interval to another such interval under the given rules. The latter result is, by the way, a fact which has never been demonstrated in a logically consistent way in musicology... And the major scale has cul-de-sacs only for the major dichotomy . Among the scales with seven tones without cul-de-sac for the major dichotomy, no European scales appear! However, there is a scale without cul-de-saces for . It is nearly a »mela« (No. 15 = {0,1,3,4,7,8,9}), i.e., a basic scale for Indian ragas. And it is very similar to the consonant half of the Fux dichotomy.