Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

A modulation which has a quantum is called quantized. The main theorem now has to guarantee the existence of quantized modulations. This is the alias of the historically grown rule canon in the mathematical model. This theorem in fact guarantees quantized modulations for all couples in $Dia(3)$ , and the pivotal degrees coincide with the pivotal degrees in Schönberg’s treatise wherever he considers direct modulations (see (Mazzola, 1990a, section 5.5.2)).

The present mathematical model has the advantage that it can also be performed on any seven-element scale, and any translation class of that scale as a modulation domain. So the modulation model immerges the classical case $Dia(3)$ in a variety of modulation scenarios which have never been dealt with in historical contexts. In Muzzulini (1995), this extension has been calculated by computer programs (including explicit lists of modulation quanta and pivotal degrees) and commented. That extension exhibits a very special position of the common scales in European harmony which we summarize as follows (see Muzzulini, 1995, for complete results):

Among the modulation domains of rigid triadic tonalities, the maximum of 226 quantized modulations occurs for the harmonic minor scale.
Among modulation domains of non-rigid tonalities, the maximum of 114 quantized modulations occurs for the melodic minor scale. Among those scales with quantized modulations for all couples of their modulation domains, the minimum of 26 quantized modulations occurs for the diatonic major scale.

Besides this »anthropic principle« for modulation, the model and its extension also apply to just tuning pitch spaces, and there, where the mathematics is quite different since one works in $n Z$ , one also has good results, see Radl (1998); Mazzola (2001b). But the model and its extension also apply to compositions of tonal character. Of course, the historical context seems to be a critical point here since not every composer would compose in the framework of Schönberg’s harmony. However, the mathematical model is not a poietic model, i.e., it does not claim that the composer has used its approach to set his/her modulations. The mathematical model is more like a model in physics: The phenomena are there (in our case: the compositions), and we have to describe their structure as adequately as possible, ignoring whether the creator of the universe has ever used our mathematics, our logic or our conceptual model of physical processes. In this spirit a number of süccessful interpretations of modulatory processes, among them the hitherto poorely understood modulation architecture of Beethoven’s op.106 (»Hammerklavier«), have been realized, see Mazzola (1990a). A reconstruction of the first movement of Beethoven’s op.106 in terms of analogous structures, replacing the minor seventh chord and its satellite structures in op.106 by the augmented triad and its corresponding satellite structures, has been realized in Mazzola (1985) and published in Mazzola (2002).