Modulation

Summary

This chapter deals with the central issue of modulation between two given tonalities. It involves explicit models of tonalities, of cadences and—even more crucial—of the transition process from one tonality to its successor. The present model involves the analogy to elementary particle physics: Modulation is viewed as a ‘force interaction’ between two ‘tonality particles’ which is mediated by a ’modulation quantum’. The model allows for a complete calculation of fundamental degrees of modulation in congruence with A. Schönbergs harmony [478]. The model is realized for diatonic tonalities in 12-tempered and just tuning. It has been extended to all 7-element scales in 12-tempered tuning and to a number of scales in just tuning. The 12-tempered extension reveals a privileged position of the diatonic scale with regard to this modulation theory. We conclude the chapter with a discussion of the basic role of modulation models and their application to optimize harmonic paths in the sense of section 27.2.

Denn das Wesentliche an einer Modulation ist nicht das Ziel, sondern der Weg.

Arnold Schönberg [478]