Bridging scale theory and geometrical approaches to harmony: the voice-leading duality between complementary chords
Daniel Harasim, Stefan E. Schmidt, Martin Rohrmeier · Journal of Mathematics and Music · 2016
Mathematical approaches to music theory provide important insights into musical structures. Despite this progress, there is at present no unified mathematical framework. The aim of this article is to present a joint model of two of these approaches, namely scale theory and geometric approaches to harmony and voice leading. We generalize Richard Cohn's voice-leading efficiency [Cohn, Richard. 1996. “Maximally smooth cycles, hexatonic systems, and the analysis of late-romantic triadic progressions.” Music Analysis 15 (1): 9–40] to a mathematical metric and investigate its properties with regard to the symmetrical set difference and the complement mapping. The voice-leading duality of complementary chords illustrates the usefulness of the model. This new fundamental principle states that certain minimal ascending voice leadings between two chords induce minimal descending voice leadings, with equal size, between their complementary chords. Consequently, this mathematical theorem explains the voice-leading duality between diatonic triads and seventh chords, and provides a novel approach to interpret the hexatonic pole from a voice-leading efficiency point of view.