Configurations, Tessellations and Tone Networks

Abstract: The Eulerian tonnetz, which associates three minor chords to each major chord and three major chords to each minor chord, can be represented by a bipartite graph with twelve white vertices signifying major chords and twelve black vertices signifying minor chords. This Levi graph uniquely determines a certain geometric configuration of twelve points and twelve lines in the Euclidean plane with the property that three points lie on each line and three lines pass through each point. Interesting features of the tonnetz, such as the existence of the four hexacycles and the three octacycles of Cohn-Waller, crucial for the understanding of nineteenth-century harmony and voice leading, can be read off rather directly as properties of this configuration.  Analogous tone networks along with their associated Levi graphs and configurations can be constructed for pentatonic music and twelve-tone music, offerring the promise of new methods of composition. When the voice leading constraints of the Eulerian tonnetz are relaxed in such a way as to allow movements between major and minor triads with variations at exactly two tones, the resulting bipartite graph has two components, each of which generates a remarkable tessellation of the plane, known to Kepler, based on hexagons, tetragons, and dodecagons. The talk will present a short survey of these and various related topics in the mathematical theory of music.