Three Tone Networks and a Tessellation

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 so-called Levi graph uniquely determines the combinatorial geometry of a remarkable configuration of twelve points and twelve lines in the real projective 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 principal hexacycles and the three principal octacycles, crucial for the understanding of nineteenth-century voice leading, can be read off rather directly as properties of the configuration. We show how analogous tone networks can be constructed for pentatonic music and twelve-tone music.