Paper
Configurations, Tessellations and Tone Networks
Authors
Jeffrey R. Boland, Lane P. Hughston
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 denoting major chords and twelve black vertices denoting minor chords. This so-called Levi graph determines a configuration of twelve points and twelve lines in 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 hexatonic hexacycles and the three octatonic octacycles, crucial for the understanding of nineteenth-century harmony and voice leading, can be read off rather directly as properties of this and its Levi graph. Analogous tone networks together with their associated Levi graphs and configurations can be constructed for pentatonic music and twelve-tone music, offering the promise of new methods of composition. When the constraints of the Eulerian tonnetz are relaxed so 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 tessellation of the plane, of a type known to Kepler, based on hexagons, squares and dodecagons. When the same combinatorial idea is applied to tetrachords of the Tristan genus (dominant sevenths and minor sixths) the cycles of the resulting bipartite graph are sufficiently ample in girth to ensure the existence of a second geometrical configuration of type , distinct from the Eulerian tonnetz as an incidence geometry, which can be used as the basis for a new approach to the analysis of the music of Chopin, Wagner, Tchaikovsky, Brahms and their contemporaries.