Configurations, Tessellations, and Tone Networks

Explore the mathematical foundations of music theory through geometric configurations and graph theory in this academic lecture. Discover how the Eulerian tonnetz, which connects major and minor chords through a bipartite graph structure, can be represented as a geometric configuration of twelve points and twelve lines in the Euclidean plane. Learn about the relationship between musical harmony and mathematical structures, including how hexacycles and octacycles crucial for understanding nineteenth-century harmony emerge as properties of these configurations. Examine the construction of analogous tone networks for pentatonic and twelve-tone music systems, along with their associated Levi graphs and configurations that offer new compositional possibilities. Investigate how relaxing voice leading constraints in the Eulerian tonnetz creates bipartite graphs that generate tessellations of the plane based on hexagons, tetragons, and dodecagons, connecting musical theory to classical geometric patterns known to Kepler. Gain insights into the intersection of mathematics and music theory through this comprehensive survey of configurations, tessellations, and their applications to understanding musical structures and composition techniques.