Tonnetz Theory, Classical Harmony, and the Combinatorial Geometry of Abstract Musical Resources

Mathematicians Jeffrey R. Boland and Lane P. Hughston have published a significant paper, 'Tonnetz Theory, Classical Harmony, and the Combinatorial Geometry of Abstract Musical Resources,' on arXiv. The work presents a formal, geometric framework for understanding music theory, treating chords, scales, and their relationships as objects in combinatorial geometry. A key finding is that the seven diatonic triads can be represented by a specific bipartite graph structure ({7_3} with girth four), which mathematically encodes their well-known harmonic relationships. Furthermore, the set of diatonic seventh chords is shown to form a Fano configuration ({7_3}), providing a complete characterization of the possible voice-leading motions between them.

The research extends this geometric lens to other musical systems, constructing a Tonnetz (tone network) for pentatonic music based on the Desargues configuration ({10_3}) and one for the 12-tone system using the Cremona-Richmond configuration ({15_3}). These structures can serve as abstract resources for musical composition and analysis. The paper also demonstrates that the relationship between the chromatic scale and major triads is represented by a D222 configuration, while minor triads correspond to a specific class of hexacycles in its associated Levi graph. This approach formally breaks the characteristic duality between major and minor triads within the network, offering a novel, non-dualistic perspective on harmonic space.