Configurations, Tessellations and Tone Networks

Technical Summary: Configurations, Tessellations and Tone Networks

Problem Statement
The paper addresses the mathematical representation of the tonnetz, a network of relations between the twelve notes of the chromatic scale traditionally visualized as a tessellation of the Euclidean plane by equilateral triangles or regular hexagons. While the tonnetz is well-established in music theory as a graph representing triadic harmony (specifically the relationships between major and minor triads via P, R, and L transformations), the authors seek to deepen its mathematical foundation. The central problem is to determine whether the tonnetz can be rigorously realized not merely as a graph or a tessellation, but as a specific geometric configuration of points and lines in $\mathbb{R}^2$. Furthermore, the authors investigate whether this geometric framework can be extended to tetrachords (specifically the "Tristan genus" of dominant sevenths and minor sixths) to provide a unified basis for analyzing complex harmonic progressions in the works of Wagner, Chopin, and Tchaikovsky.

Methodology
The authors employ a synthesis of graph theory, combinatorial geometry, and music theory.

1. Graph Theoretic Modeling: They first model the tonnetz as a finite, regular bipartite graph of degree three and girth six. In this model, white vertices represent major triads and black vertices represent minor triads. This graph is identified as the Levi graph of a combinatorial configuration.
2. Geometric Realization: The authors map this Levi graph to an incidence geometry consisting of 12 points and 12 lines in $\mathbb{R}^2$. They utilize the classification of combinatorial configurations of type $\{12_3\}$ (12 points, 12 lines, 3 points per line, 3 lines per point) established by Daublebsky von Sterneck.
3. Configuration Identification: By analyzing the cycle structure (specifically the number and type of hexacycles, octacycles, etc.) of the tonnetz's Levi graph, they identify the specific geometric configuration that corresponds to the tonnetz.
4. Extension to Tetrachords: The methodology is extended to tetrachords by constraining the transformation rules between minor sixth chords and dominant seventh chords. Specifically, they restrict transitions to those where exactly two tones change while the "dissonant" tone is preserved. This constraint is designed to eliminate tetracycles (cycles of length four) and ensure the resulting graph has a girth of at least six, allowing it to be a Levi graph.
5. Musical Analysis: The resulting geometric models are applied to specific musical excerpts from Wagner's Tristan und Isolde, Götterdämmerung, and Parsifal, as well as works by Chopin and Tchaikovsky, to analyze chord resolutions and voice-leading paths.

Key Contributions and Results

1. Geometric Identification of the Eulerian Tonnetz: The paper demonstrates that the standard Eulerian tonnetz (based on triads) is isomorphic to the Levi graph of the self-dual configuration $\{12_3\}$ of Daublebsky von Sterneck type $D_{222}$. In this realization, major triads correspond to points and minor triads to lines in $\mathbb{R}^2$, with incidence relations defining the adjacency of chords.
2. Discovery of the Tristan-Genus Tonnetz: By applying the "dissonance-preserving" constraint to tetrachords, the authors construct a new Levi graph for the 12 dominant sevenths and 12 minor sixths. They identify this graph as the Levi graph of a different self-dual configuration $\{12_3\}$, specifically type $D_{228}$. This configuration is distinct from the Eulerian tonnetz, evidenced by a different count of hexacycles (24 in the Tristan-genus vs. 16 in the Eulerian).
3. Archimedean Tessellations: The paper introduces "Archimedean tonnetze" derived from bipartite graphs where two tones change between triads. These graphs, while not Levi graphs due to the presence of tetracycles, unfold to generate semi-regular tessellations of the plane involving dodecagons, hexagons, and squares (type $\{4, 6, 12\}$).
4. Analytical Applications:
   • Tristan Chord Resolutions: The geometric models clarify the resolution of the Tristan chord. For instance, Chopin's resolution of the chord to $B\flat_7$ is shown to follow a "floppy bow tie" trajectory (a 2p-decacycle) to a polar position in the Eulerian tonnetz, whereas Wagner's resolution to $E_7$ follows a 3p-hexacycle path.
   • Wagnerian Octacycles: The $D_{228}$ configuration reveals that the three mutually inscribed squares in the geometric figure correspond to three 4p-octacycles in the Levi graph. These octacycles are shown to underlie specific progressions in Tristan und Isolde (e.g., the Liebesnacht duet) and Götterdämmerung (the funeral pyre scene), providing a geometric explanation for the "polar" relationships and resolution paths of these chords.
   • Hamiltonian Cycles: The paper notes that the perimeter of the Levi graph forms a Hamiltonian cycle encompassing all 24 triads, offering a geometric prefiguration of the twelve-tone row concept and specific progressions found in Beethoven's Ninth Symphony.

Significance and Claims
The authors claim that their work establishes a "unifying role of Levi graphs and configurations in the theory of music." They argue that the tonnetz is not merely a heuristic tool but a specific mathematical object—a combinatorial configuration that admits three simultaneous representations:

1. As a combinatorial configuration (incidence structure).
2. As a geometric configuration (points and lines in Euclidean space).
3. As a "musical" configuration (the network of audible relations).

The significance lies in the ability of these geometric models to visualize chord space unambiguously and to serve as a "road map" for analyzing complex harmonic progressions, particularly those involving tetrachords in late Romantic music. The paper posits that composers like Wagner may have been operating within multiple harmonic systems simultaneously (classical triadic, pan-triadic, and tetradic), and that the geometric configurations provide a rigorous framework to analyze these systems. The authors conclude that such configurations can serve as a basis for both the analysis and the composition of music, suggesting future work on pentatonic and twelve-tone tonnetze based on other configurations (Desargues and Cremona-Richmond).