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Generalisations of Euler's Tonnetz on triangulated surfaces

Konstanze Rietsch

TL;DR

This work generalizes Euler's tonnetz from a plane lattice to arbitrary triangulated surfaces by assigning multisets of pitch classes to vertices, edges, and faces under coherence constraints, thereby unifying geometric and musical structure through Lie-theoretic ideas. It develops a formal framework for vertex- and edge-tonnetzes, and then builds explicit, richly structured edge-tonnetzes for crystallographic tilings of types $B_2$, $C_2$, and $G_2$, including Langlands dual pairs that interchange major/minor qualities. The paper further expands to two novel tritone-edge tonnetz constructions on the $A_2$ tiling, and a spherical, sphere-based tonnetz dubbed the jazz bauble that encodes all major ninth chords, highlighting deep connections between algebraic dualities and tonal relationships. Collectively, these examples illustrate how crystallographic reflection groups and Langlands duality can model and reveal intricate chordal organizations on diverse topologies (torus and sphere), with potential implications for both music theory and geometric representation of harmonies.

Abstract

We give a definition of a what we call a `tonnetz' on a triangulated surface, generalising the famous tonnetz of Euler from 1739. In Euler's tonnetz the vertices of a regular `$A_2$ triangulation' of the plane are labelled with notes, or pitch-classes. In our generalisation we allow much more general labellings of triangulated surfaces. In particular, edge labellings turn out to lead to a rich set of examples. We construct natural examples that are related to crystallographic reflection groups and live on triangulations of tori. Underlying these we observe a curious relationship between mathematical Langlands duality and major/minor duality. We also construct `exotic' type-$A_2$ examples (different from Euler's Tonnetz), and a tonnetz on a sphere that encodes all major ninth chords.

Generalisations of Euler's Tonnetz on triangulated surfaces

TL;DR

This work generalizes Euler's tonnetz from a plane lattice to arbitrary triangulated surfaces by assigning multisets of pitch classes to vertices, edges, and faces under coherence constraints, thereby unifying geometric and musical structure through Lie-theoretic ideas. It develops a formal framework for vertex- and edge-tonnetzes, and then builds explicit, richly structured edge-tonnetzes for crystallographic tilings of types , , and , including Langlands dual pairs that interchange major/minor qualities. The paper further expands to two novel tritone-edge tonnetz constructions on the tiling, and a spherical, sphere-based tonnetz dubbed the jazz bauble that encodes all major ninth chords, highlighting deep connections between algebraic dualities and tonal relationships. Collectively, these examples illustrate how crystallographic reflection groups and Langlands duality can model and reveal intricate chordal organizations on diverse topologies (torus and sphere), with potential implications for both music theory and geometric representation of harmonies.

Abstract

We give a definition of a what we call a `tonnetz' on a triangulated surface, generalising the famous tonnetz of Euler from 1739. In Euler's tonnetz the vertices of a regular ` triangulation' of the plane are labelled with notes, or pitch-classes. In our generalisation we allow much more general labellings of triangulated surfaces. In particular, edge labellings turn out to lead to a rich set of examples. We construct natural examples that are related to crystallographic reflection groups and live on triangulations of tori. Underlying these we observe a curious relationship between mathematical Langlands duality and major/minor duality. We also construct `exotic' type- examples (different from Euler's Tonnetz), and a tonnetz on a sphere that encodes all major ninth chords.
Paper Structure (9 sections, 2 theorems, 4 equations, 12 figures, 1 table)

This paper contains 9 sections, 2 theorems, 4 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Axioms for a generalised tonnetz on a triangulated surface
  3. Edge tonnetzes of type $B_2$ and $C_2$
  4. A dual pair of major/minor edge tonnetzes of type $G_2$
  5. Vertices and Symmetry
  6. Dynkin diagrams and the Langlands dual $G_2$ tonnetz
  7. Two major tritone-edge tonnetz examples on the $A_2$ tiling of the plane
  8. An edge-tonnetz on a sphere - the jazz bauble
  9. Overview of chords appearing in our type $B_2$ and $G_2$ tonnetz examples

Key Result

Lemma 2.3

Any map ${\operatorname{tnz}}_2:\mathcal{V}_2\to {\operatorname{Multisets}}(\mathcal{N})$ with $|{\operatorname{tnz}}_2(\sigma)|=3$ extends to a tonnetz.

Figures (12)

  • Figure 1: Euler's tonnetz on an $A_2$ tiling of the plane, with fundamental alcove highlighted.
  • Figure 2: Euler's tonnetz is a vertex tonnetz in the plane that descends to a tonnetz on a torus.
  • Figure 3: A major edge-tonnetz on a $B_2$ tiling of the plane with fundamental alcove shown in gray. This tonnetz tiles the plane doubly periodically. Alternatively it quotients to a major edge-tonnetz on a torus with $8$ triangles, $12$ edges, and $4$ vertices.
  • Figure 4: A minor edge-tonnetz on a type $C_2$ tiling of the plane with fundamental alcove indicated. This tonnetz quotients to a minor edge-tonnetz on a torus with chords, $D$-minor, $F$-minor, $A\flat$-minor and $B$-minor, each appearing twice.
  • Figure 5: A major/minor edge-tonnetz of type $G_2$ with fundamental alcove shaded.
  • ...and 7 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4: Vertex and edge tonnetzes
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Example 2.7
  • Remark 2.8
  • ...and 2 more