Generalizations of Euler's Tonnetz on triangulated surfaces

We give a definition of a what we call a "tonnetz" on a triangulated surface, generalizing the famous Tonnetz of Euler (Euler, Leonhard. 1739. Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae. Saint Petersburg Academy, 147). In the modern interpretation of Euler's Tonnetz, the vertices of a regular "A2 triangulation" of the plane are labelled with notes, or pitch-classes. In our generalization, 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 the mathematical Langlands duality and major/minor duality. We also construct "exotic" type-A2 examples (different from Euler's Tonnetz), and a tonnetz on a sphere that encodes all major ninth chords.