2024/01/28 by Rietsch, Konstanze
#00A65 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Representation Theory (math.RT) #Sound (cs.SD)
paper · doi:10.48550/arxiv.2401.15692
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 `A2 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-A2 examples (different from Euler's Tonnetz), and a tonnetz on a sphere that encodes all major ninth chords.