Generalisations of Euler's Tonnetz on triangulated surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917577165373440 |
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| author | Rietsch, Konstanze |
| author_facet | Rietsch, Konstanze |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15692 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalisations of Euler's Tonnetz on triangulated surfaces Rietsch, Konstanze Combinatorics Sound Representation Theory 00A65 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. |
| title | Generalisations of Euler's Tonnetz on triangulated surfaces |
| topic | Combinatorics Sound Representation Theory 00A65 |
| url | https://arxiv.org/abs/2401.15692 |