Abstract
In this work we provide a categorical formalization of several constructions found in transformational music theory. We first revisit David Lewin’s construction of a Generalized Interval System (GIS) to show that even a subset of the GIS conditions already implies a sequence of functors between categories. When all the conditions in Lewin’s definition are fullfilled, this sequence involves the category of elements \(\int _\textbf{G} S\) for the group action \(S :\textbf{G} \rightarrow \textbf{Sets}\) implied by the GIS structure. By focusing on the role played by categories of elements in such a context, we reformulate previous definitions of transformational networks in a \(\textbf{Cat}\)-based diagrammatic perspective, and present a new definition of categorical transformational networks, or CT-Nets, in more general musical categories. We show how such an approach provides a bridge between algebraic, geometrical, and graph-theoretical approaches in transformational music analysis. We end with a discussion on the new perspectives opened by such a formalization of transformational theory, in particular with respect to \(\textbf{Rel}\)-based transformational networks which occur in well-known music-theoretical constructions such as Douthett’s and Steinbach’s Cube Dance.
A. Popoff—Independent Researcher.
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Notes
- 1.
By an abuse of notation, for a given group G, we will notate throughout this paper its corresponding single-object category as \(\textbf{G}\). All functors are assumed to be covariant.
- 2.
In the particular case of a group action, the category of elements \(\int _\textbf{G} S\) is also called the action groupoid. As discussed in the next section, this paper considers more general cases and we thus retain “category of elements” as a common unifying term.
- 3.
During the redaction of this manuscript, the authors have been made aware by Dmitri Tymoczko of his current work on groupoids. Category of elements are encountered as a common point between his approach, rooted in geometrical and topological considerations, and ours, which stems from algebraic ones.
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Acknowledgments
We thank Andrée Ehresmann for many fruitful discussions on this topic. We also thank the reviewers for their detailed and critical comments on the first version of this paper. We acknowledge the support by the lnterdisciplinary Thematic lnstitute CREAA, as part of the ITI 2021–2028 program of the Université de Strasbourg, the CNRS, and the Inserm (funded by IdEx Unistra ANR-10-IDEX-0002, and by SFRI-STRAT’US ANR-20-SFRI-0012 under the French Investments for the Future Program).
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Popoff, A., Andreatta, M. (2024). Hidden Categories: A New Perspective on Lewin’s Generalized Interval Systems and Klumpenhouwer Networks. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol 14639. Springer, Cham. https://doi.org/10.1007/978-3-031-60638-0_8
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