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Groupoids and Wreath Products of Musical Transformations: A Categorical Approach from poly-Klumpenhouwer Networks

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Mathematics and Computation in Music (MCM 2019)

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Abstract

Klumpenhouwer networks (K-nets) and their recent categorical generalization, poly-Klumpenhouwer networks (PK-nets), are network structures allowing both the analysis of musical objects through the study of the transformations between their constituents, and the comparison of these objects between them. In this work, we propose a groupoid-based approach to transformational music theory, in which transformations of PK-nets are considered rather than ordinary sets of musical objects. We show how groupoids of musical transformations can be constructed, and provide an application of their use in post-tonal music analysis with Berg’s Four pieces for clarinet and piano, Op. 5/2. In a second part, we show how these groupoids are linked to wreath products through the notion of groupoid bisections.

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Authors and Affiliations

  1. 119 Rue de Montreuil, 75011, Paris, France

    Alexandre Popoff

  2. IRCAM/CNRS/Sorbonne Université and IRMA-GREAM, Université de Strasbourg, Strasbourg, France

    Moreno Andreatta

  3. Université de Picardie, LAMFA, Amiens, France

    Andrée Ehresmann

Authors
  1. Alexandre Popoff
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  2. Moreno Andreatta
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  3. Andrée Ehresmann
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Corresponding author

Correspondence to Alexandre Popoff .

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Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Mariana Montiel

  2. Technical University of Madrid, Madrid, Spain

    Francisco Gomez-Martin

  3. Technological University of the Mixteca, Oaxaca, Mexico

    Octavio A. Agustín-Aquino

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Popoff, A., Andreatta, M., Ehresmann, A. (2019). Groupoids and Wreath Products of Musical Transformations: A Categorical Approach from poly-Klumpenhouwer Networks. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_3

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  • DOI: https://doi.org/10.1007/978-3-030-21392-3_3

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