Groupoids and Wreath Products of Musical Transformations: a Categorical Approach from poly-Klumpenhouwer Networks
| Autor: | Alexandre Popoff, Moreno Andreatta, Andrée C. Ehresmann |
|---|---|
| Přispěvatelé: | Représentations musicales (Repmus), Sciences et Technologies de la Musique et du Son (STMS), Institut de Recherche et Coordination Acoustique/Musique (IRCAM)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche et Coordination Acoustique/Musique (IRCAM)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Groupe de recherches expérimentales sur l'acte musical (GREAM), Université de Strasbourg (UNISTRA), University of Strasbourg Institute of Advanced Study (USIAS), Institut de Recherche et Coordination Acoustique/Musique (IRCAM)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche et Coordination Acoustique/Musique (IRCAM)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), Projet SMIR - Fellowship USIAS, M. Montiel |
| Rok vydání: | 2018 |
| Předmět: |
Computer science
Generalization Musical wreath product Group Theory (math.GR) category the- ory 050105 experimental psychology 060404 music [SHS]Humanities and Social Sciences FOS: Mathematics 0501 psychology and cognitive sciences Category Theory (math.CT) and phrases Transformational music theory Klumpenhouwer network [MATH]Mathematics [math] Category theory Transformational music theory Categorical variable [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT] [SHS.MUSIQ]Humanities and Social Sciences/Musicology and performing arts 05 social sciences Piano Mathematics - Category Theory 06 humanities and the arts groupoid Algebra Music theory Wreath product Computer Science::Sound Mathematics - Group Theory 0604 arts 00A65 |
| Zdroj: | Proceedings MCM 2019 Mathematics and Computation in Music Mathematics and Computation in Music, Jun 2019, Madrid, Spain. pp.33-45, ⟨10.1007/978-3-030-21392-3_3⟩ Mathematics and Computation in Music ISBN: 9783030213916 MCM |
| DOI: | 10.48550/arxiv.1801.02922 |
| Popis: | Transformational music theory, pioneered by the work of Lewin, shifts the music-theoretical and analytical focus from the "object-oriented" musical content to an operational musical process, in which transformations between musical elements are emphasized. In the original framework of Lewin, the set of transformations often form a group, with a corresponding group action on a given set of musical objects. Klumpenhouwer networks have been introduced based on this framework: they are informally labelled graphs, the labels of the vertices being pitch classes, and the labels of the arrows being transformations that maps the corresponding pitch classes. Klumpenhouwer networks have been recently formalized and generalized in a categorical setting, called poly-Klumpenhouwer networks. This work proposes a new 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 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 groupoids are linked to wreath products (which feature prominently in transformational music analysis) through the notion of groupoid bisections Comment: 16 pages, 9 figures; comments welcome |
| Databáze: | OpenAIRE |
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