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

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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 groupoid bisections. 1. Groupoids of musical transformations The recent field of transformational music theory, pioneered by the work of Lewin [Lew82, Lew87], shifts the music-theoretical and analytical focus from the "object-oriented" musical content to an operational musical process. As such, transformations between musical elements are emphasized, rather than the musical elements themselves. 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. Within this framework, Klumpenhouwer networks (henceforth K-nets) [Lew90, Klu91, Klu98] have stressed the deep synergy between set-theoretical and transformational approaches thanks to their anchoring in both group and graph theory, as observed by many scholars [Nol07]. We recall that a K-net is informally defined as a labelled graph, wherein the labels of the vertices belong to the set of pitch classes, and each arrow is labelled with a transformation that maps the pitch class at the source vertex to the pitch class at the target vertex. Klumpenhouwer networks allow one to conveniently visualize at once the musical elements of a set and the specific transformations between them. This notion has been later formalized in a more categorical setting, first as limits of diagrams within the framework 2010 Mathematics Subject Classification. 00A65.