Mathematics – Group Theory
Scientific paper
2012-04-14
Mathematics
Group Theory
20 pages, 5 figures
Scientific paper
Transformational music theory mainly deals with group and group actions on sets, which are usually constituted by chords. For example, neo-Riemannian theory uses the dihedral group D24 to study transformations between major and minor triads, the building blocks of classical and romantic harmony. Since the developments of neo-Riemannian theory, many developments and generalizations have been proposed, based on other sets of chords, other groups, etc. However music theory also face problems for example when defining transformations between chords of different cardinalities, or for transformations that are not necessarily invertible. This paper introduces a categorical construction of musical transformations based on category extensions using groupoids. This can be seen as a generalization of a previous work which aimed at building generalized neo-Riemannian groups of transformations based on group extensions. The categorical extension construction allows the definition of partial transformations between different set-classes. Moreover, it can be shown that the typical wreath products groups of transformations can be recovered from the category extensions by "packaging" operators and considering their composition.
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