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In this work we provide a categorical formalization of several constructions found in transformational music theory. We first revisit David Lewin's original theoretical construction of Generalized Interval Systems (GIS) to show that it implicitly defines categories. When all the conditions in Lewin's definition are fullfilled, such categories coincide with the category of elements G S for the group action S : G → 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 Cat-based diagrammatical perspective, and present a new definition of transformational networks (called CT-Nets) in general musical categories. We show incidently 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 Rel-based transformational networks which occur in well-known music-theoretical constructions such as Douthett's and Steinbach's Cube Dance.
Archived PDF is a preprint.International audienceThis paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientific areas. The link between these two approaches was only made recently, suggesting new interesting musical applications and opening new theoretical problems. We present some old and new results on homometry, and give perspective on future research assisted by computational methods. We assume from the reader's basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform
International audienceWe represent chord collections by simplicial complexes. A temporal organization of the chords corresponds to a path in the com- plex. A set of n-note chords equivalent up to transposition and inversion is represented by a complex related by its 1-skeleton to a generalized Ton- netz. Complexes are computed with MGS, a spatial computing language, and analyzed and visualized in Hexachord, a computer-aided music anal- ysis environment. We introduce the notion of compliance, a measure of the ability of a chord-based simplicial complex to represent a musical object compactly. Some examples illustrate the use of this notion to characterize musical pieces and styles
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