Hidden Categories: a New Perspective on Lewin's Generalized Interval Systems and Klumpenhouwer Networks
Mathematics and Computation in Music Lecture Notes in Computer Science(2023)
Abstract
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
$\int_\mathbf{G} S$ for the group action $S \colon \mathbf{G} \to
\mathbf{Sets}$ 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 $\mathbf{Cat}$-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 $\mathbf{Rel}$-based transformational networks which
occur in well-known music-theoretical constructions such as Douthett's and
Steinbach's Cube Dance.
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