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• W2728926070 abstract "FROM K-NETS TO PK-NETS: A CATEGORICAL APPROACH ALEXANDRE POPOFF CARLOS AGON MORENO ANDREATTA ANDRÉE EHRESMANN 1. INTRODUCTION RANSFORMATIONAL APPROACHES have a long tradition in formalized music analysis in the American as well as in the European tradition. Since the publication of pioneering work by David Lewin (1987) and Guerino Mazzola (1990), this paradigm has become an autonomous field of study making use of more and more sophisticated mathematical tools, ranging from group theory to categorical methods via graphtheoretical constructions (Nolan 2007). Within the transformational approach, Klumpenhouwer networks (henceforth K-nets) are prototypical examples of music-theoretical constructions unifying the three domains we just mentioned, since they provide a description of the inner structure of chords by focusing on the transformations between their elements rather than on the elements themselves. For this reason, K-nets T 6 Perspectives of New Music represent a complementary approach to the traditional set-theoretical one with respect to the problem of chord enumeration and classification. One of the main interests for the “working mathemusician” lies in their deep connections with some common constructions used in category theory. In fact, following Mazzola’s original intuition on the relevance of the categorical approach to the formalization of musical structures and process, Klumpenhouwer networks seem suitable for music-theoretical investigations making use of category theory, since they are based on concepts (such as isographies) and principles (such as the recursive network construction) which are naturally grasped by the functorial approach. However, as we have suggested elsewhere (Popoff et al. 2015), although K-nets and, more generally, group action-based theoretical constructions, such as Lewin’s “Generalized Interval Systems ” (GIS), are naturally described in terms of categories and functors, the categorical approach to transformational theory remains relatively marginal with respect to the major trend in the math-music community (Mazzola 2002; Lavelle, unpublished paper; Fiore 2011; Popoff, submitted paper). Following Lewin’s (1990) and Klumpenhouwer’s (1991) original group-theoretical descriptions, theoretical studies have mostly focused until now on the automorphisms of the T/I group or of the more general T/M affine group (Lewin 1990; Klumpenhouwer 1998). This enables one to define the main notions of positive and negative isographies, notions which can easily be extended by taking into account the affine group on ℤ12, together with high-order isographies . Since a prominent feature of K-nets is their ability to instantiate an in-depth multi-level model of musical structure, category theory seems nowadays the most suitable mathematical framework to capture this recursive potentiality of the graph-theoretical construction (Mazzola et al. 2006) or to study creativity; e.g., in music (Andreatta et al. 2013). From a graph-theoretical perspective, a K-net is a directed graph (also called digraph) where the vertices (or nodes) consist of pitchclasses (or pitch-class sets) and the edges (or arrows) are elements of the T/I group (or, in a more general setting, of the T/M affine group). As an example, we represent two K-nets in Example 1. In addition, these K-nets are 〈T2〉-isographic, in the sense that every edge of the form Tp is sent to Tp and every edge of the form Ip is sent to Ip+2. In the general case, what music theorists call the “path consistency condition” (Hook 2007) is nothing else than the composition and associativity law of morphisms, together with the definition of functors in the categorical framework. Moreover, all K-nets correspond to commutative diagrams in category theory, where the term “diagram” is taken here in a naive sense (the technical concept of a diagram will be introduced in Section 5.1.) From K-Nets to PK-Nets: A Categorical Approach 7 Commutative diagrams, together with the notion of isography as an algebraic relation between K-nets which is independent of the content of the nodes, clearly suggest that the categorical approach is the natural one for the study of any kind of networks. Moreover, category theory shows how to go beyond the K-nets commonly used in transformational analysis by defining diagrams which do not necessarily have this property and by extending the isographic relation to different levels, by considering transformations between networks..." @default.
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• W2728926070 date "2016-01-01" @default.
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• W2728926070 title "From K-Nets to PK-Nets: A Categorical Approach" @default.
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• W2728926070 doi "https://doi.org/10.7757/persnewmusi.54.2.0005" @default.
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