Thoughts on Klumpenhouwer Networks and Mathematical Models

Abstract

[1] In the past seventeen or so years since the publication of pioneering work by David Lewin (Lewin 1990) and Henry Klumpenhouwer (Klumpenhouwer 1991), the theory of Klumpenhouwer networks (hereafter referred to as K-nets) has established itself as an important topic in music theory, but it seems that it is still in its youth. Michael Buchler's recent critical examination of K-net theory (Buchler 2007) is indicative of both its standing in the discipline of music theory and the need for ongoing evaluation of its premises. In this essay I respond to one of the four central issues in Buchler's thoughtful article: K-nets and their association with dual transformation. The juxtaposition of K-nets and dual transformation leads to a broader engagement with some fundamentals and phenomenological and metaphorical considerations of mathematical models in music theory. Remodeling K-nets as dual transformations, as Buchler advocates (Buchler 2007, 7), indeed elucidates transformational voice leading, but dual transformations also dispense with the network organization of K-nets, and thus evince fundamentally different structures than K-net isographies.K-nets, set theory, and graph theory[2] K-nets are typically employed to model sets of pitch classes through internal transformational pairings of pitch classes by the canonical operations of transposition (Tn) and inversion (In).(1) While a relatively large number of K-net representations can be constructed for any set, the number of well-formed K-net representations that can be constructed is considerably smaller.(2) Selecting one K-net representation from the (well-formed) alternatives is an interpretive act; when employed in analysis, the selection is based on musical context, usually in consideration of the relationship of one K-net to its successor(s) or predecessor(s). We turn now to theoretical aspects of K-nets.[3] K-nets call upon the mathematical models of set theory and graph theory, models of different conceptual origins, as Jeffrey Johnson describes:Graph-theoretical constructs are not dependent on Cartesian coordinates, but only on the abstract relationship between dots and connecting lines. Graph theory had its origin in solutions to puzzles and games rather than more intense desires to describe motion, quantify measurement, and, bring order to scientific inquiry that impelled other branches of mathematics (Johnson 1997, 12).A graph models relations between pairs of elements, and consists of two sets: a set of vertices and a set of edges. The vertex set in a K-net consists of pitch classes, while the edge set consists of ordered pairs of vertices, each representing the transformational relation between a pair of pitch classes. K-nets are directed graphs (also called digraphs), because each edge has a specific direction or orientation.(3) (The double-headed arrow representing the inversion operation on pitch classes, identifying the operation as an involution, stands for edges with dual orientations between a single pair of vertices.) Strictly speaking, a graph is comprised simply of dots or points without content (the vertices) and connecting lines (the edges); it is an abstract construction independent of the elements that motivate and occupy the vertices. A network is a graph whose vertices have assigned values.(4) A network is thus, in a sense, an applied graph.[4] K-net theory adopts the terms node for vertex and arrow for edge, following David Lewin's definition of node/arrow system,(5) which corresponds to digraph, as described above. The distinction between graph and network likewise follows Lewin's distinction between transformation graph and transformation network (Lewin 1987), and is nicely summarized by Julian Hook: "The graph is...the more abstract of the two structures; a network adds information to a graph, bringing it to life in a real musical setting via the node labels" (Hook 2007, 7). …