Academic journal article Music Theory Online

Reconsidering Klumpenhouwer Networks: A Response

Academic journal article Music Theory Online

Reconsidering Klumpenhouwer Networks: A Response

Article excerpt

[1] My initial response to Buchler's essay is to recommend that if a reader doesn't find K-nets useful or interesting, then by all means she should use another approach. Moreover, I agree with Buchler that one should refrain from concentrating all one's efforts on K-net analysis, although not for Buchler's reasons.

[2] While this kind of reaction satisfies me, I recognize it will not satisfy others, who will expect a bit more effort on my part, and I do feel it may be worthwhile to challenge some of the problems I see in Buchler's position on K-nets, because I regard them primarily as problems of general analytical methodology and so have consequences for many contexts in addition to K-net analysis.

[3] As I see it, Buchler's criticism of K-net analysis has two elements. First, the essay finds a number of things wrong with the technological character of K-nets, and about analytical approaches that feature a sense of structural depth. Second, some of the problems Buchler has with K-nets are at root problems with the analytical philosophy K-nets seem to embody, but are by no means limited to that approach alone.

I. K-nets and Dual Transformations

[4] Buchler begins his essay with a critique of the use of /, the iconic automorphisms of the Tn/In group, as a way of relating K-nets. He sets against the theory of automorphisms O'Donnell's dual transformations as better way of relating K-nets-better because automorphisms are unnecessarily complex and because they encourage us to create structural depth in K-nets analyses, which we are later told, is a musical analytical error.

[5] From the perspective of network theory, dual transformations and automorphisms do not address the same element of structure. Automorphisms map the arrow labels of one network onto another. They do not operate on node content. Dual transformations are meant to map the node contents of one network onto another, but they do not map arrow labels onto each other.(1) One needs automorphisms (of some sort) to do that. Furthermore, dual transformations apply to different networks than K-nets; they apply to networks that contain set classes (both singletons and larger collections.) We are no longer talking about K-nets, which is perfectly fine.

[6] We are still talking, however, about networks-networks that have only two nodes and no arrows (although there seems to be some variation in how one is to think about the relationship between the two nodes). The formal nature of the relationships that map one two-node network onto another requires a great deal more theorizing than one gets here, but thinking along these lines has some intriguing potential.

[7] Nevertheless, such a system is not simply a more efficient way of doing the same things K-nets do. The style of musical logic involved in this model is almost entirely different from the musical logic brought about by K-nets. Thinking about individual pitch-classes related under Tn or In does not have the same musical meaning as arranging pitches into two pc collections internally related by intervals and related to each other by some In. The only way we can declare one style of network as more efficient than the other is to focus exclusively on the ability to relate the same musical structures one finds in the score. In other words, one can only assert the claim "the same but more efficient" by ignoring entirely the music-theoretical content of each approach, by failing to take into the account the styles of musical ideation and experience the technology involves. As we shall see later, an aversion to theoretical content is a characteristic of a number of the specific critiques Buchler launches at K-net theory.

[8] For now, it is worth pointing out that in addition to the claim (which is asserted but under-substantiated) of unnecessary complexity, Buchler's critique of the K-net technology is also motivated by a desire to render impossible recursive structuring, which gets a fairly through beating later on in Buchler's essay. …

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