Embracing Relational Abundance

[1] In his provocative article "Reconsidering Klumpenhouwer Networks" (MTO 13.2), Michael Buchler offers readers a useful analogy comparing "network structures" to "modeling clay" [par. 46]. In Buchler's hands this is cast as a dangerous similarity, but in contrast I'm excited to take a seat at the potter's wheel and throw some clay in this brief response to his essay. As with many of Buchler's observations, his analogy is equally applicable to all we do as music analysts, rather than a critique specific to "K-nets and their analytical deployment" [par. 1]. Acknowledging at the outset that K-nets "are elegant structures," Buchler goes on to state that they open "Pandora's Box of relational permissiveness" and that "clearly, the more ways that it is possible to draw equivalent relations, the less significant those relations become" [par. 2]. The alleged clarity eludes me, so I'll take a more extreme position and suggest no relations are inherently significant, that is, I propose that all relations-common or rare-are comparably context dependent. I would reformulate Buchler's criticism to state that the relational abundance generated by K-nets further emphasizes that musicality, whatever that means in a given context, is in the hands and ears of the analyst rather than the theoretical apparatus itself.(1) Of course, this is nothing new; it's true for all of our interpretive work and all of our analytical models.[2] Shifting analogies, performing music analysis is like storytelling. For example, despite his graceful precision, I doni?½t read David Lewini?½s work for his mathematical proofs of abstract relations. Instead, I read Lewini?½s essays because he was a gifted raconteur, a writer who spun compelling tales about specific musical passages, with the ability to alter my understanding of the selected music in a meaningful and permanent way.(2) A convincing analysis forges a narrative pathway through a musical passage, and individual relations or transformations only gain meaning in the context of the story. I might even suggest that relations, as well as our desire to create them, are merely a byproduct of the temporal nature of music. From this perspective, Buchler's repeated emphasis on "relatedness" as a central problem in K-net analyses, particularly the greater abstraction of potential relatedness, is off target. I will directly address some of his criticisms of K-net mechanics in the next section, but first I must note that when these abstract relations have no intrinsic meaning, the only remaining analytical benchmark is how persuasive we find any individual narrative. Again, musical relevance as the measure for our analyses is not K-net specific; it applies to all our endeavors. We are writing analytical fictions about musical gestures as they unfold in time and/or space.[3] Although Buchler's primary concern with K-nets is their potential promiscuity, he only compares them to the canonical operators transposition and inversion. Furthermore, he does so in terms of the abstract equivalence classes generated by those very operations, thus stacking the deck against K-nets. A good example is his defamatory characterization of [026]: "once [026] comes to the party, every other trichord enjoys at least one mutual relation" [par 36]. While the metaphor is entertaining, [026] can never come to the analytical party, only a member of [026] can occur in a musical work. In the literature under examination, transformations occur among pcsets, generally not set classes. In that practical context, K-net promiscuity is not nearly the serious problem suggested by Buchler. In the puritanical world of canonical transforms, {0,2,6} can only have relations with twenty-three of the remaining possible 4,094 pcsets; thati?½s less than 1% of the population.(3) This inability to communicate with other pcsets creates glaring problems of fragmentation and gaps in many post-tonal analyses.(4) In the more social circle of K-nets, {0,2,6} can have relations with approximately 3. …