Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles
David I Lewin · Music Theory Spectrum · 2002
As strongly suggested by George Perle, every Klumpenhouwer network (K-net) can indeed (essentially) be embedded in a “Perle-Lansky Cycle” (or, simply, “Lansky Cycle”), a generalization of the Perle cycles basic to Perle's theories of Twelve-Tone Tonality. This enables the K-nets to be transformed by certain spatial operations idiomatic to the nature of Perle's cycles. On the other hand, PL-cycles can be themselves regarded as certain species of K-nets, and that enables their graphs to be interrelated and organized, in certain contexts, by operations of “isography,” operations idiomatic to K-net theory that are particularly well adapted both to building and examining transformational structures at various hierarchic levels and to exploring in particular any recursive features of such structuring. By way of approach to these theoretical topics, analytic observations are made about passages from Webern and Schoenberg. The analyses demonstrate sorts of observations idiomatic to K-net theory as well as observations idiomatic to Perle-cycle theory.