ARRAYS AND K-NETS:
Gretchen Foley, University of Nebraska
In an article in Music Theory Spectrum (vol.12/1, 1990) David Lewin defined a Klumpenhouwer network (K-net) as 'any network that uses T and/or I operations to interpret interrelations among pcs.' A K-net is a graphic representation of the intervallic relationships among elements of a set. Lewin suggested that K-nets may be applicable to aspects of George Perle's twelve-tone tonality, a theory based on the conjunction of interval cycles and inversional symmetry. Perle responded in a letter to the editor (Spectrum , vol.15/2, 1993), noting several points of intersection involving trichordal and tetrachordal pc segments. More recently, David Lewin, Philip Lambert, David Headlam, and Philip Stoecker have continued the dialogue to some extent (Spectrum, vol.24/2, 2002). Their discussions only touch on the various relationships between K-nets and Perle's cyclic sets (entities formed by alternating inversionally related interval cycles).
This paper explores how K-nets are applicable at deeper levels of Perle's theory, that of array relationships. The paper first examines the relationship among cyclic set segments and complexes, then progresses to chords generated by rotating the component cyclic sets of a single array, and then arrives at the more abstract level of relationships among different arrays, observing the various types and degrees of isographies that obtain. The paper concludes with suggestions for further study at the hierarchical levels of synoptic modes and keys. Such investigations will confirm the structural integrity and cohesion that permeate all levels of Perle's twelve-tone tonality.