The Efficacy of K-Nets in Perlean Theory

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[1] This response addresses two topics in Michael Buchler's article "Reconsidering Klumpenhouwer Networks," namely relational abundance and the issue of hierarchy and recursion. In the abstract preceding his essay Buchler states that "K-nets also enable us to relate sets and networks that are neither transpositionally nor inversionally equivalent, but this is another double-edged sword: as nice as it is to associate similar sets that are not simple canonical transforms of one another, the manner in which K-nets accomplish this arguably lifts the lid off a Pandora's Box of relational permissiveness." I will confine my remarks to K-nets and their applicability to George Perle's compositional theory of twelve-tone tonality. In so doing, I hope to demonstrate that not only are K-nets "nice" to use in associating similar sets, they are highly efficient tools in a Perlean context.K-nets and Cyclic Sets[2] In order to establish the efficacy of K-nets in Perlean theory, we must first define cyclic sets, arrays, and axis-dyad chords. A cyclic set combines two inversionally related interval cycles, such as interval 7 and interval 5. When the members of the two cycles are placed in alternation, every pair of alternating pcs shows a consistent difference, representing the formative interval cycle. In Figure 1 the pcs in open noteheads show the cyclic interval 7, and the alternating pcs in closed noteheads show the inversionally related cycle 5.Figure 1. Cyclic set 2,9 generated by alternating inversionally related interval cycles 7 and 5Additionally, every pair of adjacent pcs forms a sum. Three adjacent pcs form two so-called tonic sums, which repeat throughout the cyclic set. In Figure 1 the three bracketed pcs 0-2-7 yield the tonic sums 2 and 9, which repeat with every trichordal segment. The second bracketed pcs 9-5-4 also sum to 2 and 9. The recurring pair of tonic sums provides the cyclic set its name; the cyclic set in Figure 1 is thus identified as 2,9. Finally, the difference between the tonic sums provides the underlying interval cycle (9-2=7).[3] The trichordal segments in Figure 1 belong to two different set classes, 3-9 (027) and 3-4 (015). As such, they are neither transpositionally nor inversionally equivalent. Yet a declaration of non-equivalence seems counterintuitive, since both sets derive from the same symmetrical cyclic set and thus share the same internal structure. The K-nets in Figure 2a clearly display this relationship. These networks are strongly isographic; that is, they have identical arrow configurations and associated T and I transformations. Moreover, all trichords in the 2,9 cyclic set are strongly isographic since they arise from the same source, the 2,9 cyclic set. Any three adjacent pcs from this set can be fitted into the nodes of the more abstract K-graph in Figure 2b. This K-graph efficiently displays the recursion inherent in the 2,9 cyclic set.Figure 2a. Strongly isographic K-nets of trichordal segments of cyclic set 2,9Figure 2b. K-graph of any trichordal segment of cyclic set 2,9[4] A different alignment of the ascending and descending interval 7 cycles from Figure 1 produces a new pair of repeating tonic sums, as shown in the 0,7 cyclic set of Figure 3.Figure 3. Cyclic set 0,7 generated by rotating cyclic set 2,9Although all trichordal segments within this cyclic set will be strongly isographic, they will not share this relationship with trichordal segments from the 2,9 cyclic set of Figure 1. Instead, the bracketed segments drawn from the two cyclic sets are positively isographic; that is, they share identical T-values between corresponding nodes but a consistent difference between their I-values, as shown in Figure 4. In these two K-graphs the identical T-values represent the interval 7 cycle underlying both cyclic sets, while the difference of ten (t) in both of their corresponding I-values represents the difference between the tonic sums in the two cyclic sets. …