Perle's Cyclic Sets and Klumpenhouwer Networks: A Response

Abstract

Each of the papers in this issue addresses how elements of George Perle’s compositional system and analytical methodology intersect with features of Klumpenhouwer networks (K-nets), drawing on Perle’s own communication on the subject as well as his writings.1 This commentary is intended both to clarify Perle’s position and to provide additional context from his writings. Lewin and Lambert directly refer to Perle’s materials. Lewin compares K-nets and “Perle cycles” (Perle calls them cyclic sets) as models for compositional events, and Lambert describes his K-net families in terms of Perle’s cyclic sets.2 Perle was initially in uenced by Berg’s Lyric Suite, where the row is comprised of two interlocking inversionally-related interval5 or -7 cycles and thus is comprised of adjacent dyads from two sums, 9 and 4, as shown in Example 1, where sums of 9 are bracketed on top, and those of 4 on bottom. Perle calls such a construct a “cyclic set.” A cyclic set is identiŽ ed by its interval cycles and its sums; the cyclic interval is the difference of the sums (9– 4 = 5 or 4–9 = 7; either 5 or 7 may be used). Each imbricated trichord in the set is comprised of a middle “axis note” with surrounding “neighbor notes.” The trichord is the smallest unit that can identify a cyclic set, as it contains the two sums and the cyclic interval. In K-net terms, these trichords are all related by strong isography, with the same conŽ guration of Ts and Is.3 For instance, {8,1,3}, {6,A,B} and {4,0,9} can all be interpreted as T5/7, I9/I4 K-nets. The given row-form of P5 (rows in Example 1 are rotated by one hexachord to create continuous cyclic sets) shares one sum with related forms at sum 9 (row-form I4), and sum 4 (row-form IB). These three rows vertically create the same group of K-nets related by strong isography when aligned. Cyclic sets that share one sum, such as the sums-9/2 and -9/4 sets, are called “cognate sets” by Perle. Cyclic sets are part of a “cyclic set complex” that includes all alignments of the two interlocking interval cycles. Example 2(a) reproduces Lewin’s Example 1.1, with T1 and TB cycles in an offset alignment and sums 6,7 found on the diagonals. In later examples, Lewin maintains this offset arrangement, showing trichords as triangles between the two sets, and does not represent them in one line, as Perle does in his cyclic sets. This example is rewritten as a cyclic set of sums 6/7 in Example 2(b). It is placed within a cyclic set complex, where the sets are based on interval cycles 1/B. The interval-B cycle in boldface shifts one to the right in 1See the citation list for pertinent material. 2Lewin writes that Perle suggests that K-nets “can best be viewed” in terms of cycles, but Perle only notes that they “may be deŽ ned and efŽ ciently and economically represented in this way.” Perle 1993, 300. 3Tracing this type of row would open a twelve-tone component to K-nets.