Klumpenhouwer network

[1] According to George Perle, "a Klumpenhouwer network is a chord analyzed in terms of its dyadic sums and differences," and "this kind of analysis of triadic combinations was implicit in," his "concept of the cyclic set from the beginning",[2] cyclic sets being those "sets whose alternate elements unfold complementary cycles of a single interval.

"[3] It is named for the Canadian music theorist Henry Klumpenhouwer, a former doctoral student of David Lewin's.

"[7] "To generate isomorphic graphs, the mapping f must be what is called an automorphism of the T/I system.

[8] "Let the family of transpositions and inversions on pitch classes be called 'the T/I group.

[10] Other terms include Lewin Transformational Network[11] and strongly isomorphic.

7-note segment of interval cycle C7
Cyclic set (sum 9) from Berg's Lyric Suite
Chord 1. K-net relations, inversional and transpositional, represented through arrows, letters, and numbers.
Chord 2. Inversional and transpositional K-net relations represented through arrows, letters, and numbers.
Chord 3. This chord with Chord 1 provide an example of rule #1 by way of a Network Isomorphism. [6]
Graph of graphs from the six chords of Schoenberg's Pierrot lunaire , no. 4, mm. 13–14. [ 10 ]