Klumpenhouwer network
A Klumpenhouwer Network, named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is "any network that uses T and/or I operations (transposition or inversion) to interpret interrelations among pcs" (pitch class sets). 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", cyclic sets being those "sets whose alternate elements unfold complementary cycles of a single interval."
"Klumpenhouwer's idea, both simple and profound in its implications, is to allow inversional, as well as transpositional, relations into networks like those of Figure 1," showing an arrow down from B to F♯ labeled T7, down from F♯ to A labeled T3, and back up from A to B, labeled T10 which allows it to be represented by Figure 2a, for example, labeled I5, I3, and T2. In Figure 4 this is (b) I7, I5, T2 and (c) I5, I3, T2.
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