Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families

Abstract

In 1990 and 1991, Henry Klumpenhouwer and David Lewin introduced Klumpenhouwer networks (K-nets) as theoretical tools that display transformational interpretations of dyads contained within pitch-class multisets (Lewin 1990; Klumpenhouwer 1991). Informally, K-nets are directed graphs that employ pitch classes as nodes and elements of the T/I group as edges. In order for a K-net to be well defined, its edges must commute throughout the directed graph and its nodes must map to adjacent nodes according to the corresponding edge transformations. Several types of K-nets emerged by varying the cardinalities of the underlying pitch-class multisets, the number of constituent dyads subject to transformational interpretation, the number of transpositional and inversional operators employed, and the relative positions of these operators. We will work exclusively with two common types of K-nets: trichordal K-nets and box-style tetrachordal K-nets. See Examples 1a and 1b, respectively, for representatives of these two types.