Some Structural Features of Contextually-Defined Inversion Operators

Abstract

In his analysis of the Stockhausen Klavierstiick III, David Lewin (1993, 25-30) defines and implements an inversion operator, denoted by the letter J, that resembles the inversions defined in the classical group of twelve-tone operators (TTOs), but that gives rise to a rather different mathematical structure. Example 1 shows Lewin's application of J to the opening measures of the piece. The J transformation is defined on the set class (under the usual TTO group of order 24) containing the unordered pentachordal pitch-class set P = { 2 8 9 10 11 } such that J applied to a Pform Y results in the unique P-form that: (1) is an inverted form of Y, and (2) contains the same chromatic tetrachord as Y.1 Thus, for example, J(P) preserves the chromatic tetrachord { 8 9 10 11 } and maps the pitch class 2 to its mirror image around the chromatic cluster, giving the P-form { 5 8 9 10 11 }, which Lewin calls p. Similarly, J(p) simply reverses the process, now mapping p to P, a consequence of the fact that J, like the usual inversion operators, is an involution. So with respect to P and p, J is identical to the TTO 17.2 However, consider some transposition of P, say T3(P) = { 5 11 0 1 2 }, which Lewin denotes as P3. Now, J(P3) = {8 11 0 1 2}; in this case J is equivalent to II. Thus, the behavior of J in terms of the usual TTO inversions varies based on the particular member of the set class on which it is acting.