Some Observations on P_m,n Relations within Set Classes

Abstract

It is well known (from Riemann, Lewin, Cohn and others) that two voices of a triad (set class 3–11) may be held and one moved ‘parsimoniously’ by a tone or semitone to produce another triad; or two voices of a dominant and half‐diminished seventh (set class 4–27) may be held while two voices are displaced by a semitone to produce another dominant or half‐diminished seventh (Lewin's ‘DOUTH2’ relation). This article takes 4–27 as its starting point and first explores examples of such parsimonious voice‐leading between sets of the same class in nineteenth‐century music (notably Brahms, Chopin and Musorgsky). The discussion introduces the idea of a diminution existing between forms of 4–27 when the two voices do not proceed simultaneously. The capability of tetrachords of other classes as well as set classes of larger cardinality to engage in parsimonious voice‐leading either by semitone or whole tone is then explored (with examples drawn from the music of Bartók, Mahler, Scriabin and Stravinsky). The type and degree of parsimony are indicated by the Pm,n designations developed by Douthett and Steinbach, and the article examines the ways in which the capacity of sets to behave in this manner may be predicted from interval strings and invariant subsets.