The DOUTH2 Relation as a Dramatic Signifier in Wagner's Music Dramas

Abstract

Near the conclusion of his article on 'Cohn Functions',l which is based on previous work by Richard Cohn2 and Jack Douthett, David Lewin briefly discusses the DOUTH2 relation, where two discarded pitch classes in a tetrachord are replaced with notes a semitone removed. In particular, he applies this parsimonious voice-leading paradigm to certain pairings of the half-diminished seventh chord and its inversionally related major-minor seventh [0, 2, 5, 8]. Since the interval vector of each individual sonority (0121 1 1 ) features a pair of minor thirds and one tritone, transpositions by T3, 6 or 9 likewise display a DOUTH2 voice-leading through the retention of two common notes and chromatic motion.3 As Lewin demonstrates, paired couplings of the two different chords result in no less than six instances of this same paradigm. The initial six graphs of Fig. la plot these transposition/ inversion possibilities onto a pitch matrix. The enclosed half-diminished Tristan chord (F-B-D:-G:) serves as the initial referential set for the T/I index numbers;4 dotted lines denote its inversional counterpart (the majorminor seventh).5 Since all six T/I relations cannot be demonstrated on a traditional Riemann Tonnetz, a left-to-right whole-tone diagonal replaces the usual semitone diagonal. In this way; the interval cycles in each row, column or diagonal incorporate only those interval-classes which appear in the sonorities' vector. The intervallic relation between the two invariant notes in the initial six progressions accounts for each interval-class in the vector. This property likewise holds true for the remaining interval in the first chord, as well as between the pair of new pitches which appear in the second major-minor seventh chord. These six T/I index numbers form an incomplete octatonic set: [0, 1, 3, 4, 9, 10]. In the remaining pair which completes this eight-note collection, T6/I holds three notes invariant while all of the voices in T7/I proceed by semitone.6 Fig. 1 b illustrates a continuous progression which successively utilises all six DOUTH2 relations in pairings of the two