Near the conclusion of his article on`Cohn Functions', 1 which is based on previous work by Richard Cohn 2 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 (012111) 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. 1a 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. 1b illustrates a continuous progression which successively utilises all six DOUTH2 relations in pairings of the two sonorities.Building on previous work by Cohn, Adrian Childs, 7 and others, Jack Douthett and Peter Steinbach explore some of these relationships iǹ Parsimonious Graphs: a Study in Parsimony, Contextual Transformations, Music Analysis, 20/ii (2001) 179
Numerous authors have noted those passages of common musical material in Wagner's "Traume" and the Love Duet ("0 sink' hernieder, Nacht der Liebe") which occurs about the middle of the second scene in Act II of Tristan and Isolde . 1 Aside from obvious thematic borrowings, there appear to exist deeper and more subtle relationships between the two works which suggest a kind of parody technique. In addition, it is possible that certain harmonic and linear functions embedded within the song provide the seeds for important tonal relations which appear in expanded form later in the opera.In April of 1857 Wagner was invited by the Wesendonks to live at Asyl, a cottage on their estate. It was here that the celebrated "affair" between Richard and Mathilde continued to blossom and supply the composer with much of the impetus and inspiration for Tristan. The poem of the opera was finished and read to a gathering in September of 1857. Once his new piano 1 All page references are taken from the Dover reprint of the full score of Tristan (New York: Dover Publications, 1973). The measure numbers for the Love Duet begin with the Massig langsam on p. 348. arrived, Wagner plunged into the composition of Act I, the full score of which eventually was completed the following April. During the fall of 1857 Mathilde showed Wagner some of the lyric poems she had written in imitation of the Tristan poem. He proceeded to set these "pretty verses sent over to me" at various times during the next seven months. The autograph for `Trāume, " the second of the Fiinf Gedichte fair eine Frauenstimme, is dated December 5, 1857 . 2 The first version of the song (December 4) began with the sixteenth bar; in his revision the following day Wagner borrowed the music from the piano coda to serve as an introduction, appending the first six bars with its melodic outlining of the Tristan chord (E-flat) F A-flat C-flat E-flat. Two other songs from the series furnished thematic material for the opera ("Schmerzen" and "Im Treibhause"); however, the subtitle Studie zu Tristan and Isolde was not added to ` `Traume" and "Im Treibhause" until the publication of the collection some five years later. Although sketches for the first scene of Act II date from this period, we 2Facsimile of the autograph (Leipzig: VEB Deutscher Verlag fiir Musik).
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