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The connection between the augmented-sixth chord known as the "German Sixth," and Wagner's famous "Tristan Chord," is one of creatively alternative resolution. It is not so much that either the German Sixth or the Tristan Chord is a new harmonic entity; it is that in either case we have a familiar harmonic entity used in a creatively alternative manner.
In the case of the German Sixth:
It has been mentioned already that the German Sixth is the enharmonic equivalent of a dominant-seventh chord (e.g. — Ab-C-Eb-F# = Ab-C-Eb-Gb). The usual resolution of this Ab dominant-seventh chord would be to a Db major or minor triad, or (in what is known as a “deceptive cadence”) to a Bb minor triad (in Db major), or even to a B-double-flat major triad (in Db minor). Another possible and plausible deception which has ample use in the literature is to forestall the expected resolution by shifting gears to the relative minor, by moving to the first inversion dominant-seventh of the relative minor (in this case, spelled A[-natural]-C-Eb-F, e.g.) or the fully-diminished-seventh chord on the leading tone of the relative minor (here, A-C-Eb-Gb).
All this, to illustrate that Common Practice has established ample precedent for the dominant-seventh chord’s progression to these six chords.
Although the German Sixth is the sonic equivalent of the dominant-seventh chord, it progresses otherwise still, almost always to (in this case) a second-inversion c minor triad, although conceivably to a C major triad, or even (— although this results in parallel fifths which are generally avoided in Common Practice, as poor voice-leading —) to a G major triad.
What does this have to do with the Tristan Chord? Like the German Sixth chord, the Tristan Chord (F-B-D#-G#) is really a harmonic entity familiar within Common Practice, but used in a novel fashion. It is an enharmonic re-spelling of a half-diminished-seventh chord, i.e., a diminished triad (in this case F-Ab-Cb = F-G#-B) and a minor seventh (in this case, Eb = D#). The half-diminished-seventh chord occurs naturally, as it were (that is, without chromatic alterations) on the seventh degree of the major scale (in C major, B-D-F-A). The Tristan Chord is the enharmonic equivalent of the half-diminished-seventh chord on the leading tone of Gb Major (F-Ab-Cb-Eb = F-G#-B-D#). The half-diminished-seventh chord also occurs on the second degree, and on the raised sixth degree, of the minor scale. The Tristan Chord, if Wagner had used it otherwise, would function in either e-flat minor or a-flat minor.
[I should note that I refer to the Tristan Chord as it appears at the beginning of the Prelude.]
The mysterious novelty of the Tristan Chord is that, here in the Prelude, it does not function in G-flat major, or e-flat minor or a-flat minor; it functions in a minor — or at least, it resolves to the E dominant-seventh chord which is the dominant of a minor.
In neat summation, the Tristan Chord — like the German Sixth — is not so much a new harmony, as it is a familiar harmony used in a creatively novel manner.
It has been mentioned that the Tristan Chord resolves to an E dominant-seventh chord:
F-B-D#-G# to E-G#-D-B (this is the essential harmonic motion, although we note that the melody moves chromatically from G# to B via A and A#, so that the A# is an accented non-harmonic tone against the E dominant-seventh chord).
What I want to point out is the relation between these two chords, the “before and after” of this crucial introduction to Wagner’s opera.
First, there are two tones held in common, G# and B, and they comprise the interval of a minor third.
Second, that the two tones of the Tristan Chord which do not belong to the E dominant-seventh chord, resolve by half-step (F to E, D# to D-natural).
All right then; two common tones forming a minor third, and two tones which move by half step. Before we turn to Stravinsky, let us note the similarity to the German Sixth and its characteristic resolution: the two tones of the German Sixth chord, which do not belong to the second-inversion minor triad to which it customarily resolves, also move by half-step; and the common tones form a minor third.
Consider with me now the opening of Stravinsky's Symphony of Psalms. We have a couple of punctuative e minor chords, and then in the double reeds some noodling arpeggios, the first arpeggio ascending, the second descending. Both arpeggios are dominant-seventh chords. The ascending arpeggio is a B-flat dominant-seventh chord (Bb-D-F-Ab), the descending, a G dominant-seventh (appearing in first inversion, B-natural-D-F-G).
Compare in Wagner:
F-B-D#-G# to E-G#-D-B
To Stravinsky:
Bb-D-F-Ab to B-natural-D-F-G
In Wagner, two common tones forming a minor third (G#, B) and two tones moving by half-step, in similar motion (F to E, D# to D-natural).
In Stravinsky, two common tones forming a minor third (B, D) and two tones moving by half-step, in contrary motion (B-flat to B-natural, A-flat to G).
Karl Henning