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Some musicians have written long articles or even books about the Tristan chord, they plunge into a jungle of linear connections and interpret it in a very abstract way, making a big deal of this sonority. However, apart from the multidirectional potential of the half-dim. 7 per se –something that Wagner explores to a certain extent throughout the prelude, the opening chord may have a simple interpretation. I am offering you two interpretations of it, beginning with the simpler one.
1. If we consider the tone G# an appoggiatura, then the chord is not half-dim.7, even enharmonically. Its structure only crystallizes in the next moment, when the appoggiatura resolves into the tone A. Then we can hear the chord B7-5 in second inversion as a double dominant with a lowered fifth (V7-5 of V) in the key A minor. This interpretation is confirmed by the resolution into the E7 chord (elliptical resolution). Thus the progression B7-5 – E7 will be heard as DD 4/3 alt.- D7 in the key A minor. Yes, the altered dominant contains an augmented sixth, but the geographic nickname "French" is irrelevant to harmonic function anyway.
Jazz people will hear the chord as F7-5 instead of B7-5 and will call it "a tritone substitute for V of V" but this is just another way of labeling an inverted and altered dominant of the dominant. Whenever they write bII7-5 of....they shall be aware that the true meaning of this chord is V4/3 (b5) of...because these chords are enharmonic.
2. If we do not consider the gone G# as an appoggiatura, then the sonority F-B-D#-G# coincides enharmonically with F half-dim7, and may be interpreted as an altered and inverted DVII7 in the key of A minor: VII2 (with a raised fifth). Here the half-diminished seventh chord is reinterpreted as a chord of the augmented sixth. When this chord is placed in root position, it reveals G#-B-D#-F, stemming from the VII dim7 chord, whose fifth then is raised. This interpretation is not so natural, because it ignores the appoggiatura, but the sequence DVII2 alt – DD 4/3 alt – D7 makes sense (two dominants of Am surrounding a double dominant).
Not a big deal. As for other resolutions of the half-dim. sonority that reveal other interpretations as a true altered chord, yes, one could write a whole chapter about them.
Texas State University