Twisted Tristan

Speaking of anniversaries in multiples of fifteen, Richard Wagner’s Tristan und Isolde premiered 150 years ago this month. Wagner in general, and the introduction to this opera in particular, is well known for withholding the tonic note or tonic triad of a key whose presence is made clear. A very efficient way of doing the former—with two voices, three pitch classes, and three total semitones of voice-leading work—is the progression F3D4 to E3E4 (call it what you want), which is in A although no A sounds. (The C clef, while not as user-friendly for some as other clefs are, will be useful later.) Now take this progression, and skew it by displacing the top voice forward in time. This puts the D above the E, and excludes a D above the F.

Now fill out this twisted progression in four parts, while maintaining some implication of an A tonality. An E7 is the best way to harmonize the second moment. As for the first chord, exactly two of the twelve half-diminished seventh chords have an F and also do not have a D; they are shown below. (My unusual spelling of each half-diminished (hd) chord matches Wagner’s.)

These are exactly the two solutions, allowing for transposition (shown using the roving clef below), used in the famous opening eleven measures of the opera's instrumental introduction: Wagner’s first two progressions use the first solution, and his third progression uses the second solution.

Moreover, both chords contain a note—either C# or D#—that is only a half step away from the D: a clear “not-D” note through its half-step displacement. Lastly, this derivation from a two-voice model also jibes well with the voice leading: the voices with the F or the “not-D” note in the first chord (the solid noteheads) each move by a single half step into the second chord, while the two added voices (the hollow noteheads) do not.