Two Measures of Strauss’s "Spring" Twice Reflected: R and I (Part III)

The Four Last Songs of Richard Strauss (1864–1949) were composed 70 years ago this year, and were posthumously premiered on this day 68 years ago. One of the songs, entitled "Frühling" ["Spring"], repeatedly uses a distinctive four-chord, four-voice progression at different transpositional levels. The figure below includes a transcription of the first instance of this progression in the song. Some of the notes have been enharmonically respelled from Strauss's original score so that the each chord's notation clearly reflects its harmonic quality: Chord 1 is a minor triad, Chords 2 and 3 are minor seventh chords, and Chord 4 is a major triad. The vocal line follows the top notes, but the words have been omitted. I am considering Chord 4 as essentially a B-major triad in second inversion with the F-sharp eventually on top; therefore, I am treating the G sharp, presented below using a smaller font, as a non-chord appoggiatura.

In the picture above each chord, the chord's four notes -- register ignored -- are arranged onto a circle of half steps with C# at the top and G at the bottom, and then a polygon is inscribed within the circle using the notes as points. The notes that are doubled -- the A flat in Chord 1, and the F# in Chord 4 -- are indicated with the double curves.

In short, the progression of Chords 1 and 2 is both inverted around the axis C#/G (shown with the red arrows) and retrograded (shown with the green arrows) to produce the progression of Chords 3 and 4. Even the doubling of notes is preserved in this double transformation. The figure demonstrates this visually by inserting a mirror in between Chords 2 and 3. Whereas my two visualizations of both retrograde and inversion used earlier in this blog took place in two dimensions — the vertical depiction of registral pitch and the horizontal depiction of time — here the use of circular pitch-class space accommodates both transformations as reflections in a single dimension. The double-tipped black arrows, each labeled with a duration, shows that the mirroring of the onsets of the chords is also exact in time.

Lastly, Chords 2 and 3 are related not only by inversion but also by a 60° rotation; that is, a whole-step transposition (again, with register ignored). This is the same composite relation I showed three months ago in Schoenberg's op. 19 no. 4.