Saturday, December 26, 2009

Hexatonic cycles, after Cohn

The LP transformation that has become a motif in this blog –- because the harmonic progression that expresses it is such a distinctive element in D779n13 -- is a segment from one of the four hexatonic cycles described by Richard Cohn ("Maximally Smooth Cycles")--see the left side of the graphic below.

Though the display of the cycle is neutral (the C# triads could just as well be shown on the right or "clockwise" side (in relation to A major) as on the left or "counterclockwise" side), the order chosen in the figure has substantial historical resonance, as progressions to the lowered sixth degree were common in the early nineteenth century (and later) but progressions to the major mediant were rare. (In "As Wonderful as Star Clusters," Cohn explores Schubert's use of harmonic and tonal progressions based on hexatonic cycles in the late Bb major piano sonata. The work is interesting even if Cohn's reading of the piece is ultimately more elegantly complex than it is convincing.)

We might speculate that D971n2 and D779n13 as published are the remainder of Schubert's experiments at mirroring the common move to the lowered sixth by going in the opposite direction. If he indeed tried this out while his friends danced near him, Schubert would quickly have discovered how awkward the return to A major was in the constrained context of 16 bars. The conventional return from a progression I - bVI was through an alteration that produced an augmented sixth chord, or I - bVI - +6 – V (as happens in the second strain of the Trauerwalzer).

If one attempted the same from III, one would end up on the dominant of bVI--in other words, the progression will always move one position counterclockwise along the cycle, which means that a second progression through an augmented sixth would have been necessary to reach I again. The simpler solution, undoubtedly, would be to close in the secondary key, then simply shift back to the main key--the solution Schubert used in D971n2--but the non-tonic opening of the reprise in D779n13 posed an additional obstacle. Under the circumstances, the solution Schubert used in the published version is the simplest one available.

(The other option would be to convert the tonic of C# major to a dominant seventh chord and use a deceptive resolution, or C#: I – f#: V7 – VI. Unfortunately, this would oblige Schubert to include in the final triad the pitch A (to resolve the seventh), which would greatly hinder the reintroduction of the theme's characteristic suspensions.)


Cohn Richard. "Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late Romantic Triadic Progressions." Music Analysis 15 [1996), 9-40.
Cohn, Richard. "As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert." Nineteenth Century Music 22/3 (1999): 213-32.