More to hexatonic cycles and the Riemannian Hand

I've written in an earlier post about Richard Cohn's hexatonic cycles in relation to the C# move in D779n13. I probably should have labeled today's entry another "regret post." As with Eco and Westergaard, Cohn's article on Schubert's treatment of tonality has many points of resonance with posts on this blog, probably because it has been in my mind several times already when I make reference to Neo-Riemannian historical narratives.

I will write more about the article at another time. In thinking about it this morning, however, I went back to the post with the waltz that has a close parallel to D779n13 in a direct A-C# move from first to second strain (then back again): D971n2: see the score in this post. The move is interesting in itself, of course, since D971n2 might well be another trace of a Schubertian improvisational formula, but the piece as recorded is curious in the absolute regularity of its left-hand chord shifts: in every case, as shown below, two pitches change by a half step while the third remains stationary. A rotation is required between each key area, as the register shifts of E3 up to E#4 and back reveal.

All this seems the more remarkable in light of Cohn's observation that

Theorists recognized voice-leading efficiency as an alternative basis for harmonic relations already during Schubert's lifetime and throughout the nineteenth century, but were reluctant to explore its systematic implications, primarily because to do so would have required them to relinquish their deep prior commitment to acoustic theory and tonal centricity, in favor of a less hierarchic, more networked conception of harmonic relations. (214)

I would extend that early knowledge of voice-leading efficiency to the afterbeats played by Schubert's left hand. Pragmatically, as any pianist who has played waltzes, ragtime, or stride piano knows well, anchoring the after-beats on the keyboard serves as a way of making the (frequently blind) leap down to the downbeat bass note more secure. Paradoxically then, for the improvising pianist it is the afterbeats -- and not the accented bass -- that provide a stable voice leading thread and can influence the direction of improvisation as well as harmonic formulae do.

[2-23-10: corrected text and graphic] In that light, consider the figure below. At (a) are the three Riemannian transformations R, L, and P. These are all the 1-note step-changes that will produce major or minor triads. At (b) are the diatonic 2-note changes; at (c), the chromatic -note changes. At (d) are the diatonic and chromatic triads generated through 3-note changes. In all cases except familiar inversions of V7 and ii7 or forms of viiø7, no non-triadic results are shown, and those moves that generated parallel fifths were also excluded.

It should probably not be remarkable that Schubert usually maintains close voice-leading patterns in the after-beats of his dances, given the habits of his training and the practical uses mentioned above. That such patterns are sometimes maintained even in the face of unusual or awkward chord progressions, however, is notable, an indication perhaps of how the "middle" -- always literally close to the player's eye, after all -- can guide the "edges" of the texture. In D779n7 (below), we would expect the stationary F3 in the second strain, given the T-D Ländler figures, but look at how tight are the after-beats through the odd (and not altogether elegant) harmonic meanderings of the first strain. (As usual, click on the thumbnail to see the graphic at original size.)

Reference:

Cohn, Richard. "As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert." Nineteenth Century Music 22/3 (1999): 213-32.