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.

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.)

References:

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.

more to diachrony and synchrony

This is an addition to the post on Timothy Jackson's diachronic/synchronic extension of Schenkerian analysis. There I noted that "It would be possible to use D971n2, the only other published dance that modulates to III, as the synchronic source of D779n13's distorted second half." Let's take a look at that possibility. Here is the score:

Note that the design is very similar to -- but much simpler than -- D779n13: the first strain is a straightforward 8-bar period (though Caplin would call it a hybrid theme because the pedal point tonic precludes cadential definition); the second strain consists of a contrasting middle that is a slight variation of the first strain transposed to III, and a literal and complete reprise of the first strain. All of these are elements that are present in D779n13, too, but in distorted form.

The resemblances are strengthened by the fact that the two dances come from the same time period: D 779 was published in 1825, and three German dances of D971 appeared as Schubert's contribution to a Carneval collection in early 1823.

Ternary form designs with the contrasting-middle-as-transposition may be found throughout Schubert's dances, though by no means in large numbers. In D365, only n32 is a ternary design and it follows the model of D971n2 (see graphic below). In the waltzes of D145, n7 fits the model (Eb-c-Eb); in the ländler, n2 (Eb-Bb-Eb). And so on.

It's very easy to imagine these designs originating in improvisation: one uses the same musical material and maintains the phrase design for the dancers while introducing a note of variety and extending the basic 16-bar form from 32 measures (with repeats) to 48. At the same time, for Schubert, the blocking out of key areas by 8-bar thematic units would give an opportunity to "absorb" the sound of a particular modulatory pairing, both going and coming, so to speak.

Discontinuities and synchronic/diachronic tension, after Timothy Jackson

In the MTS article, I briefly discuss Timothy Jackson's "diachronic transformation" (293-94). Jackson uses this device as a way to interpret conflicting structural levels in Schenkerian analysis by locating paradoxical moments, or ruptures, in tonal design. He says that "[a] musical work may embody in its endstate a conceptually prior state, which has become the endstate through a diachronic transformation." (Jackson, 239).

We can employ the synchronic/diachronic distinction in service of another way to relate D365n6 and D779n13. To accomplish this, we will have to intuit a diachrony by appealing to waltz design statistics, which will allow us to conceive a prior "typical" version. The existing waltzes, then, may be understood as diachronic distortions; in fact, they might literally have been the case as Schubert played for a late-evening cotillon, the kind of experience I sketched in the "improvisation history" posts.

Of the three basic design models, small binary forms dominate Schubert's waltzes. Regardless of form type, however, the second section most often opens with a dominant seventh chord, which may lead to a stable key area within the phrase but more often opens a modulating sequence. The distinctive feature of the A Major Waltz is the direct modulating shift to another tonic triad, a device that I have found in only twenty four waltzes. Of these, the majority (17) move to a diatonically related key; the remaining seven use mixture or minor/major alteration: bVI (1), VI (1), bIII (2), i (1), or III (2). Five of these seven have a stable single key in the contrasting middle (the other two make abrupt modulations in the final two bars).

It would be possible to use D971n2, the only other published dance that modulates to III, as the synchronic source of D779n13's distorted second half, but for my purpose here it will be more efficient to call yet again on the relationship between D365n6 and D779n13. The first graphic below gives the first half of D365n6, above it a traditional Schenkerian reading from ^3, and above that a depiction of the "endstate"--that is, D779n13--as a reading with an Urlinie from ^5 that takes account of the ascending cadence figure.

In the first strain, two by now familiar phenomena can be interpreted as transformations in Jackson's terms: (1) the upper-voice counterpoint of D365n6, is inverted, with the effect that parallel fourths above ii6-I6/4 become the parallel fifths; (2) the single ending of D365n6, becomes two, the first of them imperfect (upper voice remains on ^5, even if the inner voice reaches ^1), the second seeking the original register of the model via a stepwise ascent. On the other hand, the regular two-bar groups of our A Major Waltz actually smooth out a metric distortion in the model (where the 6/4-5/3 movement over the dominant harmony is repeated in bars 3-7).

For the opening of the second strain (see below), the path to the major mediant key may be more complicated, but it is plausible insofar as both small binary and ternary designs at this point typically take advantage of circle-of-fifths progressions with chromatic inflections. If we expand the four bars of the contrasting middle in D365n6, to eight bars, we can start with a C major triad and move in simple steps to V, at two bars per chord: III-vi-V/v-V. But the progression needs to move one additional step, past V to I, in order to connect with the subdominant bass of the reprise. To contain a move from III to I within eight bars requires some adjustments that would make it difficult to take advantage of the waltz's suspension motive: perhaps III-V7/III-III-vi-V7/V-V7-I-V7/IV.

Under the circumstances, the radical simplification of the key-stable version in our A Major Waltz is a much more workable solution with an effect similar to the metric smoothing out in the first strain. These "simplification paths" to diachronic distortion are an alternative to the "inspired moments"--improvisation histories can be readily understood as narratives of synchronic-diachronic transformation. In this case, I think the "inspired moments" are more plausible because they reduce D365n6 to its opening gesture and formulas, rather than dragging along layers of increasingly idealized voice leading.

References:

Neumeyer, David. "Thematic Reading, Proto-backgrounds, and Transformations." Music Theory Spectrum 31/2: 284-324. {should appear any day now]

Jackson, Timothy. "Diachronic Transformation in a Schenkerian Context: Brahms's Haydn Variations." In Carl Schachter and Hedi Siegel, eds. Schenker Studies 2, 239-75. Cambridge: Cambridge University Press 1999.