Handout for SMT presentation

I presented "Schubert's 'Riemannian Hand': An Archaeology of Improvisation for Social Dancing" during last week's joint meeting of AMS/SMT in Indianapolis. I have posted a pdf file with the handout here. [NB: This link was updated on 3 July 2016]

Here is the PROPOSAL as it was sent to the program committee last spring:

Proposal for the annual meeting of the Society for Music Theory, Indianapolis, 2010

Title: Schubert's "Riemannian Hand": An Archaeology of Improvisation for Social Dancing

Research on improvisation practices in the nineteenth century has looked primarily at the fantasia. The reminiscences of Schubert's friends provide a window into another, equally important, practice: playing music at the piano for dancing. The celebrated Schubertiades varied greatly in their formats but often consisted in musical performances followed by dancing (Litschauer 245). For the latter, Schubert was obliged to string his waltzes into "endless cotillons" (Deutsch 230; also, Litschauer and Deutsch 101). A close relative of the contredanse, the cotillon required frequent repetition of strains, particularly the principal one, which acted as a kind of rondo theme. The number of repetitions depended on the dance caller (some published instructions stretch 24 bars of music across 80 or more bars of dancing).

Using the three-layer texture of the waltz (as played on a piano) and "endless cotillons" as the design, I will demonstrate with examples and through performance (1) how strict small forms, repetition, and variation can reveal pairings and groupings among Schubert's surviving waltzes, suggesting relationships that may have arisen through varied repetition in performance; and (2) how the chordal offbeats can effect transformations with parsimonious voice leading by simply moving thumb, middle finger, or little finger, thus anchoring the more distant modulations that Schubert attempted in improvisation.

During the presentation, I will provide a handout charting and comparing the dances that make use of third relations, and I will perform a "music-stretching" cotillon that will gradually transform one Schubert waltz (as the first iteration of the principal strain) into another (as the final iteration).

An instance of the pairings that emerge from this study is given in Example 1, which shows that the A Major Waltz D. 779 no. 13 (familiar to music theorists from analyses by Schachter and by Lerdahl and Jackendoff) might easily have arisen as an improvised variation on D. 365 no. 6. [ Example 1: At the top, Schubert, D. 365 no. 6, opening; at the bottom, D. 779 no. 13; in the middle, underlying voice leading pattern -- see the presentation handout for this.]

The three-layer texture is associated with the most common ensemble in tavern or small restaurant settings in Vienna about 1800: two violins and bass. The layers are clearly differentiated in Example 2 -- the second violin's double stops would simply become the offbeats in waltzes by Lanner and Strauss. [Example 2: Beethoven, Sechs ländlerische Tänze, WoO15, no. 1, first strain. -- see the presentation handout for this.] The texture of this ensemble could be appropriated for domestic settings when the piano became popular as a replacement for the traditional violin as accompaniment for dancing. For those pieces that used the Ländler-derived "oom-pah" rhythms, the result was three functionally differentiated layers, two in the left hand, one in the right.

I focus attention on patterns of the middle layer (left hand chords) in relation to upper voice figures, particularly on those that generate third-related key areas in the second strain. See Example 3, where the left hand executes an LP transformation. [Example 3: Schubert, D. 779 no. 13, move from the first to second strain, and from A major to C# major, as an LP transformation in the left hand -- see the presentation handout for this.]

By doing multiple comparisons among dances, I try to reconstruct some sense of how Schubert, during improvised performance, may have been—in Kofi Agawu's terms—"thinking in music about music."

Works cited

Agawu, Kofi. "How We Got Out of Analysis, and How to Get Back in Again." Music Analysis 23/ii-iii (2004): 267-86.

Deutsch, Otto. Rosamond Ley and John Nowell, trans. Schubert: Memoirs by His Friends. London: A. & C. Black, 1958.

Lerdahl, Fred, and Ray Jackendoff. A Generative Theory of Tonal Music. Cambridge, MA: MIT Press, 1983.

Litschauer, Walburga. "Unbekannte Dokumente zum Tanz in Schuberts Freundeskreis." Studien zur Musikwissenschaft 42 (1993): 243-249.

Litschauer, Walburga, and Walter Deutsch. Schubert und das Tanzvergnügen. Vienna: Holzhausen, 1997.

Notley, Margaret. "Schubert's Social Music: The 'Forgotten Genres'." In Christopher H. Gibbs, The Cambridge Companion to Schubert, 138-54. Cambridge/New York: Cambridge University Press, 1997.

Schachter, Carl. "Rhythm and Linear Analysis: Durational Reduction." Music Forum 5 (1980): 197-232.

mediant blocking in D145n7

A simple diatonic mediant move (effecting the transformation R twice) is aligned with formal design in an early ternary waltz that also happens to contrast Ländler and deutscher traits (the former in the main theme, the latter in the contrasting middle) using sharp dynamic contrast to make the point unmistakably. In the example below, the mediant moves are outlined. (Another reminder that the graphics are thumbnails -- click on them to see the original-size file.)

Perhaps because of the change of figuration assisting the topic change, the left hand does not execute the transformations directly (in the manner of the Riemannian Hand). Instead, the right hand works out a pattern that combines transformation with registral shifts.

In mm. 3-4, the Hand would seem to be as at "a?": Bb4-Eb5-G5, but the direct move is from the shape at "a". The "b?" to which "a" moves is not literal, however (there is no Eb5); instead, the Eb is shifted upward to Eb6 (as at "b"). In the reprise this shape settles down as C5 goes to Bb4 (at "c?"), but the final move again takes the lowest note up an octave: G5 to G6 at "c."

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.

Some citation fixes and additions

I have made some emendations to earlier posts, adding or fixing citations.

1. Corrected a citation in the Wheeldon post.

2. Added a citation to the "music literal" post on Michael Spitzer's critique of the body analogy in musical analysis.

3a. Added a link in the first Riemannian hand post to a music file with a period-instrument performance.

3b. Added a graphic to the same post: an (admittedly whimsical) detail from Kupelwieser's watercolor showing the "Riemannian hand." See also the Gesellschaftspiel post.

4. Added comments on the F# group at the end of D365 and citation from Margaret Notley's article to the F-major post.

5. Added information and a link for the image in this Schubert-Kreis post.

6. Added clarifying information to the post on Empson.

Finally, here is a link to Franz M. Boehme's Geschichte des Tanzes in Deutschland on Google Books: link. A PDF file can be downloaded, or one can browse a scanned -- and as one might expect sometimes faulty -- plain text version in Google's reader.

The "Riemannian Hand" and Schubert's voicings

This is an update to the post on the "Riemannian hand" and the post on closed-position voicings. I have reconstructed the graphic from that post in two respects below: (1) we now see the hand from the top, as a pianist would see it, not in the "Guidonian position" of the singer putting his palm out in order to remember the sol-fa; and (2) I have charted the two stages of the transformation--the first hand for L that takes A major (A+) to C# minor (C#-) and the second hand for the P that changes minor to major (C#+).

The point of this graphic is to show that the "Hand" isn't stationary. Although L shifts the pinky from A+ to C#-, the subsequent thumb move isn't the thumb's "proper" R (on the first hand), but P (on the second hand). In other words, the abstract LPR hand-group has rotated while the physical hand has not. The separation of these two is something Schubert certainly would have noticed while playing, and that realization would very plausibly have facilitated the notion of modulation rather than "just" chord change (the latter would have led to C# as V/F#-, or V/vi, the move in many of Schubert's dances). The difference is that of harmonic thinking (in the familiar nineteenth-century sense) rather than figured-bass or continuo thinking (eighteenth-century).

Here is further explanation of the rotations. I am grateful to Steve Rings for working this out. The text is his.

It seems that the specific assignment of LPR to the fingers is based on both mode and triadic position/inversion. The table below illustrates:

5/3 6/3 6/4

M LPR PRL RLP

m RPL PLR LRP

The LPR triples here are to be read from left to right in a registral fashion as "low-mid-high," which allows one to apply them to either right or left hand. One toggles back and forth between major and minor by using neo-Riemannian letters that occupy the same registral "slot" in the triples. Thus, for example, if we begin with a major triad in 6/3 position and apply L to it, we will be moving our "highest" finger, as the given triple is PRL. This will then take us to a minor triad in 5/3 position, as it is the only minor entry in the table with L in the rightmost slot. Note the dualist patterning: the neo-R triples for the major triads are all rotational permutations of LPR, while those for the minor triads are rotational permutations of RPL. LPR and RPL are of course retrogrades of each other (a result of Riemann's dualist conceptions of major and minor).

Postscript (1): I can't show the piece for copyright reasons, but there is one instance of a mirror to the A+-C#+ transformation in D779n13: the beginning of the second strain of D980d, a waltz published in January 1828. The main key is C major, where the first strain ends. The second strain drops to A minor and repeats that chord for three bars, followed by an F minor 6/3 in the fourth bar, and G7 in the fifth. The left-hand voicing is an imperfect wedge: E-F-G in the left-hand thumb, A-Ab-G in the pinky. A- to F- is also an LP transformation.

Postscript (2): Guy Capuzzo notes a transformation series that fits the guitarist's hand in relation to the frets (183). The "Guitarist's Hand" and the "Riemannian Hand" are related to, but distinct from, the keyboard topology of Minturn and Jones (the latter according to email correspondence from Neil Minturn, 26 December 2009).

Postscript (3) [added 1-05-10]: a detail from Joseph Kupelwieser's watercolor: see this post for more information:

References:

Capuzzo, Guy. "Neo-Riemannian Theory and the Analysis of Pop-Rock Music." Music Theory Spectrum 26/ 2 (2004):177-199.

Minturn, Neil, and M. Rusty Jones. "Toward a Theory of Keyboard Topology." Paper read at the annual meeting of the Society for Music Theory, Montreal, 31 October 2009.

Schubert's "Riemannian Hand"

The modulation to C# major is an LP transformation (Hook 139): A major moves to c# minor moves to C# major. Here is that change from the first to second strain, from A major to C# major, as a direct move in the left hand (thanks to Steve Rings for pointing this out):


While thinking about improvisation, about Schubert sitting at the piano playing while his friends danced, I realized that the piano permitted the sound of the waltz that would have been most familiar to people in Vienna about 1800 -- two violins and bass -- to be transferred from tavern or restaurant to the home. The three-layer texture of melody (first violin), bass, and accompanimental chords (second violin) became right hand, left-hand accents, and the offbeat "oompahs", respectively. (Link to D790n3 played apparently on a period instrument: note the timbral differences in the three registers.) In the heat of improvisation, the latter could serve Schubert well as voice leading stabilizers -- and, as in this case, enablers of modulations. Indeed, we might speak of his "Riemannian hand" and visualize it, as below, where a simple shift of thumb, middle finger, or pinky would effect a particular transformation.

Reference:

Julian L. Hook. "Signature Transformations." In Jack Douthett, Martha Hyde, and Charles J. Smith, eds.Music Theory and Mathematics: Chords, Collections, and Transformations, pp. 137-160. University of Rochester Press, 2008. Also see these posts: (1); (2).

Transformation table

Here is a reference table with the transformations I have devised to date for use with the proto-backgrounds.

These are presented here as three groups of three: the first group results in stepwise changes, diatonic or chromatic (L, N, WEDGE); the second group adds a third note to a triad interval (DIVision, ADDINVersion, TRiadFLIP); and the third group manipulates a triad interval INVersion, EXPandUp, and TRiadTransposition). All of these assume inverses (L-1, N-1, etc.) but note that EXPU-1 contracts an interval -- it does not EXP downward (to make the point as clearly as possible, a fourth transformation, EXPandDown, is added to the third group: see the staff insert).

Caveats: (a) This is certainly not a complete list of what might be done in "triad space," much less in diatonic space; (b) these can only be regarded as informal -- I have not attempted formal definitions, here or elsewhere; and (c) to both the previous points, someone has without doubt done such work and, once I find it, this post will be updated accordingly.

Proto-background 5: the fifth ^1-^5

Also see the proto-background introduction.

The fifth space ^1-^5 re-introduces a teleological element into the reading, but by no means so radically as when we hear the background as ^1-^1. Here, the upper part of the interval receives attention at the beginning, but the interval as a whole is only defined at the end of the first phrase (see the third staff below). The contrast between upper and lower voices in the right hand, thus, is more striking even than in the reading ^3-^5 (entry to be posted later this week), in that the definition or the concrete presentation of each background tone is situated at opposite ends of the phrase. And the teleological hearing of the alto voice is a good mirror of the listening experience for a string of suspensions, which constantly push forward, ahead, towards a goal, the resolution of the last suspension in the series (here, the 4-3 suspension that brings the secure arrival of ^1).

The fifth space also coordinates nicely with the C# major section (in fact, more simply and directly than any other reading): a WEDGE transformation pulls the notes apart by a half step, to G#-E#, then its inverse pushes them back together again for the reprise/ending. (WEDGE here can be understood as inversion about C/C#; it is a variant of Lewin's transformation W -- see 1987, 124 ff.; see also 2006, 332 ff.)

In a performance of the waltz, of course, the WEDGE is distorted as the G# is G#5, not G#4. We would include the registral shift in a fourth staff/level (not shown here).

References:

Lewin, David. Generalized Music Intervals and Transformations. New Haven: Yale University Press, 1987.

Lewin, David. Studies in Music with Text. New York: Oxford University Press, 2006.

Proto-background 3: the unison ^5

Also see the proto-background introduction.

Of course, the unison ^5 forces attention away from the alto (lower right-hand voice) to the soprano, and it also (that is, like the unison ^3 from yesterday's post) aligns itself very cleanly with the formal design. The second line in the graph shows a simple harmonic transformation with one harmony for each section: first strain, contrasting middle, and reprise. In Riemannian terms, this is (L-followed-by-P) followed by the inverse (or, P-followed-by-L). L turns A major into C# minor, and P makes the latter C# major.

The third line fills in a few details for the first strain. Note that the symmetries in the harmonic patterning extend to the outer melodic voices, with the two neighbor-note figures. The inset takes this a little further by understanding the "essential" chromaticism of getting-into and getting-out-of the contrasting middle symmetrically, as well: E5 breaks up to E#5 at the beginning, but G-natural slumps down to F#5 at the end.

Signature transformations and D779n13

Jay Hook has published an essay that uses the A-Major waltz as its principal example from the era of traditional European major-minor tonality. Here is the abstract:

Two types of transposition operators may be applied to diatonic objects such as chords or melodic fragments: the familiar mod-12 transposition operators (which may be understood to transpose the underlying diatonic scale along with the object itself); and the diatonic, or mod-7, transposition operators (which shift the original object within a fixed diatonic scale). Both types of transposition are expressible in terms of signature transformations. A signature transformation reinterprets any diatonic object in the context of a different key signature. With an appropriate understanding of octave and enharmonic equivalence, the signature transformations can be shown to generate a cyclic group of order 84, of which both the mod-12 and mod-7 transposition groups are subgroups. Signature transformations therefore hold considerable theoretical potential in unifying chromatic and diatonic structures, and relate to a number of established constructions in transformation theory and diatonic set theory. Direct applications of signature transformations may be observed in the works of many composers, as illustrated by examples from composers as diverse as Schubert, Debussy, and Michael Torke.

Hook applies the signature transformations not only to the obvious case of the abrupt shift to C# major in the contrasting middle but also to the succession of four-element eighth-note motives passed back and forth between the upper voices.

Reference:

Julian L. Hook. "Signature Transformations." In Jack Douthett, Martha Hyde, and Charles J. Smith, eds. Music Theory and Mathematics: Chords, Collections, and Transformations, pp. 137-160. University of Rochester Press, 2008. Link to book page on the UR Press site.