Richard Cohn and Douglas Dempster criticize the treatment of motivic patterning in Schenkerian analysis as internally inconsistent: "In principle, motivic relationships must emerge as a by-product of voiceleading reduction; but in practice, voice-leading reductions are molded in part to optimize motivic relations" (171). In part as a way of respecting the directionality implied by this observation, they rethink the relationship of hierarchical levels in an ingeniously plural conception of a top-down hierarchy, according to which a given musical "surface"--a text (however construed)--"'holds together' underlying diversity by providing a compositional solution to multiple and disparate demands of harmonic, contrapuntal, motivic, and rhythmic operations" (177). As Robert Fink puts it, Cohn and Dempster “make a convincing case that viewing the musical surface solely as the product of a single generative hierarchy is too limiting; they prefer to consider it as the product of multiple interlocking hierarchies. Their revisionism does not, [however,] encompass the [more] radical step of detaching the surface from hierarchy altogether” (105).
Instead of conceiving analysis in terms of a top-down or chain-of-being hierarchy (after Zbikowski), they advance "a view of musical structure as a network resulting from a set of generative operations that are intertwined yet independent of one another" (172). Specifically, they suggest the use of product networks for "modeling music [in a way that is] simultaneously generative (in the sense of precisely specifying each event) and nonhierarchical" (174). They offer four examples. The first is drawn from David Lewin (204-6; Cohn and Dempster, 172-4) and allows for simultaneous generation in melodic and harmonic (intervallic) dimensions for organum. The other three examples set Schenkerian voice leading analysis against a "more loosely defined scheme of motivic generation" (174); a voice leading frame against a plan of figuration (in Bach, Well-Tempered Clavier, vol. 1, Preludes in C Major and C Minor); and "voice-leading designs [against] metric/figurational schemes" (175).
The last of these is of particular interest here because the authors point to topical conventions, which often "subsume conventions of meter, tempo, duration, and figuration into a higher-level construct" (175). If, following Cohn and Dempster, we want to locate D779n13 "at the intersection of voice-leading paradigms and topical conventions, [through a] product network [that] values the unifying potential of movement along either of the two axes" (175), we might exploit our narrative-topical conception of the piece as a portrait of a dancing couple (earlier post to this: link; also see "dancing" in the Labels sidebar). The topic is single (the waltz itself); the narrative consists of the formation of the couple, the dance figures (multiples of two-measure groups), and the separation of the couple as the dance concludes. For the voice leading generator, we could use any linear reading.
I have made an attempt at representing such an analysis in the graphic below. The upper line shows the initial courtesies (M = Man, W = Woman) and formation of the couple (F), the figures of the dance (D) and their multiples against each section of the music, and finally the separation at the end (F-1). The second line is a "primitive urlinie" ^3-^2-^3 (which, of course, could be easily renotated as a simple elaboration of the proto-background ^3-^3): here the graph shows all repetitions and thus a correct representation of the music in the time-line of the dance.
There is some virtue in writing out the repetitions -- as a reminder of the unifying power for (any) music of that simple process, but D and its multiples leave out two important features of the dance: the detail that each figure consists of a step by the man in one measure mirrored by the woman in the next, and the larger, contingent factor of the environment: how many figures one dances along any side of the room and how many figures may be accomplished in a complete circuit depends on the size of the room. Since Schubert's waltzes were played at house balls and parties, the spaces involved would most often have been relatively small.
By way of experiment, my partner and I managed 8-9 compact figures in circuits of a rectangular floor at 12 x 16 feet, and thus a couple could be expected to go around the room at least three times while dancing to D779n13. The surface represented in the graphic, then, is not a reading of the text of D779n13: it is the contingent circumstance of a dance to that music. In Cohn and Dempster's terms, we conceive this "complex . . . surface [not] as unified by an underlying structural simplicity [but] as a solution to the compositional problem of mutually satisfying the demands of several sets of independent formal operations" (176; their emphasis); that is to say, the dancers dancing and the music playing produce the dance. Note that the operations are in fact independent: unlike the extended figures of a quadrille, the figures of the waltz are coordinated with the music only up to the two-bar hypermetric level.
References:
Cohn, Richard, and Douglas Dempster. "Hierarchical Unity, Plural Unities: Toward a Reconciliation." In Bergeron, Katherine and Phillip V. Bohlman, eds. Disciplining Music: Musicology and its Canons, 156-81. Chicago: University of Chicago Press, 1992.
Fink, Robert, “Going Flat: Post-Hierarchical Music Theory and the Musical Surface.” In Cook, Nicholas, and Mark Everist, eds. Rethinking Music, 102-37. 2d ed. New York: Oxford University Press, 2001.
Lewin, David. Generalized Music Intervals and Transformations. New Haven: Yale University Press, 1987.
