File created 25 July 2008; updated 4 January 2011.

Go to part 2
Registral motif in Schubert waltzes
Summary of Proto-Backgrounds.

INTRODUCTION: LERDAHL'S ANALYSIS MODEL

Tonal frames are understood here as schemata comprising the "a" level elements of a time-span or prolongation reduction in the system of Lerdahl and Jackendoff (GTTM: Generalized Theory of Tonal Music (1983)), as amended and extended by Lerdahl (Tonal Pitch Space (2001)). An example (from TPS, ch. 1) is given below; the work at hand is one of two extant settings by J. S. Bach of the chorale "Jesus, der ist mein Leben." The upper section shows tree notation with the score for the beginning and end. The middle system shows standard notation for the time-span reduction; the lower system the "a" level elements of the prolongational reduction.

As Lerdahl explains, the fact that the standard notation for the "a" level (or basic form) of the time-span reduction looks like the background of an older-style Schenkerian analysis is merely a coincidence: "Unlike the Ursatz, which it superficially resembles, the basic form is not an a priori generating structure but a description of a common reductional state, reflecting the trajectory from structural beginning to cadence" (TPS, 25). And, later in the same source: ". . . the crutch of the constraining power of an encompassing Ursatz schema would seem too enticing to resist. I take the psychologically more plausible position that schematic prototypes arise out of a convergence of simple cognitive principles that are available at or near musical surfaces" (40).

Elsewhere Lerdahl says that "the Ursatz is construed not as a well-formed grammatical entity to which acceptable tonal pieces must conform but as a prototypical instance within a flexible underlying schema. The Ursatz need not exist a priori, nor need there even be any claim that it influences unconscious listening. It is just a particularly stable manifestation of classical tonal principles." ("Underlying Musical Schemata" (1988), 287). Go to a scratch graphic with four basic forms gleaned from analyses in TPS: basic forms.

In the dichotomous model of hierarchy schemas set up by Lawrence Zbikowski (Conceptualizing Music: Cognitive Structure, Theory, and Analysis), the analytical descriptions of GTTM fit the atomistic hierarchy, which favors a reductive mode of analysis: ". . . each level is a conformance class whose elements combine into units that constitute the elements of the next higher level in the hierarchy. This process continues recursively until the limits of the system are reached" (108). (The opposing model is a chain-of-being schema, or a top-down hierarchy emanating from and infused with mysterious (ultimately supernatural) energy.) The GTTM method is not strictly reductive (occasional "leaps" upward are permitted by certain preference rules), just as traditional Schenkerian analyses are not strictly top-down, at least with respect to the analytical task itself (the final result is another matter).

In GTTM there are two types of analyses, one focused on rhythm, meter, and form (time-span reduction) and another focused on tonal hierarchies (prolongational reduction). The former constrains the latter directly through the Interaction Principle: "In constructing a prolongational reduction, we can, for each prolongational region, simply search for the strongest prolongational connection possible among the events in the two largest levels of time-span reduction represented in the region. . . . In terms of musical cognition, this means that patterns of tension and relaxation are strongly organized by rhythmic articulation--an intuition that seems obvious" (233).

At the largest level, the basic forms of the time-span reduction ("a" at the right) and prolongational reduction ("b" at the right) are matched by a "normative (or preferred) overall prolongational structure for phrases and larger levels of grouping" (GTTM, 233) ("c" at the right). Lerdahl is willing to acknowledge these as schemata and understand them not as ideal forms but as elements of cognition: "Experienced listeners often attend to the expectations, realizations, and alterations of [schematic] units as much as they do to the unfolding of the pitch events that compose them" (TPS, 248); and "Schemata for a given domain have particular variable and default values, and can be intricately organized in ways unique to the domain. Yet schemata are fluid: we adjust or combine them creatively" (1988, 273).

REVISION OF LERDAHL'S MODEL TO EMPHASIZE REGISTER

I will use the basic forms as a starting point but will call them tonal frames in order to make a clear distinction from GTTM and TPS, because I have a stricter view of the role of register. The reduction to one staff in figures "a" and "b" above seems to me both unnecessary and misleading; there is no reason to collapse registers in the final level of a reduction other than to make the analysis resemble the pseudo-strict-counterpoint of a Schenkerian background. One might perhaps point out that these figures are abstract representations, not analyses, but the same form is used above in the time-span reduction of the chorale.

In addition to respecting registral position in so far as possible, I take the "best form" of a tonal frame to have three voices, with a harmonic bass and a clausula vera construction in the two upper voices, as in the first row of examples below. The positions of the melodic voices over the initial tonic depend on the time-span reduction. It is not necessary that they form a direct voice-leading connection to the closing cadence: the first row of examples do so, the second row do not. It is also not necessary that the final upper voice pitch be ^1 nor that both elements of the clausula vera be present (one voice could move ^7-^1 while the other stayed stationary on ^5, for example), but I focus here on the forms that do highlight the observation that rising cadence gestures derive "naturally" from inversion of the clausula vera formula: the 3-1 of the first example becomes the 6-8 of the third example, although of course no priority is implied, as one could say equally well that the 6-8 is inverted to become the 3-1.

The forms in the upper row certainly resemble backgrounds, as Schenkerian analysts overwhelmingly distribute the fundamental structure across a piece as a single extended and elaborated [prolonged] note, followed by a stepwise descent in the final or most important cadence. In my conception of a hierarchical model, however, time has priority over pitch and register has priority over line.

EXAMPLES: SCHUBERT, D366ns1 & 6

The first of Schubert's Laendler, D366, offers a simple example. The score of its first strain is below, a diagram of the tonal frame below that. In this instance, the initial pitch is unequivocal (E5) but no "alto" is expressed; at the end, the 3 in the 3-1 formula is stretched out across a bar (B5-G#5). The stretching of register (in the violinistic manner of the laendler) is a prominent surface motif in this waltz.

In the sixth waltz from the same set, the voice leading moves in 4- and 5-part block chords, and the static quality that results is reinforced by the "failure" to reach ^8 in the cadence of the first strain, but in the reprise the C5 is reached and a reinterpretation of the status of ^7 (B4) is in order.

The continuation of this essay looks at the complexity of schematic unfolding, the definition of register, and the role of lines: Go to Frames, part 2.

All original material copyright David Neumeyer 2008-2011.

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