Meter-as-rhythm (after Hasty)

Where London's theory respects a traditional distinction between rhythm and meter, Christopher Hasty actively seeks to break that distinction down by arguing that meter must be continually reinvented in listening: for example, he asserts that "meter, even when viewed from the perspective of metrical type, is fully particular and never 'the same'" (131). (London, on the other hand, cites cognition studies to argue that meter is internalized and therefore becomes a set of expectations that are maintained until strongly contradicted.)

Hasty's theory is based on projection: an attack and a duration project the possibility of another attack and duration, the simplest of which would be a repetition of the first (the solid arrow in (a) of the graphic shows a realized projection, the dotted slur that follows the potential of the same kind, the longer arrow marked Q a longer potential projection).

The simplest pattern in a 3/4 meter is shown in (b). As Hasty observes, triple meter is decidedly more complex than duple meter because

The third beat cannot function exactly like the second beat simply to continue the duration begun "before" there were any beats, for now that there is a second beat there is also a real potential for projecting a half-note duration (the potential Q in [the] example). In order to function as a continuation, the beginning of the third beat must deny this potential. In contrast, the beginning of the second beat denied no potential--rather, it created one projection and the potential for another. (132)

Given the consistency of temporal figuration in D779n13, a complete analysis on Hasty's terms would quickly become tedious, but his method may allow us a more nuanced view of the establishment of the hemiola pattern that is shown so plainly in Schachter's durational reduction. The familiar accompaniment pattern of bass and two afterbeats works out in a direct way through the first four beats the projection of triple meter we would expect from (b). In the next graphic, see P for quarter beats, R for the bar measure (the example is simplified as it shows only completed projections, not projective potential).

The repetition of R through bar 2 is sullied by some uncertainty as the entrance of the right hand figure creates an unusual two-beat anacrusis that sets up the possibility of an independent projection (Q). The completed projection of the left-hand figure through bar 2 eventually secures a two-bar hypermetric level (S), but the duple projections continue. Thus, the metric properties of the first three bars are all different: a simple development of 3/4 meter through quarter-note and bar-level projections (P, R); complicating the meter through a "superimposed" duple projection (Q); and establishment of a two-bar hypermeter (S) that includes both duple and triple projections.

References:
Justin London. Hearing in Time: Psychological Aspects of Musical Meter. New York: Oxford University Press, 2004.
Christopher Hasty. Meter as Rhythm. New York and Oxford: Oxford University Press, 1997.

Metrical reading (after London)

Today and tomorrow’s readings are based on theories with sharply opposed views of temporality in music. In Justin London's theory meter and rhythm are distinct. Meter is portrayed as cyclical, based on entrainment and other cognitive constraints. Meter is "time-continuous," patterned cyclically: once the listener is entrained through subjective rhythmization (14-5) by a minimal number of phenomenal events (an event at a beat level within perceptible range plus one set of subdivisions), meter recycles itself and thus continues "in mind" independently of phenomenal events unless undermined or contradicted by them. The issue for analysis is how well a meter is defined, how "thick" or "thin" it is ("one may characterize meters in terms of their hierarchic depth–that is, whether a meter involves a rich hierarchy of expectation on many levels at once, or only a limited set of expectations as to when things are going to occur" (25)).

I will work through an example, using not D779n13 but the well-known theme of Mozart's K. 331, I. We start with the establishment of meter through entrainment. To simplify discussion, I assume a tempo where a dotted quarter equals 60–tempo is crucial to London's theory because it determines the possible meters and the range of their rela-tionships. The first chord (a in the graphic) cannot establish meter in itself, nor can the sixteenth that follows–the latter is a subdivision perhaps, but of an unexpressed beat in a meter not yet established (b). The second chord repeats the first, and that in one respect is a weakness–the meter is still not established because the second chord is just as likely to be the second beat of a duple meter as the third eighth of a 6/8 measure (c). Only with the quarter note chord is the meter unequivocally set (as triple or compound duple, that is): we have now heard well-defined events on two beats and at least one of the triplet eighth subdivisions (itself supported by a still smaller value as a "pickup") (d).

The meter is now set, and Mozart continues to define it effectively through the quarter-eighth rhythm. Given the ubiquity of this figure across the entire theme, and the fact that Mozart even turns to continuous eighth notes in the second section, I would describe the meter in this theme as "thick," as heavily and continuously reinforced by phenomenal events. With the measure moving at 30 bpm (or 2000 ms), the upper threshold for perception of a (hyper)metric unit (6000 ms) would be reached by the end of measure 3. Thus, it is quite easy to maintain a palpable or immediate sense of hypermeter at the two-bar level, a perception that Mozart patently encourages. Beyond that, a regular hypermeter and the symmetrical proportions that can derive from it are abstractions, constructs in memory. Beyond this distinction between the immediate and the abstract, however, there is no need to separate meter from hypermeter; as London puts it, "the number of metric levels both above and below the beat can and does fluctuate, [and thus] there is no substantive distinction between meters and so-called hypermeters" (25); elsewhere he says that "having several levels of metric structure present above the perceived beat is no more extraordinary than having several levels of subdivisions below it" (19).

I find London's cyclical conception of meter appealing, in part because it respects distinctions between rhythm and meter (or between patterns of phenomenal events and processes of metric entrainment), in part because it is intuitively more satisfying than Lerdahl and Jackendoff's hierarchical model. London's cyclical conception of meter allows one, by contrast, to see that the heads of the longest time spans, "structural down-beats," "structural accents," and similar terms as Lerdahl and Jackendoff and others deploy them, are rhythmic/metric themes for hierarchy-based readings whose central task is to sort out the roles of the various temporal units that can be distinguished in hearing a musical performance.

The results obtained for K331, I, apply equally well to D779n13: meter is "thick," heavily and continuously reinforced by phenomenal events. The entrainment of meter happens in the introduction (as the dancers would require), and once the right hand part enters the metric levels of beat (quarter beats), subdivision (groups of eighths), measure, and two-bar hypermeter are maintained consistently throughout (even the right-hand hemiolas are consistent and readily subsumed in the ongoing meter). Only at the four-bar level is hypermeter inconsistent, but by now we know that trait of D779n13 well.

Reference:
Justin London. Hearing in Time: Psychological Aspects of Musical Meter. New York: Oxford University Press, 2004.