[3] Challenges to Rhythmic Theory Raised by the Analysis

[3.1] The analysis of Examples II.1-11 shows that concepts of grouping and metrical dissonance are indeed useful for distinguishing and relating different segments of the piece, and so for addressing those synchronic aspects of musical form. But it has also highlighted some features of the music that cannot be explained with reference to those concepts.

[3.2] For example, some moments appear to be designed specifically to obscure grouping structure, at least in the way that it is theorized in Lerdahl and Jackendoff (1983). Recall m. 30, for instance, which was discussed in connection with Examples II.3 and II.4. Since the right hand has just presented three similar 5-eighth-note-long groups, we would expect another right-hand group to begin on the next eighth-note beat, that is, on the downbeat of m. 30. But that beat has none of the properties of a group boundary--the solitary right-hand E4 does not produce any substantial change, but simply extends the preceding material. The succeeding cluster, on the second eighth of the measure, sounds more like a group beginning; but then so does the D4 in the next measure, which begins a long slurred series of notes. In other words, the music presents conflicting cues. Grouping theory is very helpful for identifying these conflicts, but it does not suggest any purpose for them.

[3.3] Another passage that challenges current rhythmic theory is mm. 38ff. (Examples II.5 and II.6).  The right-hand stream (Example II.5) presents a group lasting 12 eighth notes. But the duration and coherence of this material is called into question at m. 41 (Example II.6), when the left hand imitates only the last 8 eighths, then sequences only the last 6 eighths. Since each repetition omits the beginning of the preceding group, and since the groups keep changing in duration, the parallelism between the groups is obscured. Meter is also obscured, not only by the changing group lengths but also by the lack of regular phenomenal accent. For instance, even when the right hand after m. 41 is quite regular, its long and high notes do not fall on the notated downbeats, nor do they align regularly with the varying accents in the left hand. Again, grouping theory permits a clear characterization of rhythmic structure within each separate stream. But since it describes preference rules, it provides no rationale for the multiple violations of those rules, nor does it help determine a purpose for the particular coordination of misaligned grouping and accent structures in the concurrent streams. Indeed, Lerdahl and Jackendoff acknowledge that their theory is "inadequate" except for music in which "a single grouping analysis suffices for all voices" (p. 37), which is clearly not the case here. More generally, the rapidly changing group lengths, their nonalignment with the notated meter, the overlapping extents of groups in the voices, and the sporadic accents seem to violate all the "cognitive constraints" (Lerdahl 1988) on rhythm necessary to perceive large-scale grouping and metric structures.

[3.4] While metrical dissonance theory addresses some of these inadequacies, Bartók's music poses challenges to it as well. For example, mm. 26-29 (Example II.3) can readily be labeled as a combination of "grouping" dissonance, in which groups of different lengths (5 and 3 eighths) are played concurrently, with "displacement dissonance," in which the group beginnings do not coincide. But it does not explain why the groups of 5 and 3 are displaced from each other in this particular way, or even why group durations 5 and 3 are necessary at this point in the composition. Nor does it provide a basis for understanding why one type of metrical dissonance succeeds another. A related problem arises in the canonic passages (mm. 22-26 and 30-37, Examples II.2 and II.4): considering the apparent lack of metrical control of consonance and dissonance, what constraints or compositional goals dictate the choice of materials and the durational intervals of imitation?

[3.5] In sum, current theories of grouping and metrical dissonance are designed for music in which the accent and grouping of concurrent voices are coordinated consistently. They are not adequate for analyzing polyphonic music in which these basic aspects of rhythm are constantly varying.

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