1. From the essay, Rasch, reprinted in The Responsibility of Forms: Critical Essays on Music, Art, and Representation, trans. by Richard Howard (Berkeley and Los Angeles: University of California Press, 1999), pp. 299, 303.

2. Maury Yeston, The Stratification of Musical Rhythm (New Haven: Yale University Press, 1976). Kreb's first work on this method was presented in "Some Extensions of the Concepts of Metrical Consonance and Dissonance," Journal of Music Theory 31 (1987) 99-120. 

3. Example 1 is a conflation of Ex. 2.7 (33) and Ex. 3b from "Robert Schumann's Metrical Revisions," Music Theory Spectrum 19/1 (Spring 1997), 41; this article is also a fine brief introduction to the system. The labels to the right were included in the article, but unfortunately are omitted from examples in the book. 

4. The terms "grouping" and "displacement" dissonance, which replaced the more opaque terms "Types A and B," were coined by Peter Kaminsky in "Aspects of Harmony, Rhythm and Form in Schumann's Papillons, Carnaval, and Davidsbuendlertaenze," Ph.D. dissertation, University of Rochester (1989) 27.

5. See the article cited in fn. 2.

6. One meaning of metrical dissonance not emphasized by Krebs is inherent by the idea of displacement itself--that of spatial separation. A strikingly suggestive example is found in #8 of the Noveletten, Op. 21; in the second Trio, a melody marked "Stimme aus der Ferne," a quotation from Clara Schumann's Notturno (#2 from the Soirees musicales, Op. 6), is subsequently restated in the next section (labelled "Forsetzung") in a freely displaced manner that gradually floats closer to metrical alignment. Another reference to distance is implied in the Eichendorff text set as the second of Schumann's Op. 39 cycle (alluded to in the epilogue; 251). An image held in the speaker's heart stands in for an absent beloved; Schumann's setting, an essay in displacement on many levels, features an offbeat accompaniment that portrays both a beating heart and the distance it longs to bridge.

7. See, for instance, John Roeder's review in Music Theory Online 4.4 (1998) of Christopher Hasty's Meter as Rhythm (New York: Oxford University Press, 1997). Roeder has taken an approach related to Krebs in "Interacting Pulse Streams in Schoenberg's Atonal Polyphony," Music Theory Spectrum 16/2 (Fall 1994): 231-49. 

8. One such example is the recurrence of a quarter-note upbeat beginning the four-bar groups in Grillen from the Phantasiestuecke, Op. 12 (mm. 17-24 and corresponding places), which Krebs labels D12+11, or D12-1 (148). 

9. The example illustrates the neighbor figure 3-4-4-3; Free Composition, trans. and ed. Ernst Oster (New York: Longman, 1979), 73 and Fig. 80, 1.

End of Footnotes