1. A single course of projection refers here to a single periodic meter. The concept of meter as projection is taken from Christopher Hasty, Meter as Rhythm. (New York: Oxford University Press, 1997) and denotes meter 'as a process internal to the sounds themselves, an experience, rather than something already determined in advance and imposed from without.' See Joseph P. Swain, "Shifting Metre." Music Analysis 20/1 (2001): 119-41 . This author's use of the term projection is not necessarily in accordance with Hasty's, and while Hasty's concept of meter is refreshing, it is not fully assimilated here. The current model, printed as part of a thesis in 1996, was conceived outside of Hasty's paradigm.
2. William Benjamin, "A Theory of Musical Meter." Music Perception 1/4 (1984): 335-413 [391-96].
3. See Pieter C. van den Toorn, The Music of Igor Stravinsky. (New Haven: Yale University Press, 1983) chapter 8, and Stravinsky and 'The Rite of Spring'; The Beginnings of a Musical Language. (Berkley: University of California, 1987), chapter 3.
4. The model of pulse streams appears in John Roeder's "Interacting Pulse Streams in Schoenberg's Atonal Polyphony." Music Theory Spectrum 16/2 (1994): 231-249, and "Pulse Streams and Problems of Grouping and Metrical Dissonance in Bartok's With Drums and Pipes." Music Theory Online 7/1 (2001). A further application is found in an unpublished article "Rhythmic Process and Form in Bartok's Syncopation." retrieved Jan 10, 2006, from http://theory.music.ubc.ca/preprints/syncopation.pdf
5. For instance: Benjamin 1984, 390-403, Andrew Imbrie, "Extra Measures and Metrical Ambiguity in Beethoven." In Beethoven Studies. Ed. Alan Tyson. (New York: Norton, 1973) [45-66], Jonathan D. Kramer, The Time of Music; New Meanings, New Temporalities, New Listening Strategies. (New York: Schirmer Books, 1988) [98-107], Fred Lerdahl, and Ray Jackendoff, A Generative Theory of Tonal Music. (Cambridge: MIT Press, 1983) [99-104], and Carl Schachter, "Aspects of Meter." Music Forum VI (1987):1-59 [32-56]. For a comprehensive account of sources of hypermetric irregularity see William Rothstein, Phrase Rhythm in Tonal Music. (New York: Schirmer Books, 1989).
6. This definition is paraphrased from Kramer 1987, 102-3, but the broader picture is more complicated, there being a number of different terms to describe the same or similar things. For example, Benjamin, Horlacher, Kramer, and Rothstein use overlap, elision, and reinterpretation to describe roughly the same process. See Gretchen Horlacher's discussion in "Metric Irregularity in Les Noces: The Problem of Periodicity." The Journal of Music Theory 39/2 (1995): 285-309 [291-94].
7. Both van den Toorn and Horlacher prefer descriptive terms when discussing metric identity, such as the 'over-the-barline, upbeat-downbeat identity.' (van den Toorn 1987, 76). This author prefers numbers representing beats as they are provide a greater degree of accuracy in tracing irregularities and the course of projection. David Temperley in The Cognition of Basic Musical Structures. (Cambridge, MA: MIT Press, 2001) [331-33] goes further in assigning a 'metric address' to an event--a series of numbers which records the exact location of an event within all metric levels. Of course such numbers are purely abstract but they are a useful tool for analytical purposes. Such a level of abstraction is in truth no different than assigning numbers to pitches and chords, a practice few would question.
8. Benjamin 1984, 392.
10. 'Theoretically, these three operations [extension, contraction, and elision] can occur on any level, although in tonal music they most often appear as middleground phenomena.' (Kramer 1988, 106-7).
11. Joel Lester, Analytic Approaches to Twentieth-Century Music. (New York: W.W. Norton & Company, 1989) .
12. Horlacher 1995. Horlacher takes a more flexible approach to metric analysis in "Bartok's Change of Time: Coming Unfixed." Music Theory Online 7/1 (2001).
13. Horlacher 1995, 289.
14. Horlacher 1995, 290.
15. Horlacher uses the terms deletion and reinterpretation.
16. A more detailed discussion of these rhythmic/metric types can be found in van den Toorn 1983, 216-18 or 1987, 99-101.
17. Hence Kramer's perception of meter being 'resistant to change' (Kramer 1988, 83); once a steady meter is established, it tends to remain in force throughout a piece.
18. Van den Toorn 1987, chapter 3.
19. The reference to an internalized meter appears in van den Toorn, "Stravinsky, Les Noces (Svadebkas), and the Prohibition against Expressive Timing." Journal of Musicology 20/2 (2003): 285-304 .
20. Van den Toorn 1987, 67.
21. Van den Toorn 1983, 215. Concepts of polarity date back to Stravinsky's evocative remarks from the Poetics of Music but take on a more concrete form in Berger's discussion of polarity in relation to the symmetrical properties of the octatonic scale. See Igor Stravinsky, Poetics of Music in the Form of Six Lessons. Trans. Arthur Knodel and Ingolf Dhal. (Cambridge, MA: Harvard University Press, 1947) [36-40], and Arthur Berger, "Problems of Pitch Organization in Stravinsky." Perspective of New Music 2/1 (1963): 11-42 [25-26]. Van den Toorn extends Berger's work, tying in also meter, form (block juxtaposition) and melodic style, to elucidate the distinctive identity of Stravinsky's music.
22. For Berger the 0/6 axes of the octatonic scale are of 'equal and thus independent weight' which leads to the condition of polarity, namely 'the denial of priority to a single pitch class precisely for the purpose of not deflecting from the priority of a whole complex sonore.' (Berger 1963, 25). By substituting metric identity for 'pitch class' and meter for 'complex sonore', we arrive at a definition which serves van den Toorn's concept of 'rhythmic-metric-identity polarities'. The question then is whether a steady background meter is of 'equal and thus independent weight' where it is not explicitly projected. Van den Toorn appeals to Andrew Imbrie's radical/conservative distinction (Imbrie 1973, 65) to reason that it might be, because a steady background meter exists more strongly for conservative listeners who continue counting periodically in the face of foreground irregularities.
23. Roeder 1994: 232-233. The theorists referred to are Jonathan Kramer, Charles D. Morrison, and Lerdahl and Jackendoff.
24. Musical examples are drawn from the Dover edition 1989, a reproduction of the Izdatel'stvo "Muzyka" edition, 1965.
25. See Harald Krebs, Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. (New York: Oxford University Press, 1999) and "Some Extensions to the Concept of Metrical Consonance and Dissonance." Journal of Music Theory 31/1 (1987): 99-120. Krebs uses the metaphor of metric dissonance to classify a number of metric effects resulting from varying degrees of alignment between metric levels. The irregularity resulting from motive displacement, described here as a phase shift, would be classified as indirect displacement dissonance.
26. Van den Toorn 1983, 228.
27. The distinctive 1-2-3-4 1-3-4 pattern surfaces throughout the fourth tableau of Stravinsky's The Wedding. Consider rehearsal no. 91, setting aside for the moment the composer's own barring which introduces the motive as an upbeat syncopation in 4/4. The bass, taken from its first eighth-note, yields a quarter-note, quarter-note, eighth-note, quarter-note succession, proportionally identical to the half-note, half-note, quarter-note, half-note succession in the Ritual of the Two Rival Tribes, nos. 57+4-5. Over an eighth-note beat, the two quarter-notes (1-2-3-4) establish a 2/4 meter which is challenged and shifted by the following eighth-note-quarter-note (1-3-4). The listener is able to infer this pattern without reference to a fixed set of pitches or its displacement; the rhythm itself delineates the accents required for its projection.
28. Van den Toorn 1987, 104.
29. Victor Zuckerkandl, Sound and Symbol; Music and the External World. Trans. Willard R. Trask. (London: Routledge & Kegan Paul, 1956) [168-74].
30. Benjamin 1984, 375. Benjamin draws an analogy between equivalence classes in the domain of pitch and that of meter. Beats which hold an identical position in the measure are said to be 'functionally equivalent' between different measures of the same meter.
31. If by meter we mean the composer's time signature, then meter does change, and likewise if by meter we mean phase. But if by meter we mean the time signature common to the different phases, then it is the phase which changes, not the meter.
32. Already acknowledged in this respect are Benjamin, Roeder, and Krebs. Although they do not use the term phase their models of meter take phase or alignment into account. Van den Toorn also regards alignment as a significant factor in Stravinsky's music. Temperley and Bartlett discuss phase in traditional tonal music, remarking that motivic parallelism is a significant factor in defining metric periods, but not the phase of the meter. See David Temperley and Christopher Bartlette, "Parallelism as a Factor in Metrical Analysis." Music Perception 20/2 (2002): 117-49 . Temperley also mentions The Rite of Spring as a work whose irregularities are governed by 'motivic parallelisms'--the displaced motives observed by van den Toorn and Horlacher (Temperley 2001, 302).
33. Van den Toorn 2003, 295.
End of footnotes