1. Richard Bass, "Models of Octatonic and Whole-Tone Interaction: George Crumb and His Predecessors," Journal of Music Theory 38 (1994): 155-186.
2. Bass, "Models of Octatonic and Whole-Tone Interaction," 186.
3. Richard Cohn, "Properties and Generability of Transpositionally Invariant Sets," Journal of Music Theory 35 (1991): 1-32.
4. Cohn, "Properties and Generability," 4.
5. CUP is Robert D. Morriss term for the complement union property. He presented this work in the article "Pitch-Class Complementation and its Generalization," Journal of Music Theory 34 (1990): 175-245. K-net is just a shorthand term for a Klumpenhouwer Network. The general properties of k-nets are presented by David Lewin in the article "Klumpenhouwer Networks and Some Isographies that Involve Them," Music Theory Spectrum 12 (1990): 83-120.
6. IC/B is Lewins label free method of notating inversional operations. See David Lewin, "A Label-Free Development for Twelve-Pitch-Class Systems." Journal of Music Theory 21 (1977): 29-48.
7. Hearing pitch events as occupying distinct registral steams affords each stream a degree of independence from the activities or processes unfolding in another stream. For example, pitches that are members of the same pitch class can have distinct functions determined by the registral stream they occupy. Therefore, the B2 and B4 of Example 6 are not merely octave duplications, they perform different functions determined by the process unfolding in their respective streams. This point will be reinforced in the above text and the text that follows.
8. The significance of the 6-Z28 hexachord will be revealed later in the paper.
9. This method of generating aggregates was introduced by Daniel Starr and Robert Morris in their two-part article "A General Theory of Combinatoriality and the Aggregate," Perspectives of New Music 16/1 (1977): 3-35; Perspectives of New Music 16/2 (1978): 50-84.
10. As indicated earlier, a third stream unfolds in the outer registers of the piano that is initiated by the pitch B2.
11. The precedent for interpreting hexachords as incomplete members of the set class 7-35 was, of course, established at the work's opening. Hearing the hexachords within a septachordal context allows us to hear the Ab 6-32 collection as invoking the Db 7-35 collection, which in turn allows us to relate the Ab 6-32 collection to the work's opening hexachord. All of these relationship are reinforced by the immediate appearance of Db 6-32 following Ab 6-32 (see Example 29b). The union of these two collections, of course, generates the implied Db 7-35 collection.
12. Robert Morris, "Pitch-Class Complementation and its Generalization," 191-95.
13. For a full explanation of the general properties of
k-net isographies and the rules for generating network isomorphisms see Lewin, "Klumpenhouwer Networks and Some Isographies that Involve Them."
End of footnotes