In his dissertation, William Richards proposes a model of generic set-class space for music of Stravinsky that evinces “idiosyncratic expressions of serial techniques intermixed with non-serial linear constructions, and the commingling of diatonic and non-diatonic pitch objects.”39 Such a statement aptly describes the progression in “Come Lovely” and many of Crumb’s works more generally. Richards’s model relies heavily on aspects of Allen Forte’s and Richard Parks’s independently conceived theories of pitch- class set genera.40 Richards chooses three genera based upon the diatonic, chromatic, and octatonic collections, which are defined by the scs (013568A), (0123456), and
(0134679A) respectively.41 He illustrates how each of the genera may be mapped onto
39
William H. Richards, “Transformation and Generic Interaction in the Early Serial Music of Igor Stravinsky,” (Ph.D. diss., University of Western Ontario, 2003), iii.
40
Forte, “Pitch-Class Set Genera” 187–270; Richard S. Parks, “Pitch-Class Set Genera: My Theory, Forte’s Theory,” Music Analysis 17 (1998): 206–26.
41
Richards includes only seven pitch classes as representative of the chromatic genera since it is “equal or near-equal” to the size of the diatonic and octatonic genera; Richards “Transformation and Generic Interaction,” 76.
and into each other using various transformational processes.42 According to Richards, “Each genus is defined by a referential collection, a progenitor trichordal sc, a collection of primary members, and a collection of characteristic members.”43
The referential collection would be either the diatonic, chromatic, or octatonic genera listed above. Although his model favors the genera model and preference rules put forward by Parks, the “progenitor trichordal sc” is related to Forte’s theory in which each of the twelve trichordal set classes determines each of the twelve genera.44 The collection of primary members is comprised of those set classes that are subsets or supersets of the referential collection. The collection of primary members is quite expansive and may be shared between the different genera; however, the number of characteristic members is much more limited. Based upon Parks’s eighth preference rule, and expanded by including Forte’s progenitor set class, Richards lists four criteria for a characteristic member of a genus: 1) it must be a subset or superset of the referential collection and contain a well- defined progenitor set class; 2) all the characteristic members must be subsets or supersets of each other; 3) there is some uniformity of interval-class distribution within
42 Ibid., 83–85. 43 Ibid., 86. 44
Richards describes in detail his use of portions of Parks’s and Forte’s theory of pitch-class set genera in “Transformation and Generic Interaction,” 80n100. For additional similarities and differences between the theories refer to Parks, “Pitch-Class Set Genera,” 206–26 and Richard S. Parks, “Afterword,”
the interval-class vectors of the characteristic sets; 4) there is some uniformity in interval patterns within the SIAs of the characteristic sets.45
Although this may only seem to be a classification system, Richards is able to explain continuities and discontinuities in Stravinsky’s music using generic
transformational networks based upon the three genera. Whereas the transformations and networks change to reflect the different pieces he analyzes, the presence of the three genera provides a foundation for understanding more than just one of Stravinsky’s works. Richards admits, “Pitch-class set theory, however, does not predict functional
relationships among pitch classes. The potential of pitch-class set theory as a means of explanation lies in its ability to draw pitch objects into abstract associations without relying on an a priori functional model.”46
While functional models may be, at times, contrived independently from experience, there is no reason a functional model cannot be based upon experience, which Richards’s statement seems to imply. Even Richards’s generic model of three genera might initially be considered as an a priori construct, except that it was based upon his experience of the music. In the same way, the “chord of nature” and Schenker’s Urlinie and Bassbrechung might be considered a universal construct, but his theories were also an explanation of his experience with the music he admired. The transformations used to relate segments within Richards’s model, which are
45
Richards, “Transformation and Generic Interaction,” 86. See Parks’s eighth preference rule in “Pitch-Class Set Genera,” 209 and Forte’s progenitor in “Pitch-Class Set Genera,” 190.
46
Richards, “Transformation and Generic Interaction,” 97. Doerksen’s salience theory addresses, to a certain extent, the inability of set theory to determine the relative significance of events that Richards describes. See Doerksen, “A Theory of Set-Class Salience.”
based upon mathematics and logic, are a priori knowledge, whereas the model itself is representative of a posteriori knowledge gained through the a priori knowledge of mathematics and logic. Because knowledge, especially within theoretical discussions about music, seems to be entangled between “what is known” and “what becomes known,” the argument that knowledge exists somewhere between these two categories seems most plausible. Because of this, the proposed model of octatonality will
incorporate both a priori and abstract concepts.
The ordered arrays listed in Table 3.2 suggest a notion of mode. In this context, it will be used to refer to a specific ordering of an octatonic collection’s array.47 As will be discussed later, order and function are important aspects of the model and thus mode is perhaps a more suitable term than other terms considered such as referential collection or
genera. Each of the octatonic modes listed in Table 3.2 is aligned such that each array’s
SIA is equivalent; however, because the same SIA is repeated every ten nodes, indicated by shading in Table 3.2, there are four possible ways to arrange each of the octatonic modes to align the respective SIAs.48 The nomenclature used to describe the octatonic modes is as follows: R is appended to the octatonic collection designation to identify the specific Rotation of the array. The first superscript following R indicates the pitch class occupying the 1.1 node of the mode and the two superscripts in angle brackets indicate
47
The concept of mode within this dissertation lies somewhere between the concept of mode in Western music theory and the Russian concept, as espoused by Boleslav Yavorsky, Sergei Protopopov and Yuri Kholopov, discussed by Phillip Ewell in “Rethinking Octatonicism: Views from Stravinsky’s Homeland,” Music Theory Online 18, no. 4 (2012).
48
the interval between the first and second pitch classes and second and third pitch classes of the mode respectively. The use of the first pitch class and of the first two intervals is necessary to uniquely identify each mode because of the repetition of pitch classes within the array (five in total) and the repetition of the SIA every 10 nodes (four in total). For example, Oct1R1<65> is the octatonic mode which is based upon the hyper-aggregate constructed from the Oct1 collection such that pc 1 is in nodal position 1.1 and the interval between the first and second pitch, 1 and 7, is 6 and the interval between the second and third pitch, 7 and 0, is 5. Subsequent iterations will use the same SIA, which repeats every 10 nodes, of <6594564954>.
The governing octatonic mode, which will be contextually determined, will determine the octatonic key; thus, the octatonic key is the most structurally significant octatonic mode.49 A key is notated the same as a mode, but to distinguish between the two in analytic figures the key notation will be enclosed in a box. A key area is a merger of two or three octatonic modes. Whereas in tonality, the terms key and key area are virtually interchangeable locutions, in octatonality this is not the case. A reversal of the metaphorical terminology—applying an octatonic key area to a tonal key area—would be equivalent to the interpretation that a particular tonal sonata has the key area of C major, G major, and A minor. For example, a hypothetical octatonic piece may employ the modes Oct1R1<65> and Oct2R9<65>. The correlation between octatonic modes can be consolidated into one expression by the notation Oct1,2R1,9,<65>. The first two superscripts separated by commas after “Oct” identify which octatonic modes merge; the second two
49
superscripts separated by commas after “R” identify which rotations of the respective modes merge; the SIA is the same for both modes and is therefore not duplicated. If, based upon a contextual analysis, it was determined that the mode Oct2R9<65> was the most structurally significant, it would be notated first (e.g. Oct2,1R9,1<65>). In cases where the octatonic key area includes modes from all three octatonic collections, the indices of the other collections will be appended such that it follows the cyclical order 1–2–3–1... . For example, if Oct2R9<65> isthe key, then a key area might be represented as
Oct2,3,1R9,B,1<65>. The significance of keys and key areas will be discussed in detail in the following sections.
This theory of octatonic modes and keys departs significantly from Forte’s set- class genera since a progenitor set is not utilized. The theory more closely resembles Parks’s set-class genera because of the reliance on successive interval arrays, referential collections, and interval vectors. The concept of octatonic key areas aligns most closely with Richards’s model, which is limited to the diatonic, chromatic, and octatonic genera. The octatonic model primarily differs from Richards’s because of the importance placed upon the ordered octatonic array; however, it is similar in that it is an a priori
construction based upon my experience with the music.
The conceptual metaphors presented in the above discussion rely heavily upon tonal metaphors. Key areas and modes are containers isa metaphor that is featured prominently throughout the section and is similar to how one views keys and modes in tonality. In addition, the construction of the model is based upon a repeating base SIA that implies two natural metaphors: the base SIA is a germ and the octatonic key is a
is in any way organic or related to nature should be supplanted with the notion that an octatonic key is a metaphorical extension of the theory-constitutive metaphor of tonality.