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COPYRIGHT 2000 University of Washington
THE PROPOUND CHANGES that David Lewin and transformational theory have engendered in our thinking about music offer analytical possibilities that have only begun to be explored. The most compelling applications of Lewin's ideas have been by Lewin himself, in the illustrative analytic commentary on excerpts and short works of Generalized Musical Intervals and Transformations, and in the full-scale analyses of Musical Form and Transformation. [1] Analytical studies by Michael Cherlin, Henry Klumpenhouwer, Edward Gollin, and Shaugn O'Donnell, among others, have drawn inspiration from Lewin in interesting and productive ways. [2] Otherwise, Lewin's ideas have been slow to yield analytic offspring of consistently commensurate quality-a reflection not of their limits but of their depths. The basic reorientation that Lewin's approach Requires--the shift of focus away from equivalence relations among individual pitch-class collections and toward transformational processes that relate them--reaches deep into the analy tic process and adds an important, distinctive dimension to such inquiries. It also inspires an expansion and redefinition of what, exactly, a transformation is.
Perhaps the most successful, and broadly suggestive, analytical results have arisen from attempts to formulate new transformations that are sensitive to particular aspects of a given musical context. Contextual transformations help establish extensive and penetrating connections among musical events that might at first seem only unremarkably related, or not related at all. This manner of understanding transformation offers powerful possibilities, with applicability that is as rich (or as limited) as the musical ideas being examined. It deserves greater critical attention, and studied consideration of the possibilities it holds and the directions it might lead.
To explore this subject I first discuss and characterize the type of contextual transformation that has appeared most frequently in Lewin's work. Then I explore alternative conceptions and new analytic possibilities.
To start, let us take a look at a portion of Lewin's analysis of Dallapiccola's "Simbolo," the first piece from Quaderno musicale di Annalibera (MFT 1-15). Example 1a displays measures 11-16 of the work, and 1b reproduces Lewin's analytic notations for that passage through the beginning of measure 16 (from Example 1.2, MFT, 5). Lewin's analysis highlights two "configurations" of the aggregate (measures 11-14) 15- 16), each with an [F.sub.3]-[G[flat].sub.4] dyad in the left hand and a series of dyads and trichords, which he explains as products of linear motion, in the right hand. The transformation from the first configuration to the second, he asserts, occurs via operation "I," defined as "inversion about [an] odd-dyad-out" (MFT, 7). In other words, the [F.sub.3]-[G[flat].sub.4] dyad--the "odd-dyad-out"--acts as a kind of inversional hinge in the transformation of the first configuration into the second. Lewin explains,
The operation I, it will be noted, was not defined as "inversion about F and F#," or as "inversion about C, followed by [T.sub.11]." It was not defined with reference to any pitch classes whatsoever. Rather, it was defined with respect to a "contextual" feature of the configuration(s) upon which it operates .... This sort of "inversion" operation differs from those defined by pitch-class centers. (MFT, 7)
In other words, to use terminology that Lewin expressly avoids early on in his analysis, the configurations realize two forms of the piece's row that are inversionally related (at [T.sub.11]I). The odd-dyad-out is an invariant ic 1 dyad between the two forms. There is no complete, unambiguous row ordering at the beginning of the piece, but we initially suspect, and eventually can confirm, that this invariant dyad occurs in the row's first two order positions. From this we may conclude that any pair of row forms [P.sub.n] and [I.sub.(n+1)] will share an initial ic 1 dyad. Indeed, that is exactly how Lewin approaches "I" later in the piece, when configurations cease to connect via odd-dyads-out, requiring him to invoke "some stronger ordering from which both types of configurations can be derived," namely the piece's tone row, and to redefine "I" as "inversion-about-thenotes-of-the-boundary-semitone" (MFT, 12-13).
Lewin's initial "I", then, recognizes a special way of configuring any two row forms [P.sub.n] and [I.sub.(n+1)] such that the initial invariant dyad is highlighted as an inversional hinge. This "I" is typical of many of the contextual transformations that Lewin defines and describes in his analyses. The configurations being associated are inversionally equivalent, and the key contextual feature that relates them, the catalyst of transformation, is a set of common tones. A survey of other contextual transformations throughout Lewin's work reveals that invariance among inversional equivalences (in pitch or pitch-class space) is almost always somehow involved.
My analysis of "Simbolo," which was inspired and based on Lewin's-- even while drawing some different conclusions--also focuses on invariance patterns in the piece. [3] I place more emphasis on the BACH motives Dallapiccola frequently extracts from configurations, and define a new contextual transformation, K-inversion, that relates two inversional configurations in which the last note of BACH is the same. Thus the contextual relationship rests upon a single invariant pitch class. Dallapiccola plainly chooses row forms and creates configurations so as to bring out K relations, ultimately relying on that operation to bring the structure to a satisfying conclusion.
In his commentary on Stockhausen's Klavierstuck III (MFT, 16-67) Lewin carefully demonstrates the value of contextual transformations. He describes relations between several forms of the pentachord (01236) in the work but then notes the inability of conventional analytic tools to highlight the structural relationships that he finds most meaningful. Again an invariance under inversion solves the problem: he defines an operation "J" that preserves the pitch classes of (0123) subsets within (01236) sets under inversion (MFT, 26). This establishes a family of inversionally equivalent pentachord pairs that cannot easily be associated through other means, such as through index numbers.
Cherlin defines and demonstrates contextual transformations involving invariance patterns with great success in his study of Schoenberg's Moses und Aron. [4] Cherlin describes a number of different row partitions, all of which have dramatic meanings and consequences, and then defines operations that produce movement between partitions within the same row form, or between different row forms, both identically or differently partitioned. He thereby identifies a great many invariance patterns inherent in the row structure and exploited by the composer. For example, Cherlin's "BACH" operation transforms a row form into an identically partitioned RI form that holds the notes of the BACH motif invariant (adjoining the first two and last two notes of the row). [5]
One of Lewin's invariance-based contextual transformations has been applied in a variety of contexts. His RICH operation links RI-equivalent versions of a motive in a "chain" of interlocking dyadic invariances (GMIT 180-81). Lewin finds RICH transformations, and their companions TCH--the transpositional relationship created after successive applications of RI (GMIT 181)--in music ranging from the development section of Mozart's G minor symphony, where RICH interacts interestingly with the rhythm (GMIT 220-25); to Parsifal, in a series of statements of the Zauber motif (GMIT, 161-64, 181-82); to a piece from Bartok's Mikrokosmos (GMIT, 226-27). Lewin's observations about RICHed row forms in Webern's op. 27 (GMIT, 181-83) recognize a typical technique of row linkage in Webern's serial writing that can also apply to the invariant dyads among row forms in the Dallapiccola analysis discussed above.
Lewin also extends the idea of RICH to define BIND as a transformation producing invariance between the first and last notes of two RI-equivalent motives. This enables him to summarize one whole passage from Die Walkiire using a single transformational network (GMIT, 208-9). In another extension, he defines MUCH as an RI chaining in which members overlap to the maximum possible extent, showing how Bach interlocks motives in just that way in the first Two-Part Invention (GMIT,183-84).
A rare example of an invariance pattern relating transpositional equivalences can be found in Lewin's remarks on Webern's op. 5, no. 4. Three phrase-concluding motives are shown to be related via contextually defined transpositions: TLAST "transposes a series by its last interval," and TFIRST "transposes a series by its first interval" (GMIT, 188). This produces not only a noteworthy pitch-class invariance, by projecting the first note of a defining interval onto the second, but also a different sort of invariance relationship, in which a distance between notes is preserved as a distance between entire motives.
The main difference between these transformations and other types of contextual transformation found in the literature usually involves the equivalence aspect. Let us take a look, for example, at Lewin's discussion of "transformations used by Jonathan W. Bernard in studying how Varese's music expands, contracts, and displaces registral space." [6] Lewin defines two operations, FLIPEND and FLIPSTART,...
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