Tonality and transformation

Abstract

Click to increase image sizeClick to decrease image size Notes As Rings points out, there are competing analyses in 5–8 Clampitt, D. 1998. Alternative interpretations of some measures from Parsifal. J. Music Theor., 42/2: 321–334. (doi:10.2307/843882) Cohn, R. 1996. Maximally smooth cycles, hexatonic systems, and the analysis of late-romantic triadic progressions. Music Anal., 15/1: 9–40. (doi:10.2307/854168) Cohn, R. 2006. Hexatonic poles and the uncanny in Parsifal. Opera Quart., 22: 230–248. (doi:10.1093/oq/kbl008) Lewin, D. 2006. Studies in Music with Text, New York: Oxford University Press. . The passage continues to intrigue analysts: See [pp. 31–32]4 Cohn, R. 2012. Audacious Euphony, New York: Oxford University Press. [Crossref] , [Google Scholar]. See also [pp. 137–138]1 Lerdahl, F. 2001. Tonal Pitch Space, New York: Oxford University Press. [Google Scholar] for an analysis with some resonance with Rings's. ‘[D]iatonic and chromatic operations cannot combine into a single group; nor, as it turns out, can they combine into a single semigroup, due to their action on different sets. The graph underlying the Parsifal analysis … thus runs afoul of GMIT’s formal requirements’ (p. 99). Certainly, there are rhythmic components to a number of the analyses, including the analysis of the second movement of the Schubert Piano Sonata in A, D. 664 (Op. 120) and the Brahms Intermezzo (Op. 118 no. 2). However, there is no wider context for the way rhythm serves to reinforce – or contradict – our sense of tonality. See 10 Tymoczko, D. 2008. Lewin, intervals, and transformations: A comment on Hook. Music Theor. Spectrum, 30/1: 164–168. (doi:10.1525/mts.2008.30.1.164)[Crossref], [Web of Science ®] , [Google Scholar] for further discussion. I have not been able to track down the source of what Cohn calls a ‘time-worn adage’, although it is reminiscent of William Ralph Inge, who said, ‘Every institution not only carries within it the seeds of its own dissolution, but prepares the way for its most hated rival’ 17 Inge, W. R. 1922. The Victorian Age, London: Cambridge University Press. [Google Scholar]. In a more directly musical context, Eric Salzman wrote, ‘The growth of equal temperament, chromaticism, and modulation had made possible the historical rise of functional tonality in the seventeenth and eighteenth centuries and destroyed it in the twentieth; thus tonality contained within itself from the start the seed of its own destruction’ [p. 32]18 Salzman, E. 1974. Twentieth-Century Music: An Introduction, Englewood Cliffs, NJ: Prentice Hall. [Google Scholar]. The reader will recall that a GIS consists of three components: a set S of elements, a group IVLS of the same cardinality as S, and a function int that sends each pair of elements in S onto a member of IVLS. Any GIS must also satisfy two conditions: (A) for all r, s, and t in S, int(r, s)int(s, t)= int(r, t); and (B) for every s in S and every i in IVLS, there must exist a unique t in S such that int(s, t)=i [Definition 2.3.1]11 Lewin, D. 2007. Generalized Musical Intervals and Transformations, 2, New York: Oxford University Press. [Crossref] , [Google Scholar]. There are ways to focus on musical practice by looking at connected sets along the line of fifths; however, Rings invariably uses a stepwise metric for both diatonic and chromatic events. The coordination between stepwise motion and motion by fifth is effected by multiplication by 4 modulo 7 and its multiplicative inverse, 2mod7;, a fact first formalized in 19 Regener, E. 1973. Pitch Notation and Equal Temperament: A Formal Study, Berkeley, CA: University of California Press. [Google Scholar]. Lewin 11 Lewin, D. 2007. Generalized Musical Intervals and Transformations, 2, New York: Oxford University Press. [Crossref] , [Google Scholar] defines homomorphism on p. 13. There is no specific demonstration of a homomorphism of GISes, however, the material in pp. 32–37 lays out the groundwork for one. See also 20 Kolman, O. 2004. Transfer principles for generalized interval systems. Perspectives of New Music, 42/1: 150–192. [Google Scholar] 21 Hook, J. 2007. Cross-type transformations and the path consistency condition. Music Theor. Spectrum, 29.1: 1–39. (doi:10.1525/mts.2007.29.1.1)[Crossref], [Web of Science ®] , [Google Scholar]. The ‘unwieldy’ notation for the accidental function acc n is clarified in footnote 38. On the same page, a diatonically oriented heard scale degree is initially formalized as an ‘unwieldy’ ordered triple.