Friday, January 15, 2010

LINE, N, and Westergaard's Tonal Theory

In the MTS article, I present a small set of transformations that are applied to the intervals of the proto-backgrounds. I didn't realize it at the time, but Peter Westergaard had already done something very close to what I intended (an earlier, less careful formulation of it was behind the list of forms generated by a "criterion of simplicity" [Neumeyer 1987]). As Stephen Peles describes it:
Westergaard's set of generative operations that determine allowable within-line relations is small. Indeed, Schenkerian theorists will no doubt be struck by the comparatively parsimonious nature of Westergaard's typology of tonal transformations. A whole battery of Schenkerian operations, such as arpeggiation, unfolding, motion from an inner voice, and the like, collapse into a single operation category in Westergaard's system: namely, the arpeggiation operation. (77-78)
Westergaard's order of presentation in the section "Linear Operations and Constructs" (35-37) of his book Introduction to Tonal Theory would have served very nicely as a way to present the intervals and these transformations. He has four classes (he calls them "structures" (37)): (1) rearticulations (that is, repeated notes), which, coincidentally, introduce a whole range of rhythmic-metric issues, but which also lead to (2) neighbors; similarly, (3) arpeggiation [or, two consonant notes originally understood as simultaneous but arranged successively] leads to (4) step motion. The four of these "together with [the rhythmic devices of anticipation and delay can be used] to compose and to understand tonal lines."

What appeals to me is the simplicity of the pairings and the neat symmetry of the two groups, something implied in my Ex. 10 (the 30 backgrounds (301) but not made explicit: rearticulation --> N; arpeggiation --> LINE.

Westergaard does not introduce equivalents of my transformations INV or ADDINV; instead, he moves directly to chapters on species counterpoint and eventually introduces Schenker-derived operations he calls "doubling, borrowing, exchange, and transfer" (289).

Westergaard, Peter. Introduction to Tonal Theory. New York: Norton, 1975.
Neumeyer, David. "Thematic Reading, Proto-backgrounds, and Transformations," Music Theory Spectrum 31/2 (fall 2009): 284-324.
Neumeyer, David. "The Urlinie from ^8 as a Middleground Phenomenon." In Theory Only 9/5-6 (1987): 3-25.
Peles, Stephen. "An Introduction to Westergaard's Tonal Theory." In Theory Only 13/1-4 (1997): 73-94.