The appendix is a set of comments on my JMT article "The Ascending Urlinie" (338-39). Schachter spends most of the roughly 1000 words disagreeing with my interpretation of a graphic from Victor Zuckerkandl's Sound and Symbol, even suggesting that Zuckerkandl himself didn't take quite the right view of his own graphic. The last two paragraphs acknowledge that two of my readings are convincing, but only in the context of repeating his point about the exception proving the rule. In a lengthy footnote, it is understood that I misread Schenker's statements about the Bassbrechung (background I-V-I) in relation to the harmonic series.
I'll discuss the first of these three points here, the others in Parts 13b & c.
Here is Zuckerkandl's graphic, which Schachter reproduces:
I use Zuckerkandl's distinction between "acoustical space" [movement in pitch space] and "dynamic space" [rise or fall on a tension-relaxation scale] as a way of breaking through the conceptual fourth-species logjam, the style-statistics-driven assertion that descending melodic motions have priority because suspensions resolve downward. Why lines have to obey the same rules as suspensions is never explained, nor why the musics of the 19th century have to obey the rules of the 16th (we can't say it's to preserve a continuity narrative: harmonic practices and ideas about rhetoric and expression, after all, changed radically during the same time period).
Zuckerkandl points to the ^5-^8 "upper half" as potentially rising in acoustical space but falling in dynamic space, and of course I use that in the JMT article as one of the justifications for rising lines in a generative mode of linear analysis. Here is Zuckerkandl, cited by Schachter: "the tone ^6 still plays a double role, since it can be heard both as a state in the succession ^5-^6-^7-^8 and as bound to a pointing toward its comparatively stable adjacent tone ^5; the particular circumstances determine whether the meaning 'away from ^5' or the meaning 'toward ^8' preponderates in the step ^5-^6."
Schachter says that the graphic shows "implicitly a far greater bias toward downward resolution in the dynamics of scalar structure than his explicit formulation acknowledges" and "the pull of ^1 is much greater than that of ^8" (339). But Zuckerkandl doesn't imply what Schachter claims: instead, Zuckerkandl says quite directly that "particular circumstances determine whether the meaning' away from ^5' or the meaning 'toward ^8' preponderates." Schachter misreads Zuckerkandl in order to push a couple "greaters": "a far greater bias" and a "much greater" pull.
In any case, Zuckerkandl's model has firm style-statistical support in 19th century music, especially in the popular genres of dance music, where the play of ^6 and ^5 creates a kind of tonal androgyny that makes the identity of ^5 and ^6 interchangeable, in "particular circumstances," exactly as Zuckerkandl says. Eventually (that is, by around 1860 or so, but firmly and unmistakably by 1910), the two even fuse in the triad with an added sixth.
And it all starts early, in places like D779n13: see the graphic below, where I have written a narrative of the interplay of ^6 and ^5 in the first strain. The two identities of the scale degrees are plainly evident. (Recall that you can click on this thumbnail to see the graphic at its original resolution.)
References:
Neumeyer, David. "The Ascending Urlinie." Journal of Music Theory 31/2 (1987): 275-303.
Zuckerkandl, Victor. Sound and Symbol: Music and the External World (Princeton: Princeton University Press, 1956).
