Schachter's second criticism is that I confused levels. Here is what I wrote:
it is not just the direction of the motion but the goal of that motion that is important. Schenker accepts this implicitly with respect to the bass arpeggiation, in that he allows it to rise, rather than fall, from the V to the final I, despite his own rule of the obligatory register in the bass. Yet if any voice should be sensitive to motion toward the fundamental, it is certainly the bass! (280)
To which Schachter responds:
the placement of V below I . . . results from prolongations at later levels, which transform the underlying shape of the bass arpeggio. Through ascending and descending register transfers in the upper voice, Schenker also allows for octave displacements of the Fundamental Line's final tone at later levels.
Here I have to say that neither of us apparently understood what Schenker said. I was wrong to tie the Bassbrechung-as-expression-of-nature to the obligatory register, and it was Schachter who confused levels. In section 268 of Free Composition, Schenker says that obligatory register applies to both upper and lower voices and means a return to a voice's "basic register" (107), which he defines as "the register of the first tone of the fundamental line." His upper-voice examples do refer to the fundamental line; the two lower-voice examples, however, are both foreground details. Still, given the care with which he matches the first and last tonic notes in the bass in the abstract figures, it is reasonable to assume that he meant for the same to be true of the background bass.
Whether or not Schenker intended a background return to the register of the first bass note at the end of a piece, Schachter is clearly incorrect in isolating displacements to later levels. Schenker says on p. 107: "Nevertheless, the final tone of the [Urlinie] sometimes appears an octave lower or . . . higher." And the basses in his figures showing readings of pieces vary quite a bit, though more often holding to the configuration of the abstract examples, as in Fig. 7a: C#3-G#3-C#3. But already in Fig. 7b the position of V is reversed: F3-C3-F3. The well-known Fig. 21b (Schumann, "Aus meinen Tränen Spriessen") is interesting in this respect: in the foreground, A4-E3-A3; this becomes A3-E4-A3 in the two middleground graphs. (He does something similar in Fig. 39,2.)
All my theoretical maneuvering in that section of the JMT article was irrelevant, I now think, to justification of the rising Urlinie. Although Schachter's criticisms are off the mark, the specific justifications from Free Composition that he attacks were not very strong, that's clear, especially insofar as they tried to conform to Schenker's own poorly defined notions of the relation of acoustics and art and his equivocation about register in the background. At the time of the JMT article, I didn't have the style statistics in hand that I have accumulated since: see this PDF file with a compositions list. From Schenker all I really needed was: "In the service of art, the arpeggiation throws off the restrictions of nature and claims the right to assert itself in either an upward or a downward direction." From this follows a principle of symmetry (the up/down schema is not fundamental but expressive) that is neatly realized in Zuckerkandl's graphic of the major-key scale.
