Reimag(in)ing Riemann

LP intervals: here LP applied three times in succession to a given (E, +) tonic returns an equivalent (E, +) tonic, bringing the transformational process to a close. If measured in scale degrees, however, LP produces (Dx, +) rather than an equivalent (E,+) tonic: (E,+)(LP)=(G#, +), (G#,+) (LP)= (B#,+), (B#,+)(LP)=(Dx,+). Ex. 6 (page 113) paces off the LP and PL intervals of Ex. 5b in scale degrees. Overall, LP transforms (E,+) into (Dx,+), while PL transforms (E,+) into (Gb , +)! (Dx, +) and (Gb b 6 , +) are enharmonically equivalent, of course, with one another, but also with the (E, +) tonic. But to recognize (E, +), (Dx, +), and (G b , +) as equivalent tonics, we must abandon the scale-degree logic Riemann uses to define relations between triads in the first place. PL connects (E, +) at m. 1535 with (C, +) at m. 1537 and (AK, +) at m. 1617: (E, +)(PL)= (C, +), (C, +)(PL)= (AK, +). At that exact moment, however, the music arrives on (G#, +) via LP: (E, +)(LP)=(G#, +). Here, then, we must adjust the logic of scale degrees to the more abstract, functional logic of PLs and LPs, which converge on the same triad on the downbeat of m. 1617. We cannot, that is, hear that triad as both (G#, +) and (A6, +) at the same time without logical contradiction. This contradiction exemplifies what Lewin describes as a nonisomorphism between the algebraic,