If you would like to participate in discussions, please sign in or register.
I'm troubled by Tymoczko's claim that the Möbius strip models the "fundamental shape" of (representatives of) dyad (set) classes---as if this claim is akin to some objective and revelatory mathematical truth---when the resulting topology of the strip is simply a function of an a priori decision to privilege (near) maximal parsimony (as well as octave equivalence and EDO systems) while sketching the space. Privileging T6 relations, for example, uncovers the topology of a torus, which suggests there's nothing objectively "fundamental" at all about dyadic space beyond how we perforce construct and organize that space prior to modeling it. The fact that theorists have historically privileged parsimonious relations is an insufficient defense to the general indictment. That might make it the fundamental shape with respect to historical approaches to musical modeling---a description I would embrace wholeheartedly---but that doesn't make it the fundamental shape equivalent to some universal law concerning the way (representatives of) dyad classes are organized. The parsimony-strip relationship is merely one reification of a number of possible models.
[And briefly to the issue of parsimony: Why privilege semitone movement when most of the musical events we experience (e.g., melodic and harmonic motion, scales, Schenkerian Urlinie and Ursatz structures, etc.) involve  dyads? If there exists a disconnect between “closeness” in perceived musical syntax and a "fundamental" model (e.g., Lewin’s S function, the neo-Riemannian <L> transform acting on major triads, etc.), shouldn't the model be revised?]
The claim reminds me of the bias inherent in the well-known Texas Sharpshooter fallacy, where a Texan "fires his gun at the side of a barn, paints a bullseye around the bullet hole, and claims to be a sharpshooter." Within that context, then, I offer the following narrative:
That progression offers quite the non sequitur, though it does seem representative of the bias that plagues many efforts in music theory (and mathematical music theory, in particular) to provide objective models of musical passages. But I'll leave that criticism for another post.