Let Them Eat Pseudoscience

In a now-(in)famous paper published in the 313th volume of the prestigious magazine Science, Dimitri Tymoczko (DT) makes the startling claim that the Möbius strip (MS) represents the topology (i.e., the “fundamental shape”) of representatives of dyad set-classes (i.e., all the types of two-note “chords” you can play on the piano). Unfortunately, he goes one step further and suggests the MS represents a sort of Platonic mathematical truth about dyadic structures in general.

This is absurd.

From page 2 of DT’s paper in Science:

I now describe the geometry of musical chords. An ordered sequence of n pitches can be represented as a point in R^n. Directed line segments in this space represent voice leadings. A measure of voice-leading size assigns lengths to these line segments….To model an ordered sequence of n pitch classes, form the quotient space R/12Z^n, also known as the n-torus T^n. To model unordered n-note chords of pitch classes, identify all points (x_1, x_2,…,x_n) and (x_s(1), x_s(2),…,x_s(n)), where s is any permutation. The result is the global-quotient orbifold T^n/S_n, the n-torus T^n modulo the symmetric group S_n.

It should be clear, even by a cursory reading of the above passage, that the geometry of the quotient orbifold is induced by a predetermined precondition of (maximal) parsimony—as well as octave equivalence and tunings that privilege an equal division of the octave—a feature reified by the directed edges whose “lengths” represent voice-leading distances in[1] “Points” of unordered sets of pitch classes will perforce be proximate to other “points” of unordered sets of pitch classes whose distances involve minimal voice-leading perturbations. The MS (fig. 1) emerges from the decision to privilege parsimonious voice-leading principles (as a function of log-frequency) in organizing the point lattice in Euclidean space.

Figure 1: A Möbius-strip topology built from representatives of dyad classes in R^2

It is this predetermined requirement of parsimony in constructing the quotient orbifold to which I object because it represents inter alia something of a Texas Sharpshooter fallacy, which leads ineluctably to a spurious intimation of Platonic design that simply does not exist. The MS is as much the fundamental topology for dyads as the dictionary is the “fundamental design” of the English language. We don’t get to marvel ex post facto at the unadulterated “linearity” of the dictionary after we’ve decided to arrange the words according to the organizing principle that engenders such linearity. The fact that modern theorists have historically privileged parsimonious relations—e.g., the conformist Tonnetz, Power Towers, Chicken-Wire Torus, Weitzmann regions, Cube Dance, etc.—is an insufficient defense to the general indictment. The parsimonious-MS relationship is merely one reification of a number of possible topologies for dyads. Privileging T6 relations, for example, generates the topology of a (ring) torus, suggesting there’s nothing objectively “fundamental” at all about dyadic space.

The appeal to Platonic discovery galvanizes general interest in the paper and, in my opinion, explains its publication in Science. This is a problem not only because the paper fails to uncover anything approaching Platonic “Truths” about musical space but also because it is symptomatic of a certain level of self-consciousness within the subdiscipline of mathematical music theory, an attraction toward hijacking mathematical hieroglyphics (and in some cases, real mathematics) in an effort to legitimize the study of music theory and portray music-theoretic ideas as a more substantive (read: “less artsy”) intellectual pursuit. But self-consciousness transmogrifies into unshirted intellectual crisis when mathematics is banefully misappropriated to bolster subjective claims about musical objects under investigation. Such is the case here.

Musicologists would be much better to avoid such blatant non sequiturs.

Footnotes:

[1] Constructing the quotient orbifold as an n-torus modulo the symmetric group of n elements allows us to eliminate identical unordered sets with permuted elements. For example, in DT’s MS model in figure 1, we see that, which allows us to choose either {03} or {30} as the “minor third” representative. If we were modeling triads, we might haveetc., allowing us to choose, say, {037} among the 3! orderings as the “minor triad” representative inspace.