I have thus been interested in the efforts of Carol Krumhansl and her colleagues to seek out spatial representations of musical phenomena such as tonality. Consider, for example, her description of an experiment she conducted with her colleague Edward Kessler:
In the first [experiment] we obtain a quantitative measure of the degree to which each individual tone in an octave range is related to an abstract tonal center. These quantitative measurements are then used to derive a spatial map of the major and minor keys, representing the distances between different tonal centers.This pursues the hypothesis that we can abstract concepts such as tonality into a spatial locus and harmonic progression into a path from one such locus to another (possibly through intermediate loci).
The potential risk with this abstraction is that you can only talk about distance if your abstraction happens to be a metric space. As anyone with a smattering of undergraduate topology will tell you, one of the axioms of a metric space is that the distance from here to there is the same as the distance from there back to here (the property of symmetry). The problem is that, where musical progressions are concerned, distance is not necessarily symmetrical. For example, there are qualitative differences between an interval of departure, as in a progression from tonic to dominant in a Schenkerian Ursatz and an interval of arrival, which in an Ursatz is an interval of the same size. My point is that the size of the interval assumes different levels of significance depending on how it is used, and that variation works against it representing any kind of distance in the topological sense.
Of course, if all we do is look at music notation, we do not appreciate this distinction. The properties of tones abstracted into notes on a musical staff fit very nicely into the axioms that a distance metric must satisfy. However, as I have often said, the music is not in those tones but in the acts of making them. When we exchange abstractions based on nouns to abstractions based on verbs, many of our mathematical abstractions go out the window. Instead, we need to find abstractions that better capture the properties of our dispositions to act and then figure out how they apply to the acts of making music. This will not be easy, but it may finally provide a good reason for those who claim to be interest in music theory to get their noses out of the score pages!