Borrowing from the orbifold geometry of string theory (his sister is a notable algebraic geometer) Dmitri Tymoczko has drafted a new map of musical harmony, a new kind of map that not only explains the space for composers, but is sufficiently new to earn his paper a placement as the first ever paper on music theory published in Science:

"... composers have been exploring the geometrical structure of these maps since the beginning of Western music without really knowing what they were doing. If someone then showed you a map, you might say, 'Wow, I didn't realize the Safeway was close to the disco.' We can now go back and look at hundreds of years of this intuitive musical path-making and realize that there are some very simple principles that describe the process."
[ The Geometry of Music ]

Admittedly, the space defined by Timoczko's map are not all humanly harmonious, but what his example projections do show is how composers will tend to self-confine themselves into surprisingly regular forms carved in a realm of the space which suits their personal sense of harmony -- his hope is that by teaching this approach to harmony, a composer could recognize the boundaries of their personal harmonic space and step in or out of that realm as their needs require.

"What I hope to do is provide a new way to represent the space of musical possibilities. If you like a particular chord, or group of notes, then I can show you how to find other, similar chords and link them together to form attractive melodies. These two principles -- using attractive chords, and connecting their notes to form melodies -- have been central to Western musical thought for almost a thousand years."
[ Composer reveals musical chords' hidden geometry ]

As Einstein said, it isn't so much that Nature is mathematical but rather that so much of Nature can be described mathematically. Most musicians already know the geometric approaches which have been tried in varying degrees of success over the past several thousand years, the circular divisions of the Cycle of Fifths, for example, and Quartal Harmonies that characterize the Miles Davis and John Coltrane quartet years, even George Russell's lydian chromatics and the 'tonal gravity' are tools to visualize and predict the sense of harmonic progressions.

"Western music theory has developed impressive tools for thinking about traditional harmonies, but it doesn%u2019t have the same sophisticated tools for thinking about these newer chords," Tymoczko said. "This led me to want to develop a general geometrical model in which every conceivable chord is represented by a point in space. That way, if you hear any sequence of chords, no matter how unfamiliar, you can still represent it as a series of points in the space. To understand the melodic relationship between these chords, you connect the points with lines that represent how you have to change their notes to get from one chord to the next."
[ Composer reveals musical chords' hidden geometry ]