as I and other music theorists have recently shown, we need a kind of mathematics that Pythagoras could not have imagined: the geometry and topology of what mathematicians call "quotient spaces" or "orbifolds." These exotic spaces contain singularities-- "unusual" points that are analogous to the black holes of Einstein's general relativity--that can be described using only very recent mathematics. Western music can ultimately be represented as a series of points and line segments on abstract shapes in higher dimensions.
Speaking of things Pythagoras could not have imagined is ridiculous - Pythagoras was quite imaginative - but certainly these are things which rest on the thousands of years of math since Pythagoras. Despite the hubris it's a good article.
