A musical chord can be represented as a point in an **orbifold**. (An orbifold is a quotient manifold.)

Line segments join notes of one chord to those of another. Composers in a wide range of styles have exploited the (non-Euclidean) geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation.

*(via Artemy Kolchinsky)*