optic-spaces · hook2011-review-geometry-of-music

The chord spaces described here have variously been dubbed “CQT spaces” (after the groundbreaking article by Callender, Quinn, and Tymoczko, published in Science in 2008, expanding upon Tymoczko 2006), “OPTIC spaces” (after the five kinds of symmetries the theory admits, defined by equivalence under octave shifts, permutation of notes within a collection, transposition, inversion, and cardinality change), and “orbifold spaces” ... a voice leading is a path through an appropriately chosen OPTIC space, and the distance between any two chords in that space reflects the potential to connect them via smooth voice leading. As such, the OPTIC spaces use sophisticated techniques from contemporary mathematics to pull together many threads from musical scholarship of the last half-century, including set theory, scale theory, neo-Riemannian theory, and an assortment of work on voice leading and geometric representations.