On Quotient Spaces in Relation to Music with Dmitri Tymoczko
Dmitri Tymoczko Dmitri Tymoczko On Math On Topology On Symmetry On Math On Geometry of Music On Quotient Space and Symmetry in Music Music ~***~ AI Dmitri Tymoczko uses quotient spaces to model how musicians and listeners abstract away from specific musical details —such as which octave a note is in, the order of notes in a chord, or the specific transposition of a chord. By "gluing together" points in a high-dimensional space that represent equivalent musical situations, he creates complex, low-dimensional geometric shapes called orbifolds . These spaces, often called OPTIC spaces (Octave, Permutation, Transposition, Inversion, Cardinality), help visualize and analyze musical voice leading and harmonic relationships. [ 1 , 2 , 3 , 4 , 5 , 6 ] Here is how Tymoczko uses quotient spaces to model music: 1. Abstracting Musical Information (The OPTIC Relations) Tymoczko forms quotient spaces by applying five standard equivalence relations, often referred to in col...