In my talk, I explain how to translate basic concepts of music theory into the language of contemporary topology and geometry. Musicians commonly abstract away from five kinds of musical information -- including the order, octave, and specific pitch level of groups of notes. This process produces a family of quotient spaces or orbifolds: for example, two-note chords live on a Mobius strip, while three-note chord-types live on a cone. These spaces provide a general geometrical framework for understanding and interpreting music. Related constructions also appear naturally in other applied-math contexts, for instance in economics.