My talk will outline a new theory of musical representations, general enough to encompass scores, DAW timelines, and sinusoidal decompositions of audio signals. Generalizing the geometry of music, I will analyze these structures using quotient groupoids. This reveals that familiar musical transformations invariably have duals, with the dual pairs known to musicians as "symmetries" and "intervals," and to mathematicians as the duality of an orbifold's symmetry group and its fundamental group. This duality reveals a vast network of analogies connecting many different areas of music, as well as areas outside it. I will discuss some of the compositional uses for these transformations, including a theory-enabled DAW that I am building.