Music, that most mathematical of all arts! How can two disciplines continue to enjoy such an uncomfortable intimacy after so many centuries? For the Ancients, after all, music was a branch of mathematics; and, for our exquisitely quantified society, one would assume the comprehensive topology of music to be near completion. Why are we, in fact, still unable to agree on so much? We begin by exploring four relatively straightforward notions, areas where the mapping ought to be complete: intonation, rhythm, dynamics, and articulation. We explore the notation of these qualities, tracing the increasing precision with which they can be indicated. We then introduce the elements of fuzzy logic to demonstrate the deficiencies of excessively precise topologies. In addition, we chart some of the exceeding subjectivity that exists in systems commonly held to be objective.