My talk will introduce some twenty-first century music-theoretical ideas that I have found helpful musically, beginning with scales and macroharmony, proceeding through voice-leading geometry, and concluding with the idea of the “quadruple hierarchy,” a recursive nesting of collections supporting the same basic operations at each level. I will introduce Arca, a new musical programming language, and will make the case that musicians have made intelligent but tacit use of nonobvious mathematics. The core ideas will be illustrated with live improvisations that use the internet to transmit notation in real time, allowing for both spontaneity and coordination.