The logic of musical composition, representation, analysis, and performance share important basic structures which can be described by Grothendieck's functorial algebraic geometry and Lawvere's topos theory of logic. We give an account of these theoretical connections, discuss and illustrate their formalization and implementation on music software. Three issues are particularly interesting in this context: First, the crucial insight of Grothendieck that "a point is a morphism" carries over to music: Basically, musical entities are transformations rather than constants. Second, it turns out that musical concepts share a strongly circular character, meaning that spaces for music objects are often defined in a self-referential way. Third, the topos-theoretic geometrization of musical logic implies a progressively geometric flavour of all rational interactions with music, in particular when implemented on graphical interfaces of computer environments.