Mathematical Music Theory—Status Quo 2000 Guerino Mazzola www.encyclospace.org, ETH Z¨ urich, Departement GESS, and Universit¨ at Z¨ urich, Institut f¨ ur Informatik 15th October 2001 Abstract We give an overview of mathematical music theory as it has been de- veloped in the past twenty years. The present theory includes a formal language for musical and musicological objects and relations. This lan- guage is built upon topos theory and its logic. Various models of mu- sical phenomena have been developed. They include harmony (func- tion theory, cadences, and modulations), classical counterpoint (Fux rules), rhythm, motif theory, and the theory of musical performance. Most of these models have also been implemented and evaluated in computer applications. Some models have been tested empirically in neuro-musicology and the cognitive science of music. The mathemat- ical nature of this modeling process canonically embedds the given historical music theories in a variety of fictitious theories and thereby enables a qualification of historical reality against potential variants. As a result, the historical realizations often turn out to be some kind of “best possible world” and thus reveals a type of “anthropic principle” in music. These models use different types of mathematical approaches, such as—for instance—enumeration combinatorics, group and module the- ory, algebraic geometry and topology, vector fields and numerical solu- tions of differential equations, Grothendieck topologies, topos theory, and statistics. The results lead to good simulations of classical results of music and performance theory. There is a number of classifiaction theorems of determined categories of musical structures. The overview concludes by a discussion of mathematical and mu- sicological challenges which issue from the investigation of music by mathematics, including the project of “Grand Unification” of harmony and counterpoint and the classification of musical performance. 1