Type Theory for the Working Mathematical Music Theorist

Abstract

Many formal languages of contemporary mathematical music theory -- particularly those employing category theory -- are powerful but cumbersome: ideas that are conceptually simple frequently require expression through elaborate categorical constructions such as functor categories. This paper proposes a remedy in the form of a type-theoretic symbolic language that enables mathematical music theorists to build and reason about musical structures more intuitively, without relinquishing the rigor of their categorical foundations. Type theory provides a syntax in which elements, functions, and relations can be expressed in simple terms, while categorical semantics supplies their mathemusical interpretation. Within this system, reasoning itself becomes constructive: propositions and proofs are treated as objects, yielding a framework in which the formation of structures and the reasoning about them take place within the same mathematical language. The result is a concise and flexible formalism that restores conceptual transparency to mathemusical thought and supports new applications, illustrated here through the theory of voice-leading spaces.