  Mathematical music theory has assumed without proof that musical notes can be
associated with the equivalence classes of $\mathbb{Z}_n$. We contest the
triviality of this assertion, which we call the Pitch-class Integer Theorem
(PCIT). Since the existing literature assumes the PCIT without proof, the
mathematics to rigorously treat the PCIT does not yet exist. Thus, we construct
an axiomatic proof of the PCIT to support the existing mathematical models of
music theory.
