The Pitch-class Integer Theorem

Abstract

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.

Figures (5)

• Figure 1: The octaves $o_k(f),o_{k+1}(f)$ with partitions $\tau_k,\tau_{k+1}$.
• Figure 2: The partition of $o_3(f)$ together with $\Delta_4$.
• Figure 3: The partition of $o_3(f)$ together with $\Delta_4$ with respect to $\nu_{3,0}$.
• Figure 4: Two octaves of $\mathcal{T}_5(100)$ together with $\Delta_5$ defined in Example \ref{['EX.5-EDO with 100']}.
• Figure 5: The tuning set $\mathcal{T}_4(500)$ together with $\Delta_4$ defined in Example \ref{['EX.Non-uniform Step with 500']}.

Theorems & Definitions (42)

• Definition 2.1: Pitch, Lower Pitch, Higher Pitch, Same Pitch
• Definition 2.2: $k$th Octave of $f$, Base Octave
• Proposition 2.3
• proof
• Definition 2.4: Tuning Set
• Example 2.5
• Definition 3.1: Step, Step Set: Uniform, Non-uniform
• Example 3.2
• Example 3.3
• Example 3.4