In Search of the Canonical Harmony for 12-TET

Abstract

Is the specific structure of Western tonal harmony a physical inevitability derived from acoustics, or is it merely one solution among many in a purely algebraic landscape? In this paper, we strip away the physics of vibrating strings and treat harmony as the solution to a simple linear system within the cyclic group $\mathbb{Z}_{12}$. By defining a harmonic system as a partitioning of a generator interval (the "Fifth") into two complementary thirds, we derive a complete classification of all possible harmonic universes in 12-Tone Equal Temperament. We show that every such system corresponds to a specific topological structure, visualized via its Levi graph. Our analysis reveals a counter-intuitive fact: the topological structure of standard Western harmony is not unique. It is one of exactly twelve mathematically isomorphic systems. However, we demonstrate that these shadows are not created equal. We identify a "privileged quartet" of systems that preserve the global connectivity of the Circle of Fifths. Furthermore, by invoking the Chinese Remainder Theorem, we present a compelling argument that the most number-theoretically natural system in $\mathbb{Z}_{12}$ is not the Western one, but a specific "alien" system defined by the partition $(9,4)$. This classification provides a rigorous mathematical foundation for neo-Riemannian theory and offers composers a precise "translation map" to explore syntactically familiar, yet sonically distinct, harmonic worlds.