The axiom system of classical harmony

TL;DR

This paper develops a formal mathematical axiom system for classical harmony, capturing the core principles of 18th-century four-part music and providing a foundation for analyzing and constructing harmonically correct compositions.

Contribution

It introduces a complete axiom system for classical harmony based on the fundamental theorem of tonality, formalizing chord progressions and voice leading rules.

Findings

Provides a logical structure for classical harmony

Defines criteria for harmonic correctness in four-part music

Formalizes classical compositional principles

Abstract

This paper provides a new mathematical axiom system for classical harmony, which is a prescriptive rule system for composing music, introduced in the second half of the 18th century. The clearest model of classical harmony is given by the homophonic four-part pieces of music. The form of these pieces is based on the earlier four-part chorale adaptations of J. S. Bach. Our paper logically structures the musical phenomena belonging to the research area of classical harmony. Its main result, the fundamental theorem of tonality, provides a way to construct a complete axiom system which incorporates the well-known classical compositional principles about chord changing and voice leading. In this axiom system, a piece complies with classical harmony if it satisfies the formal requirements of four-part homophony and it does not violate any classical chord-changing or modulational rules.