Equal temperament vs. Pythagorean tuning: A mathematical plot line

The relationship between mathematics and music is as ancient as it is fascinating. This reciprocal contribution creates a unique synergy: while music provides “color” to mathematical abstractions, mathematics offers structural support to the most elusive of the arts. Although many arguments regarding this connection have been proposed – some profound, others tenuous – one fact remains certain: the scales of every musical culture are fundamentally grounded in arithmetic. Our analysis first axiomatizes the equal-tempered system as a geometric partition of the frequency spectrum, exploring its algebraic properties and transpositional invariance. Subsequently, the Pythagorean system is formalized through the powers of the 3:2 ratio, highlighting the inherent conflict between rational purity and the necessity of a closed harmonic circle. Finally, we discuss the “bracketing” property for a twelve-tone system, where the equal-tempered notes of the chromatic scale are encompassed by Pythagorean pairs derived from upward and downward cycles of fifths.