Elastic overtones: an equal temperament 12 tone music system with "perfect" fifths

TL;DR

This paper proposes a novel 12-tone music system using electronic tuning that achieves 'perfect' fifths by relaxing harmonic integer constraints, blending the harmonic purity of Just Intonation with the versatility of equal temperament.

Contribution

It introduces a new transposable 12-tone system with 'perfect' fifths by allowing non-integer harmonics, bridging the gap between Just Intonation and equal temperament.

Findings

Achieves exact fifths in a 12-tone system

Preserves versatility and modulating ability of 12TET

Recovers harmonic qualities similar to Just Intonation

Abstract

The impossibility of a transposable 12 semitone tuning of the octave arises from the mathematical fact that $2 \times 2^{7/12} \neq 3$ i.e., the second harmonic of the fifth can not exactly match the third harmonic of the fundamental. This in turn, stems from the whole number harmonic structure of western music, and the subsequent fundamental character of the octave interval as multiples of 2 in frequency, a property inherited by our music system from the physics of instruments with vibrating elements being to a good approximation one dimensional. In the current era of electronic music, one can relax the above assumptions to construct an analogous music system where all the structural properties of the standard music system are preserved, but where harmonics are not whole number multiples of the fundamental frequency, and the octave is no longer a factor of 2 in frequency. This now allows to construct a transposable 12 semitone music system where the second harmonic of the fifth exactly matches the third harmonic of the fundamental. The enhanced harmonic qualities of this system recover to a good approximation the musical qualities of Just Intonation, whilst retaining by construction all the versatility and modulating ability of 12TET.