Intrinsic Harmonic Spaces: A Solution to the Ancient Problem of Perfect Tuning

TL;DR

This paper introduces a novel mathematical framework called Intrinsic Harmonic Spaces that achieves perfect tuning across all keys simultaneously, overcoming limitations of traditional octave-based systems and enabling new musical instrument designs.

Contribution

It presents a new harmonic representation using vector spaces that does not rely on octave ordering, allowing for perfect tuning in all keys at once.

Findings

Mathematical model for complete harmony in all keys

Design of a physical instrument realizing the scheme

Computer implementation of perfect harmonies

Abstract

In this article we solve this ancient problem of perfect tuning in all keys and present a system were all harmonies are conserved at once. It will become clear, when we expose our solution, why this solution could not be found in the way in which earlier on musicians and scientist have been approaching the problem. We follow indeed a different approach. We first construct a mathematical representation of the complete harmony by means of a vector space, where the different tones are represented in complete harmonic way for all keys at once. One of the essential differences with earlier systems is that tones will no longer be ordered within an octave, and we find the octave-like ordering back as a projection of our system. But it is exactly by this projection procedure that the possibility to create a harmonic system for all keys at once is lost. So we see why the old way of ordering tones within an octave could not lead to a solution of the problem. We indicate in which way a real musical instrument could be built that realizes our harmonic scheme. Because tones are no longer ordered within an octave such a musical instrument will be rather unconventional. It is however a physically realizable musical instrument, at least for the Pythagorean harmony. We also indicate how perfect harmonies of every dimension could be realized by computers.