Optimal Tuning of Two-Dimensional Keyboards

TL;DR

This paper presents a linear programming approach to optimize tuning of two-dimensional keyboards for harmonic approximation, providing exact solutions for various dimensions and demonstrating the feasibility of optimal tuning with enough octave rows.

Contribution

It introduces a novel linear programming formulation for keyboard tuning and offers exact solutions for multiple keyboard sizes, advancing the understanding of harmonic approximation.

Findings

Exact solutions for many keyboard dimensions.

Optimal tuning achievable with sufficient octave rows.

Linear programming effectively models the tuning problem.

Abstract

We give a new analysis of a tuning problem in music theory, pertaining specifically to the approximation of harmonics on a two-dimensional keyboard. We formulate the question as a linear programming problem on families of constraints and provide exact solutions for many new keyboard dimensions. We also show that an optimal tuning for harmonic approximation can be obtained for any keyboard of given width, provided sufficiently many rows of octaves.