The wave equation for stiff strings and piano tuning

TL;DR

This paper analyzes the wave equation for stiff strings, computes the spectrum, and proposes a novel piano tuning method based on dissonance theory to improve consonance.

Contribution

It provides a mathematical solution for the wave equation with stiffness and introduces a new tuning approach for pianos based on minimizing beats.

Findings

Spectrum of stiff strings is slightly inharmonic

Standard 12-tone equal temperament is suboptimal for pianos

Tuning by beat minimization enhances consonance

Abstract

We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.