The wave equation for stiff strings and piano tuning
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.
Cite
@article{arxiv.1603.05516,
title = {The wave equation for stiff strings and piano tuning},
author = {Xavier Gràcia and Tomás Sanz-Perela},
journal= {arXiv preprint arXiv:1603.05516},
year = {2018}
}
Comments
16 pages, 7 figures. Version accepted for publication. Main changes: added a new appendix on sound and music (and rewriting of the introduction), several improvements in section 4 (presentation of the tuning approaches, figures)