The wave equation for stiff strings and piano tuning

Autors/ores

  • Xavier Gràcia Universitat Politècnica de Catalunya
  • Tomás Sanz-Perela Universitat Politècnica de Catalunya

Paraules clau:

wave equation, vibrating string, stiffness, inharmonic spectrum, musical scale, dissonance

Resum

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 in 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.

Descàrregues

Publicat

2018-09-27

Com citar

Gràcia, X., & Sanz-Perela, T. (2018). The wave equation for stiff strings and piano tuning. Reports@SCM, 3(1), 1–16. Retrieved from https://revistes.iec.cat/index.php/reports/article/view/139742

Número

Vol. 3 Núm. 1 (2017)

Secció

Articles

Llicència

The intellectual property of articles belongs to the respective authors.
On submitting articles for publication to the journal Reports@SCM, authors accept the following terms:
  1. Authors assign to Societat Catalana de Matemàtiques (a subsidiary of Institut d’Estudis Catalans) the rights of reproduction, communication to the public and distribution of the articles submitted for publication to Reports@SCM.
  2. Authors answer to Societat Catalana de Matemàtiques for the authorship and originality of submitted articles.
  3. Authors are responsible for obtaining permission for the reproduction of all graphic material included in articles.
  4. Societat Catalana de Matemàtiques declines all liability for the possible infringement of intellectual property rights by authors.
  5. The contents published in the journal, unless otherwise stated in the text or in the graphic material, are subject to a Creative Commons Attribution-NonCommercial-NoDerivs (by-nc-nd) 3.0 Spain licence, the complete text of which may be found at https://creativecommons.org/licenses/by-nc-nd/3.0/es/deed.en. Consequently, the general public is authorised to reproduce, distribute and communicate the work, provided that its authorship and the body publishing it are acknowledged, and that no commercial use and no derivative works are made of it.
  6. The journal Reports@SCM is not responsible for the ideas and opinions expressed by the authors of the published articles.