The wave equation for stiff strings and piano tuning
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...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2017 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/101752 |
| Acceso en línea: | https://hdl.handle.net/2117/101752 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial wave equation vibrating string stiffness inharmonic spectrum musical scale dissonance Equacions en derivades parcials Equacions diferencials hiperbòliques Classificació AMS::35 Partial differential equations::35G General higher-order equations and systems Classificació AMS::35 Partial differential equations::35L Partial differential equations of hyperbolic type Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en diferències |
| Sumario: | 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. |
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