The wave equation for stiff strings and piano tuning

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RECERCAT Home > Universitat Politècnica de Catalunya > Documents de recerca > View document

To access the full text documents, please follow this link: http://hdl.handle.net/2117/101752

Title:	The wave equation for stiff strings and piano tuning
Author:	Gràcia Sabaté, Francesc Xavier; Sanz Perela, Tomás
Other authors:	Universitat Politècnica de Catalunya. Departament de Matemàtiques; Universitat Politècnica de Catalunya. DGDSA - Geometria Diferencial, Sistemes Dinàmics i Aplicacions
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.
Abstract:	Peer Reviewed
Subject(s):	-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 -Differential equations, Partial -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
Rights:	http://creativecommons.org/licenses/by-nc-nd/3.0/es/
Document type:	Article - Published version Article
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