The wave equation for stiff strings and piano tuning

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

dc.contributor	Universitat Politècnica de Catalunya. Departament de Matemàtiques
dc.contributor	Universitat Politècnica de Catalunya. DGDSA - Geometria Diferencial, Sistemes Dinàmics i Aplicacions
dc.contributor.author	Gràcia Sabaté, Francesc Xavier
dc.contributor.author	Sanz Perela, Tomás
dc.date	2017
dc.identifier.citation	Gràcia, Xavier, Sanz, T. The wave equation for stiff strings and piano tuning. "Reports@SCM", 2017, vol. 3, núm. 1, p. 1-16.
dc.identifier.citation	2385-4227
dc.identifier.uri	http://hdl.handle.net/2117/101752
dc.language.iso	eng
dc.relation	http://revistes.iec.cat/index.php/reports/article/view/142136
dc.rights	info:eu-repo/semantics/openAccess
dc.rights	http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.subject	Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials
dc.subject	Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en diferències
dc.subject	Differential equations, Partial
dc.subject	Differential equations, Partial
dc.subject	wave equation
dc.subject	vibrating string
dc.subject	stiffness
dc.subject	inharmonic spectrum
dc.subject	musical scale
dc.subject	dissonance
dc.subject	Equacions en derivades parcials
dc.subject	Equacions diferencials hiperbòliques
dc.subject	Classificació AMS::35 Partial differential equations::35G General higher-order equations and systems
dc.subject	Classificació AMS::35 Partial differential equations::35L Partial differential equations of hyperbolic type
dc.title	The wave equation for stiff strings and piano tuning
dc.type	info:eu-repo/semantics/publishedVersion
dc.type	info:eu-repo/semantics/article
dc.description.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.
dc.description.abstract	Peer Reviewed

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