The wave equation for stiff strings and piano tuning

• Xavier Gràcia ^[1] ; Tomás Sanz-Perela ^[1]
  1. [1] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
• Localización: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques, ISSN-e 2385-4227, Vol. 3, Nº. 1, 2017, págs. 1-16
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen
  • català

    Estudiem l’equació de les ones per a una corda amb rigidesa. Resolem l’equació i n’enunciem un teorema d’unicitat amb condicions de contorn adequades. Per a una corda punxada calculem l’espectre, que és lleugerament inharmònic. Per tant, l’habitual escala de 12 divisions iguals de l’octava justa no és la millor elecció per afinar instruments com ara el piano. Basant-nos en la teoria de la dissonància, proporcionem una manera d’afinar el piano a fi de millorar-ne la consonància. Una bona solució s’obté afinant una nota i la seva quinta tot minimitzant els seus batecs.

  • English

    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.

• Referencias bibliográficas
  • D.J. Benson, “Music, a mathematical o↵ering”,Cambridge University Press, 2008.
  • H. Brezis, “Functional analysis, sobolev spacesand partial di↵erential equations”, Springer,2011.
  • J. Chabassier, “Mod ́elisation et simulationnum ́erique d’un piano par mod`eles physiques”,Ph.D. thesis, ́Ecole Polytechnique ParisTech,2012.
  • L.C. Evans, “Partial di↵erential equations”,AMS Press, 1998.
  • H. Fletcher, “Normal vibration frequencies of a stiff piano string”,J. Acoust. Soc. Am.36(1964), 203–209.
  • N.H. Fletcher and T.D. Rossing, “The physicsof musical instruments” (2nd ed), Springer,1998.
  • S.M. Han, H. Benaroya, and T. Wei,. “Dy-namics of transversely vibrating beams usingfour engineering theories”,J. Sound Vibr.225(1999), 935–988.
  • H. Helmholtz, “On the sensations of tone” (3rded), Longmans (original in German, 1877, 4thed), 1895.
  • M. Lindley, “Temperaments”,Grove Music On-line, 2015.
  • P.M. Morse and K.U. Ingard, “Theoreticalacoustics”, McGraw-Hill, 1968.
  • R. Plomp and W.J.M. Levelt, “Tonal conso-nance and critical bandwidth”,J. Acoust. Soc.Am.38(1965), 548–560.
  • J.G. Roederer, “The physics and psychophysicsof music”, Springer, 2008.
  • T.D. Rossing, F.R. Moore, and P.A. Wheeler,“The science of sound” (3rd ed), Addison-Wesley, 2002.
  • S. Salsa, “Partial di↵erential equations in ac-tion. From modelling to theory”, Springer,2008.
  • W.A. Sethares, “Tuning, timbre, spectrum,scale” (2nd ed), Springer, 2005.
  • W.A. Strauss, “Partial di↵erential equations.An introduction”, Wiley, 2008.
  • A.N. Tikhonov and A.A. Samarskii, “Equationsof mathematical physics”. Pergamon, 1963.