The Mathematical Institute, University of Oxford, Eprints Archive

Eigenvalues and Musical Instruments

Howle, V. E. and Trefethen, Lloyd N. (1997) Eigenvalues and Musical Instruments. Technical Report. Unspecified. (Submitted)

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Abstract

Most musical instruments are built from physical systems that oscillate at certain natural frequencies. The frequencies are the imaginary parts of the eigenvalues of a linear operator, and the decay rates are the negatives of the real parts, so it ought to be possible to given an approximate idea of the sound of a musical instrument by a single plot of points in the complex plane. Nevertheless, the authors are unaware of any such picture that has every appeared in print. This paper attempts to fill that gap by plotting eigenvalues for a guitar string, a flute, a clarinet, a kettledrum, and a musical bell. For the drum and the bell, simple idealized models have eigenvalues that are irrationally related, but as the actual instruments have evolved over the generations, the leading five or six eigenvalues have moved around the complex plane so that their relative positions are musically pleasing.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1310
Deposited By:Lotti Ekert
Deposited On:09 Jun 2011 08:24
Last Modified:09 Jun 2011 08:24

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