Eigenvalues and musical instruments

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 give 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 ever appeared in print. This paper attempts to fill that gap by plotting eigenvalues for simple models of a guitar string, a flute, a clarinet, a kettledrum, and a musical bell. For the drum and the bell, sample 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.