The Art and Science of Achieving Harmonics on Stringed Instruments
One may elicit the qth tone of a string by applying the `correct touch' at one of its q-1 nodes during a simultaneous pluck or bow. This effect was first scored for violin by Mondonville in 1735. Though it captured the attention of the 19th century masters,
Chladni, Tyndall, Helmholtz and Rayleigh, it remained for Bamberger,
Rauch and Taylor in 1982 to develop and analyze the first mathematical model of harmonics. Their `touch' is a damper of magnitude b concentrated at the node p/q. The `correct touch' is that b for which the modes, that do not vanish at p/q, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree q-1. We establish lower and upper bounds on the spectral abscissa and show that the set of associated root vectors constitutes a Riesz basis in the natural energy space and so identify `correct touch' with the b that minimizes the spectral abscissa.
Steven J. Cox (Computational and Applied Mathematics, Rice University)
April 05, 2006