A Frequency Error Model for Fretted String Instruments

Abstract

A mathematical model, quantifying frequency error due to string displacement and the process of fretting a note on a string instrument, is presented. The model is developed starting with the standard relationship between length, tension, density, and frequency of a perfect string. The change in string length, as a result of displacing the string to a point behind a fret, is modeled by a set of parameters approximating the resulting shape of the fretted string. The string length model is then combined with an inharmonicity model which approximates the frequency deviation of the series of partials contained in the string's frequency spectrum. Inharmonicity is weakly dependent on the change in string tension due to fretting, but to first order, inharmonicity frequency errors of a string with finite bending stiffness are independent of the process of fretting a string.