Designing inharmonic strings

• Published in: Journal of mathematics and music (Society for Mathematics and Computation in Music) Vol. 12; no. 2; pp. 107 - 122
• Main Authors: Sethares, William A., Hobby, Kevin
• Format: Journal Article
• Language: English
• Published: Baton Rouge Taylor & Francis 04.05.2018; Taylor & Francis Ltd
• Subjects: Design; Harmonic analysis; inharmonic oscillations; inharmonic sounds; Inverse problems; Mathematical analysis; musical instrument design; Musical instruments; non-uniform strings; Stringed instruments; Strings; vibrations of strings
• ISSN: 1745-9737; 1745-9745
• DOI: 10.1080/17459737.2018.1491649

Uniform strings have a harmonic sound; non-uniform strings have an inharmonic sound. Given a precise description of a non-uniform string, its inharmonic spectrum can be calculated using standard techniques. This paper addresses the inverse problem: given a desired/specified spectrum, how can string parameters be chosen so as to achieve that specification? The design method casts the inverse problem in an optimization framework that can be solved using iterative techniques, and experiments show that viable solutions are often possible despite the multi-modal character of the cost function. Three properties of inharmonic strings are studied: their behavior under changes in tension, changes in density, and changes in length. These are important to the use of the inharmonic strings in musical instruments where it is often desirable that a set of strings has consistent timbre (spectrum) over many different pitches. Several different strings are designed: one that has partials derived from the golden ratio φ, one with overtones that beat with each other, and a family of strings designed for performance in n-tone equal temperament. An Online Supplement for this article contains details of these string designs for all n from 7 to 20 and can be accessed at http://dx.doi.org/10.1080/17459737.2018.1491649 . Several of the strings have been constructed, and the predicted frequencies of the overtones are verified by direct measurement.