Designing inharmonic strings

Abstract

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.

Keywords:

• inharmonic sounds
• vibrations of strings
• non-uniform strings
• musical instrument design
• inharmonic oscillations

2010 Mathematics Subject Classification:

• 35Q93
• 90C26
• 37B35

Acknowledgments

We thank the anonymous referees and Associate Editor Emmanuel Amiot for their comments on the manuscript. We thank Co-Editor-in-Chief Thomas Fiore for corrections to the galley proof.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental online material

Supplemental online material for this article can be accessed at http://dx.doi.org/10.1080/17459737.2018.1491649. The Online Supplement contains details of the design of inharmonic strings for n-tone equal temperament for all n from 7 to 20, which extends the data presented in Table 1 of Section 5.3. It also contains complete information about the three designs of the φ-based timbre of Section 5.4.

ORCID

William A. Sethares http://orcid.org/0000-0002-0318-7638