Three-string inharmonic networks

Abstract

This paper studies the resonant frequencies of three-string networks by examining the roots of the relevant spectral equation. A collection of scaling laws are established which relate the frequencies to structured changes in the lengths, densities, and tensions of the strings. Asymptotic properties of the system are derived, and several situations where transcritical bifurcations occur are detailed. Numerical optimization is used to solve the inverse problem (where a desired set of frequencies is specified and the parameters of the system are adjusted to best realize the specification). The intrinsic dissonance of the overtones provides an approximate way to measure the inherent inharmonicity of the sound.

Keywords:

• Inharmonic sounds
• vibrations of strings
• network of strings
• musical instrument design
• inharmonic oscillations
• stretched spectra
• major third overtones
• sensory dissonance

2010 Mathematics Subject Classification:

• 00A65

2012 Computing Classification Scheme:

• applied computing

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental online material

Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/17459737.2022.2136776http://dx.doi.org/10.1080/17459737.2022.2136776.

Notes

1 Any three-string design can be rotated into this configuration since there will always be at least two angles that are between 90 and 180 degrees. (If not, there would need to be an angle greater than 180, requiring all strings to be on the same “side” of the junction, subsequently violating Newton's second law).

2 $s_{\lambda }\equiv \frac{\mathrm{\partial }s(\lambda ;q)}{\lambda }$.