Three-string inharmonic networks. (English) Zbl 1531.00012
Summary: 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.
MSC:
| 00A65 | Mathematics and music |
Keywords:
inharmonic sounds; vibrations of strings; network of strings; musical instrument design; inharmonic oscillations; stretched spectra; major third overtones; sensory dissonanceReferences:
| [1] | Bork, I., Practical Tuning of Xylophone Bars and Resonators, Applied Acoustics, 46, 1, 103-127 (1995) |
| [2] | Boyd, John P., Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series, and Oracles (2014), Philadelphia, PA: Society for Industrial and Applied Mathematics, Philadelphia, PA · Zbl 1311.65047 |
| [3] | Brahmi, Ahcène; Gauthier, Claude, Optimization of Musical Non-Symmetric 3-Strings, Romanian Journal of Acoustics and Vibration, 16, 2, 6 (2019) |
| [4] | Dewitt, Lucinda A.; Crowder, Robert G., Tonal Fusion of Consonant Musical Intervals: The Oomph in Stumpf, Perception & Psychophysics, 41, 1, 73-84 (1987) |
| [5] | Fletcher, N. H., Mode Locking in Nonlinearly Excited Inharmonic Musical Oscillators, The Journal of the Acoustical Society of America, 64, 6, 1566-1569 (1978) |
| [6] | Fletcher, Neville H., Tuning a Pentangle - A New Musical Vibrating Element, Applied Acoustics, 39, 1, 145-163 (1993) |
| [7] | Fletcher, Neville H.; Rossing, T. D., The Physics of Musical Instruments (1991), New York: Springer-Verlag · Zbl 0723.73001 |
| [8] | Gaudet, S, and Gauthier, C.. “Musical String Networks”US Patent 7763785, 2010, July. |
| [9] | Gaudet, S.; Gauthier, C.; Leger, S.; Walker, C., The Vibrations of a Real 3-string: the Timbre of the Tritare, Journal of Sound and Vibration, 281, 1-2, 219-234 (2005) · Zbl 1236.74104 |
| [10] | Guettler, Knut.2013. “Some Typical Properties of Bowed Strings.” http://knutsacoustics.com/files/Typical-string-properties.pdf. |
| [11] | Hobby, Kevin; Sethares, William A., Inharmonic Strings and the Hyperpiano, Applied Acoustics, 114, 1, 317-327 (2016) |
| [12] | Huron, David., Tonal Consonance Versus Tonal Fusion in Polyphonic Sonorities, Music Perception, 9, 2, 135-154 (1991) |
| [13] | McLachlan, Neil M., The Design and Analysis of New Musical Bells, The Journal of the Acoustical Society of America, 115, 1, 2565-2570 (2004) |
| [14] | Morse, Philip M.; Uno Ingard, K., Theoretical Acoustics (1987), Princeton, NJ: Princeton University Press |
| [15] | Oliver, D. L.; Arsie, A., Effects of Weighting the Ends of a Tubular Bell on Modular Frequencies, Journal of Mathematics and Music, 13, 1, 27-41 (2019) · Zbl 1428.74097 |
| [16] | Peairs, Evan.2016. “Designing, Building, and Testing Novel Musical Instruments.” Bachelor’s Thesis, Reed College. |
| [17] | Plomp, R.; Levelt, W. J. M., Tonal Consonance and Critical Bandwidth, The Journal of the Acoustical Society of America, 38, 4, 548-560 (1965) |
| [18] | Sethares, William A., Local Consonance and the Relationship Between Timbre and Scale, The Journal of the Acoustical Society of America, 94, 3, 1218-1228 (1993) |
| [19] | Sethares, William A., Tuning, Timbre, Spectrum, Scale (2004), Berlin: Springer, Berlin |
| [20] | Sethares, W. A.; Hobby, K., Designing Inharmonic Strings, Journal of Mathematics and Music, 12, 3, 107-122 (2018) · Zbl 1457.35083 |
| [21] | Slaymaker, Frank H., Chords From Tones Having Stretched Partials, The Journal of the Acoustical Society of America, 47, 6, 1569-1571 (1970) |
| [22] | Spall, J. C., A One-Measurement Form of Simultaneous Perturbation Stochastic Approximation, Automatica, 33, 2, 109-112 (1997) · Zbl 0867.93086 |
| [23] | Swanson, Irena., Introduction to Analysis with Complex Numbers (2021), Singapore: World Scientific Publishing · Zbl 1457.30001 |
| [24] | Thornton, Stephen T.; Marion, Jerry B., Classical Dynamics of Particles and Systems (2004), Belmont, CA: Brooks/Cole, Belmont, CA |
| [25] | Young, Robert W., Inharmonicity of Plain Wire Piano Strings, The Journal of The Acoustical Society of America, 24, 3, 267-273 (1952) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
