The Effect of String Tension Variation on the Perceived Pitch of a Classical Guitar
DOI:
https://doi.org/10.31273/eirj.v2i1.101Keywords:
guitar pitch, string vibration, perturbation model, overtones, tension modulation, geometric stiffnessAbstract
Actual motion of a vibrating guitar string is a superposition of many possible shapes (modes) in which it could vibrate. Each of these modes has a corresponding frequency, and the lowest frequency is associated with a shape idealised as a single wave, referred to as the fundamental mode. The other contributing modes, each with their own progressively higher frequency, are referred to as overtones, or harmonics. By attaching a string to a medium (a soundboard) capable of a response to the vibrating string, sound waves are generated. The sound heard is dominated by the fundamental mode, ‘coloured’ by contributions from the overtones, as explained by the classical theory of vibration. The classical theory, however, assumes that the string tension remains constant during vibration, and this cannot be strictly true; when considering just the fundamental mode, string tension will reach two maximum changes, as it oscillates up and down. These changes, occurring twice during the fundamental period match the frequency of the octave higher, 1st overtone. It is therefore plausible to think that the changing tension effect, through increased force on the bridge and, therefore, greater soundboard deflection, could be amplifying the colouring effect of (at least) the 1st overtone.
In this paper, we examine the possible influence of string tension variation on tonal response of a classical guitar. We use a perturbation model based on the classical result for a string in general vibration in conjunction with a novel method of assessment of plucking force that incorporates the engineering concept of geometric stiffness, to assess the magnitude of the normal force exerted by the string on the bridge. The results of our model show that the effect of tension variation is significantly smaller than that due to the installed initial static tension, and affects predominantly the force contribution arising from the fundamental mode. We, therefore, conclude that string tension variation does not contribute significantly to tonal response.
Photo credit: By Biblola (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
Downloads
References
Antman, S. (1980) The Equations for Large Vibrations of Strings. The American Mathematics Monthly 87, (5): 359-370.
Band, B. (2009) ‘Energy-based Synthesis of Tension Modulation in Strings’, in Digital Audio Effects, Proceedings,12th International Conference, Como, Italy: 1-8.
Carrier, F. (1945) On the Non-linear Vibration Problem of the Elastic String. Quarterly of Applied Mathematics 3: 157-165.
Coulson, C.A. (1941) Waves. London: Oliver and Boyd Ltd.
Decker, J.A. (2007) ‘Guitar Acoustics 101’.http://www.guitarmasterworks.com/about-guitar-acoustics-101.html.
Fletcher, N.H., and Rossing (1998) The Physics of Musical Instruments. London: Springer Verlag, 2nd Ed.
Fletcher, N.H. (1999) The Non-linear Physics of Musical Instruments. Reports on Progress in Physics 62, (5): 723.
Legge, K.A., and Fletcher, N.H. (1984) Non-linear Generation of Missing Modes on a Vibrating String. Journal of Acoustical Society of America 76: 5-12.
Lewis, W.J. (2003) Tension Structures. Form and Behaviour, London: Thomas Telford.
Morse, P.M.(1948) Vibration and Sound, New York: McGraw Hill.
Murthy, G.S.S., and Ramakrishna, B.S. (1965) Non-linear Character of Resonance in Stretched Strings. Journal of Acoustical Society of America 38: 461-471.
Ramsey, A.S. (1949) A Treatise on Hydromechanics. Part II Hydrodynamics, London: G. Bell and Sons, Ltd.
Rayleigh, J.W. (1894) Theory of Sound, CUP, 2nd Ed., Vol.2, reprinted by Dover, 1945.
Taylor, J. (1978) Tone Production on the Classical Guitar, London: Musical New Services Ltd. A Guitar Magazine Project.
Tolonen, T., Valimaki, V., and Karjalainen, M. (2000) ‘Modelling of Tension Modulation Non-linearity in Plucked Strings’ in Speech and Audio Processing, IEEE Transactions, 8 (3): 300-310.
Young, R. W. (1947) Inharmonicity of Plain Wire Piano Strings. Journal of Acoustical Society of America 24, (3): 247-273.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC-BY), which permits use and redistribution of the work provided that the original author and source are credited, a link to the license is included, and an indication of changes which were made. Third-party users may not apply legal terms or technological measures to the published article which legally restrict others from doing anything the license permits.
If accepted for publication authors’ work will be made open access and distributed under a Creative Commons Attribution (CC-BY) license unless previously agreed with Exchanges’ Editor-in-Chief prior to submission.
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work. (see: The Effect of Open Access)
