Abstract
The perceptual results of Plomp and Levelt [J. Acoust. Soc. Am. 38, 548-560 (1965)] for the sensory dissonance of a pair of pure tones are used to estimate the dissonance of pairs of piano tones. By using the spectra of tones measured for a real piano, the effect of the inharmonicity of the tones is included. This leads to a prediction for how the tuning of this piano should deviate from an ideal equal tempered scale so as to give the smallest sensory dissonance and hence give the most pleasing tuning. The results agree with the well known "Railsback stretch," the average tuning curve produced by skilled piano technicians. The authors' analysis thus gives a quantitative explanation of the magnitude of the Railsback stretch in terms of the human perception of dissonance.
Highlights
It is well known that the notes of a well tuned piano do not follow an ideal equal tempered scale
In this paper we describe a similar analysis using the complex tones measured for a real piano
We have calculated the tuning curves that minimize the dissonance for values of b in the range 0.1 to 10, corresponding to either decreasing or increasing Df0 by a factor of 10. (Recall that Df0 is proportional to a physiological parameter, the critical band.) We found that increasing Df0 by as much as a factor of 10 produced very little change in the predicted tuning
Summary
It is well known that the notes of a well tuned piano do not follow an ideal equal tempered scale. The octaves are “stretched”; that is, the frequencies of the fundamental components of piano tones that would differ by precisely a factor of 2 in the ideal case are separated by a slightly greater amount. This stretched tuning was noted many years ago by Railsback.. It is widely accepted that this effect is caused by the inharmonicity of real piano strings. The vibrations of an ideal, flexible string held rigidly at both ends give a complex tone whose components are perfectly harmonic.
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