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21. The reason for this seems to be that for a well-tuned piano, the difference in frequency between unison strings is much smaller than frequency differences associated with the stretch tuning.
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23. This approach for shifting the frequency of a partial is valid for all normal partials. It will not apply to longitudinal modes or so-called phantom partials (Refs. 26 and 27) which are known to be present in bass tones. Careful examination of the spectra, as in Fig. 2, allowed us to distinguish the longitudinal modes and phantom partials, and we found that they were all sufficiently weak and small enough in number that they had no effect on the calculated dissonance.
24. This approach for shifting the partial frequencies should be accurate to order α in Eq. (2). Since α was typically of order 5 × 10−5, this approximation should be adequate.
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The perceptual results of Plomp and Levelt [J. Acoust. Soc. Am. , 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.


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