Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance N. Giordanoa) Department of Physics, Auburn University, 246 Sciences Center Classroom, Auburn, Alabama 36849, USA (Received 11 March 2015; revised 28 August 2015; accepted 2 September 2015; published online 23 October 2015) The perceptual results of Plomp and Levelt [J. Acoust. Soc. Am. 38, 548–560 (1965)] for the sen- sory 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. VC 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.[http://dx.doi.org/10.1121/1.4931439] [TRM] Pages: 2359–2366 I. INTRODUCTION explains the piano tuning curve in the treble, a fact that has been known for some time.3 It is well known that the notes of a well tuned piano do However, piano tones in the bass region contain many not follow an ideal equal tempered scale. Instead, the octaves partials with significant amplitudes. When considering the are “stretched”; that is, the frequencies of the fundamental dissonance of two complex tones of this kind, it is not possi- components of piano tones that would differ by precisely a ble to achieve a “zero beat” condition for all pairs of partials. factor of 2 in the ideal case are separated by a slightly greater amount. This stretched tuning was noted many years ago by Instead, there will always be beating involving many Railsback.1,2 He also studied the tuning curves (the deviation partials, and one would expect minimum dissonance to be from an equal tempered tuning) of a collection of pianos achieved when some (presumably complicated) function of tuned by expert technicians and found that all follow approx- the beat rates of many partials is minimized. This compli- imately the same stretched tuning. It is now widely accepted cated function is a product of the human sensory system and that this effect is caused by the inharmonicity of real piano we do not currently have a theory for this function based on strings. This inharmonicity is an important factor in piano fundamental physics or physiology. A complete understand- design; it has thus been studied in great detail and there is an ing of the piano tuning curve that applies throughout the accurate theory of the inharmonicity of piano strings that has bass and treble thus requires a model or at least a description been verified through a number of quantitative studies (e.g., of sensory dissonance for complex tones. Refs. 3 and 4). Interestingly, such a description of sensory dissonance 5 The fact that all well tuned pianos appear to follow a does exist but until now it has not been applied to the prob- sort of universal tuning curve suggests that this tuning is lem of explaining the Railsback stretch. That is the purpose related to a fundamental aspect of human perception, i.e., of this paper. We will see that our analysis, which combines that the human perception of dissonance (which we will also what is known about sensory dissonance with what we know refer to as “sensory dissonance” or simply “dissonance”) is a about the inharmonicity of real piano tones, leads quantita- key factor in determining how a piano is tuned. In some tively to the Railsback tuning curve. Our result also serves cases, the dissonance of two tones is easy to understand and as a test of how the perceptual dissonance of pure tones can predict. For example, for two pure tones the dissonance is be used to determine the dissonance of complex tones that minimized when the tones have the same frequency, i.e., consist of many inharmonic components.6,7 when there is no beating. The notion that less beating results in lower dissonance also explains the perceived dissonance II. BACKGROUND of two complex tones in some simple cases, such as complex Piano tones are produced by vibrating strings. The tones that each contains only two components (which are vibrations of an ideal, flexible string held rigidly at both called partials in the case of piano tones). As we will ends give a complex tone whose components are perfectly explain, such cases apply quite well to the treble region of harmonic. That is, if the frequency of the fundamental com- the piano, so this simple notion about sensory dissonance 0 ponent is f1 , the frequency of the nth harmonic is precisely 0 a)Electronic mail: [email protected] fn ¼ nf1 ; (1) J. Acoust. Soc. Am. 138 (4), October 2015 0001-4966/2015/138(4)/2359/8 VC Author(s) 2015 2359 where n is an integer. The harmonic spectra produced by ideal strings and a number of other vibrating objects suggest why the octave plays an important role in musical scales. Two tones described by Eq. (1) and whose fundamental frequencies differ by a factor of exactly 2 have many over- lapping harmonics. The notion that tones whose components overlap in this way are perceived as “pleasing,” i.e., conso- nant, has been suggested many times over the past centuries.8 Real piano strings are not ideal; they have some stiffness, which causes deviations from the ideal harmonic spectrum in Eq. (1) (see, for example, Refs. 3 and 9). The components of real piano tones are therefore called partials, to distinguish them from ideal harmonics. The stiffness of real piano strings is small, and in that limit the partial fre- quencies are given to lowest order by4,10 2 FIG. 1. Measured spectra of notes A5 and A6. Note that both scales are log- fn nf1ð1 þ an Þ; (2) arithmic and that the spectra have been displaced vertically for clarity. where a is proportional to the Young’s modulus of the string details of the measurement will be described below.) These and the frequency f1 of the first partial is shifted slightly spectra illustrate a key property of notes in the treble; the 0 from the value f1 found for the ideal string. In practice there intensities of the different partials fall off very rapidly with are corrections to Eq. (2) that depend on the boundary condi- partial number. For example, the fourth partial of A5 is about tions of the string, the soundboard impedance, and other fac- 3 orders of magnitude smaller than the fundamental (i.e., the tors. However, these corrections do not change the fact that first partial), while for A6 the second partial is about 4 orders the spacing between neighboring partials is not constant but of magnitude smaller than its fundamental component. This generally increases with n, as predicted by Eq. (2). Indeed, strongly suggests that when considering possible beating our analysis will not rely on or even use the functional form between partials of A5 and A6, only the second partial of A5 in Eq. (2), as we will make use of the values of fn measured and the fundamental of A6 need to be considered; the com- for the tones of a real piano. The key point is that the partials bined intensities of the fourth partial of A5 and the second of two piano tones whose fundamental frequencies differ by partial of A6 are likely far too small for their beating to be precisely a factor of 2 (a perfect octave) will have few, if discernible. The second partial of A5 and the fundamental of any, partials that overlap exactly. Indeed, the second partial A6 will thus determine the sensory dissonance of these two of the lower tone will overlap with the fundamental compo- notes. nent of the upper tone only if their fundamental frequencies The situation is completely different in the bass. Figure 2 are spaced by slightly more than a factor of 2, hence giving a shows spectra for notes A1 and A2, which show that the stretched octave. partials vary in intensity in a complex manner as the partial When a skilled technician tunes a piano he or she number increases. We must then expect the beating of many presumably has the goal of making it sound as “good” as possible. This judgment may be based simply on listening, or rely on what a commercial tuning device says is the desired tuning, or some combination, but it is reasonable to assume that this judgment is ultimately based on human perception. It is also reasonable to assume that this “best” tuning is designed to minimize the dissonance of the most important musical intervals, and it is widely believed that the process of minimizing the dissonance amounts to adjust- ing the octave spacing so that the partials of notes an octave apart overlap as closely as possible. However, it is not clear what the term “as closely as possible” means when one is considering tones with many partials as is the case for real piano tones, especially in the bass region of the scale. It is useful at this point to consider the spectra of a few piano tones, which will help in understanding how pairs of partials can combine to increase or decrease the sensory dis- sonance, and why the problem is much simpler for notes in the treble as compared to the bass.