• Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance

It is well known that the notes of a well tuned piano do not follow an ideal equal tempered scale. Instead, 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.^1,21. O. L. Railsback, “ Scale temperament as applied to piano tuning,” J. Acoust. Soc. Am. 9, 274 (1938). https://doi.org/10.1121/1.19020562. O. L. Railsback, “ A study of the tuning of pianos,” J. Acoust. Soc. Am. 10, 86 (1938). https://doi.org/10.1121/1.1902080 He also studied the tuning curves (the deviation from an equal tempered tuning) of a collection of pianos tuned by expert technicians and found that all follow approximately the same stretched tuning. It is now widely accepted that this effect is caused by the inharmonicity of real piano strings. This inharmonicity is an important factor in piano design; it has thus been studied in great detail and there is an accurate theory of the inharmonicity of piano strings that has been verified through a number of quantitative studies (e.g., Refs. 33. R. W. Young, “ Inharmonicity of plain wire piano strings,” J. Acoust. Soc. Am. 24, 267–273 (1952). https://doi.org/10.1121/1.1906888 and 44. N. Fletcher, “ Normal vibration frequencies of a stiff piano string,” J. Acoust. Soc. Am. 36, 203–209 (1964). https://doi.org/10.1121/1.1918933).

The fact that all well tuned pianos appear to follow a sort of universal tuning curve suggests that this tuning is related to a fundamental aspect of human perception, i.e., that the human perception of dissonance (which we will also refer to as “sensory dissonance” or simply “dissonance”) is a key factor in determining how a piano is tuned. In some cases, the dissonance of two tones is easy to understand and predict. For example, for two pure tones the dissonance is minimized when the tones have the same frequency, i.e., when there is no beating. The notion that less beating results in lower dissonance also explains the perceived dissonance of two complex tones in some simple cases, such as complex tones that each contains only two components (which are called partials in the case of piano tones). As we will explain, such cases apply quite well to the treble region of the piano, so this simple notion about sensory dissonance explains the piano tuning curve in the treble, a fact that has been known for some time.³3. R. W. Young, “ Inharmonicity of plain wire piano strings,” J. Acoust. Soc. Am. 24, 267–273 (1952). https://doi.org/10.1121/1.1906888

However, piano tones in the bass region contain many partials with significant amplitudes. When considering the dissonance of two complex tones of this kind, it is not possible to achieve a “zero beat” condition for all pairs of partials. Instead, there will always be beating involving many partials, and one would expect minimum dissonance to be achieved when some (presumably complicated) function of the beat rates of many partials is minimized. This complicated function is a product of the human sensory system and we do not currently have a theory for this function based on fundamental physics or physiology. A complete understanding of the piano tuning curve that applies throughout the bass and treble thus requires a model or at least a description of sensory dissonance for complex tones.

Interestingly, such a description of sensory dissonance does exist⁵5. R. Plomp and W. J. M. Levelt, “ Tonal consonance and critical bandwidth,” J. Acoust. Soc. Am. 38, 548–560 (1965). https://doi.org/10.1121/1.1909741 but until now it has not been applied to the problem of explaining the Railsback stretch. That is the purpose of this paper. We will see that our analysis, which combines what is known about sensory dissonance with what we know about the inharmonicity of real piano tones, leads quantitatively to the Railsback tuning curve. Our result also serves as a test of how the perceptual dissonance of pure tones can be used to determine the dissonance of complex tones that consist of many inharmonic components.^6,76. W. A. Sethares, “ Local consonance and the relationship between timbre and scale,” J. Acoust. Soc. Am. 94, 1218–1228 (1993). https://doi.org/10.1121/1.4081757. W. A. Sethares, Tuning, Timbre, Spectrum, Scale ( Springer, London, 2005).

Piano tones are produced by vibrating strings. The vibrations of an ideal, flexible string held rigidly at both ends give a complex tone whose components are perfectly harmonic. That is, if the frequency of the fundamental component is $f_1^0$, the frequency of the nth harmonic is precisely

$$f_n=n{f}_1^0,$$

(1)

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 overlapping harmonics. The notion that tones whose components overlap in this way are perceived as “pleasing,” i.e., consonant, has been suggested many times over the past centuries.⁸8. S. Dostrovsky, “ Early vibration theory: Physics and music in the seventeenth century,” Arch. Hist. Exact Sci. 14, 169–218 (1975). https://doi.org/10.1007/BF00327447

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. 33. R. W. Young, “ Inharmonicity of plain wire piano strings,” J. Acoust. Soc. Am. 24, 267–273 (1952). https://doi.org/10.1121/1.1906888 and 99. O. H. Schuck and R. W. Young, “ Observations on the vibrations of piano strings,” J. Acoust. Soc. Am. 15, 1–11 (1943). https://doi.org/10.1121/1.1916221). 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 frequencies are given to lowest order by^4,104. N. Fletcher, “ Normal vibration frequencies of a stiff piano string,” J. Acoust. Soc. Am. 36, 203–209 (1964). https://doi.org/10.1121/1.191893310. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments ( Springer-Verlag, New York, 1991), Chaps. 2 and 12.

$$f_n\approx n{f}_1(1+\alpha n^2),$$

(2)

where α is proportional to the Young's modulus of the string and the frequency f₁ of the first partial is shifted slightly from the value $f_1^0$ found for the ideal string. In practice there are corrections to Eq. (2) that depend on the boundary conditions of the string, the soundboard impedance, and other factors. However, these corrections do not change the fact that the spacing between neighboring partials is not constant but generally increases with n, as predicted by Eq. (2). Indeed, our analysis will not rely on or even use the functional form in Eq. (2), as we will make use of the values of f_n measured for the tones of a real piano. The key point is that the partials of two piano tones whose fundamental frequencies differ by precisely a factor of 2 (a perfect octave) will have few, if any, partials that overlap exactly. Indeed, the second partial of the lower tone will overlap with the fundamental component of the upper tone only if their fundamental frequencies are spaced by slightly more than a factor of 2, hence giving a stretched octave.

When a skilled technician tunes a piano he or she 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 adjusting 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 dissonance, and why the problem is much simpler for notes in the treble as compared to the bass. Figure 1 shows the spectra for the notes A5 and A6 of a piano; in this notation, A4 has a fundamental frequency of 440 Hz. (The piano and details of the measurement will be described below.) These spectra illustrate a key property of notes in the treble; the intensities of the different partials fall off very rapidly with partial number. For example, the fourth partial of A5 is about 3 orders of magnitude smaller than the fundamental (i.e., the first partial), while for A6 the second partial is about 4 orders of magnitude smaller than its fundamental component. This strongly suggests that when considering possible beating between partials of A5 and A6, only the second partial of A5 and the fundamental of A6 need to be considered; the combined intensities of the fourth partial of A5 and the second partial of A6 are likely far too small for their beating to be discernible. The second partial of A5 and the fundamental of A6 will thus determine the sensory dissonance of these two notes.

The situation is completely different in the bass. Figure 2 shows spectra for notes A1 and A2, which show that the partials vary in intensity in a complex manner as the partial number increases. We must then expect the beating of many pairs of partials to be important when considering the sensory dissonance of these notes and others in the bass.

These insights on contributions to sensory dissonance have been recognized for some time. For example, Schuck and Young⁹9. O. H. Schuck and R. W. Young, “ Observations on the vibrations of piano strings,” J. Acoust. Soc. Am. 15, 1–11 (1943). https://doi.org/10.1121/1.1916221 used measurements of the spectra of real piano tones, similar to those shown in Figs. 1 and 2 to explore how the overlap of partials produces a stretched tuning. They did not attempt to calculate dissonance quantitatively, but instead used only the observed frequencies of a single pair of partials (one partial from each note) to estimate the optimum stretch. For example, if one assumes that the lowest partials are most important one can estimate the octave stretch by requiring that the second partial of each note be equal to the fundamental frequency of the note an octave higher. Schuck and Young found that this assumption gives a good account of the stretch in the treble (as we have suggested based on the spectra in Fig. 1) but greatly underestimates the amount of stretch found in the bass. They also showed that matching pairs of higher partials of notes separated by either an octave or by two octaves leads to different predictions for the degree of stretch, but none of their matching algorithms, which were based solely on the frequencies of a single pair of partials and ignored their amplitudes, was able to account quantitatively for the tuning curves observed in practice. For example, matching the lowest partials of notes spaced by two octaves led to an underestimation of the octave stretch in the bass while overestimating it in the treble. Schuck and Young were thus not able to quantitatively account for the magnitude of the Railsback stretch in the bass.

More recently, there have been various analyses of stretched tuning, in which purely algorithmic approaches have been used to determine the best tuning.^11,1211. F. Rigaud, B. David, and L. Daudet, “ A parametric model and estimation techniques for the inharmonicity and tuning of the piano,” J. Acoust. Soc. Am. 133, 3107–3118 (2013). https://doi.org/10.1121/1.479980612. H. Hinrichsen, “ Entropy-based tuning of musical instruments,” Rev. Bras. Ens. Fís. 34, 2301 (2012) https://doi.org/10.1590/S1806-11172012000200004. Specifically, these approaches have used algorithms that minimize an effective entropy or similar constructs, in which these functions have no relation to what is known about human sensory dissonance.

Surprisingly, the author knows of no attempt to explain the Railsback stretch quantitatively in terms of what is known about the human perception of dissonance. The purpose of this paper is to take a step forward from the analysis of Schuck and Young in which established descriptions and models of human perception are used to evaluate the sensory dissonance of real piano tones and thereby predict the Railsback stretch. Our approach allows us to account for the many partials of real piano tones and their amplitudes, using the spectra of tones measured for a particular piano when estimating the sensory dissonance of two tones. This dissonance is estimated quantitatively using the results of the perceptual studies of the dissonance of pure tones by Plomp and Levelt as described in Sec. III.⁵5. R. Plomp and W. J. M. Levelt, “ Tonal consonance and critical bandwidth,” J. Acoust. Soc. Am. 38, 548–560 (1965). https://doi.org/10.1121/1.1909741 The results are compared with the actual tuning of the piano under study and with the Railsback tuning curve. Overall good agreement is found. We also explore the sensitivity of this agreement to plausible alterations of the dissonance model.

This analysis described in this paper makes use of relations concerning perceptual dissonance that are based on the perceptual studies of Plomp and Levelt⁵5. R. Plomp and W. J. M. Levelt, “ Tonal consonance and critical bandwidth,” J. Acoust. Soc. Am. 38, 548–560 (1965). https://doi.org/10.1121/1.1909741 (see also Kameoka and Kuriyagawa^13,1413. A. Kameoka and M. Kuriyagawa, “ Consonance theory part I: Consonance of dyads,” J. Acoust. Soc. Am. 45, 1451–1459 (1969). https://doi.org/10.1121/1.191162314. A. Kameoka and M. Kuriyagawa, “ Consonance theory part II: Consonance of complex tones and its calculation method,” J. Acoust. Soc. Am. 45, 1460–1469 (1969). https://doi.org/10.1121/1.1911624). Plomp and Levelt used listening tests to determine how the dissonance of two pure tones with frequencies f₁ and f₂ depends on the difference in frequency of the tones. They showed that the perceived dissonance depends on both the frequency difference f₂ − f₁ and the lower of the two frequencies (f₁). Figure 3 shows a convenient parameterization of the Plomp and Levelt results due to Sethares,⁶6. W. A. Sethares, “ Local consonance and the relationship between timbre and scale,” J. Acoust. Soc. Am. 94, 1218–1228 (1993). https://doi.org/10.1121/1.408175 the functional form of which will be given below. The dissonance is zero when the frequencies are equal, f₂/f₁ = 1. The dissonance exhibits a maximum when f₂ − f₁ equals a value Δf₀ that depends on the frequency of the lower tone [Fig. 3(b)]. The frequency difference Δf₀ is approximately equal to one-quarter of the critical band as identified in other studies (e.g., Ref. 1515. E. Zwicker, B. Flottorp, and S. S. Stevens, “ Critical band width in loudness summation,” J. Acoust. Soc. Am. 29, 548–557 (1957). https://doi.org/10.1121/1.1908963). The dissonance then decreases smoothly to zero as the frequency separation is increased beyond Δf₀.

The dissonance curves in Fig. 3 are noteworthy for their lack of structure. One might have expected to find dips in the dissonance (and hence maxima in the consonance) for musically pleasing intervals such as the octave (f₂/f₁ = 2), perfect fifth (f₂/f₁ = 3/2), and so on, but that is not the case. The reduction of the dissonance of such intervals appears when one considers complex tones with harmonic spectra. Plomp and Levelt (and many others) have applied their result for pure tones to compute the dissonance of complex tones whose components form a harmonic series [Eq. (1)]. For such tones the total dissonance is assumed to be the sum of the pairwise dissonances of each of the components of both complex tones. One now has the possibility for the dissonance to be minimized by having some of the harmonics of the two tones overlap. For complex tones with harmonic components, such overlap does indeed produce minima in the dissonance for tones that form various musical intervals,⁵5. R. Plomp and W. J. M. Levelt, “ Tonal consonance and critical bandwidth,” J. Acoust. Soc. Am. 38, 548–560 (1965). https://doi.org/10.1121/1.1909741 thus providing an “explanation” for the mathematical structure of the 12 tone musical scale. In this picture, the intervals which minimize the dissonance (i.e., maximize the consonance) depend on the spectra of the associated complex tones. Harmonic tones lead to the standard consonant intervals of an octave and perfect fifth, etc., but other types of complex tones can give very different results (see, e.g., Refs. 1616. J. R. Pierce, “ Attaining consonance in scales,” J. Acoust. Soc. Am. 40, 249 (1966). https://doi.org/10.1121/1.1910051 and 1717. W. A. Sethares, “ Adaptive tunings for musical scales,” J. Acoust. Soc. Am. 96, 10–18 (1994). https://doi.org/10.1121/1.410471).

In this paper we describe a similar analysis using the complex tones measured for a real piano. Our main goal is to determine if the inharmonicity of complex tones found in a real piano can be combined with a model of dissonance based on perceptual studies to quantitatively predict the stretched tuning actually found for that piano. Our analysis uses results for the partial frequencies and amplitudes measured for a real piano, and is not limited to the lowest one or two partials but explicitly includes the contributions to the dissonance from all significant partials.

Figure 3(a) shows the results of Plomp and Levelt⁵5. R. Plomp and W. J. M. Levelt, “ Tonal consonance and critical bandwidth,” J. Acoust. Soc. Am. 38, 548–560 (1965). https://doi.org/10.1121/1.1909741 for the dissonance of two pure tones with frequencies f₁ and f₂. These curves are a parameterization derived by Sethares,⁶6. W. A. Sethares, “ Local consonance and the relationship between timbre and scale,” J. Acoust. Soc. Am. 94, 1218–1228 (1993). https://doi.org/10.1121/1.408175 according to which the dissonance is proportional to

$$d_2(f_1,f_2)=(e^{-b_{1}s|f_2-f_1|}-e^{-b_{2}s|f_2-f_1|})\hspace{0.17em}.$$

(3)

Here

$$s=s^{\star }/[s_1\min (f_1,f_2)+s_2],$$

(4)

$\min (x,y)$ is a function equal to the lesser of its two arguments, and b₁ = 3.5, b₂ = 5.75, s^*= 0.24, s₁ = 0.021, and s₂ = 19. These relations were derived from studies involving pure tones, and to apply them to real musical tones we must consider how the dissonances of two or more pairs of pure tones combine to give a total dissonance. This question is extremely important, since we have seen in Fig. 2 that piano tones can contain a large number of partials and for the bass notes the strongest partials may not be the ones with small n. Various authors have used the function d₂() in Eq. (3) (or its equivalent) to compute the dissonance of complex tones. Plomp and Levelt applied it to complex tones consisting of six harmonics [with frequencies given by Eq. (1) with n = 1, 2,…,6] but did not specify the relative amplitudes of the harmonics and appeared to simply add the dissonance functions for each of the components of the two complex tones. Kameoka and Kuriyagawa¹⁴14. A. Kameoka and M. Kuriyagawa, “ Consonance theory part II: Consonance of complex tones and its calculation method,” J. Acoust. Soc. Am. 45, 1460–1469 (1969). https://doi.org/10.1121/1.1911624 gave a rather extensive discussion of the issues involved in combining the dissonances of different dyads. In the end, they seem to have added the dissonance functions d₂() for each pair of components weighted approximately by the product of the amplitudes of the two components in each dyad. Sethares⁶6. W. A. Sethares, “ Local consonance and the relationship between timbre and scale,” J. Acoust. Soc. Am. 94, 1218–1228 (1993). https://doi.org/10.1121/1.408175 initially took a similar approach and assumed that the dissonance of two complex tones is given by

$$D_{\text{total}}=\frac{1}{2}\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}{B}_{i,j}{d}_2(f_{1,i},f_{2,j})\hspace{0.17em},$$

(5)

with

$$B_{i,j}=a_{1,i}{a}_{2,j}\hspace{0.17em}.$$

(6)

Here the ith partial of tone 1 has frequency f_1,i and amplitude a_1,i, and we assume that only n₁ partials have significant amplitudes, etc., for tone 2. (Note that here we omit the “self-dissonance” of a tone, which is included by some authors but is not important for our work since we will wish to find the condition in which the dissonance of two tones is a minimum. The self-dissonance may certainly be important when judging the quality of a single complex tone.¹⁸18. D. E. Farrell, private communication (2015).) We will term D_total with B_i,j given by Eq. (6) the “amplitude product” model. Sethares⁶6. W. A. Sethares, “ Local consonance and the relationship between timbre and scale,” J. Acoust. Soc. Am. 94, 1218–1228 (1993). https://doi.org/10.1121/1.408175 applied this model to complex tones consisting of seven harmonics with amplitudes that fall at a rate of 0.88 from one harmonic to the next, and showed, in agreement with the results of Plomp and Levelt, that this yields to consonances at the well-known and expected intervals, including octaves and perfect fifths.

In later work, Sethares⁷7. W. A. Sethares, Tuning, Timbre, Spectrum, Scale ( Springer, London, 2005). suggested that instead of Eq. (6) the dissonance should be proportional to the loudness of the weaker component so that

$$B_{i,j}=\text{min}({\ell }_{1,i},{\ell }_{2,j})\hspace{0.17em},$$

(7)

where ℓ_n,i is the loudness of the ith partial of tone n. In our applications of Eq. (7) in this paper we will assume that loudness is defined in the usual way [ℓ ∼ 2^SPL/10 where the sound pressure level (SPL) is related to the power by SPL ∼ log(P)].^7,107. W. A. Sethares, Tuning, Timbre, Spectrum, Scale ( Springer, London, 2005).10. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments ( Springer-Verlag, New York, 1991), Chaps. 2 and 12. We thus ignore the dependence of perceived loudness on frequency (the so-called Fletcher-Munson curves); we will consider this below in Sec. VI D. We will refer to D_total with B_i,j given by Eq. (7) as the “minimum loudness” model. One argument in favor of a model based on loudness rather than partial amplitudes is that human hearing judges the loudness of a sound on a logarithmic scale, as accounted for in the definition of ℓ_n,i. We will consider both the amplitude product and the minimum loudness model in our analysis. So far as we know, there have been no experiments that have studied which, if either, of these models is the appropriate way to measure the relative dissonance of pairs of pure tones.

This analysis described in this paper makes use of a Steinway model M piano that belongs to the author and had recently been tuned by a professional technician. No special instructions were given to the technician. Recordings were made of every note A, C, E, and G, ranging from A0 to C8. These notes were chosen so as to provide multiple notes spread over each octave and give the option of studying the dissonance of common intervals and chords (a problem left for the future). Each note was recorded at an mf level, and all of the strings for each note were allowed to vibrate. Notes A0–E1 employed a single wound string, G1–C3 had two wound strings each, and E3–C7 employed three non-wound strings each. Each note was recorded using a PCB Piezotronics (Depew, NY) model 130P10 microphone with a sampling rate of 44.1 kHz. Spectra were calculated with a standard fast Fourier transform (FFT) algorithm using data for approximately 0.75 s taken just after the attack portion of the tone. The frequencies of each partial were obtained by fitting the spectral peaks to either Gaussian or sinc² lineshapes,¹⁹19. R. N. Bracewell, The Fourier Transformation and its Applications, 2nd ed. ( McGraw-Hill, New York, 1978), Chaps. 2 and 4. which were found to give essentially identical results for the partial frequencies. The frequencies derived in this way and for sound data of this duration were sufficiently precise for our purposes since the dissonance according to Eq. (3) varies slowly compared to the frequency resolution of these FFTs. More sophisticated signal processing methods were thus not required. For the same reason, the small difference in the frequencies of unison strings²⁰20. R. E. Kirk, “ Tuning preferences for piano unison groups,” J. Acoust. Soc. Am. 31, 1644–1648 (1959). https://doi.org/10.1121/1.1907673 was not important.²¹21. 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.

This analysis yielded the frequencies and amplitudes of all partials with significant strength for each note, which were used in computing the dissonance functions of various pairs of notes. For most notes the spectra also yielded directly the fundamental frequency of each note and hence the Railsback stretch function for this piano as tuned by the technician. For the notes below G1 the fundamental component was difficult to discern from the noise; in those cases the fundamental frequency was estimated using the frequencies of the lowest partials and extrapolating f_n/n as n → 1.

We first consider the actual tuning curve of the piano under study; that is, the tuning as set by the professional tuner and presumably judged by him (according to his tuning device or simply by ear) to be the best and presumably most consonant tuning possible for our piano. We will find that it agrees well with the Railsback result. We then consider the measured spectra of individual tones and use the models of dissonance described in Sec. IV to calculate the most consonant tuning curve predicted by the models.

The results of perceptual studies of the dissonance of pure tone by Plomp and Levelt together with the measured inharmonic spectra of real piano tones have been used to predict the optimum tuning of a piano. This prediction agrees well with the way pianos are found to be tuned in practice, the Railsback stretch. This result is further confirmation that the inharmonicity of piano tones is indeed responsible for the stretched tuning of a piano. This is not a surprising result; this inharmonicity has long been assumed to be responsible for the Railsback stretch. However, to our knowledge this is the first quantitative calculation that uses what is known about the human perception of sensory dissonance to correctly predict the magnitude of the stretch. Since our analysis relies on a model of dissonance, the agreement between the predicted and observed stretch serves as a check on that model.

The model of dissonance we have used is based on the perceptual studies by Plomp and Levelt using pure tones. The extension of their results to complex tones requires an assumption as to how the dissonance of two pure tones depends on their amplitudes. Our analysis has considered two proposals of Sethares, one based on the product of the amplitudes and another based on loudness of the weakest tone, and both are consistent with the Railsback result. Further work with tones of greatly differing amplitudes will be required to distinguish which of these models gives the best description of sensory dissonance.