Course:Phys341 2020/Railsback curve

The Railsback Curve plots the typical detuning of a piano from an even-tempered scale.

The Railsback Curve, named after O.L. Railsback who first theorised it, captures the difference between observed frequencies of notes played on a tuned piano, in comparison to an equal-tempered scale. For each note, it maps out corresponding deviations from the equal-tempered scale in cents (one hundredth of a semitone). The Railsback Curve is essential to understanding the importance of human perception of harmonic dissonance in the practice of piano tuning.

Equal Temperament Tuning

In all systems of tuning, every pitch is derived from its relationship to a standard note. In the case of the piano, we usually assign to the note A4 the frequency of 440Hz. The interval between all other notes are obtained by analyzing common harmonics, which are identifiable by the beating heard when two notes are struck simultaneously.

The keys of a piano: two adjacent notes are separated by one semitone.

The equal temperament tuning system divides an octave into equal intervals, keeping a constant ratio between the frequencies of two adjacent notes. The piano uses the twelve-tone equal temperament (12-TET) system which divides the octave into twelve notes, and the ratio between two adjacent notes is equal to the 12th root of 2 (${\sqrt[{12}]{2}}$ ≈ 1.05946). The smallest interval between two notes is called a semitone.

Figure 1 and Figure 2 shows the tuning profile of a piano following a twelve-tone equal temperament tuning.

A0	B0	C1	D1	E1	F1	G1	A1	B1	C2	D2	E2	F2	G2	A2	B2	C3	D3	E3	F3	G3	A3	B3
27.5	30.868	32.703	36.708	41.203	43.654	48.999	55	61.735	65.406	73.416	82.407	87.307	97.999	110	123.47	130.81	146.83	164.81	174.61	196	220	246.94

Figure 1: Equal Tempered Frequencies in Hz, from A0 to B3 (Own Work)

C4	D4	E4	F4	G4	A4	B4	C5	D5	E5	F5	G5	A5	B5	C6	D6	E6	F6	G6	A6	B6	C7	D7	E7	F7	G7	A7	B7	C8
261.63	293.66	329.63	349.23	392	440	493.88	523.25	587.33	659.26	698.46	783.99	880	987.77	1046.5	1174.7	1318.5	1396.9	1568	1760	1975.5	2093	2349.3	2637	2793.8	3136	3520	3951.1	4186

Figure 2: Equal Tempered Frequencies in Hz, from C4 to C8. C4 is the middle C on the piano, and A4 is the standard note for tuning. (Own Work)

However, in practice, pianos are tuned to frequencies that do not strictly follow this scale: the lower notes are tuned at slightly lower frequencies, and higher notes are tuned at slightly higher frequencies. This phenomenon is often called “octave stretching”, and although it is almost negligible in the center of the keyboard, it is considerable at the treble end and at the bass. This is due to the inharmonicity of piano strings.

Inharmonicity of Piano Strings

Stretched strings of varying lengths can be seen inside a grand piano.

Ideally, in the case of a musical instrument, the wave produced on a stretched string should have a wavelength much greater than the thickness of the string. This allows the wave to have a constant velocity and to produce harmonic overtones. When the wavelength approaches the diameter of the string, the string behaves more like a metal bar: its mechanical resistance to bending adds a force to the tension of the string, which makes the pitch of overtones higher, and thus inharmonic. This is an extreme illustration however, as the wavelength does not reach the diameter of the string, but it shows the basic principle which leads to inharmonicity.

Figure 3: Spectrum analysis of notes A5-A6 and A1-A2 played on a piano. Both scales are logarithmic and the spectra have been displaced vertically for readability. (Own Work)

In the case of a piano, inharmonicity is most pronounced in the lowest and higher notes. Inharmonicity is caused by stiffness of strings, and is amplified by increased thickness or decreased length, which occurs at the two extremities of the piano due to technical constraints. The lowest strings are thicker and have higher mass density (because of the spatial restriction of having a decently sized instrument), bringing the diameter of the string closer to the wavelength. The highest strings are designed to be very short (given the need to have thin strings that resist to high tension).

If a piano was tuned strictly following the equal temperament scale, the slightly inharmonic notes, when played together, would cause the phenomenon of beating. Especially for the piano, where the player uses notes over several octaves, this phenomenon can be noticeable and irritating to the ear.

Figure 3 shows two spectrum analyses: one of A5 and A6, which are two notes in the middle of the piano, and one of A1 and A2 which are two notes are the bass end of the piano. This comparison reveals that for the notes closer to the bass end, the partials vary in intensity in a more complex manner, and we can expect to hear beating (denoted with red arrows in Figure 3), which will contribute to sensory dissonance.

To prevent this, tuners choose to tune the lower notes slightly lower, and the higher notes slightly higher.

The Railsback Curve

Figure 4: This graph shows the deviations between measured frequencies of keyboard notes and the equal tempered scale. Measurements were made between C2 and C8. (own work)

The Railsback curve, named after O.L. Railsback, captures the difference between in practice piano tuning and the theoretical equal-tempered scale. It maps out each note of the piano on the x-axis, and the corresponding deviations in cents (one hundredths of a semitone) on the y-axis.

In 1938, Railsback measured the tuning profiles of a sample of aurally tuned pianos and found that their deviations from the equal-tempered scale can be reduced to one average curve: the one we know as the Railsback curve. Subsequent scholarship, such as the study by Daniel Martin and W. Dixon Ward in 1961 has proven that tuning a piano aurally, in a pattern that follows the Railsback Curve is the method that sounds most pleasing to the ear, and listeners unequivocally reject tuning strict equal temperament tuning in favor of tuning by ear.

The deviations captured in the Railsback Curve minimize dissonance by adjusting octave spacing so that partials of notes that are an octave apart overlap as much as possible.

Figure 4 shows a curve constructed from the measurements of an electronic keyboard, programmed to mimic a piano. A spectrum analysis of every note was done to identify deviations from theoretical frequencies of equal tempered tuning. This graph only represents the frequencies of notes between C2 and C8, as the lower notes had very low fundamental frequencies, which could not be accurately picked up by our sound analysis software.

Figure 5: Spectrum analysis of an E6 played on a keyboard, using the iSpectrum software. (own work)

Figure 5 shows the spectrum analysis of the note E6: its measured fundamental frequency is 1320Hz, in comparison to 1318.5 on the equal tempered scale.

The Importance of Human Perception in Piano Tuning

Due to the specificities mentioned above, tuning a piano is a complex process which usually requires the know-how of a professional tuner, or a skilled technician. When a professional tunes a piano, they can use commercial tuning devices, they can tune the piano aurally, or most often, they use a combination of both. The tuning of a piano usually takes place in three steps:

Firstly, the tuner tunes one standard note, usually A4, using a tuning fork. Then, they tune the temperament octave, that is usually the octave between F3 and F4. Lastly, the rest of the keys can be tuned using the temperament octave as a standard.

Ultimately, the goal of a tuner is to tune piano keys to a scale that it most pleasant to the human ear. Although electronic tuning devices can make the tuning process faster and less tiring, a professional tuner wants to minimize as much as possible the dissonance of the most important intervals, and rely heavily on tuning by ear. This shows that sensory dissonance is central in guiding the tuning profile of a piano.