Piano Acoustics - The Railsback Curve

The Railsback Curve

The Railsback curve, first measured by O.L. Railsback, expresses the difference between normal piano tuning and an equal-tempered scale (one in which the frequencies of successive notes are related by a constant ratio, equal to the twelfth root of two). For any given note on the piano, the deviation between the normal pitch of that note and its equal-tempered pitch is given in cents (hundredths of a semitone).

As the Railsback curve shows, octaves are normally stretched on a well-tuned piano. That is, the high notes are higher, and the low notes lower, than they are in an equal-tempered scale. Not all octaves are equally stretched: the middle octaves are barely stretched at all, whereas the octaves on either end of the piano are stretched considerably.

Railsback discovered that pianos were typically tuned in this manner not because of a lack of precision, but because of inharmonicity in the strings. Ideally, the overtone series of a note consists of frequencies that are integer multiples of the note's fundamental frequency. Inharmonicity causes the successive overtones to be higher than they "should" be.

In order to tune an octave, a piano technician must reduce the speed of beating between the first overtone of a lower note and a higher note until it disappears. Because of inharmonicity, this first overtone will be sharper than a harmonic octave (which has the ratio of 2/1), making either the lower note flatter, or the higher note sharper, depending on which one is being tuned to. To produce an even tuning, the technician begins by tuning an octave in the middle of the piano first, and proceeds to tune outwards from there; notes from the upper range are not compared to notes in the lower range for the purposes of tuning.