Piano Tuning: A Simple and Accurate Method for Amateurs

by J. Cree Fischer

If we divide an octave, as from middle C to 3C, into three major thirds, each in the perfect ratio of 5 to 4, as C-E, E-G♯ (A♭), A♭-C, then the C obtained from the last third, A♭-C, will be too flat to form a perfect octave by a small quantity, called in the theory of harmonics a diesis, which is expressed by the ratio 128 to 125.

Explanation. The length of the string sounding the tone C is represented by unity or 1. Now, as we have shown, the major third to that C, which is E, is produced by 4/5 of its length.

In like manner, G♯, the major third to E, will be produced by 4/5 of that segment of the string which sounds the tone E; that is, G♯ will be produced by 4/5 of 4/5 ( 4/5 multiplied by 4/5) which equals 16/25 of the entire length of the string sounding the tone C.

We come, now, to the last third, G♯ (A♭) to C, which completes the interval of the octave, middle C to 3C. This last C, being the major third from the A♭, will be produced as before, by 4/5 of that segment of the string which sounds A♭; that is, by 4/5 of 16/25, which equals 64/125 of the entire length of the string. Keep this last fraction, 64/125, in mind, and remember it as representing the segment of the entire string, which produces the upper C by the succession of three perfectly tuned major thirds.

Now, let us refer to the law which says that a perfect octave is obtained from the exact half of the length of any string. Is 64/125 an exact half? No; using the same numerator, an exact half would be 64/128.

Hence, it is clear that the octave obtained by the succession of perfect major thirds will differ from the true octave by the ratio of 128 to 125. The fraction, 64/125, representing a longer segment of the string than 64/128 (½), it would produce a flatter tone than the exact half.

It is evident, therefore, that all major thirds must be tuned somewhat sharper than perfect in a system of equal temperament.

The ratio which expresses the value of the diesis is that of 128 to 125. If, therefore, the octaves are to remain perfect, which they must do, each major third must be tuned sharper than perfect by one-third part of the diesis.