It is, fortunately, possible to give quite precise directions for the calculating of string dimensions. As a preliminary, we must remind the reader of the rules that were laid down in Chapter IV, relating to the behavior of stretched strings. It will be recalled that we had occasion to observe that these rules would require certain modifications in practice, as they referred only to ideal musical strings which are of perfect flexibility and perfect uniformity, and are stretched at an absolutely constant tension.
The first modification that appears upon investigation has reference to the division of string-lengths. It has already been pointed out that, in practice, we cannot obtain the octave above the fundamental tone of a given pianoforte string by dividing it exactly in the middle. Conversely, an exact doubling of the length does not produce the exact octave below the given fundamental tone. This discrepancy occurs on account of the fact that the shortening or lengthening of a given string causes a corresponding change in the tension at which it is maintained and in the density of adhesion of its molecules.
Now if we double the length of a string in order to obtain the octave below its fundamental tone, we decrease its tension, and this causes a slowing of the frequency of vibration. Then again, the increased resiliency of the string brought about by the lengthening tends also to decrease the frequency. The frequencies of vibration of a string vary directly as the square root of the tension, inversely as the thickness, and directly also as the stiffness. These axioms being admitted, we observe that to obtain an octave lower than a given fundamental tone, we must obtain one-half the frequency that produces the fundamental. Therefore, as we see from above, the double length must be decreased by one-fourth to allow for the automatic decrease of stiffness which varies directly as the frequency. And this modification must itself be modified to compensate for the increase in frequency produced by the very act of shortening. Therefore we must consider the tension, and we find that to reduce this tends again to decrease the stiffness in exactly the same proportions as it was before increased. But frequency of vibration varies as the square root of the tension; therefore we take the square root of one-fourth, which was the fraction first arrived at. This root is one-sixteenth and is the differential factor that must be subtracted from the ideal octave lengths, in order to obtain the practical lengths.