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.
It will be found of course, as must be apparent to the reader, that the differential factor here suggested does not provide a complete solution to the problem of allowing for the exhibited differences between theory and practice. It does, however, provide a true guide to the lengths. There is of course a difference of produced frequency to be allowed for yet. Fortunately, however, this is provided for by the graduated thicknesses of pianoforte wire. By taking advantage of this almost geometrically proportioned graduation of diameter we are able to calculate a stringing scale that, if adhered to, will give the nearest possible approximation to complete harmony between theory and practice. That is to say, we can proceed with a string-length calculation based upon the differential factor already obtained, and then by arranging the distribution of the string thicknesses according to the diameters that are provided by the manufacturers of music wire, we may obtain a true estimation, not only as to the thickness of wire to be used at each place, but also as to the lengths proper to each string. Of course the reader will remember that the matter of pitch is of considerable importance in all calculations of this kind. A difference in pitch implies difference of tension when the other factors remain equal, and we therefore have calculated the following tables on the assumption that the pitch to be used is that known as the International or C 517. Attention is, therefore, directed to the following
| C5 | = 2.048 in. | = 2 1⁄25 | + ... Approx. | |
| C4 | = 2.048 × 1.9375 | = 3.968 in. | = 3 24⁄25 | + ... " |
| C3 | = 3.968 × 1.9375 | = 7.688 in. | = 7 17⁄25 | + ... " |
| C2 | = 7.688 × 1.9375 | = 14.875 in. | = 14 7⁄8 | + ... " |
| C1 | = 14.875 × 1.9375 | = 28.820 in. | = 28 4⁄5 | + ... " |
| C | = 28.820 × 1.9375 | = 55.828 in. | = 55 4⁄5 | + ... " |
[Note.—The length of the first string is chosen arbitrarily, but as given is a very close approximation to the practice of the best American makers. The vulgar fractions are calculated from the decimals and the error in no case exceed about one-fiftieth of an inch. The differential factor is, as we know, 1⁄16. Therefore we multiply by (2 − 1⁄16) or 1 15⁄16; in decimals 1.9375.]
The above table, then, affords us a reliable guide to the scaling of the unwrapped strings. At the same time, however, it is not by any means complete, for the reason that there is no method shown as yet for the calculation of the other and intermediate string-lengths. We are, however, able to accomplish this task by the aid of a very ingenious rule proposed by the late Professor Pole, F.R.S. It is as follows:
The proper length of any string may be determined from that of any other string, provided that the length and frequency of the second string be known. Given these factors: Then,
- Take the logarithm of the length of the known string.
- Multiply the number .025086 by the number of semitones that the sound to be given by the required string length is above or below the sound produced by the given string.
- If the required string is below the given string, add together the two numbers obtained; if it be above, subtract the second number from the first; the result in both cases is the logarithm of the required length.
For example, we have calculated already the proper length of C. In hundredths of an inch this length is expressed as 2882. The log. of this number is 45943. (This may be verified by any table of logarithms.)