| It is required to obtain the length of the string that, caeteris paribus, will produce one semitone above C. | ||
| 45943 | = log. of 2882 | |
| 02508 (6) | = .02508 (6) × the number (1) of semitones that required string sounds above given string | |
| By subtraction, | 43435 | = log. of 2718 = length of required string in hundredths of an inch |
| ∴ Required length for C-sharp = 27 18⁄100 inches | ||
By reversing the process described above, and adding instead of subtracting, the proper lengths for the semitone below and all others in descending progression may be calculated with accuracy.
Having thus settled the matter of string lengths, we may proceed to consider the questions of diameter. But it is first of all necessary to warn the reader that the lengths that have here been calculated refer only to such pianofortes as are capable, by reason of their size, of taking the ideal string-lengths. Very small uprights, for example, cannot be brought within that classification, except as regards the highest of their strings. In all pianofortes, no matter what their size, the higher strings are practically identical in length; but it will be found that shortness of height in an upright or of length in a grand begins, towards the middle of the scale, disastrously to affect the string proportions. As already pointed out, there are only two ways in which these disproportions can be overcome. These are through alterations in the tension or in the thickness. But such alterations necessarily disturb the whole tonal balance; and here we find a very strong reason for the poor tone that the average atrophied grand or upright possesses. Moreover, it must not be forgotten that disproportionate thickness or unduly slackened tension affect the actual nature of the vibrations that are set up within the string. And the affections are operative both as to frequency and to form. Therefore, naturally, bad tone and inability to stand in tune. This is not intended as an argument against the small pianoforte; but it is desired here to show that these little instruments, whether horizontal or vertical, must not be expected to perform impossibilities. If we are obliged to build small instruments, we must revise our calculations and tabulate the string-lengths according to a different basis of apportionment. For the purpose of the present work, however, the calculations have been made on the assumption that the standard size of pianoforte is to be designed.
Turn we then to the consideration of string diameters. The cast steel wire that is used for the pianoforte strings is supplied in definitely numbered and graded thicknesses. The numbers that are used generally run from No. 13 to No. 24. According to the tests made at the Chicago World’s Fair by the aid of Riehle Bros.’ testing machine, the wire of these numbers was of the following diameters and broke at the following strains. The wire manufactured by the firm of Moritz Poehlmann, Nuremberg, Germany, has been selected from among the various products that were subjected to these tests, on account of its superior durability and evenness of gradation.
| Number | Diameter in fractions of an inch | Broke at strain of |
|---|---|---|
| 13 | .030 | 325 lbs. |
| 14 | .031 | 335 lbs. |
| 15 | .032 | 350 lbs. |
| 16 | .035 | 400 lbs. |
| 17 | .037 | 415 lbs. |
| 18 | .040 | |
| 19 | .042 | |
| 20 | .044 |
Now it is a well known fact, and, indeed, obvious from what has already been said, that the proportional relations as to length, tension, diameter and breaking strain do not permit any other arrangement for the scaling of wire than that which is universally accepted by piano makers. That is to say, the shortest wires are taken from the thinnest numbers, and vice-versa, the whole scaling being so arranged as to secure for each tone that its strings shall be stretched at approximately the same tension. Experience and the observations of the most eminent manufacturers seem to have established that the strain upon each of the uncovered strings should be maintained, as nearly as possible, at 160 lbs. If this be done it will be found that a pianoforte so constructed will produce the proper pitch at each string when the lengths are as calculated in the tables referred to. It will, of course, be necessary to arrange with due proportion the number of strings that are to be taken from the wire of each number. It will be found that the best practice takes into account the half sizes not shown here and strings the instrument with an average of five tones to each thickness of wire, beginning at 13 or 13 1⁄2 and continuing down to the end of the unwrapped strings according to the general directions suggested. Experience and the individual ideas of the designer, assisted by such knowledge as this work aims to impart, are the best guides that can be followed. Empirical induction, based upon observation and experience, provides the only possible and practical means for arriving at the true and proper arrangements to be made for each individual instrument. This empiricism extends with particular force to all string arrangements and is seen nowhere so conspicuously as in the variety of methods that are adopted by manufacturers in determining the number of strings within the unwrapped sections of the scale. Thus, certain makers carry the wrapping over to the beginning of the treble strings and have two or three string-groups provided with wrapped wire before the overstringing is begun. The idea here is either to correct original defects of scale design or to shade down the break in tone that so often occurs at the point where the overstringing usually begins. From observation of the practice of the best makers, it may be said that the tone C below middle C is usually the first overstrung tone. Of course, when the instrument is very small it will often be found that it is impossible to give the last unwrapped strings their proper lengths. In this case these offending strings may either be covered with light wrapping or may be put bodily over into the overstrung portion of the scale, in which latter case they will be wrapped anyway.
Supposing then that the matter of the number of overstrung strings has been determined, we may proceed to the consideration of the dimensions, number and covering of the strings that are to serve here. We are obliged to confess that the problem of attaining to good tone in the bass is, indeed, difficult. It is by no means hopeless, however, as the success of more than one eminent maker has already demonstrated.
The simplest, most obvious, and easiest way out of the inherent difficulties of the scaling of bass strings is to be found in the consideration of their proper lengths. It does not require very much thought to perceive the truth that the longer the strings the less weight need be imposed upon them. If, in fact, we make the bass strings to approach as far as may be to the lengths that they would require to have if unwrapped, we shall be able to reduce proportionally the amount of artificial control that has to be exercised over the vibration speed. Not only this, but the greater length thus attained implies greater tension. That is to say that, as we saw before, the tension at which a string is stretched acts to overcome the slowness of vibration-speed induced by its greater length, and, consequently, tends to generate a more regular progression of the upper partials (as experiment has demonstrated), with resultant tendency to greater purity of tone-quality.
We may, in fact, accept it as an axiom that the bass strings should be as long and, simultaneously, as lightly weighted as possible, and that the weight of them should be strictly proportioned to the pitch of the musical tone that they are desired, at a given tension, to emit. As far as the second clause of these conditions is concerned it is well to remind the reader that limitations of space within the body of a piano usually determine the possible lengths of the bass strings. So much so is this the case, indeed, that it is not often possible to make any great difference in their respective lengths. The best makers appear to be agreed in a method of treating the problem that is at once simple and effective. They recognize the great advantage of scaling the bass strings at the greatest possible length, and then they take care that the descending increase of length is no greater than to make the lowest bass string one-fifth longer than the highest. At the same time they so graduate the weight of the wrapping material that the same results are attained as would naturally follow if they were as accurately scaled, in proportionate length, as are the plain wire strings.