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

TABLE SHOWING TRUE LENGTHS OF OCTAVE STRINGS FROM THE HIGHEST C STRING TO THE LAST C STRING THAT IS USUALLY LEFT UNWRAPPED.

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,

(1) Take the logarithm of the length of the known string.

(2) 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.

(3) 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.)

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 " 15 .032 350 " 16 .035 400 " 17 .037 415 " 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.

This equalization is, of course, only approximate. For the forms of vibration excited in two strings of the same pitch will be different whenever the various factors that govern the emission of sound by them are variable. Thus when the factor of length is varied, no counter-adjustment of tension, thickness or density can restore to the string so modified the exact form of vibration that it may have originally possessed. Consequently it becomes impossible to induce from artificially weighted strings precisely the same series of partial tones that a plain wire filament will emit, even when the tones generated by the two strings are of the same pitch.

The lesson of this is plain. As perfection of tonal quality can only be attained in part, it especially behooves us to pay strict attention to such scaling of the bass strings as will furnish a complement of sound producing agencies that may be relied upon to induce as nearly as possible the same successions of partials as are habitually emitted throughout the higher sections of the piano. Thus it becomes evident that the greatest practicable length and the least practicable weight are the chief factors that must govern the designer in laying out the scale for the bass strings.

The relative densities of the wrapping material employed in the manufacture of bass strings have been the subject of considerable study. Brass, which was the earliest object of experiment, has long been superseded by either copper or iron. As to the relative advantages possessed by these two materials, it can be said at once that the chief and almost the only advantage presented by the latter lies in its relative cheapness. Acoustically, however, copper forms by all means the most suitable material for the winding of bass strings, and this for the following reasons: The specific gravity of copper is 8.78, while that of iron is but 7.78. Again, the former metal, while inferior in tenacity to the latter, possesses, on the other hand, the great advantage of higher ductility, so that its elastic qualities are very marked. It is thus evident that copper is a more suitable material for the generation of musical sound than is iron, and the qualities which we have just noted as pertaining to it are precisely those most useful in the production of harmonic progressions of partial tones. It is therefore clear that as between copper and iron all the advantages lie with the former.

The thickest wire used for the uncovered strings is generally No. 24. In beginning the scaling of the bass strings, however, we choose No. 17 or No. 18 for the notes nearest to the treble. The covering is usually from No. 25 to No. 28 (standard, not music, wire gauge) according to the size of the piano and the practicable string-length. Of course, longer strings may be covered with lighter wire. The first covered string is generally approximately one-sixth shorter than the string immediately above it. This proportion, as suggested above, may, however, be profitably disregarded, if it thereby be possible to lengthen the bass strings. There are always two of these strings to each tone and the thickness of covering wire must be progressively increased as the scale descends. A descending increase of one number in thickness of the covering wire for each pair of strings may properly be allowed, unless the lengths are too closely alike, or vice-versa, in which cases suitable modifications may be made. But assuming that the descending lengths are arranged in arithmetical progression with a mean of 3/4 of an inch, and supposing the highest covered string to be 45 inches long; then the suggested increase of thickness should under all circumstances hold good. It may often be found, however, that the space limitations of an instrument or other practical considerations make it impossible to follow out these rules with exactitude. In any case, we must remember that all such rules are themselves the fruit of empirical observation and to such observations we must look, when it becomes necessary to revise them in order to satisfy the requirements of some

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