THE following observations on the strength of tone of wires, collected, with improvements, from a former work on the Theory of music rather than on the Mechanics, may contain something useful to the constructors of musical instruments. With the same materials and length, and when the difference of diameters is not very great, all wires break at the same pitch ; which is the necessary result of the strength of the wire, and the load required to bring it to the same pitch, both varying as the square of the diameter. If the wire of greatest diameter breaks sooner, it is because it is not treated with equal fairness at the bends ; the remedy for which is to increase the diameter of the pin round which it is bent. If the wires were of very different diameters, the increased resistance of the air on the greater diameter would make the sound somewhat flatter, and consequently require and increased tension to overcome ; which would be to the disadvantage of the thickest.
In wires of different diameter, but the same length and strained to the same pitch, the thickest will have the strongest tone ; because the vibrating area, which is what strikes against the air, will be greater, being as the diameter. In wires of different diameters and lengths, strained to the same pitch, the strength of tine will be as the diameter multiplied by the square of the length, or multiplied by the length twice over. For the diameter multiplied by the length expresses the vibrating area ; and the extent to which the vibrations will be carried laterally, will be as the length again.
Follows, that when the same pitch is produced at different lengths of the same wire by different loads, the longest, and consequently heaviest-loaded, will have the strongest tone. Whence the strongest tones at all pitches, which can be obtained from any wire, are those produced by loading it to the extend of safety, and then taking the lengths which produce the pitches desired. And from the toughest materials may be had the longest wires and strongest tones. A way to judge the toughness of steel wire without reference to its diameter, is to see how many yards of its own kind it will support. Good wire will bear the weight of 14,500 yards.
Follows from the same, that if wires of the same materials but different diameters, are loaded with equal weights, and the lengths taken which produce the same pitch, the longest and consequently the thinnest will have the strongest tone. For since the lengths will be inversely as the diameters, the vibrating areas will be the same in both ; and consequently the longest, as making vibrations of greatest extent, will have the strongest tone. Whence it follows, that if the strain upon the frame is to be of a given amount and no more, the strongest tone with which the pitch in question can be produced with the given strain, is from the thinnest wire which will bear the strain, the length being at the same time increased as demanded. Which agrees with last in showing, that all wires should be loaded to the extent of safety, if length can be afforded ; or else tone is thrown away.
These conclusions bear on the construction of pianofortes to the extent of showing that so far as other considerations fo not interfere, the most advantageous shape and dimensions for what may be called the
The advantage of multiplying wires for the same sound, is that if a given quantity of strain instead of being applied to one wire which it loads to the extent of safety, is divided among four wires of the same length and half the diameter, the pitch and the sum of the strains will be the same as before, but the vibrating areas will be doubles, and consequently the strength of tone increased in something like the same proportion. And with any other number the advantage gained will be as the square root of the number of wires ; which in round terms is with two wires, as 7 to 5, and with three, as 8 2/3 to 5. Hence the wires of the extreme bass ought to be doubled, as increasing the strength of tone in something like the proportion of 7 to 5, with the same total strain.
What is wanted, then , for improving the tone of pianofortes, is, so far as other reasons do not intervene, to increase the length and thickness of the wires. And this will end in a contest for bearing the greatest strain ; the tendency of which will be to make the frame approach to a solid shell of iron, being the form of the original lyre attributed to Mercury, which was the shell of a tortoise.
The same conclusion may be applied to strings of catgut, except that in consequence of the increasingly oblique action of the fibres in the thicker strings, the strain borne without breaking will not increase with equal regularity in relation to the diameter. But the string of catgut will bear to be strained to a higher pitch than a wire of the same diameter and length. For the loads required to bring these to the same pitch are as their specific gravities, which are as 1 to 6 ; while the loads they are able to bear are something like 1 to 3. Hence the wire to produce the final pitch of the catgut must be shortened to about five-sevenths (1 divided by the square root of 2), for no wire will bear straining to that pitch at a greater length ; and to produce the final strength of tone of catgut, the diameter of the wire must be doubled, with four-times the strain which was on it before, making twelve times the strain finally on the catgut. Which explains the superiority of catgut for the instruments in which it is found employed.
Many of these conclusions may be what everybody is acquainted with, but some of them may be new to some...Elliot Vale, Blackheath, Feb. 3, 1858.