The importance of employing proper sounding apparatus in stringed instruments is rendered manifest by
Take the violin as an example. It is, or ought to be, formed of wood of the most perfect elasticity. Imperfectly elastic wood expends the motion imparted to it in the friction of its own molecules; the motion is converted into heat, instead of sound. The strings of the violin pass from the “tail-piece” of the instrument over the “bridge,” being thence carried to the “pegs,” the turning of which regulates the tension of the strings. The bow is drawn across at a point about one-tenth of the length of the string from the bridge. The two “feet” of the bridge rest upon the most yielding portion of the “belly” of the violin, that is, the portion that lies between the two f-shaped orifices. One foot is fixed over a short rod, the “sound post,” which runs from belly to back through the interior of the violin. This foot of the bridge is thereby rendered rigid, and it is mainly through the other foot, which is not thus supported, that the vibrations are conveyed to the wood of the instrument, and thence to the air within and without. The sonorous quality of the wood of a violin is mellowed by age. The very act of playing also has a beneficial influence, apparently constraining the molecules
This is the place to make the promised reference (page 38) to Prof. Stokes’s explanation of the action of sound-boards. Although the amplitude of the vibrating board may be very small, still its larger area renders the abolition of the condensations and rarefactions difficult. The air cannot move away in front nor slip in behind before it is sensibly condensed and rarefied. Hence with such vibrating bodies sound-waves may be generated, and loud tones produced, while the thin strings that set them in vibration, acting alone, are quite inaudible.
The increase of sound, produced by the stoppage of lateral motion, has been experimentally illustrated by Prof. Stokes. Let the two black rectangles in [Fig. 34] represent the section of a tuning-fork. After it has been made to vibrate, place a sheet of paper, or the blade of a broad knife, with its edge parallel to the axis of the fork, and as near to the fork as may be without touching. If the obstacle be so placed that the section of it is A or B, no effect is produced; but if it be placed at C, so as to prevent the reciprocating to-and-fro movement of the air, which tends to abolish the condensations and rarefactions, the sound becomes much stronger.
§ 2. Laws of Vibrating Strings
Having thus learned how the vibrations of strings are rendered available in music, we have next to investigate the laws of such vibrations. I pluck at its middle point the string B B′, [Fig. 31]. The sound heard is the fundamental or lowest note of the string, to produce which it swings, as a whole, to and fro. By placing a movable bridge under the middle of the string, and pressing the string against the bridge, it is divided into two equal parts. Plucking either of those at its centre, a musical note is obtained, which many of you recognize as the octave of the fundamental note. In all cases, and with all instruments, the octave of a note is produced by doubling the number of its vibrations. It can, moreover, be proved, both by theory and by the siren, that this half string vibrates with exactly twice the rapidity of the whole. In the same way it can be proved that one-third of the string vibrates with three times the rapidity, producing a note a fifth above the octave, while one-fourth of the string vibrates with four times the rapidity, producing the double octave of the whole string. In general terms, the number of vibrations is inversely proportional to the length of the string.
Again, the more tightly a string is stretched the more rapid is its vibration. When this comparatively slack string is caused to vibrate, you hear its low fundamental note. By turning a peg, round which one end of it is coiled, the string is tightened, and the pitch rendered higher. Taking hold with my left hand of the weight w, attached to the wire B B′ of our sonometer, and plucking the wire with the fingers of my right, I alternately press upon the weight and lift it. The quick variations of tension are expressed by a varying wailing tone. Now, the number of vibrations executed in the unit of time bears a definite relation to the stretching force. Applying different weights to the end of the wire B B′, and determining in each case the number of vibrations executed in a second, we find the numbers thus obtained to be proportional to the square roots of the stretching weights. A string, for example, stretched by a weight of one pound, executes a certain number of vibrations per second; if we wish to double this number, we must stretch it by a weight of four pounds; if we wish to treble the number, we must apply a weight of nine pounds, and so on.
The vibrations of a string also depend upon its thickness. Preserving the stretching weight, the length, and the material of the string constant, the number of vibrations varies inversely as the thickness of the string. If, therefore, of two strings of the same material, equally long and equally stretched, the one has twice the diameter of the other, the thinner string will execute double the number of vibrations of its fellow in the same time. If one string be three times as thick as another, the latter will execute three times the number of vibrations, and so on.