350 Helmholtz's Resonators. them to the string by a succession of small impulses, until they become sufficiently large to be audible or even visible. It is only in the string that vibrates at the same rate as the sounding-board, that the effect of these impulses is cumulative; in the other strings, there is interference and confusion between the pulses of the board and those of the string. Any elastic body capable of vibrating readily may thus be thrown into resonant or sympathetic vibration. Bell-shaped glasses can be thrown into violent motion, it is said even shivered, by a very powerful and fine voice singing their proper tone into them or near them. So a tuning-fork at one end of a room will respond to another of the same pitch, struck or sounded by a violin-bow at the other end of the room. But the least difference of pitch destroys the resonance, as may be proved by fixing a little wax to one of the legs of the resounding fork. In like manner a stretched india-rubber membrane may be found to answer the note which accords with its natural period of vibration. A column of air is an exceedingly sensitive resonator. If a tuningfork be struck and held over a tall jar, it will be found on slowly pouring in water into the jar that for a certain depth of the water-surface, the tone of the fork is resounded, greatly strengthened, by the jar (see fig. 151). A different pitch of tuning-fork would have a different length of resonant air-column corresponding to it; so that each note has its own resonant air-column which will respond to it when sounded. Fig. 151. A 528. It is an application of this principle that Professor Helmholtz has made use of for the analysis of harmonics. His resonant air masses, or resonators, are formed by glass or brass vessels of the shapes A, or B, in fig. 152. They are open at both ends, the smaller end being inserted in the ear. A series of such resonators. is constructed so as to resound each to a note of different pitch. It is evident then that, since the inclosed aircolumn will respond only to a definite note, the corresponding note may be singled out from any mixture of notes and isolated, as effectively as the chemist singles out by precipitation any ingredient B Fig. 152. Artificial detection of Harmonics. 351 he chooses from a mixture of indistinguishable substances. Faint overtones, which would otherwise fail to be distinguished, are thus isolated and their pitch determined. "The proper tone of the resonator may even be sometimes heard cropping up in the whistling of the wind, the rattling of carriage-wheels, or the splashing of water." The proper tone of a resonator held to the ear during the playing of a piece of harmonised music, will be heard rising out of the mass in bold contrast. By means of a series of such tuned resonators every note of the piano may be analysed or resolved into a number of separate tones; and it will be found that the lower tones are more complex than the higher. Also on examining by these means the C sounded on the middle key of the piano, the same C sounded on a plate, a violin, a harmonium, &c., we should find that they all differ in one or more of three respects, namely :— (1.) In the number of overtones present; or, (2.) In the order of those present; or, (3.) In their relative intensities. The possible combinations of harmonics which may thus be created are practically infinite in number. It has been calculated that with six overtones and two shades of intensity for each, we may thus have more than 400 shades or qualities of tone. The musical sounds of strings and stringed instruments. 529. The rate of vibration in strings increases with their shortness, lightness, and tension: for if a string be long or heavy, there is a greater mass of matter to be moved than in one short or light, and thence a slower motion; and if a string be slack, the force of clasticity which pulls it from any deviation back to the straight line will be so much the less. The facts are, that a string taken of half the length, or of one-fourth the weight, or of quadruple the tension of another string, vibrates just twice as fast on any one of these accounts; a string of one-third the length, or of one-ninth the weight, or of nine times the tension of another, vibrates three times as fast; and so on for other proportions. These truths are familiarly illustrated in the violin. The low or bass string is thick and heavy, being covered with wire, and the others gradually diminish in magnitude and weight, up to the smallest or treble. The strings are tuned to each other by being attached by one end to movable pins, which, when turned, in 352 The Harmonics of a Sounding String. crease or diminish the tension; and the sound produced by each is afterwards varied to a certain extent by the performer pressing different parts of it with the finger against the board, so as to shorten or lengthen the vibrating portion. An analogous law, as to the influence upon tone, of weight and dimensions, holds with respect to bells, glasses, reeds, &c. 530. If a long musical string be made to sound, and then only half of it be made to sound, as when a movable bridge is placed under the middle, or a finger presses it there, the half will sound the note which is the octave to the first, and will therefore vibrate twice as fast; and similarly the third part sounded, will give the fifth to the last-note, and will vibrate three times as fast as the first; a fourth part, four times as fast; and so on, producing the sounds or tones thus nearly related to each other. Thus, if by means of a bridge we divide the string into two parts whose lengths are in the ratio of any of the notes of the scale, we shall obtain the two notes by sounding the two parts of the string. It was in this way that the ancients discovered the simple numerical ratios connecting the different notes of the scale. The string of a violoncello, when made to vibrate by a bow moved very gently across it, near the bridge, often divides itself spontaneously into two, three, or four, &c., equal vibrating portions, with points of rest between them called nodes. When this happens, there are heard in succession, or even together, not only the sound or note belonging to the whole length of the string, but also, more feebly, the subordinate notes belonging to its half, third, or fourth, &c., that is the whole series of harmonics as explained above. Often in such a case the subordinate sounds swell with such force as to overpower for a time the fundamental note. The same harmonic sounds may be produced still more certainly, while drawing the bow across the string, by touching the string lightly with the finger or with a feather at one of the points where we wish it to divide. Even a varied air may be thus played by the harmonics only. The sounds belonging to a single cord or string, and produced by its spontaneous division into different numbers of equal parts, constitute, when heard together or in succession, what may be called a simple music or nature herself. It is produced pleasingly, as just described, by the single string of a violoncello, but also in a very interesting manner by the instrument called the Æolian harp. 531. The Æolian harp is a long box or case of light wood, with Laws of Vibrating Strings. 353 harp or violin strings extended on its face. These are generally tuned in unison with each other, or to the same pitch, except one which is thicker than the others, and vibrates only half as fast, giving the lower octave to the others, and serving as a bass. When the harp is suspended among trees, or is placed in any situation where the fluctuating breeze may reach it, near a window partially opened, for instance, each string, according to the manner in which it receives the blast, sounds either entire, or breaks into some of the simple divisions above described; the result of which is the production of a pleasing succession of musically-related sounds. After a pause this fairy harp may be heard beginning with a low and solemn note, like the bass of distant music in the sky: the sound then swells as if coming near, and other tones break forth, mingling with the first, and with each other. In the combined and varying strain, sometimes one clear note predominates and sometimes another, as if single musicians alternately led the band: and the concert often seems to approach and again to recede, until with the unequal breeze it dies away, and all is hushed again. 532. It is to be remarked that the mere vibrations of the strings alone would be inaudible but for their connection with the extended surface of the box. The elastic wood is capable of taking up the vibrations of the strings, and by its broad surface throws a large mass of air into pulsations, and thus gives effect to the vibrations, or renders them audible. Hence the quality and value of all stringed instruments depend not on the strings, but on the proper adaptation of the elastic resonator to the production of their vibrations. In the piano, the harp, the violin, &c., everything depends on the perfect elasticity of the sounding-board. 533. The vibrations of strings fixed at their two ends may be experimentally studied by means of a long india-rubber tube, or a long spiral of fine brass wire, or even a long flexible rope. First, on swinging the rope as in AA′ (fig. 153), we find that it executes its vibrations in a definite period, depending merely on the length and weight and elasticity of the rope or tube, or wire helix. Next, on tilting up the end, A, sharply, we raise a hump or wave which will pass on to the other end, A', in the same time as the whole string, AA', takes to make one vibration. As it cannot pass the end A', the wave is reflected there, and returns towards A with its former motion reversed, till on arriving at A it is once more reflected, and passes on towards A' as at first, these reflections continuing till at last all the motion is destroyed. 354 Ventral Segments in Sounding Strings. If now we tilt up half the rope, BB', at one time, and keep tilting it at regular intervals, then, when the first wave is being reflected from B', a second is just starting from B; these will obviously meet at N in the middle of the rope, and as this point is solicited in two opposite directions at once, it will remain at rest, and the string will in consequence keep vibrating as if each half were independent of the other. The point, N, is called a node; and the two bulging pieces of the string are termed ventral segments. In like manner the string may, with a little care, be divided into three ventral segments, with two nodes, as in CC'; or into four ventral segments, with three nodes, as in DD', the divisions being distinctly visible. It is quite easy to throw a sounding string, such as a long string of cat-gut stretched over a sounding box, into ventral segments, by lightly touching with the finger, or with a feather, at some exact division, such as a fourth or a fifth of the whole string, and then sounding this part in the ordinary way with a fiddle-bow. 534. A simple method of shewing these ventral segments is to P W Fig. 154. fasten a white silk or light cotton string to one prong of a tuningfork, the tension cf the string being regulated by weights, w, in a |