But these centrifugal forces are bal anced by the components of the ten sions at the extremities, in directions perpendicular to the horizontal • line; which are, by hypothesis, the same for all the figures. Hence the time of ro tation is directly as the length of each segment. Now (see PENDULUM) any such rota tion is equivalent to two mutually perpendicular and independent pendulum vibrations of the cord from side to side of the horizontal line. Thus, a violin string may vibrate, according to the pendulum law, in one plane, either as a whole (fig.1), as two halves (fig. 2), as three thirds (fig. 3), etc.; and the times of vibration are respectively as 1, . . . Nay, more, any two or more of these may coexist in the same string, and thus, by different modes of bowing, we may obtain very different combinations of simple sounds: a simple sound being defined as that produced by a single pendulum motion, such as that of a tuning-fork, or one of the uncomplicated modes of vibration of a string.
The various simple sounds which can be obtained from a string are called harmonics of the fundamental note; the latter being the sound given by the string when vibrating as a whole (fig. 1). For each vibration of the fundamental note, the harmonics have two, three, four, &c. Of these, the first is the octave of the fundamental note; the second the twelfth, or the fifth of the octave; the third the double octave; and so on. Thus, if we have a string whose fundamental note is C, the series of simple sounds it is capable of yielding is: 0, Q., Ga, C2, E2, 0z 0350, CS, Ds, Es, &C.
Of those written, all belong to the ordinary musical scale except the seventh, which is too flat to be used in music. This slight remark shows us at once how purely artificial is the theory of music, founded as it is, not upon a physical, bnt on a sensuous basis.
To produce any one of these harmonics with ease from a violin string, we have only to touch it lightly at 4-, Sic, of its length from either end and bow as usual. This process is often employed by musicians, and gives a very curious and pleasing effect with the violoncello or the double-bass. The effect of the finger is to reduce to rest the point of the string touched; and thus to make it a point of no vibration, or, as it is tech nically called, a node.
In the case of a pianoforte wire, a blow is given near one end, producing a displace-, meat which runs back and forward along the wire in the time in which the wire would vibrate as a whole. The successive impacts of this wave on the ends of the wire (which are screwed 1,o the souudiug-board), are the principal cause of the sound. But more of this case later.
The theory of other musical instruments is quite as simple. Thus, in a flute, or unstopped organ pipe, the sound is produced by a current of air passing across an orifice at the closed end. This produces a wave which runs along the tube, is reflected at the
open end, runs back, and partially intercepts the stream of air for an instant, and so on. Thus the stream of air is intercepted at regular intervals of time, and we have the same result as in the sirene (q. v.). In this case, there is one node only, viz., at the middle of the pipe. If we blow more sharply, we create two nodes, each distant from an end by I of the length of the tube. The interruptions are now twice as and we have the first harmonic of the fundamental note. And so on, the series of harmonics being the same as for a string. We may easily pass from this to the case of an organ-pipe closed at the upper end. For if, while the open pipe is sounding its fundamental note, a diaphragm be placed at the node, it will not interfere with the motion, since the air is at rest at a node. That is, the fundamental note of a closed pipe is the same as that of an open pipe of double the length. By examining the other cases in the same way, we find that the numbers of vibrations in the various notes of a closed pipe are in the pro portions 1:3:5: 7: &c., the even harmonics being wholly absent.
There.is another kind of organ pipe, called a reed pipe, in which a stream of air sets a little spring in vibration so as to open and close, alternately, an iu the pipe. If the spring naturally vibrates in the time corresponding to any harmonic of the pipe,.
that note comes out with singular distinctness from the combination—just as the sound of a tuning-fork is strongly re-enforced by holding it over the mouth-hole of a flute which is fingered for the note of the fork. • If the spring and the tube have no vibration in common, the noise produced is intolerably discordant. The oboe, bassoon, and clari onet are mere modifications of the reed-pipe; and so are horns in general, but in them the reed is supplied by the lip of the performer. Thus, a cornet, a trumpet, or a French horn, gives precisely the same series of harmonics as an open pipe.
The statements just made as to the position of the nodes in a vibrating column of air are not strictly accurate, for the note is always found to be somewhat lower than that which calculated from the length of the tube and the velocity of sound. Hopkins showed experimentally that the distance between two nodes is always greater than twice the distance from the open end to the nearest node. The mathematical difficulties involved in a complete investigation of the problem were first overcome by Helmholtz in 1859, in an admirable paper published in Orelle's Journal. The results are found to be in satisfactory accordance with those previously derived from experiment.