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But if any repetition be made, either by periodic shocks upon a hard substance whose agitations produce agitations in the air, or by periodic interruption or modification of a current of air,-provided that such shocks or interruptions be similar in character and uniform in interval of time, and provided also that the frequency be included within certain very wide limits (from about 30 in a second of time to about 10000 in a second, or even through a wider extent),—then a musical note is produced. The more rapid is the succession of shocks, &c. the higher is the pitch of the note.

It is said that Galileo first remarked the production of a musical note by the repetition of unmusical shocks, on passing a pen rapidly upon the milled edge of a coin, which made a small snap at every roughness. The experiment has often been repeated by snaps of a quill upon the teeth of a wheel in rapid motion. But the instruments which give the most satisfactory evidence of the truth of the assertion are the following:

(1) The tuning-fork, described in Article 49. In every elastic metal, when it is disturbed from a form of rest, in a definite manner, the force tending to restore it. to its original state is exactly proportional to the extent of displacement; a state of things represented by the A. =- 2; and then the law of

differential equation

d'z
dt

displacement as connected with the time is accurately a law of sines; the solution of the equation being z = B. sin (At + C).

Here it is certain that the vibrations of air which are communicated to the ear follow the simple law of sines, but the rapidity of the vibrations is not ascertained. The note of the tuning-fork strikes the ear as being remarkably pure.

(2) The Siren. Suppose a flat disk, pierced with a great number of holes at equal distances round its circumference, to be so placed that the nozzle of a bellows can blow directly through any of the holes when by the rotation of the disk the hole is brought under the nozzle; and suppose that the disk is made, by clock-work, to rotate with a great speed which is registered by the clock-dials. Here we have a current of air interrupted very frequently, and at proper speed a powerful and sweet musical note is produced. The power is increased if, instead of having a single outlet of air, a plate similar to the rotating disk and having the same number of holes is firmly fixed near it, and the air is driven through these holes; so that, instead of a single current of air frequently created and interrupted, there are a great number of simultaneous currents of air frequently created and interrupted. By observing the character of the note produced, as known to musical ears, and by registering the number of current-interruptions, it is found that corresponding to the note c, which is that of the white key on the left of the two black keys usually next on the right of the lock of a pianoforte, and which note is thus written

the number of current-interruptions is, in modern music, 528 in a second of time. (On variations of this number we shall speak hereafter.)

(3) The Reed. This instrument, in its ruder form, (Figure 12), consists of a small pipe inserted in a larger pipe into which air is driven; the only outlet for the air in the large pipe being through the small pipe; and the only way by which air can enter the small pipe being by a long aperture that is closed by a thin plate or tongue of elastic metal which has a tendency to stand slightly open, leaving a narrow opening opposed to the incoming current of air. As soon as the current is strong, it claps the tongue close; the elasticity of the tongue opens it; it is clapped again, &c. The times of vibration of the tongue are uniform (depending on its elasticity), and a musical note, of rather harsh character, is produced. In the more refined Reeds, the tongue vibrates through an aperture without touching the sides, and then produces a sweeter musical note; this is the construction used in all instruments of the class of the harmonium.

(4) The Monochord, or single stretched string. This has usually been made, for experimental purposes, as a single wire, fixed at one end, passing over two bridges, and stretched by a weight at the other end

When it is wished to keep the wire in a horizontal position, the wire may either be led over a pulley, or may be attached to one arm of a rectangular lever, the other arm carrying the weight. But, if a vertical position is admissible, it is sufficient to suspend a weight freely to it (as the vibration of the weight corresponding to a small vibration of the string is, in practice, quite insensible), care being taken that the wire is tightly nipped at the top and bottom. When such a wire is plucked aside and allowed to vibrate, it gives a musical note: the pitch of the note does not depend on the place of plucking it, but the quality of the tone does depend on it. Upon measuring carefully the length of the wire, the weight of the wire, and the weight which stretches it, the number of vibrations made in a second of time can be computed (the theory of this will be given below). If the extending weight or the length of the string be altered by trial till the string gives a definite note, for instance, the C mentioned above, then it is found that the calculation gives the same number of complete vibrations (a motion backwards and a motion forwards being understood to mean one complete vibration) as the number of passages and interruptions of air in the experiment with the Siren.

All these experiments prove that the formation of a musical note depends on the repetition of similar disturbances of the air at equal intervals of time; but only those of the Siren and the Monochord give the means of computing the frequency of vibrations for an assigned musical note.

72. The quality of a musical note is determined by the form of the function which expresses the atmospheric disturbance.

In

The investigations of Articles 21, &c., 44, &c., 50, &c., have given us expressions for the displacement of the particles of air in the propagation of sound, in all cases represented by (at − x) or by a multiple of (at — x), where the form of the function is undetermined. the Partial Differential Equations, Article 22, it is shewn that, algebraically, the function & may be in any way discontinuous; and in Article 28, above, are explained the only limitations that physical considerations appear to impose on the generality of discontinuity. We have now another limitation, namely, that the function must be periodical (producing similar disturbances of the air after the repetition of equal intervals of time). With all these limitations, however, it will be seen that there is a very great range in the variations which may be given to the form of the function. But the condition of periodicity gives great facility for the consideration or the determination of the form. Putting v for at-x, and supposing that the equal values of (2) return when v is increased by λ, 2λ, &c., it will be seen that such a function may be represented to any degree of approximation by a series of terms, such as

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