the same kind of reasoning, that if two forks have their times of vibration nearly as 1:2, 2:3, &c.i. e., any simple numerical ratio-there will be greater intervals between the beats according as the exact ratio is more nearly arrived at.

We must now consider, so far as can be done by elementary reasoning, the various simple modes of vibration of a stretched string, such as the cord of a violin. Holding one end of a rope in the hand, the other being fixed to a wall, it is easy (after a little practice) to throw it into any of the following forms,

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

the whole preserving its shape, but rotating round the horizontal line. If the tension of the rope be the same in all these cases, it is easy to see that the times of rotation must be inversely as the number of equal segments into which the rope is divided; for the various parts will obviously have the same form; and the masses and distances from the axis of rotation being proportional to their lengths, the Centrifugal Forces (q. v.) will be as the squares of the lengths, and inversely as the squares of the times of rotation. But these centrifugal forces are balanced by the components of the tensions at the extremities, in directions perpendicular to the horizontal line; which are, by hypothesis, the same for all the figures. Hence the time of rotation is directly as the length of each segment. Now (see PENDULUM) any such rotation 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), &c.; and the times of vibration are respectively as 1, 4, 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 tuningfork, 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:

C, C1, G1, C2, E, G2 (Bb2), C3, D3, E3, &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 shews us at once how purely artificial is the theory of music, founded as it is, not upon a physical, but 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,,, &c. 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 technically called, a Node.

In the case of a pianoforte wire, a blow is given near one end, producing a displacement 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 to the sounding-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 Sirène (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 of the length of the tube. The interruptions are now twice as frequent, 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 proportions 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 opening in 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 tuningfork is strongly reinforced 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 Clarionet 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 is calculated from the length of the tube and the velocity of sound. Hopkins shewed 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 Crelle's Journal. The results are found to be in satisfactory accordance with those previously derived from experiment.

We have now to consider the subject of the quality of musical sounds; and one of its most important branches, what constitutes the distinction between the various vowel-sounds. It had long been recognised that the only possible cause of this distinction between sounds musically identical must lie in the nature of the fundamental noise, or, to express it differently, the nature of the periodic motion of each particle of air. But it appears that Helmholtz was the first to enter upon a complete examination of the point, both mathematically and experimentally, and the results he has arrived at form by no means the least remarkable of the contents of his excellent work, Die Lehre von den Tonempfindungen, recently published.

It was established by Fourier, that any periodic expression whatever may be resolved into the sum of a number of simple harmonic terms, whose periods are, respectively, that of the original expression, its half, its third part, &c. Hence any periodic motion of air (i. e., any musical sound) may be resolved into a series of simple pendulum vibrations (i. e., pure musical sounds, such as those of tuning-forks), the first vibrating once in the given period, the second twice, and so on. These notes are, as we have seen, the several harmonics of the lowest. Hence the quality of a musical sound depends upon the number and loudness of the harmonics by which it is accompanied.

Two experimental methods were employed by Helmholtz, one analytical, the other synthetical. In the first he made use of resonance cavities fitted to the ear, and giving scarcely any indication of external sounds until one is produced which exactly corresponds in pitch with the note which the cavity itself would yield. With a series of such cavities, tuned to the several harmonics of some definite note, the note was examined when played on various instruments, and when sung to different vowel sounds. It was thus ascertained which harmonics were in each case present, and to what extent, producing the particular quality of the sound analysed. The second method was founded on the fact, already noticed, that a tuning-fork gives an almost pure musical sound (i. e., free from harmonics). A

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series of tuning-forks, giving a note and its harmonics, were so arranged as to be kept constantly in vibration by an electro-magnetic apparatus. Opposite to each was fixed a resonance-cavity exactly tuned to it, and capable of being opened more or less at pleasure. When all the cavities were shut, the sound was scarcely audible; so that by opening them in various ways, any combination of harmonics might be made to accompany the fundamental note. These combinations were varied by trial, until the quality of the resultant sound was brought to represent as nearly as possible that of some vowel. The results of this second series of experiments coincided with those of the first. It appears from these investigations that the German U is the quality of a simple sound, though it is improved by adding faintly the two lowest harmonics; that O depends mainly on the presence of the third harmonic; and so on with the other sounds. It also appears, and it is well known by experience, that different vowel-sounds, to be sung with accuracy, require to be sung to different notes, the proper note being that for which the cavity of the mouth is adapted for the production of the accompanying harmonics which determine the quality of the particular vowel.

In strings and pipes, as we have seen, the higher notes are strictly harmonics of the fundamental note, and therefore the sounds of instruments which depend on these simple elements are peculiarly adapted for music. On the other hand, when, as in masses of metal, &c., the higher notes are not harmonics of the fundamental note, the mixed sound is always more or less jarring and discordant. Such is the case with bells, trumpets, cymbals, triangles, &c.; and, in fact, these sounds are commonly characterised as 'metallic.' To produce from such instruments a sound as pleasing as pos sible, they must be so struck that as few as possible of the higher notes are produced, and these as feebly as possible. Thus, for instance, to get the most pleasing sound from a pianoforte-wire, it should not be struck at the middle, as in such a case the first, third, fifth, &c. harmonics of the fundamental note will be wanting. If, however, it be struck at about 4th of its length from one end, the harmonics produced will be mainly the first five; and these all belong to the chord of the fundamental note, so that the sound produced is rich and full.