maximum sooner, and then, towards the end, descends in a longer and more gradual slope. The depression of the wave must naturally be symmetrical in form with the elevation.

Thus a perfectly new form of vibration is produced, which, again, makes a particular impression upon our ear. However, the change has not affected the pitch of this combined tone. For it is readily seen that in the time from o to I, a whole period of the new vibration must have been completed. The period of a vibration, and, therefore, the number of vibrations of both tones, is exactly the same, so that the pitch must also have remained the same. But the two notes are different.

We may, therefore, lay down the following law that the quality of a note depends upon the form of the sound

wave.

We may imagine other forms of the sound-wave besides that of c, cg, which have exactly the same periods of vibration. Let us suppose that, in addition to the first harmonic of the fundamental tone, the second were present also, performing three vibrations in the space between 0 and 1. The form of the new vibration would then evidently be more irregular than before. It might be obtained by combining the ordinates of the vibrations of the third harmonic with the vibrations, c, C3. We see that the sums and differences would necessarily always be repeated after each period. The new vibration will, therefore, consist of the same periods and have the same duration. The pitch remains unaltered but the note is again different.

The variety of forms, which a sound-wave can thus assume, is, as we may readily imagine, immense. For

not only is it possible for a still greater number of harmonics to be present, but one or other of the harmonics may be more or less strongly, or only weakly, represented, and lastly this or that harmonic may be altogether wanting, so that groups of harmonics of different kinds are associated with the fundamental tone. These variations always give the same pitch, but all give different notes, which are distinctly perceptible to our ear. Thus we may imagine the fundamental tone to be associated with the second and fourth, or the third and fifth harmonics, etc., and in each case the note will be different.

The difference between the notes of our musical instruments may, therefore, be satisfactorily explained by the fact, that the accompanying harmonics give a peculiar form to the original vibration of the tone, which form is different with each instrument. This form of vibration is only recognised by the ear as a whole, and produces in it the sensation of a note. Science has, however, proved the analysis of this whole into the fundamental tone and the harmonics.

S

CHAPTER VIII.

Analysis of Notes after the Law of Fourier-Helmholtz's Theory of the Perception of Notes - Formation of Notes by Electro-Magnetic Tuning-Forks.

THE observations in the last chapter force the question upon us, whether we are justified in imagining that all periodical sonorous vibrations, which produce a sensation of tone, are formed in such a manner that they may be separated into a fundamental tone and a series of harmonics of different strength.

Let us suppose that we have to do with a very complicated form of sound-wave, for example, that represented in the accompanying fig. 87, which may be drawn quite at will without knowing at all what kind of sound will be produced in our ear. The figure only represents one period of vibration from o to I; the elevation of the wave is quite symmetrical with the depression. It rises rapidly, and, after reaching its maximum, descends somewhat quickly, then for some distance sinks very slowly, falls rapidly once more, and finally, with a gradual inclination, passes into the depression, which is formed in the reverse direction in exactly the same manner as the elevation.

If the wave under consideration is a sound-wave in the air, we may conclude from this form, that the condensation of the air quickly attains its maximum, then

decreases first rapidly, then slowly, then again rapidly, and finally gradually; and that the expansion of the air increases first gradually, then quickly, then again gradually, and, finally, rapidly, to pass once more with great rapidity into the condensation of the second period of vibration. We have to enquire whether it be possible, that such a vibration as this can be composed of a fundamental tone, whose vibrations exactly resemble that of a pendulum, as shown in fig. 86, and of a series of har

I

Fig. 87.

monics, each of which, again, has a form of vibration like that of the pendulum, in which the elevations and depressions, in accordance with a fixed law, rise and fall with the same rapidity.

The problem which we have here to solve is evidently a purely mathematical one. In the first place, we must discover whether a simple curve could be drawn upon the axis of the supposed curve of vibration, which would have the same duration of vibration from o to I, and which would correspond to the fundamental tone. must then try whether the space remaining is sufficient to produce simple curves of half or double, or 3, 4 and 5, etc., times the number of vibrations, which will correspond to the harmonics. We should then, of course,

We

discover how many such pendulum-curves of greater number of vibrations would be necessary for the purpose; what number of vibrations they each would have; and what relation they would bear in pitch, first to each other, and then to the fundamental tone. Finally, the most important point to be observed is, that not a fraction of the surface of the supposed curve of vibration should remain over.

We here have before us a difficult mathematical problem, which fortunately was thoroughly solved by science long before anything was known of harmonics. The distinguished French mathematician Fourier (born 1768) declared:-That a vibration of any form whatever could be separated into a number of simple curves, if only it were repeated in the same period.

The proof of this proposition can only be obtained with the help of higher mathematics. We must, therefore, be satisfied with the assurance, that it may be regarded as an incontestable fact, and proceed to consider more closely its application to the theory of notes, for which we have to thank Helmholtz.

Fourier's proposition proves a remarkable accordance in the production of harmonics. He shows, namely, that when the number of vibrations of the original vibration is one, the number of the first simple vibration is also one, the second two, the third three, the fourth four, and so on. In short, we see that the first simple vibration is the fundamental tone, and that the others represent the series of harmonics. Whilst, therefore, we have been able above to construct a complicated sonorous vibration from a fundamental tone and its harmonics, we can now, by the

of Fourier's proposition, reverse the method, and