CHAPTER XII. SIMPLE AND COMPOUND TONES. 130. IT was discovered by Ohm that a simple harmonic vibratory motion produces the sensation of a simple tone, and that when several simple tones are heard together each of them is due to its own simple harmonic component, which is present in the resultant sonorous vibration. Those musical tones which we call rich are not simple. The sound produced by striking one of the keys of a pianoforte is usually composed of some four or five simple tones, due to the co-existence in the wire of so many different modes of simple harmonic vibration. The tones of a violin are still more highly compound. We have pointed out in § 112 that the periods of the several modes of simple harmonic vibration of a string are proportional to 1,,,, &c., or, what amounts to the same thing, that the numbers of vibrations made in a given time are proportional to the series of natural numbers 1, 2, 3, 4, &c. When the partial tones which together compose a compound tone have this relation to one another, the deepest of them—that which corresponds to the number 1-is called the fundamental, and the others are called harmonics of the fundamental. These terms are applied both to the partial tones themselves and to the simple harmonic component vibrations to which they are due. It was proved mathematically by Fourier that every periodic vibration—that is, every vibratory motion which exactly repeats itself in a definite period-can be resolved into a fundamental simple harmonic vibration and its harmonics. Over against this mathematical fact we may set the acoustic fact that every musical tone of definite pitch is composed of a simple tone of this pitch and its harmonics. The connection between these two facts is explained by Ohm's principle above mentioned. 131. The human ear is not by any means the only instrument that picks out the simple harmonic constituents from a compound vibratory motion. Strings mounted on a sounding-board, as in the pianoforte, will do the same thing. When the pedal is depressed, so as to remove the dampers from the wires, if a compound tone is sung or otherwise sounded in its neighbourhood, those strings which correspond to the partial tones present in the sound will be thrown into vibration, and thus the piano will echo back a compound sound very similar in its constitution to the original. To ascertain whether a given tone is present in the original sound, the pedal should not be depressed, but the key corresponding to this particular tone should be held down. In general, any body which can vibrate freely in one definite period will be set in vibration if acted on by a periodic movement which, when analysed into simple harmonic constituents, contains one whose period agrees with that of the body in question. 132. Helmholtz's resonators, one of which is represented in Fig. 48, are intended to assist the ear in detecting the presence or absence of a given elementary tone in a compound sound. They are hollow globes of brass, with two openings; the smaller one is to be applied to the observer's ear, while the larger one is directed towards the source of sound. Each resonator corresponds to one definite simple tone, whose period is the same FIG. 48. as the natural period of vibration of the body of air enclosed within the globe; and when this tone is present the resonance of the enclosed air produces a great increase in its intensity, so that the observer hears the resonator speak into his ear. These resonators are usually supplied in a series of ten, corresponding to the bass C of a man's voice and its first nine harmonics. When the bass C is sung, all or nearly all the resonators are observed to respond, thus proving the composite character of the tones of the human voice. 133. The same principles are illustrated, synthetically instead of analytically, in the 'mixture' stop of organs. When this stop is employed in conjunction with those called 'principal' and 'diapason,' the pipes which are brought into use are so combined that on putting down any key the corresponding note and a series of its harmonics are all sounded at once, each by a separate pipe. The effect, as judged by an ordinary ear, is that of a single rich note of the pitch of the fundamental. CHAPTER XIII. MUSICAL INTERVALS. 134. WHAT is called in music the pitch of a note depends upon the period of vibration, or upon the number of vibrations in a given time. A note of high pitch is a note of short period, or of a large number of vibrations per second. In comparing two notes, the interval between them depends only on the ratio of the two periods, or (the reciprocal of this) the ratio of the numbers of vibrations made in a given time. When the ratio is that of 1: 2 the interval is called an octave; when it is 23 the interval is called a fifth; when it is 3 4 it is called a fourth; when it is 4: 5 it is called a third, and so on. The origin of these names is due to the fact that, in the notes of the ordinary major scale, the numbers of vibrations are proportional to the following numbers: Do Re Mi Fa Sol La Si Do 24 27 30 32 36 |