CHAP. VII. Origin and Progress of Acoustics. THE term acoustics, unknown to the ancients, has been invented by the moderns, to denote in a succinct mauner that part of mathematics, which considers the motion of sound, the laws of it's propagation, and the relation which different sounds bear to each other. There is a striking analogy between acoustics and optics, both in their theory, and in the instru ments by which the sight and hearing are assisted. The air is the vehicle of sound. When a sonorous body is struck, it trembles, and makes vibratory motions, which it communicates to the circumambient air; and this fluid transmits them by successive undulations, arising from it's elasticity, to the tympanum of the ear; a kind of drum, at which the auditory nerve terminates. The more dense and elastic the sonorous body is, and the more violently it is agitated, the more fulness and strength has the sound which is produced. A series of sounds, succeeding each other unequally and without order, forms merely a simple noise, which is often very disagreeable. But when the sounds have measured intervals, and their relations to each other are subjected to constant and regular laws, the result is a harmony, a modulation, pleasing to the ear. This is the source of that pleasure, which all nations receive from music. On a mutual comparison of two sounds, one is more grave, one more acute than the other. This difference arises from the greater or smaller number of vibrations, which the sonorous body makes in a given time. Take, for instance, two strings of a violin, equal in thickness, and equally tense, but of which one is double the length of the other; and draw them both out of the rectilinear position, so as to make them vibrate. In this case, while the shorter string makes two vibrations, that of double it's length will make but one; and the sound of the former will be acute, that of the latter grave. say likewise, that one is an octave to the other, because they constitute the two extremes of eight notes in the musical gamut. If the tension of the two strings be greater or less, but still equal in both, the sounds produced will be proportionately more or less loud, but they will bear the same relation to each other. 5 4 3 2 3 We If you would obtain the ratios of the eight musical. notes, you have only to take eight strings, equally tense, of equal thickness, and the lengths of which shall be to each other as 1, 1⁄2, † † † † The number of vibrations, which these eight strings will make in a given time, will be reciprocally proportionate to the preceding numbers; and you will hear the fundamental or gravest note, the minor third, the major third, the fourth, the fifth, the minor sixth, the major sixth, and the octave. The The same ratios may be obtained by means of a single string, by giving it different degrees of tension, so that the forces of tension shall be as the numbers 1, 1, 75, 5, 2, 24, 25, 4. 36 25 9 6 All these proportions, and several others, spring from the following theorem: The number of vibrations, made by a string in a given time, is generally as the square root of the weight by which it is stretched, divided by the product of the weight of the string multiplied by it's length. Though this theorem was invented by the modern mechanists, I thought it proper to introduce it here, as it will enable us to appreciate the experiments ascribed to Pythagoras, the author of the first discoveries made on the subject. Nicomachus, an ancient writer on arithmetic [A. c. 400], relates, that Pythagoras, passing one day by a blacksmith's shop, where the workmen were hammering a piece of iron on the anvil, was surprised to hear sounds, which accorded with the intervals of the fourth, fifth, and octave: that, reflecting on the cause of this phenomenon, he conceived it to depend on the weight of the hammers and accordingly, having caused them to be weighed, he found the weight of the heaviest hammer, answering to the fundamental note, being represented by unity, the weights of the other three, answering to the fourth, fifth, and octave above, were as,, and . Nicomachus adds, that Pythagoras, on his return home, was desirous of verifying this first experiment by the following. Fastening a string to a fixed point, and passing it over a peg in a horizontal line with this point, he stretched the string more or less by different weights; and on causing it to vibrate, he found the weights corresponding to the fourth, fifth, and octave above, to be to each other as the weights of the smith's hammers. On applying the theorem above quoted to these experiments, we find, either that they were inaccurate, or that they are erroneously related. The length of three strings, of the same uniform thickness, which, being stretched by the same weight, would give the fourth, the fifth, and the octave above, are as the three fractions,,, but to make the same string give the fourth, fifth, and octave above, by stretching it with different weights, these weights must be to each other as,, 4. 4. Thus there is a mistake in the proportions between the weights of the hammers found by Pythagoras, or in the manner in which his experiments are related. Undoubtedly it was natural to suppose, that the three different weights, which should produce the fourth, fifth, and octave, by stretching one and the same string, would be to each other as the lengths of three different strings, which, being equally stretched, should produce these three notes: but this is not the fact. Be this as it may, it is unquestionable, that these first ideas of Pythagoras were the true source of the theory of music. As what is properly called the art of music, however, derives very little assistance from mathematics, I shall enlarge no farther on the music of the ancients; particularly as it's history may be 1 found |