| I. | (1:1) | (8:9) | (4:5) | (3:4) | (2:3) | (3:5) | (4:7) | (8:15) | (1:2) |
| II. | (1:2) | (4:9) | (2:5) | (3:8) | (1:3) | (3:10) | (2:7) | (4:15) | (1:4) |
| III. | (1:4) | (2:9) | (1:5) | (3:16) | (1:6) | (3:20) | (1:7) | (2:15) | (1:8) |
| IV. | (1:8) | (1:9) | (1:10) | (3:32) | (1:12) | (3:40) | (1:14) | (1:15) | (1:16) |
The following are the principal numerical relations which the parts in each scale bear to one another:—
| (¹⁄₂):(⁴⁄₇) | = | (7:8) |
| (⁴⁄₅):(⁸⁄₉) | = | (9:10) |
| (²⁄₃):(⁴⁄₅) | = | (5:6) |
| (⁴⁄₇):(²⁄₃) | = | (6:7) |
| (⁸⁄₁₅):(⁴⁄₇) | = | (14:15) |
| (¹⁄₂):(⁸⁄₁₅) | = | (15:16) |
Although these relations are exemplified by parts of scale I., the same ratios exist between the relative parts of scales II., III., and IV., and would exist between the parts of any other scales that might be added to that series.
These are the simple elements of the science of that harmony which pervades the universe, and by which the various kinds of beauty æsthetically impressed upon the senses of hearing and seeing are governed.
THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.
It is well-known that all sounds arise from a peculiar action of the air, and that this action may be excited by the concussion resulting from the sudden displacement of a portion of the atmosphere itself, or by the rapid motions of bodies, or of confined columns of air; in all which cases, when the motions are irregular, and the force great, the sound conveyed to the sensorium is called a noise. But that musical sounds are the result of equal and regular vibratory motions, either of an elastic body, or of a column of air in a tube, exciting in the surrounding atmosphere a regular and equal pulsation. The ear is the medium of communication between those varieties of atmospheric action and the seat of consciousness. To describe fully the beautiful arrangement of the various parts of this organ, and their adaptation to the purpose of collecting and conveying these undulatory motions of the atmosphere, is as much beyond the scope of my present attempt as it is beyond my anatomical knowledge; but I may simply remark, that within the ear, and most carefully protected in the construction of that organ, there is a small cavity containing a pellucid fluid, in which the minute extremities of the auditory nerve float; and that this fluid is the last of the media through which the action producing the sensation of sound is conveyed to the nerve, and thence to the sensorium, where its nature becomes perceptible to the mind.
The impulses which produce musical notes must arrive at a certain frequency before the ear loses the intervals of silence between them, and is impressed by only one continued sound; and as they increase in frequency the sound becomes more acute upon the ear. The pitch of a musical note is, therefore, determined by the frequency of these impulses; but, on the other hand, its intensity or loudness will depend upon the violence and the quality of its tone on the material employed in producing them. All such sounds, therefore, whatever be their loudness or the quality of their tone in which the impulses occur with the same frequency are in perfect unison, having the same pitch. Upon this the whole doctrine of harmonies is founded, and by this the laws of numerical ratio are found to operate in the production of harmony, and the theory of music rendered susceptible of exact reasoning.
The mechanical means by which such sounds can be produced are extremely various; but, as it is my purpose simply to shew the nature of harmony of sound as related to, or as evolving numerical harmonic ratio, I shall confine myself to the most simple mode of illustration—namely, that of the monochord. This is an instrument consisting of a string of a given length stretched between two bridges standing upon a graduated scale. Suppose this string to be stretched until its tension is such that, when drawn a little to a side and suddenly let go, it would vibrate at the rate of 64 vibrations in a second of time, producing to a certain distance in the surrounding atmosphere a series of pulsations of the same frequency.