These pulsations will communicate through the ear a musical note which would, therefore, be the fundamental note of such a string. Now, the phenomenon said to be discovered by Pythagoras is well known to those acquainted with the science of acoustics, namely, that immediately after the string is thus put into vibratory motion, it spontaneously divides itself, by a node, into two equal parts, the vibrations of each of which occur with a double frequency—namely, 128 in a second of time, and, consequently, produce a note doubly acute in pitch, although much weaker as to intensity or loudness; that it then, while performing these two series of vibrations, divides itself, by two nodes, into three parts, each of which vibrates with a frequency triple that of the whole string; that is, performs 192 vibrations in a second of time, and produces a note corresponding in increase of acuteness, but still less intense than the former, and that this continues to take place in the arithmetical progression of 2, 3, 4, &c. Simultaneous vibrations, agreeably to the same law of progression, which, however, seem to admit of no other primes than the numbers 2, 3, 5, and 7, are easily excited upon any stringed instrument, even by the lightest possible touch of any of its strings while in a state of vibratory motion, and the notes thus produced are distinguished by the name of harmonics. It follows, then, that one-half of a musical string, when divided from the whole by the pressure of the finger, or any other means, and put into vibratory motion, produces a note doubly acute to that produced by the vibratory motion of the whole string; the third part, similarly separated, a note trebly acute; and the same with every part into which any musical string may be divided. This is the fundamental principle by which all stringed instruments are made to produce harmony. It is the same with wind instruments, the sounds of which are produced by the frequency of the pulsations occasioned in the surrounding atmosphere by agitating a column of air confined within a tube as in an organ, in which the frequency of pulsation becomes greater in an inverse ratio to the length of the pipes. But the following series of four successive scales of musical notes will give the reader a more comprehensive view of the manner in which they follow the law of numerical ratio just explained than any more lengthened exposition.

It is here requisite to mention, that in the construction of these scales, I have not only adopted the old German or literal mode of indicating the notes, but have included, as the Germans do, the note termed by us B flat as B natural, and the note we term B natural as H. Now, although this arrangement differs from that followed in the construction of our modern Diatonic scale, yet as the ratio of 4:7 is more closely related to that of 1:2 than that of 8:15, and as it is offered by nature in the spontaneous division of the monochord, I considered it quite admissible. The figures give the parts of the monochord which would produce the notes.

I. { (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)*
{ C D E F G A B H c
II. { (¹⁄₂)* (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃)* (³⁄₁₀) (²⁄₇) (²⁄₁₅) (¹⁄₄)*
{ c d e f g a b h c′
III. { (¹⁄₄)* (²⁄₉) (¹⁄₅)* (³⁄₁₆) (¹⁄₆)* (³⁄₂₀) (¹⁄₇)* (²⁄₁₅) (¹⁄₈)*
{ c′ d′ e′ f′ g′ a′ b′ h′ c′′
IV. { (¹⁄₈)* (¹⁄₉)* (¹⁄₁₀)* (³⁄₃₂) (¹⁄₁₂)* (³⁄₄₀) (¹⁄₁₄)* (¹⁄₁₅)* (¹⁄₁₆)*
{ c′′ d′′ e′′ f′′ g′′ a′′ b′′ h′′ c′′′

The notes marked (*) are the harmonics which naturally arise from the division of the string by 2, 3, 5, and 7, and the multiples of these primes.

Thus every musical sound is composed of a certain number of parts called pulsations, and these parts must in every scale relate harmonically to some fundamental number. When these parts are multiples of the fundamental number by 2, 4, 8, &c., like the pulsations of the sounds indicated by c, c′, c′′, c′′′, they are called tonic notes, being the most consonant; when the pulsations are similar multiples by 3, 6, 12, &c., like those of the sounds indicated by g, g′, g′′, they are called dominant notes, being the next most consonant; and multiples by 5, 10, &c., like those of the sounds indicated by e, e′, e′′, they are called mediant notes, from a similar cause. In harmonic combinations of musical sounds, the æsthetic feeling produced by their agreement depends upon the relations they bear to each other with reference to the number of pulsations produced in a given time by the fundamental note of the scale to which they belong; and it will be observed, that the more simple the numerical ratios are amongst the pulsations of any number of notes simultaneously produced, the more perfect their agreement. Hence the origin of the common chord or fundamental concord in the united sounds of the tonic, the dominant, and the mediant notes, the ratios and coincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:—

In musical composition, the law of number also governs its division into parts, in order to produce upon the ear, along with the beauty of harmony, that of rhythm. Thus a piece of music is divided into parts each of which contains a certain number of other parts called bars, which may be divided and subdivided into any number of notes, and the performance of each bar is understood to occupy the same portion of time, however numerous the notes it contains may be; so that the music of art is regularly symmetrical in its structure; while that of nature is in general as irregular and indefinite in its rhythm as it is in its harmony.

Thus I have endeavoured briefly to explain the manner in which the law of numerical ratio operates in that species of beauty perceived through the ear.