Thus chord tends toward chord, and note leans toward note, and it has to be considered whether the attraction of affinity or proximity be strongest while working out the charming effects of a composition.
The relations which music has to mechanics, the sphere of centers; to geometry, the sphere of measures; and to arithmetic, the sphere of numbers, show how deeply seated music is in the nature of things, and how independent it is of the will or choice of the musician. His composition may take any form his inspiration may suggest; they are subject to him; but as to the nature of music and its laws, he must keep himself subject and obedient to them. Music is of the aesthetic; but the aesthetic is of the nature of things.
At the middle of the string the stopped note and the harmonic notes are the same; but corresponding places above and below the middle give the same harmonic, although these places when stopped give different notes.
Nine-tenths of a string, if stopped and acted on, gives a tone the ratio of 9:10, but if touched and acted on as a harmonic it gives a note which is three octaves and a major third above the whole string. If the remaining tenth of the string be acted on either as a stopped note or a harmonic it will give the same note which is three octaves and a major third above the whole string the ratio of 1:10, so that the stopped note of one-tenth and the harmonic of nine-tenths are the same. Indeed the bow acting on stopped note of one-tenth, on harmonic of nine-tenths, or on harmonic of one-tenth, produces the same note, as the note is the production of one-tenth in each case; for in the harmonic, whether you bow on the nine-tenths or the one-tenth, while it is true that the whole string is brought into play, yet by the law of sympathy which permeates the entire string, it vibrates in ten sections of one-tenth each, all vibrating in unison. This is what gives the harmonic note its peculiar brilliancy.
An elongated uniform body, e.g., a dressed lath of pine, has three primary centers - the center of gravity, the center of oscillation, and the center of velocity. The center of gravity is the center of the body; the center of oscillation is at two-thirds from the end as the point of suspension; the center of velocity is at two-thirds of one-third from the end, i.e., at two-ninths from the end as the point of suspension.
This elongated body suspended at the end, or at one-third from the end, the oscillations are the same. The one-third above the point of suspension so balances the two-thirds below that the oscillations are performed in the same time for both suspensions. When it is