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a counterbalance to it. Now, as the tending force acts in the direction BD, it may be represented by BD, which may be resolved into two, viz. BC and CD, whereof CD represents the restituent force, which is equal to the string's elasticity=E, and the other acts in the horizontal direction, and may be considered as equal to BD, because of the smallness of DC, or equal to the tending force F. Therefore F: E:: BC: CE:: L: S; putting S=the space over which the

string passes in half a vibration. Hence, E=

FS

L

There

fore in the same string, where the tending force and length are given, the elasticity is proportional to the sagittæ S, as we have seen before.

But since the sagitta or space passed over is equal to

the time and velocity of the vibration, E=

FTV
L

And

as the elastic force may be considered as uniform for the small time it acts, the momentum or quantity of

motion generated will be ET=

FVT2
L

But this mo

mentum will be as the velocity and quantity of matter

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Hence, if D and F be given, or if there be two strings of the same diameter and tended with equal weights, but of different lengths, the times of vibration will be directly as their lengths. Thus, half the monochord vibrates in half of the time, in which the whole chord vibrates; two thirds of it, in two thirds of it; and three fourths of it, in three fourths of the time: the

first is called an octave, where the coincidence is at every second vibration of the shorter; the second is called a fifth, where the coincidence is at every third vibration of the shorter; and the third is called a fourth, where the coincidence is at every fourth vibration of the shorter, &c.

If one of these strings, both of equal tension and diameter, but of different lengths, be struck, the other will vibrate with it in aliquot parts, the points of division between the parts remaining quiescent; as is evident to the sight, by pieces of paper remaining at rest on these points, while similar pieces laid on any other points will be shaken off by the vibrations. If one string be twice as long as the other, it will vibrate by halves, the middle-point remaining quiescent; and if three times as long, there will be two points at rest, while the string is vibrating in the three different lengths of it. The reason is, because the same vibrations of the shorter string are communicated to the contiguous air, and by the air to the quiescent string, which, being of equal tension and diameter, cannot vibrate in unison with the shorter, or with the contiguous air, in any other manner, than by vibrating in lengths, each being equal to the length of the shorter string. And hence, we see the reason of the construction of the double-stringed violin, whose untouched strings vibrate in unison with the others, and thereby greatly heighten the melody and harmony of the music.

Hence, also, if F and L be given, T is proportional to D; that is, if the tending force and lengths of two strings be the same, but their diameters different, the times of their vibrations will be as their diameters. If the diameters be as 2: 1, the string, whose diameter is greatest, will vibrate slowest in the same proportion,

and their coincidence will be at every second vibration of the least; and they will therefore sound an octave.

Lastly, if D and L be given, T will be as ✔F; that is, if the diameters and lengths of two strings be the same, the times of their vibrations will be as the square roots of the tending forces, inversely.

Now, as the tone of a string depends entirely on the time of the vibration, it is easy to understand, that whatever the sounding body be, or how many soever there may be of them, if they perform the same number of vibrations in the same time, they will all sound the same note and be in unison. And if the vibrations be performed in unequal times, the coincidence will be after certain intervals, and the shorter that these intervals are, so much the more agreeable is the consonance to the ear. Hence, when one string vibrates twice, while the other vibrates once, the coincidence will be most frequent, and therefore it is called the most perfect concord, as it is most agreeable to the ear. When the times of their vibrations are as two to three, the coincidence will be at every third of the quickest, and therefore, this, which is called a fifth, is in the next degree of perfection.

There are but seven whole notes or tones in the scale; for when you come to the eighth, it is no other than the octave to the first, and the octave above and below is only a replication of the same sound. And a skilful artist, in the construction of musical instruments, will compound the various proportions of length, diameter, and tension of strings, in such a manner as to produce the most agreeable consonance to the ear.

If, instead of strings of the same diameter and tension, but different in lengths, one string be taken and stopped at different places while it is vibrating, it will give different sounds, according to the length of the

string that is suffered to vibrate without interruption. The whole length of the string, whatever that may be, is called the monochord, and the sound produced by it is called the key-note. If half the string be suffered to vibrate, the sound produced is called the octave to the first note or sound; because musicians have particularly six different sounds distinguished between them, which are excited by stopping the string at certain places between these two points, while the remainder is suffered to vibrate. The proportion of length for making these sounds will be assigned presently.

The highest sound of the octave is expressed by half the line, and if this half be divided again in the same manner, a higher octave is produced by half of it, or the one-fourth of the original line; and so on, as far as we please. So that if the divisions of the first octave be once ascertained for each particular note, we may take one-half, one-quarter, one-eighth, &c. of their lengths, for the divisions of the next and succeeding octaves. The fifth of the first or key-note, is found by taking the of the whole string, and if this note be considered as the key-note, its fifth may be found in like manner, by taking two-thirds of its length, and so on, for any succession of fifths as far as you please.

Now, if you carry on the succession of octaves as far as seven, and of fifths, as far as the twelfth, they will be found nearly to coincide; but if the succession be carried farther, they will diverge farther, so that no octave agrees more nearly with a fifth, than the seventh with the twelfth fifth; and because an imperfection in the octaves is intolerable to the ear, but an imperfection in the others does not produce so disagreeable an effect, we must make use of the twelfth fifth for the seventh octave, and divide the imperfection that would

thereby be introduced into the twelfth fifth, among all the rest, which will be thereby rendered inconsiderable and almost insensible.

And if we find all the octaves of these twelve divisions of the string, we shall have twelve distinct notes in half the string, or in the first octave, to which, if we add the sound of the whole string, we shall have thirteen distinct sounds in the octave; all derived from this succession of fifths and their octaves; and this shows the reason, why no more notes are introduced into the

octave.

If the whole length of the string be supposed to be 100,000, the succession of octaves and fifths will be octaves of 100,000 fifths of 100,000

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The eight notes or sounds used in the common scale of music, are determined by the lengths of the string which gives these sounds. The whole length of the string, when vibrating, gives the first note or sound, which musicians generally call the key or first note, or C; of the length sounds the second note called D, but between this and the first they have interposed another sound, made by 1 of the string, which they call the

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