their component ratios; and having their grounds in nature itself, they cannot be disputed.

Theory of Coincidences resumed.

Upon the theory of coincidences, as explained above, a difficulty may have occurred to the imagination of the reader. If consonance arises strictly from coincidence of vibration, and two consonant strings should not set out precisely together, then how can they be consonant, when they are not coincident, as the theory supposes them to be? There are three ways of solving this difficulty, but I scarcely know which of them to prefer. 1. The vibrations of consonant strings are so exceeding quick, that if there is any error

in

* From experiments made with chords of vast length and small tension, calculations have been deduced, by which it has appeared, that if a small brass wire of nine inches long, stretched by a weight of 6 pounds, is in unison with an organ pipe of a foot long and one inch in diameter, it makes 800 vibrations in a second; and 600 in the same time, if it is stretched till it sounds an octave higher, by four times the weight above mentioned. Mr. Sauveur's experiment to find the number of vibrations in the sound of an organ pipe of a given length in a second of time, is very na

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in the coincidence, it is too minute to be perceptible in the consonance. 2. As consonance, though mechanically from the strings, is physically from the pulses of the air, the consonance may always be just, provided the pulses of the air are of the proper magnitudes with

tural and very ingenious. Musicians have observed, that if two pipes, nearly in unison, sound together, there are certain instants, and at equal intervals, when their joint sound gives a stronger pulse. This he imputed to the coincidence of their vibrations at these instants. Therefore, taking two long pipes, in which the coincidences would be slower, and consequently more observable, as being rarer, and finding that their tones were in the ratio of 45 to 46, and that they concurred six times in a second, he justly inferred that a pipe of such a tone as the longer made 270 vibrations in a second of time and that a pipe of 5 Paris feet in length had the same tone with a string that vibrates 100 times in a second. See Helsham's Lectures, Lect. XVIII. Mr. Sau veur found that a string giving the deepest musical tone which the ear could distinguish, vibrated 12 times and in a second: whence a string 12 octaves higher, which is the most acute the ear can distingiush, will vibrate 51200 times in a minute; as we find by repeating the ratio of 2 to 1 (which is that of the octave) 12 times. The thoughts are bewildered when they contemplate a motion so swiftly repeated, and compare it with the single stroke of a common pendulum yet the space passed over by the string so vibrating, is not comparable to the space which sound paffes over in the same time through the air.

To

with respect to each other, and then coincidence itself will not be necessary. 3. Although we should suppose a contradiction between the two consonant strings when their motion commences, they will immediately correct this minute difference by the power of that sympathy which will hereafter be evident from many examples.

The

To explain the effect of tension in musical strings, three things are to be taken into the account; their length, their thickness, and the force by which they are extended. This part of the subject has been carried very far by more modern authors; but the principles were touched upon by Mersennus, who has three propositions corresponding to the three parti, culars above. 1. If strings of unequal thickness are stretched with equal forces, their thickness ought to follow the duplicate proportion of the interval required. Thus, to sound the octave below, the string must be 4 times as thick; that is, it must have twice the diameter; for the thickness is as the square of the diameter. 2. If you would ascend from one tone to a higher with the same string, it must be stretched by a force, which is the square of the interval required; for the tone in all cases will be as the square root of the extending force. 3. If strings are of equal thickness, but unequal lengths, they will be in unison if the extending forces are as the squares of the lengths.

Musical proportion is a matter of fuch nicety, that these experiments which relate to tension never answer exactly; little insensible differences in circumstances make so great a difference in the effect.

The modern theory of music proposes the doctrine of coincident pulses as the best natural ground of consonance; and therefore I have taken it as such: but if we examine it critically, it will be found to labour under some difficulties not very easy to be solved. Upon the principle of coincidences, how hard will it be to assign a reason, why the ratio of

a

should give us a concord, (the minor third,) and the ratio of, which is next in order, an absolute discord. So also, the minor sixth, is concord; discord, though the two strings coincide oftener in their motions. By the rule of coincidence, an octave is more perfect than a fifth; but by the same rule, a twelfth, should be more perfect than double octave or fifteenth; which can hardly be, while we prefer an octave to a fifth: one of the two preferences must be wrong. From the difficulty of strings not beginning exactly together, which I have just mentioned, Dr. Smith concludes, that coincident pulses are not necessary, but only accidental to a perfect consonance*. And supposing the hypothesis true, it will only do for consonances in perfect tune. Depart never so little from the strict ratio, what then will become of the

* See Harmonics, p. 103.

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coincidence? Yet the interval, though not absolutely perfect, shall still be consonant and agreeable to the ear: and were it not so, we should have no music that could be endured. Concords tempered so that their vibrations are incommensurable and not expressible in numbers, are more pleasing than discords which exactly answer to their com mensurability*.

4. Of the Monochord.

To apply the theory of musical sounds to practice, and exemplify it to the ear, a monochord must be provided: which is an instrument contrived to exhibit the scale of music, and such as affords many curious and pleasing experiments. By its name, it is supposed to have only one string; but it is much better when accommodated with two, that one may always vibrate freely in its whole length, while the sections of the other are compared with it: for which purpose, one half of the moveable bridge must be cut down a little, that the string which passes over that part may have its liberty, while the

See Harmonics, p. 99, &c.

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