Instead of speaking of the lengths of the strings, we may pass to the relative numbers of vibrations in a second, which are inversely as the lengths. Thus, two strings of ten and seven feet, stretched by the same weight, vibrate so that the one of ten feet makes seven vibrations while that of seven feet makes ten.
We now proceed to consider the most simple combinations; and first, that of two to one. Let the second string be half the first, or make two vibrations while the first makes one there is then not only a joint effect which is agreeable, but a peculiar sameness of the two notes, insomuch that two instruments made to play together in such manner that the notes of the second shall always be of twice as many vibrations as the simultaneous notes of the first, would be universally admitted to be playing the same air, with no more difference than of that sort which is heard when a man and a boy attempt to sing the same air together. This perfect sameness, for so it will be called, though the two instruments never sonnd the same tone together, admits of no explanation; for though the ratio of the simultaneous vibrations is the simplest possible (two to one), there is no perceptible reason why, because simple ratios generally give harmonious com binations, the most simple of all should produce an absolute feeling of identity of character in the two tones. To this circumstance however we owe the most material simplification of the musical scale ; for let it be settled, for instance, what strings give agreeable notes between those of 20 and 10 feet long, and division by two will give all the strings which can be admitted between those of 10 and 5 feet ; thus, if it be proper to admit a string of eight feet in the former set, one of four feet must also take its place In the latter.
Again, it is observed that the relative effect of two tones is always the mime as those of other two, when the numbers of vibrations made in a given time in the first pair are in the same proportion as the corre sponding numbers in the second pair. Thus, suppose that in a given time the numbers of vibrations made by four strings are 12, 18, 40, and 60. Then, since we see that 12 : 18 : : 40 : CO or .14 =13 we may say that, according as the first and second sounded together are pleasant or unpleasant, so aro the third and fourth; also if an air beginning on the first string require an immediate transition to the second, then the same air begun on the third string will require an Immediate transition to the fourth.
A musical interval, then, is given when the fraction which expresses the proportion of the vibrations of its two notes in a given time is given. By the interval 1 we mean that of two notes, the higher of which makes three vibrations while the lower makes two. Thus, if 18, 23, and 30 be the numbers of vibrations made by three strings in the same time, and we wish to find a fourth note which is as much above the third as the second is above the first, we must not make a string of 35 vibrations in the same time (as the beginner might do), that is, not one of 90+23-18, but one of 30 x IL or 381 vibrations in the mane time.
Let us now take a string, and call the note sounded by it C, and let the string of twice as many vibrations (or half its length) have the game name, with a difference (for the reason above given); call it C'. Let us now seek for the simplest fractions which lie between 1 and 2. Take the numbers up to 6 (the ear does not so well agree with 7 and all higher prime numbers, why of course cannot be told, but situ. plicity must end somewhere, and, by the conetitntion of the ear, ratios in which 7 and higher primes occur are not agreeable), and form every fraction out of them which lies between 1 and 2 ; we have then Put these clown, with 1 and 2, in order of magnitude, and we have h 4 4 2 Take such a set of strings that while the first makes one vibration the second makes g of a vibration, the third 4 of a vibration, and so on up to the last, which makes 2 vibrations; or take a set of strings equally stretched, of which the length of the first being 1, that of the second is 1, &c., and of the last Every one of the notes thus duced will be agreeable when sounded with the first, and if the first sonnd C, the musician will have the following part of the scale before him in its meet natural form : Them intervals have the following names; why, will presently be seen, minor third. 4fifth.