Some instruments have two different keys, to make a difference in the half note at certain

to E, it should be something lowered; and that if the same is considered as A, the major third to F, it, should be something raised; but it cannot be raised and lowered without using one key for the fifth, and another key for the third. The same is the case with E; which as a fifth to A should be lower, and as a sharp third to B should be higher." Therefore, in some organs these keys are divided, and there are two pipes to express the semitone, according to the capacity in which it is used. The late learned Dr. Smith, Master of Trinity College in Cambridge, and author of a Treatise on Harmonics, above referred to, went still farther toward rectifying the scale in harpsichords, by accommodating occasional or supplemental jacks and strings to the different keys. I am not clear that music gains any thing upon the whole by such an improvement. The imperfection of the scale, if managed with judgment, may be no real disadvantage. The ear may find satisfaction from the different complexion of different keys, as the eye is pleased with the variety

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riety of light and shade, which they call the chiaro 'scuro in painting. The imperfect keys. may set off the more perfect, as concords are improved by the dissonances preceding. If imperfection is natural to the scale, and necessary, it may be wiser to moderate than to attempt to extirpate it. Some great masters have certainly thought that the more imperfect parts of the instrumental scale have their beauties; otherwise they would not have composed pieces for keys which are necessarily out of tune, which is the case with some very excellent compositions; and I may mention as an instance the celebrated Stabat Mater of Pergolesi.

If we take the pains to examine more particularly whence the imperfection of the scale arises, by summing up the intervals differently, we shall discover it first in the constitution of the diatessaron, or the degrees of the fourth. And hence it will appear, that though the imperfection in keyed instruments is in a great measure the consequence of art, yet imperfection is constitutional in the degrees of the octave, and therefore natural. For the interval of fourth includes the degrees of third major and a semitone, or of a tone major and a third minor. The third major 4 to 5 added

added to the semitone 15 to 16, give the ratio of 60 to 80, the same as 3 to 4: but the tone major 8 to 9, added to the third minor 5 to 6, give the ratio of 40 to 54; which is not the same as 3 to 4, but differs from it in the ratio of 80 to 81. So again, if we take two fourths by repeating the ratio of 3 to 4, the sum is 9 to 16: but if we compose these two fourths of their other constituent parts, the fifth and minor third, we shall have the ratio of 5 to 9. The difference is the same as before, 80 to 81. Or we may subtract 6 to 5 out of 4 to 3, and there will remain 10 to 9, a tone minor instead of a tone major ; the difference is still 81 to 80.

There is another way of discovering im perfection in the scale, by comparing the fifth with the third major. The seventeenth, or double octave of the major third, is to the unison or fundamental note as 1 to 5, or as 16 to 80: but if we consider this same interval as made up of four fifths, and repeat the ratio of four times, it gives us for the seventeenth 16 to 81: whence the error of the sharp third thus obtained is 80 to 81. This quantity of or (for the order of the terms is of no consequence) which meets us in all these instances, is the famous musical

comma, or smallest section of the octave, and, as appears, is equal to the difference between the tone minor and the tone major: for being subtracted from give . Which comma being of such great account in the scale of music, mathematicians have computed how many times this ratio is repeated ⚫ within the limits of diapason, that is, how many commas make an octave. Mersennus mistook the method, and made 584. Holder informs us that Mercator, in a manuscript work, made 55 and a little more. I find by the logarithms that there are exactly 55,8 commas in the octave: and if the reader is inclined to prove it, let him take the logarithms of 80 to 81, and raise them to the 55, 8th power, and he will have the ratio of diapason, or 2 to 1, true to five places of figures, that is, to less than a ten thousandth part.

If therefore the fifths are true, the minor third, in conformity to the rest of the scale, is naturally too flat by a comma; so that a series of minor thirds would carry us strangely out of the way. Four minor thirds (in appearance) compose the octave: but the ratio of the minor third four times repeated exceeds the octave, and gives 1296 to 625. Hence

if the minor thirds within the octave were all to be tuned perfect in consecution, and we were to go on so, we should gain a whole note in the compass of a few octaves. The octave is also composed (in appearance) of three major thirds: but as the others exceeded the octave, these fall short of it, and give us 125 to 64, instead of the true ratio of diapason: whence if the major thirds were made truly such in consecution, we should soon lose a whole note this way as we gained it

the other.

From all this it appears, that if we temper the scale, it must be done by lessening the fifths to increase the fourths and mend the minor thirds and as good fifths make bad thirds major, we must be content to widen the greater thirds, so that three of them may complete the octave as conveniently as may be. This is the shortest account I can give of so difficult and intricate a subject, which has exercised the deepest mathematicians, and might afford matter for endless calculations, with no great benefit to the science of music. That little comma would afford a man work enough for his life; and I apprehend he would find it untractable to the last.

Degrees