על בריאת העולם ל״זOn the Account of the World's Creation 37
א׳
1[107] It is however not only a bringer of perfection, but, one may say, absolutely harmonious, and in a certain sense the source of the most beautiful scale, which contains all the harmonies, that yielded by the interval of four, by the interval of five, by the octave; and all the progressions, the arithmetic, the geometric, and the harmonic as well. The scheme is formed out of the following numbers: 6, 8, 9, 12. 8 stands to 6 in the proportion 4:3, which regulates the harmony of 4; 9 stands to 6 in the proportion 3:2, which regulates the harmony of 5; 12 stands to 6 in the proportion 2:1, which regulates the octave.
ב׳
2[108] And, as I said, it contains also all the progressions, the arithmetic made up of 6 and 9 and 12—for as the middle number exceeds the first by 3, so it in its turn is exceeded to the same amount by the last; the geometric, made up of the four numbers (6, 8, 9, 12); for 12 bears the same proportion to 9 that 8 does to 6, and the proportion is 4:3; the harmonic, made up of three numbers (6, 8, and 12).
ג׳
3[109] There are two modes of testing harmonic progression. One is this. (Harmonic progression is present) whenever the relation in which the last term stands to the first is identical with that in which the excess of the last over the middle term stands to the excess of the middle term over the first. A very clear proof may be obtained from the numbers before us, 6 and 8 and 12: for the last is double the first, and the difference or excess is also double; for 12 exceeds 8 by 4, and 8 exceeds 6 by 2, and 4 is twice 2.
ד׳
4[110] Another way of detecting the presence of harmonic proportion is this. (It is present) whenever the middle term exceeds the one extreme and is itself exceeded by the other by the same fraction; for 8 being the middle term exceeds the first by one-third of the latter, for when we subtract 6 (from 8) the remainder, 2, is one-third of the first number, and 8 is exceeded by the last number by the same fraction, for if 8 be subtracted from 12, the remainder 4 is one-third of the last number.