Now these numbers aforesaid being endued with all these properties, the last of them, which is 27, has this peculiar to itself, that it is equal to all those that precede together; besides, that it is the periodical number of the days wherein the moon finishes her monthly course; the Pythagoreans make it to be the tone of all the harmonical intervals. On the other side, they call thirteen the remainder, in regard it misses a unit to be half of twenty-seven. [p. 342] Now that these numbers comprehend the proportions of harmonical concord, is easily made apparent. For the proportion of 2 to 1 is duple, which contains the diapason; as the proportion of 3 to 2 sesquialter, which embraces the fifth; and the proportion of 4 to 3 sesquiterce, which comprehends the diatessaron; the proportion of 9 to 3 triple, including the diapason and diapente; and that of 8 to 2 quadruple, comprehending the double diapason. Lastly, there is the sesquioctave in 8 to 9, which makes the interval of a single tone. If then the unit, which is common, be counted as well to the even as the odd numbers, the whole series will be equal to the sum of the decade. For the even numbers1 (1 + 2 + 4 + 8) give 15, the triangular number of five. On the other side, take the odd numbers, 1, 3, 9, and 27, and the sum is 40; by which numbers the skilful measure all musical intervals, of which they call one a diesis, and the other a tone. Which number of 40 proceeds from the force of the quaternary number by multiplication. For every one of the first four numbers being by itself multiplied by four, the products will be 4, 8, 12, 16, which being added all together make 40, comprehending all the proportions of harmony. For 16 is a sesquiterce to 12, duple to 8, and quadruple to 4. Again, 12 holds a sesquialter proportion to 8, and triple to 4. In these proportions are contained the intervals of the diatessaron, diapente, diapason, and double diapason. Moreover, the number 40 is equal to the two first tetragons and the two first cubes being taken both together. For the first tetragons are 1 and 4, the first cubes are 8 and 27, which being added together make 40. Whence it appears that the Platonic quaternary is much more perfect and fuller of variety than the Pythagoric.

1 That is, in the quaternary, ยง 11. See the diagram, p. 339. (G.)