De animae procreatione in Timaeo

To proceed by way of multiplication,—twice 3

make 6, and 4 times 9 thirty-six, and 8 times 27 produce 216. Thus six appears to be a perfect number, as being equal in its parts; and it is called matrimony, by reason of the mixture of the first even and odd. Moreover it is composed of the original number, which is one, of the first even number, which is two, and the first odd number, which is three. Then for 36, it is the first number which is as well quadrangular as triangular, being quadrangular from 6, and triangular from 8.[*](See note on Platonic Questions, No. V. § 2. Thirty-six is called the triangular of eight, because a triangle thus made of thirty-six points will have eight points on each side. (G.)) The same number arises from the multiplication of the first two square numbers, 4 and 9; as also from the addition of the three cubical numbers, 1, 8, and 27, which being put together make up 36. Lastly, you have a parallelogram with unequal sides, by the multiplication of 12 by 3, or 9 by 4. Take then the numbers of the sides of all these figures, the 6 of the square, the 8 of the triangle, the 9 for the one parallelogram, and the 12 for the other; and there you will find the proportions of all the concords. For 12 to 9 will be a fourth, as nete to paramese. To eight it will prove a fifth, as nete to mese. To six it will be an octave, as nete to hypate. And the two hundred and sixteen is the cubical number proceeding from six which is its root, and so equal to its own perimeter.

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.

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 numbers[*](That is, in the quaternary, § 11. See the diagram, p. 339. (G.)) (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.

But since the numbers proposed did not afford space

sufficient for the middle intervals, therefore there was a necessity to allow larger bounds for the proportions. And now we are to tell you what those bounds and middle spaces are. And first, concerning the medieties (or mean terms); of which that which equally exceeds and is exceeded by the same number is called arithmetical; the other, which exceeds and is exceeded by the same proportional part of the extremes, is called sub-contrary. Now the extremes and the middle of an arithmetical mediety are 6, 9, 12. For 9 exceeds 6 as it is exceeded by 12, that is to say, by the number three. The extremes and middle of the sub-contrary are 6, 8, 12, where 8 exceeds 6 by 2, and 12 exceeds 8 by 4; yet 2 is equally the third of 6, as 4 is the third of 12. So that in the arithmetical mediety the middle exceeds and is exceeded by the same number; but in the sub-contrary mediety, the middle term wants of one of the extremes, and exceeds the other by the same part of each extreme; for in the first 3 is the third part of the mean; but in the latter 4 and 2 are third parts each of a different extreme. Whence it is called sub-contrary. This they also call harmonic, as being that whose middle and extremes afford the first concords; that is to say, between the highest and lowermost lies the diapason, between the highest and the middle lies the diapente, and between the middle and lowermost lies the fourth or diatessaron. For suppose the highest extreme to be placed at nete and the lowermost at hypate, the middle will fall upon mese, making a fifth to the uppermost extreme, but a fourth to the lowermost. So that nete answers to 12, mese to 8, and hypate to 6.