The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of 1, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad in capacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter, forms the symphony diapente; 4 to 3, which is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason.
In consequence, however, of the great veneration paid to the tetractys by the pythagoreans, it will be proper to give it a more ample discussion, and for this purpose to show from Theo of Smyrna,’ how many tetractys there are: “The tetractys,” says he, “was not only principally honored by the Pythagoreans, because all symphonies are found to exist within it, but also because it appears to contain the nature of all things.” Hence the following was their oath: “Not by him who delivered to our soul the tetractys, which contains the fountain and root of everlasting nature.” But by him who delivered the tetractys they mean Pythagoras; for the doctrine concerning it appears to have been his invention. The above-mentioned tetractys, therefore, is seen in the composition of the first numbers, 1, 2, 3, 4. But the second tetractys arises from the increase by multiplication of even and odd numbers beginning from the monad.
Of these, the monad is assumed as the first, because, as we have before observed, it is the principle of all even, odd, and evenly-odd numbers, and the nature of it is simple. But the three successive numbers receive their composition according to the even and the odd; because every number is not alone even, nor alone odd. Hence the even and the odd receive two tetractys, according to multiplication; the even indeed, in a duple ration; for 2 is the first of even numbers, and increases from the monad by duplication. But the odd number is increased in a triple ratio; for 3 is the first of odd numbers, and is itself increased from the monad by triplication. Hence the monad is common to both these; being itself even and odd. The second number, however, in even and double numbers is 2: but in odd and triple numbers 3. The third among even numbers is 4; but among odd numbers is 9. And the fourth among even numbers is 8; but among odd numbers is 27.
1 2 4 8
1 3 9 27
In these numbers the more perfect ratios of symphonies are found; and in these also a tone is comprehended. The monad, however, contains the productive principle of a point. But the second numbers 2 and 3 contain the principle of a side. Since they are incomposite, and first, are measured by the monad, and naturally measure a right line. The third terms are 4 and 9, which are in power a square superficies, since they are equally equal. And the fourth terms 8 and 27 being equally equal, are in power a cube. Hence from these numbers, and this tetractys, the increase takes place from a point to a solid.
For a side follows after a point, a superficies after a side, and a solid after a superficies. In these numbers also, Plato in the Timaeus constitutes the soul. But the last of these seven numbers, i.e. 27, is equal to all the numbers that precede it; for 1 +2 +3 +4 +8 +9 = 27. There are, therefore, two tetractys of numbers, one of which subsists by addition, but the other by multiplication, and they comprehend musical, geometrical, and arithmetical ratios, from which also the harmony of the universe consists.