The Tetractys

The Pythagoreans called every number as the tetrad.

This was because the tetrad comprehends in itself all the numbers as far as to the decad, and the decad itself.

• The sum of 1, 2, 3, and 4, is 10.

Hence, they called both the decad and the tetrad as every number.

The decad was a number in energy, but the tetrad was a number in capacity.

The sum of these four numbers 1, 2, 3, 4 made up the tetractys, in which all harmonic ratios are included.

• 4 to 1 is a quadruple ratio. It forms the symphony bisdiapason.
• 3 to 2 is a sesquialter ratio. It forms the symphony diapente.
• 4 to 3 is a sesquitertian ratio. If forms the symphony diatessaron.
• 2 to 1 is a duple ratio. It forms the diapason.

The Pythagoreans greatly venerated the tetractys.

Theo of Smyrna, showed how many tetractys there are. He said:

Hence their oath was:

The 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.

First Tetractys

Of these, the monad is assumed as the first because it is the principle of all even, odd, and evenly-odd numbers. Its nature 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.

Second Tetractys

Hence the even and the odd receive two tetractys, according to multiplication.

• The even numbers receive in a duple ratio. This is because 2 is the first of even numbers, and increases from the monad by duplication.
• But the odd number is increased in a triple ratio. This is because 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 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.
• The fourth among even numbers is 8; but among odd numbers is 27.

These numbers have:

• the more perfect ratios of symphonies
• a tone

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.

The fourth terms 8 and 27 are equally equal, are in power a cube.

Hence from these numbers, and this tetractys, the increase takes place from a point to a solid.

A side follows after a point, a plane follows after a side, and a solid follows after a plane.

Plato in the Timæus says that in these numbers constitutes the soul.

But the last of these seven numbers, i. e. 27, is equal to all the numbers that precede it:

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. They comprehend musical, geometrical, and arithmetical ratios. The harmony of the universe consists in such ratios.

Third Tetractys

The third tetractys is that which according to the same analogy or proportion comprehends the nature of all magnitude.

For what the monad was in the former tetractys, that a point is in this.

What the numbers 2 and 3, which are in power a side, were in the former tetractys, that the extended species of a line, the circular and the right, are in this.

The right line subsists in conformity to the even number since it is terminated by 2 points.

But the circular subsists in conformity to the odd number, because it is comprehended by one line which has no end.

But what in the former tetractys the square numbers 4 and 9 were, that the two-fold species of planes, the rectilinear and the circular, are in this.

What the cube numbers 8 and 27 were in the former, the one being an even, but the other an odd number, that the two solids, one of which has a hollow plane, as the sphere and the cylinder, but the other a planes, as the cube and pyramid, are in this tetractys.

Hence, this is the 3rd tetractys, which completes every magnitude, from a point, a line, a plane, and a solid.