Digression On The Pythagorean Tetractys

The Tetractys are the perennial fountain of the human soul by which the Pythagoreans swore.

I think this is because between each pair of the 3 cubes 1, 8 and 27, for example between 1 and 8, there are two mean proportionals, 2 and 4.

Therefore, the Tetractys are made up of 4 numbers:

• 1, 2, 4 and 8, of which the sum is 15, or
• 1, 3, 9 and 27, of which the sum is 40

Geometers explain that:

• Pairs of cubes have 2 proportionals
• Pairs of squares have 1 proportional

Suppose the Tetractys were 1, 2, 3, 4.

• 1 is the basis of the numbers.
• 2 is the first of the numbers and of the evens.
• 3 is the first of the composites and of the unevens.

By constructing 1 at right angles to 3 a rectangle of area 3 is made, as from an uneven number.

But by constructing 2 at right angles to itself, a square of area 4 is made as from an even number, and in the construction of it, it is proper for the length and breadth to be equal.

Just as in the rectangle on 3 they are unequal.

The sum of 1, 2, 3 and 4 is 10. The human soul is accustomed to count in tens.

There are 4 numbers, therefore, 4 Unities and 4 kinds of harmonies:

1. Diapason

This is the harmony between 1 and 2, like that between 2 and 4

2. Disdiapason

This is the harmony between 1 and 4 and are equivalent to unison

3. Diapason Epidiapente

This is between 1 and 3 and is the greatest harmony in the system

, and is here the second

3. Diapente

This is between 2 and 3

4. Diatessaron

This is between 3 and 4

These are the only harmonies to the Pythagoreans, in accordance with my own thinking.

But on this same Tetractys, Joachim Camerarius thinks a little differently wrongly. In the Greek commentaries on the golden Poems of Pythagoras, he writes:

…

At first, they designated the Tenfold separately by the word Number.

Plato was taking it in that sense when he said in the Phaedd that half of a number is always uneven. For let two sets of numbers be defined alternately from Unity to Tenfold.

One series will be of unevens, the other of evens, in this way:

1. 3. 5. 7. 9. (Sum 25, which is uneven, and the square of the Fivefold, the number, that is, of the unevens.)

Or with Unity missed out, as the starting point, and the Tenfold as sep­ arately called a Number, as follows: 2. 4. 6. 8. 3. 5. 7. 9. (Sum 24, which is even.)

This is the enigma, that the unevens are even.

For the separate numbers in the series 3, 5, 7, and 9 are uneven, but taken together they are an even number, four. (And the sum is 24, which is even.)

Therefore, the Tenfold was called a Number by the Pythagoreans in a special sense.

It has the property that it is the sum of Unity and its multiples continuously up to the Fourfold.

For an equilateral triangle of numbers is constructed, of which the base is the Fourfold, and the vertex. Unity.

The Pythagoreans named every number derived from it a Tetractys.

By doubling the sides of the first Tetractys, another Pythagorean Tetractys is constructed, of the number 36, the most famous and in all respects the most useful which they possessed, that is the triangle of numbers of which the base is the Eightfold.

Thus they used the number 36 in many demonstrations, especially those concerned with harmony.

For in the patterns of the arrangement of this number are found the numbers 12, 9, 8, and 6.

They showed that all the harmonic consonances were contained within these numbers, as represented by the proportions of their intervals.

The number 36 is a square. Its side is 6. It is a triangular number, of which the side is 8. It is a rectangle of which one side is 9.

In the other case its side is 12. (For four times 9, and three times 12 both make 36.)

Lastly if 6, 8, 9, and 12 are added together into a single sum, the result is the number 35. That is called a harmony by the Pythagoreans, and if Unity is added to it again it completes the number 36.

Furthermore, of the numbers which have been formed by addition from those which precede in the natural order (that is from the triangular numbers 1, 3, 6, 10, 15, 21, 28) 36 is the first (and the only one below 1225) which is a square, and has as its side 6, the first perfect number (that is to say made up of all its aliquot parts, 3, 2, and 1).

The same 36 is also produced by the multiplication of the first 2 squares, 4 and 9.

The same is also formed by addition of, and is made up of, the first 2 cubes, 8 and 21, together with Unity which is a cube.

Because the speculation can be applied in so many ways, this Tetractys was held by the Pythagoreans to be as worthy of consideration and admiration as the foremost; and so they transferred it to Natural Philosophy, and most of all to the contemplation of the soul, and equally to Ethics.

They combined it with some Theology.

Epiphanius shows from Irenaeus Against the Valentinians, they made the Tetractys a thing to swear by; but they understood it to mean these 4 things:

1. Foundation
2. Silence
3. Mind
4. Truth

In the golden poems, the formula for swearing is not the Tetractys itself, but he who through the Tetrcwtys showed the permanence of the essence of the Soul.

Plutarch explained the spiritual Tetractys in physical terms, as being:

1. Sensation
2. Opinion
3. Knowledge
4. Mind

Yet the cosmic Tetractys may be more precisely viewed in the following way: from Unity, set out in a threefold way, taking Unity to fill the gap in the middle, and with Quaternaries enclosing it like straight lines, it turns out that this Tetractys produces the tenfold, since on this showing that is the third of the triangular numbers in origin.

(For after Unity, the first triangular number is 3, of which the base is 2; the second is 6, of which the base is 3. If you draw three lines enclosing these, through the two points in the former and three in the latter, sketching out a triangle, nothing is left in the middle; but if the third triangular number, 10, with base 4, is given lines enclosing it on the outside, in each case in the positions of its sets of four points, a single point will be left in the middle, which belongs to none of the lines which form the figure but sketches out the space inside, like a heart or kernel.) For this reason the Pythagoreans called the Tenfold All-embracing Mother, that encloses all things.

Unyielding and indomitable and pure as Proclus tells us.

This very completion of ten Units, that is the Tenfold formed by addition from this Tetractys, was reported by the Pythagoreans as containing and accomplishing, or completing, the embellishment of the entire universe

Plato also follows them. For:

1. the universe has become material and sensible;
2. it retains all those things which are in it, indissolubly, by the bond of similarity, or commensurability;
3. it is a whole, inasmuch as it is formed from whole elements;
4. its substance is round in shape; 5. it is that which suffers in itself, and from itself, all that there is to suffer;
5. it moves in a circle;
6. its body is animate;
7. it is the creator of time by means of the revolution of the stars;
8. it indicates certain stars as sacred: they are included in the number of the gods, and make up the Great Year, which is perfect;
9. in every way it is the perfect completeness of things, having in itself all living things, representing four forms (stars representing heaven, birds air, fish water, four-footed creatures earth). On this showing from Unity (as the Pythagoreans say, “from the cave of the monad”j, there is a progression up to Four (as they say, until reaching the divine Tetrad itselfj, and thus it gives birth to the Tenfold, the mother of all things as we have said. Now the progression of Unity is as follows.

One is the world. The Twofold signifies the first multiple contained in it. The Threefold signifies the bond or knot, necessary for the linking together of things.

It is not possible for two single things to combine into one in the absence of the Third.

The Fourfold is the number which marks out and enumerates the elements.

For the world is a solid body, and two solids always require two intermediates, to correspond in continuous proportion. Now their sum (that is, of 1, 2, 3, and 4) is the tenfold, of which we have been speaking all along. For this is the apparel of the completeness, this is its dowry, with which its maker endowed it.