Digression On The Pythagorean Tetractys
by Johannes KeplerThe 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
and8
, of which the sum is15
, or1
,3
,9
and27
, of which the sum is40
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:
 Diapason
This is the harmony between 1
and 2
, like that between 2
and 4
 Disdiapason
This is the harmony between 1
and 4
and are equivalent to unison
 Diapason Epidiapente
This is between 1
and 3
and is the greatest harmony in the system
, and is here the second
 Diapente
This is between 2
and 3
 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:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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:




 (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:
 Foundation
 Silence
 Mind
 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:
 Sensation
 Opinion
 Knowledge
 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 Allembracing 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; 6. it moves in a circle; 7. its body is animate; 8. it is the creator of time by means of the revolution of the stars; 9. it indicates certain stars as sacred: they are included in the number of the gods, and make up the Great Year, which is perfect; 10. in every way it is the perfect completeness of things, having in itself all living things, repre senting four forms (stars representing heaven, birds air, fish water, fourfooted 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; for 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. The philosophy of Hermes Trismegistus on numbers. So quotes Camerarius from the ancients. Most of what Hermes Trismegistus (whoever he was) impressed on his son Tatius agrees with it. His were the words^^: Unity embraces the Tenfold on the basis of ratio, and again the tenfold embraces Unity. Next he makes up the faculty of the soul which is responsible for desire from the twelve avengers, or ethical vices, in accordance with the number of the signs of the zodiac, and makes the body and this power of the soul which is closest to the body subject to it; whereas the same man makes up the rational fac ulty of the soul from the tenfold ethical virtues. Thus while the Pythago reans celebrate the Tetractys as the source of souls, and Camerarius says that there is more than one Tetractys, not only that which from the fourfold as base rises to a total of 1 0 , but also above all the other which from the eightfold as base up to its vertex adds up to a total of 36, the said Tatius also hints at the same thing from the teaching
of his father Hermes when he says it was the time when he himself was still in the Eighth Level, the Eightfold.’^^ Indeed the father sent the son back to Pimander singing of the eightfold. There in fact occur the eightfold ethical casts of soul, seven corresponding with the seven planets, as is apparent, starting from the Moon; but the eighth, more divine and more at rest, to the idea, I think, of the sphere of the fixed stars. Furthermore everything is carried out through harmonies. There is much impressing of silence, much mention of mind and truth. Also the cave, the foundation, the inner sanctum, the mixing bowl of spirits, and many other things are evinced, so that there can be no doubt that either Pythagoras is playing Hermes or Hermes Pythagoras.^^ For there is the additional fact that Hermes expounds a particular theol ogy, or cult of a divine power. Often he paraphrases Moses, often the Evangelist John in his sentiments, especially on regeneration. He im John 3 . presses on his disciple certain ceremonies; whereas the authorities declare the same of the Pythagoreans, that part of them were given over to theology and to various ceremonies and superstitions, and Proclus the Pythagorean locates his theology in the contemplation of numbers. So much by way of digression. Let us now return to the Pythago rean demonstration of the harmonious proportions. For the Pythagoreans^’ were so much given over to this form of The error of the philosophizing through numbers that they did not even stand by the Pythagoreans about the judgment of their ears, though it was by their evidence that they had number of originally gained entry to philosophy; but they marked out what was harmonies. melodic and what was unmelodic, what was consonant and what was dissonant, from their numbers alone, doing violence to the natural prompting of hearing. This harmonic tyranny of theirs lasted up until Ptolemy, who was the first, one thousand five hundred years ago, to uphold the sense of hearing against the Pythagorean philosophy, and accepted as melodic not only the proportions stated above, and the proportion of one and an eighth to one as equivalent to a Tone, but also admitted the proportion of one and a ninth to one as equivalent to a minor tone, and that of one and a fifteenth to one as equivalent to a semitone.’^’ He did not only add other proportions of one and a single aliquot part of one, which were sanctioned by the ears, such as one to one and a quarter or one to one and a fifth, but he also added some of the proportions of several aliquot parts, such as the proportion of 3 to 5 and 5 to 8 and others. On this showing Ptolemy did indeed correct the Pythagorean specu lation on the origin of the harmonic proportions as forced, but did not completely eliminate it as false; and the man who restored the judgement of the ears to its rightful place in words and doctrine never theless deserted it again, as even he adhered to the contemplation of abstract numbers. For the cause of the number of the harmonic proportions and of the individual proportions is not, even so, ade quate for its effect; but in designating the consonances it falls short, in the case of the other melodic intervals it goes too far. Ptolemy still denies that the thirds and sixths, minor and major (which are covered by the proportions 4:5, 5:6, 3:5 and 5:8) are consonances, which all musicians of today who have good ears say they are. On the other hand he accepts the proportions 6:7, 7:8 and others among the melodic musical intervals, so that if a tune proceeds from UT to FA, a note is constituted, intermediate between RE and MI, in the proportion in which 7 is the middle term between 6 and 8 . Let this note be RI, so that we can refer to it. Then it is possible to sound^^ just as it is possible to sound UT, RE, Ml, FA, which is utterly abhorrent to the ears of all men and the usages of singing, even though it may be possible for strings to be tuned in that way, seeing that as they are inanimate they do not interpose their own judgement but follow the hand of the foolish theorist without the least resistance. Furthermore if both the cause which was sought in abstract num bers, and the effect, that of consonance, were as far as possible equal in scope, and it could without absurdity be seen as the archetypal cause, bearing witness that it was from the contemplation of those numbers that the Father of things, the Eternal Mind, took the idea of notes and intervals, and so that they should be pleasing to human spirits He had to arrange them in the shape of those spirits, yet it would still not be very clear why the numbers 1, 2, 3, 4, 5, 6 , etc., conform with musical intervals, but 7, 11, 13, and the like do not conform. Also, the cause of this fact would not be revealed by the numbers, as numbers, from within themselves. Eor the cause drawn from the Threefold basic prin ciples, and the family of squares and cubes derived from them, is no cause, since the Eivefold is foreign to it, although it refuses to have its rights of citizenship in the origin of musical intervals torn from it. Yet not even this is satisfactory to the theorist, for he knows that the numbers 1, 2, 3 are symbols of the basic principles of which nat ural things consist. Eor an interval is not a natural thing, but a geo metrical one. Hence unless these numbers number something else, which is more akin to the intervals, the philosopher will not be able to put any confidence in this cause but will suspect it of not being a cause. For these reasons,^^ then, for the last twenty years in order to work this out fully I have set myself the task of illuminating this part of Mathematics and Physics, by discovering causes which on the one hand would satisfy the judgement of the ears, in establishing the number of the consonances, and the other melodic intervals, without trespass ing beyond what the ears bear, but which on the other hand would set up a clear and overt criterion between the numbers which form musical intervals and those which have nothing to do with the matter, and lastly which, with respect both to the archetype and to the Mind which uses the archetype to shape things to fit it, would have a kinship with the intervals, and so would rest on the clearest probability. For since the terms of the consonant intervals are continuous quantities, the causes which set them apart from the discords must also be sought among the family of continuous quantities, not among abstract num bers,^" that is in discrete quantity; and since it is Mind which shaped human intellects in such a way that they would delight in such an inter val (which is the true definition of consonance and discordance) the differences between one and the other, and the causes of such inter vals’ being harmonious, should also have a mental and intellectual essence, that is that the terms of the consonant intervals are properly knowable, but those of the dissonant intervals either cannot be prop erly known or are unknowable. For if they are knowable, then they can enter the Mind and into the shaping of the archetype; but if they are unknowable (in the sense which has been explained in Book I) then they have remained outside the Mind of the eternal Craftsman, and have in no way matched the archetype. But more will be said on these points when we describe the actual theory Chapter by Chapter; and may we embark on it with God’s help. Throughout we shall indeed speak of melody, that is harmonious intervals which are not abstract but realized in sound; yet to the educated ears of the mind the under lying reference throughout will be to the intervals abstracted from the sounds. For it is not only in sounds and in human melody that they yield their charm, but also in other things which are soundless, as we shall hear in the fourth and fifth Books.