De animae procreatione in Timaeo

Plutarch

Plutarch. Plutarch's Morals, Vol. II. Goodwin, William W., editor; Philips, John, translator. Boston: Little, Brown, and Company; Cambridge: Press of John Wilson and Son, 1874.

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.

Now the more readily to find out these means Eudorus hath taught us an easy method. For after you have proposed the extremities, if you take the half part of each and add them together, the product shall be the middle, alike in both duple and triple proportions, in arithmetical mediety.

But as for sub-contrary mediety, in duple proportion, first having fixed the extremes, take the third part of the lesser and the half of the larger extreme, and the addition of both together shall be the middle; in triple proportion, the half of the lesser and the third part of the larger extreme shall be the mean. As for example, in triple proportion, let 6 be the least extreme, and 18 the biggest; if you take 3 which is the half of 6, and 6 which is the third part of 18, the product by addition will be 9, exceeding and exceeded by the same proportional parts of the extremes. In this manner the mediums are found out; and these are so to be disposed and placed as to fill up the duple and triple intervals. Now of these proposed numbers, some have no middle space, others have not sufficient. Being therefore so augmented that the same proportions may remain, they will afford sufficient space for the aforesaid mediums. To which purpose, instead of a unit they choose the six, as being the first number including in itself a half and third part, and so multiplying all the figures below it and above it by 6, they make sufficient room to receive the mediums, both in double and triple distances, as in the example below:—

Now Plato laid down this for a position, that the intervals of sesquialters, sesquiterces, and sesquioctaves having once arisen from these connections in the first spaces, the Deity filled up all the sesquiterce intervals with sesquioctaves, leaving a part of each, so that the interval left of the part should bear the numerical proportion of 256 to 243.[*](Timaeus, p. 36 A.) From these words of Plato they were constrained to enlarge their numbers and make them bigger. Now there must be two numbers following in order in sesquioctave proportion.

But the six does not contain a sesquioctave; and if it should be cut up into parts and the units bruised into fractions, this would strangely perplex the study of these things. Therefore the occasion itself advised multiplication; so that, as in changes in the musical scale, the whole scheme was extended in agreement with the first (or base) number. Eudorus therefore, imitating Crantor, made choice of 384 for his first number, being the product of 64 multiplied by 6; which way of proceeding the number 64 led them to having for its sesquioctave 72. But it is more agreeable to the words of Plato to introduce the half of 384. For the remainder of that will bear a sesquioctave proportion in those numbers which Plato mentions, 256 and 243, if we make use of 192 for the first number. But if the same number be made choice of doubled, the remainder (or leimma) will have the same proportion, but the numbers will be doubled, i.e. 512 and 486. For 256 is in sesquiterce proportion to 192, as 512 to 384. Neither was Crantor’s reduction of the proportions to this number without reason, which made his followers willing to pursue it; in regard that 64 is both the square of the first cube, and the cube of the first square; and being multiplied by 3, the first odd and trigonal, and the first perfect and sesquialter number, it produces 192, which also has its sesquioctave, as we shall demonstrate.

But first of all, we shall better understand what this leimma or remainder is and what was the opinion of Plato, if we do but call to mind what was frequently bandied in the Pythagorean schools. For interval in music is all that space which is comprehended by two sounds varied in pitch. Of which intervals, that which is called a tone is the full excess of diapente above diatessaron; and this being divided into two parts, according to the opinion of the musicians, makes two intervals, both which they call a semitone. But the Pythagoreans, despairing to divide a tone into equal

parts, and therefore perceiving the two divisions to be unequal, called the lesser leimma (or defect), as being lesser than the half. Therefore some there are who make the diatessaron, which is one of the concords, to consist of two tones and a half; others, of two tones and leimma. In which case sense seems to govern the musicians, and demonstration the mathematicians. The proof by demonstration is thus made out. For it is certain from the observation of instruments that the diapason has double proportion, the diapente a sesquialter, the diatessaron a sesquiterce, and the tone a sesquioctave proportion. Now the truth of this will easily appear upon examination, by hanging two weights double in proportion to two strings, or by making two pipes of equal hollowness double in length, the one to the other. For the bigger of the pipes will yield the deep sound, as hypate to nete; and of the two strings, that which is extended by the double weight will be acuter than the other, as nete to hypate; and this is a diapason. In the same manner two longitudes or ponderosities, being taken in the proportion of 3:2, will produce a diapente; and three to four will yield a diatessaron; of which the latter carries a sesquiterce, the former a sesquialter proportion. But if the same inequality of weight or length be so ordered as nine to eight, it will produce a tonic interval, no perfect concord, but harmonical enough; in regard the strings being struck one after another will yield so many musical and pleasing sounds, but all together a dull and ungrateful noise. But if they are touched in consort, either single or together, thence a delightful melody will charm the ear. Nor is all this less demonstrable by reason. For in music, the diapason is composed of the diapente and diatessaron. But in numbers, the duple is compounded of the sesquialter and sesquiterce. For 12 is a sesquiterce to 9, but a sesquialter to 8, and a duple to 6. Therefore is the duple proportion composed of the

sesquialter and sesquiterce, as the diapason of the diapente and diatessaron. For here the diapente exceeds the diatessaron by a tone; there the sesquialter exceeds the sesquiterce by a sesquioctave. Whence it is apparent that the diapason carries a double proportion, the diapente a sesquialter, the diatessaron a sesquiterce, and the tone a sesquioctave.

This being thus demonstrated, let us see whether the sesquioctave will admit a division into two equal parts; which if it will not do, neither will a tone. However, in regard that 9 and 8, which make the first sesquioctave, have no middle interval, but both being doubled, the space that falls between causes two intervals, thence it is apparent that, if those distances were equal, the sesquioctave also might be divided into equal parts. Now the double of 9 is 18, that of 8 is 16, the intermedium 17; by which means one of the intervals becomes larger, the other lesser; for the first is that of 18 to 17, the second that of 17 to 16. Thus the sesquioctave proportion not being to be otherwise than unequally divided, consequently neither will the tone admit of an equal division. So that neither of these two sections of a divided tone is to be called a semitone, but according as the mathematicians name it, the remainder. And this is that which Plato means, when he says, that God, having filled up the sesquiterces with sesquioctaves, left a part of each; of which the proportion is the same as of 256 to 243. For admit a diatessaron in two numbers comprehending sesquiterce proportion, that is to say, in 256 and 192; of which two numbers, let the lesser 192 be applied to the lowermost extreme, and the bigger number 256 to the uppermost extreme of the tetrachord. Whence we shall demonstrate that, this space being filled up by two sesquioctaves, such an interval remains as lies between the numbers 256 and 243. For the lower string being forced a full tone upward, which is a sesquioctave, it

makes 216; and being screwed another tone upward it makes 243. Which 243 exceeds 216 by 27, and 216 exceeds 192 by 24. And then again of these two numbers, 27 is the eighth of 216, and 24 the eighth of 192. So the biggest of these two numbers is a sesquioctave to the middle, and the middle to the least; and the distance from the least to the biggest, that is from 192 to 243, consists of two tones filled up with two sesquioctaves. Which being subtracted, the remaining interval of the whole between 243 and 256 is 13, for which reason they called this number the remainder. And thus I am apt to believe the meaning and opinion of Plato to be most exactly explained in these numbers.

Others, placing the two extremes of the diatessaron, the acute part in 288, and the lower sound in 216, in all the rest observe the same proportions, only that they take the remainder between the two middle intervals. For the base, being forced up a whole tone, makes 243; and the upper note, screwed downward a full tone, begets 256. Moreover 243 carries a sesquioctave proportion to 216, and 288 to 256; so that each of the intervals contains a full tone, and the residue is that which remains between 243 and 256, which is not a semitone, but something less. For 288 exceeds 256 by 32, and 243 exceeds 216 by 27; but 256 exceeds 243 by 13. Now this excess is less than half of the former. So it is plain that the diatessaron consists of two tones and the residue, not of two tones and a half. Let this suffice for the demonstration of these things. Nor is it a difficult thing to believe, by what has been already said, wherefore Plato, after he had asserted that the intervals of sesquialter, sesquiterce, and sesquioctave had arisen, when he comes to fill up the intervals of sesquiterces with sesquioctaves, makes not the least mention of sesquialters; for that the sesquialter is soon filled up, by adding the sesquiterce

to the sesquioctave, or the sesquioctave to the sesquiterce.

Having therefore shown the manner how to fill up the intervals, and to place and dispose the medieties, had never any person taken the same pains before, I should have recommended the further consideration of it to the recreation of your fancies; but in regard that several most excellent musicians have made it their business to unfold these mysteries with a diligence more than usually exact,—more especially Crantor, Clearchus, and Theodorus, all born in Soli,—it shall suffice only to show how these men differed among themselves. For Theodorus, varying from the other two, and not observing two distinct files or rows of numbers, but placing the duples and triples in a direct line one before another, grounds himself upon that division of the substance which Plato calls the division in length, making two parts (as it were) out of one, not four out of two. Then he says, that the interposition of the mediums ought to take place in that manner, to avoid the trouble and confusion which must arise from transferring out of the first duple into the first triple the intervals which are ordained for the supplement of both. --- But as for those who take Crantor’s part, they so dispose their numbers as to place planes with planes, tetragons with tetragons, cubes with cubes, opposite to one another, not taking them in file, but alternatively odd to even. [Here is some great defect in the original.]

--- Which, being in themselves permanently the same, afford the form and species; but being subject to corporeal division, they become the matter and subject to receive the other’s impression, the common mixture being completed out of both. Now the indivisible substance, which is always one and the same, is not to be thought to be incapable of division by reason of its smallness, like the most minute of bodies, called atoms. But as it is unmixed,

and not to be any way affected, but pure and altogether of one sort, it is said not to consist of parts, but to be indivisible. By means of which purity, when it comes in any manner whatsoever to approach and gently touch compounded divisible and differing substances, all their variety ceases and they crowd together into one habit by sympathy and similitude. If now any one will call that substance which admits corporeal separation matter, as a nature subject to the former and partaking of it, the use of that equivocal term will nothing disadvantage our discourse. But they are under a mistake that believe the corporeal to be blended with the indivisible matter. First, for that Plato does not here make use of any one of its names; whereas in other places he calls it the receptacle and nurse, capable of both receiving and fostering the vast infinity of created beings; not divisible among bodies, but rather the body itself parted and divided into single individuals. Then again, what difference would there be between the creation of the world and that of the soul, if the composition of each proceeded from both matter and the intelligible essence? Certainly Plato, as endeavoring to separate the generation of the body from that of the soul, tells us that the corporeal part was by God seated and deposited within it, and that it was outwardly covered and enveloped by it; and after he had thus wrought the soul to its perfection out of proportion, he then proceeds to this argument concerning matter, of which he had no occasion to make mention before when he was producing the soul, as being that which had not its existence from matter.