De animae procreatione in Timaeo

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.