The Commentaries on the Timaeus of Plato — trans. Thomas Taylor

If, therefore, we also wish to condense these terms with harmonic and arithme¬ tic midrib's, which I join;; inserted, make sesquialtcr and sesquiterlian intervals, the intermediate numbers will 1m* 384, and 788, the double of 381; 512, which produce* an harmonic, and 578, which makes an arithmetic middle. But if we

1 The number 61 has not a sesquitertian in whole numbers. Fur as 3 is lo 4 <o is 01 to SjJ.

* For as fl is to 8, so is 243 to 216.

* For as b is to 0 so is 576 to 618 ; auil also so is 64b to 7 -9*

p. 64

PROCLUS ON THE

[book III wish to assume the above-mentioned middles of the triple interval, viz. of 304 and 1152, then 576 will preserve the harmonic ' middle, which filled for us the arithmetic * middle in the double 1 interval; and 760 will be the arithmetic middle, which was the greater extreme of the double interval. Again, if we wi-di to assume the same middles of the duple and quadruple, i. e. of the middles between the terms 760 and 1536, the former of which is the double of 301, and the latter the quadruple, the harmonic midtile will bo 1024, ami 1152 t ic arithuietic middle. If also we widi to condense the second triple, the terms of which are 1152, and 3456 [the former being the double of 576, and the latter the triple of 1152 ] then 1720 will give us the harmonic, and 2301 the arithmetic middle. And if we wish to condense the third double, w hich consists in the terms 1536 and 3072, then 2 ) 16 will be the harmonic, and 2301 the arithmetic middle. But if we wish to condense the third triple, with similar middles, but I mean the fifth and seventh part, the extremes will be for us 3456, and 10360; but the harmonic middle will be 5101, and the arithmetic 6012. If again, we should condense each of the sesquin rlians which present themselves from these mid¬ dles, and sesquialters, with sesquioctaves and a leimina, this will lie manifest to us after the whole exposition, when we exhibit the whole diagram with all the terms in a consequent order, which has indeed 24 s ( squioctaves hut 9 leimmas.

These things therefore, having been elucidated by us, we shall observe thus much concerning tlie leimmu, that as it is not possible to divide any superpartitular into equal ratios, a semitone cannot be assumed in numb* rs ; hut taking the ratios which are contiguous to each other, vi/ the seventeenth and the sixteenth part, and demonstrating that the seventeenth part is greater than that which is called the leimina, and vvh eh is h ss than an accurate M initone, it is inferred that the leimmu and also the seventeenth part are less than a semitone. But that it is h s S than a semitone, is demonstrated as follows; I>‘t there lie given the term 16 and the sesquioctave of it 16. Betw een these placin': 17 it will divide the scsquioctave into unequal ratios, vvhii i will he near to the semitonic interval, since 17 dillirs from the extremes by unity alone. And it is evident that it will make a greater ratio with the less t< rm; because in all arithmetical proportion, the ratio is greater w hieli is in the lcssteims, so that the seventeenth part is less than a semitone. .Moreover, the leimmu is less than the seventeenth part, as is evident from the terms exhibited by IMato. For since 256 has to 243 the ratio of the leimina, as we shall demonstrate in what

» For aottlfiiTTui)*' here. m the orijiinut it i» necessary lo read npfioy^y.

’ For mj» alto here, we mini re-d apiD^ntttv.

1 And for rp*»Xa<riy, il is requisite U> read }iw\avi{-"'•] TJMyEUS OF PLATO. , ;5 follows where we shall show that the radical ratio of the leimma is in these nuinhers , anti since 256 exceeds 243 hv less than the seventeenth part of it;

for if exceeds it hy 13 unities, hut the seventeenth part of 2 13 is more than 13;_ tills bring the case, much umre is the ratio of the h iinma less than the seinitonie interval. Hence the ratio which remains to the completion of a tone, and which is called the ratio oi an apotome, is necossanlj greater than a semitone.

I arther still, this may also he deinonstrated after another manner, as follows: Lctthe iiuiiiIhts 2-i 6 and 243 Ire given, and let there lie assumed three numlMTS in a conseipient order, in a ratio of this kind ; from 256 indeed, 63336, but Irom 243 39049, and from both 6>Z08. These three numlien, therefore, are analogous in the ratio of the leimma, which, if it is a semitone, will be the tonic ratio ot the extremes. lint if it is greater than a semitone, that also wi ! he greater than a tone; hut if less, that also will he less. The sesrpiioctaTe how ever of 3309, is 664310 i. But this is greater than the greater term.

lint if it is greater than a semitone, that also wi ! he greater than a tone; hut if less, that also will he less. The sesrpiioctaTe how ever of 3309, is 664310 i. But this is greater than the greater term.

Afu r another and a third way the same thing may also Ire demonstrated, viz. that a tone cannot he divided into two erjual parts, hating the same ratio as that of 230 to 243. For if we take the eighth part of 213, w hicli is 30 \ and add this i irf to it, we shall make 73 w hich has a sesquioctave ratio to 243. You see therefore that 230 has to 2 43 a less ratio than 273’, to 230. For 250 has to 243 a superpartient ratio, exceeding it hy -Vj ; hut 273 } exceeds

2oO, hy ,\\\ __ But the ratio is greater which exceeds hy seven¬ teen and more, than that which alone exceeds hy thirteen, according to the ratio of excess. A tone, therefore, cannot he divided into equal parts, hut this is ts < leimma , as I lato also calls it, and that which has the greater ratio is apotome, as musicians arc accustomed to denominate it. For let 273 have to 213 a srsquioctave ratio, hut 230, to the same 213, the ratio of the leimma, which has a loss ratio than that of the seventeenth part, it is evident that 273’, which hasthe ratio of a tone to 213, will bate to 250 the ratio of the apotome, which is the remainder of the leimma, lreing greater than the seventeenth part, which we have demonstrated to lx* less than the ratio of the leimma. If therefore we multiply these right times, we shall find the first numbers winch in perfect unities have the ratio of the apotome. For the octuple of 243 is 1914, of 25G is 2048, of 273j, 2187. Hence the ratio of the apotome in radical numlrers (tv Wj}it<Tiv ’) is that of 2107 to 2040. And we shall he in want of these three terms which arc in a consequent order, in the diagram. Let then these terms be, 243, 2«>6, 273 m . But on account of [the fraction] let the octuple of these be the

II .-fyiijr is a primary ratio, being as it were a bottom or root, from which other ratios arise.

Tim. Flat. Vou II. j

I'KOCLUS ON THE

p. 111

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