The double and triple intervals therefore are filled with middles, and with wwjuioctaves and leimmas. The whole likewise of this diagram has nine lcimmas, and twenty-four sesqnioctaves. Tor the intervals are le^s in number than the terms by one. It also proceeds as far as to a quadruple diapason, and a diapente ami tone. Adrastus how ever, who was a lover of the arts, makes the figure, as we have said, in the form of the letter 7 ,; and places the terms in certain triangles. And in the interior triangle, indeed, he places the ratios that are in monadic imml>ors [i. e. that consist in the numl»ers within t* nj ; but in the trian¬ gle next to this, the sextuple of these numbers, w hich have two middles according to each duple or triple interval. And in the outermost triangle he places the terms winch make the whole of the lieforc-mentioned diagram. What we have said, however, will become manifest from the delineation. But between the double and triple intervals, lie inscribes all the above-menlioiii d numbers, which we have not thought tit to add. being unwilling to introduce a [needless] multitude of terms. Tor such a disposition of terms, and the insertion of the same numbers twice, is unmethodical. Tor many of the same media are found between the duple and triple intervals; since the triple intervals themselves consist of duple terms and sesquiallers. What is s,*ul by Plato, therefore, has been elucidated by ns. Tor two media have been discov* red between all tb*‘ duple and triple inter¬ vals. And from these media scsquialter anil sesquiterlian ratios having lieen produced, these are divided by the sesquioctave; a portion being left in both, which has the ratio of the leimma. From these likewise, assumed in an orderly
* Fur tiaTatrvr here, it 11 necessary lo read iir\nirtoy.
* The word* within the brackets arc supplied from the version of Thoma-us, where however it ■■ necessary to read scsquioctavuin instead of sesqutallerum.
[book in-
manner, the terms which comprehend the whole diagram will be found to be thirty-four only.*
Siuce, however, the Pythagoric Timieus says that the terms of the diagram are thirty-six, and vet assumes the same extremes as Plato, viz. 301 and 10308, in order that these philosophers may not appear to Ik* in any respect discordant with each other, let us show how the other two ti rm.s are inserted. These tin n there¬ fore li. e. the Pythagoreans] were willing that there should not only he the ratio of the leimma in the diagram, hut also that of the apotome, whieh they twire dis¬ covered, both m radical iuuiiIrts, and in those alone which are the triple of these. Adding likewise one term to each, they introduced this into the diagram. But Plato makes no mention of the apotome-; whence aKo we being satisfied with the leimma, have alone employed the above-enumerated terms. Pur how, since lie assumes the diatonic genus, could he make use of the apotome, the sesquioetave not being divided in this genus; the apotome being produced when the sesqui- octave is divided ? For the part of the sesquioetave which remains idler the leimma, is the apotome. Hence, since Plato does not mention the apotome, and it is not possible for it to occur in the dialotiii genus, it would lx ridiculous in us to end« avuur to insert other terms, in ordt r that we may h ive the apotome, the thirty-four terms Ih mg sufhi i« lit to the completion of the sesquioct; ves and lcimmas. It seems also, that the number 3 1 is adapted to the diatonic genus, in
* Thu will he evident from the following diagram, which also will he found to contain a quadruple diapason, together with the diapente and tone.
Sesq. ^esq Lemma, ‘'esq. Sesq Sesq l eimma. JS4, 432. 4Sb. 512. 576. 648. 779. 768. Tlie first duple interval.
Sesq. Sesq. Leimma. Sesq. Vsq. Sesq. Leimma. 86 4. 972. 102 4. 1152. 129<i. 1 45S. 1536. The second duple interval.
Sesq Sesq Lviinma. Sesq. Sesq. Leiuima. 1728. 194 4. 2187.* 2304 2592. 2‘Jl6. 3l>72.
The third du|ile interval.
Sesq. St-sq Leimma. Sesq. Sesq. la-imma. 3456. 3SS8. 437 1. 4008. 5184. 5832. Mil. The fourth duple interval.
* 2187 •» the octuple of 273
Sesq. Sesq. Ijcimma. Sesq.
I lie third triple interval.
lu this diagram it must be observed, that the last term of each interval forms a sesquioetave with the first term of the interval that it next m order The first triple interval likewise begins with tin term 384, and ends at the term 1152. The second triple interval begins at 1152 and ends at 3456. And the third triple interval begins at 3430, and ends at 10308.
Hook Ii!.]
which alone the sesquinctave ratio is found. For it consists of the terms 10 and 10, which are to eacli other in a sesquiortave ratio. For the sesquialter and sesrjuitertian ratios, and leiinmas, are also in the other genera; hut the sesqui- octaves are found in this alone of the three genera. Hence tliis ratio of the sesquinctave, very properly produces hy composition the niunher of the parts: and this being the second, is adapted to the second progression of the soul from the first intelligible principles.
It there lore \ie assume the less- term of the third double, viz. 1.13#!, and airun the se>quiorta\e of this 1728. and afterwards the sesquioctave of this- I‘M1, and again the tritone of this 2187, there will he one interval of the extremes. Because however 2018 h is a sesquilertian ratio to 1550, hut 11)14 has to it the ratio of the leimmn.it is necessary that 2187 should make an apotome to 2048. For an apotome is, as we have before said, that which remains to a tone, after the lcimma. In a simi¬ lar manner also, hv assuming in the third triple, 4800, which contains the tritone * 8501, * and also assuming 8144, which makes a sesquitertian ratio to 4608, hut to 5882 lias the ratio of the leimma, we shall necessarily have the apotome in the ratio of 0501 to 01 14. which are triple of the radiral terms that were before discovered hv us m the third double. For it is evident that the ratio of the apotome is radically in those terms. For 2187, and 2018, are demonstrated to
}>o first terms to each other hv the theorem of subtraction: first terms- bein°*
necessarily such as arc least. The multitude indeed of the terms descrilied hy Tinneus, is demonstrated l»y Philol.nts; hut the diagram of Plato proceeds without tin* ratio of the apotome. And thus much concerning these particulars.
Since however we have lie fore observed, that if of one term two numliers are assumed, one of which is the double, hut the other tin* triple of it, the mean which between the duple terms is arithmetical, is between the triple terms harmonic, hut the duple term is between the triple terms, an arithmetical mean, we will now concisely elucidate and at the same time demonstrate this theorem. I> t then b he the double of a, hut c the triple of it, and between a and /;, let the arithmetical mean he d. I say that will happen which is enunciated in the proposition. For since b is the double, hut c tin triple of a, of sueli numbers as a is two, b will lie four, and c will be six. Hence of sm h as b is four, c will bo six. By so much,
’ For TfttTor here, it is neccs«ary In re lit rptrovuv.
* According In t)u moderns, a Inlouc is a dissonant inters si, oliierssisc tailed a «iipcrfluous fourth.
It is also a kind of redundant third, consisting of two tones, and two semitones, one greater and one less. And the ratio of the tntoue i a, |j to J2. Tins however does not accord with the ratio of the tritone given by I’roclus, both in this plate, and above. For I fins is uot to 0561 as 32 to 45, bul as 32 to 45 IJJJ. Nor in the other instance above, is lyii to 21 b7 as 32 to 45.
Proclus Os the
therefore doe* c exceed b as b exceeds a. Hence b is an arithmetical mean between a and c. Again, Iveeause of such as a is two, of such b is four, hut the arithmetical mean between them is d; hence d will |e three of such nuinhers, as a is two, and b four.' But of such as b is four, of such c is six. Of such there¬ fore, as a is two, of such d is three, and c is six. ! Ieuce d compared to a and to c, will produce an harmonic midtile/ l or by the same part of the greater it is ex¬ ceeded bv the greater, and by the same part of the less exceeds the less. And thus much concerning this particular.
Seierus, however, thinks that this diagram should not end in a tone, but in the leimma, because Plato terminates in this all the discussion concerning the division of the soul, lu order, therefore, that it may terminate in the leimma, Severus transfers some of the terms, and makes all of them to he thirty-four. But as in the thirty-fourth term, the half of unity occurs, lie doubles the terms, and makes the first part to !>e 788, vvhii h is the double of 381. <M this, therefore, lu* places the sesipiioctave 881, and of this again the sesipiioctave 872. l'o this also he adapts according to the leimma, 1021. But ot this he takes the sesipiioctave 1152; of this the sesipiioctave 1288; and ol this again the sesijuioclaie 1158. But to this he adapts according to the leimma 1538, and places the sesipiioctave of this, 1728, and of this again the sesipiioctave 1811. To this likewise he adapts according to the ratio of the h iiuma, 2187. And of this lie assumes the sesqui- octave 210 1; of this the sesquioctave 2002; and of this again the sesipiioctave 2818. To this also he assumes 3782, which has the ratio of the leimma to it, to this the sesipiioctave 3158; a id to this in a similar manner 3888. 'l'o this likewise he adapts as the leimma 4371; of this he assumes the sesquioctave 4808; of this the sesipiioctave 5181; and of this again the sesipiioctave 5832. ‘l'o this also he adapts according to the ratio of the leimma, 8144 ; and ot this he assumes the ses¬ quioctave 8812 ; of this the sesipiioctave 7778; and of this again the sesrpeoctave 8718. To this likewise he adapts as a leimma P218. But of this he makes the sesquioctave 10388; of this also 11881; and of this, agon, he makes the sesqui- octave 13122. 'l’o this he adapts as a leimma 13824 ; ol this also he assumes the sesquioctave 15552 ; of this the sesipiioctave 17 188; and of this, again, the sesipii¬ octave 18783. And to this he adapts 20838, having the ratio of the leimma. As far as to this, therefore, he gives completion to the diagram, making the leimma to 1 m* the end ; except that in these terms, there is first the sesquilei tian, after¬ wards the sisqnialter, then the sesquitertian, and afterwards the sesquialter ratio.
* Hence a* 2, 3, V are in arithmetical proportion, so likewise will their eijuiinulliples, 2x. in. lx.
* For 2, 3, anil 6 arc id harmonic proportion, and therefore their eijnimuitiples also are in the same proportion. For 6 ex* eeds 3 bv (he half of 6, and 3 exceed* 2 bv the half of 2.
TIM /El'S OF PLATO.
Arxl again the sesqniterlian, afterwards the sescjuialfor, and then three sesquialter* in a following order, as is evident from tin* above description.