it is necessary in the next place to premise those things which ought to l»e readily known hy us concerning numbers, and the harmonic ratios of the soul, in order that we may not attempt in vain the interpretation of what follows It is neces¬ sary then-fore to premise such things as are usually mentioned m harmonic dis¬ cussions, viz. what sound, inter*al, and system are, and that the Pythagoreans did not assume the symphonies in harmony from any thing else than numbers, and not from all these, but from multiples and super-particulars. 1 or they said that the diatessaron is in a sesipiitertian ratio ; hut the diapente in a sesquial- ter; and the diapason in a duple ratio. And again they said, that the diapason and -it the same time diapente is in a triple, lnit the disdiapason in a quadruple ratio. .For the diapason and at the same time diatessaron, did not appear to them to be symphonious, because it consists in a multiple soper-parlienl ratio, \ i/. in the ratio of It to 3. For b is a medium between the two, producing with the less numlx-r a duple, but with the greater, a su! isesqiiitcrtiau ratio. These tilings therefon*, must he premised, and also that the si-squioetave is m the ratio of a tone; that the sesipiitertian ratio consists of two tones and a leimma; and the sesquialter of lliree tones and a leimma. 1 lint we shall afterwards learn what the ratio of the leimm a is. Moreover, the Pythagoreans said, that there arc- three genera of harmonies, the diatonic, the enharmonic, and the chromatic. Likewise, that the diatonic consists of a semitone (hut this which I now call a semitone is not properly so, hut a leimma), and of a tone, and another tone. But the enharmonic consists of a diesis, another diesis, and a ililone. Ami the chro¬ matic of a semitone, another si untune, and a trisemitone. But diesis is as it were the fourth part of a tone, not hi ing in reality a fourth, as neither is a leimma accurately a semitone. These things, however, we shall demonstrate in what follows.
But a 4 there are three genera, each of which is a certain division of the tetra- ehord, Plato appears to ha* e used the diatonic genus alone. I’or he thinks lit to divide the seMjuitrrtian ratios, into sesquioctaves and leimmas, hut not into enharmonic dieses; since some of the ancients called a semitone diesis. Plato, likewise, seems to have assumed this genus, 1 mean the diatonic, as more grand, simple, and generous, than the other genera; though the enharmonic: appears to lie more adapted to erudition And if it he requisite to declare my own predic¬ tion on this subject, the enharmonic genus, presides over all tin- hie which is
• After Xti/j/zarti 111 tlie original, it is necessary to su|>|dv from tlie version of riioma-us, the word* to Xe i' ik Tfutuv rdiur k«i Xti/iftfiTB i I relcr llie reader slio is desirous of thoroughly understand¬
ing what is here, and fat (her on, said, to in> Theoretic Arithmetic.
Hook Iii,]
TIM.tlS* OF PLATO.
divisible about bodies, 1 just as llie diatonic presides over the rational lie Hence the enharmonic genus, is adapted to instruct and discipline llie «iiv isihb life, lint the cliromatic genus presides orer tin corporeal idea itself. Helici: it is e/l'emuiate and ignoble. The enharmonic genus therefore is deseneiilv tbs i- plinativc, • Hence, Socrates in the lb public thinks lit to mention it particularly, in w hat he says about harmony. And Timauis knowing this, anil having beard Socrates asserting these things on the preceding day, at the same time constitutes the essence oi the soul through the diatonic, and not through the enharmonic genus; the latter, as we have said, being adapted to erudition. Tor on tln> account, the ancients called the leadt i s [or preceptors] of these diseiphm s Ilarmnnici |or skilh d in music]. Aristoxenus therefore, in the first hook of his Harmnnir Klemeuts, says, it happened that those wen: truly called Hannonici, who formerly employed themselves in what pertains to music. Tor bring solely engaged in bannonv they neglected every other pursuit. In which Aristoxenus also asserts what is wonderful, vi/. that the ancients had no knowledge of the diatonic diagram. Tor lie thus writes: “ As an indication of the truth of this, their diagram alone exhibits enharmonic systems, but no one ever saw a diatonic or chromatic dia¬ gram delineated by them." It is worthy of admiration, however, that lie should assert these things, since Plato exhibits a diagram according to the diatonic genus, and also Timauis himself. Perhaps therefore what Adrastus says is true, who derides Aristoxenus as a man of not very elegant manners,* but studious of appearing to say something new.
Plato, therefore, in the diatonic genus, makes a division of letrachords, and proceeds not only as far as to the diapason, hut also as far as to a quadruple diapason and diapente, adding likewise a tone. Or according to Severiis, Plato did not produce the tetrachords without a tone, but ended m a leimrna, and not in a tone. If, however, some one should doubt, how Plato produced the diagram to such an extent, let him attend to the words of Adrastus. Tor he says that Aristoxenus, extended the magnitude of hw multiform diagram, as tar as to the diapason and diatessaron, and the symphony of these, in consequence of prefer¬ ring the information of the ears to the derision of intellect. Hut the more modem musicians extended the diagram a-* far as to the fifteenth mode, vi/. to the thrice diapason and tone, in so doing looking solely to our utility, and thinking that those who contend in singing could not exceed this, nor their auditors judge clearly beyond
1 Instead of rou wrpi tom irw/i im firpiarcut m this place, il is neeewary lo read rqi *-«p« rmi empimw • * For m *«£k>« here, it u requisite from llie version of Thomarus, to read ro
hm. Plat.
PKOCLUS ON Tin:
it. Plato, however, looking to nature, constitutes the soul from all these, in order that it might proceed as far as to solid tuuuliers, as it ought to preside over bodies. For the progression a> far as to the quadruple diapason and diapeiite, necessarily follows the seven terms [or hounding number*]. llut tliis is evident from tlie greatest term liciug twenty s* veil. And thus much in answer to the doubt.
In short, then are these three things into which the consideration of harmony inav be divided. One of these is the « \posilion ot the seven parts. The second is the insertion of tin* two media. I'he third is the div ision of the sesipiitertian and sesquialter ratios, into sesquimtaves and leimnias. Hence some, a** Ad rust us, are acciistomed to make three triangles, and in one of them, v\ hit h is the least, to describe the seven parts, making the summit ot the triangle to be one of die parts, and distributing the other six about tins. In one ot the sides also, they describe I lie whole duple order, hut in th< other, the wind triple order. Murcov* r, in the other triangle, which is greater, ami contains the former, they increased the immhers, and attain in a similar manner in** ited two nu-dia, arranging the duple separate from the triple numbers ; and placing one of the parts at the summit. But in the third triangle, which comprelicnds liotli tin* others, tiny described after the same maim* r the whole di r, r rmi. Olliers attain, adopting a description in tlie form of the I* tier X, arrange the numbers successively, as m the s* etion of a rule, according to three centers, assuming the first, second, and third immhers, as we aFo shall do. I his method likewise is adopted by Porphyry and Severus. And such are the particulars which ought to In premised, ami also licit Plato divides this h ail into three parts, in the lii"t ot the three, discussing the seven parts, in which there are three duple, and three triple intervals, according to the geometrical middle, i c. according to the same ratios. But in the serum! part, he discusses the iiiseition of the other two media, viz. the harmonic and arith¬ metic, into each interval ot the duple and triple numbers. \nd m the third part, he considers the division of the sesipiitertiau and sesipiialter ratios, into scsijuioe- taves and leimnias, and as far as to these • Mends the discussion ot the parts of the soul.
It is necessary however to he well acquainted with such things as are said about the three media, and to know tin ir dilh-rcnccs, ami what the methods are through which they are discovered. The arithmetical medium, therefore, is that in which the middle t* rin exreeds and is e\e* eded by an equal 1 quantity, as tuav In* seen in all the numbers that are in a consequent order, conformably to the definition of Timams himself. But tlie harmonic medium is that in which the
* \ocv is omillttl in the original.
TIM /HUS OF PLATO.
middle term is exceeded by the greater, I»y tin* same part of the greater, liy which it exceeds the loss tmn, as in tin* mimlirrs 3, 1 ami 3. For here 1 is exceeded l>v ti 1»v *2, \\ is tin* tliird part of 3, ami it exceeds 3 the less term l»y 1, which is the third part of 3.’ Ami the geomi trical medium is that, in which there is the same ratio of the greater to the middle term, as there is of the middle to the less term.
The methods however of discovering these, must in the next place he unfolded hy us. Id two terms, then fore, he given, hetwivn which it is proposed to find an harmonic, and also an aritlum tie medium ; and let the terms have a duple ratio, as for instance 1*2 and 0. I take, therefore, the excess of the greater mini her above the less, which is evidently 3, and dividing it into two equal parts, 1 arid the half to the less nu miter, and make this the middle term. Hence is the arith¬ metical medium In-tween 12 and 3. For the excess is three, hotli of the greater ahovc the middle, and of the middle above the least term. Again, taking the difference of the extremes, w hich is 0, I multiply this by the less term, and the product is 30, and dividing this l»y the sum ol the extremes, i. e. hy 13, the quo¬ tient 2 is produced, which is the breadth ol the comparison * lo tins also, I add 0, ami I have the harmonic middle 3. y For hy that part of the greater term 1*2 hy which 3 is exceeded hy it, hy this part of the less term 0, 0 exceeds the less. For it is exceeded h\ the third pari of 1*2, and hy a third part ol 0 it exceeds 0. Again, let there lie a triple interval, as for instance lit and <»,’ adding these together I make 24, of which taking the half, 1 have the arithmetical middle 12. Again, taking the i \c< ss of 13 above 0, i e. 12, I multiply it hy the less term 0, and the product is 7*2. This 1 divide hy 24 the sum of the extremes, and 3 the breadth of the comparison is produced. Afterwards, 1 add this to 0, and 1 have ft for the harmonic medium, which exceeds ami is exceeded hy the same part of the extremes. Thus also, if 1 and *2 were the extremes, hy adding them together, and taking the half of both, 1 shall have 1 and the half of I, for the middle term ol the aritlm.etieal middle. Hut taking the excess of the greater term above unity, and multiplying it hy the h*ss term, viz. unity hy unity, I have t from both. Afterwards dividing* this hy 3, the sum of the extremes, I shall
' Il.immmr proportion mav also lir d, liool In lie that, in which the difference between the greatest anil niiddic li-rin, ;■> l<> the dilVcrrnic IkIwicii Ihe iniilillc and least term, as (lie greatest term is to the l< ait 4 Inis in i!ie iionihi rv (>, l, 3, as ti— t t—3 :: (i: 3 ; via. as 2 is In I, so is C to 3.
1 Fv the brtndth cf the con/iari-rn, Prochis means the ratio of the terms first proposed to etch other, winch m this iii'l nne i'duple.
* er is mmltcil ill the orii,uidl.
* For TapaXafjtir here, it i> obviously necessary to read firfuiur.