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 I

INTRODUCTION TO THE TIMyEUS.

411

and a ſemitone ; and the difdiapafon, or quadruple proportion, i. e. four to one, which contains twelve tones. But it is neceffary to obferve further concerning a tone, that it cannot be divided into two equal parts ; becaufe it is compofed from a fefquiofitave proportion, and 9 cannot be divided into two equal parts.

Hence, it can

only be divided into two unequal parts, which are ufually called femitones ; but by Plato Asmara, or remainders.

But the lefifer part of a tone was

called by the Pythagoreans diefis, or divifion; and this is furpaffed by a fefquitertian proportion by two tones ; and the remaining greater part, by which the tone furpaffes the lefs femitone, is called apotome, or a cutting off-

But as it is requifite to explain the different kinds of harmony, in order to a knowledge of the compofition of fymphonies, let the reader take notice

that harmony receives a triple divifion, into the Diatonic, Enharmonic, and Chromatic.

And the Diatonic genus takes place when its divifion conti¬

nually proceeds through a lefs femitone and two tones. monic proceeds through two diefes.

But the Enhar¬

And the Chromatic is that which

afcends through two unequal femitones and three femitones; or Tpivi^troviov, according to the appellation of the antient muficians.

And to thefe three

genera all mufical inftruments are reduced, becaufe they are all compofed from thefe harmonies. But though there were many different kinds of inftru¬ ments among the antients, yet the Pythagorean and Platonic philofophers ufed only three—the Monochord, the Tetrachord, and the Polychord ; to which three they refer the compofition of all the other inftruments.

From

among all thefe, therefore, Plato affumes the diatonic harmony, as more agreeable to nature ; in which the tetrachord proceeds through a lefs femi¬ tone and two tones; tending by this means from a lefs to a greater femi¬ tone, as from a more (lender to a more powerful matter, which pofteffes a fimple form, and is at the fame time both gentle and robuft.

And hence,

as all inftruments are converfant with thefe three kinds of harmony, Plato, fays Proclus, in confequence of preferring the diatonic harmony, alone ufes two tones when he orders us to fill up the fefquitertian, (efquiodtave and femitone intervals. With refpeft to the firft numbers, which are evidently thofe defcribed by Plato, the firft three of thefe, 1, 2, 3, as Syrianus beautifully obferves, may voL. 11.

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be