Now that there is something of majesty, something
great and divine in music, Aristotle, who was Plato's
scholar, thus labors to convince the world: ‘Harmony,’
saith he, ‘descended from heaven, and is of a divine,
noble, and angelic nature; but being fourfold as to its
efficacy, it has two means,—the one arithmetical, the
other enharmonical. As for its members, its dimensions,
and its excesses of intervals, they are best discovered by
number and equality of measure, the whole art being contained in two tetrachords.’ These are his words. The
body of it, he saith, consists of discording parts, yet concording one with another; whose means nevertheless
agree according to arithmetical proportion. For the upper
[p. 120]
string being fitted to the lowest in the ratio of two to one
produces a perfect diapason. Thus, as we said before,
nete consisting of twelve units, and hypate of six, the
paramese accords with hypate according to the sesquialter
proportion, and has nine units, whilst mese has eight units.
So that the chiefest intervals through the whole scale are
the diatessaron (which is the proportion of 4:3), the diapente
(which is the proportion of 3:2), and the diapason (which
is the proportion of 2:1); while the proportion of 9:8
appears in the interval of a tone. With the same inequalities of excess or diminution, all the extremes are
differenced one from another, and the means from the
means, either according to the quantity of the numbers
or the measure of geometry; which Aristotle thus explains,
observing that nete exceeds mese by a third part of
itself, and hypate is exceeded by paramese in the same
proportion, so that the excesses stand in proportion. For
by the same parts of themselves they exceed and are exceeded; that is, the extremes (nete and hypate) exceed
and are exceeded by mese and paramese in the same proportions, those of 4: 3 and of 3: 2. Now these excesses are
in what is called harmonic progression. But the distances
of nete from mese and of paramese from hypate, expressed
in numbers, are in the same proportion (12:8 = 9:6); for
paramese exceeds mese by one-eighth of the latter. Again,
nete is to hypate as 2:1; paramese to hypate as 3:2; and
mese to hypate as 4:3. This, according to Aristotle, is the
natural constitution of harmony, as regards its parts and
its numbers.

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