23. 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 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. 24. But, according to natural philosophy, both harmony and its parts consist of even, odd, and also even-odd. Altogether it is even, as consisting of four terms; but its parts and proportions are even, odd, and even-odd. So nete is even, as consisting of twelve units; paramese is odd, of nine; mese even, of eight; and hypate even-odd, of six (i.e., 2 x 3). Whence it comes to pass, that music—herself and her parts—being thus constituted as to excesses and proportion, the whole accords with the whole, and also with each one of the parts.