The same may be said of Tyrtaeus the Mantinean, Andreas the Corinthian, Thrasyllus the Phliasian, and several others, who, as we well know, abstained by choice from the chromatic, from transition, from the increased number of strings, and many other common forms of rhythms, tunes, diction, composition, and expression. Telephanes of Megara was so great an enemy to the pipe made of reed (called syrinx), that he would not suffer the instrument maker to join it to the flute (pipe made of wood or horn), and chiefly for that reason forbore to go to the Pythian games. In short, if a man should be thought to be ignorant of that which he makes no use of, there would be found a great number of ignorant persons in this age. For we see that the admirers of the Dorian composition make no use of the Antiginedian; the followers of the Antiginedian reject the Dorian; and other musicians refuse to imitate Timotheus, being almost all bewitched with the trifles and the idle poems of Polyidus. On the other side, if we dive into the business of variety and compare antiquity with the present times, we shall find there was great variety then, and that frequently made use of. For then the variation of rhythm was more highly esteemed, and the change of their manner of play more frequent. We are now lovers of fables, they were then lovers of rhythm. Plain it is therefore, that the ancients did not refrain from broken measures out of ignorance, but out of judgment. And yet what wonder is this, when there are so many other things necessary to human life which are not unknown, though not made use of by those who have no occasion to use them? But they are refused, and the use of them is altogether neglected, as not being found proper on many occasions. Having already shown that Plato neither for want of skill nor for ignorance blamed all the other moods and casts of composition, we now proceed to show that he really was skilled in harmony. For in his discourse concerning the procreation of the soul, inserted into Timaeus, he has made known his great knowledge in all the sciences, and of music among the rest, in this manner: After this, saith he, he filled up the double and treble intervals, taking parts from thence, and adding them to the midst between them, so that there were in every interval two middle terms. Plato, Timaeus, p. 36 A. See the whole passage in the treatise Of the Procreation of the Soul as discoursed in Timaeus, Chap. XXIX. (G.) This proem was the effect of his experience in music, as we shall presently make out. The means from whence every mean is taken are three, arithmetical, enharmonical, geometrical. Of these the first exceeds and is exceeded in number, the second in proportion, the third neither in number nor proportion. Plato therefore, desirous to show the harmony of the four elements in the soul, and harmonically also to explain the reason of that mutual concord arising from discording and jarring principles, undertakes to make out two middle terms of the soul in every interval, according to harmonical proportion. Thus in a musical octave there happen to be two middle distances, whose proportion we shall explain. As for the octaves, they keep a double proportion between their two extremes. For example, let the double arithmetical proportion be 6 and 12, this being the interval between the ὑπάτη μέσων and the νήτη διεζευγμένων ; 6 therefore and 12 being the two extremes, the former note contains the number 6, and the latter 12. To these are to be added the intermediate numbers, to which the extremes must hold the proportion, the one of one and a third, and the other of one and a half. These are the numbers 8 and 9. For as 8 contains one and a third of 6, so 9 contains one and a half of 6; thus you have one extreme. The other is 12, containing 9 and a third part of 9, and 8 and half 8. These then being the numbers between 6 and 12, and the interval of the octave consisting of a diatessaron and diapente, it is plain that the number 8 belongs to mese, and the number 9 to paramese; which being so, it follows that hypate is to mese as paramese to nete of the disjunct tetrachords; for it is a fourth from the first term to the second of this proportion, and the same interval from the third term to the fourth. The same proportion will be also found in the numbers. For as 6 is to 8, so is 8 to 12; and as 6 is to 9, so is 8 to 12. For 8 is one and a third part of 6, and 12 of 9; while 9 is one and a half part of 6, and 12 of 8. What has been said may suffice to show how great was Plato’s zeal and learning in the liberal sciences. 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. 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., 2×3). 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. But now as for the senses that are created within the body, such as are of celestial and heavenly extraction, and which by divine assistance affect the understanding of men by means of harmony,—namely, sight and hearing,— do by the very light and voice express harmony. And others which are their attendants, so far as they are senses, likewise exist by harmony; for they perform none of their effects without harmony; and although they are inferior to the other two, they are not independent of them. Nay, those two also, since they enter into human bodies at the very same time with God himself, claim by reason a vigorous and incomparable nature.