But these remarks have been extended somewhat beyond what the occasion requires. However, it is clear that men make Five an attribute of the god, which at one time of itself creates itself, like fire, and at another time out of itself creates ten, like the universe. And in music, which is especially pleasing to him, do we imagine that this number plays no part? For the main application of harmony, so far as it can be put into words, is concerned with chords. That these are five, and no more, reason convinces anyone who wishes, by perception alone without employing reason, to pursue these matters on the strings and stopsCf. Plato, , 530 d - 531 c.; for they all have their origin in numerical ratios. The ratio of the fourth is four to three,Cf., 1018 e. that of the fifth is three to two, and that of the octave two to one; that of the octave plus the fifth is three to one,Cf. 429 e, infra. and that of the double octave four to one. The extra chord which the writers on harmony introduce, naming it the octave and the fourth extra metrum, does not deserve acceptance, since we should be favouring the unreasoning element in our sense of hearing contrary to reason, which is as much as to say, contrary to law. Now if I may omit any discussion of the five stops of the tetrachord,Cf. 430 a, infra, and , 1021 e and 1029 a. and the first five tones or tropes or harmonies, whatever be their right name, from the changes in which, through a greater or a less tension, the remaining lower and higher notes are derived, I must ask whether, although the intervals are numerous, or rather of infinite number, yet the elements of melody are not five only,Cf. 430 a, infra. quarter tone, half tone, tone, a tone plus a half tone, and double tone; and there is, in the range of notes, no additional space, either smaller or greater within the limits set by the high and the low, which can yield melody.

There are many other examples of this sort of thing, said I, which I shall pass over. I shall merely adduce Plato,Plato, , 31 a. who, in speaking about a single world, says that if there are others besides ours, and ours is not the only one, then there are five altogether and no more.Cf., 421 f, 422 f, 430 b, and 887 b. Nevertheless, even if this world of ours is the only one ever created, as Aristotle, i. 8-9 (276 a 18). also thinks, even ours, he says, is in a way put together through the union of five worlds, of which one is of earth, another of water, a third of fire, a fourth of air; and the fifth, the heavens, others call light, and others aether, and others call this very thing a fifth substance (Quintessence), which alone of the bodies has by nature a circular motion that is not the result of any compelling powrer or any other incidental cause. Wherefore also Plato, apparently noting the five most beautiful and complete forms among those found in Nature, pyramid, cube, octahedron, icosahedron, and dodecahedron, appropriately assigned each to each.