and this may be measured by the weight depending from the string, by means of which it is extended. Pythagoras here found that if one string were weighted with twelve pounds, and another with six (_λόγος διπλάσιος_, 1 : 2) it would produce the musical chord of the octave (_διὰ πασῶν_); the proportion of 8 : 12, or of 2 : 3 (_λόγος ἡμιόλιος_) would give the chord of the fifth (_διὰ πέντε_); the proportion of 9 : 12, or 3 : 4 (_λόγος ἐπίτριτος_), the fourth (_διὰ τεσσάρων_).[45] A different number of vibrations in like times determines the height and depth of the tone, and this number is likewise proportionate to the weight, if thickness and length are equal. In the first case, the more distended string makes as many vibrations again as the other; in the second case, it makes three vibrations for the other’s two, and so it goes on. Here number is the real factor which determines the difference, for tone, as the vibration of a body, is only a quantitatively determined quiver or movement, that is, a determination made through space and time. For there can be no determination for the difference excepting that of number or the amount of vibrations in one time; and hence a determination made through numbers is nowhere more in place than here. There certainly are also qualitative differences, such as those existing between the tones of metals and catgut strings, and between the human voice and wind instruments; but the peculiar musical relation borne by the tone of one instrument to another, in which harmony is to be found, is a relationship of numbers.