“FD, therefore, will be two-thirds as well of the whole CD as of the whole AB. Let the tactus now be placed in F, and let AB and FD be struck at the same time, and a consonance very sweet and perfect will indeed be heard, yet not so sweet as the diapason. This the ancients called diapente (i. e. through five chords), because the first and the fifth chord produce this consonance. But according to proportion it is called sesquialter, because the chord AB is sesquialter to FD, and consequently the sounds of these chords also are in the same ratio. But sesquialter ratio is when the greater quantity AB contains the less FD once, and the half of it besides. It is, indeed, commonly called the fifth, because it is composed from the first sound _ut_, and the fifth, _sol_. “Again, let the same chord be cut into four equal parts in the points H, E, I, [Illustration: A——————————————————————————B C——————————————————————————D | | | | | | | | | K L H F M N E G I] “so that the chord HD, may be three-fourths of the whole CD. The tactus, therefore, being placed in H, let AB and HD be struck at one and the same time, and a consonance will be heard, indeed, yet more imperfect than the preceding two. This was called by the ancients diatessaron, i. e. through four chords or sounds, for a similar reason to that by which the former were denominated. With reference, however, to the ratio of the chords and sounds, it is called sesquitertian, because the greater AB contains the less once, and a third part of it besides. But it is now commonly called a fourth, because it is found between the first sound _ut_, and the fourth _fa_. If now the point F be added in the preceding figure, and at one and the same time two chords HD and FD are compared in arithmetical ratios, we shall find that the greater HD will have to the less FD a sesquioctave[105] ratio, and the sound of the greater HD to the less FD will have the same ratio, i. e. in modern terms, that between _fa_ and _sol_ there is a sesquioctave ratio. But if these two sounds are heard together, they will be discordant to the ear. Again, the distance between these sounds _fa_, _sol_, or between the chords HD and FD, or between the two harmonic intervals HD and FD, the ratio of which was sesquioctave, was called by the ancients a tone. Afterwards they divided the whole of CD into nine equal parts, the first of which is divided in K, so that the whole CD may have to the remainder KD, which contains eight of those parts, a sesquioctave ratio. This, in like manner, will be the interval of a tone, the first sound of which, i. e, of the whole CD, is now called _ut_, but the second sound of the rest of the chord KD is called _re_. Afterwards they in a similar manner divided the remainder KD into nine parts, the first part of which is marked in the point L. And for the same reason between the chord KD and the chord