“Now again, let the tactus be placed in I; but I is the fourth part of the whole CD. Let, also, AB and ID be struck at one and the same time, and the sweetest consonance, called bisdiapason, will be produced; which is so denominated, because it is composed from two diapasons, of which the first is between AB or CD, and ED, but the second is between ED and ID; for the ratio of these is double as well as of those. The ratio, also, of the bisdiapason is quadruple, as is evident from the division; and is commonly called a fifteenth, because from the first _ut_ to this sound, which is also denominated _fa_, there would be fifteen sounds, if the interval EI were divided after the same manner as the first CE is divided. “Farther still, let GD be a third part of the whole CD, and let the tactus be placed in G. Then at one and the same time let AB and GD be struck, and a sweet consonance will be heard, which is called diapasondiapente, because it is composed from one diapason contained by the interval CE, or the two chords CD, ED, and one diapente, contained by the interval EG, or the chords ED, GD. For the chord ED is sesquialter to the chord GD; which ratio constitutes the nature of the diapente. The proportion, also, of this consonance is triple. For the chord AB or CD is triple of GD; and it is commonly called the twelfth, because between _ut_ and _sol_, denoted by the letter G, there would be twelve sounds, if the interval EG received its divisions. From all which it is manifest by the experience of the ear, that there are altogether five consonances, three simple, the diapason, the diapente, and the diatessaron; but two composite, the bisdiapason, and the diapasondiapente.” In the last place, it is necessary to observe that those ancient Greeks differently denominated these sounds, _ut_, _re_, &c. For the first, i. e. the gravest sound or chord, which is now called _ut_, they, denominated hypate, and the others in the following order: Ut, Hypate, i. e. Principalis. Re, Parhypate, — Postprincipalis. Mi, Lychanos, — Index. Fa, Mese, — Media. Sol, Paramese, — Postmedia. Re, Trite, — Tertia. Mi, Paranete, — Antepenultima. Fa, Nete, — Ultima, vel suprema. P. 109. _I swear by him who the tetractys found._