Life of Pythagoras — Iamblichus, trans. Thomas Taylor (1818)

p. 233

or AB, and the chord ED. But from the intervals which are between the sounds there are two semitones, viz. one between m1 and fa, denoted by the letters L, N, and the other between the last mz and fa, denoted by the letters N, E. The remaining five intervals are entire tones. It must, also, be observed, that from ut to the first sol is the consonance diapente, which contains three tonic intervals, and one semitone; nevertheless in all there are five sounds, ut, re, mt, fa, sol.

Again, from sol to the last fa there are four sounds, sol, re, mt, fa, which are perfectly similar to the first four, ut, re, m1, fa. Nevertheless these are more grave, but those are more acute. And as from wt to the first fa is the diatessaron, so likewise from sol to the last fa is another diatessaron; from which, in the last place, it must be observed, it follows that the two consonances diatessaron and diapente canstitute the whole diapason; or that the diapason is divided into one diatessaron, and one diapente. For from wt to sol is the diapente, but from sol to the last fa is the diatessaron. ‘This will also be the case if we should say that from ut to the first fa is the diatessaron, as is evident from the division of the chord; but from the first fa to the last fa is the diapente, as is evident from the four intervals of the chord, three of which are tones, and the remaining interval is a semitone, which also in the other diapente were contained between ut and sol.

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

234 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 1s 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, lLychanos, — Index.

Fa, Mese, — Media.

Sol, Paramese, — Postmedia.

Re, Trite, — Tertia.

Mi, Paranete, — Antepenultima.

Fa, Nete, — Ultima, vel suprema.

p. 235

P. 80. I swear by him who the tetractys found.

The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of I, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad ncapacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter, forms the symphony diapente; 4 to 3, which Is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason.