“Again, from sol to the last fa there are four sounds, sol, re, mi, fa, which are perfectly similar to the first four, ut, re, mi, fa. Nevertheless these are more grave, but those are more acute. And as from ut 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 constitute the whole diapason; or that the diapason is divided into one diatessaron, and one diapente. For from ut 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 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.
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 1, 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 in capacity. 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.
In consequence, however, of the great veneration paid to the tetractys by the Pythagoreans, it will be proper to give it a more ample discussion, and for this purpose to show from Theo of Smyrna,[106] how many tetractys there are: “The tetractys,” says he, “was not only principally honored by the Pythagoreans, because all symphonies are found to exist within it, but also because it appears to contain the nature of all things.” Hence the following was their oath: “Not by him who delivered to our soul the tetractys, which contains the fountain and root of everlasting nature.” But by him who delivered the tetractys they mean Pythagoras; for the doctrine concerning it appears to have been his invention. The above-mentioned tetractys, therefore, is seen in the composition of the first numbers 1. 2. 3. 4. But the second tetractys arises from the increase by multiplication of even and odd numbers beginning from the monad.
Of these, the monad is assumed as the first, because, as we have before observed, it is the principle of all even, odd, and evenly-odd numbers, and the nature of it is simple. But the three successive numbers receive their composition according to the even and the odd; because every number is not alone even, nor alone odd. Hence the even and the odd receive two tetractys, according to multiplication; the even indeed, in a duple ratio; for 2 is the first of even numbers, and increases from the monad by duplication. But the odd number is increased in a triple ratio; for 3 is the first of odd numbers, and is itself increased from the monad by triplication. Hence the monad is common to both these, being itself even and odd. The second number, however, in even and double numbers is 2; but in odd and triple numbers 3. The third among even numbers is 4; but among odd numbers is 9. And the fourth among even numbers is 8; but among odd numbers is 27.
{ 1. 2. 4. 8. }
{ 1. 3. 9. 27. }