De institutione musica V.16: Quomodo Aristoxenus vel tonum dividat vel genera eiusque divisionis dispositio.
This diatessaron, then, Aristoxenus divides according to the genera by the following reasoning. He divides the tone into two parts and calls that the semitone. He divides it into three, and calls the third part the diesis of the soft chromatic. He divides it into four, and the fourth part together with its own half, that is, with the eighth of the whole tone, he calls the diesis of the hemiolic chromatic. Since these things are so, and since according to him the division of the genera is twofold, one genus is softer, the other more spirited. The softer is the enharmonic, the more spirited the diatonic. Between these stands the chromatic, partaking of spiritedness and softness. According to this order, then, there arise six differences of the mixed genera: one of the enharmonic, three of the chromatic, that is, of the soft chromatic, the hemiolic chromatic and the tonic chromatic, and the remaining two of the soft diatonic and the spirited. The division of all these according to Aristoxenus is as follows. Since the fourth part of the tone was said to be called the enharmonic diesis, and since Aristoxenus does not compare the voices themselves with one another but measures the difference and interval of the voices, according to him the tone is of 12 units. Its fourth part, the enharmonic diesis, will therefore be 3. Since the diatessaron consonance is made up of two tones and a semitone, the whole diatessaron will be constituted of twice 12 and 6 units. But since it often happens that, if we wish to carry the parts down to eighths, we fall not on whole numbers but on certain fractions, the whole diatessaron is to be made 60, the tone 24, the semitone 12, the fourth part, which is called the enharmonic diesis, 6, and the eighth 3. The eighth joined to the fourth, that is, 6 with 3, to make the diesis of the hemiolic chromatic, will be 9. With these things thus established, the three genera, enharmonic, chromatic and diatonic, seem to Aristoxenus to have these properties: that some of them are called dense, others not. Dense are those in which the two lower proportions do not exceed in magnitude the one that is added toward the high; not dense are those in which the two proportions can exceed the one remaining. The enharmonic and chromatic are dense, the diatonic not dense. So the enharmonic is divided, according to Aristoxenus, thus: 6, 6, 48, so that between the lowest string and the one near the lowest there is the fourth part of a tone, which is called the enharmonic diesis, the tone being constituted of 24 units. Likewise, let the second interval, from the second string from the lowest to the third, be the same fourth part of a tone, 6. The rest, which remain of the 60 that belong to the whole proportion, are placed between the third string from the lowest and the highest fourth, 48. And the two proportions placed toward the low, that is, 6 and 6, do not exceed the one remaining toward the high, that is, 48. The soft chromatic makes this division: 8, 8, 44, so that 8 and 8 are two thirds of tones. For the tone is, as has been said, of 24 units, and the third part of the tone is called the diesis of the soft chromatic. Likewise the diatessaron of the hemiolic chromatic is divided thus: 9, 9, 42. For the diesis of the hemiolic chromatic is the eighth part of the tone with the fourth, that is, from 24, 6 with 3. Likewise the division of the tonic chromatic according to Aristoxenus is this: 12, 12, 36, so that in the two intervals it sets a semitone each, and what is left in the last. And in all these the two proportions that are nearest the lower string by no means exceed in magnitude the one that is placed toward the high. For they are, as has been said, of the dense genera. The dense genera are the enharmonic and the chromatic. The diatonic division is itself also twofold. The division of the soft diatonic is as follows: 12, 18, 30, so that 12 is a semitone, 18 a semitone and the fourth part of a tone, and 30 what remains. Of these 18 and 12 make 30 and are not exceeded by the part that remains. Likewise the division of the spirited diatonic is such that it has a semitone and two whole tones, that is, 12, 24, 24, of which 24 and 12, that is, 36, are not exceeded by the remaining part toward the high, but rather exceed it. According to Aristoxenus, then, there is the aforesaid division of the tetrachords, which the description below will show.