De institutione musica IV.7: Ratio superius digestae descriptionis.
VII. Three tetrachords have therefore been described by us by such a method. For every tetrachord sounds the consonance of the fourth. Nete hyperboleon and nete diezeugmenon contain the symphony of the fourth in the three kinds, that is, in the diatonic, in the chromatic or in the enharmonic. The consonance of the fourth consists of two tones and a lesser semitone. This has been divided in the following way through the three kinds in the tetrachords written above. In the diatonic kind, which is the first, paranete hyperboleon, that is 2,592, holds to nete hyperboleon, that is 2,304, the distance of a tone, which we have marked with a little sign of this kind. Again, trite hyperboleon of the diatonic kind, which is 2,916, holds to paranete hyperboleon of the diatonic kind, which is 2,592, again the difference of a tone, which we have marked with a like little sign. But nete diezeugmenon to trite hyperboleon, that is 3,072 to 2,916, yields a semitone, which we have marked with such a little sign. And this is the whole space of nete diezeugmenon and nete hyperboleon, of two tones and a semitone. But the same two tones and a semitone are divided in the chromatic kind by this method. The second kind, which is the chromatic, has been described in this way. For paranete hyperboleon chromatice, which is 2,736, compared with nete hyperboleon, which is 2,304, contains the space of paranete hyperboleon of the diatonic kind to nete hyperboleon, which is one tone, that is two semitones, the greater and the lesser, and it divides again the space of paranete hyperboleon of the diatonic and nete hyperboleon. For it was so made, which is half a tone, but not exactly, because, as has been most fully shown above, a tone cannot be divided into two equal parts. We have therefore marked this space of three semitones, that is of a tone and a semitone, in this way. Again, paranete hyperboleon chromatica to trite hyperboleon holds a part of a tone, that is a semitone, which was left of the two tones contained between trite hyperboleon of the diatonic and nete hyperboleon. When four semitones have been taken away, what remains of the whole tetrachord is the space of a semitone, which is contained between nete diezeugmenon and trite hyperboleon. This tetrachord consists, therefore, also of two tones and a semitone, divided in one space into three semitones, and in two spaces into two semitones. The three spaces are contained by four strings. In the enharmonic kind, however, the means of knowing it is the easiest. For from that which is nete hyperboleon, that is 2,304, paranete hyperboleon enarmonios, that is 2,916, is distant two whole tones, which we have marked in this way. There remains, then, of the two tones and a semitone of the whole tetrachord, one semitone, which is contained between nete diezeugmenon and paranete hyperboleon enarmonios, which we have divided into two dieses, with trite hyperboleon enarmonios interposed between, and the space of the diesis we have marked in this way. Thus, then, the hyperboleon tetrachord has been described by us. This done, let us come to the diezeugmenon tetrachord, nor need we linger on the same remarks in the rest, since from this description an example can be taken also for the others.