De institutione musica V.7: Quem numerum proportionum Pythagorici statuant.
Those voices are called consonant which, when joined, produce mixed and pleasant sounds; dissonant, those which do not. And this is Ptolemy’s judgment on the difference of sounds. Now we must say in what he departed from the others in the placing of the consonances. The Pythagoreans reckon the diapente and the diatessaron simple consonances, and from these they join together the one diapason consonance. They also hold that the diapente with the diapason, and the double octave, are the former of the triple, the latter of the quadruple. But they do not reckon the diapason and diatessaron a consonance, because it does not fall under a superparticular or multiple comparison, but under a multiple superpartient. For this proportion of voices is as eight to three. If one places four in the middle of these, he will make the terms 8, 4, 3. Of these, eight to four make the diapason consonance, four to three the diatessaron. But eight to three is constituted in the multiple superpartient. What a multiple superpartient comparison is must be learned from the arithmetical books, and from what we set out in the second book of this treatise. The Pythagoreans place the consonances in the multiples and superparticulars, as was said before in the same second and in the fourth book, but separate the consonances from the superpartients and multiple superpartients. By what means the Pythagoreans link the diapason to the double, the diatessaron to the sesquitertian and the diapente to the sesquialter is to be sought from the second and fourth books of this treatise on music.