De institutione musica V.11: Quibus modis Ptolomaeus consonantias statuat.
Such is Ptolemy’s judgment on the opinion of the Pythagoreans. By what means he himself traces the proportions and numbers of the consonances must be begun from this point. Voices, he says, are among themselves either unisonous or not unisonous. Of the voices that are not unisonous, some are equisonant, some consonant, some emmelis, some dissonant, some ekmelis. Unisons are those which, struck singly, give one and the same sound; equisonants, those which, struck together, make from two a single and, in a way, simple sound, as is the diapason and its doubling, the double octave. Consonants are those which make a compound and mixed sound, yet a pleasant one, as the diapente and diatessaron. Emmelis are all those which are indeed not consonant, yet can rightly be fitted to melody, as are those which join the consonances. Dissonants are those which do not mingle their sounds and strike the sense unpleasantly; ekmelis, those which are not admitted in the joining of consonances, of which we shall speak a little later in the division of the tetrachords. Since, then, the equivocal comparisons are nearest to the univocal, it is necessary that to equal numbers be joined that inequality of numbers which is nearest to equals. Now the inequality nearest to the equality of numbers is the double. For it is both the first species of multiplicity, and the greater number, when it exceeds the smaller, exceeds it by an amount equal to that smaller itself, as two exceeds one by one, which is equal to that same unit. Rightly, therefore, the double proportion is fitted to the equisonants, that is, to the diapason, and the double-double, that is, the quadruple, to the double octave. The proportions which divide the double proportion, being the first and greatest, are to be fitted to those consonances which divide the equisonance of the diapason. Hence the diapente is linked to the sesquialter comparison and the diatessaron to the sesquitertian. The consonances joined with the equisonants produce other consonances: the diapente and diapason in the triple, the diatessaron and diapason in the proportion of eight to three. Emmelis are those which divide the diapente and diatessaron, such as the tone and the other proportions of which we shall speak a little later in the division of the tetrachords, that is, their simple parts.