De institutione musica I.10: Quemadmodum Pythagoras proportiones consonantiarum investigaverit.
This, then, was chiefly the reason why Pythagoras, leaving the judgment of the ears, passed to the weight of rules. He trusted no human ears, which are changed partly by nature, partly by external accidents, and varied partly by age itself; nor did he give himself to instruments, in which much variation and inconstancy often arises, since, if one wished to examine strings, now a damper air would dull the stroke, now a drier would sharpen it, or the thickness of the string would make the sound lower, or its fineness thin it, or some other way would change the state of its former constancy; and as it was the same with the other instruments, he judged all these unsettled and of little trust, and, long in suspense, he inquired by what reason he might learn the moments of the consonances firmly and constantly. Meanwhile, passing by the workshops of smiths by a kind of divine prompting, he heard the hammers beating and from different sounds one concord, in a way, resounding. Astonished, therefore, he approached what he had long sought, and, considering for a long time, concluded that the diversity of sounds was produced by the strength of the strikers; and to make this clearer, he ordered them to exchange the hammers among themselves. But the property of the sounds did not cling to the men’s arms; it followed the exchanged hammers. When he had noticed this, he examined the weight of the hammers, and as there happened to be five hammers, two were found to be of double weight, and these answered each other according to the diapason consonance. He also found that the same one, which was double the other, stood in sesquitertian relation to another, to which, that is, it sounded the diatessaron. To a certain other, which was joined to it by the diapente consonance, he found the same upper one to be sesquialter. As for the two, to which the upper double was proved to be sesquitertian and sesquialter, when weighed against each other they were found to preserve the sesquioctave proportion. The fifth was rejected, since it was dissonant with all. Though, therefore, before Pythagoras the musical consonances were named, some diapason, some diapente, some diatessaron (which is the smallest consonance), Pythagoras first found in this way by what proportion this concord of sounds is joined together. And to make clearer what has been said, let there be, for example, four weights of hammers contained in the numbers written below: 12, 9, 8, 6. The hammers, then, of 12 and of 6 weights sounded the diapason in the double concord. The hammer of 12 weights to the hammer of 9, and the hammer of 8 weights to the hammer of 6 weights, were joined by the epitrite proportion in the diatessaron consonance. The hammers of 9 and 6, and of 12 and 8, blended in the diapente consonance. The 9 to the 8 resounded in the sesquioctave proportion the tone.