On the Institution of Music (De institutione musica)

It is in vain, however, that anyone attempts to investigate what they truly desire through varied judgment. Thus, the Pythagoreans Followers of Pythagoras, who believed the universe was governed by numerical patterns travel a middle path. For they do not surrender all judgment to the ears, yet certain things are explored by them only through the ears. Indeed, they measure consonances original: "consonantias"; musical intervals that sound pleasing when played together themselves by ear. But as to the distances by which these consonances differ from one another, they no longer entrust this to the ears—whose judgments are dull—but permit it to rules and calculations. Thus, sense original: "sensus"; physical perception acts as a kind of obedient servant, while reason original: "ratio"; mathematical calculation is the judge and commander. For even though almost all the significant moments of the arts and of life itself are brought forth through the occasion of the senses, there is no certain truth in them, no comprehension of the true, if the power of judgment is absent. For the sense itself is confused by both the very large and the very small. It cannot perceive the smallest things because of the tiny size of the objects themselves, and it is often overwhelmed by the largest. Just as with voices: if they are very small, the hearing can barely capture them; if they are very large, the hearing is deafened by the very intensity of the sound.

How Pythagoras Investigated the Proportions of Consonances

This, therefore, was the primary cause why, having set aside the judgment of the ears, Pythagoras moved toward the weight of fixed rules. He did not trust human ears, which are changed partly by nature, partly by external accidents, and partly by age itself. Nor did he devote himself to instruments, within which much variety and inconstancy often arise. For instance, if you were to look at strings: either more humid air might dull the strike, or drier air might sharpen it; or the thickness of the string might make the sound coarser, or a thinner string might make it finer, or in some way change the state of its former consistency. And since he found this reliability lacking in other instruments as well, and considering all these things to be least

worthy of trust, and burning with desire, he inquired by what method he might firmly and constantly explain the weights of consonances. Therefore, by a certain divine providence, while passing a blacksmith's shop, he overheard the struck hammers echoing a single concordance from their diverse sounds. Stunned by this, he approached the work he had long sought. After considering it for a long time, he thought that the strength of those striking the hammers created the diversity of sounds. To prove this more clearly, he ordered the men to exchange hammers with one another. But the property of the sounds did not remain with the strength of the men's arms; instead, it followed the exchanged hammers. Once he noticed this, he examined the weight of the hammers. And as there were by chance five hammers, those that responded to one another in the octave original: "diapason"; the interval spanning eight notes consonance were found to be double in weight. He also discovered that the one which was double another was a sesquitertian original: "sesquitertium"; a 4:3 ratio of yet another hammer, to which it sounded a fourth original: "diatesseron". To another, which was joined to it by the consonance of a fifth original: "diapenthe", he found that same double of the higher hammer to be sesquialter original: "sesquialterum"; a 3:2 ratio. Those two to which the aforementioned double hammer was proven to be sesquitertian and sesquialter were found, when compared to each other, to be in a sesquioctave original: "sesquioctaua"; a 9:8 ratio, corresponding to a whole tone proportion. The fifth hammer, however, was rejected, as it was dissonant with all the others. Therefore, although before Pythagoras musical consonances were called partly octave, partly fifth, and partly fourth (which is the smallest consonance), Pythagoras was the first to discover by this method the proportion by which this concord of sounds was joined together. And to make what has been said clearer, let there be, for example, four weights of hammers contained in the numbers written below: xii. viii. viiii. vi. 12, 8, 9, 6 Therefore, these hammers which were of weights 12 and 6...