On the Institution of Music (De institutione musica)

...does not produce a certain strike and percussion. Indeed, a strike or percussion can in no way exist unless motion has preceded it. For if all things were immobile, one thing could not run into another so that one might be struck by the other. But with all things standing still and lacking motion, it is necessary that no sound be produced. Therefore, sound is defined as a percussion of the air that remains unbroken until it reaches the sense of hearing. original: "sonus perculsio aeris indissoluta usque ad auditum"

Of motions, however, some are faster and others are slower. And of those motions, some are more infrequent original: "rariores," referring to low frequency and others are more dense original: "spissiores," referring to high frequency. For if one looks at a continuous motion, one must perceive therein either speed or slowness. If, indeed, someone moves their hand, they will move it with either a frequent or an infrequent motion. And if the motion is slower or more infrequent, it must produce low-pitched sounds by that very slowness and infrequency of striking. But if the motions are swift and dense, high-pitched sounds must be produced.

For this reason, if the same string is tightened further, it sounds higher; if it is loosened, it sounds lower. For the greater the tension, the faster the strike it returns; it recurs more swiftly and hits the air more frequently and densely. However, a string that is looser produces relaxed and slow strikes; these are infrequent due to the very weakness of the striking, and it does not tremble for long.

Nor should it be thought that as often as a single string is struck, only one sound is emitted, or that there is only one percussion in these instances; rather, the air is hit as many times as the trembling string strikes it. But because the speeds of the sounds are joined together, no gap is perceived by the ears. Thus, a single sound strikes the hearing—whether low or high—even though each consists of many strikes: the low sound from slower and more infrequent ones, and the high sound from swifter and dense ones.

It is just as if someone were to carefully fashion a cone—which they call a top original: "turbo," a spinning top—and draw upon it a single small streak of red or another color, and then spin it with as much speed as possible; then the whole cone seems to be stained with the red color. This is not because the whole thing is actually red, but because the speed overtakes the red parts and does not allow them to appear [individually]. But more on these things later.

Therefore, since higher sounds are produced by denser and swifter motions, and lower sounds by slower and infrequent ones, it is clear that by a certain addition of motions, a high pitch is stretched from a low one; whereas by a subtraction of motions, a low pitch is relaxed from a high one. For quantity already consists of a plurality of motions. In those things where a plurality of motions creates a difference, it is necessary that they consist of a certain numerical relationship. Indeed, every small amount relates to a plurality just as a number compared to a number. Of those things which are viewed according to number, some are equal to each other, and some are unequal. For this reason, sounds are also partly equal and partly distant in their inequality. But in those voices which do not disagree by any inequality, there is no consonance at all. For consonance consonance is the concord of voices dissimilar among themselves brought back into one.

On the types of inequality. Chapter 3.

Those things which are unequal maintain the measures of inequality among themselves in five ways. For either one is surpassed by the other by a multiple; or by single parts; or by several parts; or by a multiple and a part; or by a multiple and parts. And the first kind of inequality is called multiple. It is a multiple when the larger number contains the smaller within itself entirely, either twice, or three times, or four times, and...