Of Continuous and Discrete Quantity

The Consonances Repeat in Natural Order

Musical consonances in multiplying — or to say it better, in being enumerated — follow almost that same order that is found in the numbers placed before the Decenary and arranged in natural order; beyond which one sees that no new number is added, but rather that those numbers come to be repeated: for just as after the Decenary follows the Undenary, and after that the Duodenary, and similarly the others in order, so likewise after the Diapason and the Diapente — which are placed in their natural order without any mean — all the other consonances go on repeating according to the order shown, almost to infinity. For placing the Diatessaron after the two named consonances, the Ditone is immediately added to it; then the Semiditone; and to this the Diatessaron is again added; and so they go on repeating and multiplying in that order.

And although in this way one could proceed to infinity, should it be necessary, as is manifest — nonetheless Music does not receive the infinite, because of it one has and can have no science whatsoever; and the intellect is not capable of it. So that when it is necessary to know the reason for some thing, one makes use only of a determinate quantity, and by that means comprehends and ascertains the truth of what one seeks.

Quantity Divided into Continuous and Discrete

But all things necessarily falling under number and being gathered — whether they be one or more — under this name of Quantity, which philosophers, on account of its excellence, have judged equal to, and eternal together with, Substance: therefore they immediately divided it into two parts, that is into Continuous and Discrete.

Continuous they named that whose parts are conjoined at a common boundary — as the Line, the Surface, the Body — and beyond these, Time and Place, and all those things which are attributed to Magnitude.

Discrete they said to be that whose parts are not conjoined at any common boundary, but remain distinct and separate — as is Number, Speech, a Flock, a People, a heap of grain or of anything else — to which things the name of Multitude is fitting. For many separate parts compose themselves in their extremes, as is seen in Number, which beginning from Unity — below which there is no other lesser number — when multiplied to infinity without encountering any impediment, comes to generate the other numbers. So that its nature is very much in conformity with the Multiplex genus in proportions: for considered in numbers, it is finite in any number whatever, but renders itself infinite through increase — since it can be multiplied to infinity, as we shall also see in the Multiplex, which is finite in its species, even though those species can extend themselves to infinity.

The Analogy to Proportional Genera

Continuous quantity, on the other hand, which begins from a finite quantity, receives an infinite division, losing the quantity of the measure in the increase of the parts and multiplying it in the decrease. For if a line ten feet long were divided into eight, and these into four, and thus the remainder were always divided into two parts, that line would be found to be infinitely diminished, and the number of its parts multiplied to infinity. Such nature serves the Superparticular genus in proportions: for the more it proceeds toward greater numbers continuing the natural order, the more it shows itself diminished — since the difference of the terms containing its species is always of lesser quantity, so that its species being infinite, each individual species is found to be finite.