On the Different Ways in Which the Relationship of Sounds Can Be Known to Us

In order to know the relationship of sounds, a string was chosen, stretched in such a way that it could render a sound; this string was then divided into several parts by means of moveable bridges, and it was found that all the sounds or intervals which could accord together were contained within the first five divisions of this string, by comparing reciprocally each length that resulted from this division.

Some have sought this relationship in the ratio borne between the numbers that mark these divisions; others, taking apart the lengths that result from these divisions, have sought this relationship in the ratio borne between the numbers that mark these different lengths; others again, having remarked that the communication of sound to the ear could not occur without the participation of air, have sought this relationship in the ratio borne between the numbers that mark the vibrations of these different lengths; and without stopping at several other ways in which this relationship can be known to us — such as in the different thicknesses of the

string, in its different tensions by means of weights, or in wind instruments, etc. — it has been found, in a word, that all the Consonances* are contained within the first six numbers, with the exception of the thicknesses and weights, for which one must use the square roots of these radical numbers; which has given occasion to attribute all the force of Harmony to numbers, doing nothing thereafter but making a just application of them to the operation upon which one wishes to found one’s system.

It must be remarked at present that the numbers which mark the divisions of the string, or the vibrations thereof, follow their natural progression, and that everything there is founded on the rules of Arithmetic; whereas the numbers that mark the lengths of the string follow an inverted progression of the first — which destroys a part of the rules of Arithmetic, or rather obliges us to reverse them, as we shall see in its place. But if the choice of these operations must be indifferent with regard to Harmony, we shall attach ourselves only to those where the numbers follow their natural progression, because the whole is far more intelligible there.