Jean-Philippe Rameau · section 5 of 109

CHAPTER TWO. On the Different Ways in Which the Relations of Sounds May Be Known

To discover the relations of sounds, a string was chosen and stretched so that it could produce a sound. The string was then divided into several parts by movable bridges. By comparing with one another all the lengths resulting from this division, it was found that all the sounds or intervals capable of agreeing together were contained in the first five divisions of the string.

Some sought this relation in the relations between the numbers that express these divisions. Others considered separately the lengths resulting from the divisions and sought the relation in the numbers expressing those different lengths. Still others observed that sound could not be communicated to the ear without the participation of air, and sought the relation in the numbers expressing the vibrations of these different lengths. Without dwelling on the many other ways in which this relation may be known—through different thicknesses of string, different tensions produced by weights, wind instruments, and so forth—it was found, in a word, that all consonances* were contained in the first six numbers. String thicknesses and weights are exceptions: for these, the squares of those root numbers must be used. This led to attributing all the power of harmony to the power of numbers. Thereafter, all that remained was to apply them correctly to the operation on which one wished to found a system.

We must now observe that the numbers expressing the divisions of the string, or its vibrations, follow their natural progression, and that everything there rests on the rules of arithmetic. The numbers expressing the lengths of the string, however, follow a progression that reverses the first. This destroys some of the rules of arithmetic, or rather obliges us to reverse them, as we shall see in the proper place. Although the choice among these operations should make no difference to harmony, we shall attend only to those in which the numbers follow their natural progression, because everything is much more intelligible there.

* See the Table of Terms.