Jean-Philippe Rameau · section 25 of 109

CHAPTER ELEVEN. On the Fourth and Eleventh

It is appropriate here to distinguish the fourth from the eleventh. The latter interval has not yet been known under that name, but has always been confused with the fourth. Hence opinions have divided: some maintain that the fourth is consonant, others dissonant. Those following the order of ratios could not imagine it dissonant; those considering practice found it difficult to treat it as consonant. Their disagreement arises only from a failure to understand one another.

First, Zarlinoᵃ treats it as a consonance in practice, gives examples, and invokes the authority of the Greeks and, more strongly, that of ratios. He even maintains that two successive fourths have approximately the same effect as two fifths, because, he says, the fourth is a perfect consonance—and, according to our observations, because it is an inversion of the fifth. He does not add these last words, but unknowingly proves them in his examples, besides mentioning the matter in his Dimostrationi Harmoniche.ᵇ Does he not make its full power felt when he says that the modern Greeks of his time used the fourth in the lowest parts without putting another consonance beneath it as a base? Notice that he cannot help recognizing a base in harmony and shows that he desires it when he does not hear it. It is therefore only by understanding this base as implied—a base always residing in the fifth's lowest sound—that we can prove inverted chords pleasing to the ear.

a Third part, chapters 60 and 61, fols. 291, 292, 293, and 294; chapter 5, fols. 177 and 178.

b Ragionamento secondo, definition X, fols. 83 and 84.

He also cites the eleventh as a replication of the fourth and the ninth as a replication of the second. Certainly every interval has its double, triple, quadruple replication, and so forth, as we have said several times; but intervals with different properties must not be confused. If the second and ninth have been distinguished, why not distinguish the fourth and eleventh, whose difference is much greater? The second and ninth are both dissonances, whereas the fourth is consonant and the eleventh dissonant. The second arises from inversion of a fundamental chord; the ninth, conversely, is formed by adding a sound to that chord, cannot be inverted, and is prepared and resolved differently from the second. This obliges us to distinguish them. Likewise, the fourth arises from inversion of the perfect chord and, as a consonance, has unrestricted progression. The eleventh, conversely, is formed by adding a sound to the seventh chord, cannot be inverted, and must be prepared and resolved. Ratios supply further proof, as observed in Book I, Chapter VIII, Article IV, p. 39. Is this not enough to convince us that these intervals must be distinguished according to the different chords they compose, without fastening upon their relationship when considered in isolation? The name of the interval always identifies a chord primary in its kind, composed only of sounds within that interval's compass. Such are the seventh, ninth, and eleventh chords; the latter two names are implied in the augmented-fifth and augmented-seventh chords. Thus the fourth, found only in an inverted chord where it represents the fifth, is consonant. The eleventh identifies a chord primary in its kind, whose constituent sounds must lie within that eleventh, and is therefore dissonant. If we figure it with 4, this is to follow ordinary usage.