La Iniciación · Año V, Nº 45 · January 1946

1CHAPTER III. Law of consonances. The word consonance signifies: "sounds with".

2Two sounds are, then, the more consonant the more frequently their vibrations coincide — or, in other terms, the simpler the relations of their vibrations are.

3The octave consounds with the fundamental: their vibrations meet once in 2; in the twelfth, once in 3; in the fifth, 2 in 3; in the fourth, 3 in 4; in the major third, 4 in 5, etc. The relation 1-2, given by the first division of the string, forms, then, the most consonant interval — so consonant that, as we have seen, the two sounds have the same harmonics and make but a single sound.

4The relations 2-3 and 3-4, given by the first subdivision — by the introduction of a middle term into the octave — form the fifth and the fourth.

5The coincidence of the vibrations is still fairly frequent; that is why the fifth and the fourth (with the octave) are called perfect consonances1.

6The second subdivision, by the introduction of a mean into the fifth, gives the intervals of thirds and their inverses, the major and minor sixths.

7These intervals still consound often enough not to be disagreeable to the ear, but dissound enough to excite it — and even to please a trained ear more than the perfect consonances2.

8This exciting character of the thirds and sixths is very well felt.

9It is what makes striking the mediant (third) and the leading note, seventh note of the scale.

10We could carry the divisions further: introduce an arithmetic mean between 3 and 4 and obtain the relations 6-7-8; between 4 and 5 and obtain the relations 8-9-10; between 5 and 6 and obtain 10-11-12; between 8 and 9 and obtain 16-17-18.

11But it is not the case: 1) because limits must be set to variety — it must not make us lose sight of unity; 2) because we can obtain many of these sounds, and others more, in an indirect manner, simpler and more practical; 3) because the signs necessary to represent sounds corresponding to these relations would become very complicated and, consequently, the writing, reading and execution of music composed with these elements would be very difficult and, moreover, of little effect.

12All the intervals different from the perfect and imperfect consonances form part of the category of dissonances.

13And they are the more dissonant the more complex their numerical relations are. (To be continued) 1 Musicians are not yet in agreement respecting the fourth — notwithstanding that Descartes long ago resolved the difficulty in his Compendium Musicae.

142 It is not the sweetest things that are most agreeable to the senses, but those that titillate them in a more tempered manner; thus salt and vinegar are frequently more agreeable to the tongue than fresh water.

15And that is why music receives the thirds and the sixths — and at times even the dissonances — as well as the unisons, the octaves and the fifths. (Descartes – Traité de l'Homme, Paris edition, 1729, p. 143.)

Our translation, period-faithful. Citable as our translation, never as the source.