La Iniciación · Año V, Nº 45 · January 1946

1The octave is the limit within which the other sounds must be sought, in order to introduce variety into unity.

2The numbers l and 2 are the beginning and end of music; they are its alpha and omega; they constitute the two extremities of the system, the two ends of the staff. [Sic: the Spanish prints alta, for alfa; and the one is printed with a letter ell.]

3The octave is the circle in which all the sounds of music are contained, and from which we have to extract them.

4We come to obtain this by introducing into it, between the two terms of the octave (1 and 2), an arithmetic mean.

5We thus obtain 1-1 ½-2 and, by reduction, 2-3-4 — that is, do sol do: the fifth do-sol, and the fourth sol-do. (N. B. — In all that follows the notes must be read as ascending, going from left to right.)

6The note 3 (sol), thus obtained, is the fifth, or twelfth, already given by the sonorous body and constituting the timbre.

7This new note is no longer confounded, like the octave, with the fundamental.

8The reason is that its harmonics are no longer the same.

9Those of the fundamental, do, are: do sol do mi sol.

10Those of the fifth, or twelfth, sol, are: sol re sol si re.

11This first subdivision of the string gives us, then, a little variety; but evidently it is not enough.

12Let us carry this same principle of division further.

13Let us introduce a new arithmetic mean between 2 and 3, do and sol.

14We shall thus have 2-2 ½-3 or, by reduction, 4-5-6: do mi sol.

15In sum, the division of the octave has given us the fifth (do sol) and its complement, the fourth, sol do, whose relations are 2-3 and 3-4.

16The second division, that of the fifth, has given us the major third do-mi, the minor third mi-sol, whose relations are 4-5 and 5-6; and their inverses, the minor sixth, mi-do (relation 5-8), and the major sixth sol-mi (relation 6-10, or 3-5).

17We have thus obtained, by subtraction, the same series that sound had already given us through its harmonics.

18In effect, the series of the overtones is 1, 2, 3, 4, 5, 6, etc.; that of the divisions of the octave is likewise 1, 2, 3, 4, 5, 6 — and the following figures, if one wished to continue introducing arithmetic means into each new relation.

19The term 7 would come from the introduction of a mean between 3 and 4 (sol and do), which would give 3-3 ½-4, or 6-7-8.

20We have not adopted this division, nor that of the major third — which would give the relations 8-9-10 — because, as we have just said and as we shall see further on, the same or analogous sounds can be obtained by simpler means from the theoretical point of view and, above all, from the practical point of view.[Sic: the Spanish prints les mismos.] · 8

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