"The next fundamental law of the universe is the law of seven or the law of octaves.
"In order to understand the meaning of this law it is necessary to regard the universe as consisting of vibrations. These vibrations proceed in all kinds, aspects, and densities of the matter which constitutes the universe, from the finest to the coarsest; they issue from various sources and proceed in various directions, crossing one another, colliding, strengthening, weakening, arresting one another, and so on.
"The principle of the discontinuity of vibration means the definite and necessary characteristic of all vibrations in nature, whether ascending or descending, to develop not uniformly but with periodical accelerations and retardations. This principle can be formulated still more precisely if we say that the force of the original impulse in vibrations does not act uniformly but, as it were, becomes alternately stronger and weaker. The force of the impulse acts without changing its nature and vibrations develop in a regular way only for a certain time which is determined by the nature of the impulse, the medium, the conditions, and so forth. But at a certain moment a kind of change takes place in it and the vibrations, so to speak, cease to obey it and for a short time they slow down and to a certain extent change their nature or direction; for example, ascending vibrations at a certain moment begin to ascend more slowly, and descending vibrations begin to descend more slowly. After this temporary retardation, both in ascending and descending, the vibrations again enter the former channel and for a certain time ascend or descend uniformly up to a certain moment when a check in their development again takes place. In this connection it is significant that the periods of uniform action of the momentum are not equal and that the moments of retardation of 'the vibrations are not symmetrical. One period is shorter, the other is longer.
"In order to determine these moments of retardation, or rather, the checks in the ascent and descent of vibrations, the lines of development of vibrations are divided into periods corresponding to the doubling or the halving of the number of vibrations in a given space of time.
"Let us imagine a line of increasing vibrations. Let us take them at the moment when they are vibrating at the rate of one thousand a second. After a certain time the number of vibrations is doubled, that is, reaches two thousand.
"It has been found and established that in this interval of vibrations, between the given number of vibrations and a number twice as large, there are two places where a retardation in the increase of vibrations takes place. One is near the beginning but not at the beginning itself. The other occurs almost at the end.
"The laws which govern the retardation or the deflection of vibrations from their primary direction were known to ancient science. These laws were duly incorporated into a particular formula or diagram which has been preserved up to our times. In this formula the period in which vibrations are doubled was divided into eight unequal steps corresponding to the rate of increase in the vibrations. The eighth step repeats the first step with double the number of vibrations. This period of the doubling of the vibrations, or the line of the development of vibrations, between a given number of vibrations and double that number, is called an octave, that is to say, composed of eight.
"In the guise of this formula ideas of the octave have been handed down from teacher to pupil, from one school to another.
"The seven-tone scale is the formula of a cosmic law which was worked out by ancient schools and applied to music.
"A study of the structure of the seven-tone musical scale gives a very good foundation for understanding the cosmic law of octaves.
"Let us again take the ascending octave, that is, the octave in which the frequency of vibrations increases. Let us suppose that this octave begins with one thousand vibrations a second. Let us designate these thousand vibrations by the note do. Vibrations are growing, that is, their frequency is increasing. At the point where they reach two thousand vibrations a second there will be a second do, that is, the do of the next octave.
"The period between one do and the next, that is, an octave, is divided into seven-unequal parts because the frequency of vibrations does not increase uniformly.
"The ratio of the pitch of the notes, or of the frequency of vibrations will be as follows:
"If we take do as 1 then re will be 9/8, mi 5/4, fa 4/3, sol 3/2, la 5/3, si l5/8, and do 2.
"The differences in the acceleration or increase in the notes or the difference in tone will be as follows:
between do and re 9/8 : 1 = 9/8
between re and mi 5/4 : 9/8 = 10/9
between mi and fa 4/3 : 5/4 = 16/15 increase retarded
between fa and sol 3/2 : 4/3 = 9/8
between sol and la 5/3 : 3/2 = 10/9
between la and si 15/8 : 5/3 = 9/8
between si and do 2 : 15/8 = 16/15 increase
"The differences in the notes or the differences in the pitch of the notes are called intervals. We see that there are three kinds of intervals in the octave: 9/8, 10/9, and 16/15, which in whole numbers correspond to 405, 400, and 384. The smallest interval 16/15 occurs between mi and fa and between si and do. These are precisely the places of retardation in the octave.
"In relation to the musical (seven-tone) scale it is generally considered (theoretically) that there are two semitones between each two notes, with the exception of the intervals mi-fa and si-do, which have only one semi-tone and in which one semitone is regarded as being left out.
"In this manner twenty notes are obtained, eight of which are fundamental:
do, re, mi, fa, sol, la, si, do
and twelve intermediate: two between each of the following two notes:
and one between each of the following two notes:
"But in practice, that is, in music, instead of twelve intermediate semi-tones only five are taken, that is one semitone between:
"Between mi and fa and between si and do the semitone is not taken at all.
"In this way the structure of the musical seven-tone scale gives a scheme of the cosmic law of 'intervals,' or absent semitones. In this respect when octaves are spoken of in a 'cosmic' or in a 'mechanical' sense, only those intervals between mi-fa and si-do are called 'intervals.'
"What precisely does happen at the moment of the retardation of vibrations? A deviation from the original direction takes place.
"All this and many other things can be explained with the help of the law of octaves together with an understanding of the role and significance of 'intervals' which cause the line of the development of force constantly to change, to go in a broken line, to turn round, to become its 'own opposite' and so on.
"Such a course of things, that is, a change of direction, we can observe, in everything.
"The law of octaves explains many phenomena in our lives which are incomprehensible.
"First is the principle of the deviation of forces.
"Second is the fact that nothing in the world stays in the same place, or remains what it was, everything moves, everything is going somewhere, is changing, and inevitably either develops or goes down, weakens or degenerates, that is to say, it moves along either an ascending or a descending line of octaves.
"And third, that in the actual development itself of both ascending and descending octaves, fluctuations, rises and falls are constantly taking place."