In Search of the Miraculous

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.

"The principle of dividing into eight unequal parts the period, in which the

vibrations are doubled, is based upon the observation of the non-uniform increase of

vibrations in the entire octave, and separate 'steps' of the octave show acceleration and retardation at different moments of its development.

"In the guise of this formula ideas of the octave have been handed down from

teacher to pupil, from one school to another. In very remote times one of these

schools found that it was possible to apply this formula to music. In this way was

obtained the seven-tone musical scale which was known in the most distant antiquity,

then forgotten, and then discovered or 'found' again.

"The seven-tone scale is the formula of a cosmic law which was worked out by

ancient schools and applied to music. At the same time, how-

ever, if we study the manifestations of the law of octaves in vibrations of other kinds

we shall see that the laws are everywhere the same, and that light, heat, chemical,

magnetic, and other vibrations are subject to the same laws as sound vibrations. For

instance, the light scale is known to physics; in chemistry the periodic system of the

elements is without doubt closely connected with the principle of octaves although this

connection is still not fully clear to science.

"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.

do ———————————————————————— do

FIG. 9

"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.

, do, re, mi, fa, sol, la, si , do,

FIG. 10

"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 3/2 , si 15/8, and do 2.

1 9/8 5/4 4/3 3/2 5/3 15/8 2

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do re mi fa sol la

si do

FIG. 11

"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 again retarded

"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 semitone 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:

do-re

re-mi

fa-sol

sol-la

la-si

and one between each of the following two notes:

mi-fa

si-do

"But in practice, that is, in music, instead of twelve intermediate semitones only

five are taken, that is one semitone between:

do-re

re-mi

fa-sol

sol-la

la-si

"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 a 'mechanical' sense, only those intervals between mi-fa and si-do

are called 'intervals'

"If we grasp its full meaning the law of octaves gives us an entirely new

explanation of the whole of life, of the progress and development of phe-

nomena on all planes of the universe observed by us. This law explains why there are

no straight lines in nature and also why we can neither think nor do, why everything