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
'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.
1000 2000
fig. 7
'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
'Approximately:
1000 2000 fig. 8
'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
'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
, 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
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
'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
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
'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 with us is
'What precisely does happen at the moment of the retardation of vibrations? A deviation from the original direction takes place. The octave begins in the direction shown by the arrow:
'But a deviation takes place between mi and fa; the line begun at do changes its direction
and through fa, sol, la, and si it descends at an angle to its original direction, shown by the first three notes. Between si and do the second 'interval' occurs—a fresh deviation, a further change of direction.
| Fig. 14 |
