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Musical
Numerology
By
Don Robertson, 1970 with revisions in 2003
Copyright
1970, 2003 by Don Robertson
(First appeared in the book Kosmon)
I wrote the article Musical
Numerology in 1970 and
published it in Denver, Colorado in a book called Kosmon,
a book
created by myself and two other people. It
was in this book that I first published my findings about the
duochord and proclaimed
that rock and roll music was taking a downward turn.
One
day, after we had started distributing the book to local
bookstores, I received a letter from a young man who lived
nearby and wanted to meet the author of this article. I went
to see him and looking around the one-bedroom apartment, I saw
that everywhere there were large pieces of white paper on
which he had been making mathematical calculations: on tables,
chairs, and taped to all of the walls. He was ecstatic,
"I had to meet the author because I have applied what was
stated in the article to every known science and found that
the same formula proves itself in every case."
I
went back a week later to discuss the matter with him more thoroughly
, but the apartment was empty and he was gone.
Now
after 23 years, I have revised the article, and am publishing
it again on DoveSong.Com.
I hope this information can be as useful to others as it was
for this young man and has been for me. I believe that musical
scales and harmony are based on alchemical realities that can
be proved by math. Math demonstrates the formation of the laws
of the universe, and music, color, and all manifestation
follow the same laws.
Don
Robertson, May, 2003
*
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
*
To
our knowledge, there is no religion on the earth that does not
incorporate some kind of symbolic use of the quantities three
and seven. Our objective is to illustrate some of the
mysterious relationships between the quantity three, the
quantity seven, and what these quantities have in common with
the basic physical structure of the universe.
The
direct relationship between three and seven is as follows: If
we have three objects, which we will call 1, 2, and 3, we will
observe that the absolute number of different combinations of
these 3 objects is 7; that is, we may arrange them in no more
than seven different combinations, regardless of the order in
which they are arranged.
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Combination
1
|
1,
2, 3
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Combination
2
|
1,
2
|
|
Combination
3
|
1,
3
|
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Combination
4
|
2,
3
|
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Combination
5
|
1
|
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Combination
6
|
2
|
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Combination
7
|
3
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There
are three primary colors in the color spectrum, plus four more natural colors that are
created from the combinations of the
three colors similarly to the three objects above. All
colors that exist are either: a) One of the primary colors, or
b) A resultant mixture of two or three of the primary colors.
The primary colors cannot be created by mixing other colors,
so they are rather special. The three primary colors are red,
yellow, and blue.
The natural, or secondary
colors are those colors that are created by mixing equally the
three primary colors according to the chart above. The
resultant seven natural colors the colors of the rainbow, a
naturally occurring phenomena that can be created using a
prism to break down light. They are arranged in the spectrum
as follows:
The Color Spectrum
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Red
|
1
|
Red
|
|
Orange
|
1,
2
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Red,
Yellow
|
|
Yellow
|
2
|
Yellow
|
|
Green
|
2,
3
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Yellow,
Blue
|
|
Blue
|
3
|
Blue
|
|
Indio
|
1,
2, 3
|
Red,
Yellow, Blue
|
|
Violet
|
1,
3
|
Red,
Blue
|
Thus
we see that the color spectrum is based on the relationship
between 3 and 7.
There
are three primary tones of the musical scale and seven natural tones:
The Musical Scale
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C
|
1
|
C
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|
D
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1,
2
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C,
E
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|
E
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2
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E
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|
F
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2,
3
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E,
G
|
|
G
|
3
|
G
|
|
A
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1,
2, 3
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C,
E, G
|
|
B
|
1,
3
|
C,
G
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The
notes that correspond to the primary colors are C, E, G: the major triad,
which is the most important chord in music,
and the groundwork of the scale. It is important to realize
the correspondence between the colors of the spectrum and the
notes of the scale, as the musical scale is based on the same principals of nature (the natural overtone series) and
it has
been adapted in some form or another most all cultures.
There
are three root-position triads contained in the major scale (C,D,E,F,G,A,B). These are:
The major triad that is based on the
correspondence with the primary colors (C, E, G) and two minor
triads, based on three of the natural, or secondary, tones (D, F,
A and E,G,B). Major and minor triads are
the fundamental underpinnings of harmony used in the music of
Western cultures.
Three Forces at the
heart of Nature
The
simplest physical structure in the material universe is the
hydrogen atom. It has the least number of forces at work:
three. These three forces are:
1) At the center of the atom resides the nucleus, composed
of a positive charge of electricity called a proton.
2) Orbiting around this proton in an elliptical orbit is a
negative charge of electricity called an electron.
Since this electron has a negative charge, and since a
negative charge is attracted to a positive charge, the
electron is attracted to the proton.
3) Since the electron is attracted to the proton, there
needs to be a force that causes equilibrium between the two,
as the electron would collide with the proton if no such force
were in place. This force then is the force that governs
bodies in motion, causing them to travel in an orbit. It is centrifugal
force. In the atom, this is the neutron.
Some Aspects of Three
If
there is a given object that is fixed in space, this object
has, as its location, three coordinates. An example of this
would be the relative position of an airplane above the earth.
It is calculated by longitude, latitude, and altitude, its
three coordinates.
If
someone looks into the corner of a room, where the room meets
the walls, they will see three intersecting planes that are
considered to extend into infinity. These are the three
dimensions of space.
There
are three time zones that man senses, past, present, and
future.
If
there is space with no motion, then time does not exist, only
space. But if there is no motion, then there can be no space,
for there has to be space to contain the motion. The motion through space causes sequentially. The
earth’s orbit around the sun and revolution on its axis
causes a time sequence on earth. Thus we have three aspects:
time, space, and motion.
The
Law of Three
The
fact that events, laws, and all creation result from three
principals is referred to as the "Law of Three" or
the "Law of the Triangle." This law is illustrated
by the example that every effect has two causes: one passive,
and one active.
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What
does the Russian philosopher Georges Gurdieff have to
say about the Law of Three? In the book In Search of
the Miraculous, PD Ouspensky quotes him as follows:
“This
is the ‘Law of Three’ or the law of the three
principles or the three
forces. It consists of the fact that every
phenomenon, on whatever scale and in whatever world it
may take place, from molecular to cosmic phenomena, is
the result of the combination or the meeting of three
different and opposing forces. Contemporary thought
realizes the existence of two forces and the necessity
of these two forces for the production of a phenomenon:
force and resistance, positive and negative magnetism,
positive and negative electricity, amel and femail
cells, and so on. But it does not observe even these two
forces always and everywhere. No question has ever been
raised as to the third, or if it has been raised it has
scarcely been heard.
“According
to real, exact knowledge, once force, or two forces, can
never produce a phenomenon. The presence of a third
force is necessary, for it is only with the help of a
third force that the first two can produce what may be
called a phenomenon, no matter in what sphere.
“The
teaching of the three forces is at the root of all
ancient systems. The first force may be called active or
positive, the second, passive or negative; the third,
neutralizing. But these are merely names, for in reality
all three forces are equally active and appear as active, passive and neutralizing, only at
their meeting points, that is to say, only in relation to one
another at a given moment. The first two forces are more
or less comprehensible to man and the third may
sometimes be discovered either at the point of
application of the forces, or in the medium, or in the
result. But speaking in general, the third forces is not easily
accessible to direct observation and understanding. The
reason for this is to be found in the functional
limitations of man’s ordinary psychological actively and in the fundamental
categories of our perception of
the phenomenal world, that is, in our sensations of
space and time resulting from these limitations. People
cannot perceive and observe the third force directly
any more than they can spatially perceive the forth
dimension.
“The
first fundamental law of the universe is the law of
three forces, or three principles, or as it is often
called, the law of
three. According to this law, every action, every
phenomenon in all worlds without exception, is the
result of a simultaneous action of three forces: the
positive, the negative, and the neutralizing. Of this we
have already spoken…. The next fundamental law of the
universe is the
law of seven or the law
of octaves.
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Some Aspects of Seven
During the late 1700’s, some chemists began identifying
certain chemical elements. About 1800, they began to determine
the atomic weights of some of them. In 1808, John
Dalton suggested that atoms were physical objects with
specific weights. In 1829, Johann Doebereiner showed that
particular elements could be arranged in groups of three. In
each group, the elements had similar properties and the weight
of the middle atom of the group was close to the average of
the other two. These were said to be chemical equivalents to
the musical triad.
The Law of Octaves
In
1864, John A. Newlands grouped all the known elements in the
order of their atomic weights. He then divided them into
groups of seven elements each. He showed that when the atoms
were put into order with their weights increasing, there was a
repetition in the similarities according to the musical
octave, the same found on the keyboard. When he talked to
chemists of this time about his newly found law
of octaves, they laughed.
But
octaves are the substance of the musical scale, the same
musical scale that was demonstrated by Pythagoras and used
today, with slight modifications according to culture. It is a
naturally occurring phenomena. The basis of the musical scale
is the octave, the repetition of the vibration of the
set of seven notes that occurs as the fundamental vibration of
the first note, the basic note of that scale, is doubled.
In
1869, Mendeleev compiled a “Periodic Table of the
Elements.” This table was arranged according to atomic
weights. Mendeleev found, as had Newlands, that the chemical
properties of the elements reoccur at definite intervals. He
concluded that these were the periodic functions of their
atomic weights. According to Daniel Morris, in his article Music of the New
Spheres in the December, 1969 Chemistry
Magazine, Mendeleev’s arrangement of the elements came to
him while listening to a performance of Schumann’s exquisite
piano quintet. He was seated upon a sofa, apparently mulling
over Newland’s and Doebereiner’s arrangements of the
elements, when all of a sudden he jumped up, sat at his desk,
and arranged all of the elements accordng to a new plan. Morris'
feeling was that there was a definite relationship
between the quintet and the periodic table in the repetition
and development of the seven-note melody of the quintet.
Vibrations in
Different Octaves
It
was in 1925 that French scientist Louis
Victor Pierre Raymond duc de Broglie discovered that the
electron was actually a wave of vibration. These waves
properly follow the same property and characteristics of
vibration as the
vibrating string of a musical instrument, the waves being
complete with overtones. (We are familiar with the presence of
overtones on the vibrating string from the experiments of
Pythagoras and his monochord – a single-stringed
instrument he built to display how musical scales were
produced by the natural overtone series).
An atom such as
uranium has more that 90 electrons, each capable of its own
harmonies, and all this can be recorded by a spectroscope.
From this, one can conclude that all matter is made of waves.
Therefore, we, and the world we live in, are vibrations…just
as the children of the 1960s discovered while taking large
doses of LSD! But this should not be too surprising. Our
understanding of the world around us is through perception of the five senses. What we
experience is actually taking place in our
brain, interpretations of waves of light, taste, smell,
feeling, and sound. Are
we really living in a world, as we suspect, or is this simply a world
created in our brain, similar to that when we dream? And if
spectral and aural vibrations obey the same law of octaves
that applies to the physical makeup of the universe, then
perhaps it is all just a part of one large keyboard of
vibrations, with particular sensations and manifestations
taking place in particular octave ranges. Light, heat,
magnetic and chemical vibration are all subject to the law of
octaves.
The
great German mathematician Gottfried Wilhelm Leibniz wrote:
“I am able to prove that not only light, color, heat, and
the like, but motion, shape and extension too are mere
apparent qualities.” Many scientists and philosophers have come
to the conclusion that everything exists in the mind: stars,
planets, atoms. Einstein showed us that even space and time
were forms of intuition.
Georges
Gurdjieff, the important Russian spiritual teacher said that
the law of octaves was based upon the discontinuity of all
vibrations in nature, as they do not develop uniformly, but
with periodic accelerations and retardations. The original
impulse becomes alternately stronger and weaker. This could be
a phenomenon similar to the AC cycle. This cycle of
alternating electrical vibrations can be viewed by means of an
oscilloscope or with a computer program such as Sound
Forge. Called the duty cycle, it appears as such.
All
sounds are made up of these cycles. The cycles take place in
the time domain. By using a brilliant mathematical
formula known as the Fourier transform, these cycles can be
translated into the frequency domain, where we find that the
timber of the sound depends upon the strength of overtones,
the higher harmonics of the fundamental tone.
The
duty cycle can be equated to a circle broken at its diameter
with the bottom half twisted 180 degrees or to the continuous
movement of a spiral. Alternating current (AC) obeys the laws
of the duty cycle: energy alternates with no energy. Home
electricity in the United States alternates sixty times per
second, thus it is called 60 cycles, or 60 Hertz. An electric
light bulb does not continually burn, but flashes sixty times
in one second, faster than our eyes can follow, so we perceive
the appearance only of continuous light. This would
not be the case when light is powered by a battery, as
batteries produce direct current (DC).
Musical
tones can be represented using the mathematical symbols of trigonometry, signs and
cosigns, and the imaginary numbers of
calculus. Computer systems that generate musical tones, such
as synthesizers, do so using these mathematical formulas. All
of this is based on the alternation of the duty cycle.
“Shocks”
While teaching in Russia during the first world war,
Gurdjieff described
something about octaves that had not been publically known before. He
stated that there were two places in the octave where
retardation of the vibrations takes place. These places are
between the notes E and F and between the notes B and C. These
are the two places on the keyboard where there are no sharp or
flat note (black) keys between the white keys. Octaves,
according to Gurdjieff, develop according to whether or not
these places where retardation of vibrations take place are
“filled.” In the right development of these octaves, these
places are filled by the energy of octaves that run in parallel to
the given octave. This process of filling in Gurdjieff calls a
shock. Processes,
institutions, laws and other things all progress through time
according to the law of octaves, and therefore these two
shocks in the octave are a part of the development process. Gurdjieff stated that
if development continues
without the shocks being filled in, then the line of
development constantly changes. The way he described it, if
octaves – as a principal of nature – develop unimpeded,
the development looks as if it is proceeding along a straight line,
but every time one of the two points in the octave is reached
where a shock occurs, even though development of the process
that the octave represents appears to be progressing forward
along a straight line, the development of the process
actually jogs a little, causing what would be compared to a jog
in the line. The development continues and after the progress
through the series of octaves has gone on for a particular
amount of time, what was considered to be development along a
straight line has actually turned into a development in the opposite
direction! Further continuation will bring the development
back to the same direction that it started in. This is
actually a full duty cycle, or one full revolution on a
spiral.
The law of octaves, with this intrinsic circular
motion generating a spiral, governs all activity and shows how
organizations, religions, and governments -- if allowed to
develop without conscious knowledge concerning the application
of the ‘filling in’ of these shocks -- always develop into
something that is opposite to what was originally intended.
The Octave with Two
Added Shocks
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5
x 1 = 5
5 x 2 = 1
5 x 3 = 6
5 x 4 = 2
5 x 5 = 7
5 x 6 = 3
5 x 7 = 8
5 x 8 = 4
5 x 9 = 9 |
4
x 1 = 4
4 x 2 = 8
4 x 3 = 3
4 x 4 = 7
4 x 5 = 2
4 x 6 = 6
4 x 7 = 1
4 x 8 = 5
4 x 9 = 9 |
|
6 x 1 = 6
6 x 2 = 3
6 x 3 = 9
6 x 4 = 6
6 x 5 = 3
6 x 6 = 9
6 x 7 = 6
6 x 8 = 3
6 x 9 = 9 |
3 x 1 = 3
3 x 2 = 6
3 x 3 = 9
3 x 4 = 3
3 x 5 = 6
3 x 6 = 9
3 x 7 = 3
3 x 8 = 6
3 x 9 = 9 |
8 x 1 = 8
8 x 2 = 7
8 x 3 = 6
8 x 4 = 5
8 x 5 = 4
8 x 6 = 3
8 x 7 = 2
8 x 8 = 1
8 x 9 = 9
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1 x 1 = 1
1 x 2 = 2
1 x 3 = 3
1 x 4 = 4
1 x 5 = 5
1 x 6 = 6
1 x 7 = 7
1 x 8 = 8
1 x 9 = 9
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7
x 1 = 7
7
x 2 = 5
7
x 3 = 3
7
x 4 = 1
7
x 5 = 8
7
x 6 = 6
7
x 7 = 4
7
x 8 = 2
7 x 9 = 9 |
|
5 x
1 = 5
5 x 2 = 1
5 x 3 = 6
5 x 4 = 2
5 x 5 = 7
5 x 6 = 3
5 x 7 = 8
5 x 8 = 4
5 x 9 = 9 |
4 x
1 = 4
4 x 2 = 8
4 x 3 = 3
4 x 4 = 7
4 x 5 = 2
4 x 6 = 6
4 x 7 = 1
4 x 8 = 5
4 x 9 = 9 |
| order:639 |
order:369 |
order:369 |
order:639 |
order:639 |
order:369 |
We will
now show each number and the order of its series. This
order, as you have just seen, was determined by the inner triangle
of each series (the points of 3, 6 and 9 on the
circle) and number present at those point turn out to be
either a sequence of 639 or 369. 639 represents energy
moving counter clockwise and 369 is the energy moving
clockwise. For purposes of demonstration, we will fix
polarity designations to the 639 and 369. We will call 369 positive
and 639 negative. The numbers 3,6 and 9, not being
part of the periods, we will call neutral (=):
| Note |
Number |
Order |
Polarity |
| C |
0 |
|
= |
| D |
1 |
369 |
+ |
| E |
2 |
639 |
- |
| (shock) |
3 |
|
= |
| F |
4 |
369 |
+ |
| G |
5 |
639 |
- |
| (shock) |
6 |
|
= |
| A |
7 |
369 |
+ |
| B |
8 |
639 |
- |
| C |
9 |
|
= |
We will
turning again to the sequences that we created above. For
simplicity sake, we will just use the forward sequence from
each set. These sequences were: 123456789 (from 8-1),
135792468 (from 7-2) 639639639 (from 6-3) and 159483726
(from 5-4). Applying the polarities we assigned in the chart
above to these sequences we get:
| 8-1 |
1
2 3 4 5 6 7 8 9 |
| +
- = + - = + - = |
| 7-2 |
1
3 5 7 9 2 4 6 8 |
| +
= - + = - + = - |
| 6-3 |
6
3 9 6 3 9 6 3 9 |
| (all
neutral) |
| 5-4 |
1
5 9 4 8 3 7 2 6 |
| +
- = + - = + - = |
|
