INPUT OPTIONS:
(Note: In 1870 there was a discussion as to the correspondences of color and musical tone. The following is that discussion as found in Nature February 10, 1870, taking up Newtons experiment, tones and scales):
..."I do not know how far Newton's measurements are correct; but I find
that Professor Zannotti, of Naples, gives for the diameters of the rings
from red to red the cube-roots of the numbers 1, 8/9, 5/6, 3/4, 2/3,
3/5, 9/16, 1/2
The intervals between these, taken successively, are 9/8, 16/15, 10/9,
9/8, 10/9,16/15, 9/8; that is- major-tone, semi-tone, minor- tone,
major-tone, minor-tone, 1/2-tone, major-tone. Calling the major-tone M,
the minor-tone m and the semi-tone x, for the sake of brevity. I will
give the 5 different forms of which the musical scale is capable-
expressed by the succession of intervals- and show that the above series
of intervals is one of them:
.
D
^
E
^
F
^
G
^
A
^
B
^
C
^
D
Comments
(1)
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m
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x
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M
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m
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M
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x
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M
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Or Sintono
(2)
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M
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x
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m
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M
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m
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x
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M
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Or Newtons Scale
(3)
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M
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x
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M
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m
.
M
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x
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m
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.
(4)
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m
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x
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M
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M
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m
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x
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M
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(5)
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M
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x
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m
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M
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M
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x
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m
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.
Varieties depending upon the permutation of the quantities M, m and x. The 1st contains the imperfect fifth, DA; the 2nd two such fifths, EB and FC; the 3rd GD; the 4th A sub. 2, E sub. 2; and the 5th the imperfect fifth,
C sub. 2, G,- all of course with their corresponding augmented fourths.
Thus, Newton's scale of color is one of a series of 5 scales of sound, all requiring a modification by a comma of one, or at the most two- fifths; but all are found of perfect major and minor tones and major semitones. If the correlation between color and sound exists, I think it will be found here. If this be admitted, the colors and notes corresponding are as follows:
D- Red
This is from a discussion page linked below. I've clipped only the first
bit of the discussion as is of interest to the group. Secondly is a
piece on the relationship of PHI & PI and the magic and similarities of
(1+0) 10.
This is a repost of the Phi Theory
Two numbers are 1.6180339887... and Ð0.6180339887...
I hope it helps.
From Infinity a one dimensional line has, through the Phi relation
become a length having two extremes. Any length will have two extreme points and therefore will also have mean points between these extremes.
Three of these mean points are significant and are naturally generated
by the flow of number.
These three are the ARITHMETIC, GEOMETRIC and HARMONIC means.
From these principles a model of the ATOM can be derived which is based
on the same principles of number that life forms vividly show.
A Theory of Dimensions and Geometry.
The inherent triplicity of nature manifests throughout all spectrum
areas. At the root point is a triangle with its hypotenuse equal to Planck's
Length. This can be calculated to precision in Spiral Energy Transformation Theory.
Each Dimension is displaced in the greater universe by a velocity factor of the speed of light c. c is a universal fundamental constant. Each Dimension is structured by number in the form of ratios which
approximate the perfect Golden Section Ratio (PHI). Phi-Golden is a universal and fundamental constant.
Basically, a revealing study of dimensions and light reveals the ATOM
in its inmost structure and origins, as well as the nature of forms
arising within the domain of other dimensions. The Pi relation of Diameter to Circumference is displayed in the universe in all dimensions and is a universal and fundamental constant.
MODIFIED PLANCK THEORY
The Quantum of Action 1/2h is a complex constant composed of the phi
relation, the pi relation and the speed of light relation which all
exist in perfect accord in energy exchanges.
The constant related to Planck's Constant which reveals the true
structure of matter / energy is derived from the Phi relation at the heart of Planck's Constant and is equal to c (squared) x the square root of 1/2h. This then as a phi ratio is the true Planck Constant.
This is PhiX and this number is extremely important in the universal
generation of fundamental properties of the atom. From it in combination with 2Pi and powers of the speed of light, can be formed the important fundamental constants, (by a simple progression).
PHI PHYSICS
The Relation of the Golden Ratio to Quantum Physics: Phi Physics.
The Golden Section Ratio is a perfect mean number having the property of
rotation inherent within its perfect value.1.618033988... inverted becomes 0.618033988...[PhiG-1 = 1 / PhiG]. Another number which displays this perfection in a similar way is the number.1. Inverted it remains one.
However, when utilized as 10 one can see the deep symmetry in its
inversion to 1/10 or 0.1=1/10. The digits remain the same as in the phi relation. A deeper relation still can be seen in the value of the root of 10.
The square root of one is one but the square root of 10 is a number with
very similar properties to PhiG. /10=3.16227766 and the inverse of this
number is the same number divided by 10. 1/3.16227766 = .316227766, all the digits are the same but place value is displaced. It is the property of these numbers and their inverse which gives manifestation its opportunity to occur.
This is why the base 10 system works so well and is quite natural to
use. From this point of view where the flux of the universe becomes
manifest by means of number in certain perfect, harmonious and
symmetrical forms. The flux state has a direct relation to the Phi
ratios created by perfect numbers.
This is true mathematically and philosophically. The manifested state has a direct relation to the powers of 10 and the properties of its square root.
Flux and mass [Phi and 10] are related together by motion at the
velocity of light and powers of that velocity. Flux and mass in relation and manifestation through number generate lengths of flux lines.
The next three interesting facts and figures on the Phibonacci brick
were first pointed out by Donald Seitz in The Mathematics Teacher, 1986,
pages 340-341 in an article entitled A Geometric Figure Relating the
Golden Ratio to Pi.
What is the surface area S of the brick?
Above we saw that the sum of the 2 smaller face's areas equals the
largest face's area, and that this is Phi. Since there are 2 faces with smallest area, 2 of middle-sized area (which total 2 times the largest face area, that is 2 Phi) and we also have two other faces of the largest area (Phi) , then:
The surface area of the brick is 4 Phi
How long is the diagonal across the brick?
Another surprise awaits us when we calculate the length of the diagonal across the brick.
The formula is a 3-dimensional analogue of Pythagoras Theorem. For a rectangle of sides x and y, its diagonal is sq.rt(x2 + y2). For a 3-D brick with sides x, y and z, its diagonal has length sq.rt (x2 + y2 + z2).
So how long is the diagonal of our Phi-bonacci brick?
Since its sides (x, y and z) are 1, Phi and phi, the length of its diagonal is :: sq.rt (12 + Phi2 + phi2). I'll leave you to check the algebra but the
surprisingly simple answer is
The diagonal of the brick has length 2
A relationship between Phi and Pi
Even more surprising is that the brick shows a simple relationship
between Pi and Phi, using the values for the diagonal D that we have
just found and the surface area S.
If we imagine the brick tightly packed into a sphere, the centre of the
sphere will be half way along diagonal D. So the radius of the sphere will be 1 and its surface area will be 4 Pi. We showed above that the surface area of the brick, S, is 4 Phi.
Putting these two together we have:
The ratio of the surface areas of the Phibonacci brick and its surrounding sphere is Phi : Pi
This is off the excellent PHI pages
PI denotes a relationship (ratio) of circumference of a circle to the
dia-meter = 3.14159....3.14159 X the line (wavelength) give the size of the sphere.
It is the sixteenth (16) letter of the Greek alphabet Phi is the
twenty-first (21) letter. Just for fun...Together they add up to 37 ,
3+7=10 the base for the metric system as well as the digital method.
www.heartbeat2000.com The Power and Beauty of Compassion.
Heart Intelligence and the Wisdom its connected To.
This is from Wayne Kirby's web sight link listed below. Note: Wayne's
sight has information on sound and brain disorders. Please note ** at
the end of the piece & PHI-Pentatonic (1.1010259) Scale. Also the
similarities with this interesting fractal scale found at:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
The Golden String
The golden string also referred to as the Infinite Fibonacci Word or the
Fibonacci Rabbit sequence.
Fibonacci Rabbit Sequence
There is another way to look at Fibonacci's Rabbits problem that gives
an infinitely long sequence of 1s and 0s, which we will call the
Fibonacci Rabbit sequence:- 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ... which is a close relative of the golden section and the Fibonacci
numbers. You can hear the Golden sequence as a Quicktime movie track
too! The Fibonacci Rabbit sequence is an example of a fractal.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
"Understanding Sonic entrainment, you may wish to consult the following sources. For introductory information on the neuropsychology of binaural beat frequencies, consult Gerald Oster's article entitled Auditory Beats in the Brain (1973). You will find relevant information on the theory of sonic entrainment in Jonathon Goldman's article entitled Sonic
Entrainment (1991). Information on the phenomenon of hemispheric
synchronization can be found at the World Wide Web site of the Monroe
Institute (1998). "
Here is a listing of some of the tunings that are included with the
http://www.phi-music.com/default.htm
Phi-Musicª Microtone/Brainwave Generator by SERIOUS COMPOSER Music Software by Wayne Kirby
ALPHA (W.CARLOS)*
ALPHA PRIME (W.CARLOS)*
ARABIC (AFTER ZALZAL)
ARABIC (18 TONE OCTAVE)
ATOMIC-HYDROGEN LOWEST TO PHOSPHORUS HIGHEST**
ATOMIC RECIPROCALS-RUBIDIUM LOWEST & HYDROGEN HIGHEST
ATOMIC RECIPROCALS-LAWRENCIUM LOWEST & HYDROGEN HIGHEST
BARNES-BACH TEMPERAMENT
BETA (W.CARLOS)*
CHROMATIC GENUS
DIATONIC GENUS
EARTH HARMONICS (W.KIRBY)
EIGHTH-TONE EQUAL TEMPERAMENT
ENHARMONIC GENUS
EQUAL TEMPERAMENT
GAMELON (EXTENDED PELOG)
GAMMA (W.CARLOS)*
HARMONIC (AFTER W.CARLOS)*
HARMONIC SERIES (48)
22 INDIAN SCRUTIS (NO Bb KEYS)
JUST INTONATION (MAJOR)
JUST INTONATION (MINOR)
MEANTONE TEMPERAMENT
PARTCH (43-TONE SCALE)
PELOG & SLENDRO COMBINED
PELOG OCTAVE
PHI (1.6180339)
This scale is generated by multiplying the lowest frequency by the
square root of. Phi. Then, as described above, the resulting product is
also multiplied by the square root of Phi.
PHI-Pentatonic (1.1010259)
This scale is generated by multiplying the lowest frequency by Phi to
the power of 0.2. Then, as described above, the resulting product is
multiplied by the same ratio, and so on. This scale is the same as the
"Fifth Root of Phi," a scale used by some alternative therapists and
health practitioners because of some various esoteric properties it is
believed to possess.
PHI-Decatonic (1.0492978)
This scale was generated by multiplying the lowest frequency by Phi to
the power of 0.1.
PURE FIFTHS CYCLE (48)
PURE FIFTHS SCALE WITH PYTHAGOREAN COMMA
PYTHAGOREAN
QUARTER-TONE EQUAL TEMPERAMENT
QUARTER-TONE SHIFTED EQUAL
SCHOLAR'S LUTE
SLENDRO OCTAVE
VALLOTTI AND YOUNG
WERCKMEISTER
You can create any tuning imaginable with Phi-Music! Just type the
frequencies you want to hear into a text file and load them up! You can
also create new tunings by simply entering a ratio in the "Enter
Constant Ratio" selection under the "Tunings" menu.
*SPECIAL THANKS to Wendy Carlos for permission to include these scales which she invented and discusses in her 1987 Computer Music Journal article, "Tuning: At The Crossroads."
**These atomic tuning systems were derived from a list of frequencies based on the periodic table of the elements. Created from a table of the X-ray atomic energy levels of the elements expressed in ev/s, Dr. Wayne Kirby calculated the desired frequencies by dividing these values by Planck's Constant in ev/s. After transposing these values into the audible spectrum, text files of these frequencies were created. The
"reciprocal" scales invert the frequencies so that hydrogen is the
highest note.
Copyright (c) 1996-2000 by Serious Composer, Inc.
Copyright © 1998-2000 Serious Composer, Inc.. All Rights Reserved
Worldwide.
An excerpt from THE AMAZING SYNCHRONICITY OF CHATELAIN'S MAYAN CALENDAR BY DAVID WILCOCK
..."13 appears to be the last number that carries a unique vibration before the vibratory properties again duplicate themselves.
We are reminded that many ancient cultures including the Sumerians would use base-12 as a counting system at various times, and in this system the vibrations of the number 13 would represent an octave - the first counting unit on the next "level," just as 10 is the next "level" of the
number 1 when using a base-10 system. The vibrations of the number 13 as an octave can be seen on a conventional piano if you were to play a
"chromatic" scale, where you go from C to C playing both white and black
keys. Each scale will have 12 notes before resolving to the Octave on
the 13th. So again, we can see that 26 is a harmonic number in its own
right, only it appears that the Maya were much more aware of it than the
Sumerians".
Pythagorean Scale and Pythagoras' Comma
One of the greatest of the mysteries of the science of sound is the strange phenomenon known as Pythagoras' comma, which since time immemorial has been to man a symbol of his fallen state of imperfection.
Given that different pitches of tone arrange themselves into octaves
(which have the ratio of 2:1), man from the earliest times needed to
find a method whereby the notes within each octave could be calculated. This could be done, it was discovered, by using the next most fundamental
and harmonious tonal ratio, that of 3:2, or the interval known in music
as a pure fifth. If one pictures the octave as a circle, then the interval
of a fifth represents a seven-twelfths circuit of the circle, from 12
o'clock, moving clockwise, to 7 o'clock. This gives one pitch. By continuing
around clockwise again, another pitch is produced, but not at 2 o'clock
on the same circle, for the 'circle' is actually not a circle at all,
but a section of a spiral which spirals upward into higher octaves. By going around clockwise in a series of perfect fifths, a series of twelve notes can be produced.
However, these twelve notes could not practically be used as the scale
of a system of music, since they extend over several octaves with wide
gaps in between each note. Therefore the ancient philosopher-musicians
devised schemes whereby intervals of a fifth could be used to calculate
twelve notes within one octave. For instance, the Chinese went up in
pitch by the ratio of 1:3, then down by 3:1, up by 1:3, and so one.
However, given these twelve notes, if a thirteenth note was attempted,
in order to complete the octave with a series of twelve intervals of a
fifth, it was found that the octave could not be perfectly completed! This is a universal law of the physics of tone and, indeed, of mathematics: a
cycle of twelve fifths completes seven octaves (note the mystical numbers: twelve and seven) plus a little more. Expressed mathematically:
(3:2) 12 > (2:1) 7 by a slight excess interval. This interval, since
Pythagoras was one who noted it, has become known as Pythagoras'
comma. The difference is a ratio of about 80:81, the extra 1 being the
comma.
The comma produces huge cosmological, as well as practical,
implications and results. Since this system of calculating twelve notes
does not perfectly complete the octave, the specific twelve pitches it
produces are imperfect for use together in harmony. They do not
perfectly harmonise since they do not divide the octave into perfectly
accurate divisions of twelve as, say; the hours of a clock do perfectly
divide a clock face. Yet to adjust their slight pitch discrepancies in order to produce perfect harmony would render each note imperfect as an interval of a fifth, or as a ratio of 3:2. The perfecting of their musical system was paramount to the ancients, for their music had to harmonise with the eternal laws of the universe. Therefore the perfect ratio of 3:2 and its interval of a perfect fifth were regarded as sacred and inviolable. If the system resulted, as it did, in the harmony of mortal music being imperfect, then this had to be borne as a manifest symbol of man's fall from grace, and of the inherent imperfection of the non-heavenly realm of time and space. Indeed, perhaps the ancients were correct in this, for the comma is a strange and wide-ranging phenomenon, being literally 'written into' the physical and mathematical laws of the universe. We are entering here into regions of thought difficult for the Western mind to grasp, but the possibility seems to be that it is by the phenomenon of Pythagoras' comma that the very nature of our lives within the realm of mortality is arranged.
Though the comma may be a symbol of the imperfection of man's mortal
state, the very same comma simultaneously provides the way back to the
original state of perfection. For the comma is not a slight interval less than seven octaves, but in excess of them. In the ancient world this fact was widely conceived as a symbol of renewal. The cycle of twelve perfect
fifths did not close and finish a cycle of seven octaves, but exceeded it,
and thus, as it were, spiralled upward. There is evidence that this
upward spiral of renewal was mystically associated with the widespread ancient myth of the phoenix, the archetypal 'bird' which is resurrected from its own ashes. Pythagoras' comma, then, can be seen as being God's own anagram written into the very laws of the universe and of physics....
The months of the year were associated in antiquity with the total
twelve notes of the musical octave. But there are, of course, two slightly
different kinds of months: the solar or calendar month which averages 30.44 days, making a total of 365.256 days per year, and the synodic lunar month of 29.5306 days. The solar month is exactly one twelfth of the time it takes for the earth to orbit once around the sun. The synodic lunar month is the time that the moon takes to pass through all of its phases - say, from new moon to new moon. Though the lunar month is not exactly the same as a solar month, it comes remarkably close, with the curious result that there are, approximately twelve lunar months per calendar year.
Indeed, since solar months offer less obvious signs of their coming and
going than do those of the moon, it has frequently been by the phases of
the moon that cultures have measured the passing of time. The reader will probably not have missed the noteworthy fact that this phenomenon of two different divisions of the year into two slightly different lengths of months strikes a very close parallel with Pythagoras' comma, with its two slightly different sets of twelve notes around the circle of an octave. That the parallel may be more than academic is suggested by the fact that according to the ancient wisdom, the year
does represent the passage of the earth through an octave of twelve notes, or Cosmic Tones. Taking the parallel to its logical conclusion. The solar
year, which totals the exact and accurate length of 365.256 days,
equates with a cosmic form of equal temperament. By measuring time by the solar year rather than by the lunar year, man not only displays an obvious practicality, but also opens for himself the possibility of renewal and self evolution.
The two ratios - between the two years (solar and lunar) and the two
types of musical scale (of perfect fifths and of equal temperament) -
are astonishingly close to each other. The ratio of Pythagoras' comma is
531441 to 524288, the former figure being larger than the latter by the
factor of 1.01364. The ratio between the two years is 365.256 days to
354.3672, the former being the larger by a factor of 1.03073. To express the closeness of the two ratios in a more graphic way, we can translate the ratio of the comma into calendar terms. That is, 531441 is to the solar year of 365.256 days as 524288 is to the time period of 360.340 days - which is only six days out from the precise lunar year of
354 1/3 days.
As might be expected, the cultures of antiquity seem not to have missed this phenomenon of a 'cosmic comma'. Traditionally, the measurement of the year by means of solar months was considered holy and righteous; whereas to keep time by the moon was both incorrect and evil. This was one of the chief differences of opinion between the Pharisees and the Sadducees. Whereas the Sadducees marked time by the sun, the Pharisees marked time by the lunar year. Interestingly, however, this lunar year was not stated as being 364 1/3 days long, which it is, and which the Pharisees must easily have been able to observe that it was. Rather, it was dogmatically stated to be 360 days in length. This figure was a direct but coded reference to the cosmic Pythagoras' comma since, as we have calculated above, 360 is the nearest whole number of days to what an absolutely perfect incidence of the comma would turn out to be, at 360.340 days.
The Musical Octave By Thomas Vaczy Hightower viewed in an acoustic, mathematic perspective.
HIGHTOWER:
Music, being the representation of the relationship between heaven and
earth, must quite naturally have this confirmation of a center or tonic
(gong) surrounded by four notes assimilated to the four directions of
space, the four perceptible elements, the four seasons, and so on.Ê
The pentatonic scale thus presents a structure that allows it to be an
adequate representation of the static influence of heaven on earth. But
a static representation of a world in motion could not be an instrument
of action upon that world. It is necessary, if we want to act upon the
represented elements, to evolve from the motionless to the moving, from
the angular to the circular, from the square to the circle. To express
the movements of the universe, the sounds will have to submit to the
cyclic laws that, in their own field, are represented by the cycle of
fifths."
The spiral of fifths
As we have already seen, the fifth is the third sound of the series of
harmonics, the first being the fundamental and the second its octave.
According to the formula of the Tao-te ching ("One has produced two, two
has produced three, three has produced all the numbers"), we can
understand why the third sound, the fifth, must necessarily produce all
the other sounds by its cyclic repetitions. The first to be produced
will be the four principal sounds, which form comparatively simple
ratios with the tonic. They are:
I, C (Sa)
II, G (Pa) = 3/2
III, D (Re) = 9/8 = (3/2)2
IV, A+ a comma sharp (Dha+) = 27/16 = (3/2)3
V, E+ a comma sharp (Ga+) = 81/64 = (3/2)4
To these five primary sounds, whose disposition represents the
elementary structure of the perceptible world, the pentatonic scale, can
be added the two auxiliary sounds:
VI, B+ a comma sharp (Ni+) = 243/128 = (3/2)5
VII, L+F# sharpend a major half tone (MaL+) = 729/512 = (3/2)6
Thus is formed the seven-note scale.
C D E+ (F)*L+F# G A+ B+ C' Sa Re Ga+ MaL+ Pa Dha+ Ni+ Sa 1/1 9/8 81/64 (4/3*)729/512 3/2 27/16 243/128 2/1
The two auxiliary sounds should not be used as fundamentals, though they are needed for transpositions, because they belong to the scale of
invisible worlds, and so we can neither perceive their accuracy nor
build systems upon them without going out of tune.
*Let us note here that the most striking difference between the system of fifths and that of harmonic relations to a tonic resides in the
perfect fourth, which is an essential interval in the scale of
proportions. The scale of fifths has an augmented fourth as its sixth
fifth, (3/2) 6 = 729 / 512. Instead of starting from C (Sa), we could had
begun one fifth below, that is to say, from F (Ma), and we would have
obtained this essential note without changing anything in our scale,
except that, since we begin with a masculine interval instead of a
feminine interval, the character of the whole system is modified.
The 5 successive fifth, whether in an ascending or a descending series,
represents the limit of consonance in modal music also. Beyond this
limit, no interval can appear harmonious, nor can it be accurately
recognized. A rule originating from the same principle was also known in
medieval Europe, where the tritone was prohibited as diabolical, that
is, as connected with forces that are supernatural and therefore
uncontrollable.
After these seven notes, the next five notes generated by the series of
fifths are:
VIII, bDb lowered a minor half tone (Re komal),
IX, bAb lowered a minor half tone (Dha komal),
X, bEb a minor half tone lower (Ga komal),
XI, bBb a minor half tone lower (Ni komal),
XII, F+ a comma sharp (Ma+).
We now have twelve sounds, which divide the octave chromatically into
twelve half tones.
The twelfth fifth (note 13) brings us back to the fundamental, but with
a slight difference. It is higher than the fundamental by one comma, the Pythagorean comma (312 / 219 = 531,441 / 524,288, 5.88 savarts* or 23.5 cents). It is, therefore, in our notation, C+ one comma sharp (Sa+).
*Savats named after a French physicist and cents are logarithmic systems developed to make it easy to compare intervals on a linear scale instead of using fractals or frequencies ratios. Savats is calculated as the
logarithm (base 10) of the frequency ratio and, for convenience,
multiply with 1000. We shall then have the interval expressed in terms
of a unit call savats. The logarithm of 2 being 0.3010.. the interval of
the octave is equal to 301 savats.
The more widely used American system, cents, is allso a logarithmic unit which make it easier to operate with pitch intervals or frequencies
ratios, since the size of a pitch interval is proportional not to the
frequency ratio, but to the logarithm of the frequency ratio. It is a
simple relationship: pitch interval (in octaves) = log2 (f2 : f1). Since
there are chosen to have 1200 cents in an octave, this is equivalent to:
pitch interval (in cents) = 1200 x log2 (f2 : f). In practice,
logarithms to base 10 (log10) are normally used.
If we proceed further, the twelve following fifths will place
themselves one comma above the former ones, and the 24th fifth (note 25)
will be one comma above C+ (Sa+), that is, C++ 2 comma's higher (Sa++).
In this way, successive series of twelve fifths will be placed one above the other at one-comma intervals, up to the 52nd fifth (note 53). However, after the 52nd fifth, the octave is filled up and the 53rd
fifth (note 54) comes out of the octave and inserts itself between the
octave C (Sa) and the twelfth fifths C+ (Sa+), thus forming, above C
(Sa), a small interval of 0.84 savarts.
Thus begins a new cycle, which in its turn, with a period of 53 fifths,
will divide the octave into small intervals of 0.84 savarts. The next cycle will appear during the seventh series of 53, when the 359th fifth (note 360) comes out of the octave and forms, with C (Sa), an interval of 0.47 savarts.
The next cycle is of 666 notes, with a basic interval of 0.035 savarts.
Then comes a cycle of 25,524 notes with a basic interval of 0.0021174
savarts. This cycle is very near to that of the precession of the
equinoxes, or the Pythagorean great year, which is of 25,920 solar
years. The small difference between the twelfth fifth and octave,
similar to that of the lunar and the solar year, leaves the door open
for further cycles.
HIGHTOWER AGAIN:
In practice, for reasons that are symbolic as well as musical, after the
52nd fifth (53rd note) the Chinese follow the series only for the next
seven degrees, which place themselves above those of the initial
seven-note scale, and they stop the series at the 60th note. The reason
given is that 12 (the number of each cycle) x 5 (the number of the
elements) = 60.
This scale of fifths, perfect for transposition because of its extreme
accuracy, also allows the study of astrological correspondences and of
terrestrial influx, provided one possesses knowledge of the hierarchy of
its intervals.
"In the same melody, the prime, the third, the fifth, the sixth, are
chosen independently because of their connection with diverse
constellations and, consequently, with heavenly spirits, earthly
spirits, and manes."
Quote:
"Proton, Neutron and Electron and the mysterious constant that governs
their actions, fine-structure constant (137.035999935), are derived in
mathematical detail by using only 2 number constants, the height and
base leg of the Cheops pyramid at Giza. This is as simple as it gets".
"The following formula shows the orbits of the Pi variables that
influence the action of the electron (.510998902 Mev...NIST 1998) on the
shells of the proton mass (938.271998 Mev...NIST 1998):
(938.271998*10)^[((.510998902^(-1/Pi))/Pi]= Cos 137.0360015...in radians.
The shells are guarded by the gate-keeper only allowing possession of a fine-structure key (angle) to gain entry. 137.035999935 is entry key,
the inverse of fine-structure. The square of this inverse (distance) is
the collective unconscious link to spiritual ...mind. This is the metric
37 degee Celsius heat of sentient mind-body, the amplitude of a real
electron to emit or absorb a real photon, or electron shell changing
key. The action allowed by fine-structure on the charged +1 , base
weight 938.271998 is only through the total weight of the electron to
the power of negative inverse Pi. This total weight .510998902^(-1/Pi)
showing -1/Pi units of freedom is then divided by the ratio of least
time Circumference divided by Diameter or Pi. This energy matter form
involving Pi actions on the electron can be reduced to a harmonic
derivitive that is entwined in the human mind. For example the Mayans of
Meso-america and the Druids of Stonehenge England both used the same
harmonic (288) in representations of their respective
calendars.(Pictun = 288 = Mayan); (double light = 288 = Druid)":
GRAVITY RECTIFICATION
"... In a message to the Quantum-Mind list of 24th March, Stuart Hameroff once again replies to a critic, J. Sarfatti, who draws attention to the apparently unbridgeable discrepancy between the very weak gravitational force as compared with (electron) energy -- the point being that unless these two forces can be shown to directly correlate, then the case for Orchestrated Collapse of the wave function within microtubules (as constituting the "neural correlate of consciousness") cannot be
sustained as a viable solution to the 'hard problem'.
Hameroff then relates the following exchange between himself and
mathematical physicist, Jack Sarfatti:
"[Jack]
2. Gravity is simply too damn weak. Penrose [i.e. Hameroff's
collaborator in the Orch OR model] has lost his marbles. How can a
distance of 10^33 cm influence chemical processes in the brain at 10^-7
cm? That is a difference of 26 powers of 10!"
At this point, Hameroff, after giving his and Penrose own calculations
for the forces involved, suddenly concludes thus:
"I gave a talk on this at Starlab in Brussels not too long ago, and was
asked this same question by some Russian mathematicians, and I gave the
same answer. They objected: "But the energy is instantaneous, not over a
Planck time". I replied: "Even better!". Dick Bierman then said,
jokingly, "Then the power is infinite!"...............
From the book the " The God Particle" by Richard Lederman, page 93......
" Compared to the electrical force that holds matter together and gives
its rigidity, the gravitational force is extremely weak. How weak? I do
the following experiment in a physics class I teach. I take a length of
wood, say a one foot long piece of two by four.......... Using the
equation for the law of gravity, we can calculate the difference between
the two gravitational forces. It turns out that the earths gravity on
bot wins out by being more than one billion times stronger than top's
gravity on bot. Conclusion; the electrical force of top on bot before
the saw cut was at least one billion times stronger than the
gravitational force of top on bot. That's the best we can do in a
lecture hall. The actual number is (10^41) or a one followed by
forty-one zeros!! Let's write that out:
100,000,000,000,000,000,000,000,000,000,000,000,000,000
The number (10^41) can't be appreciated, no way, but perhaps this will
help. Consider an electron and a positron one hundreth of an inch apart.
Calculate their gravitational attraction. Now calculate how far apart
they would have to be to reduce their electrical force to the value of
the gravitational attraction. The answer is some thousand trillion
miles(fifty light years). This assumes that the electric force decreases
as the square of the distance--just like the gravitational force......
(end quote)
This "unbridgeable discrepancy" can be bridged through the master
harmonic of modulation (37) and the fine-structure constant, the
amplitude of an electron to emit or absorb a photon.
Letting 1.000027792*(10^41) equal the differences in strength between electrical and gravitational forces, then it can be shown:
[1.000027792*(10^41)]^-(1/37)=.077963542744
The .077963542744 form is the key to the "collective unconscious" number 144/37 as follows:
.077963542744*(10^5)=7796.3542744
The log of this form = (log 7796.35427744)*37=143.9999879
Fine-structure can be shown as:
(10^(143.9999879/37))/666=11.70623765=137.0359994^(1/2).........where 137.0359994 is the inverse of fine-structure.
This shows a "direct correlation" between the strength of the the
gravitational force to the electrical force(fine-structure) .....10^41
through the "collective unconscious" ratio 3.891891891 (this is the
"neural correlate of consciousness")... the form that links all of the
ancient and modern religions of the world and the giant earth structures
Teotihuacan pyramids, Cheops and Stonehenge".
J.Iuliano
"The 12 Lynus Tonal System or the cycle of fifths of China and the 22
natural intonation Scruti scales of India, are arithmetical
progressions... the musical scale systems of different peoples can be
mathematically oriented into a spiral complex when translated into
digits 1, 2, 3, 4 and 5. This means, as already noted, that a "spherical
music" may eventually express the evolution toward a world-wide
acceptance of a common tonal system, perhaps the Mercator,
equal-tempered 53 tones-to-the-octave scale within the overall
mathematical perimeter of the circle of fifths"... 13^3 = 13 x 2 x 2 = 52
(Correspondences, File, "30 Unique Divisions").
..."the binary aspect of the octave intervals of the tonal spiral
requires a Boolean algebra".
Dr. Andrew Pikler, International Congress on Acoustics, Tokyo, Aug.
1968, Generation and Plotting of Musical Tone System with the Digital
Computer.
..."possibility of a "double helix" in the domain of the musical tone
system...the "sharps" and "flats' furnish helical formations, each
separately split with the Pythagorean comma"...
Dr. Andrew Pikler.
Music of Pi (Link is down)
More about the Music of Pi Project
The musical system we use consists of 12 semi-tones:
A, Bb, C, C#, D, D#, E, F, F#, G, G#.
The numeric system we most frequently use in everyday life is base 10
which consists of the numbers:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Mathematics allows us to express numbers in other bases. Base 2, for
example, uses just 0 and 1 which is used in the digital world which
includes computers.
By converting Pi into base 12 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b), one
can directly assign semi-tones to numbers as follows:
A = 0, Bb = 1, B = 2, C = 3, C# = 4, D = 5, D# = 6, E = 7, F = 8, F# =
9, G = a, G# = b.
To see the actual project mapping for part 1, click here. To see the
actual project mapping for part 2, click here.
On Rabbits, Mathematics and Musical Scales By John S. Allen
But what does this rabbit formula have to do with music? Lots! If we
only substitute x0= 2 and x1 = 5 for the first two terms, we get the
series 2, 5, 7, 12, 19, 31, 50... which bears an important relationship
to the structure of musical scales. Any musician will recognize 5, 7 and
12 as the numbers of pitches per octave in pentatonic, diatonic and chr
omatic scales; and also as the number of black keys, white keys and keys
per octave of the traditional musical keyboard.
Now let's see how these numbers build musical structures. The simplest
musical interval is the octave, a 2/1 frequency ratio. The next simplest
are the fifth, a 3/2 ratio (more or less, in various tunings) and its
inversion, the fourth (4/3, more or less). The numbers in the musical
Fibonacci series 2, 5, 7, 12, 19 ... all are generated by increasingly
long series of musical fourths and fifths, as Sir James Jeans described
and shown in the table below.
scale no. calculation number of intervals Series of musical fifths
Two-tone scale
Our first division of the octave (scale no. 1 in the table) is into a
scale of two tones a fifth apart, say F and C, with a frequency ratio of
3/2. The next fifth after our two tones is a G, and the resulting
sequence of two fifths, 3/2 x 3/2, takes us to 9/4 (2.25), which brings
us close to an F a musical octave (2/1) higher than our first F. We
think of the G as the same as the F, and stop with a two-tone scale.
With two tones per octave, it is already possible to play music which
distinguishes a tonic and dominant -- like a typical tympani part of the
early Classical era.
We may also lower the fifths to produce a 2-tone equal temperament. In
fact, we can play this temperament on any conventional musical
instrument by lowering the C of our two-tone scale by a half tone to B.
But assigning different roles such as "tonic" and "dominant" to each of
the two pitches of our scale is difficult. The intervals between them
are equal, and so we can't tell which pitch has which role.
Five-tone scale
Now if we prolong the series of fifths, F-C-G-D-A-(E), the E at the end
is close to F again. If we leave the fifths mathematically exact, we can
now play pentatonic music ("Mama makes shortnin', shortnin bread ...")
in one key signature, the way we do on the black keys of any
conventional keyboard instrument. As you can see by looking at the black
keys, the pentatonic scale consists of groups of three and two pitches
separated by two larger steps. This pattern repeats in every octave,
allowing us to identify the role of the tones of a pentatonic scale.
But suppose that we raise all of the fifths a little, until the final E
is at the same pitch as F. In this way, we construct a 5-tone equal
temperament. We can't play this temperament on a regular keyboard, but
we can play it on any synthesizer which lets us set nonstandard
intervals: set the semitone or octave interval for 2.4 times the
standard one (or 1.2 times and play on alternate keys). The resulting
scale will sound somewhat out of tune to our ears which are used to
standard 12-tone equal temperament, but its melodic and harmonic
properties are clear enough. It is not only a theoretical temperament;
it is the standard Javanese tuning called slendro (see for example
Helmholtz, p. 518.)
In the 5-tone temperament, we can distinguish tonic and dominant of
two-tone scales and modulate freely among five key signatures - but
pentatonic melodies do not work. Even though there are five pitches per
octave in this scale, ordinary pentatonic music sounds oddly rootless --
we can't determine the keynote. With five equal intervals, there is no
differentiation between wide and narrow intervals to identify the
pentatonic key signature. This is the same problem we had earlier in
distinguishing the tonic and dominant in our 2-tone temperament.
And onwards...
Now that we have established the first two terms of our Fibonacci
series, succeeding terms follow logically. Each new, extended series of
fifths is the sum of the previous two. The reason for this not the same
as for the numbers of Fibonacci's rabbits, but is just as simple: since
sequences of two or five fifths approximate whole numbers of octaves,
sums of these numbers of fifths also approximate whole numbers of
octaves and allow us to close a circle of fifths by filling in the wider
steps in the scale which the earlier series of fifths built. So now
let's increase the number of tones per octave, just as Fibonacci did
with his rabbit population.
Seven-tone scale
Continuing from 5 on to 7, the series F-C-G-D-A-E-B-(F#) once again
brings us close to F. If we leave the fifths exact, we can now play
seven-tone diatonic music (like playing on the white keys of the piano),
in one key. We can also play two-and five-tone music.
If, on the other hand, we lower all of the fifths of our seven-tone
scale a little, we arrive at a seven-tone equal temperament. Set the
semitone or octave on a synthesizer to 1.7142 (that's 12/7) times
standard to play in this temperament. In a seven-tone temperament, we
can play in the 2 and 5 tone scales and also play any ordinary
pentatonic music with full freedom of modulation in seven keys.
The seven-tone temperament is not just a theoretical construction. It is
the standard tuning used in traditional Siamese music (see for example
Helmholtz, p. 556). You might think that, with 7 tones, you could also
play standard diatonic music in this temperament.
But if you try, it sounds oddly rootless. Though we have the seven
pitches needed to play diatonic music, the intervals between them are
all the same. There is no distinction between large and small intervals
to tell us what step of the musical scale we are on -- the same problem
we had playing pentatonic music in the five-tone temperament in the
previous step of our experiment, or two-tone music in the step before
that.
12-tone scale
Next, we proceed to the series of 12 fifths,
F-C-G-D-A-E-B-F#-C#-G#-D#-A#-(E#). If we lower all of the fifths very
slightly, we arrive at our familiar 12-tone equally-tempered tuning. Now
we can play all of the music we played in the earlier 2, 5 and 7-tone
scales, and we can also play diatonic music using 7-tone scales with
full freedom of modulation through 12 keys.
The progression from 7 to 12 tones occurred in European music between
the Middle Ages and the Baroque era. Composers have explored the
potential of the 12-tone scale over several centuries, and these
explorations have culminated in the highly chromatic music of the later
Romantic era and "atonal" music of the 20th century.
The term "atonal" reflects the oddly rootless quality of music that
moves freely among all 12 of the tones of the scale. Does this begin to
sound familiar?
19 tones: Joseph Yasser's proposal
Each new temperament generated by extending the series of fifths
encompasses the melodic and harmonic possibilities of all of the earlier
ones. The mathematical logic of this progression of equal temperaments,
and the way it generates new musical possibilities at each step, lead us
to ask "what if we went on to the next step of this progression, a
19-tone temperament?"
In the 1930's, music theorist Joseph Yasser published a book, Theory of
Evolving Tonality, exploring that question. Yasser suggested that
certain composers of his time, notably Scriabin, were already composing
in a 12-tone "supradiatonic" scale with distinct tonalities identifiable
by the wide and narrow intervals a 12-tone scale would have in a 19-tone
temperament. Yasser also suggested that a logical next step in musical
instrument design would be to adopt a 19-tone temperament, which he
contended would open up possibilities in music similar to those which
followed the transition from the 7-tone to the 12-tone scale.
Why don't we use Yasser's system?
The advance to a 19-tone scale which Yasser proposed has not yet
occurred. Why? I think that there are a number of reasons.
Ingrained habits
One is that conventional musical instruments, music theory and practice
are so deeply ingrained in our musical culture that breaking away from
them would be difficult. The perception is grounded in the traditional
design of musical instruments, and particularly of keyboard instruments,
that D# and Eb (for example) are the same pitch, even though we spell
them differently in musical notation.
Keyboard difficulties
Another problem is that a practical keyboard to play in a 19-tone
temperament does not evolve simply from the traditional keyboard.
A 19-tone keyboard based on the traditional keyboard would have pairs of
black keys (Db and C#, Eb and D#, etc.), and an additional black key
between the semitone intervals E-F and B-C. Every diatonic semitone
would, then, become a two-step interval, and every diatonic whole tone a
three-step interval. The octaves would have to be lengthened to make
room for the fingers, with all of the additional keys.
An alternative, favored by Yasser, would be exchange the positions of
the black and white keys, front to rear. This keyboard might be somewhat
easier to play, but it would still have long octaves and require
difficult relearning.
Poor intonation of diatonic intervals
Another fundamental reason that the 19-tone system has not developed
naturally out of the 12-tone system is that the 12-tone system works
rather well. The 2, 5, and 7-tone systems which lead up to the 12-tone
system have very impure fifths. The fifths in the 12-tone system are
quite good.
But going from 12 to 19 tones requires that we make the fifths worse
rather than better.
In 19-tone, the minor third is very pure but the major third, though
more harmonious than in the 12-tone temperament, is slightly flat,
compromising the melodic role of the leading tone.
Poor intonation of additional intervals
We should also hope that a system with more than 12 tones would provide
good approximations to additional harmonic intervals used in non-western
and experimental music (and discussed by Yasser), but the 19-tone system
does poorly at this. The 19-tone scale is, for example, not capable of
distinguishing between the 7th and 11th harmonics, or of distinguishing
the 11th from the augmented fourth.
The rule of seven plus or minus two
The seven-tone scales in the twelve-tone system approach the limit of
what the human mind can assimilate. The rule of "seven plus or minus
two" in sensory psychology states that for any sensory continuum, humans
describe between five and nine different categories: to give an example,
we describe the gray scale using the categories white, off-white, light
gray, medium gray, dark gray, near black and black. Though we can
discriminate more shades of gray when they are placed side by side for
comparison, we do not give names to them, or use the discrimination
between them as part of a conceptual structure based on unaided
observation and memory.
The musical octave is regarded by sensory psychologists as a continuum
like the gray scale, and it follows that 12 distinct pitch classes may
be more than we can easily assimilate as elements of a musical
structure.
It's not as if 19-tone wasn't tried...
Historically, the evolution toward a 19-tone scale would in fact have
succeeded if not for the problems discussed here. The mean-tone
temperament used from the late Renaissance through the early Classical
era had unequal semitones; it is possible to play Yasser's theoretical
12-tone scale in one key signature on any keyboard instrument tuned in
the mean-tone temperament. Not only this, many keyboard instruments,
especially organs, had extra keys -- for example, separate keys for Eb
and D# (see for example Jeans, p. 173). The extra keys let the
instruments play in several more key signatures than a standard 12-tone
keyboard and still avoid the discordant "wolf fifth" which resulted from
the mean tone series of fifths' not closing at 12. Handel, among others,
had an organ with such a keyboard. The logical culmination of such
keyboards with doubled sharps and flats would have been a 19-tone
system.
But unlike the transition from the 7-tone to the 12-tone keyboard, the
transition to 19 tones never occurred. Instead, 12-tone equal
temperament gradually replaced mean-tone temperament beginning around
1700 AD and culminating around 1850 AD.
What if we leap over 19 to 31?
Suppose, however, that we leap past the 19-tone temperament and examine
the next step of our Fibonacci series, with 31 tones.
This temperament turns out to be extremely close to the historical
mean-tone temperament, with major thirds that are nearly pure. In
addition, it gives a nearly perfect natural seventh, and a natural 11th
which is slightly flat, but distinct from the augmented fourth.
If we put aside the idea of a 12 or 19-tone diatonic scale and regard
the 31-tone tuning primarily as a tool to improve the intonation of
traditional scales and intervals, then it looks like a sensible idea. I
have tried it on a guitar, which sounds very pleasant, although it is
difficult to play. The question of how to construct keyboard instruments
which use this temperament has been addressed by Fokker. In forthcoming
articles, I will address this tuning, among others, as they relate to
modern, computer-controlled instruments. The next article in this series
describes musical tunings as mathematical series.
[Top: John S. Allen's Home Page]
Rediscovering Gravity and Force By John Paily
Address
John Paily
Introduction
A turning point in science occurred when Quantum and Uncertainty
Principles were discovered. This turning point is well explained in the
words of Heisenberg and Einstein
"The violent reaction to the recent developments of modern science
[hinting to Einstein's outburst at Copenhagen meeting] can only be
understood when one realizes that here the foundation of physics have
started moving; and that this motion has caused the feeling that the
ground would be cut from science."
'Heisenberg'
"All my attempts to adapt the theoretical foundation of science to the
new type of knowledge failed completely. It was as if the ground has
been pulled from under the foot, with no firm foundation to be seen
anywhere upon which one could have built"
'Einstein'
Einstein spent last part of his life, desperately searching for a new
platform and the Unification Theory. During this period he introduced
Anti- gravity but for some reason best known to him he quickly retreated
from it. In his biography he suggested that science and its method has
limitations to express nature. Personally, I feel he came close to
discovering the Ultimate Theory, when he suggested that constants of
physical science should be replaced with ratios, for example one mass to
another, one speed to another and so on.
The implications of quantum mechanics led to the conclusions that
physical phenomena's are closely and intimately related with mind and
consciousness. Science has its base in the isolation of mind and matter.
But through many complex evolutions it came one circle back to the
original point and proved that mind and matter is inseparable.
The collapse of science's base led many acknowledged scientist to look
back on the Great Eastern Philosophies such as "Veda's", "I Ching" and
so on for the secret of nature and its oneness. Every Scientist
conscious of his profession knows that the reckless exploitation of
science, without knowing the inter relationship and the oneness of
nature is leading us to disaster possibly even to the complete
annihilation of mankind. The technologies that the scientist built on
'part knowledge' are now in the hands of the "mass"[Public]. The mass
fuelled by competitive and conquering mentality is geared up against one
another there by against nature. What guides these developments is the
ignorance about the relationship and oneness in nature.
The scientists are helpless. They are limited by the boundaries they
themselves laid. If the "mass" should be stopped from heading to
disaster then the scientist should show in clear terms the drawbacks in
science and its limitations. They have to discover a new base on which
all developments in science can rest and at the same time it should have
potentials to show the relationship between various phenomenon of nature
that science has discovered and exploited. In other words scientist
should show in clear terms the Unification Theory. It has to link mind
and matter. In other words scientist have to clearly show that nature is
not conquerable and that any attempt to conquer it by physical means
only leads to his own destruction.
This article is a humble attempt to present a new base over which the
oneness can be built up and explored in great detail. It is a product of
nearly two decades of relentless struggle to understand nature. What you
encounter in this article is the filtered essence of this exercise. I
believe if the essence is understood (the) rest would flower by itself. The
basal idea on which science was constructed and all the transformation
that occurred in it leading to the bifurcation of macro and micro can
dissolve into this new vision. The new vision not only unifies micro and
macro but also brings up the connection between physical and biological
world. It would end in the unification of Science and the great Eastern
Philosophies. The word, Gravity, Force, Quantum are the three corner
stones of science. Therefore I begin here.
[A request with the Reader: Please approach the write up with an empty
slate. Keep aside all preconceived thought and visions. You are welcome
to pass the judgement in the end. This article was written in a hurry,
very briefly, basically to project the basal vision through the free place available on the Internet. I hope to follow it with explanations as
to how all the scientific developments can fit into it. I am new to the
'inter-net'. So please bear with me for any lack of clarity in my
presentation..]
Discovering Gravity, Force and Quantum Phenomenon
The basic force of nature is "gravity". Newton quantitized force. This
quantification of force gave the thrust to science and its development.
The theory of gravitation is built on a fundamental assumption-
non-equilibrium. This fundamental assumption escaped the attention of
scientist as they set about predicting and conquering nature. It is
important that we analyze the concept of equilibrium and non-equilibrium
in order to rediscover gravity and force. It is also important for
discovering anti- gravity and the oneness of nature.
[There are many flaws at the foundation of science that becomes apparent to our senses the moment we review it without any attachment to it. Many fundamental questions such as what causes gravity, is left behind.]
Understanding the Concept of Equilibrium and Non-Equilibrium
A Path to Discovering the Fundamental Particle of Nature
Fig.- 1 shows an absolute equilibrium system [A possible precursor to
the fundamental particle]. It has 8 equal components distributed in
space. Each component has three triangular flat surfaces, which joins it
with the adjacent components and a curved outer surface open to the
outside. These eight components, we note, are obtained by cutting the whole system on three possible left and right schemes. See Fig.-1
Positive and negative sign given to it, gives better understanding of
the system. Each positive component is covered by 3 negative components
and has a negative component as directly opposite pair. This means our
system is absolutely neutral. [This is a non-existent situation we will
understand it as we go on.]
The components are distributed in two planes. I call them certainty plane
and uncertainty plane See Fig.-2.ÊEach of these planes has horizontal
and vertical component. As it is there is no difference between the
certainty and uncertainty plane. They come into effect the moment an
external force acts on any component. The horizontal and vertical plane
on which the force acts becomes the certainty plane, the other becomes
the uncertainty plane.
An important point to be noted is that each of these components is
distributed equally in the certainty and uncertainty plane. They are
flexible by a unit to on all the three directions and also to the center
or away from the center. In other words they exist in three dimensions;
the fourth dimension is towards the center or away from it. See Fig.-3
and 4
Each dimension has two choices as far as the external force is
concerned; it can take a left or right path. The choice is linked to the
direction of external force. This choice is also linked to the fourth
dimension that is it can take motion to the center or away from it.
[Classical physics dealt differently with the two direction, that is,
left or right vice-versa and towards the center or away from the center
vice -versa. The modern physics however has linked it. Let us not
explore this line at this stage.]
The neutral state of the system means our system has no attractive
property. This also means that our system cannot attract any other
system in space. This means that our system has no gravitational
property. In other words it has no mass. Let us now observe the
following figure [Fig.-5].
In the first case there is no force. In the second case too the systems
do not possess the gravitational force. In the third case the first
object has force but the second object has no force of its own nor does
it yield to the external force. In the fourth case the first and the
second object are equal and has equal and opposite distortion in space.
The systems move towards each other simultaneously on two different
paths, one left to right the other right to left towards a center. At
the center they fit into each other.
The fifth case is the real case here the systems are unequal and
produces unequal force on each other. There is resultant displacement
towards the bigger object. The bigger object will move in a smaller
curved path and the smaller will move in a bigger curved path [we will
see this aspect in detail as we go on]
We have now discovered a possible origin of gravity in the distortion of
the system in space. This distortion produces the field thus we have
also discover the origin of field. When we say distortion then it is
constriction in one plane and expansion on the other. This invariably
means the system exist in bigger area. See Fig.- 6
Now this distortion of the system in space needs the application of the
external force. Forces we know are created in pairs. Thus there should
be minimum of three systems, two systems on which force is applied and
the third system, which applies the force or causes the distortion. Thus
it becomes important that we analyze the concept of force and discover a
third system that produces the force. This third system should in its
essence be different from the first.
Old and New Vision of Force
Old Vision of Force:
Science represents force by means of a straight-line arrow. The head of
the arrow represents the direction of the resultant force. See Fig.-7
Fig -7 represents the force and direction in a simple manner. A closer
look says that the force has to be applied over an area to obtain the
desired result. The area has a minimum, optimum and maximum that is
related to the system. Beyond the minimum and the maximum it cannot
produce the desired result, that is the mechanical displacement of the
body. In other words force is most effective when it is applied over an
optimum area and its efficiency decreases as we go to the limits. [A
little confusion might occur here because we do see in nature, object(s)
responding to force below the described limit. This would disappear if
we assume that our system in reality is a series of layer(s) of (the) system. A
clear picture could be visualized if we extend our picture to a minimum
point. We will come to this aspect]
The vision invariably means that four points covering a circular area
should represent force. See Fig.-8
The fact that the force has minimum, optimum and maximum value
invariably means that the four points of the force is flexible between
two limits; one to the outside the other to the inside See Fig.-8. This
means it exist(s) flexible between 12 points. If it goes beyond minimum in
falls to 'zero' and if goes beyond maximum into falls to 'one'. In other
words beyond the limit the force breaks down. [We will understand the
statement as we go on]
The Fig.- 9 brings out how I imagine these four forces. 'A' and 'B' are
one pair and 'C' and 'D' are a second pair. When 'A' and 'C' on the
vertical plane tends to the maximum simultaneously 'B' and 'D' on the
horizontal plane tends to the minimum level. The possible link between
the two pair is also shown in the figure.
The force we visualize is understood from the action it renders on the
system. We know that our system opposes the force acting on it. The
force with which the system opposes depends on the force with which the
left and the right are held. Simple reasoning tells us that, one has to
apply an external force, which is more than half of the force with which
the system is bound.
Classical Science has its beginning in the above base. It assumes that
there is a minimum force required to move an object. The displacement it
causes on the system depends on the force applied beyond this minimum. It deals with the optimum situation described, the magnitude of the
external force and the opposition rendered by the system on which force
is being applied. The optimum here means a situation where the vertical
and horizontal components are equal and force takes perfect cyclic form
on perfect cyclic surface. The force according to science acts through
the center point "O" sees Fig.-3.
This is a non-existent case because non-equilibrium is a basic
requirement. By our Fig-5, an important point emerges here, that is, if
our force recipient system is in absolute equilibrium, then it is not
amenable to the external force and no amount of force can produce any
change or displacement in the system. This invariably means all system(s)
exists in a non-equilibrium in respect to one or the other left and
right frame equality].
Force as visualized by science is inadequate to answer our day to day
experiences. We know that all force exerted on a system does cause some
type of stress on the system. Little force produces little actions.
These little actions add up before a visible action and reaction takes
place. Science has quantitized force and has developed methods to
predict the external action and reaction [Motion of the system]. But it
has failed to figure out underlying process. There is a qualitative
change within the system before any quantum actions are witnessed. To
understand the basic process of action and reaction we must develop a
new vision of force.
We did see a form of flexibility in force. That is, it can expand on one
plane by contracting on the other. The ruling out of the optimum
invariably means the force should be in favor of the vertical or the
horizontal plane. However this flexibility alone is not sufficient to
explain nature we witness. Our nature is basically a revolving one. It
is a system that is some how winding and unwinding in three dimensional
space. This winding and unwinding picture becomes quite evident the
moment one cares to observe nature. This means that basically the force
also should be structured in a curved manner and it should exist in
three dimensions, such that it adds or negates the winding. When we
speak of winding then there are two choices; one left to right, the other
right to left. The fact that forces are created in pairs invariably
means both these choices are mutual and acts on two different system;
one is wound left to right and the other is wound right to left. [We
will see more about it later on]
New vision of force [Concept of Ratio's and Dimensions in Space]
From our basic vision we understand that force has its origin in
non-equilibrium. In other words force has always a left or right
orientation. We must try to imagine it as winding or unwinding one.
Imagining force as winding or unwinding one automatically links the left
or right winding motion to another motion, that is a displacement to the
center or away from the center.
Thus in the new vision, force is viewed as curved one capable of
producing a unit motion from left to right or right to left accompanied
by a unit motion to the center or away from the center. In other words
the force is basically a winding or unwinding one. The process of
winding is a three step process that leads the system to the critical
state by destroying all the three left and right frame equilibrium. The
process as we note, basically proceeds in a stepwise or quantum manner.
See Fig.-10
The first step is a collapse into 4:3 ratio on the left and right. The
second step is collapse into 2:2 on the left and right. The third step
is a collapse into 3:1 ratio on the left and right frame. The third
ratio is unstable. If we speak in terms of dimensions our absolute
system has four dimensions. Three left and right frame the other to the
center. The moment it gets into 4:3 ratio, three left right frames
reduces into two left and right frames. This is accompanied by the
formation of a new unstable(ility) of left and right frame. This new frame is
directed to the center and away from it. [Centripetal and Centrifugal
frame].
When it enters the second critical point, one more left and right frame
is sacrificed in favor of the centripetal and centrifugal frame. Here it
collapses into one left and right frame and one stable centripetal and
centrifugal frame. These two frames are equal and opposite. [We will
understand it in detail as we go on]. In other words, in reality the
system gets into two frames, which are equal and opposite. The idea of
equal and opposite is non-existent. It means it has no direction. [We
will understand it as we go on.] The action of an external force
suddenly actualizes a third dimension to it.
From the Fig.-10 we can make out a fact that our force creating system
in reality is made up of four spheres one around another. Some unknown
action [we will discover it later] is differentiating these spheres. At
the first critical state the four circles merge into three. That is,
they lose one left and right dimensions and acquires a new dimension to
the center or away from it [2+1]. At the second critical state the three
circles merges into two, distributed into one left and right dimension
and gets flexibility of winding and unwinding by a unit [1+1+1]. [We
will see this in detail as we go on]
When the system is pushed further to the third critical state the(n) all
the left and the right dimension breaks down in favor of the winding or
unwinding or centripetal or centrifugal force. That is, to the center or
away from it. Here the ratio becomes unstable 3:1 in favor of winding or
unwinding.
It is important that we recall that a non-equilibrium is a necessity for
any system to produce force. In other words its left and right frames
should have a non-equilibrium. In the Fig.-10 the left part is bigger [I
call it active or dominant] and the right part is smaller [I call it
passive or recessive] in the ratio 4:3. Further the force component 'A'
and its pair 'B' exist in a 4:3 ratio. This is also so true for the
passive right component. This ratio 4:3 is stable. It is independent yet
relative. It is passively opposite. A 2:1 ratio forms an attachment. It
is, no longer opposite, it co-exists within a limit. A 3:1 ratio means
the small system looses its existence. This statement is a vital one. As
we go on we will understand its depth.
The component 'A' is the lead component. This determines the direction
of winding. [What necessitates winding is an important question. We will
discover this vital question as we go on. By simple reasoning of cyclic
existence, it points to the fact that it lies in the past action.]
Let us now sketch the ratios between different components of the whole system.
Initial state.
[First state of winding]
AB = 4: CD = 3
Second state of winding
A:B = 5:2
Third state of winding
A:B = 6:1
It is important to note that the force produced is a process of
twitching (ratcheting?) of the component forces. The process differentiates the
underlying four component spheres. At the third critical state of
winding the four component spheres goes into a process of differentiation and then collapses into a new order by joining on a
different plane. In other words the system breaks their relation
temporarily and rejoins on the second plane. I request Reader's to put
some effort to understand this aspect. If you could follow the changes
in the strength of the charge and one can follow the twitching process.
This would help visualizing this vital process in space. [Help to
project this vision more clearly using computers is also welcome]
Here the movement of "A" away from its center to the center of the
system on which it is producing the force, causes the winding force.
This is accompanied by equivalent and opposite motion by its passive
pair "C" away from the center on the opposite side. This movement is
related to the other component "B" and "D". We must note that the whole
process of (the) force producing system, by way of our above visualization is
nothing but transmission of winding. It should be noted that this
winding on one plane is accompanied by unwinding on the second plane by
the same magnitude. Imagining a drill at work would help us understand
the process.
Action and Reaction
Let us now visualize the force and its quantum action in depth on a
system. In other words let us visualize action and reaction process. To
understand the action and reaction let us return to the action of force
on the equilibrium system.
We saw that a circle with four points represents force. We can now
graduate to a vision with a sphere. This would give us a spatial vision.
In our new vision these points are oriented to produce a left or right
winding action. There are some important aspects to be noted here.
Equal and opposite force cannot produce an action. This means our force
creating system should be bigger than the force receiving system. That
is it should exist in a ratio 4:3 ratio for action and reaction to
proceed. This means that there should be two types of non-equilibrium
for an action and reaction to proceed. One distortion in respect to (a) systems own left and right schemes and second is non-equilibrium in its
size, thereby the curvature or the field of the two systems.
So far we saw how (a) force-creating system produces the winding force on
the external system. Let us now try to visualize how the external system
reacts to the force.
Action and Reaction - First type
The force in nature we noted is basically produces a winding or
unwinding. In other words it causes a spinning force on the component
system of the system on which it acts. This spinning force can take a
direction to the center or away from the center. [Positive on positive
wound produce winding(. ) (P)ositive on negative would produce unwinding]. This action is simultaneously accompanied by equal and opposite
displacement in its pair on the opposite side. In other words when one
component wind(s) left to right to the center the other unwinds right to
left away from the center on the opposite side. The force is transmitted
through (the) other six components. The other pair of components works against
the former pair yielding to the external force. This means an opposite
force originates. The ratio of force winding and opposing (is) 1:3. This
invariably means winding cannot proceed. So our visualization of winding
in Fig -10 is wrong.
An alternate way to look at it is that forces are applied in pair, one
dominant [active]-lead force and another recessive [passive]- supportive
force, which exist in the ratio 4:3. [The term is borrowed from biology
intentionally] See Fig -11. The angle between them now determines the
efficiency of the force. Near to 45 degrees is the most efficient above
and below it the efficiency decreases. The maximum is 67.5 and the
minimum is 22.5 degrees. The fact that force can exist as left or right
winding means an absolute 45 degree angle does not exist.
We must note that there is an orientation to the left or the right. So
there is nothing called a strict optimum angle. The root of uncertainty
exist here. We are already noted that winding is a three step process.
The fact that there is a fundamental non-equilibrium invariably means
the action of force starts at the second ring from inside or second ring
from outside. See Fig.-12. The moment the winding or unwinding force
touches the third critical point the system breaks down.
The Fig.-12 for all-purpose could be rewritten as shown in Fig.-13.
Here we note that our four components are flexible by a unit on either
side.
What this means is that the real system is not distributed in 12 point
space or four dimensions but in a 9 point space that is three dimension.
In other words, it exists wound and linked up with other systems
capable of winding and unwinding by a unit. When a system winds or
unwinds to readjust to the external force it passes the four-dimensional
state temporarily. Each system thus has three worlds. An inner world, an
outer world and middle world. The inner and outer word is linked by the
middle world. When the outer world winds the inner world winds in
relation. And when the outer world unwinds the inner world also unwinds
in proportion. The full implication of the statement will flower on its
own. For the present let us forget these worlds.
Let us return to the winding aspect and note the process step by step.
1] The use of a pair of force(s,) active and passive can wind the system to
the third critical state. Here the system gets compressed to the minimum
on the horizontal plane and it expands to the maximum on the vertical
plane. A reason why the system breaks (omit?could) and results in motion could
be understood from the figure -13. When the system tends to the third
critical point the external force gets strengthened. The force of
opposition gets directed in the same direction as the external force.
2] The break down involves inversion and rejoining into new order.
Externally it would appear as a spinning motion. If the force applied at
the critical point is more than the required magnitude to take it to the
third critical state then the system is displaced in proportion. In
other words the unwinding process is slowed down. We are here
visualizing a motion that could fit to the mechanical motion we witness
in nature. E.g. kicking a football.
3] The external force applied here is released back to the outside on
the opposite side.
Action and Reaction - Second Type
The second type of winding is possible if we apply winding force active
and passive [4:3] on the vertical and horizontal plane. See Fig.-14
Let us list some prominent difference(s) between the first case and the
second case
1] In the former case, force had to be applied continuously till it
reaches the third critical state. If we remove the force in the middle
the system returns to the original state. In contrast here as the system
reaches the first critical state it collapses into a relative
equilibrium state with the force producing system. If the force is
removed the system does not return to the original state. It is flexible
to winding and unwinding force by a unit. If the system has to be
returned to the original state then one has to apply an equal and
opposite force. That is if the system was wound by right winding force
then a left winding force has to be applied on it. Fig.-15 and 16
explains these facts. One has to visualize it as a system wound in space
with a pair of equal opposite arms each consisting of an active and
passive arm in the ratio 4:3. Here for every left wound system there
should exist right wound system. The positive and negative charge and
the repulsive and attractive force would help us visualize such a
picture.
2] When the winding force is increased and it is pushed to the third
critical state. The system breaks down spins and rejoins. An important
difference exists between the first and the second case. Unlike in the
first case where the force is released to the opposite side here it is
concentrated to the center. Which means one part [1/6] more than (¹, pi?) of
the total matter is wound to the center. This gives the matter two
layers, an inner layer and outer layer with a ratio 4:3. Another
three-step winding will further wind the system, adding another layer
over the previous layer. This layer constitutes one part [1/6] more than
(¹, pi?) of the remaining matter. The process continues adding layers to the
system. See Fig.-17 [Note we are leading ourselves to an atomic picture]
3] Since forces are created in pairs it invariably means that for every
right wound system a left wound system also exist.
4] To unwind the system, one has to apply an opposite force in an
increasing manner. [Doubling]
5] Force or the action of the first type can be applied on the relative
equilibrium system. [The system resulted from (the) second type of action].
Here the system winds on one plane and unwinds on the second plane. This
means increasing opposition to the external force. One has to double the
force to increase the speed [Recall Einstein's work - mass of a system
increases with a speed In our vision mass is related to distortion of
the system in space]
The systems we deal (with) in nature are essentially of the second type which
wind and unwind in relation to the environment and changes in it. In
other words the system has a relative existence with the external system
which produces the force. If the right wound system experiences a left
winding force and reaches the critical point to the left [first critical
point] it unwinds to the previous wound state. If in contrast it
experiences further winding on the right it reaches the critical point
on the right [third critical point] it winds matter to the inside.
Some important points that we should bone(bear?) in mind
1] There exist(s) no absolute system. The fact that the force producing
system is distorted to produce 4:3 ratio of force [active and passive]
(i)nvariably means that our force receiving system is oppositely
distorted.
2] Since equal and opposite forces cancels and cannot produce a winding
action, it invariably means that the force producing system should be
bigger than the force receiving system. [4:3 relationship] This
invariably means there is two type(s) of non-equilibrium in action and
reaction process. The first is the non-equilibrium in its own left and
right frame. The second is that there should be non-equilibrium in the
size of the two systems that are interacting. [To put it another term
the curvature of these systems should be different]
3] This also means that the force producing system and force receiving
systems are intimately connected by an attractive force(,) (l)ike body is
connected to soul. We will explore these statements in detail later on.
Action and Reaction - Third Type
The third type is one where a left winding of the second type is
accompanied simultaneously by right winding force of the same type
[second] or vice versa in the ratio 4:3. This ratio means that our
system gets an orientation into left and right wound system. See
Fig.-18. It also means that there should exist a recessive partner to
the first to create this force.
This type of winding winds ² of the matter in one plane and the ¹ of the matter in the (second plane?????????????????).
Force can wind the system to the third critical state. Here 'A B' is the
active force, which determines the winding direction, and 'C D' is
passive force, which supports the winding. This type of winding can also
occur on the relative equilibrium system discussed in the second type.
When it touches third critical state the system collapses to the center.
In the process ¹ if the matter which was caught with in the pair of
bigger arms is ejected from the opposite side. The whole process is
compression and expansion process. The compression is accompanied by
ejection of matter in one plane and is simultaneously accompanied by
sucking of matter on the second plane to regain the state prior to
compression. We must not forget that all the system is linked because of
the basic non-equilibrium. [Here we can imagine it as the suction of the
field or the unwinding force on the external system]. The process is
thus repeated endlessly. They can principally cause a never-ending flow.
The ejected matter because of the structure of the system and the process of breakdown and emission has 3:1 ratio in the arms. The emitted
matter unwinds in a stepwise manner (ratcheting?). To gain 4:3 ratio and exist
flipping between two alternate states. The whole system spins and moves
in relation to the non-equilibrium in the ejected matter. See Fig. 20
A brief review of the above exercise:
Our thinking thus presents to us three different types of action and
reaction process. The first one accounts for the mechanical motion. The
second one gives us the picture of an atom as well as the heat induced
compression and expansion process (heat=gravity?-Arch. Plutonium). The third type accounts for stellar
and radiating systems.
The first, the second and the third type of action can be effected on
the type of system that results in the second type. [Action and reaction
on the second type of system is the action and reaction we deal in our
day to day life]
The mechanical force is winding force on one the horizontal plane and
unwinding on the vertical plane vice-versa. Heat induced phenomenon's
are winding or unwinding on both the vertical and horizontal planes. The
third type is unwinding on certainty plane and winding on the
uncertainty plane.
Discovering force pairs and relative motion.
Forces are created in pairs. They are equal and opposite. This means our
force producing system should produce another pair active and passive
force. Which is equal but opposite in direction. This also means there
exists another system on which it applies the force. Let us take the
example of a person kicking a football. He should have applied an equal
and opposite force on the earth system on which he is standing. Simple
reasoning tells us that he should have moved the earth system by the
same magnitude of force by which he moved the ball. But the displacement
is very small and is not evident.
To understand the above statement we must understand the relationship
between the earth and the ball system. We know that both are material
systems. Could it be that the two apparently independent material
systems has its origin in one system? What is the relationship between
them? To answer these questions let us explore the possibility of
creating such a situation.
The best way is to imagine a situation where an external system twitches
a portion out of the parent body. Some conditions exist for such action.
1] The external system should be bigger than the system on which it is
producing the force by a ratio 4:3. This also means the two systems
exist (as a) distorted or wound state around one another.
2] It can only twitch a maximum of 1/6 of the parent body. This is
agreeable from the fact that the system exists in 2 dimensions left and
right center and away from center. The creator has to choose one of
them. Left or right and between to the center or away from the center
simultaneously.
Let us look at this phenomenon in greater depth.
All systems have a center. If we imagine an absolute system we have a
central point. The moment we think of left or right wound system in the
first order decribed above it invariably means that this center is left
or right twisted area. If we take relative equilibrium system then it is
an area, which is flexible between two limits.
The moment there is intervention by twisting a new center begins its
birth. Let us call the new system being formed as sister system. It has
its direction from the maximum distortion to the minimum. The Fig - 22
explains these facts.
The twitching is a three-step process. The first step is reversible(.) (T)he
second step gives the sister system a relative existence with the parent
body. If one has (to) reverse the system one has to apply equal and opposite
force on the system. The third step leads to the critical state and
break down of the system. The sister system is temporarily isolated
spins and moves in relation to the parent body. The parent body too
moves by the same magnitude but the difference in size 1:5 results in
rolling of the sister body around the parent body.
We must note that the winding force applied on the parent system unwinds
the system in proportion to the winding it causes on the sister system. But the magnitude of the force is not sufficient to unwind the parent
system by a quantum unit. We must recall here the saying that - a chain
is as strong as its weakest link. The system rolls on top of the parent
system because the parent system is stronger than sister system by
certain ratio 5:1. The field of its influence is proportionately bigger.
We must not forget that the parent system is unwound by the same
magnitude as the sister system and the process of reversal has equal and
opposite actions.
Suppose the action of the external force provider at the third critical
point is six times stronger, then we can imagine a break down situation
that can hurl the sister system beyond the influence of the parent
system. We can also imagine a situation where the sister system(,) pushed
out with certain fixed force(,) exists revolving round the parent system by
an equal and opposite attraction and repulsion force. Here 4:3 ratio
alternates in favor of the parent and sister system. We can imagine the
above situation with a relative equilibrium system.
Let us review the three states of non equilibrium again and the
revolution of sister body around the parent body.
When we say our system is in first order of non-equilibrium it
invariably means it is constricted by a unit on one plane but expanded
by the same proportion in the second plane. This means its effective
area for revolving has increased. Suppose we take a circle with a radius
4-cm. Then its effective area of spin is 5 in the first order of
non-equilibrium. At the third critical point the effective area is 7.
For a relative equilibrium system it is 5 plus or minus 1. Here 6 denote
unwinding and 4 denote winding.
In our above case our parent system is the initial state is in 6:0. In
the first step of winding the ratio becomes 5:1. When it goes into
second step of winding it falls into a ratio 4:2 state. In the third
step it reaches 3:3 ratio that is equal and opposite. The system breaks
down but since the sister system position is below the middle of the
parent field the superior strength of the parent field pulls it down to
it own center.
We noted that earlier that only 1/6 portion could be twitched(.) The
creator of twitching force is applying equal and opposite winding force.
In other words he is trying to unwind the parent system. By twitching
1/6 of its mass. Because of the 1/6 restriction for twitching a sister
system, the sister system is taken to the third critical point but the
force applied on the parent system would not be sufficient to unwind the
parent system even by a unit. It would have decreased the effective
revolving area of the parent system by 1/6. A slot thus is created for
the sister system to exist along with the parent system.
But when the force applied at the third critical by(it?) is increased to 6
times the original force required to push it to the critical state then
the parent system collapses into a new order where left turns right.
[See Fig- 23] The sister system exists in relation to the parent system
by an equal and opposite force. Suppose the force is further doubled(.) (I)t
would escape the fold of the parent system.
In contrast(, the) sister system is thrown out at the critical point with a
force that is 5 times higher, then it still remains in the field of the
parent body but the strength of the field is equal and opposite. Thus it
remains in the orbit. See Fig.-23
Discovering Anti-Gravity
Introduction
So far we dealt with rediscovering gravity, field and its interaction in
a new simple way. Gravity is always directed to the center. It is the
instinct of all material system.ÊWe noted that gravity has its origin in
the external force acting to distort the system but we never touched
upon the external system creating the force. Simple reasoning tells us
that this system should be different from the system, which is receiving
the force. The system that receives the force has no choice but to yield
and react to the external force by winding or unwinding. In contrast the
external force provider has a free will to wind or unwind the system. So
by principle this should be different from the first.
Gravity we all know is centripetal force it is directed to the center.
Much of early physics was bent on giving a picture of steady state. But
the time was not stopping for this view. (The) most evident factor of nature
is the evolution and thereby a direction to nature. Advancing Science
had to give scope to changing time and evolution. Thus relativity theory
extended in time gave us big-bang origin. But it failed to link the end
of time and the beginning of it. If every thing was to collapse to a
center point then it is reaching a dead end. Gravity we know is
attractive. If all the mass in the universe condenses into point it is
reaching the final state to which it is directed and therefore looses a
need for origin. The only alternative way is to invent anti-gravity, such
that gravity is supported between two limits. Einstein was quite aware
of the dead end to which his theory would lead. His dilemma is very much
evident in his biographies. He thus tried to introduce the Anti-gravity
into science, but failed. Possibly he failed to comprehend a base for
it.
My path to Anti-gravity was a simple one. Nature led me in to it. That
was more than a decade back when I was forced to quit the competitive
world of science. I was working as a bio-technologist at that time. My
entry to the wor(l)d of research was unique. I began my research in (an)
ill-equipped lab isolated from (the) academic environment. I was handling some
of the toughest problem in my area. I had no experienced guide to guide
me. (The) only advantage I had in the lab was freedom and enthusiasm to
achieve something. When nearly a years struggle, day and night, produced
no results, enthusiasm turned into desperation. Just when I was
contemplating on giving up, I happened to build a sort of communication
with nature. That opened a floodgate of ideas. In every thing I touched
results came with absolute ease.
The communication with nature opened not one path but two. One gave me
the advantage to go ahead and exploit living system(s), but the second path
was restricting me from doing it. In my communication with nature,
nature was responding to my question but I was failing to respond to
nature's questions. I had no answers why nature mixes its information?
Why nature brings up diversity? Why gene is a triplet code? Why active
and passive gene? Why chromosome pairs? Why DNA is double helical? Why
mitotic and meiotic division and so on. Slowly I got ashamed to go to
her fold with my greed to conquer nature.
When I was forced to quit (the) competitive world, I started pursuing these
questions. This took me to the foundation of biological science, then to
the foundation of physical science and then to the foundation of science
itself. I discovered the science has grown leaving behind vital
questions. At the foundation of science right in the first experiment
done by the father of science [Galileo], I discovered anti-gravity.ÊWe
must recall here the first experiment. Galileo lifted balls of different
weight on to the top of a slanted plank and rolled it and measure
showed that the acceleration in the gravitational field is the same for
all bodies. An important point that escaped our attention is that, if
not for Galileo the balls would go up the slanted table! Galileo's
action here is anti-gravitational- what a simple way to discover
anti-gravity!
No body expects the world to stand up to a discovery of this type. Thus
I was left with a new challenge - develop a unified picture of gravity
and anti-gravity. The second law (of) thermodynamics applied to living and
non living systems, clearly showed the opposing flow of time in these two
systems.
There are two clear aspects to nature we witness, that is
1] Existence
2] Evolution
Simple argument tells us that the second aspect some how is enclosed in
the first.ÊIn other words evolution should be cyclic not a directed one.
At maximum and minimum they should change the direction. One of the
prominent observation(s) I made in nature is that living systems in nature
tends (to) both ways. The direction is guided by the condition in the
environment against which it actually is surviving. But the environment
and the living system together seem to have direction that is towards
increasing fluctuation.
It became apparent that the increasing fluctuation has its direct origin
in mankind and his exploitation of nature out of ignorance. It was
crystal clear that somewhere along, the system would collapse or
reverse. I had (a) complete explanation for a possible method (of) how nature
would maintain it self. But that picture never gave scope for (a) cycle of
oneness and my search remained incomplete - I lived in the fear of
destruction.
The final answer came to me when I stood in prayer in total submission
to nature and its powers. [Scientist please excuse me for taking you
along (a) non-scientific path] With that ultimate link the truth of the
religious philosophies came revealing it self in all splendor. I could
comprehend Veda's, I-Ching, Bible and such great text. The truth existed
clearly written in these and other great books. But we failed to
comprehend it. We will come across these paths in detail in our future
talks. Let us return to gravity and anti-gravity
From our above argument nature beyond doubt appeared supported by the
opposition of gravity and anti-gravity. That is matter and life.Ê
Gravity is supported by anti-gravity and anti-gravity is supported by
Gravity. Two opposing forces, which are equal, could never produce any
action.
(The) next best option is to assume a ratio 4:3 or 3:4 of anti- gravity and
gravity. This meant there is a direction in favor of one of them. That
gave scope to the concept of evolution, for the 'whole' as well as the
'individual' components. But the question remained where is the
evolution taking us to? How does the ratio change its direction? It is
important that we should understand this reversal phenomenon if we are
to understand the nature to its core. In other words it simply means the
big-bang and big-crunch becomes an inescapable reality of nature.
To understand nature it thus becomes important that we understand life,
its structure function in detail. In other words we must re-visualize
our knowledge pertaining to life on the new base we discovered. In fact
the new base I presented above was (a) reflected image from a living system.
You will understand it in greater detail in future talks.
[Dear brothers and sisters. I am sorry for breaking my write up here. I
hope to write and load the next part in a week or two. I am new to the
net. Please bear with me for any shortcomings in the presentation. I
live in an interior village. I have to travel over 100 kilometers to be
on a rented net facility. As jobless dependent on an agriculturist
family, in a country where government makes agiculturists life
miserable, it is beyond my means to spend more time on the net at a
stretch.
In the past I have spent lot of my precious time and resources
communicating to the temples of science through out the world seeking a
chance to discuss the idea that took shape in me. But all these efforts
fell into deaf ears. In a competitive world the best option open to a
crank like me, who left a job with a multi-national company is "Inter-
net". I promise to be on it, more and more at all cost.
I am only a student seeking a master to evaluate. You should feel free
to comment, correct and iron out any wrinkles in my thoughts and its
presentation.
Any form of help to reach out to you all more elaborately is welcome.
"In the beginning was God with power through speech. God said, 'May I be many... may I be propagated.' And by his will expressed through subtle
speech, he united himself with that speech and... Praja pathi and
Saraswati were then created. And Prajapathi is called the progenitor of
all beings." Sata patha Brahmana
The gospel of John and other scriptures present more or less the same
truth. John writes, "The same (the Word) was in the beginning with God.
All things were made by him; and without him was not any thing made that
was made. In him was life; and the life was the light of men." John
1:2-4
What exactly is the Word? Physics is discovering that light, sound,
matter, etc. are all vibrations of more or less density. Vibration
requires movement. In Genesis, we read that at the moment of creation
"the Spirit of God moved upon the face of the waters." The Word is the
original vibration or sine wave from which sprang all other vibrations.
According to the Vedas, this Word is "Aum." Let's examine this mantra
written out in longhand.
AUM
The circle in "A" symbolizes the self-contained, unmanifest, absolute
Godhead and the tail represents an emanation at the moment of creation.
In the preface to "The Science of the Spoken Word," the Messenger,
Elizabeth Clare Prophet writes, "The A comes forth from Alpha (our
Father)-- as the initiator, the creator, The Beginning-- the origin of
spirals of consciousness, of being."(51)
The next letter, U, shows the full sine wave of the Word as God or
Elohim begins to move or vibrate. This is the Son, Narayana, who sings
the song of creation. The French word, "son," means sound. The person of
the Son, and the "son" or vibration, are one and the same.
And the "M" means manifold creation, the many folds or waves (the
"mmmmmm..."), of the resulting creation as the Word multiplies and
becomes denser. R. J. Stewart in "The Spiritual Dimension of Music,"
writes that "the bridging tone of M flows either way. For us it acts as
resonance that attunes our consciousness to inner reality, while for the
source of creation, it acts as a medium whereby energy flows out to
generate the outer and ultimately the material worlds."(52) M also means
Mother as in the Divine Mother, Saraswati, shakti or power of Brahma, or
the Mother of the world of "mater realization."
The Word is the cosmic hum. David Tame in "The Secret Power of Music," reminds us that it is the sound of "the ongoing process of creation and preservation of all atoms and all worlds... To the yogi, the OM is as
immediate as the air around him, sounding out in the eternal present. It
beats the rhythm of all hearts, and speaks to the soul having ears to
hear. The great mystics of all time have felt the Word to be imminent in
and around themselves; have known that it was the elixir of light."(53)
You don't have to be a yogi or mystic to hear the light of God. In a
moment of inner and outer quiet, such as in deep meditation or right
before falling asleep, listen for a faint, high-pitch hum. John Beaulieu
in "Music and Sound in the Healing Arts," describes his experiences
listening to the cosmic hum. "The pitch of the sound would change,
depending on the events of the day and my internal state... One day,
after having an argument with a subway attendant I noticed that the
sound became much higher, louder, and that it wavered radically. Over a
period of several months I began to associate physical, mental and
emotional states with the sounds of my nervous system."(54)
Beaulieu states that the sounds we can hear in silence are our bodies
resonating with the cosmic hum. When we are in tune with it, we resonate
with it like the strings of resting violins resonate when one is played.
We can tell when we are out of tune by listening carefully. We get back
in tune through the power of the Word (i.e., prayers, mantras, decrees
and song.)
THE MANDALA OF SOUND
The Great Divine Director instructed us to visualize his mandala over
the throat chakra, where thought and intention are expressed into
physicality. When I meditated on the mandala, my thoughts led me to the
stump presentation on the Science of the Spoken Word that includes
slides of Hans Jenny's sound forms or cymatic pictures. A cymatic pic
ture is a pattern created when various media such as fine sand or clay
are placed on a steel plate and vibrated with a crystal sound
oscillator. Jenny found that different sounds create different patterns,
many of which look like yantras or mandalas. Figure 1 is an example of a
cymatic picture that looks remarkably like the Great Divine Director's
mandala as seen on the home page. The thought occurred to me that this
mandala is not just a symbol of the new earth grid but a sound form of
the higher vibration that will create the new earth grid.
Figure 1 - Cymatic Sound Form
Hans Jenny discovered that when Sanskrit vowels were pronounced, the
sand took the shape of the written symbols for these vowels. Stan Tenen
reproduced these results with Hebrew vowel sounds. Did these two
ancient languages spring from a more ancient language?
A friend of mine who was at LaTourelle told me that the Messenger, Mark Prophet used to teach that the ancient Lemurian language of Sansar was the root of Sanskrit and that the priests of Lemuria were able to manifest objects by reciting their names in a certain way. Can Sanskrit
and Hebrew, like Sansar, influence and transform physical reality,
create or destroy, or heal a person who is out of tune?
David Tame writes, "in Egyptian cosmogenesis the first god, Khepera, or Ra, emerged into being by pronouncing his own name. Thereafter he began to create other forms and other gods by the combined process of
visualizing them and uttering their names. The gods which had then been
created by him were also able to create through a similar process,
engaging in what might be called a 'stepping down of the frequency' of
the One Vibration... The lesser gods were accredited with the knowledge
and the use of words of power which they could use to create and destroy
form, heal the diseased and give life to the dead.... In precisely the
same way that the gods created by combining visualization with speech,
so too did the priests believed it was possible for man to work change
within the physical world."(55)
In the last issue, I deciphered the word "Angoulmois" in Nostradamus'
X.72 quatrain as an anagram for "Logos Manu." However, instead of using
the common definition of "the Word" for Logos, I used the other
definition of "ether" and interpreted it to mean the etheric realm of
the Manu of the seventh root race. In retrospect, I think that "the
Word" is the correct translation, meaning the sacred sound or word of
power from which will be created this etheric realm.
Sir John Woodroffe in "Principles of Tantra," wrote, "Like the Greek
word Logos, mantra means thought and word combined as an expression of
creative will."(56) Hans Jenny, when he uttered the mantra, "Aum" into
his tonoscope, found that the medium created the concentric circles,
squares and triangles of the sri yantra associated with that mantra. The
Great Divine Director's mandala also has concentric circles, squares and
triangles. If the Great Divine Director's mandala is the sound form or
visualization that will create the new earth grid, what is the mantra
that corresponds with it? I can't answer that question, but I have found
more general information about mantras or words of power.
R.J. Stewart wrote, "Closely connected to the throat centre and the
energized utterance of tones, is the ancient concept of the 'word of
power'... In Eastern monastic practice, as in Western, sacred words are
uttered musically, upon drawn out and regulated tones... The vocal
utterance is quite a unique ability that relies upon activation of the
throat centre. This extension of the vocal tone is equivalent to the
extension of energy in the creation of the universe, but expressed
through the limited form of the human consciousness in the human world."
(57)
Swami Maheshwarananda taught that "vibration forms the basis of the
ancient language of Sanskrit. One cannot simply 'speak' Sanskrit but
must combine the sounds with a particular tone or vibration and observe
the pronunciation precisely. The words and word combinations have to
have a particular sound.(58)"
THE TONES OF ATONEMENT
Soon after reading an article about people gathering in churches and
temples to ring bells and pray for atonement (see "For Whom the Bell
Tolls), I heard an interview with Dr. Leonard Horowitz who wrote
"Healing Codes for the Biological Apocalypse." He and his co-author, Dr.
Joseph E. Barber, found coded passages in the Book of Numbers and other
books of the Bible that refer to the musical scale and certain key
frequencies that they think can be used for creation, destruction,
healing and working miracles. They discovered these code numbers in
verses that refer to feast days, sacrifices, and atonement.
Horowitz claims that atonement meant "a tone" in addition to meaning "at one."(59) That makes sense because when you're at one with God, you are attuned to his will or vibrating at his tone. One verse in particular,
Lev 25:9 connects atonement with the sound of a musical instrument:
"Then shalt thou cause the trumpet of the jubile to sound on the tenth
day of the seventh month, in the day of atonement shall ye make the
trumpet sound throughout all your land." The numbers 10 and 7 add up to
17 which can be reduced to 8, the number of musical octaves. The word
day is italicized, indicating that it is a code word. Converting the
letters d-a-y to their corresponding numbers, using the Pythagorean
skein, you would get 417. According to Horowitz, the number 417 and five
others: 396, 528, 639, 741, and 852 are found throughout the Bible in
coded verses.(60)
Horowitz said that each tone has a certain effect. The 396 tone was the tone of creation. The 528 tone was for working miracles as in healing. It is the same frequency used by geneticists to repair damaged DNA. The 417 tone may be used for destruction as when Joshua made the priests blow trumpets for six days to crumble the walls of Jericho(61). I connected these qualities of the tones with the three Hindu deities of the Trimurtu: Brahma, the creator; Vishnu, the preserver; and Shiva, the destroyer. Each of these dieties has a complementary shakti or power, without which creation would only be an idea in the mind of God. Likewise, each tone has a complementary tone: 639 with 852, correspond to Brahma and Saraswati, 396 with 528,correspond to Vishnu and Lakshmi, and 417 with 741, correspond to Shiva and Kali.
In the Hindu Pantheon there are seven lesser dieties under the Trimurti.
Five of them: Indra, Vayu, Agni, Varuna, and Kshiti are active and two
are concealed. Since these are lesser dieties that act as step-down
transformers of the vibrations coming from the Trimurti, they should be
at the next octave down. David Tame points out that the five active and
two hidden gods are like the five major whole tones: C, D, F, G, and A;
and two semitones: E and B of the diatonic scale. He writes, "Many of
the world's religions agree with the idea that there are seven 'first
Gods' which are the living embodiment of the first differentiation of
the One Tone into seven. The Hebrews call these Gods the Elohim..., the
Gods of the seven Tones."(62) Kshiti would be C two octaves down from
Lakshmi; Varuna, the next note, D; Agni, F; Vayu, G one octave down from
Vishnu; and Indra, the next note, A. I decided to try to devise a divine
scale that would include all the frequencies for the Trimurti and lesser
devas.
THE GREAT TETRAKTYS
While researching the musical scale in a book on mathematical patterns, I came across two patterns of numbers that relate directly to the six Bible code frequencies: the magic square and the tetraktys. The
tetraktys or decad is a triangular number that looks like this:
Figure 2: The Tetraktys
Pythagoras taught that the tetraktys shows the major consonant intervals in music. They are the numerical ratios involving adjacent rows of the tetraktys (octave 2:1, fifth 3:2, and fourth 4:3). Note that the code
numbers 528 and 396 are in the ratio 4:3 and the ratio of 396 to 264,
middle C one octave below 528, is 3:2. Also 639, 852 and 426 have these
same ratios. Music therapists Kay Gardner states that musical intervals
with ratios that have small whole numbers like 2:1, 3:2 and 4:3, are
best for healing since they produce the greatest consonance.(63) Perhaps
they're also keys to creation. Pythagoras implied this when he wrote,
"Bless us, divine number, thou who generate gods and men! O holy, holy
Tetraktys, thou that containest the root and source of the eternally
flowing creation!"(64)
Another important ratio is the one between 639 and 396, and 852 and 528. That is, between Brahma the originator and Vishnu the perpetuator of
spirals of creation. It is 1.61363636, which is what you would get if
you averaged the golden ratios of Fibionaci number (1.5, 1.67, 1.625,
and so on).
Each number of the tetraktys represents a stage of creation. The 1 is
the one Word of the one Being that was in the beginning. In music, this
is the fundamental tone from which all other tones are created. In
Chinese symbology, it is the wu ch'i.
The 2 is the first division of the one Word or tone into polarity. In
the story of creation, the Word creates polarities: light and darkness,
heaven and earth, land and water, male and female, etc. Polarities make
possible all life. These are not opposites but the complements of yin
and yang. In music, according to "Pistis Sophia," the yang interval is
the fifth while the fourth is the yin interval. Out of their mystical
union, arises the Pleroma, which is the octave.(65) The polarity of yin
and yang is symbolized by the tai ch'i.
The next number, 3, also symbolized by the tai ch'i, represents the
spirit of harmony which condenses to produce all things. Kay Gardner
writes that "The Greek root, harmos refers to the process of joining
together objects, people, concepts, etc. previously having a separate
existence. Universal harmony is Nature's laws determining the
interdependent relationships of all living things, as well as the
cycling of the seasons, the movement of the tides, of birth and death,
the cycles of the sun, moon, stars, and planets in the cosmos."(66) When
harmony is established, then you can have the various combinations of
the four elements of fire, air, water and earth. Then all things, animate and inanimate, are created. This is the 4 of the tetraktys and is
symbolized by the squaring of the circle. This is taken from Figure 14
of "The Spiritual Dimension of Music" by R. J. Stewart.(67)
Much more can be said about the tetraktys. Cabalists used it as a symbol of God by placing the four letters of his name, meaning I AM THAT I AM,
in place of the numbers. The gematric sum of this tetraktys is 72, a
fundamental gematrian number. Some believe that gematrian numbers are
creation frequencies used by God. In a future issue, I will devote an
entire article on Gematria.
THE MAGIC SQUARE
The other pattern that relates to the Bible code frequencies is the
magic square. A magic square is a grid of nine numbers from 1 to 9 that
are arranged in such a way that three rows, three columns and two
diagonals of three digits all add up to the same sum. Magic squares get
their name from the belief that they possess magical powers. People in
various cultures wear them around their necks as talismans. In Figure 3, the code number 852 is in one of the diagonals and
parts of two others are in other diagonals.
4 9 2
3 5 7
8 1 6
Figure 3: The Magic Square
I decided to make a grid of four magic squares and, lo and behold, all
six code numbers appeared in the diagonals going from bottom left to
upper right.
The magic square in Figure 3 has a counterpart called the heavenly magic square which is formed by subtracting each number in the earthly magic square from 11. The result is shown in Fig. 4
7 2 9
8 6 4
3 10 5
Figure 4: The Heavenly Magic Square
I noticed after expanding this to four magic squares that all six code
numbers were present in this magic square as well, but going in the
opposite direction from upper right to lower left.
The magic square forms the basis of Feng Shui and Nine Ki astrology
which put great importance on placement and direction. Each direction
has a different quality or frequency of ki or ch'i, the intelligent life
force, associated with it. The southwest-northeast axis that the code
numbers fall on is associated with creative earth energies of the old
mother ch'i and the youngest son ch'i.
Also present in the heavenly magic square were the numbers 279, 297, 330 and most of the other numbers needed to make a just musical scale with 264 as C, 528 as C' and 396 as G. However, you have to treat the number 10 as a flexible number. In some cases, the digit 1 is used to get a
scale number as in 417. In other cases, 0 is needed as in 704. In the
case of 660, the 10 is multiplied by the 6 to get 60 and the third
digit, 2, is ignored. 924 is the only number in the scale that is found
only in the earthly magic square. The code numbers are the only scale
numbers that can be found in both the heavenly and earthly magic square.
Could they be links between heaven and earth?
I tried to fit all six code numbers into two octaves of this magic
square scale but found it impossible. The 417, 639, 741, and 852
frequencies were either too flat or too sharp, creating dissonance. Then
the thought occurred to me that each pair of frequencies should have its
own scale. One scale would contain 396 and 528 for Vishnu and Lakshmi.
Another scale with a C of 511.2 would contain 639 and 852 And a third
scale with C at 278 and C' at 556 would contain 417 and 741. It is
interesting to note that the interval of a fourth from 556 to 741 is
flat, creating dissonance, as would be expected for a tone that destroys
matter.
Rather than showing all the frequencies in the three scales, Table 1
shows just the code frequencies, the C notes and important octaves of
these frequencies. These tones represent the Trimurti and their shaktis,
the five active devas, and the two nameless devas. The other tones
correspond to unmanifest or manifest creation.
Table 1 - The Divine Scale
* Listen ** Sanskrit name of object
Also shown in Table 1 are the elements, chakras, and bija mantras
associated with each diety. Sir John Woodroffe wrote that "bijas are the
archetypal seeds, the actual mechanisms of the One Mind that underlie
manifestation...seeds, out of which grow the fruit of the universe..."
(68) Bija mantras are more powerful than other mantras.
The E flat note at 639 Hz is close to the frequency of the kung note in
the Chinese system of music. According to David Tame, "this foundation
note was considered to be the earthly manifestation of divine will, and
a sacred, eternal principle, upon which was based the proportional
system (the weight, size and tone of all things) of the entire state."
(69) If the kung was wrong, civilization would decline. Unfortunately,
the ruler of each dynasty changed the value for the kung because they
reasoned that if the previous dynasty had been right, it wouldn't have
ended. The last Chinese dynasty put the kung at 601.5 Hz, a D. They were
wrong.
The central role of the E flat note was pointed out by Barbara Hero in
her collection of musings and sketches on the mystical aspects of music,
Eyes + Ears = Ideas. She found that it was at the beginning, end, or
other key locations in 18 different musical, mathematical or
metaphysical patterns. She wrote, "It's the place at which we find the
beginning of life, and the fastest musical pulse of the thirty second
note... It becomes the boundary of the end of the time where space
begins... it is at the limit of the macrocosmos."(70)
To fully explain the significance of what Barbara Hero wrote above is
beyond the scope of this newsletter. If you are intrigued, I would
recommend that you get the book. It is available to read on-line at ,
but unfortunately the illustrations are illegible. Suffice it to say
that the E flat note at 639 Hz corresponds to the kung note and to
"Aum," the cosmic tone of Brahma, the creator. I am not saying that 639
Hz is the actual frequency of the tone of creation. It is the octave of
that tone which the human voice can recite comfortably.
AN EXERCISE IN SOUND ALCHEMY
When I arrived at the divine scale in Table 1, I wondered how the tones
and mantras could be played to create, destroy and heal. The great
adepts and alchemists of all ages knew the secret. One of them, in
Volume 3 of "Life and Teachings of the Masters of the Far East," was
reported to have said that, " The vast sea of God's creative, unlimited,
moving space... is completely full of vibrating, emanating energy... in
which all substance or elements are in soluble form or suspended in
harmonious relation, ready to respond to the call of the vibratory rate
that will allow them to coalesce into form. When the proper vibratory
influence is set up through the thoughts of the human unit, cooperating
with the whole, the elements, having no other course, rush in and fill
the mould set by the desire. This is absolute law and none can stay its
true course."(71)
In this teaching, we see that the law of creation has several
requirements: vibration, thought or visualization, the elements, desire,
and the call. In my research, I found other requirements. One is an
instrument that has consciousness, that is a part of God. The only
instrument that meets these qualifications is the human voice or throat
chakra. David Tame wrote that "the voice of man is particularly
intimately associated with the OM, the voice of God. Human speech is a
lesser, stepped-down aspect of the OM itself, since man is a Son of, and
a part of, God Himself. Therefore, through the use of his vocal cords in
speech or in singing, man is thought to be a co-creator with God."(72)
I have pieced together a seven-step process for creation, destruction
and healing from researching the teachings of the Ascended Masters, the
Bible, and other books. These steps, in brief, are: purification,
desire, invoking light, visualization, vocalization, repetition, and
protection. I don't claim that this process is complete or correct; it
is what I have been able to surmise from various sources at the time of
publication. I am open to constructive criticism and additional
information. For example, the frequencies of creation and destruction
may have been intended for musical instruments like the trumpet and
other wind instruments through which prana or the life force can flow,
and not the human voice. It may be that mantras can be said at any pitch
and still be effective.
I would like to propose an alchemical exercise in the use of sound,
love, and vision to create, destroy, and heal: Create or precipitate
rain for the drought-stricken Sahel region of Africa. (This region
includes Ethiopia, Sudan, Chad, Kenya, Somalia, Uganda, Niger, Nigeria,
Mali, Mauritania, Burkina Faso, and Senegal). Destroy the meteor
mentioned in the article "Three Monsters." And heal someone you love of
a minor disease. Ravi Shankar, in speaking about the great saint
musicians and singers of India, said that some could, by their singing,
heal, bring rain and melt stones.(73)
CREATION
1. Purification. The lower chakras must be purified and stilled so they
won't interfere with the proper functioning of the higher chakras. Only
a purified heart can keep a pure desire and the highest vibrating
emotion of love. The throat chakra must be pure enough to sound the
mantras at the correct pitches. A purified third eye chakra is needed to
activate the will and visualize the object. And the intellect must be
free of the illusion of duality and know that I AM God and the universe
resides within. The best way to do this is through the violet flame. In
"Intermediate Studies in Alchemy," St. Germain states that, "Our first
step, then, is to abort and transmute the negatively qualified substance
in the world of every would-be alchemist. The power of the violet
transmuting flame, as an agent of the Holy Spirit, can be called forth
from God for the purification of man's world."(74)
A simple and effective violet flame decree for purifying the chakras is
number 20 in the booklet, "Mantras of the Ascended Masters for the
Initiation of the Chakras."(75) In case you don't have this, just recite
"My ____ chakra is a chakra of violet fire, my ____ is the purity God
desires!" for each chakra, starting with the heart and going to the
throat, solar plexus, third eye, soul, crown, and base. You should also
recite the mantras associated with each chakra 108 times, beginning at
the base of the spine and working your way up, pausing for a moment
between chakras. When you reach the crown chakra, instead of using the
voice part of the throat chakra to recite a mantra, use the ears,
another part of the throat chakra, to listen to the All Sound resonating
in your nervous system. The All Sound is to the sound spectrum as white
is to the color spectrum. If you can't be in a totally quiet place, put
earplugs in your ears. This process not only purifies your chakras and
attunes them, like the frequency receivers and energy processors that
they are, with your Holy Christ Self and Mighty I AM Presence, it also
charges them with the elements necessary for the creation process. After
chanting "Ram" 108 times, you may feel the heat of the fire element in
your solar plexus.
2. Desire. The next step is to feel a strong desire to create from the
depths of your heart. But the desire must be founded on positive, not
negative, emotion. If we fear that people will continue to suffer and
die if our alchemy doesn't work, it won't. In order to create rain for
Africa, we need to feel it with the joy of a child playing in the cool
summer rain. Feel the mud oozing between your toes. Smell the cleaned,
cool air. See the lush crops growing tall and healthy as in figure 5.
The feeling you want to evoke is one of fearless compassion. Only
fearless compassion is pure enough and strong enough an emotion to work
this alchemy. Your desire has to be totally selfless or the petty human
ego will interfere with your attunement with our Holy Christ Self and
Mighty AM Presence. Jesus never performed a miracle out of a
self-centered desire, even when his life depended on it during the
temptation in the desert. Similarly, in the destruction process, any
negative emotion of hatred or fear of the object will create duality and
prevent its destruction.
Figure 5 - Sugar Cane Field in Africa
3. Invoke Light. The third step is to invoke the Light through a fiat
such as "In the name of Brahma, Vishnu and Shiva, let there be Light
where I AM THAT I AM!"
4. Visualization. The fourth step is to formulate a thought or image of
the rain or water vapor to be created, using your clarified third eye
vision. St. Germain, in "Creative Abundance: Keys to Material and
Spiritual Prosperity," states that "this should incorporate definite
size, proportion, substance, color and quality in detailed picture
form...Determine where you wish the object to manifest [the Sahel]. If
you know the material substance of which it is composed, memorize its
atomic pattern. If not, call to the Divine Intelligence within your
Higher Mind to register the pattern for you from the Universal
Intelligence and impress it upon your memory body and mind... Call for
light to take on the atomic pattern you are holding, to coalesce around
the pattern and then to 'densify' into form... Call for the
multiplication of this atomic structure.. . Ask for the full lowering of
the atomic density into three-dimensional form and substance within the
pattern established by the matrix of your mind."
Figure 6 - Water Molecule
"When the visualization of the blueprint within your mind is complete,
immediately seal it. The words 'It is finished!' are therefore the
second fiat of creation... The call of beloved Jesus at the hour of his
greatest testing, 'Nevertheless, not my will but thine, be done,' when
spoken at the moment of the sealing of the matrix, ensures that the
guiding forces of power, wisdom and love will amend the precipitated
pattern where necessary in order that the more perfect designs of the
Creator may come forth... Protect your experiment by guarded action and
guarded meditation. Your visualization of a blue light around yourself,
your matrix and its manifestation serves to focalize the desired
protection."(76)
Figure 7 - Drought-stricken Africa
In your visualization of rain, see a gentle steady rain, not a sudden
torrential downpour that would wash away seeds. The best visualization
aid for this exercise is to view a real-time satellite image of clouds
over Africa.
5. Vocalization. The fifth step is to invoke the creative power of
Brahma and Vishnu and their shaktis through your throat chakra by
intoning their mantras. Also call forth the object by sounding its
Sanskrit name. Rain is "varshham." This is done at the correct pitch for
each mantra given in Table 1. Because the pitches span the range of the
human voice from bass to soprano, this exercise requires the combined
efforts of a couple. The woman would intone the high-note mantras while
the man recites the mantras with low notes. The couple should sit back
to back with the woman facing the southwest to line up with the
nurturing creative energies of the old mother ch'i, and the man facing
northeast to align with the playful creative energies of the youngest
son ch'i. I will present a method of getting close to the correct
pitches, even for tone deaf people, later on. But, in any case, you
should also call to your Holy Christ Self to register the precise tones
in your throat chakra for you and impress it upon your memory body and
mind. Also, call for the full momentum of the mantras from thousands of
years of repetition by millions of people. Since we are going from
spirit to matter, the sequence is from highest to lowest. That is, from
"Aim" (pronounced "I'm") at 852 to "Aum" (Ah Oom) at 639. Then recite
the name "varshham" (varsh hahm) at one octave below "Aim" at 426 Hz.
This is because the fourth from 852 to 639 is the yin interval of the
mother, and the fifth from 639 and 426 is the yang interval of the
father. The union of the two gives the octave which contains all
potential objects in unmanifest creation. So any desired object would be
intoned at a pitch of 426. However, this is only half the sequence; the
rain is not physical but etheric. Vishnu and Lakshmi create and nourish
physical matter out of akasha. So, we next recite "Shrim" (Shreem) at
528, "Hrim" (Hreem) at 396, and "varshham" again at 264 Hz. to
materialize it. Each mantra should be preceded by "OM" to tune the voice
to the proper note. This also activates our will which is centered at
the Third Eye.
6. Repetition. The sixth step is to repeat step 5 six more times. On the
seventh and last repetition, however, instead of reciting the mantras
out loud, meditate on them and recite them in your mind subvocally. Thus
did God rest on the seventh day.
7. Gratitude. The seventh step is to give thanks to God for the rain,
for you have created it. You can shout the fiat, "Hear, O Universe, I AM
grateful!" over and over with joy. And remember, tell no one about this
alchemy lest their negative thoughts interfere with it.
DESTRUCTION
The process of destruction is necessarily different than that for
creation. The first two steps are the same. We still need to purify and
charge our chakras and we want to have the highest motive and emotion
for destroying, which is fearless compassion for those who would be
killed and injured by the meteor. But we don't want to invoke light; we
want to return the light that composes the atoms and molecules of the
meteor back to its source. Therefore in step 3, we say an appropriate
fiat such as "In the name of Shiva, I command the dissolution and
disintegration of this meteor into pure light that it may be returned to
God!"
Step four involves visualization of the meteor using the steps of
precipitation up to and including calling to the Divine Intelligence
within your Holy Christ Self to register the atomic pattern of the
meteor. However, instead of calling for the densification and
multiplication of this pattern and its materialization into
three-dimensional form and substance, call for its etherialization and
disintegration into formless fourth-dimensional substance. When your
visualization is complete, seal it with the same fiats.
The following image of the meteor is taken from the Millennium Group's web site.
In step five, the sequence of mantras is reversed because, instead of
going from spirit to the physical, you are going from the physical to
the unmanifest. Thus you would first intone the name of the object to be
destroyed, "giri dyau" (gear ree d'yow) at 370.5 Hz. Then intone "Haum"
(How'm) at 417, "giri dyau" again at 556, and finally Krim (Cream) at
741 Hz.
The destruction process is different also in that instead of meditating
during the seventh repetition, you would dance with joy in a circle as
Shiva dances on Mara and as the Israelites danced and clapped their
hands on the seventh circuit of Jericho.
HEALING
The healing process should be used to heal someone of a minor condition such as a rash or wart. If you have a loved one who is seriously ill, I would advise that you first seek the help of healing professionals.
For healing, use the creation process steps, only in step 5 just recite
the mantras for Vishnu and Lakshmi together with the Sanskrit name for
the organ or body part affected. Do this in an ascending sequence from
the organ to Vishnu to Lakshmi, since we want to raise the vibration to
a healthier state. Then use the destruction process with the Sanskrit
name of the disease and the mantras for Shiva and Kali to raise its
vibration out of existence. A Sanskrit-English dictionary is on-line at
Alkhemy.com
TUNING
Before we can use the instrument of our voices to sound the mantras at
the correct pitches, we need a musical instrument to tune our voices.
Those who have been trained in singing will be able to do this easily
and will likely have a piano or other musical instrument to give the
tones. However, there are some who have no training and others who are
tone deaf. For these people, I have a solution. Purchase a chromatic
tuner. Korg makes an inexpensive CA-20 model that gives a digital
readout of the note you are intoning plus a dial readout of how many
cents sharp or flat. They still would need a keyboard to sound each tone
for a second. Mark each key (e.g. mark 528, one octave above middle C,
"Shrim"). The readings you want to get on the tuner would be C +15 cents
for 264 and 528, G +17 cents for 396, G# +7 cents for 417, G# +20 cents
for 426 and 852, Db +5 cents for 556, Eb +20 cents for 639, F# +2 cents
for 370.5 and 741.
This exercise takes about one hour to do. Be sure to take the phone off
the hook and put a "do not disturb" sign on your door. Those who want to
discuss this exercise on-line before and after doing it can do so at
Mandala Forum. You'll have to register first with www.delphi.com.
A VISION OF CREATION
One night after staying up late researching for this article, I went to
bed tired but still mentally active. I could not resolve in my mind the
two seemingly contradictory ideas that Vishnu is the Word or son of
Brahma, and that Vishnu as Narayana gave rise to Brahma. As my body
drifted off to sleep, my mind kept up the debate. Finally, my body fell
asleep, but I was still awake. I was out of my body and felt light. The
vastness of space opened up before me and I felt I could fly anywhere in
the universe at the speed of thought. But instead, I chose to lie still,
and, in a sudden stroke of inspiration, recited mentally the mantra, OM
Namo Narayanaya! Instantly, light exploded before me, emanating from a
central sun of incredible size. The concentric circles of light swirled
and became a vast expanding spiral. Then eddies formed in the curves of
the spiral as it expanded outward and these became smaller spirals that
grew and sprouted even smaller spirals until the entire far-flung
universe of galaxies and star systems was created. This vision only
lasted a few seconds, but I felt that I had just witnessed billions of
years of the creation and evolution of the cosmos fast forwarded a
trillion times the speed of light.
Mark Prophet, as recorded in "The Lost Teachings of Jesus: Volume Two," said, regarding the effect of chanting this mantra, that "your eyes are
actually imbued with a spiritual vision and you begin to sense your
existence less subject to the confinements of time and space. All this
through the power of the spoken Word -- and it's the same Word flowing
through you now by which the worlds were framed..."(77)
One of the few Western mantra experts, Thomas Ashley-Farrand, wrote
in"Healing Mantras: Using Sound Affirmations for Personal Power,
Creativity and Healing," that "Om Namo Narayanaya takes the sayer to
sublime spiritual realms where spiritual questions are answered and
great truths are revealed."(78)
The great truth of creation that was shown to me is this: Narayana is
the divine spark in every part of creation. At the highest level of the
cosmos, it is the Great Central Sun. In man, it is the three-fold flame.
It is the abode of Brahma, Vishnu and Shiva. Just as it is sometimes
called the Holy Christ flame, so too is Narayana identified with Vishnu,
the second person of the Trinity, because, out of the three, he is the
closest to us. His avatars, Jesus and Krishna, are loved by millions.
The Relation of the Golden Ratio to Quantum Physics
The Golden Section Ratio is a perfect mean number having the property of rotation inherent within its perfect value. 1.618033988... inverted
becomes 0.618033988...[PhiG-1=1/PhiG]. Another number which displays
this perfection in a similar way is the number 1. Inverted it remains
one. However, when utilized as 10 one can see the deep symmetry in its
inversion to 1/10 or 0.1=1/10. The digits remain the same as in the phi
relation. A deeper relation still can be seen in the value of the root
of 10. The square root of one is one but the square root of 10 is a
number with very similar properties to PhiG. /10=3.16227766 and the
inverse of this number is the same number divided by 10. 1/3.16227766 =
.316227766, all the digits are the same but place value is displaced. It
is the property of these numbers and their inverse which gives
manifestation its opportunity to occur. This is why the base 10 system
works so well and is quite natural to use.
From this point of view where the flux of the universe becomes manifest by means of number in certain perfect, harmonious and symmetrical forms. The flux state has a direct relation to the Phi ratios created by perfect numbers. This is true mathematically and philosophically. The manifested state has a direct relation to the powers of 10 and the properties of its square root. Flux and mass [Phi and 10] are related together by motion at the velocity of light and powers of that velocity. Flux and mass in relation and manifestation through number generate lengths of flux lines.
Qabalistic text excerpts from "The Fountain of Wisdom." With commentary in parentheses showing relation to Phi Physics:
The first attribute, long and true and straight..(the first principle is
the first dimension, a representation of an infinite line) Each
attribute is called a flame..(these attributes are connected with light,
heat etc. from the electromagnetic spectrum) Each flame or line is also
an eye.. (an eye receives light) Each eye or line divides into 5
matters.. (compare with Luke 12:51,52, "Suppose ye that I am come to
give peace on earth? I tell you, Nay; but rather DIVISION: For from
henceforth there shall be FIVE in ONE house DIVIDED, THREE AGAINST TWO,
AND TWO AGAINST THREE.)
From the flame the ether springs forth, and the ether informs
STRUCTURE.. (the infinite lines of the one are the root of the ether
which is here taken as two dimensional field, and this is the foundation
for three dimensional geometric structures) also ether informs ACTION,
RESTORATION, MECHANICS, CALCULATION, UTTERANCE, AND GROUPING: These together are the essentials of all being. (Seven in number) The holy one is the first existent.. it is incumbent for us to understand and
contemplate an examination of His reality, and how He began creation..
In which path did He emanate..Did they divide into parts or extended
entities.. channels are narrow and short, paths are excessive, roads are
wide, also called children, mothers and androgenous.. From the flames
the thread extends outward.. the threads thicken, grow stronger to
become scepters.. Everything dissolves and returns to the ether which is
the essential element..
Before the celestial world, known as the 377 compartments, there was an ether, an essence from which sprang a primordial light refined from
myriads of luminaries; called the Holy Spirit.. (377 is the 13th
Fibonacci number counting from 1. The number of the Heavens 377 is the
number SHaMaYiM = 390 minus the 13 attributes, thus connecting 13 with
377. 13 is the 6th Fibonacci number itself).
After the first light, another fountain was drawn out from which flows
darkness. This darkness is a combination of three hues. 1st. green, 2nd.
green and light blue, 3rd. green, light blue and red.. And this is the
primeval darkness which springs out of the ether. (this is very clear;
first the middle of the solar spectrum is considered, then the portion
towards the Ultraviolet or the blue end of the spectrum, then the red
towards Infrared.
Green, Blue and Red are indeed the three primaries of color called
variously the additive primaries, the light primaries, the physicists
primaries and also the physiological primaries. Pigment primaries or
subtractive primaries are Red, Yellow and Blue.
The point that the text then makes is that these light hues cannot be
seen directly by any means, which is correct for only the absorbing of
light can detect it, we see reflected and partially absorbed light
producing color around us, not the etheric light itself which cannot be
known). Therefore God said, "You shall see My back, but My face shall
not be seen." "But from that point you may know all." Moses began to
observe the primeval light, the root of all.. composed of two entities,
from two sources, one light and one darkness.. This flow extends and
gushes forth by way of channels, and the flow again becomes weak like a
stream, ant the stream again becomes minute, turning into a thread..
till it becomes tiny droplets.. these join, mingle, expand and a sap p
ours forth from them.. The colors are akin to a flame extending from
ether.. The two flows are really one matter coming from the primeval
darkness, alluding form.
When this form is altered, it paints itself through its courses and the
hues of its colors into ten, and each separate hue contains the number
ten, arriving at a total of one hundred. The hundred returns again and
again into utterance, calculation, and the grouping of entities.. until
they return to the sum of one, and one is the essence.. (this is
explicit about 10 and 10 squared being significant at the level of the
first manifestations of form, i.e. at the root of dimensional geometry).
The thirteen attributes can be related to the Fibonacci series of
numbers directly, 1 2 3 5 8 13 etc. and also can be related to the
series of ratios formed from them approximating phi. 1/1, 2/1, 3/2, 5/3, 8/5 ,13/8, 21/13, 34/21, 55/34, 89/55, 144/89, 233/144, 377/233, 610/377. These numbers are encoded into all of nature. It should be no wonder that they are at the root of quantum events as well in the heart of the atom, setting up the rules that the formation of geometric entities follow, in contrast to what modern quantum physics portrays,which is non
geometrically based. Einstein always believed in the geometry of the
microworld and he was right. From the one does all of the geometry of
phi come.
THE ENDS JUSTIFY THE MEANS:
From Infinity a one dimensional line has, through the Phi relation
become a length having two extremes. Any length will have two extreme
points and therefore will also have mean points between these extremes.
Three of these mean points are significant and are naturally generated
by the flow of number. These three are the ARITHMETIC, GEOMETRIC and
HARMONIC means. From these principles a model of the ATOM can be derived which is based on the same principles of number that life forms vividly show.
NUMBERS AND MUSIC
A modern equal tempered scale for music generation is one among many
methods of doing so. Most pleasant to the ear and voice could be C=512,
the scientific scale which is numerically superior to others as C also
will equal one cycle per second. A=440 is commonly used as the American
Standard Tuning. However, we need to see music as more than notes and
scales but as chords and harmony, melody and rhythm. The 12th root of 2
is the multiplier for the equal tempered scale. This is based on
irrational numbers. It proceeds in perfect fifths. 1/1, 3/2, 9/4, 27/8,
81/16. Basic Pythagorean tuning proceeds in major thirds. This ratio is
5/4. 4/1 x 5/4 = 5/1. This = 80/16 = 5. Certainly, this is in accord
with the numeric roots of PhiG which is derived from /5.
THE COSMIC SPECTRUM
The Center of the Universe. Is there a center to the Universe? This
sounds like a simple enough question but it turns out it is much more.
Spatially, it is a question of the type of Geometry one imposes on the
universe in the first place. Negative or positive curvature geometries
and Euclidean geometries give different centerpoints. It is not yet
known if the universe is one or the other, although the empirical
evidence shows that the universe is Euclidean to a high degree. That
General Relativity is accurate within virtually all empirical methods of
research shows that positive curvature does yield the mathematics of
gravity extremely well. While negative curvature showed itself to be a
geometrical statement of magnetic forces as in the work of Gauss. This
is interesting and significant. It would appear on the face of it that
it would require ALL THREE forms of geometry to encompass the universe
and there would also be THREE CENTERPOINTS in our universe. Is there a
center? INDEED! A gravitational center, a magnetic center, and an
Electromagnetic center representing the three geometries. This can be
thought of as the FUTURE (magnetic, formative forces with NEGATIVE
curvature) arriving in the PRESENT become Euclidean to the avenues of
perception. When our attention is focussed onto the microrealms of atoms
and subatoms, things appear to be positively curved, forming ever
smaller substructures. These structures are connected with the PAST
through "world lines."
PAST POSITIVELY CURVED GEOMETRY PRESENT EUCLIDEAN GEOMETRY FUTURE NEGATIVELY CURVED GEOMETRY PAST GRAVITATING MASS PRESENT ELECTROMAGNETIC FORCES FUTURE MAGNETIC FORCES
The centerpoint of the past is the origin point in spacetime as is put
forward in the Big Bang Theory of the origin of the universe. This
theory relies too much on the past-positive geometry of gravity and
mass.
It is probably not correct for this reason.
The centerpoint of the present is nowhere to be seen because we can only look at signals sent some time in the past from stars and galaxies.
However, there is much to be said for locating the centerpoint of the
EM-present in the narrow band of visible light rather than in some
geometrical segment of space we cannot see in its true present state.
This means just this; the visible spectrum contains most of the
Electromagnetic energy in all the universe. The Infrared, Ultraviolet,
gamma ray, radio and the other bands of the Electromagnetic spectrum
contain only a tiny fraction of that contained in the visible light
spectrum. The reason is that most stars emit strongly in visible colors,
evolving from the blues through the yellows to red stars etc. In the
whole EM spectrum the visible part is a very narrow portion, similar to
a laser being a focussed narrow band of visible light. This indicates a
centerpoint which should come in handy when investigating aspects of the
universe.
The centerpoint of the future is something else.
If you want to go to the main index page "Let There Be Light", just
click on the icon to the left and you will be taken there.
Or if you want to go to a fantastic "Page of Links" click on the icon to
the left and you will be taken there.
And if you want to register this page with an "E-Mail Reminder" service
so that you will know when this page changes (because it will every
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PLEASE TELL THEM ejlight REFERRED YOU
If you have comments or suggestions or just want to talk, email me at
Teacher-ji@Light-Mission.org
Sympathetic Vibratory Physics - It's a Musical Universe!
NEW SCIENCE-NEW ENERGY by Ben Iverson
Studying the advanced paradigm of Science is not so difficult as it at
first seems. First we consider the air around us and how it transmits
sound, by vibrations. Energy is really only vibration. The colder the
air the more dense it becomes. Cold air being more dense will transmit
the vibrations of music faster and further than will warm air. Again,
air at lower altitudes is much denser from the pressures above. It too,
will transmit musical energy faster and further.
The air is an analog to the aether. The air is matter but the aether is
metaphysical. Aether is a progenitor of matter and of course, air.
Aether condenses creates all matter, which takes organized form as we
know it, including the individual molecules of air. The molecules
vibrate internally and are essentially condensed aether. We need
consider only the vibration, and not the medium, whether it is air or
aether.
Aether continuously vibrates. When aether transmits energy it transmits it in three phases for each wave. Each wave has its positive high phase, its negative low phase and its neutral phase. In the neutral force, the positive force and negative force are both present but will balance each other. These are generally shown as sine waves. They transmit gravitational force, magnetic force, and electrical force.
We generally picture a wave beginning at its place of propagation as a
positive wave which is always at the NEUTRAL CENTER. Every aggregated
body has one neutral center for itself, and lesser neutral centers for
each of its parts down to atoms and smaller. We will deal only with the
main NEUTRAL CENTER at this time. It appears that the NEUTRAL CENTER
receives its aboriginal energy from the surrounding aether and transmits
the vibrations outward to the skin surrounding the object in question.
Let us say the transmission is along a musical string. The wave starts
out in its positive phase. On reaching the other end of the string the
wave is reflected back in its exact opposite phase, from the far end of
the string. This far end can be considered a second NEUTRAL CENTER, but
an exact opposite from the original NEUTRAL CENTER..
The other extreme of shape is that of a sphere. Here the original
NEUTRAL CENTER is at the center of the sphere. The other (reflecting)
NEUTRAL Center is at the shell or the skin of the sphere or body, where
the wave becomes a reflection of itself. It always appears in the
opposite phase from that from the origin, which is being reflected.
In either case, the energy of the wave is not depleted in a single
traverse of the string or body. If additional energy is continually
added to the wave in the proper timing, according to its phase, this
energy becomes great enough that it will strain and eventually explode
the sphere. The timing is important. This is where the mechanics of
harmonics comes into play. One can easily see how this can happen in a
sphere. Harmonics will be described below.
But let us take a second look at the string. Instead of a string, let us
look at it as a bridge over a stream. A marching army over the bridge
will find the step will coincide with some harmonic of the bridge. The
bridge will gain so much energy that it will eventually fail. That is
why an army should break step, when marching over a bridge.
One end of the bridge becomes the positive Neutral Center and the other
end becomes the negative (reflecting) neutral center. Long slender
objects will have its neutral centers at the ends. A wider and more
compact body will develop a neutral center at its center and another at
its outer skin.
We can see and visualize this structure of energy in matter. Let us
transfer it in analog form to the aether. The aether fills all space.
This energy is considered to be generated at the NEUTRAL CENTER of the
universe (THE CREATIVE ONE), and radiated outward to the edge of the
Universe. At this outer skin of the opposite NEUTRAL CENTER, it is
reflected back in its opposite phase. Both phases will essentially
cancel each other and be undetectable by present means. There will be
points at which one of the phases is more dense than the other. The
aether will densify at these points and ultimately collect as matter,
which is around us and of which we are made.
The original generation of energy at the center will be in the range of
quadrillions of vibrations per second. Through the mathematics of
harmonics certain waves will coalesce in groups of longer and slower
vibrations to create the longer waves of energy, in what is called a
waterfall. There are six waterfalls with which we must deal. Between
waterfalls all is quantum and are called Myriads. Matter is created
after the third waterfall, in the Myriad of Chemism. The myriad above is
the Myriad of Light energy. Above that is the metaphysical Myriad of
Thought energy, and then the Creative Myriad. There are "waterfalls"
between the Myriads, as the energy flows downward into longer and lower
frequency waves. Science calls it "entropy."
These Myriads go over a waterfall through harmonics, in which several
different waves are attracted together. They coalesce and congeal to
make a longer wavelength. It is very much like a chord as played in
music. In order to better understand the mechanics one must go to the
mathematical formation of music and musical chords.
HARMONICS: --
As is well known, music is divided into octaves Seven octaves of
frequencies make up one Myriad. Before it goes over the next waterfall.
The octaves in turn, are divided into single notes. These notes are all
based on one keynote value. They are composed of the keynote and
"partials" of the keynote. These partials can be added to the Keynote or
subtracted from it. The following table shows the formation of six of
the seven octaves.
These are six of the seven octaves which make up the Myriad of Music.
The octaves are taken as the first four octaves in one group. The first
two octaves are presumed to start in positive phase. The Third and Fouth
octaves are presumed to start in negative phase, because they are
doubles of the first two octaves. The Fifth and Sixth octaves are again
in the positive phase and the Seventh octave (not shown), would be in
the negative phase.
Notice that all notes are whole numbers except for the Fifth partial. In
spite of the decimal values, the values are exact without any more
residual decimals. The frequencies are all relative to each other. These
note values can be multiplied by any number and still maintain this same
relativity. For instance if they were multiplied by "5" they will all be
whole numbers.
Research has indicated that if multiplied by 11, for one scale of
octaves. The other scale of octaves can derived by multipling all notes
by 19. Then if these two scales are intermixed and played together, they
will produce an emotional reaction of Unconditional Love.
This is what harmonics is all about. In the first place, the original
energy is multiplied and increased by the harmonics. In the second
place, each harmonic chord has a certain message or information to
transmit. This is broken down to a mathematical formula. We hear only
the portion which is in the audible range but there are other aetheric
ranges up through the higher Myriads, in the aether, which are also
brought into play. One of these ranges will be in or near the value of
the spectroscopic values which have been assigned to the various
elements of the Periodic Table of Elements.
It produces its "Love" effect, in the case given, through the Myriad of
Thought Energy. The main point to be put across is that the aether plays
a part in all of Nature. The analog of the aether to the air of our
atmosphere, is the only manifestation which is detectable by our human
senses. As Edgar Cayce said, "We will find all higher science can be
reached through Music."
For anyone who is capable of composing music using MIDI, it is
recommended they utilize two of these scales, (one multiplied by 11 and
the other multiplied by 19), to see what conditions are needed to
produce the projected message and feeling of "Peace" and "Love". How are
the two scales to be blended?
There will be other effects as well but one can only guess how drastic
they may be. In my case, it burned out 5 different computer monitors
before I identified the source. Even at low volume the music does carry
messages which can cause of nerve interaction, and may interfer with
sleep patterns.
Aether continuously vibrates. When aether transmits energy it transmits it in three phases for each wave. Each wave has its positive high phase, its negative low phase and its neutral phase. In the neutral force, the positive force and negative force are both present but will balance each other with one travelling outward and the other travelling inward. These are generally shown as sine waves. They transmit gravitational force, magnetic force, and electical force.
Our problem in harvesting this energy is to unbalance it. If we are
working with the energy of music, we can unbalance between the positive
and negative phase of the energy waves. This is done through introducing
a dischord in the music, by using an "enharmonic" chord.
If we create this dissonance as the energy goes over a waterfall we can neutralize the positive phase within this lower Myriad. In doing so we
must be able to retain its higher harmonics in the higher Myriads of
Ultra Sound and Chemism. Within a sphere we will have more of the
negative phase remaining. It is made negative by going through a
reflection which changes the phase of each wave from positive phase to
negative phase. This is done by introducing chemical subliming
substances which will have pushed these higher harmonics into an even
higher Myriad. This substance must be able to resonate with the higher
harmonics and thus, let the higher harmonics sublimate and carry the
energy to the higher scale. In this specific case the sublimates will be
powdered carbon on one half of the interior of the sphere, and powdered
quartz on the other half. These will sublimate the carbon and the
quartz, which sublimate passes as ultra-fine matter, (a plasma) through
the metal of the sphere. It enters an outer ring which will resonate
with the specific frequencies, which are upper harmonics of the original
entered frequencies. The lower frequencies are minimized by another
sublimate of tin, tungsten, tin, and / or zinc.
Attached to the outer ring are resonators which cause the selection of
the proper frequencies. These frequencies are again elevated by a
sublimate, (powdered copper and zinc / or tin) within the outer ring which
carries them further outward into the Myriad of Light.
The light frequencies radiate outward still further until they encounter
the outer (invisible) skin of the device. This outer skin is an analog
of the human aura on its outer surface. In both cases the outer skin is
a thin slice of condensed aether like the film around a soap bubble.
Here the energy wave is reflected back, ion opposite phase, along the
route to which it followed on its outward journey. It will retrace its
steps. It travels inward again until it reaches the original Neutral
Center in a reverse process at each step. The sublimates will return to
their original state at each step of the journey.
Energy is available at each step, but is collectable only within the
sphere, On the outward journey the waves press against the inner half
surface of the sphere which is covered with powdered quartz. On the
inward journey the wave is attracted to and pulls against the inner half
of the sphere which is coated with powdered carbon. This is supposed to
cause the sphere to rotate.
Let us review parts of this scheme. Going inward, between Myriads the
energy waves go over a waterfall through harmonics. As they do so,
several different waves from the upper Myriad are attracted together,
forming a single modulated wave in the lower Myriad. As they travel
inward they coalesce and congeal to make a longer wavelength. It is very
much like a chord as played in music, and listening to the undertones.
On their outward journey the waves are broken down into their original
higher harmonics from the waveform creating waveletes. In order to
better understand the mechanics one must go to the mathematical
formation of music and musical chords. As they travel outward this
process is reversed and each wave is broken down into its higher
harmonics.
Notice that all notes are whole numbers except for the Fifth partial. In
spite of the decimal values, the values are exact without any further
decimals. As the waves go outward or inward they take up a different
unit-of measure, but the numbers do not change. They adopt the new
unit-of-measure of their new Myriad and remain as integers in that new
unit-of-measure. If one has multiplied the original notes by 11 or 19,
that same mutiplication applies to the higher or lower Myriads alike.
The analog of the aether to the air of our atmosphere, is the only
manifestation which is detectable by our human senses. The one place it
is manifested is in the outer skin before the return journey. Here will
be seen a scintillation of light and colors. As Edgar Cayce said, "We
will find all higher science can be reached through Music." How is that
done? It will be explained again in a different format.
Consider this device as some sort of analog to the atom but on a much
larger scale. It is also an analog to our Solar Planetary System but on
a much smaller scale. In the first case the Neutral Center is an analog
to the protons at the center of the atom. In the second case the Neutral
Center lies at the center of our Sun. In this last case, the planets are
all kept in perfect alignment by harmonics between themselves. Note that
the Earth also has its own Neutral Center in the 500 mile core at its
center.
In the case of the atom, the electrons maintain a balance, but this
balance can be changed by input or output of energy from the electrons.
That is therefore a closer analog between the Musical Sphere and the
atom. So far, in our current science, the energy of the atom has been
released only by the destructive sledge-hammer method. The Musical
sphere releases this energy photon by photon by a much gentler method.
As the energy is input from existing earth energies, it is augmented by
continuous input of Earth energies, and by the two different sets of
resonators, through sympathetic vibration. These begin with the Myriad
of Music; Go through the Myriad of Chemism; Then arrive at the Myriad of
Light. So this Musical Sphere is a two-stage device whereas the atom is
only a single stage natual conservitor of energy. So also is the
planetary arrangement of our Solar System. We are able to draw energy
from the intermediate between these two stages. This intermediate is the
globe itself, which appears to rotate without input of visible outside
energy. There is continuous input of natural earth energies.
Will it work? Yes! One need only consider what it was, that caused the
collapse of the suspension bridge over Puget Sound at Tacoma Narrows,
nearly 60 years ago. It was only a continuous input of discrete
ultrasonic energy. This input happened to be a higher harmonic of this
particular bridge. There are other cases other cases where harmonics
were encountered to cause severe destruction.
On the other hand, in 1948 in England, Professor Searl built his first
disc which was self powered. He went from here to make a two stage (SEG) device after that. This powered an electric generator for his home. From this later device he learned that it put out extraordinary healing
powers and actually cleaned the environment of his home. This device,
with its two stages, was more like the Musical sphere.
Then 30 years ago, Profesor Searl constructed a 3-stage device which he named "Disc I". It had levitaton powers and propulsion, such that it
could fly through the air faster than any of today's airplanes.
Professor Searl then constructed a second disc which was 20 feet in
diameter. He called it "Disc II". He learned that it had its own
built-in gravity which was repelled by Earth gravity, hence its
levitation powers. These were all unmanned remote controlled devices.
Professor Searl drew up plans to construct a still larger device about
150 feet in diameter with facilities to allow a person to operate it
from within. It all came to an end when Professor Searl was charged with
the crime of producing his own electrical power. He went to prison for
more than a year. When he was finally released he discovered that all of
his previous work and most of his data had been destroyed.
The musical sphere was first constructed by John W. Keely more than 100 years ago. There are numerous accounts of his work as written in the
newspapers, journals, and by individual authors, scientists and
engineers. He was successful, but there were many detractors who had no
knowledge of the inner workings as simply as described above. These
detractors looked for weak points at which additional energy was being
fed into Keely's Sphere. They were successful in destroying him by their
know-nothing airs, and impossible conditions which they dallied in their
own vacuous minds. It was enough that support funds were withdrawn
unnecessarily. Most of his work was destroyed but some of it did
survive, these 100 years.
Not enough information is given above to enable one to construct their
own Musical Sphere.
1.) The harmonic musical scale,
2.) the description of a Myriad,
3.) the waterfalls between Myriads, and more-so,
4.) the exacting rules which are imposed from harmonics,
5.) sympathetic vibration, and
6.) chemical applications, are over simplified.
These subjects are further described in two books entitled, "Pythagoras And The Quantum World" Volume 1 (1993), and Volume 2 (1995). These five subject areas are described in these volumes and all have been used in the present Musical Sphere. A third, Volume 3 (1997), will be ready to be offered for sale very soon. It will take time just to learn the new paradigm which concerns all energy.
For now, the reader must content himself / herself with playing around
with the musical scale as given. It must be done with a digital musical
synthesizer.
It will give many surprises when slight modifications are made, and the changes in the amount of energy produced. One must proceed with caution because there are things which can cause the release of harmonic energy
beyond anything dreamed. One must become aware of these dangers through a slow and cautious learning of the process.
SUBLIMATES:
The two preceding articles set up the basis of New Science in
preparation for this portion of the application of advanced current
science for practical use. We prepare the Earth energies, much as
gasoline is prepared from crude oil to extract its most useable
energies.
In this case, we use chemical formulation of the elements, mainly
metals, to select from the input Earth energies and modify them to our
needs. In outline, these energies are amplified and refined for their
greatest energy content.
The Earth energies which are selected are in the subsound range. Each
wave has a waveform which is shaped by five to seven wavelets of which
it is composed. They are sent through a series of tuned musical strings.
These strings will activate in the audible musical range through
sympathy with the wavelets which form the earth energy frequency. This
is a strictly mechanical conversion. The wavelets, of course, are what
create the waveforms of the original Earth energies.
Having done that these refined energies in musical tones are fed into
the device as wavelets. They become a series of waves in their own right
as musical tones. They also each have their own waveforms which are
composed of wavelets from a higher Myriad of energies. These waveforms
are separated out as wavelets through a chemical process. This gets us
into the nitty gritty of the energy conversion and refinement. Each type
of elemental metal has its own resonance frequency. We must pick out the
metals which have the frequencies which are needed to match the present
wavelets in the selected train of energies.
When the low frequency Earth energies are fed into the device, each
single frequency will have up to seven different wavelets. (If three
different frequencies are utilized there will be up to twenty-one
wavelets).
It does not require twenty-one strings to handle and convert this
because every string is capable of vibrating in several different modes,
or frequencies. This is the first phase, and/or, mode change as the
energy is transferred through the structure of the instrument to the
center of the globe above.
The second phase change occurs within the device. It is a reflection of
the sounds within the resonant sphere. The reflection is a change of
phase of the wave from positive to negative. The phase change occurs as
it is reflected from the inner surface of the globe.
It is then passed on through the metal of the brass sphere, by the
attached tuned resonator tubes to form a plasmic aetheric field around
the outside for those frequencies which are more precise. The resonators
must be tuned to an accuracy of one or two hertz, but we are now in the
low megahertz range of frequencies. But some of the longer frequencies
remain in the globe.
Those frequencies which are imprecise are retained within the sphere and are picked up by the three actuators, for further refining. Specific
metals must be selected which are capable, through their electron energy
states, to handle the corresponding aetheric frequencies. These metals
will be those which will conform in energy states which are most closely
to the precise aetheric energies of the now refined and elevated Earth
Energies. One of these metals must be nickel. It is combined with at
least two other metals to pick up the much wider range. There now
somewhere between 49 and 343 different specific wavelet frequencies from the several waterfalls over which we have established a reverse flow of energy upward on the waterfalls.
Passing the energy on, within the sphere, to the actuators to convert
the wavelets to be processed. The frequencies are now in the Myriad of
Chemism. The vibratory range of the metals of the Periodic Table of
Elements will bring the remaining energies back into the precise
vibratory rate which it should be, and in accordance with the energy
states of which these metals have energy states to do so. Other elements
composing the actuators may be zinc, tin, silver, tungsten, and such
others, including the rare earths. When the right proportions of the
other metals are used and applied, their "Lawful" energy states will
refine the residue energies within the sphere, correcting each one to
their proper and precise frequencies before passing them through the
walls of the sphere. These corrected frequencies are picked up by the
resonators and passed through the sphere in a plasmic state to the outer
surface.
These frequencies on the outside of the sphere are all well beyond our
sensory range but probably can be picked up through Kirlian photography,
as used to detect the aura surrounding the human body.
The plasmic frequencies are then picked up by an outer stationery ring
ready to be advanced to the higher Myriad which is in the seven octaves
of the nano-hertz range. The outer ring is filled with materials which
will sublimate to elevate these frequencies into a more true refined
aetheric form. The sublimate here is powdered copper in a matrix of
shellac.
This outer ring may be considered as an analog to the current atom
smashers of conventional science. It can also be likened to the Van
Allen Belt which surrounds Earth It breaks the frequencies down to true
aetheric form beyond its plasmic state.
The energy has now been taken from its first "earth" state through its
"liquid", its "air", and its "fire" states through different forms of
phase / mode changes. It is put into its photonic state within the seven
octaves of the Light Myriad.
The energy, as a sublimate of copper and organic shellac, then migrates
to the "outer skin" of the device which is the outer bounds of the aura.
These many waves, being wavelets above light will then drop back over
its last waterfall and create a scintillation ring around the device.
Here it again comes within our sensory capability and the rainbow of
scintillating light can be seen. At this point the energy in a different
form from that of its outward journey now begins to traverse its journey
in reverse direction and in reverse phase. It goes, step by step,
returning the sublimates to their original origins, and back to the
Neutral center of the sphere.
The Neutral center is densified aether being more dense than anything we
have ever created. It can be the analog of the theorized "black hole" of
space. Some of the energy will go back through the structure of the
instrument to the base plate and cause the musical strings to hum
audibly.
Its round trip completed, the energy is now ready to begin another
traverse of its outward journey followed by its inward journey to repeat
the process through the established Laws of Harmonics. The whole process
is nearly instantaneous, and the obtainable energy is nearly infinite in
quantity. One can readily observe that extreme caution must be used in
operating the device. A sudden release of this energy can be disastrous.
The globe is designed to rotate from the attraction / repulsion of the
sound when reflected on its inner face. Estimated rotation speed is
about 1,500 r.p.m. and estimated horsepower output is approximately 30
HP. But this is not designed to be a motor or an engine. It is designed
to be a device through which to learn the parameters of its operation.
There is so much new material, beyond what is given here. The device is
to be used only for the considerable education which we presently lack.
Several of the pieces of education which contribute to the building this
device, (The Musical Globe), comes from the newly developed mathematics
of Quantum Arithmetic. It comes mostly under the proven Laws of
Harmonics as developed in Quantum Arithmetic. The design of the strings
applies the Music of the Spheres which is found as a part of Quantum
Arithmetic. The metals used as sublimates are determined and
mathematically checked by quantizing the energy states of all metals.
The progress of the energy over the various waterfalls, and the seven
Myriads, is found only in Quantum Arithmetic. The balancing of the
inward and outward flows of energy are explained in the part of Quantum
Arithmetic described under "Synchronous Harmonics".
The two flows of energy, balance each other throughout their journey to the extent that they are undetectable by ordinary means or
instrumentation. We have gained information and learned how the energy
travels in vortexes and tori. Some of the details are described by
Quantum Arithmetic, in Volume 3 of "Pythagoras And The Quantum World."
But all of this, (even Quantum Arithmetic), is not of human origin. It
occurs through spiritual input from the metaphysical Myriad. A
consortium of the spirits of long dead scientists is actively guiding
today's scientists through the routine which must be utilized. This
consortium has also guided others in the past in conventional science,
but conventional science has chosen to ignore them, not realizing that
their "creativity and enlightenment" by spiritual participation, is what
lead them to their conclusions. There is still so very much more to
learn, and it will take time to do so.
Some of the parts of the information come from current science.
Especially so, is the adaptation of a very rudimentary "Wavelet Theory"
and finalizing that theory into Law. This was done, again, through
Quantum Arithmetic, and the actual synthesizing of single tones by using
wavelets. This is the only current theory which is applied to any
degree. The dogma of current science is dismissed entirely in the
Musical Sphere. The truth emerges and the validity of the aether is fi
nally justified.
B.I. 2/27/97
Sympathetic Vibratory Physics - It's a Musical Universe!
QUATERNIONS by Ben Iverson & Dale Pond, 1993
Quaternions were developed in 1843 by W. R. Hamilton (1805-1865) in
Dublin, Ireland. The background of these is rather hazy and is rather
moot.
These were a precursor of the actuality and reality of force and energy
as manifested in the form of longitudinal, transverse and vortexian
vibrations (Energy manifests rhythmically over Time). Quaternions are
not an invention but parts of the procedure had to be invented in order
for quaternions to be applied. The use of quaternions has a profound
effect on science because: 1) It recognizes a greater scope in physics;
and 2) they add recognition of the dynamism inherent in energy. However,
in nature there is not the necessity of having imaginary numbers. Nature
is real and hence it uses real quantities. Quaternions resemble to no
small degree the well established arthimetical methods used in the study
and development of music and sound (as music) dynamics. In fact one
could almost say that what are known as computations of "summation and
difference" tones between musical notes from the primary through the
higher chords are in fact a subset of a quaternion-like math. These
computations are a form of arithmetic / algebra hybridization.
Putting together this 'virtual' relationship and making it workable is
quite another thing. That is for putting it together into a workable
mathematics for use in science. A large part of the backdrop is now in
place but more explanation is needed.
Everything in science and nature is quantum; i. e., discrete quantities.
That is everything works and is numbered by whole numbers except (on
first view) quaternions in themselves. They are the undercurrent of the
whole number system or a "look behind the scenes".
In addition to being in whole numbers, all values must be based within
the system of prime numbers. This much is covered by the books on
Quantum Arithmetic1. Essentially, a Quaternion is the square root of a
prime number. They never actually appear individually in any
calculation, but are combined in various combinations of them. The
quaternion represents the latent unmanifested, unseen potential locked
into a given triple vector system. Because this triple vector system is
in reality a compound vibration this latent energy or potential may be
quantified in terms of "polarization" or "degrees of polarization" (when
dynamic) as opposed to a raw individualized and independent quantum of
energy (when latent). John Keely referred to this undifferentiated
triple vector complex as the Full Harmonic Chord of vibratory energies.
The ancients referred to this as "white light" or undifferentiated
(unrefracted) light from whence comes all degrees and forms of
manifestation; i.e., Ra, Om, Yod, etc.
The involved prime numbers are combined in various and sundry groups
such that their products in each group is a whole number. This
combination takes place until there are only four whole numbers, some of
which are prime and some of them are composite numbers. These four whole numbers must then be able to fit into a Fibonacci type arrangement. They must also be formed of not more than seven prime numbers for all four whole numbers. This is easier to demonstrate than to describe.
If one takes the combination of whole numbers to be 15, 7, 22, 29 for
the Fibonacci configuration. The prime numbers here are 5, 7, 11, and
29, and the quaternions are the square roots of these six integers, plus
the square root of 1, 2, and 3, to make the set of seven quaternions:
Integers Square Roots Square Roots
(Common) (Quantized)
1 1 1/1
2 1.41421... 14169/10019
3 1.73205... 18985/10961
5 2.23606... 7900/3533
7 2.64575... 10583/4000
11 3.31662... 13031/3929
29 5.38516... 19015/3531
Now, these seven prime numbers will always contain the square roots of 1, 2, and 3 in every set of quaternions. These are what sets the
unit-of-measure for nature and creation. The other prime numbers, in
their relationships of one to another, are what sets the parameters of
the energy of the thing created.
Going back to the set of four integers, they show the deficiency in
Fibonacci's postulate. The configuration is that given by Euclid in Book
VII Proposition 28, in what we call "sum and difference numbers". (But
these were in use more than 2000 years before Euclid). Of the four
numbers, in that configuration, the first and last are the extremes. The
third is the mean, and the second is the variation as they apply to
quaternions. Here we can begin to see the computation of "beat
frequencies" or chords (whichever the case may be) as is done in musical
tones. Having seen this musical connection to higher physics we can
begin to understand the heavy emphasis John Keely placed on music
throughout his entire system of vibratory physics.
In the example, 15, 7, 22, 29 the 15 and 29 are the extremes. 22 is the
mean, i. e.,
(15 + 29)/2 = 22.
7 is the variation from the mean, i. e.,
29-22 = 7 and 22-15 = 7.
The quaternions will be expressed as:
The square root of (15 x 29) x the square root of ((22^2 x 7^2)-n^2).
This product of square roots is then divided by 7.
The value of n is the actual distance from the mean in integral units,
from either extreme for the point being measured. If the extremes are
designated b = 15 and a = 29; the mean is designated as d = 22, and the
variation is designated e = 7, then the formula will be for y and x:
y = ((ab)^1/2 (d^2e^2-n^2)^1/2)/e
x = nd/e
In this formula, the quaternions are used but the exact value of them
will be the square roots of a, b, d, & e. In essence, the formula will
be the product of three square roots: a, b, & (d^2^e2-n^2), or the
product of three quaternions divided by the variation. Broken down
further, each quaternion will be the square root of a prime number which
is a factor of the product of the four larger numbers, a, b, d, & e.
The only variable in the equation is the value of n. When n is equal to
de, the result will be zero. Then n = 0, then the value of the equation
will be at its maximum. The value of n will progress from de to zero and
then back to de. In this case, de = 7 x 22 = 154 and d2e2 = 1542.
The term d^2e^2-n^2 as part of the value for "y" in the above formula
for the scalar is what Hamilton called the quaternion. In his formula,
"the square root of -1" is a reintroduction of our number which we call
The One, or The Ubiquitous One. 4 That "One" originates in the higher
dimensions and it is suspected that Hamilton put his math in and through
the higher realms and had to then call these numbers imaginary numbers.
Another major difference is Quantum Arithmetic and Sympathetic Vibratory Physics does not recognize nor use negative numbers in the conventional sense. All quantities are real amounts. In lieu of this we use a natural phenomena called polarity. In many instances this polarity may be referred to as male or female numbers or even as positive or negative.
All discrete quantities have real significance but are generally phase
matched or opposite in polarity just as no coin can exist without two
opposite sides. Conventionally a negative number is generally considered
as "less than 0" which is a misconception and can lead to tremendous
illusions. In Hamilton's system the -1 caused the equation to rotate 90¡
and further discussions of "negative Time". In QA and SVP a change in
polarity also causes a 90¡ rotation. So to derive a full 360¡ sweep
there must be a positive / positive, a positive / negative, a
negative / negative, and a negative / positive orientations which of course
there are.
This will make little sense to the experienced mathematician or
scientist in contemporary terms. The background behind it will be
explained in Quantum Arithmetic, Vol. 2, Bk. 2, pages 25-32(2) and also
in Pythagoras and the Quantum World, Vol. 1-4.
The numbers derived from the working of this equation will give, not a
sine curve, but an equivalent elliptical curvature, which is the true
formation created by the quaternions. This might be considered as the
epi-sinusoidal curve. Each point so calculated will be a quantum point
in which most of the values will be irrational numbers when reduced. (3)
However, at least two of the derived answers will be whole integers and
they will be located at the latus rectum of the elliptical curve.
These two equations are for calculating the coordinates of a point on a
2-dimensional surface. Three or more dimensions may be calculated
individually and then superimposed on this 2-dimensional surface.
What makes this new type of calculation possible is the replacement of
invented calculus, along with other extraneous invented features of
conventional mathematics, with an entirely natural Quantum Arithmetic.
In this new method, there are no "imaginary" or "irrational" numbers and
all answers are simple and entirely logical. Children love the
simplicity of this arithmetic and geometry when exposed to it.
The above description is a bit difficult to follow. A repeat of the
description in a different frame will help to understand quaternions
from a different angle. There is a change from the accepted information
on quaternions, much like there is a definite change from the way
Fibonacci described the parameters of Fibonacci numbers and the way in
which Euclid described them. By these changes we can see that Quantum
Arithmetic is evolving from older ideas into a more complete whole.
In the case of quaternions, there are major changes. Quantum Arithmetic takes quaternions out of the field of the virtual and imaginary numbers, and their background is changed in many ways. But this new method is only the basic beginning of Quaternions.
The approach to quaternions is through Quantum Arithmetic. The current contemporary texts describes quaternions as: "An invented associative, noncommutative algebra based on four independent units or basal elements." This is changed in two ways. In the first place it is now
based on four interdependent variables. In the second place, it is
changed from a noncommutative algebra to simple arithmetic.
The four interdependent variables are dependent on any two of
themselves, and the four will be in Fibonacci configuration.
The adaptation of simple arithmetic precludes the necessity for use of
calculus to obtain a resolution. This is the simple arithmetic of
Quantum Arithmetic.
The origin of quaternions lies in the series of elliptical equations for
the Quantum Ellipse. So it is first necessary to define the difference
between an empirical ellipse and a quantum ellipse.
Only the quantum ellipse need be described. A Quantum Ellipse: In the
Quantum Ellipse, the measurements are exact, and most of the
measurements, 16 of them, are always measured in exact whole integers. For every empirical ellipse there is a comparative Quantum Ellipse. The
measurements of this Quantum Ellipse will be within 0.001%(5) of the
comparative values for the empirical ellipse, but the unit-of-measure
will be changed to quantum units in every case.
Once the Quantum Ellipse is derived, the four numbers will be obtained.
These four interdependent integers will be coprime 6 between themselves,
just as Euclid stated in Book VII, Proposition 28. These four integers
will involve from five to seven prime numbers as their factors. The
quaternions, per se, will be the square roots of these prime numbers.
In the Quantum Ellipse there are four measurements along the major
diameter. These four measurements are in the proportion of the four
integers which generate their factors, and the square roots of the
factors are the quaternions.
The four measurements are the perigee and the apogee of the ellipse.
These are the "extremes" or 15 as "b" and 29 as "a" in our example. The mean of these extremes is the semi-major diameter of the ellipse. It will be designated as d or 22. The mean will vary from the extremes by the
fourth integer which is the "variation" and will be designated as e or 7
as given. These are the "basal measurements" which are termed above. To
obtain the actual quantum measurements of these four segments, they will again be multiplied by d, to give the linear measurements along the
major diameter.
The actual measurement of the major diameter will then be:
db = J = the perigee;
da = K = the apogee;
de = the variation; and
d^2 = D = the mean, or the semi-major diameter.
ab = the semi-latus rectum;
d^2 + e^2 = the vector from either focus to the opposite semi-latus
rectum;
d^2 = D = the vector from either focus to the midpoint n of the ellipse
There are two of each of these above the major diameter and two more of each below the major diameter. All sixteen measurements will be
integers.
This provides the backdrop of the quantum ellipse behind the
quaternions.
For the purpose of constructing a Quantum Ellipse on the basis of
rectangular coordinates, the major diameter is divided into quantum
segments. Each segment will be d/e in length, along the major diameter.
There will be 2de such segments along the major diameter which is 2d^2
in length, (d/e)(2de) = 2d^2. These segments provide the x-axis for
plotting by rectangular coordinates.
This brings us to the origin of the quaternions which relate to the
y-axis coordinates at each of the x-axis segments.
The segments of the major diameter are numbered, (n), beginning with
zero at the center and increase incrementally to the value of de at
either end of the ellipse. This value is the amplitude of the wave.
Then:
y = ((ab)^1/2(d^2e^2-n^2)^1/2)/e
This gives the scalar value for the vertical vector (n). This combines
the quaternions into a list which is handled like an ordinary grocery
list.
At the midpoint of the ellipse the equation for y is simplified when n=0
and the scalar is also simplified to:
y = d(ab)^1/2 for the minor semi-diameter
(ab)^1/2 is the square root of (F), the height of the triangle which
generates the given ellipse. The calculation gives the scalar which is
always an integer when converted from conventional irrational terms back
to quantum whole number ratios.
so
y = 22((15 x 29)^1/2 = 458.84638 = 310639/677.
This is the origin of quaternions in 2-dimensional calculations. To
simplify the definition of quaternions it can be said that quaternions
are not a thing but are mathematical terms. Quaternions then are the
number roots of the numerical terms used to define an ellipse. In the
case of other calculations which get into higher dimensions another
calculation, using the quantum number of the next dimension is
superimposed on this calculation giving the j, k values of the
quaternions superimposed on the above calculations. These calculations
will apply in cases of the lissajou, ellipsoid solids, etc.
This is not the full story of the origin and usage of quaternions but it
is a significant beginning. As stated earlier, Hamilton's quaternions
are an invented system. The system which is given here is not invented.
It is derived from natural mathematical functions and relationships,
which the remainder of Quantum Arithmetic consists. In due time, others
will derive the extension of this usage of quaternions, in which the
whole idea of quaternions disappears, along with invented imaginary
numbers and Gaussian integers. These were a step in the right direction
and served their purpose for a time.
One feature which may confuse many mathematicians is the necessity for
conversion to quantum units-of-measure in order to utilize the
equations. That conversion is a linear coefficient which is obtained in
the conversion process. After quantum answers are derived this same
conversion coefficient is again applied to return to conventional
units-of-measure. In following Quantum Arithmetic procedures absolute
answers are obtainable.
1 An interactive computer program exists for QA/SVP. For a complete list of the QA and other books request free catalog from Delta Spectrum
Research.
2 At the time of this writing this book is in the hands of the printer
and will be available 12/93.
3 For purposes of creating harmonically balanced systems the answers of such equations must never be reduced or allowed to become irrational.
All irrational numbers can be 'quantized' to a ratio of whole numbers to
any desired degree of accuracy.
4 In Quantum Arithmetic and Sympathetic Vibration Physics there is a
process we call "Quantizing to One". Basically this is a method whereby
all values are set as relative to each other in a parametric fashion.
This can be easily done by using algebraic type notation or even music
symbols as representations of values and their relationships to each
other.
5 Any degree of accuracy can be easily established in these equations.
One of the seeming discrepancies shows up when a simple ratio is reduced
to an irrational number. In these equations and most others developed
within QA and SVP all ratios are used as they occur. These ratios are
never reduced because a whole number is a discrete quantity (quantum)
and an irrational number is not.
6 COPRIME: Coprime Numbers are two different numbers which are not
divisible by any single prime number. Examples: 8 & 9; 15 & 22; 6 & 35.
7 Ben Iverson; Institute for Technically Applied Music; 11466 SW Royal
Villa Dr., Tigard, OR 97224.
For greater details see Quadrature of the Circle and Iverson's Quantum
Arithmetic.
Spirit, Inc.
Ben Iverson 5/26/98
I want to recap some of the uploads and pull them together. Some of
these may appear to be religion based, but they are not. Under New
Science, religion and spirituality are two separate subjects. For each
person, religion is an entrained part of the Belief System, and is
different and unique for each person.
Spiritual and spirituality is based upon scientific methods as they have
been developed over the many years. That is not to say that the
"Scientific Method" is infallible It is in fact very fallacious,
unsound, and often illogical. In New Science a new qualification is
added to the "Scientific Method". That qualification is "mathematical
truth" -- not the mathematics with which we are all familiar but -- the
natural mathematics of all Creation of which "Quantum Arithmetic" is a
beginning. It is the sieve which sifts out all but the truth, and leads
into Sacred Geometry of ancient times.
This sets up a mathematical analog to every situation which can be
imagined. Find the analog and one has found the truth. One cannot say
the same for common math.
Everything is composed of four parts and those four parts begin with the Quantum Number of a thing , of an action, or of a thought. Quantum
Arithmetic identifies the four parts of the Quantum Number and then
identifies the chacteristics of the thing, action or idea, and gives a
full description of it and its characteristic parameters.
The four parts of a human are (1) Soul; (2) Spirit; (3) Mind, and; (4)
body, in that order and in descending importance.
(1) The Soul is the "God-particle" of a person. The soul can be
considered as a spark of the Creative force, separated from that
Creative force, but hopeful of returning to that Creative force in more
full measure than it began. It is imperishible and everlasting, except
it can be destroyed, or "wasted" by #4 the body.
(2) The Spirit is non-material, but is a cohesive energy field. It is
the important part of a person. It is a virtual analog of the body,
except the Spirit preceeds the body and the body is an analog of it. The
Spirit is in search of "oneness" with the Creative force, and
perfection, which it can gain through utilizing several bodies in a
progressive way to that goal. That is why it is necessary for a Spirit
to reincarnate.
The Spirit is that which is really in charge of #3 & #4. It is sometimes
called the "unconscious". The Spirit is all knowing but it can be
missled by the body or by the mind through faulty input information from
them. This faulty information comes from the Belief System which the
person has accumulated over the present lifetime. Young childen do not
have this problem. That is why the bible admonishes, "Listen to the
children!". The Spirit is traced out in Volume 3, Chapters 5 through 9,
of "Pythaagras And The Quantum World" (1997)
(3) The Mind is concerned with memory, and with autonomous functions of the body. The mind is compartmentalized into Memory; the Belief System; that which controls the body maintaining the harmony of health through the Autonomous system; and a system for "feeding" the Spirit.
The Mind can be seen through kirillian photography, as an energy field
or aura about a body. Three parts of the brain of a body are programmed
to read / write to the aura. They are the pineal gland, the pituitary
gland, and the hypothalamous. The Third Eye is the modem to make this
information connection.
Many people are developing this Third Eye today, and it can be
identified as a rounded spot on one's forehead. Under certain light and
certain angles it appears to be 2 to 3 inches in diameter and seems to
protude ever so slightly, and have a slight variation from the color of
the surrounding scalp. The Spirit communicates with the body through
this Third Eye and we call it "Intuition". This Third Eye is to be found
on all experienced Dowsers, and on persons who have psychic ability.
There is some correlation between the prominence and physical appearance of the Third Eye and the state of development of that entity.
(4) The Body has been studied extensively by current science but only in its physical context. The one thing which the body has, but is lacking
in the other three parts is Free Will. We continually over-rule messages
from Spirit when we use our Free Will to ignore the intuititive messages
it sends. That is what leads to the many mistakes which we make in daily
living, invariably causing us to pay for the mistakes. However, we learn
mainly from those errors and it enhances our judgement. We will return
to the Transition which takes place when a Spirit enters a body.
Each of these four parts is more than they seem through our senses. They are all "energy fields" formed originally from positive protons,
negative electrons, and neutrons for the body. The other three parts are
strictly energy fields made up of male and female energy. They are all
bound together by the attractive forces of harmonics, under the natural
"Law of Harmoncs". as described in Volume 2 of " Pythagoras and The
Quantum World" (1995) . Likewise, they are pulled apart and destroyed by
the negative forces of enharmony and dischord.
So, a person is actually formed of four parts of which the body is
simply the housing for the other three. Plato described the body as an
overcoat, which the Spirit discarded when the body is old and worn, or
when it has served the Spirits ppurposes. Then the spirit finds another
body and puts it on as one would put on an overcoat. Reincarnation is a
valid concept according to Quantum Arithmetic; According to Sacred
Geometry; and According to the Grand Unified Field.
TRANSITION:
It is an important event in which the Spirit enters the body of a child.
Many Spirits may watch the growth of an embryo. Each Spirit may check
the projected Akashic Record which is dependent upon each Spirit's
eventual occupation of that body. The body will go to the highest and
best Spirit which wants it. But the needs of each Spirit for its
personal growth is different, partly through its karmic needs for
balancing out its karma. If a Spirit's karma is balanced then its
objective is for further growth, and for that they will need a most
perfect body to occupy. If the karma is too badly out of balance that
Spirit may be relegated to occupying an animal body.
When a choice is made, a Spirit does not enter a new body until some
time after birth but it usually occurs within a few hours after birth.
In one case in India, the spirit waited six years before entering the
body of a boy. It seems that the Spirit and accompanying Mind, (halo),
entered and activated, and the boy suddenly, at age 6, was cognizant of
having lived in that same village in his immediately preceeding life.
Sometimes a Spirit and Mind will enter the body of a much older person
to gain a very specialized characteristic in a much shorter time. These
are generally experienced in what are called "Near-Death-Experience,
(NDE), and the Spirit is called a "Walk-In". One such experience is
discussed in Chapter 6, pages 94 through 103, of Pythagoras And The
Quantum World, Volume 3. It was a personal experience of this author at
age 68, and the visiting spirit stayed for only 9 years, and one month,
when his goal was reached.
TIME SPACE
The body is the only one of the four parts that must comply with our
concept of time-space. The mind, spirit, and soul exist only in the NOW.
The information on time-space is much like a movie film. The one frame
we are viewing is the NOW, and the film can move forward or backward at
a rate of four quadrillion frames per second. That is to say that Time
is "quantum". The film from the past is developed and stable. The film
of the future is already exposed with all that is to happen under
optimum circumstances. If there are changes in the forces of the future,
the film behind that frame are developed, and the frames ahead of the
NOW are reexposed to take this into account. The one thing which can
make this change in causative effects is the Free Will. If we would all
forego our Free Will and let Intuition be our guide, no revision of the
future film would be needed. The future is not set. It will change in
accordance to the forces which are encountered.
The mind and its memory of the past is engraved on this film. To recover a memory we must scan the film of the past of the NOW frame. This is done by the Spirit or Unconscious and the information flows at a speed much faster than the speed of light.
The same thing occurs when we dream and view the film ahead of the NOW frame. Dreams can be very revealing and prophetic. They are difficult to interpret because we do not receive all the information on the causative forces or scenerios, leading up to the dream picture. This film of past, present, and future is called the Akashic Record.
The Mind, Spirit, and Soul are not confined to given spaces, and
excluded from otherwise occupied space as is the body. They are much
like the waves on the surface of a pond. New waves can be introduced in
the pond without first displacing a given set of waves.
SYNCHRONICITY.
When waves are not synchronized, (do not occur at the same time between frames as other waves), they are separate from all other waves and can exist in their separateness within the ONE space. If a set of waves is teamed up with another wave at the same instant the composite of the
newly formed wave with the original, will form and entirely different
complex wave.
RESONANT FREQUENCY OF THE AETHER
Letter From Greg Hodowanec
From: NEN, Vol. 6, No. 8, March 1999, pp. 21-22.
New Energy News (NEN) copyright 1999 by Fusion Information Center, Inc. COPYING NOT ALLOWED without written permission.
ALL RIGHTS RESERVED.
Sacred Geometry , music and a possible correlation with
quantum-mechanics
Version 1.2 , July 1997 by Martin Euser.
In this article I will introduce a simple formula of Sacred Geometry
which hasinteresting properties. I will not provide an extensive introduction to the concept of Sacred Geometry since there are many websites where you can find introductory material into this field.
Instead of that I will, however, provide some links to interesting
material (new developments in this area of research).
The basic idea of Sacred Geometry is summed up by Plato: 'The Demiurgos geometrizes'. Now, the Demiurgos, often erroneously translated as "God", is simply the collective host of beings that is sometimes referred to as 'the Architects' or the third (manifest) logos of our cosmos.
In the Biblical Genesis we find the term 'Elohim', which again is
erroneously translated as "God". Instead, it can be better translated as
'host of angelic beings' which formed mankind in their image.
This, of course, is a story in itself and more details can be found in
Ralston Skinner's work 'Source of Measures', Gottfried de Purucker's
'Fundamentals of the esoteric philosophy', Alvin Boyd Kuhn's writings,
Gerald Massey's writings and Helena Petrovna Blavatsky's writings such
as 'Isis Unveiled' and "Secret Doctrine'(the last book being rather
mystical in nature and generally being little understood). See my other
articles for links to websites with writings from some of these authors.
Now, the question is, 'How does the demiurgos geometrize?'
What are the ways and methods which are used in or through the Minds of
the intelligences that have prepared the design for our earth, man,
flowers, animals, minerals, etc.?
This question has occupied many minds , from the old Hindus to the
Babylonians, Greeks, Jews, to the modern student of Sacred Geometry.
Some (partial) answers seem to have been found and laid down in
architectural designs to be found in Hindu temples, Egyptian pyramids,
etc., and coded into scriptures as the Bible, the Hindu Puranas, etc.
See e.g. the above mentioned work of Skinner. For the keen observer it
is very clear that there is an inherent order in this universe although
there may appear to be chaotic phases due to the interactions of a
multitude of factors, wills, energies, etc.
The growth and development of a human embryo is a good example of the harmonious workings of nature. In the beginning stage, the cells are
configured in a beautiful geometrical pattern, reminding one of the
structures that are found in the mineral kingdom. Later on in the
developmental stage one can observe structures reminding one of plants,
lower animals, etc. culminating in the fully developed foetus. Michael
Schneider has written a beautiful book on Sacred Geometry which
illustrates features such as the above.
He wrote "A Beginner''s Guide To Constructing The Universe"
(HarperPerennial) Some websites relating to the book & Michael:
www article:
Cosmic Dozens: Twelve-Fold Designs of Society and Art Workshop:
Sacred Harmony: The Patterns of Nature and the Ways of Knowing
Book Reviews:
A Beginner's Guide to Constructing the Universe: The Mathematical
Archetypes of Nature, Art, and Science (A Voyage From1 To 10)
Book review by David Fideler, reprinted from PARABOLA (Winter 1995)
Michael's workshops in northern CA
Another classic on Sacred Geometry is Doczi's book 'The power of limits' which richly illustrates the harmonies in nature, art and architecture. Manifestation means limitation and thus the title of the book was enough incentive for me to buy it and have a look at it. Doczi develops the idea of dinergy and patterns of sharing. Dinergy refers to the working
of opposites united in a harmonious proportion. An example is the
working of the minor and major parts of the golden proportion. He
invented the word 'dinergy' to refer to the universal pattern-creating
process. As a prefatory remark to what follows I want to add that I don't believe that one magical mathematical formula exists which explains all and everything. The Boundless has an infinite number of attributes and its
emanations vary accordingly. Yet we are told that analogy seems to be
the guiding principle in the emanation and building of worlds, forms,
etc.
Infinite variation along analogical lines (nature repeats itself in
endless variations) may be the answer to the question of the operations
in / of the Divine Mind and more specifically pertaining to the operations
of the Builders , but more research is necessary. The emergence of
fractal theory seems to be an interesting development in this respect.
Fractals embody the principle of repetition so abundantly present in
nature.
The theosopher Gordon Plummer has shown that the dodecahedron is capable of generating the icosahedron, and these two Platonic solids together, the other Platonic solids (tetrahedron, cube, octahedron) through the working of the well-known number Phi (the golden section number).
The dodecahedron and icosahedron can generate each other in an infinite series through a repeated geometrical process where the Phi proportion plays a dominant role.
His "Mathematics of the Cosmic Mind' is interesting food for thought.
Some independent researchers have followed up on his ideas and also try
to combine ideas from Sacred Geometry with chaos theory and fractals. An
interesting example of this can be found at : San Graal School for
Sacred Geometry
Another example of Sacred Geometry , applied to the Platonic solids and the Kabbalistic Tree of Life , can be found at: Sacred Geometry
discovery
Now I come to my own , thus far limited research in the area of Sacred
Geometry.
While studying the Secret Doctrine of Blavatsky, I discovered her
mentioning something about seven being the root number of formation.
Being engaged in a little study of Greek geometrical concepts, like
arithmetic mean, geometrical mean and harmonic mean, I decided to do a
little research on the root of seven ( 7 ) and this proved to be the
start of some interesting discoveries. Figure 1 shows the relation of these means to each other:
harmon~1.gif
I will sketch some highlights of my findings, which form only the
proverbial tip of the iceberg. The next section presupposes a basic interest in and acquaintance with
high school mathematics. I'll skip the derivations of the results (which
are very easy to obtain, however).
First of all, I started with the idea that Pythagoras' famous musical
proportions might be found by using the idea of arithmatical,
geometrical and harmonic means generated from a pair of numbers (two
poles or a pair of lines) , thus:
A - n .V(7) and A + n .V(7) where A is the arithmetical mean of this
pair and V is the symbol for taking the square root (in this case of the
number seven).
I found out that the following two poles had a relation to the major
third (which is a proportion of 4:3) :
16 - 4.V(7) and 16 + 4.V(7) ( the '.' is a symbol for multiplication)
This pair of numbers yields the following means:
A = 16 ; G = 4 * 3 = 12 ( '*' is symbol for multiplication) and H = 9
G means the geometrical mean for the two poles and H indicates the
harmonic mean.
The formulas for G and H are in this case:
G = V( 162 - 16 * 7) and H = G2/ A
Now, 16:12 is 4:3 and 12:9 is 4:3 , so the 4:3 proportion is implied.
This result encouraged me to research further pairs of numbers and I
found some interesting poles with properties as follows:
9 - 3. V(5) and 9 + 3.V(5) have A= 9, G= 6 and H= 4 which imply a relation of 9:6 = 3:2 and 6:4 which again is 3:2. This is the proportion for the quint or fifth in music!
I'll spare you some details of research and present the abstracted
general formula capable of generating musical proportions:
The poles N2 - N * V(2N-1) and N2 + N * V(2N-1) generate the following
means:
A = N2 , G2 = N2 * (N -1)2 , H = (N-1)2
G = N * (N -1)
Note the beautiful interplay of terms here: substitute N + 1 for N and H
becomes N which is the value of A for N proper.
For N=2 we get : A = 4, G = 2, H= 1 which implies 4:2 = 2:1
In other words, the idea of an octave!
N =3 yields the proportion of the fifth as I've shown already, N = 4
yields the major third proportion in music. N = 5 yields the 5:4
proportion which Pythagoras seemed to dislike, but was incorporated in
the musical scheme of another Greek School of philosophy as being a
natural proportion.
However the case may be, the famous 9:8 proportion (the distance between two notes on the major scale) is generated when N=9.
So far for music. As an aside: the root-scheme includes all the roots of
uneven numbers including the famous ones in Sacred Geometry. The only
root missing is 2 which is symbolical for a square. I suspect that there
is another principle to be synthesized into my little theory - some
formula containing this root-number and maybe some squaring function.
Systems interacting, so to speak. (a small part of this has been solved
by me now, but that's material for a second article on Sacred Geometry).
I did some more research on this generative set of poles and discovered
some more interesting properties. When researching the presence of
cyclicity in the generated means by use of this type of scheme (I call
it the 'root-number scheme ') I discovered that it seemed to have a
relation to the famous periodic shell-number in quantum-mechanics:
2 * N2
First electronic shell for N =1 has 2 'positions' for electrons (1S-
shell)
Second electronic shell for N=2 has 8 'positions' for electrons
(2S-shell allows two orientations of spin) and 1P shell (allows 3* 2 or
six orientations)
And so on.
I discovered was that by varying on the above mentioned scheme by
introducing a factor m a new feature arose:
N2 - m.N * V(2N-1) and N2 + m. N * V(2N-1)
where m is an integral number.
Now, G and H depend on m besides N as to their value. I discovered that
by setting m = N (which is the maximum value of m that is possible
without making G negative) and substituting 2.N2 for N yields integral
values of G and H :
(2*N2)2 +- N.2.N2 * V(4N2-1)
which renders:
A= (2N2)2 ; G = (2N2)*N ; H = N2
which again shows an interesting relation between A and H: H has now the same numerical value as A has in the previous scheme. Somehow there
seems to be an interplay between the schemes in such a way that the
arithmetic mean and the harmonious mean take each other's value.
This makes sense to me, however, as I think that everything in nature is
connected to each other and the above mentioned interplay would fit
perfectly into nature's wondrous works. BTW, if you care to research the
values for A, G and H in this case you will find musical harmonies
again!
I have not as yet deeply investigated the relation between the above
scheme and quantum-mechanics (QM) except for noting the interesting
results for H being equal to N2 for certain values of N , namely where
A=(2N2)2, and discovering the limited allowed values for m: m >= -N and
m <= N which reminds me somewhat of the number of allowed values for the projection of orbital angular momentum of a particle, say an electron,
on the z-axis of a Cartesian coordinate-system (although we need a third
variable to establish this connection - the angular momentum quantum
number l). This is speculative, of course, but not as weird as it may
seem to be. Motion might be described by a new kind of formalism which
connects natural phenomena in a clearer way than has been established up
to now, i.e. underlying forces, patterns, etc. may be supposed to exist
in order to describe the connections between the rich variety of
phenomena that we observe (and who says that we are aware of all these
phenomena?)
Now, N2 is a term which appears in the equation for the energy states of a hydrogen atom :
EN = constant/ N2
Spectral emission occur when an electron falls from a higher energetic
level to a lower one. The frequency of the spectral line (for hydrogen)
is simply given by:
frequency = C ((1/ N)2 - (1/ M)2) where N and M are integral numbers > 0 and M > N.
Actually, I suspect that relations such as I discovered may be connected to orbital motion, be it of electrons or of planets and possibly with rotations around the own axis of these bodies (which are called 'spin'
in the case of subatomic particles such as electrons).
Time and inspiration permitting I may do some more research in this
area.
I need to mention another relation between G2 and A:
(G2 )2 / A 3 = ¹ which is interesting since it reminds me of the famous
law of Keppler: the quotient of squared time of revolution and long
semi-axis of orbit of the planets around the sun to the third power is
constant. He derived the elliptic motion of the planets from this
empirical fact and some other ones. Don't bother about the number ¹(Pi): the original form of the root-scheme can be multiplied by a constant without any essential change as to the results. Admittedly speculative, but a whole field of research opens to the open-minded researcher. After all, titanic intellects such as those of Pythagoras spoke of the music of the spheres and this is just what the root-scheme formulas seem to be about: harmonic proportions in nature.
The same proportion for G and A can be derived for subsystems of the
root-scheme.
I leave this as an exercise to the mathematically inclined readers.
Schemes as the above are known (in a general sense) in the algebraic
theory of groups. I've not yet researched all the mathematical
properties (as to group properties, subgroups, cyclicity, existence of
unity element, etc.) of this type of system. The system will probably
have to be extended with other factors, elements of interaction, etc.
To the student of Sacred Geometry: there seems to be a beautiful element of squaring involved here. Finding a root is related to squaring, since the squaring of a root yields the original number from which the root has been extracted. Squaring and taking the square root of numbers are inverse functions, which very much seem to have a place in
sacred-geometry. The play of nature (according to the pattern in the
mind of the Demiurgos), I suppose.
This squaring and finding a root must be brought into relation with
circles and spheres, or circular / spherical motion to make it produce
natural phenomena such as harmonic motion, spirals, etc. This should be
the subject of deep investigation, I think, the more so since scientists
have got stuck in their paradigms and may need to get an impulse from
alternative researchers to establish new paradigms pertaining to nature.
I almost forgot to mention that the ancient Greeks seemed to attach much
value to surface and perimeter of geometrical figures. I have related
these two with the harmonic mean as follows (for square and rectangle,
but the same deduction can be made for circle and ellipse):
Let a be the side of a square, b and c the sides of a rectangle.
Demand the invariance of proportion between surfaces and surfaces and
perimeter to perimeter of square and rectangle.
Then it follows that: 4a / (2b + 2c) = a2/ (bc)
Elaborating this gives: 2bc/(b+c) = a which is equivalent to:
a = bc/((b+c)/2) which is the formula for the geometrical mean for the
poles:
¸(b+c) +/- ¸(b-c)
So, ¸(b+c) is the arithmetical mean here, being the ¹ part of the
perimeter of a rectangle, so to speak an average of the four sides..
H = a = 2bc/(b+c) which is the side of a square
and G2 = bc which is the surface of a rectangle, which means that G =
V(bc) which starts one thinking about the relation of square to
rectangle again.
Could it be that nature prefers certain numbers a, b and c so that
integral numbers and proportions of integral numbers are preferred in
its work? Certainly I see no way in which a transcendent number by
itself could appear in manifested form, the perfect circle being a
mathematical idea and not a physical fact. What else could be meant by
"squaring the circle" than the operation of a principle of limitation in
nature?
Remember that pi is a transcendent number which means that it cannot be written as a proportion of two integral numbers. Nature (the divine mind to others) must have found a way to manifest itself (by limitation),
thus using approximations of the famous numbers pi and phi (and other
numbers). Ralston Skinner has decoded a system for the use of
approximations of pi in his ~Source of Measures~ and I think he is right
in many of his assertions.
Madam Blavatsky has confirmed some of his central assertions and she
also noted that:
'the whole of the work of Euclides was meant to demonstrate the
properties of the dodecahedron' (a quote from memory). A quote which is
certainly memorable when viewed in the light of some discoveries in
Sacred Geometry (see Gordon Plummer's book referred to previously).
So, I suspect that the principle of quantization is inherent in nature,
even more than scientists have already discovered in the realm of
quantum-mechanics. Much research has still to be done in this regard and
I would invite others to engage in this type of research!
An interesting link in this regard is:
Cycles in the universe
To end this article I want to express my hope that geometry will one day again play a dominant role such as it once had in mathematics. Clerk
Maxwell (from the famous Maxwell equations in electromagnetism) used
Thomson's geometrical proofs of his equations, but nowadays this
geometrical knowledge and these skills seem to be lost to the scientific
community to a large degree. Add to this an integration of Phenomenology
(as Goethe very much wanted) and Sacred Geometry with science, and, a
new science, more true to nature, may be born!
Martin Euser euser@euronet.nl
Bibliography Ralston Skinner: The source of measures; Wizards bookshelf, San Diego, Cal., USA ISBN:0-913510-47-5
Gordon Plummer: The mathematics of the cosmic mind; Theosophical
Publishing House; Wheaton, Ill., USA ISBN: 0-8356-0030-0
An Egyptian Geometry Lesson - Part One
One, the Monad, the beginning of all counting, individual by nature, the
God of all numbers. Alone. Indescribable, since there is nothing outside
of unity to act as observer. To the Greeks, One represented Khaos, the
primeval state of the universe before the separation into positive and
negative, which was the Dyad, the first stage of Creation.
Two, the Dyad (Gk. Duas). The first feminine, even number in the
classical numerological litany of the Cabalists. The division between
heaven and earth and symbolic of movement away from Khaos towards
symmetry. Yet it also produces conflict, it is two sides of the same coin that will never see each other. It needs an observer to see both
sides, that's where the first real number makes its appearance.
Three, the Triad, the Watcher. The first masculine, odd number and the
natural product of one plus two. It is the beginning of dimension and
represents the second stage of creation, fertility. The mathematical
interactions of the first three numbers produce every subsequent number.
Religious history is littered with the symbolism of three and it's role
in Creation. Egypt was no different to Hindu or Christianity in its
recognition of the power of the Trinity.
Four, the Tetrad, the corner posts of the Earth, the squaring of the
circle, two plus two or two times two. It represents symmetry and
surveying and is the number of foundation and land measure. The square
is man's attempt at rationalizing the irrational circle. It can be symbolized by the cross. (X)
Five, the Pentad, symbolic of Man and the Golden Mean, the Divine
Proportion. The pentagram has been invoked as a symbol of protection
throughout history, especially in Egypt. The five pointed star enclosed
in a circle was their symbol for the Duat. The Pythagorean theorem
states that in a right triangle, five is the square root of the sum of
the squares of the two previous numbers, three & four. 3 x 3 (9) + 4 x
4 (16) = 5 x 5 (25)
Six, the Hexad, the perfect divider of the circle. The radius of a
circle divides its circumference into the six points of the hexagon to
form the Seal of Solomon, the hexagram. The mathematical properties of
six are abundant in nature, in crystals, honeycomb and flowers.
Seven, the Heptad, the virgin. The magical heptagram was known as the
seven-horned beast. It is the only number of the first ten that cannot
be divided by any of the others except one, it is the only one of the
decad that will not divide evenly into 360 degrees. Seven is the number
of mystery, it does not integrate with the geometries of the other
numbers, or so it is said. However, there is a way, and Giza describes
the path.
Eight, the Ogdoad, symbolized by the cube and it's eight corners. It
represents stability and the Earth. Eight is the square root of the
enigmatic sixty-four, seen in the chessboard and the I Ching. It is the
natural divisor of the square and the beginning grid of the Giza
geometry.
Nine, the Ennead, the Gods of Heliopolis. Re-Atum, Shu, Geb, Nut,
Tefnut, Osiris, Isis, Seth and Nephthys. Representing completion. The
Egyptian mnemonic to calculate the area of a circle was incredibly
simple and accurate to.06%. It goes like this. Subtract the ninth part
of the diameter of a circle and square the result. If the diameter is
nine, minus one ninth = eight. Eight squared is sixty-four, therefore,
the 8 x 8 square represents the "nine" diameter circle of completion.
Ten, the Decad, the cycle of the Monad begins again but now it
represents the all-encompassing Universe. The Decad ties together all
the number systems of the Giza Pyramids, it is the underlying mainframe
of the computer
That was a very brief outline of the symbolism associated with the first ten numbers, the Decad. I don't intend to give you a complete history of sacred geometry but some background is essential to the understanding of what we are about to explore. Geometry is Man's humble attempt at
describing the universe he exists in. There are immutable laws that
nature follows by instinct, as is demonstrated in the spirals of the
sunflower. They are determined by the numbers of the Fibonacci sequence,
the basis of cellular growth. This series operates by progressively
adding the previous number to the sum, beginning :-
1+1= 2+1= 3+2=5+3 = 8+5=13+8 =21+13= 34+21=55+34= 89+55 =144
We can see the outcome of this sequence in the growth of the Nautilus
shell. The ratio that appears between the successive numbers of the
sequence is the irrational number (phi) or 1:1.618039. This is the
natural ratio incorporated in the pentagon and the "Golden Section". It
was also a fundamental of Egyptian mathematics and essential to their
"Canon of Proportion". It is encoded in the geometry of the Great
Pyramid, along with the Pi ratio for determining the circumference of a
circle.
Before we proceed to the next level of the Giza geometry it is probably
wise to insert a quote from a man I admire very much. He has written
several eloquent books on the subject of sacred geometry and number
symbolism, John Michell. In his book "The Dimensions of Paradise" (p.65)
is a passage that sums up everything that the pyramid builders were
trying to say in stone.
" Sacred geometry is the essence of the geometers art. Its use is for
making descriptions of the universe by combining together in one
geometric composition the basic figures which represent the different
orders of number and underlie the manifestations of nature. It is the
art of synthesizing diverse elements. The tradition survives today in
the patterns of Islamic art, but it is no longer thought worthy of
scientific study on the grounds that it is irrational. That epithet is
applied to mathematical ratios which cannot be defined in terms of
whole / number integers.
Ever since the legendary renegade Pythagorean let the cat out of the bag by revealing that 'pi', like many of the other important ratios in
geometry, is irrational - and that one can not therefore define the
relationship between the diameter and circumference of a circle or
between the side and diagonal of a square by any pair of integers - the
procedures of the old geometers, in squaring the circle and uniting the
different orders of geometry in one scheme, have been considered a
mare's nest. Rationally indeed, the problem of constructing a circle of
the same perimeter or area as that of a given square, or of fitting
together geometric shapes developed from the numbers 5, 6 and 7, is
insoluble because these shapes are not commensurate.
Yet there is one system in which all numbers and proportions and every
disparate or incommensurable element co-exist and function together in
perfect harmony, and that system is the universe. It can never be
represented perfectly and literally by any artist's or geometer's model
because, although it is said ultimately to be a creation of reason, the
paradox in its nature is too deep rooted for human resolution. The
geometer's aim, therefore, is to imitate the universe symbolically,
depicting its central paradox by bringing together shapes of different
orders, uniting them as simply and accurately as possible and thus
creating a cosmic image which, as Plato claimed for his Magnesia
plan,'most nearly resembles the original' and is 'second only to the
ideal'. And within that limit of ambition it will be found possible, to
all practical intents and purposes, to square the circle, unite the
different orders of geometry and depict adequately that most perfect
expression of the Monad which the Greeks called Cosmos." John Michell
copyright 1988 Thames & Hudson Ltd. London.
I wish I'd said that!
The Giza pyramids are the Ancient Egyptians version of that model of the Cosmos. All the numbers and principles of sacred geometry are marked out by the dimensions and positions of the three major pyramids and their
subsidiary satellites, temples and walls. The outcome of their model is
the uniting of all the number geometries of the Decad, including seven.
But that is just the beginning. Their system also includes the numerical
values of music, the courses of the Great Pyramid follow a sequence in
their differing thickness that relate directly to musical notation, the
Great Pyramid actually plays an entire song. The length in cubits of one
side of the G.P is 440. The note A440 is the tuning note for an
orchestra and has a frequency of 440 cycles per second.
The thickness in inches of the base course, divided into the speed of
sound in inches per second, gives a result of 220, its resonant
frequency. 220 is also the half base dimension of the G.P. and one
octave below middle "A". That A220 becomes the starting note of the
Great Pyramid's tune, and the predominant 35th course (5x7), which is
two sacred cubits deep (50p"), resounds to middle "C", 261.63 cycles
per second. The diagonal of the G.P. is 622.22, corresponding to an E
flat. Because we are dealing with an integral number system it soon
becomes obvious that there is a pattern emerging. The pyramid builders
tried, and possibly succeeded, to incorporate all their numerology in
one grand design.
There are 20.6066 p.inches in the Royal cubit and 25" in the Sacred
cubit. (See notes) Therefore, all measures quoted are in Royal cubits
(r.c.) unless stated otherwise. The royal cubit equals 20.625 British
inches or.52375 meters, but we will not be needing either of those
systems from now on, though once in a while a comparison will be given
to make you aware of the scale of things and their accuracy.
This musical aspect is also apparent in the second and third pyramids.
The side of Khafre's measures 411 r.c. The length of two sides, 822,
divided by Pi also results in 261.63, middle "C". Menkaure's pyramid,
the third and smallest measures 196 x 200. 196 corresponds to a "G".
200 x 200 is the size of the squares which comprise the fundamental
framework of the Giza geometry. As we will see later, each deviation
from the norm has a purpose. For example, the third pyramid measures 200
x 196r.c., where it should be 200 x 200, but 200 x 196 produces a
diagonal of 280r.c., the precise height of the G.P., and it makes its
volume one eleventh of the Great Pyramid. Do you see how it all
interrelates on many different levels? Somebody had a great time working
out this incredible picture.
Just as we sent our mathematical message into space carried by our
ultimate achievement in technology, so did the Pyramid Builders send
theirs into history on board the Giza pyramids. All it takes to
understand their epitaph is to give them the credit for the knowledge
they obviously had.
When Petrie measured the Pyramids he concluded that the centres of the three pyramids bore no mathematical relationship to each other, and yet he measured them with fantastic accuracy considering that in 1881 the sand was covering a lot of the detail of what we now see, especially the corners of Menkaure's pyramid.
Ironically, if he, or anyone else in history, had placed more importance
on the satellite pyramids of Menkaure, the so-called Queens pyramids,
then the problem could have been solved long ago. Remember how in the
Overture we talked about decoys? We have to give the designer credit for
having a sense of humour. Whatever insight he had into the future you
can be sure he could second-guess how long it would take until someone
looked at the relatively insignificant detail of his masterpiece, the
Coda.
This is the perfect place for a musician's joke, it goes like this: -
A famous, but notoriously bad, orchestra leader died and went directly
to hell. There he was welcomed warmly and enthusiastically by Old Nick
himself, who expressed his happiness at finally getting the leader he
had always wanted for his orchestra. On expressing his lack of
understanding of the situation, the musician was led into a magnificent
concert hall and there, on the stage, was the largest orchestra he had
ever seen. Old Nick led him to the podium and handed him the baton. "You
shall be my conductor," said the Devil. The musician broke down in tears
of joy and said, " How can I possibly deserve this? This is my idea of
Heaven, not Hell " Old Nick smiled." You haven't seen the music yet! "
The musician opened the pad on his music stand and read the title "
Strangers in the Night ", he laughed, "One of my favourite pieces, I
don't suppose you've got 'Ole Blue Eyes ' himself down here as well?"
The Devil smiled again, " You had better check the arrangement, the show
begins in two minutes. " The musician began furiously scanning through
the music and soon a look of panic appeared in his eye and he turned to
see Old Nick wearing a tuxedo and top hat. " I can't find the Coda!! ",
he screamed. The Devil's eyes lit up as he swaggered to the microphone.
" There isn't one you fool, this is hell remember. Now are you ready?
After four....., a one ana two ana three ana........"
I hope you see the analogy. Trying to solve the Giza geometry can be a
nightmare if you don't find an exit. The Pyramid builders played number
games just as we play word games today, it was the pastime of the
intellectual ruling class, the priests, and possibly of the whole
society at some time, if Plato's Magnesia ever existed.
The geometrical paths that can be constructed in the "Rostau" circle are inconclusive; they can lead you round and round the circle for an
eternity and only replay the same piece over and over. It will still,
eventually, teach you the tune of its sacred geometry, but the Coda has
to be found and played to give the composition a Finale, so that in the
silence, at the end, one can savour the entirety of the piece and
possibly, for a moment, see God.
For a musician, repetition leads to memorization, as it should. Why read the music on paper when it can be translated directly from the mind to the fingers and limbs? After a while the pathways become engrained in
the matrix of the mind and can be recalled, intact, upon demand,
assuming of course that the memory is exercised once in a while. As we
all know, there can be many variations on a theme, but all have to be
contained within the original framework or it becomes unrecognizable as
the original. Even jazz has to stay within certain constrictions of the
original metaphor. The Giza symphony has a set of well-stated metaphors
that have to be included in any rendition, but once learned they can be
the key to an infinite variety of expression.
The Great Pyramid Khufu (G1)
We must begin with the prime statement itself, the Great Pyramid and its measurements. We have to use the measure utilized by the Builder
himself, the Royal Cubit and the Pyramid, or Primitive Inch, since that
is the only system in which the encoded equations actually take on their
true numerical values.
The Great Pyramid is 280rc. high and 440rc.along its base, giving a
perimeter of 1760rc.and diagonal of 622.22.rc. Divided by 40 its
proportions are 7 high by 11 wide. Consequently it represents the
approximate formula for (22 (2 sides) -:- 7 = 3.14285714...) However,
there has been much deliberation over the years as to its true measure,
each side is fractionally different to the others and the consensus
comes down to an intended base measure of 439.823 r.c., which produces a
true Pi of 3.141592654.. The G.P. is an approximate scale model of the
northern hemisphere of the ratio 1 : 43,200.
1,760 r.c. x 43,200 = 76,032,000r.c = 24,754 miles (approx. equatorial circumference)
The height of 280 is the approximate degree of flattening at the poles,
(1/ 280th) caused by the bulge of the equator. The Golden Mean
proportion, phi, is incorporated in the dimensions of the half base
(220) and the apothem (from apex down centre face to base centre). This
works thus,
* 280 -:- 4 = 70 (ht.)*................* 220 -:- 4 = 55 (half base) *
therefore: -
70 squared + 55 squared = 89 squared ** 89 -:- 55 = 1.61818 = (phi)
55 and 89 are consecutive Fibonacci numbers.
89 x 4 = 356 ( lengthÊof apothem )
The Egyptians called that ratio of 70 : 55 or 7 : 5.5 a Seked angle of 5
1/2 palms. A royal cubit was divided into 7 palms, or 100 n, so a
pyramid structure one cubit high (7 palms) and a half base of 5 1/2
palms generates an angle ofÊ51* 51' and a tangent ratio of 1.2727, as is
found in the G.P.. Now we can expand the visualization process a little.
The proportions we have just been working on have some intriguing
properties. 70 : 55 : 89 are the measurements of the top quarter of the
G.P. if one imagines it in four horizontal slices. This we can call one pyramid unit and it measures 70 high and has a side of 110; it constitutes 1/64th. of the volume of the whole pyramid. The formula to find the volume of a pyramid is,
* Base squared x Height -:- 3 = volume (cubic cubits) *
The courses that are missing from the apex of the G.P. constitute 1/16th of its height or one quarter of the top quarter, or 1/64th of 1/64th of the volume of the whole. It gets better. The volume of the missing piece
is approximately 4400 cubic cubits, a mnemonic for the whole that runs
like this.
* (4400 x 64 x 64 x 3) -:- 280 = 439.4282 squared (440 squared)*
or we can use (4411 x 64 x 64 x 3) -:- 280 = 439.9771 squared.
It has to be emphasized that this is a teaching system that was to be
memorized by initiates, so small discrepancies will occur from the
mnemonic numbers, but they are always within.05% of the real value.
Back to the missing 1/64th and the next mnemonic trick. This "One
Pyramid Unit" we have established which comprises the top seventy cubits
of the G.P, when multiplied by 7, results in the volume of the next
lower quarter slice, which is 7/64th of the whole. Add that to the top
unit and it becomes 1/8th, therefore the top half, 140 r.c. high, equals
1/8th of the whole, again multiplied by 7 and we have the volume of the
lower half, 7/8th. And just to take things to extremes, the specific
gravity of the limestone of the G.P. is 2.75, multiply that by seven and
the result is 19.25, the specific gravity of pure gold.
This coincidence could well be intentional because there is another
strange correlation to gold that appears from this "pyramid unit"
sequence, not only in the pyramid proportions but also indicating the
"Gateway", as we will see later. For now, to complete this visualization
process, we will mentally construct the whole G.P. from One Pyramid
Units (1/64th). We will find that we can use 44 whole units by beginning
with 16 units (4 x 4) as the base and then inverting others to fill the
spaces, like an egg carton. When we have stacked the 44 whole units we
are left with 40 tetrahedral spaces, which equal the remaining 20/64th (5/16th) of the whole. Therefore each tetrahedral space = 1/128th. That
crystalline tetrahedron is identical to a gold crystal.
Also to be noticed here are the numbers 44 and 40, both additional
mnemonics. If you look at the drawing you will also see a series of
equations based on the sq.rt. of 41,100,000, this aspect of the Great
Pyramids measure has never before been examined by Pyramid numerologists and is based on the measure of the second pyramid of Khafre, 411. Before leaving the G.P. there is one more dimension to be noted and one that is essential to the final picture. If the side of the G.P. is 439.82 then its diagonal is precisely 622 cubits. If it is 439.9771 the diagonal is 622.222.
* 622.222 -:- 2.2222 = 280 (G.P.ht.). *
* 4,411 x 64 x 64 x 3 -:- 280 = 439.9771 squared.*
Mnemonics !
Khafre ( G2 ) The Movable Feast
The second pyramid of Giza is 274 rc. high and 411 rc. along each face,
making its volume 6/7ths. of the Great Pyramid. It encapsulates the
other Seked ratio favoured by the Egyptians, 5 1/4 palms. For every 7
palms (1rc.) of height the base steps out 5 1/4 palms, this generates a
3-4-5 triangle.
E.g.: - (274 -:- 7) x 5.25 = 205.5 x 2 = 411 (see ratio sketch)
( 4Ê-:- 7 ) x 5.25 = 3 x 2 = 6.
* 205.5 -:- 3 = 68.5 * 274 -:- 4 = 68.5 * 68.5 x 5 = 342.5 (apothem ) *
Petrie's measurement of the pyramids places the centre of "G2" 929.6rc south west of the G.P., or 675.63rc south and 638.5 west. Therefore, by adding or subtracting the half base measure of 205.5, we can determine the positions of the four sides of G2 relative to the G.P.. The southern edge is 881.13rc south of the G.P.'s east / west meridian and the western edge is 844rc west of the G.P.'s north/south meridian. 881 equals two widths of the G.P. + 1, ( 440 x 2 +1 ), it seems to have been placed one
cubit beyond its logical symmetry, 880. As it happens, 881.4760348 is
the square root of 777,000, so it looks possible we have been given
another mnemonic, and the likelihood is more apparent when we look
deeper into the "illogical" placement of G2.
The eastern edge of G2 is 433rc west of the G.P.'s N / S meridian, 440
would have been the logical position so as to fit the overall
geometrical picture we are working toward, but we are still in the early
stage of understanding the designer's purpose. For now it is easier to
accept that G2 is 7 cubits east and 1 cubit south of where it should be
for geometrical precision and then we can look at why it is so. Consider
that on a scale drawing, seven cubits is only the width of a pencil line
and you will see that it makes very little difference to the overall
picture, put it down to artistic license by the master architect. There
are several other mnemonics encoded into G2 that make it obvious that
its offset is intentional and not a mistake by the builder. If you
consider that the designer or any other geometer were already aware of
the grand geometrical design then the problem becomes one of solving the
hidden equation. We'll get there soon enough.
As mentioned earlier, two sides of G2 total 822, divided by Pi that
results in middle "C "(261.6). The length of its diagonal, 581.3rc,
multiplied by Pi is equal to the length of the year x 5 (1826.21). The
measure of its centre, south of the G.P., is the square root of 456456
(675.615).Ê881 minus 844 is 37, this is the horizontal displacement of
G2 away from the diagonal meridian of the G.P., therefore the diagonal
displacement (N.W to S.E) is precisely 26.163, middle "C " divided by
10. The south side of G2 is 661 r.c. south of the south side of G1.The
prime number 37 also contains some other nice relationships, including a
mnemonic for the volume of G2 involving the two displacements, 37 and
661: -
* 3766.1 x 64 x 64 x 3 -:- 274 (G2 ht.) = 411 squared (base)*.. and
then-
* 37 x 3 = 111 * * 37 x 6 = 222 * * 37 x 9 = 333 * * 37 x 12 = 444 *
* 37 x 15 = 555 * * 37 x 18 = 666 * *37 x 21 = 777 * * 37 x 24 = 888 *
* 37 x 27 = 999 * * 37 x 33 =1221*
Before we move G2 to its " real " theoretical position there is one more
measure to examine, its perimeter of 1644rc. The western edge is 844
west of the G.P. north / south meridian. The eastern wall of the Sphinx
temple and Khafre's valley temple is 800rc east of the G.P meridian. 844
+ 800 = 1644. Get it? One complete rotation of G2 eastwards aligns it
with the north / south meridian of the Gateway.
It is an apparently deliberate clue to lead to the other meridian and
the gateway, if the novitiate has not already found, or been made aware
of, its existence. But since we started with the solution we have to
backtrack to the question. Here's a good one. Did the solving of the
Giza geometry constitute some kind of exam for entry into the upper
echelons of the priesthood? It seems like an adequate problem to pose to
evaluate the mental capabilities of the aspirant. We have enough modern
evidence of this type of education in the oriental religions like
Buddhism, where initiates are pointed in the right direction and then
left to find the Truth for themselves, to construct their Mandala. That
is how I began this search, so I could finish my Mandala and know where
I should be standing to see a 90-degree angle at the apex of the Great
Pyramid and draw the final line for completion.
(See Mandala)
And is it not possible that much of what became ritual and ceremony in
Dynastic Egypt was a corrupted version of a science, which had its
origins in a society where a disciplined education in mathematics and
astronomy was the standard? It is quite possible that Pharaonic Egypt
had lost the original premise of its founders and had simply become a
power structure to control the population, an all too familiar scenario
in recent history. Enough digression, it's time to move a pyramid seven
cubits west.
You're asking why we have to move this pyramid and the explanation is
that we have to put it back to where it started. We have enough
numerical clues to say where it should be geometrically, and we will
soon explore them, but first let's examine symmetry. If G2 were the same
size as G1 (440 x 440), then the position of its southeast corner in the
geometry, relative to the centre of G1, would be 880 south by 440 west,
a double square proportion. This will be explained in detail later, for
now we will see what happens with the equations when we examine the
fundamental geometrical picture.
As we can see on the drawing, G2, in its new symmetrical position, does not fill its "box " completely, it becomes 29 cubits short of the north
and west sides of its "box ", since it is only 411 x 411 and not 440 x
440. The true position of the southern edge of G2 is 881 cubits south,
not 880, however, that additional cubit can be added or subtracted, as
in the rules of gematria, to complete the equations that tie the entire
mathematical picture together.
By moving G2 seven cubits west we have now located its western edge 851rc. west of the G.P. meridian (440 + 411 = 851). This now reveals
several important number clues. 851w. + 881s. = 1732. Move the decimal
point three places and we have the square root of 3, (1.732050807). That
means that the s.w. corner of G2 measured from the G.P.centre would be
1224.89rc. 1224.744871 is the square root of 1,500,000, so it seems we
have another mnemonic, but this one runs right into the cabalistic
"magic" numbers 1224 and 1225.
1224," the number of Paradise, and the 153 fishes in the net " (Michell)
This number has a great significance in Christian lore and its "gematria
" (see note). One rearrangement in Greek translates as "divine circle",
which is, mathematically, very interesting. Remember that the aim of the
geometer is to merely represent perfection, then follow this next
mnemonic equation: -
(Square root 6 -:-2) = 1.224744871 x 1000 = 1224.744871
(1224.744871 x 2) squared = 6,000,000
Six is the number of perfection, the hexad.
There are many more correlations to be made to confirm the movement of Khafre's pyramid from the norm, but they involve measurements within the composite geometries and from the "gateway ". Suffice to say, for now, that G2's n.w. corner is 2000rc from the gateway and the s.w. corner is 1800rc from the gateway. The centre of G2 is also now 933.33rc from the G.P.'s centre, or three times the half diagonal of G1, 311.111, or
musically a E flat. One more number before we move on. The s.w. corner
of the 'box' that G2 does not quite fill is 1244.4 r.c. from G1's
centre, twice the diagonal of G1.. It all becomes clearer, believe me,
but it does take patience. We must not move too quickly through the
process or none of it makes sense. This is a jigsaw puzzle after all,
and even though the solution has been shown already, all we have are the
pieces, and they have to be assembled in a logical way so as to
appreciate the final composition. So now we must interrogate the next
witness.
Menkaure's Pyramid ( G3 )
The smallest and southernmost of the three pyramids. Its height is
124.444 rc. and its base is 196 x 200. (Lehner). The diagonal measures
precisely 280rc, equal to the height of the G.P..
(196 squared + 200 squared = 280 squared).
124.444.. is 4/9th of the height of the G.P., 280, and 1/5th of its
diagonal. The level of the granite layer of G3 is 31.11..r.c., one
quarter of the height of G3 and 1/9th the height of G1 and 1/20th of its
diagonal.
196 corresponds to a musical "G" and 200 is one tenth of the radius of
the 2000rc radius circle that encloses the entire geometry, around which
is the 4000 x 4000 square that forms the underlying framework. By the
designer reducing two sides to 196 and leaving two at 200 he created
many intriguing possibilities, one being that it represents almost
precisely one eleventh of the volume of the G.P, -
* (4411 x 64 x 64 x 3) -:- 11 = 4927488 -:- 124.444 ( G3ht.) =198.98
squared (200 x 198)*
However the true ratio is far more interesting, it involves the sq.
root of 123.45678987654321, which is 11.11111 or 11 & 1/9th.
*(4411 x 64 x 64 x 3) -:- 11.11111... *
= (196 x 200 x 124.444) Exactly!
* 196 + 200 x Pi = 1244 *
Musically, 1244.4 is the frequency of E flat.
Once again, Menkaure has been moved a little from its true theoretical
location and again that movement opens up a vista. We'll begin with its
position relative to the G.P. centre. It is 1,411rc south and 1,096.765
west, (1244 squared -:- 1411 = 1096.765., also 1411 minus 1096.765 =
314.235..or Pi x 100)
The direct hypotenuse to G1 measures 1787.125 and that is the first to
draw attention. The diagonal of a double square rectangle 1600 x 800
measures 1788.8543.. (Sq.rt. 3,200,000), very close considering how many
more clues are to be found, and even more relevant when that is also the
distance from the G.P. centre to the Gateway, 1788.8543 rc. The centre
of G3 falls on the root 5 circle, as does the "gateway".
Sq. root of 5 (2.236067977) x 800 = 1,788.8543 * (1,789)*
In gematria,
Alpha has a numerical value of 1.
Omega has a numerical value of 800.
It doesn't take a great stretch of the imagination to suggest that there
may be a link in that number symbolism to the Great Pyramid and the
"Gateway".
So far we have used only the measurements made by Petrie in the 1880's for reference. There have been several surveys of the Giza plateau since then, all of which came up with different measurements and with huge discrepancies between them. For example, the measurement from the east side of G1 to the east side of G3 ranges from 1180 rc. up to 1270 rc.,
that's 90 cubits (150 ft./47 m.) difference, an enormous gap. The ideal
measurement should be 1220, putting the eastern edge of Menkaure's
1000rc west of the G.P. meridian. 1220 is also the uncorrected measure
from the G.P. centre to the s.w. corner of G2. In Petrie's day the base
of G3 had not been cleared of sand and it was difficult to establish the
corner points, as a result he gave the side values as 201.5 x 201.5. We
now know that it is 196 x 200, which compromises Petrie's measure at
this point, however his reading for the centre of G3 is accurate in the
grand design. It places the eastern edge of Menkaure's 997.8rc west of
G1's meridian, 1000 would have been better, but as usual the designer
had something up his sleeve. The measure from the northern edge of G1 to
the southern edge of G3 is once again 1732 (sq.root of 3,000,000).
If we now calculate the distance from G1's centre to the S.W corner of
G3 it comes out as 1930.873997. This is the radius of a circle with a
circumference of exactly 250,000 p." or 10,000 Sacred Cubits of 25 p."
or one thousandth of the polar radius. The S.W. corner of G3 is 2000 rc.
from the gateway. The horizontal displacement of G3's centre from the
G.P. diagonal is 1411 minus 1096.765 = 314.235 as mentioned previously.
This makes the diagonal displacement, NW - SE, 222.22 r.c..
There are more equations to describe, but once again they involve the
higher-level geometries, and include the causeway and enclosure walls of
the Menkaure complex. They will be included as they become relevant to
the construction.
One last important piece of the puzzle before we begin the building
stage. The centres of G3's satellites are the only three that fall on a
straight line, as it happens, that line runs east-west and is precisely
1600 rc. south of the G.P. meridian, and that's where we take our first step into Rostau, by composing the Bassline.
Putting the Band Together.
Excuse the analogy but it's drums 'n' bass, rhythm and fundamental
harmony. It is the basis of the composition and the framework. The time
signature is 4/4 and the first four notes are A, C, E flat and G, an A
dim. 7 chord. See it like this; -
The Great Pyramid is the inspiration for the melody, it encodes in its
numbers the possibility of a beautiful tune, yet it is alone, a one-man
band. A great composer without an orchestra or audience, the Monad, One.
The central thought. Alpha. A.
Geometrically we have the logical starting point, a centre to place the
compass point, but we must determine a radius for the circle that can be
constructed from that centre. The composer / geometer would see those
other musicians standing nearby and, always thinking he may be able to
get a gig together, would invite them to join in, as long as they didn't
want paying. He would need to see if they could play in tune first, and
the first to be auditioned, since the composer plays the drums, is the
bass player. He has to keep perfect time and always stay within his
octave.
He stands between time and melody, holding both together, parallel,
forever separated but intrinsic to the structure of the harmony. The
three satellites of Menkaure, from east to west; -- G3A, G3B and G3C,
look like a base / bass section, they're small, unnoticed and they have a
lot to say about who joins the band.
The "base-players" are pointing straight at the "pianist", the
"gateway", metaphorically speaking. The gateway is the final member of
the "rhythm section ", the one member who can substitute for the whole
band, except possibly for the drummer, but that's a personal problem and
we won't go there. He has at his command seven octaves on which he can
create an infinity of melodies, he completes the trio. So we shall heed
the wisdom of the bass-players and listen closely as they play their
baseline. We draw the line parallel to the E-W meridian of the G.P. and
through the centres of G3A, G3B and G3C. As we know already, these two
parallel lines are exactly 1600 cubits apart, and even without that
knowledge of measure this is still the first step to take. We are draw
ing the stave on a blank sheet of paper.
Using the baseline as our radius from the G.P. centre we draw the first
circle, it does not enclose Menkaure's complex, but the square drawn
around that circle, utilising the baseline, does. By drawing in the
diagonals of the square we can begin "decomposing the square". The first
division creates 4 squares, the second creates 16, and the third
division produces 64 squares. * ( 8 x 8 )*. Mathematically we now have a
3,200 x 3,200rc grid containing 64 smaller squares, each divided
diagonally and measuring 400 x 400rc. All of these measures are 4 based;
a square does not divide naturally into fifths.
Looking closely at what we now have it is apparent that what at first
seemed to be random placing is suddenly a little more cohesive. The
Sphinx temple and the adjacent Khafre's valley temple form the edge and
corner of one square. Campbell's Tomb pinpoints the corner of a double
square, 800 south and 400 east of the G.P. The eastern wall of the
Sphinx temple is 800 rc. east of the G.P. n-s meridian and its north
wall is 800 rc. south of the G.P. e-w meridian. The western edge of G3
is defined by the vertical line 1200 rc. west of the G.P. and that same
line runs down the length of the wall of the "Royal Workshops", west of
G2, and then intersects the baseline at the second step on the west side
of G3B, the stepped, middle satellite of Menkaure.
The second step on the eastern side of G3B is 1600 south by 1200 west, 4 down, 3 across and, yes, you've guessed, five (2000) back to the centre. Since our first circle did not enclose Menkaure's complex it is
irresistible not to draw that "5" radius and so get everyone into the
"Magic Circle ". All except poor little G3C, who sits alone, outside.
But G3C is in the bass percussion section of the orchestra, he plays the
Tympani in the Finale. He's sitting outside smoking, waiting to creep in
at the back of the band at the end and play the enormous crescendo that
ends the symphony. We should leave him to finish his cigarette in peace
for now, he'll have enough to say later. Having drawn that irresistible
" 5 " circle with its intriguing 2000 r.c. radius, we must construct the
square around it, which measures 4000 x 4000 rc., or 10 x 10. By simple
extension of the framework of the first 8 x 8 square, we can subdivide
and absorb the larger square, around the 4000 diameter circle, into our
final framework.
We could leave our grid at this stage of development with its "pixel "
size of 400 x 400rc, however Menkaure seems to be saying " Come down to
my size ", which is 200 x 200. One more subdivision of the master
framework achieves just that, and makes our grid 20 x 20 and consisting
of 400 squares the same size as G3, 200 x 200rc. This also divides the
circle radius into ten. The original "4" radius becomes "8" and the "5"
becomes "10 ".
Looking again at the map we find that several more "hits" have been
made. G3 is now boxed on two sides and the northern edge of its causeway
defines the "7" (1400rc) line south of the G.P.. The southern wall of
Khafre's valley temple sits on the "5 " (1000rc) line south.ÊIt is
becoming obvious that there is an underlying pattern of placement of the
structures of Giza within this fundamental framework, but musically it
is rather "stiff ", it is still only drums 'n' bass. So far Khafre is
still not in harmony with the tune. Back to the analogy, we need the
pianist.
G3B is the predominant player on the circumference of the outer circle,
it marks the corner of the 3, 4, 5 triangle and is 2000rc from the G.P.
centre. It is " the bass player " and it becomes involved in the
movement through several complexÊ"time signatures", as we will see.
Imagine the circle around the G.P. as the drummer - composer's sound
wave, expanding out from the central point. We also need to draw the
"sound wave" circle for the bass player, with the same radius, so that
he can be heard. The intersection at the western second step of G3B is
the centre where we place the point of the compasses and the GP. centre
is our radius for the "bass groove". It's a simple statement that stays
in the background but it gets the whole band dancing together. As the
"bass circle" is drawn its circumference passes through only two
significant structures, one "hit" is the centre of the Great Pyramid,
obviously, but the only other hit is the head of the Sphinx. Wow! What a
bass line!
Suddenly harmony appears, out of nowhere, as the "bass groove" circle
intersects with the "baseline", its natural "repeat sign". The pianist
joins in with a powerful theme, " Sokar, the Maestro", has appeared.
This is the "Gateway"; here is where all the equations coincide. We now
have the prime reference point from which to measure and confirm all the
intricacy of the composition. We have to draw its sound wave to be sure
the "piano" is in tune. Drawing the "Sokar circle" with the same 2000
radius back to G3B we see an amazing thing happen. First it passes
through the S.W. corner of G3, then the N.E. corner of G2, Khafre is now
becoming involved. The next hit is the eastern meridian point of the
G.P.'s 2000 radius circle. We have just constructed the Eye of Horus, or
Vesica Piscis," the fish shaped vessel ". To confirm that we have found
the right rhythm section, the Architect has left a wonderful note to be
played right here.
Sitting in the middle of the "eye" is Campbell's Tomb. Campbell's Tomb
is a shaft about 60 cubits (100 ft.) deep and about 20 cubits square. It
is made of massive construction blocks that could, virtually, never be
removed. The shaft could be filled in but its footprint would never be
erased. In the middle of the framing stonework on the top of the western
side of the shaft is an obelisk pit, the obelisk long since gone, but
its melody lingers on. This is the precise geometrical centre of the
eye. It marks the halfway point on the line that leads from the Great Pyramid to the Gateway. Halfway between Alpha and Omega.
Join the Dots
The physical location of the "gateway" is a major clue in itself as to
what may be found beneath the sand. The hill, Gebel Ghibli, is not very
impressive, until you stand at the "gateway". At that point, on its
northern side, it rises as a vertical cliff behind you as you align
yourself north to the eastern wall of the Sphinx temple. It rises
stubbornly from the debris that has accumulated over the centuries and
poses a question. Why was it not quarried away during the construction
period, why was it left with its vertical face?
All the pyramids of Giza have their entrance on the northern side. The
entrance passageway of the G.P. is at a 1 : 2 ratio angle, the same
angle as the Grand Gallery and the same angle as from the G.P. centre to
the "gateway ". All the Kings were buried in underground tombs, and it
has already been demonstrated (note) that Akhnaten, grandson of
Thutmosis IV, built his capital, Amarna, geometrically aligned to focus
on his tomb in the hills to the east, where the sun rises. We have every
reason to believe that his grandfather, Thutmosis IV, taught him the
principles of Atenism, the monotheistic belief in which God is represented by the disc of the Sun. But that belief has to incorporate
geometry as its fundamental model, as Akhnaten so definitely
acknowledged, but possibly he got a little too carried away with it.
Fundamentalism needs to be appealing to, and be understood by, the
masses. As in Plato's Republic, educating the population into conformity
has to be a gentle process; it must comprehend the reasoning behind the
"Philosopher King" and understand the societal benefits of a truly wise
leader. It appears that Akhnaten lost the plot. The model he was using
was the right one, historically, but he tried to implement it in his own
lifetime, he tried to run before he could walk, and take the glory for
himself.
After two millennia, Akhnaten tried to reinstate a belief system that
had been dormant in Egypt all that time. Why? Where did the inspiration
come from? We think his grandfather found Sokar and, if he did find
Sokar, what did it tell him? More interesting is who told Thutmosis IV
where it was. He was the one who cleared the sand from around the Sphinx
and erected the "Dream Stelae" between its paws. He is the one recorded
as having found, as a youth, a stone in the shape of a divine hawk,
which sounds suspiciously like an Omphalos stone. Was he shown an
ancient archive and instructed in the Truth, World Order, Maat? Did the
Knowledge get passed on to his offspring? Simon will elaborate on that
theme, but we can decide later, after we see Thutmosis IV assume the God
Form.....
Copyright © 2001 Kheraha. All Rights Reserved
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An Egyptian Geometry Lesson - Part Two
The Form of God
If geometry is the representation of the Cosmos then what is meant when the King "assumes the God form", does it mean he assumes a geometrical pose? It may sound a little too literal to presume that, until we take into account the philosophical aspect of sacred geometry and then take a look at the pictures of King Thutmosis IV. (Drawings 24 -32). Yes, the King actually did strike a geometrical pose for his portrait, and that
portrait alone describes a life that could only have been founded in the
strict self discipline and hard study of an initiate destined to be
King.
Frequently we read in the Egyptian texts that "the King became a star",
what does that mean? Usually it is assumed that the King's soul was to
take its place in the sky and literally " become a star ". However there
is a small visualization problem that needs to be addressed before I can
happily go along with that premise.
If you were asked to step outside and look at the night sky, and then
asked to draw a true representation of a star in the sky, what would you
draw? Let's assume that you had never before seen anyone else's version,
that you are the first person to ever do this peculiar task. Be
perfectly honest now, how many points would you put on your "star". The
truthful answer is none, stars don't have points, they're a white shiny
spot on a black background, no stretch of the imagination would take you
into the area of geometrical constructs that we colloquially call "stars".
Do you see the problem? We have lived with the image for millennia, it
was inherited. We see a symmetrical geometrical shape with a number of
points and we call it a "star", and assume it is representative of an
astral body. Is it more likely that "stars" are depicting the God form,
a mathematical rendition of the Cosmos, on the ground? The hieroglyphic
symbol for the Duat is a five pointed star enclosed in a circle, a
complex construction, as you are about to find out for yourself.
The Pentagram, the five pointed star, has been a geomantic symbol of
protection since the beginning of history, and was especially endemic to
Dynastic Egypt. In earlier times Egypt had held reverence for the
heptagram, the seven-pointed star, as we can see on the Narmer Palette
(note). The heptagram was also synonymous with Sumer. The "Book of What is in the Duat " guides the initiate to" build a replica, on the ground ", of an area of "the Sky " known as " the hidden circle of the Duat "."
Whosoever shall make an exact copy of these Forms, and shall know it,
shall be a spirit and well equipped both in heaven and earth,
unfailingly, regularly and eternally ". " The hidden circle of the Duat,..Êin the Body of Nut (the Sky ).. Whosoever shall make a copy... it shall act as a magical protector for him, both in heaven and on earth ".
Taken at literal value, an architect's plan is "in the sky ", looking
down on the elements of the picture in plan form. The plateau at Giza is
not level, yet its measurements are those of a flat surface, so
presumably the literal instructions are telling the initiate to make a
map first and then draw the geometries that reveal the hidden circles
and the Gateway to the Underworld. The body of the goddess Nut is always
depicted arched over, balancing on toes and fingers. (See picture). The
outline she forms with her belly and breasts is virtually the shape of
the hill Gebel Ghibli, as seen from the Sphinx. Nut depicts the path of
the stars through the night sky and she acts as a canopy over the tomb
of Osiris. This same hill shape, in geomantic lore, is representative of
Hermes / Mercury. (Pennick). Shall we "build a replica".
We have our framework, and the prime circles and vesica piscis,
constructed already, but we now need to add the other "hidden circles".
The "bass circle" need not be developed at this stage, it has made its
statement for now, but the Rostau and Sokar circles need completing.
Rostau consists, so far, of two concentric circles, one of radius "8",
our original circle, and one of radius "10". (1600 & 2000 rc). Sokar, so
far, has only one "10" radius circle. If we now draw the "8" radius
circle from the Gateway / Sokar centre it creates symmetry for the
"eye", or vesica piscis, formed by the interlocking "10" radii circles
centred on the G.P. and the Gateway.
Looking at the picture being generated we notice that Menkaure is
"floating" between the "8" and "10" circles of "Rostau", and its centre
is almost precisely halfway, or "9". As we mentioned earlier, its
precise distance is one cubit short of the distance to the Gateway from
the GP, 1,788.85 rc., call it 1,789 rc (sq.rt.5 x 800 ). It seems to
warrant a circle of its own, so we draw it with the G.P. as centre.
Symmetry demands we duplicate that circle from the Gateway centre, and
when we do just that we are led to the next hidden circle. Here's why.
That 1,789rc. (rt.5) circle circumference passes very close to the S.E.
corner of G3, the S.W. corner of G2 and then the precise centre of G1.
If the radius had been 11 cubits larger, 1800 rc., or "9", it would have
touched the corners of both G3 and G2, or rather, the "corrected" s.w.
corner of G2, which, if you remember, was the point that produces 1244.
The obvious move to make is to draw the "9" radius from the Gateway and
then duplicate it in the G.P. / Rostau circle. The "as built " measure
to the S.W. corner of G2 from G1 is 1220 rc., so, as a result, its true
position falls halfway between the 1789 and 1800 circles, just as the
"uncorrected" N.W. corner is a little short of the theoretical 2000 rc
from the Gateway, being 1994.9 rc. However, the designer left a lovely
note behind to say " don't worry, all is in tune". Here's how he did it.
The half diagonal measure of G2 is 290.65 rc. and the measure to G2's
centre from the Gateway is 1710 rc.. Added together they make 2000.65,
restoring the harmony, and please don't forget how many other equations
G2 is describing, and there are many more to be found as it unfolds its
manuscript. It has just given us the "9" circle, the mnemonic for the
squaring of the circle. Subtract one ninth of the diameter and square
the result ; diameter 9 minus 1/9 = 8 x 8 = 64. The "9" radius circle
has the same approximate area as our original 8 x 8 square and it's
circumference is equal to the perimeter of the square inside the 2000
circle.
* 8 x 8 = 64 *
* (64 -:- Pi. r. sq.) = dia. 9.02703333+ *
There is more than one purpose to the "9" radius circle, it is not only
a mnemonic for the squaring of the circle, it is the framework for
something far greater. But once again we are rushing ahead too quickly,
we must allow ourselves to be led down the path. All we have at the
moment is the "flesh" of Rostau, we need to give it "bones". The
pentagram is the symbol for the Duat and also the symbol of Man, it
therefore seems appropriate that we move from the square of the Tetrad
and build our "bones" from the "5" of the Pentad. The 2000 rc radius
circle was our original "5" radius, and it encompassed all except G3C
within its bounds. So we shallÊconstruct our pentagram within it.
(Construction method in appendix.)
If you were very good and read the appendix you now know how to divide the circumference of a circle evenly into five, it is not simple like
the perfect Hexagon, yet it does have a perfection all of its own, in
the Golden Section proportion. We will begin with one pentagonal
construction, although this will soon become four constructs, marking
out 20 points on the circle.
Once again, the architect has given a clue, and a short cut, to it. The
622.222 rc diagonal of the G.P. is 1/20th of the circumference of the
2000 circle. Setting the compass points on the diametrically opposite
corners gives the perfect division, but, for now, assume we missed that
"grace note" completely, it will be repeated again later, at double
forte.
If we begin at the northern cardinal point of the G.P. circle and divide
the circle into five points, the first "hit" is once again G3B. The
southwestern point on the circumference falls on that same second step
as does the corner of the 3-4-5 triangle, only this time on the southern
flank of G3B, not the western. The "bass player" has just turned up his
volume a little.
When we join the northern point to the S.W.. point the line passes
through the centre of Menkaure (G3) and a point on the western edge of
G2, that becomes important as we develop the pentagram. By continuing to
join every second point we build the first pentagram. The points which
are roughly N.W and N.E. are joined and form a line parallel to the
G.P's east / west meridian and north of it by exactly 618 cubits, it
forms the northern wall of Khufu's valley temple. This is one factor of
proportion of the Harmonic Mean, defined by its real number times 1000.
(.618 x 1000 = 618). By then joining each point in order we create a
pentagon around the pentagram. Notice that the "baseline" of the
pentagon runs alongside the original "baseline", it is 1,618rc south of
the G.P. meridian, another ratio of the Harmonic mean times one
thousand.
* 1.618 x 1000 = 1618 *
To establish symmetry once again, since the pentagram is asymmetrical, we now construct three more pentagrams and pentagons, beginning from each of the other three cardinal points, South, East and West. Now we have produced 20 points on the prime circle, 4 pentagrams and 4 pentagons. Menkaure and Khafre are beginning their harmonious duet, they are now going to be locked together in beautiful synchronicity, in 5/4
time.
Remember that the diagonal of G1 was 622.222rc, and that measure divides the circumference into 20 ? This is why. There is a classical system for squaring the circle that can be achieved by drawing lines through the points on either side of the meridian points. This is a reasonably accurate system, which produces a of the value 3.1752.
As you can see in the drawing, the line drawn from west to south forms
the precise diagonal of G3; it marks the edge of the square that has the
same area as the 2000 rad. circle. Amazing! But as we said, this is
merely a reasonably accurate squaring of the circle, and since we have
found this correlation it seems we should explore its accuracy.
If we calculate the distance along the S.W. diagonal from the G.P.'s centre to the midpoint of that side of this new square, where it is
bisected by Menkaure's diagonal, we get a nice surprise. Using this
system, and its value for Pi of 3.1752, that measure should be
1781.90909+, which is the square root of 3.1752 x 1000. (See 18a detail)
If G3's diagonal marks the perimeter of a truly accurate square then
that measure would be the sq.rt. of true x 1000, which is 1772.453851.
Using Petrie's measurements, as we have done so far, we now find another
wonderful message. The centre of G3 is 1411rc south and 1096.765.rc
west. Subtract 1096.765.. from 1411 and we have a horizontal
displacement from the S.W. G.P. diagonal of 314.2345 r.c., which is x
100. With a little trigonometry we can calculate G3's parallel
displacement away from the G.P.'s S.W. diagonal as 222.2 rc., and then
calculate that the intersection measure, which, for absolute precision,
should be 1772.4538, is actually 1773.25 rc. A difference from
perfection of 16 inches, if Petrie's measurements were correct, but did
you see the mnemonics.
* The centre of G3 is 411+1,000 south of G1*
* G2's base = 411 x 411.*
* G1'sÊvolume = 4411 x 64 x 64 * * Diag. of G1 = 622.2 x 2 = 1244 *
* 1411 minus 1096.765 = 314.2345 (Pi x 100)*
* sq.rt. 314.2345 x 100 = 1772.666 *
* 1244 squared = 1411 x 1096.765 *
* 1244Êis the measure from G1 to the S.W. corner of G2's "box" and also G3's height x 10 *
Before we continue, there is something else to examine about this square constructed through the points of the pentagrams and its ratio of
3.1752. The length of the side of that square is 1781.909 x 2 =
3563.818, which makes its diagonal measure exactly 5040, the ultimate
magic number. For those of you who have no idea what that means, here is
a clue. 5040 and its half value, 2520, are both divisible by all the
numbers of the Decad, and there is more, as usual.
* 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 *..........* 7 x 8 x 9 x 10 = 5040 *
It appears to be that Giza is the place where that magic number was
first used, and from which the pentagram square can be constructed, but
this is merely the opening of the Menkaure and Khafre duet, there is a
lot more happening in the internal geometries of the four pentagrams and
pentagons. The whole system replicates itself inside on a smaller scale,
(drawings 15-19) and produces another 20 points where the four 'stars'
intersect. One of these points falls on the western edge of G2 and
another falls on the 'box' built onto the southern flank of the Sphinx.
Having drawn a circle through all 20 points, we now construct another
square through the points of the internal pentagons, which now defines
the diagonal of G2. Notice also that each of the points used for the
internal square construction fall on the '5' (1000rc) grid lines. This
enables the calculation of the radius of this internal circle and its
area.
The radius calculates as 1051.5 r.c. (sq.rt. 1105656) = (1000 squared + 325 squared).
Another set of mnemonics to remember that number 1105656
* 110 = 440 -:- 4. *
* The diagonal of the 4000 x 4000 square is 5656.854.*
* 5656 = sq.rt. 32,000,000. *
*1789 (G1 Gateway) = sq.rt. 3,200,000. *
The Square Dance
Now the music is really getting hot and there are notes popping up in
every place we look. By our fine tuning of Khafre's pyramid and
constructing the concentric pentagram squares we have established that
Khafre's position defines the south east corner of a "box" which is of
the same area as the G.P. (440 x 440). It does not fill the "box" since,
in its 'adjusted' position, it is 29 cubits short on its north and west
sides. (440 - 411 = 29). However it does define the perimeter of the
inner pentagram square and this gives us a geometrical measure of the
parallel N.E. to S.W. separation of the diagonals of G1 and G2. This
becomes 933 r.c., three times the half diagonal of G1 (622 -:- 2 x 3).
932.33 is the frequency of E flat. Once again, if we look for
confirmation of our progress down the right path it can be found.
The internal pentagram square has a side measure of 1866 r.c. (933 x 2)
and an area of 1866 squared (3481956sq.rc.), which is precisely eighteen
times the base area of the G.P. and nine times the square of its
diagonal. (622 squared x 9 = 3481956)
Now watch this equation ;-
* 3481956 -:-20.6066 ( p" in 1 r.c.) = 411.063 squared (G2 base) *
There is a direct mathematical relationship to the 1866 square's area in both G1 and G2, and since G3 is the volume of G1 divided by 11.11111.,
we can check to see if it also bears a relationship. The base area of G3
is 39200 sq.rc. (196 x 200), so ;-
* 3481956 (1866 squared) -:- 39200 (196 x 200 ) = 88.8 *
We have been given confirmation that the 1866 square is what we are
supposed to find, each of the three pyramids bears a direct proportional
relationship to it. It appears there is a theme to be developed, and
there is still more inner structure to be explored within the pentagram.
We have only constructed on the first inner circle of twenty points and
that has given us the 1866 square, defined by the diagonal of G2.
The pentagram constructed from the north, on the first inner circle,
placed its S.E. point on the "box" on the southern flank of the Sphinx,
which we can now calculate as 618rc east of the G.P. meridian. (drawing
18b) We can calculate its position south of the G.P. by noticing that
one of the inner circle points falls on the western edge of G2, in its
"adjusted" position of 851rc west of G1, therefore the "box" is 850
south by 618 east of the G.P.'s centre.The angle from the G.P. to the
"box" is the pentagram angle of 36 degrees and the direct measure is the
inner circle radius of 1051.5. Since the "box" aligns with the central
E.W. axis of the Sphinx temple we can now deduce that the western edge
of G2 is the same measure west from the G.P.'s centre as the axis of the
Sphinx temple is south, 851 rc..
To complete the pentagonal geometry we now move to the "core" pentagon. This is the structure that appears inside the original singular
pentagram of the 2000 circle. Once again, this now has 20 points but the
radius of its circle is 764rc. (2000 -:- 2.618) When we draw a pentagram
from the eastern point of this circle its S.W. point marks the doorway
into Khafre's pyramid. In drawing 18b it is highlighted in yellow.
Once again we can plot the precise coordinates of this point as 618 west
and 449 south (618 sqrd. + 449 sqrd. = 764 sqrd) and place G2's doorway
as 21 rc north of G2 and 21 rc east of its central axis. And guess what,
that is exactly its location in the northern pavement of Khafre's
pyramid. (Edwards ; Pyramids of Egypt. p.133) At last we have
confirmation that the geometry indicates gateways.
By completing the entire internal pathways of the inner 20 points we see the next ensemble join the refrain. The violins have begun a high
harmony to complete the Menkaure / Khafre duet. The satellites of Khufu,
G1B and G1C, become "boxed" by the inner structure and are now playing a
major role, unfortunately G1A, the northernmost of the three, seems very
quiet. By now, logic should be telling us that it must have a part to
play in the grand design, the composer could not possibly let us down
now, and you're right, he doesn't. If we take our straight edge and lay
it along the N.E. - S.W. diagonal of G1A it begins a new theme entirely.
That line runs at 45 degrees from the pentagon point on the south side
of the N. East diagonal, through the center of G1A; the S. East corner
of G1; the S.E. corner of G2; the division point between granite and
limestone on the S.E. corner of G3, (see Ratios & Harmonics) and the
second step of G3B. It seems we have been here before, but now we know
why. G3B, the "bass player", is the "back sight". Different locations on
that second step provide perfect alignments to points on the horizon.
This line we have just located is the precise bearing to the obelisk at
Heliopolis, 17 miles to the N. East. A small hill to the south west of
G3b is the perfect place to stand to sight this angle, G3b and G3a
forming a 'notch' to align the corners of the main pyramids.
Because that line is parallel to the G.P. diagonal and touches its
corner, we know it is separated by 311 r.c. (622 -:- 2), which is one
sixth of 1866. Symmetry demands we join the dots. Now we have a square
around the diagonal of the G.P. and the 1866 square has been divided
evenly into nine (3 x 3) squares with sides of 622. The Khafre / Menkaure duet has produced the Ennead, the magical 1866 square, and now it is ending, to set the stage for the next movement, the Dance of the Spirals.
The Fibonacci spiral is the hieroglyph for 100 and forms part of the
Wedjat Eye, where it represents the fraction 1/32nd. It is also
incorporated in the crown of Lower Egypt (sketch). The spiral
construction is synonymous with the pentagram and the proportion of 1 :
1.618, so it is probably wise for us to look for its inclusion by the
composer, as the conclusion to the 5/4-time duet of Khafre and Menkaure.
That fine solo by G1A has led us straight back into the arms of the
"bass player", G3B, the fundamental seems to have been emphasised once
more.
We have completed the double decagon (20 pointed) geometry and now we have to find our way into the next movement, and the clue seems to
be,"question the bass player ". The bass player played one amazing
"lick" when he first joined in, metaphorically punching the Sphinx in
the nose before running into the arms of the pianist / gateway. It seems
that we have to question the Sphinx next, if a graceful, harmonious
transition is to be found.
The Spiral Dance
There is a system for constructing a spiral utilising the points of a
singular pentagram star. We, however, have four stars and many more
points to choose from as our starting point. The bass player was right
in his choice of the pianist, he could be right about the Sphinx. There
is an intersection at the "box" on the south flank of the Sphinx, as
there is also one on the western edge of G2, we need more clues. The
bass player was also saying this is the "fundamental", did he mean that
the clue is in the base grid?
If we mentally erase the pentagrams, or, more easily, look at the
drawings of the 20 x 20 base grid, we can begin to see a relevance to
the position of the Sphinx. We have to concentrate on the double square
within the "eye". One corner is the G.P. centre and the diametrically
opposite corner is the Gateway. It measures 1600 x 800 (8 x 4) and its
diagonal is 1789 ( sq.rt.3,200,000). Campbell's Tomb marks the centre of
the double square, and the vesica piscis.
The beginning sequence of the Fibonacci series " whirling squares" is ;-
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.
Looking at the "composition" of the double square we can see a rectangle within it that measures 3 x 5, and within that rectangle are all the major valley temples, Khentkawe's mastaba, Campbell's Tomb and the
Sphinx. The S.E. corner is the Gateway and exactly three units north of
that is the S.E. corner of Khafre's valley temple. It seems obvious that
this is the 3 x 3 square of the Fibonacci series, since the western edge
of it is the edge of Khentkawe's mastaba. Using the S.E corner of
Khafre's valley temple as a centre and the Gateway as a "3" radius, we
can describe a 90 degree arc. This arc is the link between the two
systems of constructing the spiral, the pentagram and the "whirling
squares". We need our spiral to sing a sweet melody if it truly belongs
in the symphony, and it begins by forming a "bridge" between all the
players.
Here's how the "bridge" is built. We are heading for the "box"
intersection on the Sphinx, because it is obviously a "logical" starting
point. But we didn't begin our construction from there, we used the 3 x
3 square in the 3 x 5 "golden mean" rectangle. Here's the anomaly and
how the composer solved it. Because the "3 square" is close to the
beginning of the Fibonacci sequence, to work our way back to the "start
point" we can only pass through the next three lower squares, one of w
hich measures 2 x 2 and then two which are both 1 x 1.
As you can see, we have lost the proportion, but it is still leading us
towards the Sphinx, we need a fine-tuning to complete a conclusive
proportional spiral. The solution is to increase the scale as we
approach the Sphinx. The higher numbers of the Fibonacci series provide
the true spiral ratio of 1 : 1.618 and they run thus ;- 55, 89, 144,
233, 377, 610, 987, 1597.
The 3 x 3 square measures 600 x 600rc., very close to 610 and close
enough to allow a transition from the base grid into the true proportion
of the pentagram spiral, and enabling us to pinpoint the precise start
point of the spiral, a small hole on the left hand side of the "box" on
the side of the Sphinx. Now the Sphinx begins its song.
The Song of the Sphinx
The spiral that begins at the "hole in the box" is left handed, it
rotates anti- clockwise, as does water spiral down the drain in the
northern hemisphere, this is probably not a coincidence, more likely it
is another aspect of the composer's knowledge of the Earth and Cosmos.
The Sphinx spiral takes us on a pathway through the elements of the
composition. Leaving the Sphinx it passes through, in sequence, Khafre's
valley temple the Sphinx temple Khentkawe's mastaba the western
doorway of Menkaure's valley temple and then the Gateway. This is the
first part of the spiral sequence and we have been led from the Sphinx
to the Gateway via the Valley Temples. The "bass groove" circle took a
more direct route, but that's bass players for you!
The last arc to the Gateway had a radius of "3" ( 600rc.), or 610 in the
higher Fibonacci ratio, and was centred on the S.E. corner of Khafre's
valley temple. To continue the sequence we must determine our next,
larger radius, which is a choice between using the lower number sequence
or the higher. The lower sequence numbers next in line above 3 are 5 and
8, which actually measure on the ground 1000rc and 1600rc.. The numbers
which are next in line in the higher sequence are 987 and 1597, so we
can see that our "8" measure arc, when we get to it, is within 3 cubits
of 1597, at 1600rc.
It seems we can happily continue with the lower sequence, 3-5-8-13-21, and utilise the base grid to calculate the radii of the progressively
larger arcs. The centre for the "5" radius arc is therefore 1000rc.
north of the gateway, and if the complete circle of this arc is drawn it
passes through the G.P.'s centre, the Gateway and a pentagram point.
(Drawing 19 red circle)
As we expand the spiral, the "13" arc intersects the diametrically
opposite point of the spiral's start point, in the N.W. quadrant, before
exiting the circle. Symmetry again says we must construct another spiral
to keep the sequence contained within the circle. As this second spiral
adds to the melody it touches the S.W. corner of G2 and the centre of
G2's satellite, G2A. It then re-enters the hole in the "box" on the side
of the Sphinx. Isn't symmetry wonderful?
We have two more spirals to draw, to harmonize with the four pentagrams. One begins on the western edge of G2, loops around and through its centre and then goes off through the N.E corner of G3C, the centre of our friend, the bass player, G3B, the S.W. corner of G3A, the S.E.
corner of Menkaure's valley temple and the S.E. corner of Khafre's
valley temple. The "13" arc passes through Khufu's valley temple to the
northeast. For total symmetry we can also construct the right-handed
spirals, and again these tie into the total harmonic picture. There is
no need to describe the rest of the picture, explore the one we prepared
earlier.
The Song of the Sphinx brings to a close the geometry of the Pentad and
the spiral and square dances that interlace it with such intricacy. We
now encounter a short passage in 6/4 time, a quick trip into the Hexad
that will provide a motif for the Finale and the Coda.
The Fifth Division
The six-pointed hexagon was intrinsic to our initial 8 x 8 square, its
geometries forming some of the base grid lines. We could have subdivided
our first square and circle by utilising its perfect harmony. But we are
now constructing our "stars" on the "10" radius circle and the Hexad is
not needed in building the fundamental framework anymore. So shall we
assume it has a statement of its own to make instead?
Six is perfection, the ultimate divider, a really cool sixophonist. He
has a very short, but precise, solo to play. He doesn't need to go on
and on like the Pentad, he'll impress you anyway. We need only two
hexagons and hexagrams to make the statement, and we begin in the East.
Using the 2000,1800 and 1600 radii we divide the perimeter of the
circles into six, from the eastern meridian point, and join the dots to
form a series of concentric hexagons. (drawings 10 &11)
The process is repeated from the north and this time the southern
"crossbar" in the '10' circle forms the southern wall of Khafre's valley
temple, one thousand cubits south of the G.P. centre and an internal
square 2000 x 2000rc., which interlocks with the internal pentagram
points that fall on the "5" (1000rc.) grid lines. Five and Six have
joined in harmony. The inner hexagrams delineate the Sphinx temple and
the boundary between the Sphinx and Khafre's valley temples. A diagonal
of the '9' circle intersects G2's centre and Campbell's tomb obelisk is
bisected.
There is only one more subdivision to make to complete the Hexad solo.
We have twelve points on our circle which can be doubled to 24 by
drawing lines through the mid - point intersections. As you can see in
drawing 11 the southern hemisphere has been divided into twelve from
east to west.This places the hill, Gebel Ghibli, and the Gateway,
completely within the fifth division, or fifth hour of the southern
night sky. (highlighted)
The twelve hours of the Duat place the entrance to Sokar, the Splendid
Place of the Beginning of Time, in the fifth hour. The Hexad was a brief
soloist, as we said he would be, but the theme established by him will
not be forgotten.
We have now explored the first six movements of the Decad, Monad to
Hexad and, by association, have encountered the Ogdoad (2 x 4 = 8) in
the Tetrad. The Ennead appeared in the 9 divisions of the Pentads
internal 1866 square and once again in the "9" circle, which is equal in
area to the 8 x 8 square. Essentially we have harmonized all the Decad,
but I assume that you have noticed that the "virgin" Heptad is missing,
c'est la vie, seven has to be searched for, but you can bet that she
will obey the rules of the game.
The Web of Athena
To misquote an old phrase, the fat virgin is about to sing, and the
show's not over until she's finished. She has been sitting at the front
of the orchestra from the start, in full view, but completely unnoticed
until now. Subconsciously, we all knew she had to stand up eventually,
her name is written in stone on the Giza billboard, seven has a page to
herself in the symphony program. We'll pretend that she's a little shy
until she begins to sing, but she has the voice of the Diva and knows
it. She can bewitch all who hear her aria.
So you want to know where she was sitting that she should be so
unnoticed ? She sat quietly between the square around the "9" circle and
the "10" circle itself. Where the edge of the "9" square bisects the
"10" circle a chord is produced which has the beauty of being one
seventh of the circumference of the "10" radius circle. We can set the
compass points on both intersections of the chord and produce seven
points around the circle, each line of the heptagon being tangent to the
"9" circle.
It is the legendary "seven lengthed circle", the Ennead embraced by the
Virgin. It ties together the Ogdoad "8" square, the Ennead "9" circle
and the Decad, "10", Cosmos framework. Remember the mnemonic ; 7 x 8 x 9 x 10 = 5040. The magic seven has just begun to sing.
She knows more than one variation on the theme of constructing the
Heptagram, we have heard the opening lines and the first word was Seven.
She came straight out with it, not shy at all, but she was also very
secretive about the depth of her virtuosity. The first seventh division
was accomplished by using the compasses to measure the "chord", and then
stepping around the circumference to create seven points, not an
entirely accurate method, but it sets the mood. The secret of the Virgin
is that she can provide all seven points with only the aid of the base
grid, in fact, she can produce 28 points (4 x 7) by merely joining the
dots within the framework. Don't fall asleep now folks ! This is where
it starts to get really exciting.
We shall listen to the Diva sing the opening lines again, with the
understanding that she was merely teasing us with her innocence when she began. The Khafre - Menkaure duet was an intricate piece and, for now,
they will remain silent, we don't need their geometry until the Finale,
when the whole ensemble joins in again. We shall return to the rhythm
section, the base grid and circles, and watch Athena weave her web.
We begin at the eastern cardinal point (E), and we know that on the
opposite western side of the circle, the vertical "9" grid line bisects
the "10" circle to produce two points, one seventh of the circumference
apart. We now draw the line C - C, which becomes the baseline of our
eastern pointing heptagon, and connect each intersection with the '10'
circle to point EÊto form the eastern pointing triangle. We still need
to locate four more points of the seven, and it is extremely easy. From
point 'E' we draw two lines that intersect the G.P. north - south
meridianÊat the "8" points (A)north and south. We extend these lines to
intersect the "10" circle and create points K & K.
Athena now hits her first high note, the line drawn from K to K defines
the eastern edge of Khafre's pyramid and is 440rc west of the G.P.
meridian and parallel to it. The angle generated by E - A - G1 centre is
51*20.2' (tan. 1.25); this is only 5.5 minutes of arc short of the true
seventh division angle of 51*25.7' (360*-:- 7), an accuracy of one in
four thousand.
On the ground at Giza that amounts to only 3.2 cubits error in the
placement of the point on the circle with a circumference of 12566rc. A
man with his arms outstretched would cover the actual discrepancy over
the 4 mile / 6.5 Km circumference Remembering that on our human drawing scale the pencil line is 7 cubits wide. The N.E. corner of G3C, however, marks the exact seventh division ratio on the 2000 circle. Can you see how precise is the tuning of the irrational Diva ? So far she has sung five notes ; A,C, D, E & F., she has two more to sing to complete her
soprano's octave.
We follow the same formula to find the remaining two points. From the
G.P. centre we move north and south to the edges of the "10" square,
then we move 8 units east along the perimeter to the points 1600rc east
x 2000rc north and south, points B - B, and repeat the same 4 : 5 ratio.
A line drawn to these two points from the G.P. centre bisects the circle
to complete the seven points of the Heptagram. The octave is complete
and now we can proceed to create the Web of Athena and test the virgin's
expertise.
First we connect each point in sequence, to create the heptagon,
wrapping around the "9" circle. There are two types of heptagram stars,
one formed by connecting every second pointÊand the other by joining
every third point.
Once the points are in place it is possible, with practice, to construct
the web with one continuous line. The soprano can sing the first verse
in one breath. She has created an inner heptagon with an inner circle
diameter of 880rc and an inner heptagram with an outer diameter circle
of 1414.21rc, in other words, the sq.rt.2 x 1000. What a voice!
A short digression is required at this stage. 1414.21 has been integral
to the picture from the beginning, as it is one quarter of the diagonal
of the 4000 x 4000 square, 5656.854 (sq.rt.32,000,000) The square inside
the 2000 radius circle has sides measuring 2828.427 ( sq.rt.8,000,000),
and consequently the half base is 1414.21 (sq.rt. 2,000,000). The
distance from the G.P. centre to the southern side of Menkaure's
causeway is 1414rc, the causeway therefore defines the edge of the
square inside the 2000 circle on its south side, and the 1400 ("7") grid
line on its north side. The 1414 diameter or 707rc radius circle has a
circumference of 4444rc, another nice mnemonic. The same measure also
appears between the G.P. n.-s. meridian and the western boundary wall of
G3.
Symmetry has to be satisfied once more to reveal the true nature of the Heptad. We duplicate the construction from each cardinal point to create 28 divisions of the circle and four complete webs. It takes great
concentration and patience, but the effort is worth it, the score for
the Song of the Virgin is a beauty to behold. The square that appears in
the centre measures 880 x 880, it is four times the base area of the
G.P. and 2/9ths. of the area of the 1866 square inside the Pentad.
The diagonal is 1244 r.c. and consequently the same measure as from G1 to the S.W. corner of G2's theoretical 440 x 440 "box", which now
becomes integral to both pentagonal and heptagonal geometries. Notice
that Khafre's causeway is now tangential to the 707 radius circle on its
northern side and the southern edge is aligned to the central "5" point
of the S.E. quadrant square.(dr. 03) The angle of G2's causeway is a
ratio of 1 : 4, or 14* 02' south of east.
The Heptad can be subdivided beyond 28 points into 56. It provides the
geometer with tremendous angular calculating possibilities and, if the
geometry is limited to connecting every second point of the individual
heptagrams, a pronounced circle is described by the construction lines
and it has a radius of 1244rc..
This circle, as we know, touches the S.W. corner of G2, but it also
coincides with the centre of Khentkawe's mastaba and one other curious
place. In the internal eastern wall of Khafre's valley temple is one
huge, black granite construction block. It is in the base course and is
slightly proud of the surface of the rest of the pink granite wall. In
our first construction of the Web of Athena the line drawn from point E
to A to K in the southern hemisphere passes directly through the black
granite block, which is therefore 1244rc from G1's centre.
Nobody has yet explained the significance of the black stone or why it
is not flush with the rest of the structurally perfect wall. It appears
to have been slid into place at a time after the temple wall was built.
Is this one of the Virgins secrets ? After all, it is this same eastern
temple wall that lies precisely on the northern meridian of the Gateway.
Khafre and Menkaure now join the melody of the Virgin, as we overlay the
Pentad geometry onto the Heptad. The intricacy of the harmony is
emphasized by the bold interlocking of the 440 and 880 vertical squares
of the Heptad with the subdivisions of the 1866 diagonally aligned
Pentad square. The interlocking of the geometrical harmonies is now
complete, though the song is not over, the tenor has been waiting for
his moment to sing. The harmonic structure has led to this moment, the
last movement of the Symphony, Menkheprure makes his dramatic entrance
to the stage and begins the Finale sequence.
The Dream of Menkheprure
Menkheprure became King Tuthmosis IV because he had a dream that he
would become King. Sounds like the stuff of myth and fairy tales and yet
it is part of Egypt's history. Little is known about him other than the
fact that he erected the "Dream Stelae" between the paws of the Sphinx
to commemorate his dream, and that he was the grandfather of Akhenaten.
And then we have his portrait, his song that was last heard 3,500 years
ago. His melody has been available for quite some time now, but nobody
has found the orchestra with which to play it, until now. That same
orchestra has now reformed for the first time in history, and it's all
the original cast. His portrait was found overlaid on a grid of squares,
but see what happens when it is placed in the right context, the Symphony.
Menkheprure takes the first God Form with his head to the North and
facing the Eastern horizon, the place of rebirth. In this position he
becomes the King,and demonstrates his power. His ring is the G.P. centre
and Khafre's pyramid becomes the "Stone in the Throne". His flail
intersects the head of the Sphinx and terminates at the southern wall of
Khafre's valley temple. The throne stands upon Menkaure's causeway and
mortuary temple, with the plinth terminating at G3B and the western edge
of G3.
His right foot arches over Menkaure's valley temple and his Achilles
tendon is protected by Khentkawe's mastaba. The crossing of two pentagon
lines forms the bow in his headdress and a diagonal of the Heptad
terminates its "filet".
(drawings 30 & 31)
His elbow rests on the Great Pyramid.The proportions of his face, the
bend of his arms and legs, are all determined by "the Canon" and "bones"
of Rostau.
In the second God Form (drawing 26) he faces the West to consult his
ancestors for their wisdom, his head is still to the North, and his ring
stays as the centre for each of these rotations. In this form his staff
touches the N.W. corner of G2, rests on theÊcentre of G3, and is tangent
to the 1414 diameter circle. (Drawing 31) The flail is a heptad line and
intersects G2's centre, his toes touch G3 and its mortuary temple. G1's
three satellites form his erect phallus and its causeway forms the line
of his belt. The inner framework of the throne encloses the Sphinx and
it's temple, then turns south at Campbell's Tomb to terminate at
Menkaure's valley temple. In each rotation the bow in his headdress will
be an intersection of the pentagon lines.
The third form he takes is with his head to the West, facing South.
(Drawing 27) It is the hour before the Equinox dawn. The tip of his
staff aligns his eye along the wall of the Royal Workshops to the
southern morning star, Canopus, pinpointed by G3c. The staff hand rests
on G2 and the staff itself forms G2's causeway. G1's valley temple sits
inside the plinth and the flail terminates at the 1244 circle.
As Menkheprure takes the fourth God Form his head is to the South and he faces the East. (drawing 28) The King holds the Sphinx in his hand and
his staff forms a cross with G2's causeway, then touches G3's valley
temple with its tip. Khafre forms his heart chakra and its satellite
mastaba his throat chakra. Menkaure's mortuary temple is the crown
chakra and the causeway forms his headband and the line of sight of his
"third eye" to the equinox sunrise, through Menkaure's valley temple. It
is the beginning of the King's enlightenment, he faces Sokar and
judgment.
In the fifth and final form his head is to the South and facing the
West. (drawing 29) He kisses the Tomb of Khentkawes and again G1's
satellites become his erect phallus. The fingers of his staff hand rest
on G2 and its satellite, his elbow still rests on G1. The Sphinx is his
throat chakra, his voice. Menkaure's valley temple becomes his pineal
gland, his "third eye", and the causeway is the line of its sight to the
equinox sunset, behind Menkaure's pyramid in the West, the domain of the
dead. The Gateway becomes his Crown chakra, and now that he has honoured his Ancestors his spirit is free to enter the underworld to await
judgment and rebirth.
The song of Menkheprure has described the entire philosophy of the
Architect Archetypes, he has assumed the God Form in the Duat and left
no note of his melody unplayed, except the Coda, the exit from Rostau
that can only be the entrance to Sokar.
The Coda sign is the only piece of the Giza jigsaw puzzle that has not,
so far, been included in the melody. It will restate the original
problem of,"where do IÊstand to see a right angle at the apex of the
Great Pyramid ?" It ends the Symphony with its beginning. There is one
wall to the south of Menkaure's complex that is not parallel to anything
else on the site, it is at an angle of 7*07' south of east, a ratio of 1
: 8. None of the geometries have included it within their frameworks,
so, it is time to strip away the entire fabric of the Decad and reveal
the illusion that was the fundamental question, and then we can ascend
into the sky and another dimension.
The Coda, at last.
In most musical compositions the Coda will frequently return the
musician to the beginning, to restate the introduction once more before
the Coda sign moves him to the final bars that make the closing
statement which resolves the composition. To most musicians, the final
bar is a little place on the corner, within thirty seconds of the stage
door, but that's another story.
Giza is no different in its musical structure, we have to re - examine
how we began, with all its simplicity and understated complexity. If we
draw a picture of the Great Pyramid with a right angle at the apex,
where do we draw the line that leads to the third corner ? That's how I
began this visualization process, so let's go back to it. We know where
to stand to see it at Giza, the Gateway, and we found that in the base
grid and its three interlocking "10" circles, the "Triskele".
So we take the first step again, with the 8 x 8 grid.
The southern diagonals of the G.P. are extended to the lower corners of
the prime 8 x 8 square, the "baseline". Now we have a pyramid with a
right angle (90 degrees) at its apex, represented by the G.P. centre. We
now draw the line to the Gateway, and we have drawn our vertical plan
elevation of the G.P. as it appears when viewed from the Gateway itself.
There it is, a simple problem solved simply. Does this mean it's over,
can we close the music and head for the pub ?
You should know better by now, remember the bandleader in hell, that's
how he got caught. We still have the problem of Menkaure's enclosure
wall to solve. Look again. We lay the straight edge along its length and
see where it goes. If we extend its line westwards it would intersect
the baseline at the western cardinal point of the "bass circle" centred
on G3B.(not shown) It intersects the G.P.'s south western diagonal at a
point 2520 r.c. (5040-:-2) from the G.P.'s centre. Extending it
eastwards it coincides with the southern cardinal point of the Rostau
circle. Now its purpose is becoming obvious, the composer is painting
his final illusion three dimensionally.
We continue extending the enclosure wall line to the East, moving out of the Rostau geometry and into Sokar territory, south of the Gateway. Now we extend the G.P - Gateway 26.5 degree line until it intersects our new line and we have drawn one face of a three-dimensional pyramid. A
45-degree diagonal drawn from this new intersection to the S.E. corner
of the 8 x 8 square completes the illusion, almost.
The composer saved the sweetest melody for the end. The new, extended part of the G.P. Gateway line is exactly one third longer than it was originally, therefore the extension equals one quarter of the total
length back to the G.P.. Remember that "one pyramid unit " that equals
one sixty-fourth of the volume of the whole? Come on now, the mnemonics
weren't there for nothing !! Yes, that's right, it is one quarter of the
height, but nobody said it had to be on the top of the Pyramid, it can
just as easily become the cornerstone, the one the builder rejected.
(drawing 36) So we draw the diagonals south from the Gateway to the
baselines of our 3D pyramid and build the cornerstone pyramid, whose
apex is the gateway and whose body is the hill of Gebel Ghibli, the
missing 1/64th. has reappeared, and we see it from above.
The Eleventh Division.
From the Gateway to the centre of G2 is a precise angle of 32.7272..
degrees, one eleventh of 360. Eleven is the master number, one beyond
the Decad. G2 is triangulated precisely to the Gateway. From the
northern cardinal point to G2 is 1 radian, that section of arc equals
the radius of 1710 rc. . Multiply that number by 11 and it gives the
circumference in Egyptian feet (1rc-:-7 x 4 = 1 foot of 11.7752 ins.).
That circumference measures 18810 ft, the Grand Gallery measures 1881
inches. The measure to G3a, 1900rc., multiplied by one eighth of G3's
perimeter (99) = 188100. The length of the year between spring equinoxes
(365.2424 days) x 1.881 = 687, the length of the Martian year. Is this
the music of the spheres or what!
When the 11 geometry is drawn in the Rostau circle once again the
architect has left his mark and shown how to construct eleven by sacred
geometry. The intersections of the 5 x 7 rt. triangle and the 5 x 4 rt.
triangle, basis of the Heptagon, provide the points and the two
horizontal connections box G3 with precision. The connecting line
between the intersections of the construction triangles bisects the
centre of G2. (Drawings 33a, 33b, 33c)
At this very moment, the Tympanist, G3C, is stubbing out his cigarette,
he has just heard his cue. See, there he is, creeping in behind the
cellist and the bass player, his measure is coming. The sun is beginning
its descent into the west, the shadow of the cellist, G3A, has begun to
play her closing theme. She has another secret to disclose before the
equinox sunset.
Behind the cellist the bass player G3b is bowing a sonorous fundamental, he still stands between melody and time, as he has so capably demonstrated throughout the Symphony. Now he calls out with his true measure. His centre is 1183.216 r.c. from the G.P. meridian, the square root of 1,400,000. 11.8316 is the root of 140, half the height of the G.P. and half the diagonal of Menkaure. His western edge is 1213.2 west of the G.P. meridian, 25,000 inches, 1000 Sacred Cubits, one ten
thousandth of the polar radius. Now, in the distance, the start of the
Tympanist's crescendo can be heard.
The measure of the one who sits outside ends our symphony of numbers.
The second step on the eastern side of G3C from the Gateway is 2060.66
cubits (sq.rt. 4246319 ). 2060.66 is also the number of inches in 100
rc., one side of the Setat or Aurora. (100 sq. rc.). 2060.66 cubits is
42463.19 p.inches, 20.6066 p.inches are in one royal cubit. 42463.19rc x
1800 is the circumference of the planet (24,880 miles). The centre of
G3C is 2069r.c., the radius of a circle of circumference 13000rc., half
the precessional cycle of 26,000 years. The tympani roll is reaching its
zenith, the sun is approaching the horizon, to enter the Underworld.The
last note resounds from the cello as the Sun descends into the abyss,
the Symphony ends. God is in the silence.
The shadow of G3A is now 1900 cubits long and touches the Gateway, the shadow in the indented face of the Great Pyramid is disappearing, the
tenor kisses the Virgin and waves goodbye to the audience and orchestra.
The orchestra members close their music and head for the pub, leaving
their instruments behind, nobody will steal them, they never do,
besides, they'll be back shortly to play it all again, in another 13,000
years,.. but that's show business.
Seventeen
As the dusk of the equinox gathers over Giza there is one statement left unplayed. The composer sits alone, smiling to himself. He knows how
impressed the departed audience are and he has the satisfaction of
knowing that they never got to hear his greatest composition. The line
from the centre of the Great Pyramid through the Gateway made the most
remarkable statement as it passed out of the '10' circle. (drawings
36-40)
From the Eastern cardinal point of Rostau (a) to that point on the 2000
circle (b) measures 3/17ths of 360 degrees, from that point to the
corner of the 3-4-5 triangle at G3b (c) is the next 3/17ths. By
continuing around the circle in 3/17th steps the compass points
eventually fall back on the Eastern cardinal point having divided the circle into 17ths. Seventeen is a division of 3.1416
http://mathpages.com/home/kmath487.htm
The mathematician Gauss discovered a method of constructing the
Heptadecagon. He was so proud that rumour has it that he wanted it
carved on his tombstone. It is a very clumsy method and does not provide
an accurate division. The Pyramid composer left the perfect system
behind on the music stand and it is accurate to 3 inches in 4 MILES. All
it takes is the double square and the 3-4-5 triangle.
The angle of G3 is 1/12th + 1/17th of 360
You'll have to travel a long way before you will ever hear that sweet
melody played. Stonehenge is exactly 1/17th. north of Giza, which is
1/12th north of the Equator. I'll see you there later, just as soon as I
can get the band out of the pub. And you thought it was all over, ha !! I hope you have a sense of humour, it's only just beginning, remember,
this is HELL.
Everything!
Copyright 2001 David Alan Ritchie
Copyright © 2001 Kheraha. All Rights Reserved Website designed by Artifice Design
Just as all colors can be derived from a mixing of the 3 primary
colors, so can all forms be derived from the progressions and ratios of
3 primary lengths of unity, square root of 2, and Tau, also know as the
Golden Mean. ÊThese primary lengths can be combined into 7 primary
elements which then can then be ordered into 5 female-male progressive
pairs with and additional non Tau pairing. The above drawing is of what
might possibly be the "Fire Crystal of Atlantis".
Diagram- pyramid2.gif
The reason for this hypothesis is that the pyramid of Geops, also
sometimes considered a relic of Atlantis culture also imbodies the same
3 elemental length combinations. These combination of elements form the
basic Platonic elements of the Great Pyramid as shown below. Platonic
elements:
"The simplest solid would be a pyramid that consists of four
triangular surfaces" (written by Plato) The Great Pyramid can be broken
down into 8 basic Platonic elements.
Musical Scale based on Cheops Pyramid
The significance of the musical scale based on the Cheops pyramid is
that Plato, who studied the writings of Pythagoras as well as the
culture and science of Egypt, mentions Atlantis. In Platos time, music
was one of the principal sciences. A 12 note scale can also be derived
from the Nature's elements. Primitive people generally use a pentatonic
scale, a 5 note scale.
The 12 structural elements of nature are composed of structurally
stable, 6 edged, 4 sided pyramids. The internodal lengths all being of
proportions and progressions based on the three lengths; unity, Tau
(((square root of 5) plus 1)/2), and square root of 2. These elements in
turn can be combined to produce larger element shaped compounds with the progressive sizes being of the fibonacci series. These elements can then be used to form the Platonic Solids in progressive sizes in line with
the fibonacci series.
Elements of Structures
element6.gif
PART TWO: HOLOTOMES & THE HOLOTOMIC SEQUENCE
Syndex Number Theory
SYNDEX NUMBER THEORY
A. Universe is inherently resonant. Resonance is a complex of
intertransformative frequencies of miniintertensioned systems.
B. The inherent resonance of Universe is caused by nature's never
pausing at, and only forever transiting, exact equilibrium.
C. The union of Universe is a differentially complementary regenerative- production wedding of inherently, uniquely prime numbers 1, 2, 3, 5, 7, 11, 13, and all of their successive primes. The prime numbers are numbers divisible only by themselves and by 1, representing in
syergetics unique system behaviors.
D. The prime numbers impose an eternal disquietude--transformative
adjustings and omniintertensional resonances eternally
interaccelerating.
Fuller, Cosmography (1992)
Eliciting the structural profile of the "BASETEN" system of
quantitative notation naturally comes under the jurisdiction of
geometrical display. Since an enfoldment occurs at the end of the base
digits when numbers start to be empowered by the occurrence of the
second digit (ten), we are inclined to enspiral number about a circle of
ten radial increments, but this strategy does not work. We are left with
a radial array that is void of any significant patter: 10 - 20 -30 - 40
- etc.
This uneventful cyclation of numbers produces nothing remotrely
suggestive of a "base wave" that would manifest a synchronicity between
number of a mutual class such as primes or square numbers which, when
played out in a linear sequence show no interactive relationship of any
significance.
The same goes for the enspiralment of nine except that when we look at the axis representing 9, we note that numbers reverse between 45 and 54 and go backwards. Until 81 is reached that is the reverse of 18 the
first multiple of 9. This is a solid clue and one that leads to the
first and genuine synchrograph, or a graph that synchronizes the
fundamental classes of number in a highly meaningful context.
When we use the 12-fold clock face or zodiac forms, we are overwhelmed with a series of synchronicities that deal with primes, composites and squares.
First we discover that the primes fall in only four of the 12 possible
axial locations: it means that eight axes are prime free. We've taken
our first step to numeracy, the scientific discipline that we call
synchrographics. This, in turn, legitimizes the term numeronomy as a
genuine scientific discipline in that it has produced a systematic
quality.
The synchronicity of quality and quantity (number and form) is the gist
of numeronomy, which is the proliferation of the Holotomes, or
symmetrical cicular unities which are also the series of minimal sums
which accomodate the maximum amount of consecutive factors of division.
The sequence of all sequences is the series of circular unities that are
generated by doubling twelve, then by multiplying the sun by three, and
then by multiplying each resultant sum by the next prime as the primes
occur in their natural order:
[Insert:THE NUTSHELL DOCUMENT]
This sequence, aptly designated as the Holotomic Sequence, represents
the minimal plural quantities that accomodate the maximum amount of
consecutive factors. Anyone who has ever pondered the source of 12 and 360 by ancient metrologists will investigate the total rationality and serious implications of the foregoing statement.
Dr. Fuller's superscheherizade number of 71 integers is vastly
superceded by the holotomic sequence whose 16th member is divisibile by
all of the first 16 primes with merely 19 integers: 7378677391061900000, (52 less).
And I am not a fraction of the mathematician that Bucky was -- just a
better numeronomist.
This is the primary member of an exemplary pattern of finite (or
transfinite) moduals of circular unity that each contain all of the
informational data of the preceeding modual plus one additional
information datic identity. They retain original information while
adding new information into the original context without sacrificing any
of its significance or meaning.
NUMBER 12: Holotome A contains four base digits: 2, 3, 4, and 6. (One,
singularity is not a number in the same sense that 2, 3, 4, 5, 6, 7, 8,
and 9 are numbers. These base digits are componentational unifications
of the magnitudinal identity of singularity (One).
NUMBER 24: Holotome B contains five base digits: 2, 3, 4, 6, and 8,
which increases its permutational capacity by one more degree...
NUMBER 72: Holotome C contains six base digits: 2, 3, 4, 6, 8, and 9
which again increases its permutational capacity by another degree...
NUMBER 360: Holotome D contains seven base digits: 2, 3, 4, 5, 6, 8, and 9, which still again increases its permutational capacity by another
degree...
NUMBER 2520: Holotome E contains eight base digits: 2, 3, 4, 5, 6, 7, 8,
and 9. This raises its permutational capacity to the limit of the
base. In that one, singularity is not a number, Holotome E is regarded
as the Auric Node. This is, in every sense of the term, a full octave,
and totally synchronetic with the NINELEVEN RETROGRADE OCTAVE, which
also being of an exemplary nature, synchronizes via 4 forward and 4
reverse event octaves with a 9th null event.
Definition of Mathematics with Description of the Holotomic Sequence
Ordinary number, the psychological mechanism which we commonly refer to as the base ten concept of qualitative notation, is the ultimate
foundation of science and technology. In order to state this proposition
in the proper perspective, we must understand that number and geometry
are two interdependent aspects of a singular mechanism and that the four
fundamental operations of arithmetic are also an intrinsic component of
these interactive disciplines. They cannot possess an independent
existence of their own without the presence of each other in a totally
unified context.
This is to suggest that base ten number, which also involves the zero,
geometry and the fundamental operations of arithmetic are a structural
complex of interactive functions. Taken all together, they are a
premathematical compound multiplex. Through the interactive mechanisms of their interdependent composition, they permit those secondary operations that we refer to as mathematics.
The extensive realm of mathematical disciplines cannot occur without the complete aforementioned complex of synchronetic functions that operate as a holistic system of geonumerical mechanisms.
It is precisely through this descriptive syntax that we draw the line
between the tools we call number / geometry / arithmatic and the secondary operations we refer to as mathematics.
The common definition of arithmatic calls for four fundamental
operations: addition, subtraction, multiplication, and division. A more
exact regard tells us that only two fundamental operations exist, since
addition is reverse subtraction, and no multiplication can occur without
division.
This perspective reduces the actual complexity of the componentry of the basic numerical and mathematical tools.
We note that a triangle is a qualitative expression of the quantitative
number THREE, and likewise a square is a qualitative example of the
quantitative number FOUR. Number and geometry are two aspects of the
very same physiomathematical function. One cannot exist independent of
the other.
In practice, however, in the discipline of mathematical operations,
these distinctions merge in such a way that the line between the
geonumerical tools and the mathematical products of those tools become
merged in such a manner that number, geometry, and mathematics appear to be inextricably united in a common field of inseparable fusion. With
such a perspective, it is nearly impossible to analyze their actual
interrelations.
For the sake of a dialectical analysis, we will suppose that number and
geometry, the two ambidirectional operations of arithmatic, are
premathematical. All of the varied operations of mathematics are the
secondary products of the geonumerical operations of arithmatic.
In consideration of the foregoing suggestions, we have engendered a
syntactical format. We have a clearly finite array of terminological
componentry of physiomathematical epistomography. It successfully avoids the unspoken levels of distinction that confuse the dichotomy of quality and quantity that emerge in the discriptions of all mathematical
definitions. They involve the separation of number, geometry, and the
operations of mathematics.
The primary confusion that occurs in the synchronization of number and
geometry is due to the fact that the first three signs for plurality
(one, two, and three) do not represent spatial geometry at all. It is
not until we reach number four that spatial geometry becomes involved in
the form of a tetrahedron, as the ideal example.
In the first place, one is not even a number in the sense of the other
eight digits. One is not unity, having no componentry to unite -- it
must be deemed as singularity.
Number two may, at best, represent a line which cannot involve
volumetric space.
Three, on the other hand, may only represent a plane, its ideal form
being a triangle.
Four, then, is the first of the base digits that can represent spatial
geometry with the ideal form being a tetrahedron.
This means that only six of the base digits can deal with spatial
geometry: 4, 5, 6, 7, 8, and 9. They are points of reference in
geometrical space.
SIX is the first perfect number in that one plus two plus three
represents the harmonious interaction of the first three number
signs. Six also represents the three ambidirectional axes of the TRI
which denotes the six directions of spatial geometry.
It is through this logic that we begin the true synchroncity of number
and geometry.
The T.R.I. is the cornerstone of real numbers that formulate the
geonumerical series of spatial forms that produce an exemplary series of
geometrical forms that synchronize numerical values with geometrical
forms that involve numerically synchronous spatial integrity.
The Holotomic Sequence begins with the dodecahedron or twelve spheres that surround a 13th nuclear sphere, producing the first quasi-spheroid of six square windows and eight triangle windows.
The Holotomic Sequence begins by doubling the first perfect number and then multiplying each resultant sum by the series of prime numbers as
they naturally occur.
1 x 2 x 3 = 6
A x 2 = 12
B x 2 = 24
C x 3 = 72
D x 5 = 360
E x 7 = 2520
F x 11 = 27720
G x 13 = 360360
H x 17 = 6126120
In this way the minimal sums with the maxiumum amount of consecutive factors are produced. Each represents both a planar and cubic mode of a harmonious and symmetrical retrograde octave unity which synchronizes number and geometry in a qualitative and quantitative geonumeric modual (dual module).
No other sequence can accomodate these aspects in an order of
mathematical exactitude.
The Holotomic Sequence represents the series of minimal numerical sums that accomodate the maximum amount of consecutive factors of division.
The science of mathematics, the alphanumeric system of notation and
industrial civilization, in general, has resulted from the metrological functions derived from this sequence.
The transpalindromal function of the exemplary 99 retrograde octave
wavecycle accounts for the synch / desynch mechanism of the continuity of the baseten system of quantitative notion of which the discipline of
geometry is an interdependent ambifunctional unification of number and
form.
To recap, spatial geometry does not play a role in number until the
number four is involved. Three can only represent a surface or plane
with the triangle as the minimal regular form.
Number four may represent spatial volume in the form of a tetrahedron as its minimal regular form: 4 windows and 4 vertices. So, already we have a half positive and half negative octave situation: windows opposed to vertices.
This means that only six of the basic numbers can involve spatial
volumetric geometry.
The next important number in spatial geometry is twelve.ÊThis occurs
through the dodecahedron (an irregular solid by Platonic description),
which evolves from the closepacking of 12 spheres about a 13th nucleus
sphere. This produces six square windows and eight vertices. This also
produces #14 as a geometrical factor (2 x 7).
The dodecahedron, then, can represent a quasi-sphere. In constrast to
the tetrahedron, (which is the most surface for the least volume), the
step to the dodecahedron is a great volumetric increase.
Ignoring the regular solids of Plato and considering the whole range of
Fullerean geodesic configurations with a distribution of pentagons and
hexagons we find a series of quasi-spheres, which become more spherelike as they grow in size / number. Still, they will never achieve perfect sphericity, in nature or the abstract.
When we consider this sequence of geodesic spheroids, (or quasi
spheres), we are struck by an analogy that occurs in number sequencing.
The Fibonacci Series grows irrationally until the 17th member (6180339), then stabilizes for the first eight digits and continues on. It is at that point where, when represented as a spiral, it grows in size but
does not change in shape.
Number sequences that are deficient shrink faster than they grow,
reaching a sort of circularity while their counterpart proficient goes
wild into an arch that approaches linearity, (but never quite attains
it).
The Holotomic Sequence is totally different than all the other
sequences. It becomes somewhatÊerratic in its initial phase especially
the first three members which are two digit: 12, 24, and 72. Then it
grows one digit at a time until its 8th member: 6126120, which is seven
numbers long, then it skips to a nine digit number, (omitting an 8 digit
Holotome) From then on, the series is very slightly dificient and skips
over an integer length every so often..
The remarkable thing about the Holotomic Sequence is that the hard
digits quit at fourteen positions and continue to build on with
accumulative zerosity. That is, after 14 hard numbers, we get nothing
but zeros.
The Holotomic Sequence is more representative of geometrical space than planal geometry. Like the polygons - the 56 Aubrey Holes of Stonehenge give us a pi value of 3.111, repeating-- that means a polygon of 56 circumference is commensurate with a diameter of 18 of the same units.
3.11111108280 - 2520 = 0.123456789
We find an association here with Holotome E: 2520. Inasmuch as 45 x 56 = 2520, the 45 representing also a right angle (1/2 of 90 degrees) to 360 circular unity.
At this point, the question to Neil J. Sloane is just what logic is
there in finding the ideal distribution of points on the surface of a
sphere, when (as Fuller asserts) there is no ideal sphere to be found in
nature? Only quasi (geodesic) sphere, which can never reach a finite
sphericity, even though they come close. Using pi, we never come up with
an end to an irrational sequence, and nature simply doesn't operate this
way.
The geodesic is the modus operandi of nature herself, and by regarding
the sequence of holotomes as a series of rational (finite) steps, we
have a genuine logical dialogue with nature.
The Platonic solids were an attempt to find ideal forms, but some of
these have never been found in nature, but only in the abstractions of
the imaginations of mathematicians and geometers.
The Fullerean geodesic quasi-spheres, on the other hand, are prominent
in nature's scenarios and structural forms. Even viruses have recently
been discovered, through high magnification, of geodesic
design. Dodecahedral seed pods are also prevalent topological formations
in biological stereotypes of nature. Even the apparent sphericity of
soap bubbles is geodesic.
Fuller's geodesic modelling works on the atomic and molecular
level. Extending these spherical models into chains is what led to the
architecture of the D.N.A. spiral.
Fuller presented a model of DNA with the chain of tetrahedrons that
requires 22 to make a full revolution. Qabalist, Stan Tenen discovered a
correspondence with the Hebrew alphabet with these 22 tetrahedrons
(tetrahelix). Actually, it was the Akkadians who first used the 22
letter alphabet by adding Aleph to the Sumerian 21 letter alphabet (six
of which were vowels). It is my opinion that the Akkadians were the
first to corrupt a perfect alphabet of 21 letters--an alphabet capable
of producing an ideal syntax.
The infrastructure of such an ideal syntax can be seen in the
proposition:
THE SYNDEX PRETZEL
12
+108
= 9 x 12
= 9 x 21
Transpalindromic composite roots producing transpalindromic composite squares.
This is the means whereby transpalindromic squares are derived from
transpalindromic roots...with a palindromic nave 252.
2520 or Holotome E is the first and lowest number divisible by all eight base numbers:
12 x 2 x 3 x 5 x 7 = 2520
The interrelationship of number and geometry begins by doubling the
first perfect number: 6, then multiplying the primes in their natural
order of occurrence. This produces the series of minimal sums that
accomodate the maximum amount of consecutive factors.
Having studied the nature of this sequence for the past 30 years, I am
totally confident that the Sumerians were programmed by this system in
order to produce the metrology that created not only the alphanumeric
system of notation, but the whole mathematical foundation of industrial
civilization or civilized culture itself.
The first regression of this perfect system was the Akkadians addition
of Aleph.
The second regression was when the ancient Hindus divided the
astrocalendric aspects of the original Sumerian plan into two systems by
the introduction of : OM:
THE YUGAS 4320
Then the Greeks came along with their improper geometry from the
sequence of "ideal" solids and cut us off from the last traces of the
ancient wisdom which paved the way for the social chaos of the dark
ages.
Later, Sir Issaac Newton formulated the laws of motion, the most
significant of which is the retrocity involved in action vs.
reaction. This put us on the track to modern industrialization. His mind
could not deal with the triangle, but he made the most of dualism.
The return to valid numeronomy did not recur until 1980, when Marshall discovered the Holotomic Sequence through an innovative discipline termed Synchrographics.
It was because I was not programmed by mainstream scholasticism that I had to disbelieve the fiction that prime numbers were an example of
chaos. In my semi-literate way, I knew better and set about devloping a
method of rediscovering the truth which is that geometry and number,
quality and quantity are two sides of the same coin. But to find the key
to their synchronicity, I had to realize that only six of the base
digits could involve spatial geometry.
That I have discovered a strange and wonderous aspect of the natural
sequence of number there can be no doubt. And that these patterns of the
continuum of number were responsible in extreme antiquity for
formulating the basic metrology that established and proliferated
industrial culture is beyond dispute.
The question that recurs in my mind with great regularity is this: Of
what use is this knowledge or how can it be further employed in the
realm of science in a world where metrology is as natural as our
breathing and for the operations of technology just as essential, like
correcting the ways that technology is misused.
The Holotomic Sequence is physically interrelated with these geometrical geodesic forms that look like perfect or ideal spheres. On certain levels we have some legitimate questions that require talented answers.
Vieta used a polygon of 393216 sides to obtain pi within 10 correct
decimals. Whether these polygonal sides are ideally straight is
inconsequential.
Einstein did away with the ideal straight line, but it had no effect on
the practicality of the ideal straight line. Bucky Fuller did away with
the ideal sphere, but its purpose still remains as a useful element in
our conceptual process.
It therefore seems that the rediscovery of Numeronomy and the Holotomic Sequence might be applied as a unifying field of reference that in its ideal non-existence was here before the beginning, and so it shall
remain, even in a world of erroneous thinking.
(Editorial Note: There are several pages of as-yet-unscanned number
tables which support these allegations. We will post them as they become
available in a presentable form besides xexored snail mail. They include
graphics of the first Holotomes; Respective Divisors of the Holotomic
Sequence; Reflexive Nature of Holotomes; Extended List of Holotomes;
Anatomy of the Syndex Pretzel Sequence; In Consideration of Boxcars;
Brief Clarification of Grid Sixteen as 2-D Basis of Holotome A; Nature's
Key Ratio, etc.)
THE HOLOTOMIC SEQUENCEÊOF CIRCULAR UNITIES
1 x 2 x 3 = 6
A x2 = 12
B x 2 = 24
C x 3 = 72
D x 5 = 360
E x 7 = 2520
F x11 = 27720
G x13 = 360360
H x17 = 6126120
:Robert Marshall, Master Numeronomist : Iona Miller
ANATOMY OF THE STAR GODDESS by Iona Miller
Hyperchannels for crosstalk between dimensions are known as "magic
windows". These interdimensional nodal points have a naturally tuned
frequency of a good hyperchannel between orthogonal frames where scalar
wave anenergies crosstalk readily.
Crosstalk normally means the transfer of energy or signal from one
channel to another, by cross modulation or cross coupling between the
channels. In this new approach it refers to the virtual energy exchange
between orthogonal universes or frames--that is, between different 3- or
4-dimensional slices of an infinite-dimensional universe.
Magic windows are frequency dependent. Some magic windows are 38-40 kHz, 150-160 kHz, 1.1-1.3 MHz, 1.057 (Lamb shift), and in the near
ultraviolet frequency. These frequencies represent enhanced channels
between subquantal (virtual) and spatial (observable) states. A
particularly good magic window exists when the infrared and ultraviolet
bands being utilized are phase-locked so that the ultraviolet represents
a first harmonic of the infrared.
Also known as the Tesla wave, this standing scalar wave can be seen
alternately as a time wave or a gravitational wave. It is a wave of pure
potential. It is a longitudinal scalar potential wave in massless
charge, in the vacuum charge flux itself, and in pure spacetime. Since
there are no spinning charged mass particles, it does not form a vector
electromagnetic wave, as conventional theory would predict. It breaks
into shadowy, virtual vectors which are not integrated.
JAMES FURIA
© Copyright. Robert Grace. 2001
of Colors
E- Orange
F- Yellow
G- Green
A- Blue
B- Indigo
C- Violet
D- Red
http://www.alkahest.com/archive/1999/april99/PHI-factor.htm
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/propsOfPhi.html
This scale covers all 128 MIDI note numbers. It is a constant ratio
scale based on the ratio 1.0460838
This 128 note tuning is based on the ratio 1.227824
This is a 128 note tuning was introduced by an Arabic lutenist around
the end of the last millenium.
This ancient scale was derived from a series of pure fifths. This scale
starts on MIDI note 60 (middle C).
This 15 note tuning begins on MIDI note number 36 (C two octaves below
middle C).
This 37 note tuning begins on note number 36.
This scale contains the reciprocal frequencies of all 103 elements of
the periodic table beginning at note number 0.
This tuning was developed by John Barnes in 1979 as a probable tuning used by Bach in the performance of the Well-Tempered Clavier. It is a full 128 MIDI note tuning.
This 128 note tuning is based on the ratio 1.0375393 BETA PRIME
(W.CARLOS)* This 128 note tuning is based on the ratio 1.0185968
This is a seven note per octave ancient Greek tuning laid out on the
white keys.
This is a seven note per octave ancient Greek tuning laid out on the
white keys.
This tuning was derived from the Schuman Resonance. This
ever-fluctuating electromagnetic earth resonance was arbitrarily fixed
at 7.83 Hertz for the purpose of this study. The first 128 harmonics of
this fundamental frequency were calculated for this tuning. Starting
MIDI note number is 0.
This is a full, 128 MIDI note tuning based on the ratio 1.0145453, which yields 48 eighth-tones per octave.
This is a seven note per octave ancient Greek tuning laid out on the
white keys.
This is the standard constant ratio tuning of 1.05946. This is the
tuning most often used in contemporary Western music (this is the
standard MIDI tuning).
This is a two octave Pelog scale laid out on the white keys starting on the C key two octaves below middle C. It extends to the E a tenth above
middle C.
This 128 note tuning is based on the ratio 1.0204815
This is one of several scales developed and discussed by Wendy Carlos in her 1987 Computer Music Journal Article, "Tuning: At The Crossroads." It starts on MIDI note number 0.
This is a tuning based on the harmonic series starting at 65.41 hertz
assigned to MIDI note number 48 (one octave below middle C).
This is the 22 tones per octave tuning that serves as the basis for the
Indian ragas. It starts on MIDI note number 0 and the tones are assigned
to all but the Bb keys.
This 128 note tuning is designed so that the diatonic intervals of the
major scale are pure.
This 128 note tuning is designed so that the diatonic intervals of the
minor scale are pure.
This tuning allowed the performer to play in a wider range of keys than earlier tunings. It is a 128 tone tuning beginning on MIDI Note Number
0.
The great microtonal pioneer Harry Partch devised this 43-tone scale
(the 44th tone is an octave above the starting tone). This tuning starts
on the D key an octave and a seventh below middle C. I extends to the A
key an octave and a sixth above middle C.
The seven-tone Indonesian Pelog scale is laid out on the white keys
beginning at middle C. The pentatonic Slendro scale is laid out on the
black keys in the same octave.
The seven-tone Indonesian Pelog scale is laid out on the white keys
beginning at middle C.
This is the "Mother" of all Phi scales! All the other Phi Scale presets
are derived from this one. The fundamental frequency is multiplied by
Phi (1.618). Then that resulting product is multiplied by Phi, and so
on. You will notice that there are only a few notes in this scale.
That's because the large ratio (1:1.618) defines very wide intervals.
PHI-Square Root of Phi (1.2720)
This is a 20-note tuning derive from a cycle of pure fifths starting at
32.7 hertz assigned to the C below middle C.
This chromatic octave, derived from a cycle of pure fifths begins on
middle C (at 261.6 Hz.). Notice the "out of tune" Pythagorean comma
between the fundamental and the octave!
This is the traditional Pythagorean tuning derived from pure fifths (the fourth is calculated a perfect 5th below the fundamental then transposed up an octave). The octave is a 2:1 ratio from the fundamental. This 128-note tuning starts at MIDI note number 0.
This full-keyboard equal temperament tuning yields a 24 note-per-octave scale derived from the constant ratio of 1.0293022.
This is a traditional 12 note-per-octave equally tempered scale that has been shifted up a quarter tone.
This 7-note ancient Chinese tuning is laid out across the keyboard with the notes of each octave assigned only to the white keys.
The pentatonic Slendro scale is laid out on the black keys beginning on
the C# above middle C.
This is a variation on the Pythagorean tuning, devised to allow
modulation to any key.
This is one of Andreas Werckmeister's (a contemporary of Bach) tunings designed to allow performance in any key (each with its own unique tonal character).
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SERIOUS COMPOSER, INC.
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1 2 + 0 = 2 F-C (G)
2 5 + 0 = 5 F-C-G-D-A- (E)
3 5 + 2 = 7 F-C-G-D-A-E-B-(F#)
4 7 + 5 = 12 F-C-G-D...G#-D#-A#-(E#)
5 12 +7 = 19 F-C-G-D...Gx-Dx-Ax-(Ex)
6 19 + 12 = 31 Gbb-Dbb...Gx-Dx-Ax-(Ex)
and so forth...
Shirady Village- Post
Puttur Tq. Karnataka, India
PIN- 574 229
Phone: 08251 53204
E- mail: Johnpaily@ yahoo.com
A:B = 4:3
A:C = 4:3
A:D = 5:2
C:D = 4:3
C:B = 4:4
B:C = 4:3
A:C = 4:3
A:D = 6:1
C:D = 5:2
C:B = 5:3
B:C = 4:3
A:C = 4:3
A:D = 6:0 [maximum-Lead Component]
C:D = 6:1
C:B = 6:2
D:B = 0:1[minimum]
Holotomes A, B, C, D, E
Definition of Mathematics with Description of Holotomic Sequence
2
: 144
12 x
21 = 252 palindrome
+189
21^2
: 441
#108 (3 x 36).
THE AGES 6480
(and an undetermined amount of zeros)
Under Synergetic Constrution...
Created 7/10/99
Last Updated 10/06/99
Web Design by Vickie Webb
A# = 57.29578 = Radian (180/pi)
B = 240.17358 = One half the height of the Great Pyramid
D = 288 = Diameter of the outer circle of Stonehenge
D# = 152.89924 = Multiplied by Pi = Height of Great Pyramid
G# = 101.93282 = Difference in height of Chephren and the Great Pyramid