This illustration shows one way we can interpret thoughts from a written page, and translate these ideas into pictorial symbolic form. Three suggests three objects. Since the world is a sphere, circular forms become the next step. Since elements, number, weight, measure, planets, place, motion and nature and time all fit into one sphere, we can almost conceive of wholes within wholes, the sphere of our visible world. The music of instruments, however, suggest the invisible world of vibrations, solid hitting solid, to give off vibrations in the spaces between the objects. It is here where we find patterns of sound, always following the same sequence of fundamental, fifths, fourths, sixths, thirds, sevenths and seconds, which when rendered pictorially give us much to think about.
The music of humanity in essence is the body, the visible, and the soul, the invisible. The connection between them is the reaching of the visible, or material, for the invisible, or spiritual.
This describes what we are trying to do, specifically in life, no less than in art, uncompleted beings searching for fulfillment.
The lines are the five lines of the music staff at right angles. The right angles assumes importance as we go on, as do the differing degrees of angles, as we shall see.
So do the circles within the circles, and the rims of the circles meeting at the center. If we can imagine the circles all rotating, we know, from mechanical laws that the inner ones must go in the opposite direction from the outer ones.
Light and darkness also can be depicted to show opposites, by making what was light in one, dark in the next, symbolizing the juxtaposition of the known and the unknown. In all three cases the inner circles were different, rims meeting in one case, then turned at right angles in the next, and finally both coming into the whole at the same time, superimposed over one another and giving the impression of a backward forward motion, pulsating.
The smaller to larger spheres in the foreground stand for the notes on a keyboard. (Do) or the tonic, becoming the middle (C), one side of middle (C) representing time, the other side representing space. The descending scale represents contraction, as we shall see later. The earliest known music, incidentally, had the scale descending step by step, rather than ascending as we know it today1 . Could it be that in an expanding contracting universe, ancient civilizations were in a contracting phase, whereas now, with present emphasis on overtones, ascending, time and all we know and experience is expanding?
The ascending scale represents expansion, not only because of the patterns of notes when laid out on a numbered grid, but because of decreasing radii (the overtones can be reproduced by shortening the string length or radii) and the fact that we are dealing with an infinite, in the sense that we have an ever increasing set of numbers, rather than a given length. Thus the descending scale represents the finite, the ascending scale the infinite.
The N. S. E. W. directions were used because we could then determine where the notes were in relation to one another.
The Chinese are apparently the only culture which assigned direction to their musical scale. Since they also assigned color and taste in some instances, that is another reality, we can make visible.2
This Illustration does not deal with specific instances of making musical ideas visible as much as showing the relationship of an art to everything around us, material and spiritual. It suggests a receptive frame of mind which the viewer or reader might assume to gain the most benefit from the work.
A work of this sort is not aimed only at those who see or hear, but the physically blind and the deaf should also be able to understand and experience these abstract truths, and benefit with the same esthetic enjoyment of discovery.
Pluto's orbit is in an ellipse. While all other planets have orbits with the sun as center, the orbit of Pluto has its center off from the sun. We know that we need two points to draw an ellipse, and in space two forces are needed to create an elliptical curve. The sun is at one focus, the other is empty. The idea was intriguing so this was the model used for relating the notes of the scale to the planets. Pluto represents the limit to the circle.
The smaller, more dense rings symbolize the four planets nearest to the Sun, Mercury, Venus, Earth and Mars or (Fa), (Do), (Sol) and (Re). The asteroid zone, or (La) is between Mars or (Re) and Jupiter or (Mi), the largest planet, but the limit musically, which we shall discover later.3
The five outer planets, Jupiter, Saturn, Uranus, Neptune and Pluto, or musically (Mi), (Ti), (Fa#), (Do#), (Sol#), (Re#) and (La#), represent unknown planets 4 then become almost an inversion of the planets nearest the sun, and enlarged inverted mirror images.
By far the most significant aspect of this is that there might be a relationship between the planets and sound, that the furthest planets from the sun might be an enlargement in scale of the planets nearest the sun. We know from astronomy that there is a tendency in the universe to change its scale by expanding or contracting.5
What is important is the laws of proportion. No matter what the scale, the objects we are measuring are always in the same relation to one another. Could we apply the same laws of measurement to the macrocosmos and the microcosmos? No. The universe is finite or infinite depending on the measuring rod. In the case of a contracting universe we could use the finite measuring rod of dividing a given whole. In the case of an expanding universe, we could use an infinite set of points, always adding or multiplying an imaginary radius.
Musically, Pluto as Sol# could be taken to mean the earth. Musically, Jupiter, the largest planet, symbolizes the limit at (Mi). (Mib) could mean a contracting to Mars or (Re). Jupiter is, at the same time one of the largest and the lightest of the planets. Pluto is, at the same time the smallest and the most dense. Are vibrations faster in increasingly dense material? Would the most dense of all musically be the (La), whose ratio to (Do) is 32000:19683? If asteroids are most dense, could not the ancients have been valid in thinking of the asteroids as far away stars?
I bring all this out to show how by focusing on one aspect of any problem leads to conjectures about other problems, conjectures which are stimulating to the mind but harmless in that they are not applied to astronomy or physical planes, but reflect perhaps a motivating creative state of mind that is necessary to all creative thought. Let us see what would happen musically if we took the notes in order. Up to (Fa), down to (Do) up to (Sol), down to (Re), etc.We have an ascending, descending pattern, which also resembles a zig zag which according to P. D. Ouspensky symbolizes the fourth dimension.6 Since ascending represents expansion and descending represents contracting, we have a picture of time, expanding and contracting, ascending and descending in a zig zag.
Musically, the note on a keyboard of middle (C), represents the zero point; a balancing of ascending and descending steps, a see saw like image, the (Sol) ascending and the (Fa) descending making the widest angle, whereas the (Re) ascending and the (Ti) descending make the smallest angle. Think of the see saw as the diameter of a circle. Think of the circle as being a limit of the universe. Think of the half of the diameter as a radius, and the rate of acceleration or deceleration being dependent on the degree of the angle. Think of life processes as going forward in a leap of five steps, then going backward in a leap of ten steps. The octave between (Sol) and (Sol), then all the other oscillations are shorter and shorter leaps, until there is finally at (Re) and (Ti) one step up and one step down. Every effort has it backsliding, it is a natural process, activity and passivity or receptiveness, the giving and the taking.
If we imagine (Do) as the center of a circle, and draw rings, one for (Re), one for (Mib), a larger radius, one for (Mi), still larger, one for (Fa), (Sol), (La), TIb), (Ti) and (Do), and imagine all of these rotating at different speeds, along a flat plane, we have a very good picture of the solar system, or even the rings of Saturn. As for the construction of the rings, an optical illusion occurs when concentric rings are drawn around a center point.
They either seem to be solid/space or vice versa, space/solid. We can utilize this in drawing, by rendering it three dimensionally, but we do not even have to do this as our own vision supplies this balance optically.
The spheres are given a three dimensional effect by shading , size difference, and shadows so that they appear to be standing on a drawn surface, as an extension of the surface into another dimension. The numbers 4, 7, 10, 15 represent the number of units of each planet's distance from the sun, and are an approximation of Bode's law. In the system presented here the relationship is that the smallest planet Mercury as (Fa) which is FOUR units from the sun, Venus ((do), seven units from the sun, Earth (Sol), ten units from he sun, Mars (Re), fifteen units from the sun, Asteroids (La) twenty eight units, Jupiter (Mi) fifty two units, Saturn (Ti), one hundred units, Uranus (Fa#), one hundred ninety six Units, Pluto (Sol#) three hundred eighty eight units and Neptune (Do#) should not be there according to bode's law. Certain questions come up, such as why do we come to a limit at (Mi) and (Mib) jupiter & mars? (See illustration 48 Concentric Circles). Why is it that in the entire universal ladder of sounds we always hear only the notes we can attach to seven planets, and when we are dealing with one portion of a contracting field of the circle of fifths, we don't find or hear the note in question.
(Do) would be the fundamental in its field, (Sol) the fundamental in its field, etc.
We know that overtones can be produced on a string by holding the string at midpoint, for the first overtone of an octave above a plucked open string, and halving each section until it is no longer possible to differentiate the note, and the finger reaches the end of the string. This occurs in about four or five steps on a violin string, for example. Since the undertone is a doubling instead of a dividing of the string length it is impossible to physically reproduce. In the case of a vibrating string, every other harmonic in succession does not vibrate, so what we hear are the first harmonic, the third, the fifth, the seventh, etc.
I would like to put forth the idea that what is heard in the overtone series, is not necessarily the whole, but only a fragment, that there are vibrations occurring which we do not hear, but which we know are there because their numerical ratios appear on the framework we have set up.
If we take the idea of ascending and descending fifths, fourths, thirds, seconds,we find that their counterparts descending are fourths, fifths, sixths and sevenths. This pattern of taking ascending and descending notes on a keyboard fits exactly into the pattern of notes as set up on the grid.
The visual pattern of this maneuver is a zig zag, a shape integral to our thinking. The slanted diagonal lines fit into our puzzle, as being the fold lines created when we fold each circle to its corresponding size. They represent the slanted lines which originate from multiples of ratios on a fraction grid, where the numbers fan out on each side of the 1/1, 2/2 etc., diagonal lines.7 Lines drawn from the most dense galactic sources to other galaxies also have the same type of arrangement and slant, as do spectral lines. They appear to fit into a 60 degree angle, an angle which is germinal to our thought. They also form the basis of triangular shapes, which are sectors of circles. The circles are to be thought of as moving orbits. The angles are shorter or longer radii, contracting or expanding. Vibrations are to be thought of as traveling along adjacent circles, up and down, ascending and descending. Every circle is part of a wave motion. Every spiral represents a falling out of one orbit into another.
How does this apply to our lives, or other phenomena? If many waves can occur on a fixed length of a vibrating string, many universes could exist within a given finite space. Astronomers are unable to find what exists beyond the galaxies beyond our own Milky Way. Let us say that beyond our galaxy is the tonic or fundamental tone of the universe, the (Do). The next layer, a smaller sphere, our own galaxy would then be the dominant, the (Sol), the plane on which our star exists. The third harmonic, descending to a smaller scale, would be (Fa), a musical inversion or (Sol), or the earth. The next overtone is (Mi), or mankind. The next is (Tib), or organic life, or microbes. Then comes (La), or space. Finally at the end of the cycle comes (Reb), or energy, expanding. After the expansion we would come back to the fundamental (Do), the largest circle, the all inclusive wave, and begin all over again the reductive process of contracting to different scales. If we imagine each sphere lying on a different plane, and that each one as viewed from another dimension is flat, we can picture an extension of dimensions, from each plane.
We would also have to assume that the tones heard in each sphere would all be present except the tone, the fundamental tone of each. So that in each sphere would be missing the sound of its own tone, and would be incomplete in that sense. A conscious intelligence in any form would long for the missing vibration, the missing sound. And we as human beings are searching for the missing piece, to the puzzle.
The most significant thing about the Mayan Katon Wheel relevant to this study is the curious shape the counter-clockwise motion takes across the diagonal of the wheel. The wheel was divided into thirteen sections. The thirteen sections represented a 20 year period. The pattern of notes on our grid up to the 15/16 point, a completed cycle of (Do) to (Dob), an octave, can fit into this circle. In other words we can take the thirteen notes on the grid, and place one cycle on another, to come to rather similar conclusions.
If we look closely at the structure of the zig zag eleven pointed star, we can easily determine that three of these pointed figures form a 60 degree angle, and if magnified to the scale of the previous illustration (P3), would also correspond to the fold lines in that illustration in both angle and number of lines. This wheel might help to explain the recurrence of (Mi) on the grid, because at this point a reversal of direction appears to have taken place. If we turn back to illustration P2, we see how this happens. The snake-like pattern of planet to planet progresses until we reach the asteroid zone where perhaps a missing planet area causes a reversal of sorts. This may seem very arbitrary and unjustified speculation now, but when the same phenomena appears to occur at the exact same point in every system that we use, there must be some reason for its recurrence (See Circle of Fifths, Trapezoids).9
As we have already shown, the (Mi) and the (La) represent ascending and descending properties of the same note. (La) is a descending third, (Mi) an ascending sixth
One is a stepping up from the bottom step. The other is a stepping down from the same step on the ladder.
Again, the significance of this is applying a cultural analogy to an abstract theory.
However, on a grander scale, it represents spiritual powers of a search for absolutes, and making these absolutes concrete. The importance of it lies in the search of a meaning to the universe, a meaning to our lives. A placement of all of our values on something greater than ourselves. Symbolically it represents a shedding of material values, to a putting on of values that correspond to a universal harmony. It makes us acutely aware of the knowledge and the search of other ages. It gives us an insight into the thought processes of other civilizations. By focusing on a fragment of an unknown puzzle from another age, we respect the mind, and feel a link not bound by time.
In fact, it makes us almost think that civilization is slipping downward, that perhaps the golden age of man did have all the answers, that somehow mankind forgot some of the basic motivating and universal laws, that the Garden of Eden was a reality, and mankind has slipped backwards ever since.
(When this was drawn, it was an example of how vibrations might be depicted on a string. I have set it up according to the grid sequence rather than the acoustical way it is heard, as it is depicted in most cases.)10
I would like vibrations to be thought of as waves traveling along in any space until they reach an object and must bounce back. When they bounce back the waves would seem to fall in the spaces between the original wave. Where there was originally a trough, there is a crest, Even in stars, scientists have found that vibrations from small to large cannot be reversed, and that in the interior of starts are found the short wave lengths, at the surface of the star the length is longer.
As the outside of a star or any body is a positive curve, and the inside of any body is a negative curve, if we draw adjacent circles we can see that lines drawn perpendicular to the slant of the surface implode on the inside, come together, and explode on the outside. A star, as an example of a closed body of negative curvature on the inside, any sound or pulse or striking of object against object would produce, it seems, waves which would produce the whole overtone system as they bounced back and fourth, becoming multiples of the original wave.
If we apply this idea to the notes as set up on our grid, we find that the first space between comes between the (Fa) ,Mercury and the (Mi) Jupiter at 5/3 or (La) asteroids, so we can imagine a wall of sorts at the asteroid zone where sound bounces back, travels in a reverse direction, as if it were on the inside of a star and hits other waves so that the whole line of vibrations becomes increasingly complicated because of the reverberations.
These reverberations can also be depicted in another way by setting up circles along a line, then dividing them by halves so that within the radius of one, another is drawn, or two circles within one for the (Mi) yellow within the (Fa) green. Within the (Mi) is the (La) lavender and within the (La) is the (Ti) black and within the (Ti) is the (Re) orange. While to go bigger we find two (Fa's) within one (Sol), and finally a gigantic (Do) enclosing all. Just to give some idea of the dimensions we are dealing with in a rough approximation of inches, the smallest (Re) would have a radius of 3/8ths of an inch. (Ti) would have a radius of 3/4ths of an inch. (La) would be 1.5 of an inch. (Mi) would be three inches. (Fa) would be six inches, and finally (Do) would have a radius of twenty four inches. In another way of putting it, each succeeding radius of a smaller circle becomes the diameter for the radius of another circle. We have reached very huge dimensions, proportionally taking in just seven steps. These seven steps represent an example of an entire scale, a universe of the physical things within a human's grasp, so to speak. The objects we deal with, in a day to day existence, the familiar world.
We might say that within a given length of 48 inches the diameter of (Do), we have a universe of vibrations which is complete within itself, a finite measuring rod of proportions such as these, in our houses, the objects we use could all be designed with proportions such as these, in differing scales, we should actually be living in environments of harmonic proportions, which would undoubtedly influence our states of mind.
We could be enveloped in an esthetic harmonic law, which could have unparalleled benefit, purpose and usefulness, for our activities as productive individuals.
Each of us has a special tonal signature, the tones are different, like the twelve months, the overtones are different, but all follow a similar pattern. It is this pattern which we are attempting to isolate, analyze, and put to use, as a means for fulfilling our destinies as individuals and as members of an evolutionary process which is unfolding in harmonic order. Even the membrane of the human ear takes certain wave-lengths and distorts them to a different scale in order to interpret for the brain the meaning of certain sounds. Perhaps we, as individuals, are cosmological membranes to a greater order. Perhaps the asteroid zone is a cosmological membrane of our solar system.
This image suggests inverted cone shapes, which we obtain when trying to encompass expanding circles. If we deal with diminishing circles, by taking a 60 degree angle radius of each circle, we find the inverted cone shapes becoming increasing smaller. Also they become increasingly diminishing progressions by squares placed end to end.
This is one reason we think of squares being two triangles fitted together,or even better, two sectors of tangent circles. This is part of our puzzle, that we see each shape as part of a larger whole. We also reach our limit at Mib as expected. In illustration P6 the notes were set as planets, in the progressing small to large scale as their distance increased from the sun. In this illustration they are not inverted at the point where the cones meet, but are placed in the opposite order, to show that if the paper or space were folded over, we would have a true reverse.
It is at the zero point where the object seems to contract, in order to expand at the other side. In music, as described here, we find (Mi), the most contracted, the zero point of all the other notes. As (Mi) is Jupiter, in this system, the largest planet, perhaps we see a magnification of a nearer planet, say Mars. The asteroids would then become a magnifying mirror to our first four planets, Mercury, Venus, Earth and Mars. The important thing is not all of these conjectures but the idea that when space is folded over, all the small orbits fit into the larger.
Space is supposed to curve more in the vicinity of more dense matter, Pluto, as the most dense planet, would be in the most curved space, perhaps even turning over itself. Pluto, is the furthest planet from the sun, the periphery of our solar system, yet it is the focal point or center of our drawing. Could we imagine space to be curved over to such an extent that it becomes a 60 degree angle.? If so, we can picture a funnel shape.
The funnel shape, in turn suggests that contracting principal, as opposed to a diamond shape, its opposite, an expanding shape. If we can imagine six funnels radiating from the focal point, a hexagon shape, this gives another dimension to our thinking. Even, more so if we think of the center point being at Pluto. If we fold over 1/2 the distance from 1/2 to 1/32, Pluto and Mercury become one and the same. Is Pluto a mirror image of Mercury? Mercury is (Fa) in music. (Fa) is the starting point of the circle of fifths, the "I CHING" diagram where (Fa) 81 (9 x 9) coincides with number one in the sequence of unfolding Hexagrams.11
We know that density is proportional to the frequencies of vibration. We also might consider that the degree of angles determines the speed. When we apply an idea of density to a diagram of galaxies, we can take what is thought to be the most dense source in the universe, and draw lines from this to the other galaxies, to see what patterns might appear. Also if we consider that at right angles motion stops, we can start with (Do) at right angles to the most dense source in the drawing. We find very regular intervals, which lead us to compare these with the pattern of the grid. The galaxies seem to be as pulses of differing intervals apart,much as the pulses we would find by multiplying the fractions of 2/3, 3/4, 4/5, etc. The galaxies are spaced in neat little rows, six to be exact. If we can imagine these galaxies as notes of a musical scale sliding to a focal point, imploding perhaps until past the zero point where implosion becomes explosion, perhaps some of Einstein's ideas of relativity might apply,13 i.e., the faster an object goes the smaller it becomes. Even on a diagram as rough as this, we can tell that each parallel section appears to be either a doubling or a halving of the one either preceding or following it. This might be a principal which we could use in designing a structure, alternately multiplying and dividing, or adding and subtracting, it would be built in measuring rods wholly dependent on adjacent parts, and balancing in proportion of one to another, its neighbor. If we can imagine an accordion pleated fold at each dotted line, we in essence have a straight line, extending diagonally across a given minimum of space when folded and fanning out two dimensionally at right angles to the diagonal when unfolded to its maximum. Again a right angle is extremely important to this study, as an immobility of motion. To go back to Ouspensky, any angle greater than a right angle is impossible motion. These folds cannot become greater than a right angle.This is why we assume that the right angle is the limit, and that it fits perfectly into this particular diagram. The fact that there are thirteen folds reminds us of the Mayan Katon Round Calendar.
If we imagine the planets, or notes,swinging around in a circular line, in a chromatic color wheel, we have (Do#) as Venus, (Re#) as Mars, (Mi) as Jupiter, (Fa#) as Mercury, (Sol) as Earth, (La) as Asteroids, (Ti) as Saturn, and (Do) as Venus again.14 We not only have a serpentine-like form biting its tail, but it might be easier to imagine why ancient people thought there were two Venus', the morning and evening stars, and why music and the planets might have seemed more connected then than today. If we carry our imagination on even further to imagine that this circle is an ever bigger inner tube, what would the tube universe look like to those on the inside versus those on the outside? Does the inner eye look inward to the tube?
CHINESE COLOR MUSIC SCALE 15 vs. PYTHAGORAS COLOR MUSIC SCALE. 16 (RE: ILLUSTRATION # P-9).
When comparing the notes and their colors it is interesting to study the differences, and wonder why, with the exception of (Do) orange and (Do#) acra violet, the colors the Chinese assigned to notes are the exact opposites (Complementairies). When taking each row individually, the Chinese sequence follows a complementary pattern, of radical change from (Do) orange, to (Re) green, from (Mi) lavender, to (Fa) red, then the sequence becomes one of adjacent colors after (Sol) yellow to (La) blue. (Ti) is white the opposite of the Greek (Ti) black. It is only the two (Dos) which are the same in both systems. The Greek system from the same notes of the scale follows a chromatic color pattern, one of gradual change to neighboring notes, rather than the sudden dramatic change of the Chinese. The Greek being red to orange to yellow, to blue-green to blue, to lavender, to blue-black, to acra violet.
The curved lines within the circle show how we can interpret the ratios two to three by having three crests and two troughs. The idea that the sound varies from rhythmic below 10 vibrations per second, to melodic above 15 vibrations per second, is akin to the idea that the same string length encompasses all the multiplications of vibrations. In other words melody and rhythm have the same base, the only difference being the speed or the number of vibrations.
If we maintain a constant length for each note and whatever the ratio is, such as an eight to nine, (Re), if the whole length is taken to be nine, we measure nine equal segments, and then count off eight on the length. When we arrange each segments in an order of gradation, and make one an inversion of the other by hanging it upside down, as it were, on the opposite side of the paper, we find that the so called unstable notes cluster together, as a measure of space, perhaps, and the others, the (Do), (Sol) and the (Mi), the stable notes, which might represent time.
In the center of the drawing, the tiny spheres are arranged as if on a keyboard, with (Do) being five steps from one end, and eight steps from the other end. It is this keyboard arrangement which points out very clearly the ascending-descending quality of the notes, as if they were being played. One significant thing about this is that the shape that is formed in the center of the paper assumes the configuration of a pseudo-sphere, a bone-like shape which to our way of thinking becomes the space between two circles. This again, is an important aspect, as either one can be alternately space or solid, but not both at the same time, as emotionally our reactions to the latter is very unpleasant and confining.
We should imagine the lengths vibrating, as if the crests were electric energy, the troughs, magnetic energy.
The vibrating strings in illustration P10 should be seen as being placed together so as to form a circle, a half circle, more precisely as the other part represented here is but a mirror image. If one part represents an ascending scale, and the other a descending scale, we already have a concept which is most important to understanding reversals, and another integral facet of music. One of the things that has stood out in our study is that we really only have to deal with one half of a circle, in almost all cases. here, the eight segments form a half circle. Within the vertical segments, we must imagine a wave traveling up and down. What is outside becomes inside as we can very clearly see, if we just draw lines extending from the crest, and as we follow it, we see that the lines must become inward and at different angles at each point. We now have a frame of reference as to the originating point of lines radiating outwards from the outside of a curved surface. We have explained to ourselves visually the reason why we don't come to a point when determining the slant of the multiples on a grid. We can see a picture in our minds of curve of a wave of vibration from which radiate other vibrations at different angles. The reverse of this is, of course, the trough. The crest might be considered as an extension, an expansion, the trough must be thought of as a quickening, a contraction. Why not make these pulse beats? Rhythms? From space we have time. The line is very thin. Space can be translated into time.
Can we apply a universal field theory of octaves of music to a flat two dimensional surface? In nature we find a leaf flat, the entire solar system flat, the rings of Saturn flat, galaxies flat. A whole field of sound waves is like two pebbles dropped in a pond. An octave, a (Do) at the beginning, red. A (Do) at the other end, acra violet, double the vibrations, and a shorter length, the rainbow, a spectrum of red through the full range of color to red again at the violet end.17
When we arrange fractions18 in an order at right angles, and each point representing a fraction spaced equidistantly apart, if we study carefully the differences between the horizontal 1/2, 1/3, and 1/4 to the vertical 2/1, 3/1, 4/1 , we find that there is an immense conceptual difference between the value of the fractions which lie along the diagonal 1/1, 2/2, 3/3, (representing a whole), one being much less 1/2 than a whole, the other being much more than a whole 3/2 or 1-1/2. Yet the large difference diminishes towards the end, and 8/9 being closer to a 9/8 in distance. So, in other words we picture simultaneously a diminishing cone in both directions, or a complete picture of an expanded body, two triangles base to base. This can be described three dimensionally as an enormous fold diminishing as it runs diagonally across a sheet of paper, or space itself. If we think of the notes as planets, we imagine the fold with a reflection of a planet on the other side, as on another dimension. Perhaps the square paper or space does not exist as a square but only as two triangles fitted together to form a square. If we imagine other folds, at right angels to the diagonals between each note, we will have thirteen folds in one cycle of (Do) to (Do). Only two pebbles, two planets, thirteen folds, thirteen reflections of mirror planets, the Mayan Katon Wheel.
There is another thing we can do in analyzing this field of notes, if we try to apply some of Einstein's theories. Since we have established a pattern, let us see what might result if we take the large spaces, such as the empty area at the end of the series. Einstein, in effect, brought out that at vast distances, (relatively speaking) speed increases, length decreases, mass increases and time slows. The idea of objects shrinking in vast spaces, we can apply here, because the area of the empty space is twice as big as the whole cluster of notes in the areas up to 8/9. The ratios also become smaller, as we have seen. As 15/16 is the end of the first cycle, we can imagine the pulse slowing, as the speed increases. It is much like our view of waves breaking in slow motion on a shore, when viewed from a plane.
The idea of mass increasing is also proven by the note (Do) being both 1/1 and 1/2 and 15/16 being closer in size to 1/1 than to 1/2. It is important to state here that Helmholtz believed the descending notes to be 2/3, 3/4 etc., while ascending notes became 3/2, 4/3 etc.19 The idea that the more massive an object the more sharply spaces curves around it, could be illustrated by the orbits ever expanding in both directions from (Do) 1/1 to (Do) 15/16. The fact that the ratios go from 1 to 1/2 is a rather drastic shrinkage, then to lesser and lesser shrinkage until they reach 32000/19683 end of the cycle at (La). If it is possible that the snake-like progression of the wave are at right angles to the diagonal in ever increasing crests and troughs until we reach 1/8 on the grid, which includes the (Mi) Jupiter area of notes (the mediant, the halfway point between (do) and (Sol)), we might have a little picture of the universe forming before our eyes.
According to some music theory, at the 1/8 point we have an undertone, rather than an overtone, in which case the notes which lie on the vertical line of 1/8, 3/8, 5/8, 7/8, 8/9, and 9/16, we have the notes (Fa), (La), (Re) or (Reb) and (Tib) our unstable notes, no less
We are able to construct another focal point, by taking 9/16 (Re) (Tib), and by drawing with the compass lines from the 1/8, and through each note which lies on the plane of 8, we may construct a rainbow-like structure,which could be proved mathematically to be an increasing - contracting radii, according to a system we already set up for measuring lengths, with the largest space being near the focal point. Compare this to the series of rings we put around the five notes, (Do), (Sol), (Fa), (La) and (Mi), the cluster which represents mathematically an expanding/diminishing syndrome, because of the rings clustered toward the focal point. As mass increases, time slows, speed increases, length decreases, the more sharply space is curved. Perhaps it is at this point where space folds back upon itself. Think of this tiny cluster of notes forming an octave, which is a universe in itself, and might mirror the universe in which we find ourselves. If matter, or length, diminishes, and decreases as time slows, might not the (Ti) notes, with their astrological counterpart as Saturn, a contracting element, be a rather logical conclusion. One might picture the pattern of the (Ti) notes as four black holes in space in the shape of a trapezoid, as the ultimate end of contracting matter.
If we can find a measurement, a scale, into which each contracting cluster of notes, octaves, actually fits, the implications become enormous. We would be able to predict growth patterns, physical phenomena, possibly behavioral patterns, the list is endless. We have discovered that we can fit the whole universal ladder of sounds into chunks of octaves, decreasing by halves, and in a definite order. "See circle of Fifths, Article "Painting Based on Relative Pitch in Music."20 If this ladder of sounds does mirror our universe, we might have found a key to unlock its mysteries.
A possible explanation for the reversal at (Mi) 5/6 or (Mi) 27/32 (33/24) might be that in the progression by squares, the fifth square 25 (625 is a cubed image in the drawing) encloses the entire space of five trapezoids, (see illustration). In a circle divided into nine sections the 5/4 also becomes a reversal.21
Do the vibrations roll along the diagonal parallel to it, or at 60 degree angles to the diagonal,which form equilateral triangles at any point we choose? As the area gets bigger we find that the base of the trapezoids progressively curves more and more. Would we not find that on increasing even further we might come to a full circle? This illustration shows roughly how, if we put our finger on a point along the vertical, we can stop the harmonics of that note. The fact that each sphere represents an octave of a note, that each sphere can be (Do) in its own little realm that each sphere is constantly moving and shifting, not only within its own 60 degree segment, but that the whole circle is moving, each band representing a different speed, or a reverse motion, that some come in conjunction at times, in opposition at other times, how easy it becomes to find analogies with planets, electrons, calendric calculation.
It is this page which is the key to all of our calculations to determine a visual pattern of notes. This is the the grid, so often mentioned. The sequence of overtone and undertone notes is laid out at the bottom of the drawing. The fold lies represent space folding between each note, so that the image is not of notes lying on a flat surface as much as of notes on the surface of rippling water, or even deeper folds of unknown depth.
If we think of these dots as representing galaxies, although the spaces involved is, of course, much greater in reality, (it cannot be depicted, in fact graphically) and fold the space over a point between the dots, and then repeatedly fold at the base of each succeeding triangle six times, we end up with one galaxy. As we unfold, we find we have three galaxies, in the next triangle, six in the next, twenty eight in the next, or if we overlap into the other side we have thirty six, the whole number of (Sol). The total number of dots or galaxies is sixty four, the number for (La). What is important here is that we have established an order, and a proportion from which we can determine many things. The scale can be any size. What ever scale it is, big, medium or small, the same order applies.When we examine the dots along each side of the diagonal, we find that on one side of the diagonal there is only one dot at the mid point. On the other side, however we find three. One of Einstein's theories was that time is represented as the fourth point of space, space having three dimensions, time one. When we go back to the symbols which represent the notes, (La) is the entire half of the diagonal folded over. (La) represents one of the I Ching deductions, which we go into later. When we fold the remaining triangle over one half again, we have sixteen dots, representing 9/16 or (Ti), followed by (Re), Mib), (Sol) and (Do). These fold lines are remarkably similar to those which we discovered when laying our a grid pattern. The importance of this is in applying a logic of a different sort to a pattern of dots, and making analogies to other things which in turn suggest other analogies etc. We have the whole feeling of chinks of space folding over and disappearing, diminishing, until we arrive at (Do), where the process reverses and begins to unfold, until it again reaches (La), the void.
Here, again, is an example of an ascending scale representing mathematically a diminishing, (going from the outside to the core), and a descending scale increasing (going from the inside outwards). When these two rings are superimposed on one another, coming from both (Do's), in the center we find a large core, the musical note being at this point a (Re), an expanding factor. Even the notes follow a curved surface in this case, as a curved grid was used rather than a grid at right angles. In essence we have what appears to be a hollow sphere. The orbit on which no note falls is cross hatched, adding still another optical dimension to the sphere. The things which appear to happen when circles intersect is curious. On a larger scale there would be a shifting from one circle to another giving an impression of expanding and contracting or exploding and imploding. If we make it appear that there are spheres or lines traveling along these wave-like intersecting circles, we have indeed a sense of motion, journeys in space. It also gives a visual picture of both particle and wave motion simultaneously.
This illustration shows how we can take one order, such as a triangle, and by using the numbers as axii on a square framework can come up with other patterns, other shapes. The most significant part of this diagram is to show in detail the mathematical cycle of the circle of fifths in music, to show how the numbers for the notes originated.22 It was sometimes thought that the numbers might have been the number of strings wound around one another to yield a certain musical note. So much of the origin of this is obscure, that we can only conjecture more. The interesting thing about the cycle is that we must increase by adding and alternately diminish by subtracting the actual number of strings. Thus a perfect balance is set up for the entire cycle. The pattern of notes becomes a zig zag, our symbol for time.
George Arnoux also brought out the interesting fact that 80/81, the end of the cycle, representing (Fa) or when put into decimal form yields a descending series of numbers, or we could call it a descending scale, an infinite series. Whereas an 81/80 yields a finite series of 1.0125. We have determined that a descending scale is finite, since it represents a given length. Perhaps it is the inversion of 80/81 to 81/80 which we should consider. Actually the drawing to illustrate this principal was upside down from the other.23 This is a pictorial way of illustrating an inversion. The sequence of numbers and notes is extremely important. As we have already mentioned if we know sequence we can determine growth patterns. Remember that this sequence is the same for the I Ching sequence of the second world order, and that it is also the same sequence as the natural order of the planets in our solar system from the sun.
Some of the cardinal points of sequence would be therefore, five Steps Ascending, or 1 to 5, or one plus five, five Steps Descending, or 1 minus 5, alternating in a balance like a huge pendulum swinging wildly , and then running down to no motion. If we investigate pendulum motion, we come upon some fascinating principals which we can apply. The first might be the wide swing of the pendulum which would take on an elliptical elongated curve, we come across this type of curve in the areas of intersecting circles, which we have called cores. We are not speaking in terms of energies, as the widest swings incorporate the most energy. We are also incorporating the image of circles. If we then incorporate some of the ideas of angles or radii of circles determining the energies, to the diameter of circles, as representing the distance covered, we have a motion. The perfectly round circle represents motion as rest, and we can easily think of the planets as nodes of a gigantic universal string, as the points of rest. This is similar to the same string we press to arrest certain harmonics. This might explain why we could not hear our own tone, why each field in the universal ladder of sounds does not include its own fundamental tone.
Trying to translate the fractions represented by each note into time values, through a comparison of mathematical proportions, is perhaps a way of coordinating distance with time. By taking a line from one to sixty indicating seconds, we do not begin to have fractions which correspond to the music ratios until we count 30 seconds, or points. 1 .............. 15 ............... 30 ............... 45 .............. 60. 30/60 would be 1/2 or (Do). 40/60 can be reduced to 2/3 or (Sol). 45/60 can be reduced to 3/4 or (Fa). 48/60 can be reduced to 4/5 or (Mi). 50/60 can become 5/6 or (Mib). 60/60 becomes 1/1 or (Do), an octave difference.
Why this procedure should follow the exact same sequence as our music analogies perhaps requires a looking into the basic structure of our mathematical system. The question comes up, is it the system of mathematics or is it perhaps some more basic universal laws that we are encountering here?
The sequence in the illustration is different from the sequence here because of some miscalculations. While it is not valid in specifics, the basic idea could be explored further, therefore I have kept it in, so as not to disrupt the sequence of lines of thought involved in the whole process of this work. It is the principal, not the details (which are subject to error) which are most important in understanding this effort.
An eight by eight grid was set up. A diagonal line was drawn, so that the line did not fall on any of the dots, or small spheres. We called it the evolution axis because the line went from the corner of the perimeter to the other side, marked space fold. In other words it traveled away from our point of reference. It did not become an evolution motion until it had passed half way through a section of the circle.
At right angles to this evolution motion we drew a diagonal and called it the involution axis, since it proceeded inward to the center of the circle. At the point where the involution axis meets the evolution axis, involution becomes evolution. It is at this exact orbit that the note (Sol) representing Earth moves. We can picture space moving in sine curves around each orbit, with the planet notes becoming the nodes, the place on the vibrating space string where the concave becomes convex, and motion ceases to exist.
The orbits of these two rings of circles mesh. There is a blow up of this in the next illustration, with all of its implications hinted. The circles mesh because we took as the center point the first sphere on a diagonal along the involution axis. This center point can be considered a sun, or Venus (Do) point. We took as a radius the two points lying close to the diagonal of the evolution axis (Sol). We then took twice the radius for the perimeter of the circle or (Do). We then proceeded to halve each orbit. This is explained more fully later. By halving each orbit, first going in toward the center, then halving by going towards the rim, we obtain two different patterns, one expressing the contracting energy (bottom) the other expressing expanding energy (top). This fits in with our finite, consisting of a descending scale, and the infinite, the ascending. The infinite orbits are on the space side. The finite orbits are on the time side. Time contracts. Space expands. What we find when we contract by halving inward is that the notes are the first (Sol), then (Fa), then (Mi), then (Mib) which dissolves into the core, on the time side. On the space side, we have first (La). banded by (Ti) on the inside, (Re) on the outside.
The way we arrived at this was to count the proportional spaces on the half of the circle which we did not use in the beginning, the half which we left open when dividing by half in an inward motion towards the center. If we should examine the grid in illustration P13 we shall see this open space to the right on the lower side. If we now divide the full length of the radius from the center to the rim into eight sections (the sections are diminishing and unequal in the center area because of the progressive dividing by halves whereas there are equal sections in the rim because of the dividing by halves first on the one and then on the other), we can determine all the ratios needed. By counting three from the perimeter, we have five out of eight or 5/8 (Ti). To obtain (Re) 7/8 or seven out of eight, we count seven from the center. Counting five from the center gives us (La) 5/8. The pattern of all these rings are remarkably satisfying and symmetrical and balanced visually. We find that in constructing the orbits in sequence, that there is a sucking inward motion first, where speed apparently increases, length shortens, matter disappears, then perhaps blossoms from an implosion to an explosion, where it might curl up in petal-like forms on the rim in four layers, first the center, then inward, then outwards, with the outer rim being the all inclusive (Do), 1/1.
It is very important to mention here that if we set up all the degrees within a circle (according to the number of sides of a polygon) (See illustration P38), and if we stretched the angles out on a string marked with the degrees in their appropriate places, we would have the exact same movement, sequence, and visual form as these orbits. As we mentioned before, the two orbits were juxtaposed for clarity, and even more importantly, to show how orbits meeting in this way have many different properties, as we would discover if we followed single lines from one circle to the next in wave-like lines. As we pointed out earlier, one line at any point on a series of circles either expands together, or contracts together. That is the line runs either along the outside rim, or towards the center part of each circle, and they do this alternately, much as we would expect from wheels going in opposite directions next to one another in a series of gears.
The conjunction of two rings, where one would be going in one direction, the other in the opposite, we put as the beginning of life, as motion must slow at that point where matter is formed. This concept leads directly to illustration #19.
Here again we come to our germinal core, where many of our intersecting circles meet. The shape of the core is in reality a whole circle with the middle folded inward. So we may imagine a fold in space running through the length of this form. If we use the principals we just discovered of halving, we find that the line or fold when divided in half yields the exact measurement of a radius of a whole circle. This assumes enormous implications if we remember that the radius of a circle divides the circumference into six equal parts, forming a hexagonal shape, an I Ching manifestation of one becoming six. The 60 degree angle has relevance because it is the angle of the sides of the hexagons, or six triangles. In other words, the halfway point becomes the center for another circle. We also find an involution to an evolution sequence, where the entire radii of one side, in essence, becomes the motionless zone, a whole note, a (Do). The other side is divided into halves representing 2/3, 4/3, 4/5, and 5/6 time wise, a half note, quarter note, eight note, and sixteenth note, where it moves at a faster and faster pulse until it intersects with another circle, where the motion in the opposite direction halts where at the halfway point it again becomes a radius to another circle. Where these two centers find themselves in opposition, another core of a diminished size is formed. In this case it is the whole core that is the concave radius of the whole series. The size of the radius has differing center points, the first at the center point of the opposite circle, the other at the conjunction of the circumferences. The center notes were determined by utilizing the idea of double fifths which make up an ascending scale of (C), (D), (E), (F#, Gb), (Ab, G#), (Bb, A#), (Cb, B#).
If the ascending scale is an approximation of double fifths, this establishes another rule, of doubling, the measuring rod being the same, that of fifths. Double fifths give an impression of a contracted octave. The B# being closely related to (C), was placed at the half way point on each circle. The notes would descend on the other side. We again would have little trouble in imagining a larger fold down the center of the smaller core. The angle remains the same. There is simply an increase of fold.
This diagram is almost identical to the orbits described for illustration eighteen. The differences are that the entire field of nine notes or points represent numerically (Fa) 81, rather than (La) of eight points (64). Forty five dots on one side of the diagonal symbolizes (Re#) 45. Thirty six on the other side stands for (Sol) 36. (La) at 60 overlaps some of (Sol), and all of the (Re#). A traditional music staff was drawn, and notes placed along the orbit, in the order they fall, (Do), (Re), (La), (Ti), (Sol), (Fa), (Mi), (Mib) and (Do). The entire diagram should be considered as a moving diagram, the rings pulling out in negative tension, pushing in, intersecting, revolving around each other, when touching going in opposite directions, one in centrifugal motion, the other in centripetal motion, macrocosmos, microcosmos, out, in.
From our vantage point between the microcosmos and the macrocosmos, time seems speeded up in the microscope, slowed in the telescope. Does a magnification of large areas in a large scale decrease speed? Is time acceleration and deceleration? Is time merely speed? These are the sorts of questions we ask when we think of motion.
If we can imagine this drawing to be a close up of what might happen when the cores of the musical orbits meet, it shows a possible reason for the layers of different dimensions resulting in a stop, a reversal, a going in a different direction at the (Mib). Since this grid is made up of only eight points along the ladder of sounds, and if we imagine waves spreading simultaneously from point 1/1 and from point 8/8, they could meet at 4/4, (the midway point). It could be at this point that the shock of stoppage could generate a change of scale, a regeneration, nodes, or planets, visible worlds. The proof for this is offered in the following illustration.
"When circles overlap... the forces seem to reverse and the cycle begins again..."
This image might indicate how a core can be conceived as the contraction leading to an explosion, whereas the outer circles surrounding a square projection of a square would led us to think of the expanding forces leading inward to an implosion.(Drawing to the right of spheres diagram). The curved square-like form might lead us to consider a square form as symbolizing a contracting force, as opposed to the core form symbolizing expansion. If in the square extensions (drawing to left)24 we say that the limit of one is time, then one symbolizes time, three symbolizes space (three dimensions of space as opposed to one dimension of time).Four symbolizes the visible world, which is counted at right angles to time and space. five is Man,25 forming an imperfect square, but on both sides of the perfect square seven symbolizes perfection. nine squares represents the moon, which comes at the junction of the two ends of the core. Thirteen represents immortality, the limit of the extension of the square. It is this diagram which is the key to the numerology used in this work. (Also see Music Graphs Solo Orchestra).
This may appear at first glance to be merely a blow up of the previous illustration, but there is an important difference. The grid was set up diagonally within the square, so that each corner progressed by one, then three, then five, then seven. seven squares form a perfect square within the larger square framework. Because of this we are able to find a proportional difference between square one (the overall) and its tilted smaller one within it. At the same time we have a pattern very similar in triangular form to our original seed pattern of notes, this triangular form is, of course, a right angle, rather than a sixty degree angle.
Within the circular form bounded by the smaller square, the arrows represent a direction of motion, of different orbits which form expanding or contracting spirals, as each either goes towards the center or away from the center. The bar represents the point at which the arrows must shift either to the right or to the left. Musically, the line would look like the score to the right, when following the outer rim of the larger square (See darkest arrows). Direction becomes valid with a spiral as the notes can be arranged on this framework using the ancient chinese direction note correspondence.
This center portion could also explain why (Mib) reverses motion, and how the positive curve becomes a negative curve.
In this diagram we took the framework of the spirals in two different forms and imposed our note system on them.26 We coupled this with direction. We have also combined our system of numerology on the left. The diagram at the left shows how by starting with a rectangle,27 (Rectangle d'or), with the first unit the equivalent of two squares, we can use the first unit as a smaller section of the following, the next whole the smaller section of the next, etc. The notes were laid out on this pattern conforming to the sequence of the grid. The other spiral was used much the same way, one ring symbolized (Do), two rings (Sol), three rings (Fa), four rings (Mi), Five rings (La), Six rings (Mib). The following notes became (Lab), (Tib), (Re), (Reb), (Ti), (Do), (Re), as the end of the cycle is a very dubious note, because the ratio of the vibrations to the original (Do) show it to fall in an area of three notes, (Ti), (Do), and (Re). By this time the spiral has completed itself as it comes back to its original direction, and lateral point in space, but so close to the center orbit that it could dissolve into it, at a dimension quite beyond our visual perception. We will note that there are seven larger rings, and that the fifth sequence is (Mib). The notes which visually seem to be on a greater scale are further apart, for example. The next six notes are clustered together in a much smaller area, at least half the length of the first five notes. There is some similarity to the planets here, as one can picture the planets closest to the sun, Mercury, Venus, Earth and mars, as orbiting around the smaller closer rings, while Jupiter, Saturn,Uranus, Neptune and Pluto are those further out.
If one applies a numerology to the rings, it is rather interesting to see the consequences:
1. Venus or (Do) would be being and time.
The directions of plant growth as indicated to the left resembles very much the direction of movement of the Mayan Katon Wheel, as the direction of the plant growth is counterbalanced by criss crossing diagonally movements back and forth in an ascending spiral.28 That is the reason the bottom leaf is (Do). The fact that the growth is in an octave enables us to place our sequence of notes, on each leaf, and in the process we have a spiral movement on a vertical rather than a horizontal plane. This fits some of Ouspensky's theories of the sun spiraling within the galaxy, and we can think of the plant's growth as a miniature galaxy. The sets indicate how the directions were determined. Since we used the progression from the grid, in order to fit the Chinese order of the notes into this, we had to set up four different systems, much as the I Ching is thought of as moving Hexagrams. Therefore, we picked from the four sets the notes which complied with the Chinese order. If we substitute North, South, East and West for the bars and broken bars of the I Ching trigrams, we see how this works.
Since we know the ratios of the notes to (C), we might take numbers which are found in those fractions, and use them to determine the slant of certain multiples of the numbers. For instance, 2 would equal (C), because 1:2 is (C), and 3:4 (2 x 2) is (F). The combination of the dotted slanting lines with the more heavily drawn slanted lines (D) and (G) bear a striking resemblance to the dotted lines extending from the point of the squares in the progression by squares (Illustration P46).
While the placement of the notes is inaccurate in this diagram, since it was one of the first, and the notes had not been narrowed down to those found on the universal ladder of sounds compiled by Alain Danielou, nevertheless, the principals behind the patterns can be applied, if the scale is proportionately the same. The scale of the galactic patterns can be enclosed in a square, and can become that of an octave, as can be plainly seen. The numbers 38 to 45 comprise eight steps. The notes falling within the circumference of the circle represent an octave, a (Do) to (Do), because the second (Do) is found at the 15/16 point of the diagram. The idea of the big dipper analogy is purely a visual one, but one which does appear to bear a remarkable resemblance to the big or little dipper. Where the planets fall at each note, its symbolic planet orbit was drawn.The idea of the space being folded was applied to this pattern, the folding representing either a folding inward (implosion) or a folding outward (explosion). The right section down represents the shape each note joined with its counterpart would take. (Fa) forms a cone, or perhaps more literally a trapezoid, as (Do), (La), and (Ti), (our symbol for the universal ladder of sounds).
More importantly, (La) also forms a zig zag, our symbol for TIME. 45 degrees show the apparent angle of (Fa) drawn from the widest mark within the octave (Ti) to (Ti) (9/15-15/9) to (32/45-45/32). Since there are twelve tones found within the octave, and the number of the notes found along one side of the diagonal are twelve, this might be an indication of the logic of these theories. The notes might move inward from the large (Ti) point at 9/15 and then outward from that point in the opposite. This was the same point, we remember, that we constructed another circle to represent the orbits of the physically impossible undertone series with its notes all falling on the eights. This was also imagined as one of the unknown holes in space, where (Ti) or Saturn represents a contracting force. The reasons this section was labeled trace paths were because of the similarities of sound and its nature with electrons.
Of the two octave patterns, and triad patterns, the latter seems to hold the most interest as being similar to the pointed zig zag shape, in the progression by squares, trapezoids, etc. The points in any of these cases could indicate the place where one would hold a vibrating string to eliminate some of the harmonics. (A 1/7 eliminating the seventh harmonic, for example).
We have already noted that one of the puzzling aspects of music is the middle (C). When we count up from the middle (C) we come to (G) after five steps, however, when we count down five steps we come to (F). The ascending measurement seems to have been based on a different numbering system, than the descending measurement.
If ascending is expansion, while descending is contraction, and one is time, the other space, one is forward, one is backward, could we not say that space and time are joined like the fingers of two hands. In time we can only go forward, in space we can move in any direction, but preferably in the direction opposite from the direction of time. It is this imbalance of space and time which makes the inversion notes zig zag.
The notes are not optical mirror images in these inversions but are counting first up and then down in narrowing zig zags until we reach the zero point of middle (C).
Danielou, "Traite de Musicology Commparee" (Paris, Hermann,
Actualities Scientific et Industrielles, 1959) p. 86
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