The preceding discoveries might appear to be very inconclusive. They should, because we are, at this point, in a situation of having amassed much data, and have it laid out before us, as if we were primordial creatures, recreating a civilization. It is a step by step process, a detective story, of a sort, where writer and reader are both struggling to fit together the bits and pieces of insights and growing awareness of the significance of being.
We should think of the part of this study as being the time in history when all the sciences, arts, religions, musics, magic, were one, before they were separated into their respective roles, codes, information, symbols and particular languages of form. It is when we lay it all out that we can begin to sort and speculate as to its meaning, its usefulness, its credibility. The language is strange, the symbols are unfamiliar, the thought process does not fall under traditional logic, because there is no tradition to follow. It is a gathering of small pieces of apparently unrelated unknowns to try to make sense of the physical and metaphysical world. It is putting imagination and fantasy in the provinces usually dominated by fact and practicality, to try to see what kind of a world we would perceive around us. In essence, then, we are substituting imagination for reason. We might call this an age of imagination. It is an example of the primitive mind of mankind striving to understand, and musing about existence. Or, it might be an artist's or poet's or composer's thoughts while in the process of creating. In any case, its a focusing on the unknown rather than the known, so that it has little to hamper it's style, has nothing with which to compare its fantasies, and nothing of what has been learned before to cramp its thoughts.
We might wonder, at this point, why a search for universal values has not germinated or yielded much fruit in the human race? Why the emotional side of nature was cramped, and the rational side extolled? Why the mystical side was shunned and the pragmatic lauded? We will, then, try to turn the pages back in the history of time and by accentuating those facets of imagination, feeling and looking at the world as a new born spirit, we shall attempt to reconstruct everything it has perceived with almost all of it's senses. And progress we will, perhaps, develop a heretofore unknown, unnamed sense, which might be closer to what is known as abstract reasoning. But this is abstract reasoning with a difference, a visual dimension. It is perhaps mystical in the sense that it is trying to make the seen heard and the heard seen, and we might be surprised at the results.
So far, this whole study has tried to take number and play with it, to discover what it means, not in terms of numbers of objects exclusively, but number of vibrations within a given length. this same number becomes a proportion not only on the scale of space, but on the scale of time.
It is a deceptively simple tool to use numbers representing both space and time. We are trying to take on the perhaps impossible task of trying to coordinate space, the visual, distance, shape, place, size with time, the pulse, sequence, vibration, motion, direction, change, cycle. We find that we can make analogies substituting the abstract concepts of space and time to angle, form, shape, color, etc. It is these analogies which enable us to shift more easily from one dimension to another. This whole work might be considered an exercise in analogies, both visual and abstract. The purpose for these analogies is to extend our perceptions, and consciousness to other dimensions. We are giving in detail a process that might lead to this. We are followers of our inner nature, our intuition, which we believe is stronger than learned knowledge for this purpose.
This section implies concepts of perception. The idea behind this being one which suggests that the worlds which we hear, touch and see are merely the response of our own organs to themselves. Is the universe not light years from us, but within us? Do we have our faces pressed up to the universe, and don't even known it? Could all of our knowledge come from elaborate conjectures built up in closed systems? Why are our mathematical systems based on symmetry of inertia, one side of an equation balancing the other at zero.
It may be dangerous treading on the toes of established theories, but the human being at some point must become as an Adam of Humanity, to get back to the essentials. There will always be criticism of those who threaten the system of others. Ideas are still the most dangerous instruments of all, in terms of influencing others. Yet, to be in a system, to have grown out of any system, pregnant thoughts about the unknown might have been crushed in the beginning. It might be being without blinders, one can see the most. Any system of study creates its own blinders of language or code, as an ancient secret society, to keep all other except the initiate out. The non-expert being is searching too, without specialized tools. The average human being has much to say and think. If all ideas were presented to include this ubiquitous being, "the average" person, all sorts of barriers might break down and esoteric knowledge might no longer be hidden. Perhaps the arts are the best way of presenting such things to the people. The arts which deal with symbols, emotions, esthetic pleasures, which all can react to.
This thesis, the aim of this study is to create another alphabet of visual symbols which relate to the fragments of known bits about many subjects, so that all may participate in the discoveries of many fields. all these fields use diagrams to translate and recreate these diagrams three dimensionally so that when a certain shape, or color, or space, or interval, or ascending, or descending, or angle, or number, or fold, or framework is used, they all have a common meaning which can serve to add another piece to the puzzle of our universe, ourselves, our responses, our nature, physical, mental and emotional.
And so that this set of postulates doesn't fall into the trap of other disciplines, it is not intended to be a definitive, absolute. No one should try to put the blinders of these ideas on permanently, but should put as many blinders on as possible, and then take them off, to put others on. The point is to be constantly curious, and inquiring about everything known that no one thing holds the answers. It is the questions which are always of the same nature. It is the questioning spirit, the search of impossible absolutes, which give meaning to our otherwise dull life. If we try to focus our entire being on some one thing, it is true that it becomes an obsession. But we can try to aim the telescope or microscope of our lives so that it is pointed at whatever gnawing we feel, in order to understand ourselves and our place in space and time.
So this is aimed, at the poor, the weak, the old, the deaf, the blind, the lame, the physically, mentally, emotionally crippled, the young, the rich, whose wealth does not satisfy, the unlearned, the untaught, and the learned who wish to fantasize and be reminded of things they once learned and forgot, the doers, to whom the doing does not satisfy. It is aimed at those who like to lie on their backs and look up at the clouds or stars, and wonder what it is all about. It is for those who search in the dim collective memory of the past for the lost Garden of Eden of Mankind.
We can determine the relative lengths of proportions of a string, very quickly and easily by this method. The proportions are determined by the numerical ratios of the notes in music. By setting up two axii at right angles to one another, and then counting off 1/2, 2/3 etc. , we not only have a picture of the different lengths, but also the spacing between each interval. Visually the result of this resembles a fold, as if two areas receded towards the middle. The diagonal 15/16 (Do) divides our square almost in two parts. This is very reminiscent of the diagonal drawn to one side of a square so as not to connect points (time side, space side since the diagonal is the harmonic diagonal of the grid, any point on it is taken as a whole 1/1, 2/2, 3/3, etc. It also follows the same sequence as the vibrations of the harmonics on a grid, however the pattern of notes does have a different spacing in this case. We are dealing with wavelength, which is the reciprocal of frequency in order to portray musical ideas visually. (Mi) in frequency would have the ratio 5/4, 6/5. The two (Mi's) (4/5, 5/6) appear to disappear into the center of the bar. If we can remember that the entire ladder of sounds, within an octave lies along the diagonal, we can imagine the 15/16 line being this ladder, enclosing 52 notes at increasingly high ratios. If we can also imagine folding any square over at the one half mark, and then proceeding to fold at the next half, etc. , we will have accordion-like folds of differing depths, the most shallow being at the (Mi) area. The top web-like pattern shows how different the structure looks when the axis are reversed in number. We obtain a whole new concept of depth. We have already thought of this webbed structure as simulating time, as it seems to equate distance with sequence, and interval.
Space as symbolized by the lines of length, distance from one another, balance time on the other side using exactly the same for, a 1 to 15 structure in an inversion. The space side is less than a whole, the time side is more than a whole. Each letter should be given as multiples of 2/3 according to "the circle of fifths" formula, where one side of a triangle is doubled and the hypotenuse is tripled (harmonic-diagonal).
If we take just the physical appearance of the eye and the ear, we can attempt to draw conclusions about expanding versus contracting, inner versus out, the circular form of the orbits versus the thee dimensional spiral. We speculated previously about a descending scale representing a contracting. We also speculated about time contracting to space. We have also discussed implosion and explosion. Here, at the heading time contracts to space, is an imploding from the top of the cone, changing direction, at the implosion point, where matter becomes most dense, and smaller, and finally explodes on the other side of becoming the base cone.
We speculate from the present astronomy that the new blue stars are the rim of former galaxies. If we say that the (C) or (Do) is the newest star, we progress in terms of our music theory sequence to our familiar pattern, and complete an octave, we would find our own universe to be the lower half of the cone, where we would be progressing in a descending motion. However, in order to maintain a continuity in motion we must make a loop to continue in the same order. At the point where implosion becomes explosion, the image might look like the body on the left. (space extends to time.) The eye, then, in shape alone resembles the expanded point of space, a perfect measuring rod for space? The ear, in shape alone, resembles the contracting point of space, a perfect measuring rod for time? The ear, unlike the eye, multiplies one sound into harmonics which do not actually exist. 1
It is here that we wonder if the universal ladder of sounds is nothing more than the multiplied vibrations of the ear? The membrane of the ear also seems to shorten wave lengths which are coming from outside, in what might be considered a macrocosmos to the ear, and transforms these to a microcosmos signaling the brain. If in the study of music, we try to reduce each octave to its particular scale, we are doing what nature is doing, giving each scale the ability to measure itself within its own field. If man measures the sound environment around him by reducing large wave lengths of sound to short vibrations, does the solar system measure itself with its planets, and perhaps its asteroid membrane? The galaxies measure themselves by what?
If we think of the highest pitch in hearing vibrating at 40, 000 vibrations per second, and the highest pitch has its waves within an inch of space, and that the lowest pitch in hearing has 20 vibrations per second, which are found within a space of 50 feet, 2 it would take us approximately six steps to reach even fifty inches, based on previous construction of modules, and many more to reach fifty feet, unless we reduce each step to encompass a larger field. This is exactly what we find we must do.
We are very lucky to have an octave to work with. For as long as we encompass approximately a whole octave into each reduction, we have a very good measuring rod, which takes into account the multiple of vibrations very much as electrons have always a multiple as a unit.
This is not the place to go into detail the complexities of this reduction of scale, but it is important to that in the process of reduction of scale we go from a microcosmos-like scale of implosion, to a macrocosmos-like scale of explosion. This makes us think that the outside world of the sounds implodes into our senses, and explodes into our consciousness.
Specifically, the diagram here presented is a rough attempt to link mathematics with a sense, that of hearing, and an idea of the emotional response of linking number with a word symbol, which conjures up a host of image, and mental tones together. At first glance this may seem superficial. The diagrams were taken directly from another source, with the notes superimposed over this framework. 3 The reason this diagram was chosen is that the shape of the area of hearing was quite fascinating, very similar to our familiar half circle, whereas the square progression of the falling object was also familiar, so that obvious analogies were found.
The idea of time and distance co-existing, seems to relate time and space. The formula y=16 x 2 also coincides with the end of the first cycle, (15/16). It isn't as though these images and thoughts are definitive in themselves, but that they lead to other ideas, which in turn engendered others. It is this mind stretching type of reasoning which is important, not a conclusive statement. From the previous pages we are aware of the origins of the number symbols. They, too, could have been different. The one's we finally chose have been used consistently throughout. It is the consistency of a few numbers which gives this work its framework. It is because we have decided to use a few unchanging numbers, that we can spread over so many subjects. We simply want to see what will happen if we apply these numbers to anything we can think of. To our knowledge, this is not usually done, in mathematics. That is why we like to think of it as a new form of reasoning where we have specific numbers, we just have to fit everything else into their pattern.
The numbers were chosen for two reasons. One, they were the ratios of intervals known in music in Pythagoras' time up to the present day. Two, they were chosen because of the ease with which they could be applied to intuitive flashes of what might best fit the numbers. In intuition something inside says, yes! That something inside is sometimes called innate knowledge, a knowledge which is barely recognized today as significant, but which exists in all of us, so that when we are presented with an idea or image, or sound, an affirmation nods yes within us. If we should take these concepts of essentially single words, and analyze them as applying to our innate knowledge, let us see what we might discover. Time appears to be equated with being. Time is movement, traveling a distance. In our bodies all is movement. Being is movement. Thus time is being. Space is distance. In space objects are either attracted to one another in a type of gravity, implosion, or repelled in a type of anti-gravity, explosion. Thus space describes the area of explosion or implosion. Space taken on a larger scale denotes an enveloping, of other space, our bodies enclose their space, the world encloses its space, the solar system encloses its space, the galaxy encloses its space, and the universe encloses its space. To some the universe is God. To some the universe is made up of natural forces. Thus, space can be both God and nature. The visible world is a world of objects we can see, feel, touch. The invisible world is a world of perceptions emanating from an object, smelling, hearing. Organisms seem to be the receptors for these objects and emanations. The earth is an organic whole. Between the human and the visible world is the invisible world of longing for perfection. Perhaps the pattern of growth to this perfection is by means of an octave development. We have already established that the octave deals with eight directions, N, S, E, W, NE, SW, NW, SW, eight tones, (or twelve, or fifty two), eight steps, seven vibrations, the entire color spectrum of the three primaries, blue, red, yellow, black and white, and the secondary colors of orange, lavender and green. The octave deals with sequence, and with cycles, with time and space. The octave deals with whole numbers and fractions, with shapes, ascending and descending motions, and with reductions and expansions.
The moon creates the tides, growth of trees and plants, a generator to the earth. Within the hollow of the moon would be a reverse perspective. (Mi). The moon would be the rim of the universe to whatever is inside. A new cycle is a half of a circle, with life on one side, death on the other side, one half of a wave imploding on one side, exploding on the other side, and reversing as it travels from one half circle to the next in vibrations of the earth in time. Beyond the skin of any object is immortality, and the ability to choose the positive or the negative, or good or evil.
This then brings us to the problem of inside, outside. As we know, the only way to go from inside to outside without actually turning over a piece of paper is by way of the half twist of the Mobius strip. A music score inscribed on one side of a strip of paper, shows what happens when the whole is twisted and flattened out. It forms a triangle. When it is cut down the middle and pressed flat again it forms a square. one half of the square cut along the diagonal represents the inside, the other half represents the outside. This is extremely important as it recalls our grid and one half of the square becomes space, the other side becoming time. What might be significant is that the diagonal might be the place where we could begin our angles. The diagonal represents 180 degrees. All the other angles could be determined on the outside section of the strip (one half of the diagonal. ) To go even further, cutting again through the middle of the strip, we find we have two pieces, linked. One can be pressed flat to form a rectilinear shape, with the staff running the whole length of the shorter side (outside) and the other (inside) running the entire length of the other side. On one half of the other two sides, each outside is directly over its counterpart. The other loop can also be formed into a square by flattening out four corners, and again we have one half separated by a diagonal forming outside and inside, space and time. It is the metaphysical significance of this operation which we wish to convey. It illustrates the apparent incommensurability of space and time. It indicates that what separates space from time is the fold. It reveals that what is inside the fold is time, what is outside the fold is space. This procedure also shows us the importance of angle. In the first instance the three sides form 60 degree angles to form an equilateral triangle. In the second case, the four sides form angles of 90 degrees in a perfect square.
In the third cutting of the Mobius strip, the eight sides form two interlocking rectangles. The fact that a physical manipulation of one unit, so to speak, can yield so many different shapes, so easily, gives us much to think about, especially if we remember that it is the flattening of the loops out into polygons which we can analyze. We have previously discussed the angle of the radius as indicating the degree of motion. In the ninety degree angle, we remember the motion is stopped, whereas in the sixty degree angle the motion has the greatest acceleration.
The musical ratio axis developed by Alain Danielou 5 was drawn to show how the multiples of notes on different axii could conceivably fit into the Mobius strip. His patterns of triangular shapes which suggest folds in space could apply. They all fit into a basically square shape. This is only one analogy, the imagination could supply many more.
The I Ching Hexagrams are pictured here to indicate the possible analogies to the concept of inside, outside, or opposites such as the creative and the receptive powers of change of the Hexagrams. 6 The chart also shows exactly how the placement of the notes was arrived at. The whole number or its equivalent ( a multiplying number ) was placed at the square of that number in a spherical form, and given its appropriate color from our music color analogy. The pattern becomes a fixed pattern visually. But if we imagine each sphere to be on an orbit with the center, the center of the chart, we have the exact same progression arrived at by halving the circle first toward the center four steps, then back toward the rim oscillating. It becomes a moving diagram close to Ouspensky's via Gurdjieff's ideas. 7
If we wish, we can give each note its corresponding Hexagram ( a combination of the horizontal and vertical trigrams) and by its number determine chapters of images, dealing with psychology, philosophy, change, attitude, a bible of thoughts.
The images of light, well, increase, small, thunder, lake, fire, wind, wood, eye, ear, hand, mouth, foot, thighs, belly, the images of merely eight notes of an octave give free rein to our imaginations, where we can wander in even greater expanding realms of thought.
To get back to Danielou's diagram, the inside core, is simply a detail we worked out to show what could be happening at the center of his. Ours included points one through six. This is applying our theory of six steps or points of activity to complete a cycle. A cycle implies movement and change, so we are not too far removed from the I Ching book of changes.
The drawing of the element #45 happens to be the number for (Re#) or (Mib) in our system. (Re#) or (Mib) is the core which disappears into another dimension, as if an energy pulled it to the inside with such force that it missed its point of inertia and went on to swing back and forth at the outer rim of the circle. (Mib) or (Re#) is also the hexagon and the 60 degree angle, THE GREATEST FORCE. This also shows how we can apply the numbers of the notes to the periodic table of elements, and in so doing conjure up images based on what is known about these elements in terms of numbers alone. 8
Of all the ideas in this book, this one is perhaps the most important to us. It is important, because if the radius of space is considered to be a function of time, we might be close to solving the riddle of SPACE TIME. 9 the radius is integral to our thinking. The keyboard itself can be considered as diminishing and increasing radii, as each note is played.
An octave would be the limit of the increased radii, whereas (D), or (B) would be the diminished radii. The ascending could be considered positive, the descending negative. Just as the clock face of a watch has radial arms to count time, angles as determined by polygons are the radii of the circles around the polygons. A way to determine angle as a point of reference is to take a circle, and starting from (Do) or (C) with a ratio of 1:2, one half of the circle if 180 degrees. If we start then at 180 degrees and plot out our notes from this reference point, we will have notes representing three sides, 120 degrees, (G) (2/3), four sides, 90 degrees (F) (3/4), five sides 72 degrees, (E) (4/5), Six sides, 60 degrees (Eb) (5/6), etc. The limit is reached at twenty two sides, which brings us back to the full circle, and there is no need to go further.
The other thing which is extremely important is that if we take a string the length of a closed circle, and lay it out on a straight line and section it according to the same fractions, and then compare this to the degree marked on the string, the two coincide. What is remarkable about this is that the systems used were very different, angles forming polygons vs. a folding or sectioning of a string. But not only does the string correspond to the angle measurements, when the two are laid out together in a straight line, but it also corresponds to the diminishing orbits of a circle by halving toward the center, and then towards the rim. What we seem to have is a picture of a section, and a picture of an elevation, almost as an architectural version of a building, floor plan and view from the eye level. But this different because the height, as such does not exist here. But what exists is on a different dimension. We are using a new perception to recognize it. If the radii are a function of time, we might have a spatial and temporal picture, a co-joining of the organs of our ear and eye.
If we can imagine the circles with their inscribed polygons as patterns of the forces of life, and as the radius swings around the circle, the patterns change, ascending, descending scales are heard, accelerations take place, and we find ourselves in a cosmological universe of imagination.
The next four illustrations show how we can apply a philosophical system to the scale of octaves. One of the reasons the I. Ching or Book of Change was chosen as a framework was that it is based on number (1 to 64). It also deals with beginnings of the world or universe in very visual, abstract and interchangeable symbols. The numbers represent steps on a scale, and a sequence of becoming. Starting with number six, they represent the elements of water, fire, wood, metal, and earth. According to this, the beginning was water, which turned to fire, which became wood, which transformed to metal and finally became earth. Musically, it is (A), (F), (D), (C), (E), (B), (Eb), (G), (A). We were able to arrive at this scale by translating whole numbers of multiples into the hexagram symbols, and then re-translate these back into a sequence of numbers up to ten. This whole idea represents the five stages of change, as it was in the beginning of time. We merely set up the numbers as objects joined by webs of lines, representing time.
A square was divided into four equal sections. On each section equidistant dots were put. A diagonal was drawn through the space between the dots, and a time-space dimension was formed. A circle was drawn with its center at the middle of the original square. . The notes were laid as objects around this circular orbit, as they would appear as objects instead of numbers, objects which symbolize musical notes, and represent what is known as the Prima Axis in the I Ching. Within this circle were laid, as if on radii, the note spheres representing the inner world axis of the I CHING. These radii indicate the concept of time, as the outer circle represents the concept of space. If we fold diagonally across, along each of these axii, the whole takes on a spiral shape. The folds must change direction at the fourth step, which is (D#) or (Eb). This we can not help but notice is the same place the direction changes in the Mayan Katon Wheel, a dividing of the orbits by halves, the dividing point of a string when we shift from one half of a string to the other half, the focal point of the time contracting to space. (Illustration P-34).
In the patterns of multiple of 2, 3, 5, 7, (Eb) is the angle of most acceleration (Illustration P-26 dotted line). It is the end of the first SPACE CYCLE of 9/8 on the grid (illustration P-13), the point at which space become time. It is the point of reverse in the contracting spiral (Illustration P-24). It is the beginning and the end note of explosion becomes implosion (Illustration P-23), as well as being the tip, or limit, to the four corners of the square in that diagram. In illustration P-22 it is the balanced seven squared framework which holds the implosion within the square, the explosion on the outside of the square, where we can trace the forces seeming to reverse are close to the shaded square area. In illustration P-21 it is the area which appears to be a square bounded by triangles on two opposite ends ( a significant shape of a hexagon, and one which matches nature's hexagonal shapes).
In illustration P-20 it is the middle section of a music staff set up on an orbital relationship of one note to another, where the notes were placed depending on where the orbit fell, and, of course, the orbits were hardly arbitrary, as we have seen. In illustration P-19 its the place at which we find the beginning of life, and the fastest musical pulse of the thirty second note. It is at the end point of the involution axis of illustration P-18. It is the end of the mathematical cycle of the circle of fifths in illustration P-16. In illustration P-15 it becomes the boundary of the end of the time where space begins. It is the area where the inner eye looks inward to the infinite in illustration P-8. It is the approximate 60 degree angle on the chart of galaxies in illustration P-7. It is at the limit of the macrocosmos illustration P-6. It is the spot at which the vibrations on a string are reversed in illustration P-5. It is the only note lying outside the triangular form in illustration P-3.
We are making these analogies illustration by illustration to show you how we could take any statement and trace it back on itself. Whether they would yield so much that is similar would have to be experimented with. The question raised, is why do so many things fit? Is it the way we have to set up the whole, so that it becomes only a function of our own reasoning? Or is it a universal phenomenon which leads us to greater truths?
This diagram merely shows how the preceding diagram was constructed in accordance with the Chinese symbols Chien, Sun, Kan, Kun, Chen, Li, and Tui. If we examine closely these bars and broken lines, we discover that they are direct opposites of one another not only in direction (at each end of a diagonal) but in subtler ways... three broken bars... opposite... three unbroken bars, etc. The fact that these are pictured as see saw type forces, balancing one another is an important facet to understanding he implications of this image. It is reasoning at a very abstract level.
If our reasoning processes could be so balanced a new astonishing world might open unknown vistas to our minds. In a sense, they are inversions. One order is called theprimal order, not the order as we know it now, but the first order of the universe. If we examine the second order, the inner world axis, we find an even more asymmetrical balancing. for example, Chen is balanced by Sun, three bars... to bars, one broken. Kan and Li are the same in both systems, one bar in the middle, versus a broken bar in the middle. Ken is balanced by K'un, two broken bars, one bar, versus three broken bars. Chen is balanced by Tui, one bar, two broken, versus one broken two bars. When we translate these symbols back into another vocabulary of notes, we have in the primal order (Fa), balancing (Re), (Mib) balancing (Mi), (La) balancing (Sol). (Ti is a mistake in the drawing!)
While the inner world axis we have musically (Fa) balancing (Mib), (La) balancing (Sol), (Ti) balancing (Re) (Mi) Balancing (Do).
Numerically, this becomes in the primal order six balances two, four balance three, one balances nine, eight balances seven. While in the inner world axis, numerically, we have six balance four, one balances nine, eight balances two, three balances seven.
Could we not take an equation for universal phenomena in the visible world. The primal numbers, and for the equations of the organic or inner world the inner world axis numbers?
An equation might be:
X = One Axis
If we actually construct objects for this equation, one square stands for a unit of one, two squares stands for a unit of two, etc. , the length increasing while every other dimension is held constant, applying the first order to these objects and arranging them in order, we have a very non-symmetrical form of varying lengths. However, if we apply the laws of the inner world axis to this, our objects maintain a perfectly balanced symmetry of four bars, each segment of which adds up to ten units. The shape then becomes either a square or a cross. Further, if we add our planet correspondence to both orders, we find that the earth (nine, (G)) and the asteroids (one, (A)) remain constant in both operations.
There are four rows, then, of eight interchangeable pieces, the ratios being 4/6, 1/9, 2/8, 7/3. In Chinese mysticism the odd numbers symbolize heaven, the even numbers symbolize earth. If we say that earth means the material world then Mercury and Jupiter (four and six), and Saturn and Mars (eight and two) represent the material world, or musically (F), (E), (B), (D). Whereas, the asteroids and earth (one and nine), and Venus and Jupiter (seven and three), or musically, (A), (G), (C), (Eb), represent heaven, the invisible.
Are we, then, part of the invisible worlds? Are we then invisible, and to whom? Perhaps we are invisible to the creative forces of the universe, just as those forces are invisible to us. Musically, too (Eb) and (F) and (D) and (B) would be the material, yet these are considered to be the unstable note in music, the second and the seventh. It is the (A) (G), (C), (Eb) which signify heaven, which, with the exception of (A) and (Eb), are considered the stable notes. Are then, these the notes which we long for, to complete our existence? The (E) and (Eb) assume such a dichotomy, being both in the material and the immaterial world, a connector, so to speak, between the visible and the invisible.
Even the bars we have constructed are arranged in such a fashion as to create a material world surrounded by the immaterial, which is, in turn, surrounded by the material, a progression from the material... immaterial... material... immaterial. This, then, is the sequence.
In order to see exactly how this was constructed, it is necessary to go back to illustration P-37 of the I Ching hexagrams. This diagram is simply an enlargement of the first, with the addition of orbits drawn to each note from a center point. When the resulting orbits are studied, it is perhaps not so surprising to find that they fit exactly into the same orbits we have been dividing by halves, sectioning angles, and a given length of string. This pattern also indicates where we might start, if we began rotating each note on its own orbit. To determine the rotation of each orbit, we might take the rotation of the rings of Saturn, or the rotation of the planets around the sun, or of electrons around a nucleus, etc. The fact that the notes do not correspond to the natural order of the planets around the Sun, we have already discussed. If the motion is in a zig zag as if there were folds in space, this order would then fit into the pattern.
Something which should be brought out here is if we analyze the steps necessary in determining the placement of the notes, we have fourteen steps. We can also see the balancing of the Chinese symbols at work. To analyze these here would take much extra time and work, but anyone could work this out and probably find a meshing of the primal order and the inner world axis. The little idea at the left has much importance, as it might be an indication of how the numbers of the notes originated.
If we divide 360 degrees by five (which is Man) we arrive at the figure 72 (Sol), we could either divide this in half to arrive at 36, or we could divide 360 by 10. 360 is, of course, the number of degrees in a circle. These degrees are not arrived at by simply sectioning a circle into quarters, eighths, sixteenths, thirty-seconds, sixty-fourths, one hundred and twenty eighths, two hundred and fifty six, and five hundred and twelve seconds. Why 360 was chosen is an unknown we cannot discover. 12 Perhaps it was done to facilitate a quartering of a circle into 90, and 180 degrees, etc. which idea seems dependent on polygonal shapes. (See illustration P-38).
This illustrates the reasoning behind the construction of an axis with its numbers running in inverse directions. This is important because to fit into everything else with which we are dealing, we know that one axis goes inward, contracting, as it were, while one axis travels outward, expanding and increasing. In the very beginning of this study we noted that different ways of measuring yielded very different results. Here, we see that these different measuring rods, the finite and the infinite, fit together as right angles. Before, we did not realize this. Therefore, a whole is beginning to take form in our minds. If we, at the same time, combine the numbers with which we have been working up to this point, we will obtain slightly different configurations in terms of the shape of the curve. For instance, the number ten indicates completion. We can indeed plot notes (Do), (Sol), (Fa), (La), (Mi) on this web by simply joining the two numbers in a ratio. One is joined to two for (Do), three is joined to two for (Sol), four is joined to three for (Fa), four is joined to five for (Mi), five is joined to six for (Mib), three is joined to five for (La), and seven is joined to eight for (Re). This, we recall is the space dimension.
The time dimension folds over into the area which is unseen, invisible. When we apply the number seven to indicate perfection, the last one of the space dimension, and the number 11 to indicate a completion of a cycle, we see very visually how this can be possible. Fitting all of these together into a perfect whole is a task beyond the present capacities of our perception. However, we like to think that in the future this will be done as simply as adding or subtracting or dividing and multiplying are done now.
This diagram was indeed an attempt to link the numbers of the I Ching together in a web. At the center of the diagram is a square made of four spheres with a sphere in the center, as five symbolizes Man. Each of the symbols stand for earth, metal, water, wood and fire. Each has direction and season, and color, and number. (La) then, is North, water, winter and the number one and the color lavender. (Ti) is eight, Northeast and earth and the color black. (Mi) is three, East, yellow green, wood and spring. (Mib) is four, wood, yellow orange, Southeast. Nine is (Sol), South, blue, fire and summer. The Chinese color is different from the above. See illustrations P1 and P9. Seven is (Do) West. Metal, red. Six is (Fa), metal green, Northwest.
The factor which makes these numbers translatable into notes is the fact that notes can be changed for numbers. The numbers of the I Ching interchange so that nine becomes seven, becomes six. Eight becomes four and three. Six becomes seven and one. Five becomes two and eight. Seven becomes nine and two. Or as the I Ching says "is complemented" rather than becomes. Or it could be stated in words, metal becomes fire and water, wood becomes earth and wood, water becomes metal and water, earth become earth and earth, fire becomes metal and earth. Translated into notes we have (Sol), (Do), (Fa), (Ti), (Mib), (Mi), (Fa), (Do), (La), (there is no note for five in the I Ching System) (Re), (Ti), (Do), (Sol), (Re).
This series on the I Ching, give a sort of flesh and bones to this work. It brings the symbols back to direction, movement, color, seasons, elements, parts of the body, the everyday world of lake and mountain, journeys, thunder and wind, fire, and things of earth. It, also, is carried further, by reading each chapter signifying the note numbers it would give one a whole philosophy of behavior, a psychology of human response, fears, loves, hates, adventures, states of being required for different times, and extolling of CHANGE, which is as natural as the waxing and waning of the moon, and how to meet it. It gives a spiritual direction to those who turn to it for that end. It is a bible of meeting change spiritually and psychologically. It is not really, as so many think a fortune telling sort of book. It can be read with great enrichment as one would read any book on spiritual growth. As I mentioned before, it is the number system which makes it of great use in symbolic analogy.
The I Ching diagram is very similar to a chess board. We could have used a chess board to yield analogy, with music, except that the motion on the chess board does not have a definite sequence. Its laws are based on skill, and chance. The element of chance in the I. Ching is found with the use of yarrow sticks or coin oracles. But one can completely eliminate this part of it, as we have done.
This drawing indicates a progression not only by square shapes, but also a progression of some of the ideas portrayed in this book. In illustration P-6, for example, just before the folding and unfolding square the macrocosmos and the microcosmos is portrayed. This could be interpreted as a point of expanding or contracting before a folding. This change of scale occurs at the sixth square, the (Mib) square. The first square is Illustration one, the seed within which is found the totality of existence. Time folding backwards into the seed. It incorporates the idea of the universe as being basically a music. We have dealt with the music of the worlds, in the aspect of number and measure, at a very elementary and basic level, which does not preclude the omission of many essentials in mathematics. But briefly, we dealt with number as if each number were on its own orbit, with the distances of the orbits from one another paralleling the distance of the planets from one another, as if (Do) or (C) were the sun. (Illustration P-2). We saw how one right angle can create triangular lines of force, that one side of a triangle is a dividing axis, the other a multiplying axis. (Illustration P-3).
We saw how certain numbers can rotate in a circle, and come back to themselves, after thirteen steps. (Illustration P-4). We saw how numbers could be counted as vibrations on a stretched string. (Illustration P-5). (If analyzed, these show an adding to three notes, until we come to the ratio (4/3 (F)) 1024, at which point we go back to 256, and add to obtain 1280). And we find that 512 and 768 also equal 1280. From this step, which is the fifth, we must add 256 again twice to the preceding 1280 to obtain 1792, (Eb). We saw how we could set up a crossed diagonal, the multiplying element being on one side, the dividing element being on the other side. (Illustration P-6). We saw how number could be extended by rays (Illustration P-7), and divided by segments (Illustration P-10). We saw the possible relationship between time values of seconds, as laid out along a line, as in a music score, and fractions representing notes. (Illustrations P11, P17, P19). We took a whole field of note numbers and laid them out in an array of ascending and descending fractions, (Illustrations P13 & P14), and wondered at the patterns formed, as we put these on orbits representing overtone and overtone. We pretended that groups of numbers could be folded in such a way as to yield a sequence which would fit into a pattern of dots as laid on a grid. (Illustration P-14). We determined which notes would fall on which orbits when opposing circles were drawn from an end of a cycle, and from the beginning of a cycle (C to C). (Illustration P-15).
We found that dividing by halves on radii signifying circular orbits yields results that correspond to the sequence of notes, but the changing direction of movement is what caused this correspondence. (Illustrations P18 & P20). We made number correspond to direction. (Illustration P-25). We gave lines lengths and direction by using the patterns of multiples as the determining factor. (Illustration P-26). We extended lines standing for octaves and thirds to make certain patterns. (Illustration P-27). We assigned to each number of dots within an orbit a physical and a metaphysical meaning. (Illustration P-28 & P-32). We tried measuring proportions as if from two right angles, divided proportionately into divisions which correspond to the musical ratios, and then we tried reversing the proportions (signifying time) at the other right angle, opposite. (Illustration P-33). We show how points on spirals can be either expanding or contracting, representative of the eye and ear in actual form. (Illustration P-34). We showed how seconds could be coordinated with distance, a known law, and how this fits in visually with the total picture of the progression by squares, and showing how each musical note fits into this picture. (Illustration P-35).
We proved that the patterns of notes, numbers, laid out on the I Ching square grid, fits into the same sequence of number-note relationships as other systems we tried. (Illustrations P36 & P43 compared to illustrations P48, P18, and P20).
We saw how we could create shapes based on the polygons (number of sides determined by the ratios), and we could also determine angles of each note in terms of acceleration (based on Ouspensky's theories).
We saw from other I Ching ideas how numbers could be related and balanced by seemingly asymmetries. One to six, two to seven, three to eight, four to nine, the five being central to each set (i. e. 1 + 5 =6). (Illustration P-40). This idea could be applied to the first order rotating around the circumference of a given circle, the second inner order rotating as radii, pointing to the outer order. (Illustration P-41). We see how each set must have a round, odd and square, even number, a merging of heaven and earth, according to the I. Ching. (Illustration P-42). We saw that each number must have a set of steps to complete a cycle. (Illustration P-43). We found that axii must have direction, called in this case evolution and involution axii. (Illustration P-44).
We have not dealt with weight to the extent we dealt with number, as it should or would have to be dealt with to illustrate the sounds physically. Each string, (if we used strings) would have certain tension to it to reproduce the wanted sound. We have made only very sketchy analogies to planets, but have used extensively the idea of orbits. We have determined direction and pattern as if we begin with a static planet universe. We made conjectures about the character of Jupiter, the note (Mi) and (Mib), which occupies a crucial spot in our system as either a reverse of motion, or a change of scale, or a mirrored illusion of the planets nearest the Sun. Motion was never specific speed, but the idea of motion is constantly with us in either rotating spheres, orbits, spirals, see-saws, interlocking circles, vibrations of all sizes, expanding, contracting, zig zags, angular accelerations, folding, flows, oscillations, pulling and pushing motions, ascending and descending, growth, galaxies drawn into a source, implosion, explosion, rate of acceleration by falling, and transformation of one number or material or note into another.
In terms of time, we discussed the Mayan Katon Calender, with its intervals of 20 year periods, one unit of 13 sections equaling 20 years. We discussed time as vibrations per second, even melody becomes pulse, while pulse becomes melody above 15 vibrations per second. We looked at the possibility of denoting space in terms of time by equating note ratio to time as having the same fractions. (Illustration P-11). We saw how time could be denoted by a line of dots for each second. Each note could signify any unit of time, such as 20 year periods, if applied to the Mayan Katon Wheel. As an example, 260 years would be the length of the first octave cycle of 13 notes. (We determined a cycle, if it encompassed an octave of sound).
Therefore, the entire cycle of the 53 notes on the ladder of sounds would be completed in 640,000 years. We determined that the time side of a square is the inverse of the space side, and that we have to use different measuring rods for each, as the numbers included in the squares vary. In a different sense, we decided that an angle between two right angles would compose the time dimension. We tried to equate a given note on the scale with a quarter, eighth, sixteenth, etc. duration of time. We decided that where two forces overlap, the acceleration must decrease. We dealt with the seasons as time in the I Ching, with units of four.
To go fully in to the time aspects of this would mean another book so for now these are just suggestions of this aspect. We would have liked to speculate, however, on the idea of 1/10 of a second being a crucial element of time in regards, at least, to our perceptions. For it is not only the alpha rhythm of the brain, but also the crucial interval in the perceptual process as the minimum quanta of time experiences. 13
1/10 of a second is the sweep of ascending and descending wave in sound perception. In the I Ching, we could apply the unit of ten as being a 1/10th of a second, as it found in the four bars of ten units in the modules constructed on the principals of the inner world axis. This modular unit could also very easily form a four sided square, instead of a rectilinear solid. The four sided square would remind us of the folds of a Mobius strip cut once down the middle, or the four seasons.
The music of humanity, we have left almost untouched. This again, could be expanded to another book. Life and death, body and soul, are the studies of a lifetime of living. We conjectured that the infinite is inward, the eye which looks on our world is blind. We took as humanity, the earliest beginning of the Chinese, the Greek, the Mayan, to discover how they related music to the senses of color, and to see if we could fit our system into theirs. We could. Rhythm is only based on the heart beat, we said. These are the rhythms which we can actually use and reproduce. The larger rhythms and the smaller rhythms we can only consider abstractly for the present, as lying outside the domain of our senses, but nonetheless there. We have really only drawn a possible map which extends to the beginnings of number, music.
Human, God, immortality, perfection, being, good and evil, the visible world, imploding, exploding, the void, cycles, changes, these all were only suggested, But if we can draw anything from all of this, it is that our visible world is the point at which motion on the surface has ceased. Within our bodies there is motion, without our bodies there is motion. Are we and the entire visible worlds of, organic life, the world, the stars, the galaxies, the nodes of creation, the points of non-motion on the vibrating string of creation? Are we, then, and our entire visible world the realms of the dead, the motionless, the stagnant, the static, where, in essence time has ceased to be, so we must create it, so that it encompasses our still state? Does this then mean that there is no beginning, and no end, no space and no time, but only vibrating motion and nodes of silence and stillness? Therefore, seconds, minutes, hours, days, weeks, years, centuries, millenniums are all one. Why are the stars so fixed in the sky since the beginning consciousness of mankind, if this were not so? The stars and galaxies, could they be the fixed points on tensely drawn invisible strings, plucked by invisible forces, each vibration causing a different quality of sound and shape, and sequence and pattern. So the music of our earthly instruments mirrors as closely as anything in this visible world of ours the universal harmonies which exist in the universe. Again, the specific study of the music of the instruments would take another book, dealing with the science of acoustics, coupled with the science of esthetics, harmonics, overtones, undertones, emotions, intuition. I do not mean to imply that this has not been attempted. It has. But if it could be redone with the idea that all is one plan, one puzzle, that each part fits into, to create the whole design, it might elucidate our concept of the universe, our concept of our place in the universe, and even our purpose within it.
Therefore, this study proposes that we restudy everything from the beginnings, as though we knew nothing of what we think we know at this moment. It proposes that we concentrate more on the unknown, rather than the known aspects of physical phenomena. Perhaps it means reinventing the wheel. Perhaps it appears to be going backwards. But we must reassess everything. It is the only hope, as we see it, of realizing our oneness with the universe and humankind. It is time that our lives became not fragmented bits of separate egos competing for shadows of security acceptance, power, wealth, prestige, but organic parts of a totality of which we, as thinking feeling organisms, have a major part. The consciousness of the human race is at stake. Not only the consciousness but the happiness and the longing of the human race for a fulfillment of a spiritual nature is imminent. We are the conscious nerve endings of a power of creation which is trying to use us as instruments of a higher calling, instruments by which the force of creation can know itself.
The outside world exists only in so far as our imaginations conceive it, or our perceptions acknowledge it. If we extend the boundaries of our consciousness and awareness, then the outside world expands with it. What is outside and what is inside are the same, the same patterns, the same structures, the same cones, trapezoids, spirals, diagonals, vibrations, squares, triangles, circles, colors, lights, darks, textures, intervals, all on a smaller scale.
The functions of an organism exist in a circular pattern. The harmony of synchronization of parts make a whole. We as individuals are both whole and part, ideally synchronized with the rest of humanity, as well as with the overall cosmic harmonic structure of the universe itself. Isolation begins when the individual separates the self from the natural universal order of these harmonies, and considers the part in itself, the whole.
To plug each nerve of each of the senses into the outlet of the underlying creative force of the universe, we are carried along to participate in the destiny of the human race, as instruments of a higher power. Are we able to think of ourselves as only sections of huge dimension which extends beyond the ability of the present imagination to grasp, a section like that of a slice of an invisible body? Just as a slice of an apple give as a pattern entirely different from the found whole we see, a slice of our "spectral" body gives a pattern of ourselves as seen only with the organs of our eyes, an imperfect and misleading whole, which some of us mistakenly call the "real world". If a circular form most of us could acknowledge is that of the half circle from birth to death, if we go further and turn it around to be the circle from death, our present state, to life, we could still stagnate in still pools of waiting whether for life or death. But as each of us consciously tries to fulfill a purpose, which each of us was innately given, if each of us consciously tries to submit to our destiny, by turning all of our being and our thought to this end, then surely what we search for will be found.
If we can approach the next few illustrations with all of this in mind, realizing that what we seek is an answer to all sorts of questions, and to try to see if what we have so far discovered relates in any way to fitting the parts of separate knowledge together, so that without being specialists, we may have at least a sense of the searching for wholeness, we will continue with this most important sequence, a progression being likened to a falling body.
Let us try to imagine that we are on this falling body. We have been involuting, so to speak. We have taken six steps in the involution process, as though we were accelerating like a falling body.
When the center of this involution motion is reached, and the velocity is greatest, we find ourselves in the area of dissolving into nothingness. (On the musical scale Eb).
The pendulum swings back and forth in an oscillating motion. We find ourselves on a radius of a circle, where we would hear repeating tones (Do) to (Sol) to (Fa) to (Mi) to (Mib). If we look outside this radius we would see something zig zagging into the circle. These zig zags are as a sound cone on the universal ladder of sounds. We originated from the first, and as we swing into the center of the circle, we see the others touching each expanding square on our radius of life. So we might realize that the vibrations travel along these jagged lines leaving the nodes or points of stillness at right angles to one another. We see the pyramid shapes of Egypt, and see that the limit is reached at the note (Fa)/(Mi) (81/80). Then, we are pulled from inside to the outside, as if we were a glove being pulled off. The primal arrangement has been transformed to the inner world axis. The end has become the beginning in an astonishing series of folding into a triangle. It is this triangle which becomes the generator or the seed pattern of the clusters of sounds, which we see as spheres, the diagonal line, being, of course the harmonic line on the breadth of the triangle. The folding backwards on the dark strips, only indicates how the ever expanding cone, by folding backwards on itself remains the same size always. Only the density changes, as it expands and folds, it becomes less dense, and slows down. If density and weight have anything in common, we then have, dealt with weight.
At the rim of the cone acceleration almost ceases. Is this the rim of the galaxy, the milky way, our sun, our earth, ourselves, our motionless state? Do we now have a visual image of the vibrating strings of our galaxy? Is it a section of a funnel with invisible strings of force pulling it towards a black hole point in space?
If we old see this galaxy, as if it were a flat disk, this diagram might give us a clue to the visual interpretation of universal harmonies as applied to our galaxy. Remember, that it is in eight contracted octaves which we have discovered to enclose the entire universal ladder of sounds. We arrived at this by the discovery that the origin of the circle of fifths was by no means arbitrary, but was based on mathematical laws. It is in this diagram, which can actually be played as music on the staff lines across the diameter, the we can apply the principals of the Mayan Katon Wheel, the zig zag motion from one end of the circle galaxy to the other. In this case, however the radius (time) constantly changes as it runs in a counter-clockwise direction across the expanding and contracting surface. As we mentioned before, the motion appears to be inward first, to an accelerated speed and increased density, to the point where it disappears in the center only to reappear as force in an oscillation expanding, contracting motion at the rim. All of this motion is base on the principals of halving the radius, or call it time, or call it harmonics. We found that the mathematical ratios of sounds in music did correspond spatially to the same ratios, undivided this way, in terms of angle, string sectioning, etc. In fact, the staff, indicated here, could represent strings, of a sort, plucked at specified points by the stars within the galaxy.
The inner tube image of the oscillating area around the rim, represented by the notes (Ti), (La), (Re), (Do), suggest a visible three dimensional world, a balance of little acceleration, a little density. Within this tube, the angles indicate an accumulating acceleration. From (Do) to (Sol) to (Fa) to (Mi) to (Mib), we have an equilateral triangle, a Mobius strip of the first twist, pressed flat. A terraced layer suggest that different dimension of the tube. The second terraced layer suggest the limit of the tube, whereas, each successive layer suggests more space, less density, and a folding backwards, like an accordion fold, springing from the center of the circle, and in essence obliterating half of it, if it were to be actually folded toward the center.
This illustration suggests how the radius of progressing square might look, as it swings in an S type curve across a diagonal. The seed pattern, then, is generated on another plane to that of the square progression. The I Ching symbol of primal and inner world orders indicate the two different ways order is originated, and provides a key to unravel certain of its mysteries. Perhaps we could think of the primal order being the first stage of evolution, and the inner world axis being theinvolution axis, the point at which the glove of the universe is pulled inside out. So, the key can only be regarded in relation to itself. If the whole work is contemplated as a moving diagram, a pulling in and a pushing out of forces, a study in the different balancing of number relationships, it may yield its secrets.
Integral to the Chinese philosophy of number is the number five. Five is the center which balance each of the nine numbers. One to Nine, Two to Eight, Three to Seven, Four to Six. As we remember, the inner world axis when put together as a unit becomes a whole, very different from the fractured look of the primal axis. The number five, as well as signifying MAN also the five senses, five fingers, toes, could also mean steps or sequence toward fulfillment. Usually we have dealt with sequences as a progression from small to large. Even though (Do) is both the small and large, the beginning and the end. In this case (Do) is the smallest, as if it were the puncture in a tube to be turned inside out. As the inside is gradually pulled through, we should have more area for each note, until we reach a point where we had pulled half of the area through, and the size of each area representing a note would correspondingly diminish, an expansion contraction.
If we go back to the beginning of this work where we decided that one set of measuring rods measured an expanding set of points, and another set of measuring rods measured a contracting of a given length, we could take a gigantic step of the imagination to visualize, that the two apparent opposites are really part of the same process, the process of pulling the universe inside out. What is more is that in five pulls, five stages we could reach our natural limits.
Now if we imagine we can cut through sections of this process of pulling inside out, we might have something which looks like a circle, divided into six equal parts. This division is one inherent in the construction of the circle itself. In other words the unit by which a circle can measure itself is by its own radius, which divides its circumference into six equal parts. When we draw a circle, starting from the smallest at row one, or (Do) and with our compass, set each point on the circumference of the circle, each end equidistant from an imaginary horizon line, or diameter, we find that as we progress, step by step, to each row, and mark off each rim of each circle, we have a wavy line representing either contraction, going in, or expansion, going out. This not only ties in with what we have been thinking about just before, but gives us some visual proof of vibrations which we see move along the radial axis. We see very definite sine curves taking shape in different scales. The sine curve is the curve which expresses each harmonic note in music. At the same time we have somewhat jagged lines which would connote noise. We also, carrying our logic still further, felt that the adjacent areas must be the inverse of one another what was white in one, being black in the other, symbolizing either space vs. object, or degree of acceleration. The latter is probably more likely. If we apply this idea of acceleration to the diagram, we could take as a guideline the acceleration of the rings of Saturn, for example.
The first five, we will say, revolve at a given speed. Then twice as far out from the center the next will revolve at 1/2 the speed of the first. Then at three times the distance of the first from the center, the next revolves at 1/3 the speed of the first. Then, four times the distance of the first rings from the center, the last revolves at 1/4 the speed of the first. The three outer rings, then would be in unison at the 12th time. The first of the three outer rings would combine every fourth time. The first two of the three outer rings would combine every sixth time.14
If each of the three rings represents four, six, and eight seconds, minutes or whatever unit of time we calculated, we would have a moving form of synchronizations which would have its own speed, and would stay within the limits of its own speed path. If the first cluster of rings, close to the center, represented the cluster of seed notes up to either the eight or sixteenth rows in our system, the entire octave cycle of (Do), the subsequent notes might be those which fell on the rows in their spatial proportions from the center. Interestingly enough, the notes thin out, as they move further from the center, less dense, slower motion, longer length, less curvature of space. time is an expansion of space space is a contraction of time. It may be that our physical senses can only apprehend the first wedge, which would be our three dimensional world. The five other sections could well be the dimensions beyond our physical perceptions, existing yet unbeknownst to us.
We play with the idea of the different patterns on a curved field of notes. It is remarkably obvious that on the curved field we have great differences in expansion where the notes are, and contraction on the rows where they are not. It is very obvious that from a space filled with sixteen small balls, each representing a point, that it would take more balls than seventeen in the seventeenth row, and eighteen in the eighteenth row to fill up the triangular shape of the wedge, wherein is contained our first octave of notes. Curiously, at the nineteenth row we have a sudden expansion, from a contraction, in an area of no notes. Obviously, a limit of sort has been reached, visually, and it is no surprise to find it the natural limit of the octave.
We could carry this through the 32, 000 balls necessary to construct a total picture of the universal ladder of sounds, if we had the time, the patience or a computer. And we would see from this the ever increasing scope, or length, the increasingly less dense field of notes, inevitable contractions or larger and more energetic chunks of space, which we would of course be unaware of, as a whole field expands and contracts to it's own scale. It is not merely a coincidence that this illustration bears a marked resemblance as the inner workings of the circle of fifths, earlier mentioned, the spiral like opposites of space and object on either side of the pseudo-sphere, the space curving back upon itself, at its most dense and fastest speed.
If we fold the sheet of paper into a tube, with the opposite ends of the circle meeting, we might have an idea of how the beginning become the end.
This brings us directly to the problem of concave shapes versus convex shapes. The whole concept of concave versus convex shape, is one in which we might have to consider different factors working, a different pattern forming. The convex, or going outward, might seem to suggest an expanding surface, while the concave, or going inward would lead us to the idea of a contracting surface. In the one case, we would have the ascending scale, the increasing radii, the expanding sphere, the infinite number of points. In the other, we would have the descending scale, the diminishing radii, the contracting sphere, the finite measuring rod, such as our common ruler, but with an important difference that it would measure fractions not inches, and what is more, fractions based solely on the ratios of the universal ladder of sounds.
Paradoxically all appear as they would be drawn on a negative curve, on an involution and an evolution axis. So what we might have concluded is, that it is the spaces between the circles which might be as stretched strings pulled across the universe, in the spaces between the visible round objects we call planets, suns, galaxies.
Each of these webbed structures would be composed of the number of strings complying with the number of notes, based on Arnoud's mathematical cycle of dividing 81 (9 x 9, God in numerology and Fa/Mi in music 81/80), and alternatively subtracting and adding the subsequent notes. When rearranged in a certain order we might have a total picture of the inner workings of universal forces. And I leave this up to the fertile imagination of the reader and others to complete.
If we go back in our minds to the time space charts we might remember that the time side and the space side each had different numbers, when we counted the equidistant dots, on triangles drawn from each side to form slightly disjointed squares. If we remember that the numbers on the space side were 13, 11, 7, and 3 and the numbers on the time side were 15, 9, 5, and 1 we would be able to determine time with one set of numbers representing a contraction (time is a contraction of space), and space with a set of numbers representing an expansion (space is an expansion of time). The numbers in these cases we will note are entirely odd. That in itself is remarkable. Remember we took as our limit, eight dots, totaling 64 dots, the number for (La) our void, our asteroid zone, and within this framework we were able to find each note in the octave with the exception of (Fa), although (Fab) could be considered equivalent to (Mi), in which case we would even have (Fa).
We should also remember that odd numbers represent heaven in ancient Chinese mysticism, and even numbers represent earth. So we have a set of numbers representing time and space, and things of this world and things not of this world. If we analyze these numbers we come up with some fascinating thoughts. Firstly, both sets are formed by addition of four for two steps. then the third, or last step is formed by adding two on the time side, and subtracting two on the space side.
In terms of musical ratios, one of the notes on the space side is the (Re) at 8/7. (13, 11, 7, and 3). Oddly enough this is the note which symbolizes expansion, the rim of the circle, the largest expansion when ripples meet. If we say that an expansion equals space, we then have one link with the spatial dimension on one side of the diagonal. The equation balances because the expansion has to be cut down to balance with the time side. (Ti), the contacting factor, whose ratios to (Do) are 5/9, 8/15, 16/9, (15, 9, 5, and 1) are on this time side, so adding two would balance this side which diminishes.
As for the practical application of any of these theories, they must be left to those who can manipulate computers, and higher mathematics. In fact, there might not even be any practical application. In any case if these ideas can stimulate any kind of creative thinking about any one of the myriad subjects we have merely touched upon, the aim of this book has been met. Its purpose is to stimulate a different way of looking at familiar phenomenon, a different way of reasoning, a different way of approaching complex problems, not with ready made formulas, but with a fresh viewpoint. Of course, all of this will be disregarded by some, and accepted by some. That is what it is meant to do. As we said before, in no way is it meant to be definitive. It is both fantasy and fact. It is a creation of a whole world from nothing, with its own imagery, its own mathematics, its own logic, its own vocabulary, its own symbols, peopled by abstraction, dream and reality combined.
It has formulated its own science, its own literature, which continuously refers to itself. It has conceived its own seemingly soundless musical scale, and music theory. It has constructed its own view of the origin of matter, the stars. It has developed its own concepts of time and space, of man and mortality, perfection, good and evil, which are really nothing but number. It has invented its own mechanics, its own philosophy. It has made up its own charts and graphs, its own numerology, and astrology, botany and theology.
In the process of doing all of this, it has mirrored civilization, on a very small scale. It has emerged from the ooze of beginning consciousness and tried to reinvent language, number relations as though it were the first being trying to manifest itself and reinvent the world, in the hope discovering what might have gone wrong. It mirrors a search for the phantoms of our consciousness.
This final illustration shows how number was the KEY to this whole work. It bases this on the idea that there are 53 harmonic intervals in contracting octaves. This means there are fifty three fractions which, when analyzed and arranged in certain orders, can give us a key to unlock the mysteries of sound, object striking, plucking, blowing, bowing, clashing object. Thus the mysteries of sound, in turn can reflect the workings of all phenomena, both physical and metaphysical. By turning these fractions into shapes, by setting them up on grids, we are able to see certain patterns with the organ of our eyes, instead of hearing them with the organ of our ears. In other words, we make an invisible structure visible, we see the heard. Granted that the gird structure has its limitation, and does not necessarily reflect reality, still it enlarges our powers of perception. Ideally the grid might have been more accurate if we had actually used spaces not equidistant, but of three unequal sections, reflecting in themselves the inequality of the fractions as they progress in an ascending descending pattern. Only the middle section would be whole, a one. The one above would be greater, the one below less than the whole, in terms of having either more or less space between the dots. This then might have created a truer image in terms of the values for the fractions. With a vaster knowledge much more could have been included. If everyone in their own field of interest could explore the possibilities inherent in this method it might prove very fruitful. It is obviously beyond the scope of one person to handle adequately. However, one should not ever be afraid of encroaching on another field because of lack of experience. All knowledge should be available to everyone. Number is available to us all.
This whole effort is, of course, nothing less than trying to express number spatially, which was abandoned in the time of the early Greeks, who thought space was incommensurate with number. Therefore, we are trying to prove that number is not incommensurate with spatial symbol. That, in fact, spatial perception is perhaps more accurate than an abstract number alone. We believe that spatial perception has not been developed highly enough, and has therefore been considered inapplicable to deal with the problems we have been concerned with. We believe that a spatial thought sense is the organ of the future. We believe that this spatial thought sense, when developed, will raise the consciousness and perception of the human race. We believe that thought has form and shape, and balance and color, and sound, texture, and rhythm, and dark and light, and motion, and interval, and that the highest form of thought is pure abstraction with visible form and structure.
And so we have named this work "Glimpses of the Bridges of Art". For art is a microcosmos of all that goes to make up life. It is not a specialized field relegated to its special corner, but it has license to encroach without impunity on everything which falls within the mind's grasp. And with it comes imagination, emotion, fantasy, structuring, creating an order where none existed before, an ecstasy of Glimpses available to everyone.
The End Is The Beginning.
Sir James Jeans, "Science of Music" p. 2
Albert Rousitit, "La Prophetie Musicale dans L'Histoire de L'Hunamite, Precedee d'une etude sur les nombres et les planetes dans leurs rapports avec la musique, apres l'archeologie la musique confirme la bible". Preface d'Olivier Mesiaen, (42 Roanne, Editins Horvath, 1970)
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