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Each of these strands are twisted of three strands and each twist is at a certain pitch, to which we can assign a quantum number. So, while the strings fill the Universe, they are of many different sizes and are twisted at different rates. The finest twist is at a rate of 4 quadrillion per second (of possibly the speed of light). This 4 quadrillion is a very small distance and far beyond our capability of measurement. But let us call this small distance the "one unit." All other measurements of twists and of cord sizes are measured in this unit.

Now, so far so good, but let me digress to a fill-in. Most people know a little about prime Pythagorean triangles, but very few know about the parallel set of equilateral triangles which go with them. These equilateral triangles are derived from exactly the same roots that the right triangles and all the rest of Quantum Arithmetic come from. That is, from a four-integer group of Fibonacci numbers in which the two center ones are prime to each other. These four roots are called b, e, d, and a, respectively in the group.

The smallest permissible equilateral triangle has a side which is 8 units in length and is derived from the roots 1, 1, 2, 3. It corresponds with the 3-4-5 right triangle. The specific feature of these equilateral triangles is that one can drop a line from one angle to the opposite side and that line will be measured in integers and the opposite side will be divided into two integers. In this smallest equilateral triangle the dropped line must be 7 units long and the opposite side is divided into 3 units and 5 units.

In this case b, e, d, a, = 1, 1, 2, 3, respectively. To find any triangle, one should start with two coprime roots, e, and d. Then b= d-e and a = d+e to give the four roots. When roots have been selected, a triangle may be calculated as follows:

Sides of triangle equal d squared + 2de. - d2+2de

The side dropped line equals d squared + ae. - d2 + 2ae

The opposite side is divided into two parts, one of which is ab, and the other is e squared + 2de. - ab and e2 = 2de
It is not necessary to memorize these unless one wants to. The point I wish to get across is that there is a set of such triangles and they have a mathematical basis as unique as the right triangles.

Now having said that, let me go to the second part of this fill-in. This part selects the tetrahedron, the octahedron and the icosahedron. These three Platonic solids are similar in that they are all formed from faces which are equilateral triangles as described above. Two of these, working together alternately, the tetrahedron and the octahedron, will make a space-filling combination. It is used in a practical way as a support beam called the tet-octa truss, and is the main continued on support structure for the San Mateo bridge near San Francisco.

So with this fill-in, let us replace all those strings and ropes above with tet-octa trusses of different sizes, and use them to fill the universe. With this we will have a crystal Universe because these strands will all interlock in all directions to fill the entire universe. It is filled with tet-octa strands of different sizes but they crisscross in all ways under strict mathematical control.

So, how does this quantum work? Anyone who has encased one Platonic solid within another will realize that this crystal universe will contain many different shapes depending which lines one wants to follow. But remember, we cannot see these lines. They are essentially, the "ether" which fills all space. The triangles are of all permissible sizes but measured by that original "one-unit measure" lumped together in quantum groups.

This is where the four forces come in. We can imagine one edge of each equilateral triangle representing one force. So let us color the edges of every equilateral triangle. One edge will be red to represent the electrical force. One edge will be yellow to represent the magnetic force, and one edge will be green to represent the gravitational force. These three forces will be represented in every equilateral triangle in the universe, and these three forces will always work together. They will be in perfect balance and undetectable until one of the forces is disturbed by an outside influence. When disturbed, the force (color), which is disturbed will become apparent and will do work in its attempt to regain its original stability. If the red line is broken we call it "electricity." If the yellow line is broken we call it "magnetism." If the green line is broken we call it "gravity."

So now enters the fourth force, - the blue force. This fourth force can be any name we wish to apply. I call it the "mental" force, but some may call it the "organizing" force, or the "force of natural law," or even the "Godhead" force. This is the force which gives differentiation and change to the universe and all within it.

This blue force is represented in each triangle or in specific triangles as the line which is dropped from one angle to the opposite side, breaking that opposite line, whatever its color, red, yellow, or green. In doing so, the crystal is "fractured," so to speak, along the blue line, (and thus is the mathematical basis for crystal fracture), and the appropriate force is released and made apparent.

Now, these equilateral triangles can be any quantum size. Depending upon their size, they can break into specific mathematically controlled lines. They will break on the blue line and the blue line, (mental force) will outline and become a membrane around any such crystal so fractured. continued on page 17
continued from page 14

How they can break is shown on the cover picture of Volume III, of Pythagoras and the Quantum World. The fractures will be able to "fract-ize" any creation, as Mendelbrott might say.

These crystals can form anything from a galaxy to a planetary system, to an electron, and possibly a quark. It all depends upon the size of the triangles and how they are combined. When an object is formed, it is formed of the pure energy which we can see, hear, touch, feel and smell because we are each, one of those crystalline formations. Thus the saying "all is energy" and "matter is energy."

So here we have a universe which is frozen in time like a statue. How then does everything move? So far I have been speaking only of energy and linear distances. There is another basic foundation of the universe and that part is called "time."
Like space, time is also divided into four quadrillion parts, and we will say four quadrillion per second. Time acts like a lighthouse flashing at 4 quadrillion times per second, and nothing exists except at the time of a specific flash. Let us say that our dimension of the universe exists every millionth flash, that is, we exist only every precise billionth of a second, since a million billion makes a quadrillion. At the other 999,999 flashes it is possible for other dimensions to exist. But during this time of "nonexistence" the blue lines can change ever so slightly, and be ready for the next flash. This can cause an object, and its membrane, to change shape or to appear to move, just as a television picture moves on the tube.

I should leave it at that. The picture described above gives a possible picture of the universe which is based on mathematical certainties. But I have used only the feature of the Fibonacci numbers and projected them into the equilateral triangles, - triangles which have the property of being split only in certain ways depending on their quantum size. It does help us to understand the basis and beginnings of nature, even as incomplete as this picture may be. Everything we do must be based upon a picture of how we consider things fitted together. Usually such a picture is based upon empirical experience. Here is a picture based on solid mathematics. Perhaps it will help.

If one wants to produce electricity just "twang" a red line. To produce magnetism "twang" a yellow line, and for gravity, "tang" a green line. But if you manage the blue membrane it must be done "mentally," and in conformity with the Godhead.

The use of the words "harmonic" and "en- harmonic" by John Keely has been confusing because we do not understand harmonics as John Keely did. That new understanding may be forthcoming.

It was first thought that these were words describing human perception, but when one goes over to the truly mathematical concepts of harmonics it seems to lay harmonics at our feet. In understanding harmony, we must first establish the mathematical basis behind music, and this is quite different from the commonly accepted understanding as we derive this basis empirically.

To derive our present descriptive terms, one must listen to a chord of music and determine its acceptability to our ears, for pleasantness or unpleasantness. Playing in the center of the musical scale there are seven different major keys and five basic minor keys. It was found that certain mechanical placements of the fingers, by counting halftones that each key would have certain placement positions to obtain certain chords. Again by counting halftones, the finger placement was the same, or nearly the same for whatever key we chose. These various placements were defined by terms such as thirds, fifths, sevenths, and major, minor, augmented, etc. These apply most specifically to the use of the chromatic scale of the twelve halftones per octave.

Now when we drop the empirical methods and enter directly into mathematical composition of the whole scale, (The MoO or Lemurian Scale), some of these terms still apply but their meaning is completely different. There are still thirds, fourths, fifths sixths and sevenths, but the fraction is the fractional part of the keynote in strict mathematical terms. Now, there is no longer just the seventh but there are six of them - they are 1/7, 2/7, 3/7, 4/7, 5/7 and 6/7. Of the sixths, there are 1/6, 2/6, 3/6, 4/6 and 5/6. Of the fifths there are 1/5, 2/5, 3/5 and 4/5. There are 1/4 and 3/4. And finally there is the keynote itself. Each of these is an individual note making 6 sevenths, 5 sixths, 4 fifths, 2 fourths and a keynote, making eighteen notes for any keynote. Since the natural scale, (called the MoO scale), is composed of eight different keynotes, none of which are primary fractional parts of any other note, nor are their fractional parts duplicated in any fractional part of any other keynote. There are 144 notes within the full musical scale, and each keynote is designed, for a mathematical value such that it is in harmony with every other keynote.

While the keynotes are in harmony, the fractional parts may or may not be in harmony with others. That is to say, any two notes of the full scale will be in different degrees of harmony with each other. It is here where we get the idea of mathematical harmony as opposed to empirical harmony.

One can imagine each of the eight keynotes as a comb with eighteen teeth. One end tooth is marked with its mathematical value for the keynote, and the other seventeen teeth are marked and spaced according to their fractional part. Each tooth is marked with its fractional value, in hertz or in wavelength.

Then, as a second step, all eight combs are intermeshed to put every tooth of every comb in their mathematical sequence. Then each tooth, (note) of whatever comb, (or keynote) will be spaced at various distances from the two adjoining teeth, (notes). What we derive is what appears to be a hodgepodge until one understands.

It is very important for everyone, musician, or scientist, to understand the combination. To the musician, this arrangement is the essence of natural harmonics. To the scientist, this is also the essence of natural harmonics as he or she finds them. Let the scientist consider these eight combs as representing eight elements. Each comb is an element and each tooth on that comb is an energy state. And now, the meaning of harmony begins to fall into place. Each chemical compound becomes likened to a chord of music.

Now, that is all well and good. But there are seemingly insurmountable obstacles to overcome.

(1) What are the relative values of those eight keynotes? Luckily, we can go back into ancient history and see where these eight keynotes first occurred in the ancient Chinese book, "The Book of Permutations" written probably some 7,000 or more years ago. This book is popularly called "I-Ching," but I-Ching is, in reality the philosophy behind it, and the eight characters of "the I-Ching" refer to eight different cyclical things. These eight things are probably different forces, or cyclical inclinations. But the Chinese do not give values. They give only philosophical relationships.

But again we can go back into ancient history some 5,000 years later to the time of the enlightenment between Pythagoras and say, 100 BC. There are certain obscure and mystic writings, (which I am not at liberty to disclose at this time). Which do give us tentative number values to assign to the various keynotes, and I give these values as hertz values for the eight keynotes. In ascending order they are:

55.461
77.064
78.137
79.782
103.505
117.495
138.559
163.528 hertz respectively.

(2) If these are the keynotes, and we obtain the partial notes which are derived from the various keynotes according to the various fractional parts, they can be fitted together for use. These keynotes and their fractional notes can be raised or lowered proportionally to present more desirable music, but this music will not necessarily, be applicable to scientific usages. There is a critical piece of information which is still missing, and that is the exact unit of measure to be used. In this case the question is, "What is the exact length of the second?" in order to derive vibrations per second.

(3a) How do these notes work to develop harmony between themselves? They work precisely by developing a "beat frequency" between themselves when played together in groups. That beat frequency can be heard but it cannot be measured as precisely as is presently required. It is this beat frequency which catalogues and categorizes our terms of which "harmonic" and "enharmonic" apply. It seems that when the beat frequency is between a low limit of 3 to 6 hertz and a high frequency of approximately 25 hertz, the relationship would be called enharmonic. If the beat frequency is below 3 hertz or above 25 hertz it will be termed harmonic. It is this latter category where present empirical music lies.

(3b) Precisely how the mathematical equivalent of the beat frequency is derived is unknown. The question of how the beat frequency is derived can be answered with another question. That second question is, "How does a violin string know to divide itself into an integral number of segments, as it plays a harmonic of itself?". It appears that by some feature of quantum nature, [later on called the "mental" force], the various notes factor each other into their common prime factors. They do this by adopting a particular and specific unit of measure. This unit of measure which is used will be super critical in finding the factors. The quantum ratio does not change but the magnitude of the integers is changed and will have entirely different factors.

(3c) Considering the values originally designated as the keynote values, any number values can be used. The very special feature of the values given above is that these numbers, and their fractional notes will divide each other, (in pairs), into prime factors common to both, which are 2, 3, 5, and/or 7, and powers of these numbers. There will be no higher prime numbers as factors. [This is contrary to Raleighs findings.] If number values as designated for the keynotes are proportionally different than those given, then there will almost invariably be higher prime numbers involved in the factoring. This is the difference between the musical scale and the accepted empirical chromatic scale. The note pairs within the chromatic scale will not factor in this way. Each higher prime number, (higher than 7), in the factoring will produce a higher "overtone", (a summation tone), to correspond with that or those higher prime numbers. When the values for the keynotes are used as given above, there will be no summation tones, (overtones). They will all be "difference-tones", (undertones), less than the individual notes used. This is the very critical difference between this scale and any adopted chromatic scale. It is the one feature which makes this music directly applicable to scientific usage.

(4) If these keynote numbers, as given are tentative, then how do we find the correct numbers for ourselves? Within the framework of Quantum Arithmetic, there is a procedure for determination of "quantum" ratios, (See NOTE at end of this report), between any two given numbers. That is to say, the probable quantum ratio between any two empirically derived real numbers, can be derived. This was done in the previous article, "Quantum Numbers", for the earth/moon orbit around the sun, and for Haleys comet. It was also done for the various energy states for the hydrogen atom and the oxygen atom. In each of these cases, a harmonic resonance between energy states, and between planetary orbits was achieved.

As an extension of this process, a serial analog of this procedure can be designed for any and all number combinations. This will give a tentative list of permitted and prohibited orbits. The problem with putting this procedure into operation is that the serial application will involve millions if not billions of discrete operations, each of which may consist of several thousand operations in itself. Were this done, we could dispense with the assist given us by the Greeks.

But this operation, alone, does not completely solve our problem because one must then go into an arithmetical second derivatives. In this second derivative* there would be only 10,440 major operations, as we refine the number system. Then having derived this, the complete Periodic Table of Elements, and isotopes, could be classified according to all of the possible energy states, and this empirical table could be matched with the quantum table of permitted and prohibited orbits. It is a very complicated process and subject to so many possible errors that the final outcome could be doubtful. *(Quantum ratios between previously derived quantum ratios.)

(5) There are other problems which arise but they are minor, in comparison to the above questions. Most of these minor problems have been solved or are in the process of being solved.

So in the end, it now comes to the probable definition for the word "enharmonic" as used by John Keely. It means those inaudible vibrations which fall between 6 and 25 hertz. We cannot hear them directly but we receive them in our music by the "tremolo" effect we hear in listening to music. When the tremolo is below 6 hertz or above 25 hertz we call it "harmony." When the tremolo is between 6 hertz and 25 hertz it can be devastating for certain musical chords.

The information given here may not be sufficient for some to establish this musical scale. For those who wish to proceed, EXTREME CAUTION IS ADVISED. It appears that the energy of certain accidental "enharmonic" values can feed into certain electronic orbits of various atoms, sympathetically, and raise their energy state higher than the specific atom is able to withstand. Theoretically, an explosive transmutation can occur if enough atoms are involved. It is believed to have been this feature which caused many of Keelys disastrous accidents 100 years ago.

NOTE: Exact detail of "Quantization", (Item 4, above) is given in volume III of "What is Quantizing?."

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