GRAVITY EQUATIONS

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Average Gravity

The acceleration we call gravity varies with the altitude above the surface of the celestial body (attracting mass). As an object falls, it passes through these various magnitudes of gravity. In falling, it spends more time in the zones of lower gravity than it does in the zones of higher gravity. So the average gravity that an object experiences as it falls cannot be found by a simple linear equation. Instead, there are two non-linear equations used to find the average gravity experienced by an object in free-fall.

re = radius of the earth (for example)
ra = distance from the center of the earth to a point above its surface
gave = average gravity experienced by an object when falling

1. Gave = Bge + (1 - B)ga, B = 1/[(ra/re) + 1]

2. Gave = [(ve - va)/H] [(ve + va)/2]
H = (ra - re)

 

Derivation of Escape Velocity (explanation)

Simple potential energy, "Ep", may be the product of an object's weight and its distance, "d", above the surface of the earth. This product is expressed here in English measurement as foot-pounds. A pound is actually a force, "F", rather than a mass unit, "m", and is equal to the acceleration of gravity, "g", multiplied by the mass of the object which is expressed in slugs.
So: Ep = Fd = mgd.

Kinetic energy, "Ek, for an object that has been dropped is equal to the Ep which existed before it was dropped.
Ek = mv2/2 where v is velocity when an object impacts upon the earth. Ek at impact is the way the object discharges what was its Ep.

Gravity is a "force" (acceleration) which extends to infinity (or for as far as the universe extends).

If we were to cause an object to escape from our gravity funnel, it would have to leave the earth's surface with a velocity at least equal to that found in its kinetic energy equation, were it to fall from an infinite distance to the planetary surface. In other words, gravity extends to infinity so the maximum kinetic energy a body can have, due to the earth's gravity alone, is what would be given to it if it were to fall to earth from an infinite distance.

In the absence of any other force, an object at an infinite distance from Earth would not be subjected to Earth's gravity, so the gravitational force from Earth would be equal to zero at this distance. Once having been given a very small shove toward Earth, an object would begin to accelerate along with its surrounding nether that also started at zero velocity toward Earth.

So the nether which passes through the earth's surface should have the same kinetic energy as an object dropped from an infinite distance. Since the two Eks are equal, the velocity found for the object's Ek should be the same velocity as that of nether passing through the earth's surface.

So Ep will be equal to the weight (ma) of the object multiplied by an infinite distance. Ek will be equal to the square of its instantaneous velocity multiplied by its Mass, "m", divided by two. In the process of discovering the velocity, both "m" and the infinite distance divide out and are no longer part of the equation.

The equation we use with the energy equality is that used to find the average gravity, "gave", experienced by an object falling from an infinite distance. Bear in mind that the average gravity varies with the velocity experienced during each part of the fall. The time experienced while at lower gravity will be greater than the time experienced while at higher gravity. Thus, we find that gave is a function of B.



   
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Derivation of Escape Velocity (equation)

We know that for an object in space not to be affected by earth's gravity, it must be at an infinite distance, ra, from the earth's surface, re. To fall to earth, this object must be pushed slightly. It will eventually strike the earth with an impact velocity, ve, which can be found because we know that its kinetic energy, Ek, on impact is equal to its original potential energy, Ep. Average gravity during the fall is gave. Gravity at the earth's surface is ge. Gravity at the infinitely remote distance is ga. F is force.

1. B = 1/[(ra/re) + 1], from average gravity equation one.

2. When ra is infinite, ra/re is so great that "+1" is negligible.

3. So: B = 1/(ra/re) = re/ra

4. gave = Bge + (1 - B)ga. From average gravity equation one.

5. At distance ra, ga is zero.

6. So: gave = Bge = (re/ra)ge

7. Ek = Ep, Ek = mve2/2, F = ma or mgave, and Ek = Fra. These are known physics equations.

8. mve2/2 = Fra. Substituting.

9. mve2/2 = mgavera. Substituting.

10. ve2/2 = gavera. Dividing both sides by m.

11. ve2 = 2gavera. Multiplying both sides by 2.

12. ve2 = 2(Bge)ra. Substituting.

13. ve2 = 2[(re/ra)ge]ra. Substituting.

14. ve2 = 2rege. Dividing out like terms.

15. ve = (2rege)1/2. Taking the square root of both sides.

The energy needed for a rocket to escape from Earth is the same as its kinetic energy from falling from an escape distance, so ve is its escape velocity. Since the rocket must have fallen at the same rate as the nether surrounding it would have fallen, the velocity of the nether at the earth's surface would also be ve.

The equation ve = (2rege)1/2 can be generalized as
v = (2rg)1/2 or g = v 2/2r.

So the escape velocity for any celestial body at any point is the same as the instantaneous velocity of the nether moving past at that point.


Usually, escape velocity is found with the equation:

v = (2Gm/r) 1/2 where G is the gravitational constant.

The gravitational constant comes from the equation:

F = Gm1m2/r 2 where F is force, m1 is the mass of one attracting body, m2 is the mass of a second attracting body, and r is the distance between the two bodies.

Gravity, g, is usually given by the equation:

g = Gm/r 2 where m is the mass of a planet and r is its radius.

By substituting g for Gm/r2 in the usual equation for escape velocity as follows, the equation becomes the one derived above:

v = (2Gm/r) 1/2
v = (2Gmr/r 2) 1/2
v = [(2r)(Gm/r 2)] 1/2
v = [(2r)(g)] 1/2
v = (2rg) 1/2

The derived equation is a more direct means of showing why escape velocity equals nether velocity at any point within a gravity funnel.



   
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Proof for Nether Velocity Equation

Hypothesize two spheres (funnel cross-sections), one above the other, and use the earth's surface as the lower sphere (called "e") with an upper sphere (called "a"). We want to find the average gravity between the two spheres. The difference between Ve and Va is the increase in nether velocity between the two spheres. Gravity is an increase in nether velocity which is normally given as an increase per second.
[(ve - va)/H], where H is the distance in feet between the two spheres, is the increase in nether velocity per foot. To find the increase in nether velocity per second, we must multiply
[(ve - va)/H] by the average number of feet in one second.

vave = [(ve + va)/2] gives us the average number of feet per second.
H = (ra - re) gives us difference in the radii of the two spheres.

1. gave = [(ve - va)/H][(ve + va)/2] for the two spheres. From average gravity equation two.

2. gave = [(ve - va)/(ra - re)][(ve + va)/2]. Substituting for H.

3. gave = [(ve - va)(ve + va)]/[2(ra - re)]. Rearranging the equation.

4. gave = [ve2 - va2]/[2(ra - re)]. Multiplying term on the right side.

5. 2gave(ra - re) = (ve2 - va2). Dividing both sides by the term on the right.

6. va2/ve2 = re/ra. Each spherical cross-section is an energy level.

7. va2 = ve2(re/ra). Multiplying both sides by the bottom term on the left.

8. 2gave(ra - re) = {ve2 - [ve2(re/ra)]}. Substitution into equation five.

9. 2gave(ra - re) = ve2[1 - (re/ra)]. Removed term from parentheses on right side.

10. 2gave(ra - re) = ve2[(ra - re)/ra]. Multiplied one by bottom term on right side and rearranged.

11. 2gave = ve2/ra. Divided both sides by the same terms in parentheses.

12. 2ragave = ve2. Let H approach and become zero, the ra becomes re and gave becomes ge.

13. 2rege = ve2. Substituting.

14. ve = (2rege) 1/2 and ge = ve2/2re. Taking the square root of both sides and reversing sides.
For the general equations:

15. v = (2rg) 1/2 and g = v 2/2r.



   
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Derivation of Nether Velocity from Orbital Velocity
This derivation was added to the website on March 14, 2005.
Copyright (C) 2005 by Lew Paxton Price
 

Back to The History of Nether Gravity Theory

From the above, we know that nether velocity inward is equal to the escape velocity at any point in a gravity funnel. We may call this velocity "vi". The best form of equation for our use here is:

(1)     vi2 = 2rg

For instance, to find the incoming nether velocity toward the sun at our orbital distance from it, "vi" is the velocity of nether moving toward the sun, "r" is our distance from the sun, and "g" is the value of solar gravity where we are.

When an object like the earth is in orbit, it has an orbital velocity which we may call "vo". A planet in orbit about a sun remains in orbit when the centrifugal force of its motion is equal to the centripetal force of the sun's gravity at the planet's orbital distance from the sun. If "Fc" is centrifugal force, "m" is the mass of the earth, "a" is acceleration, and "r" and "g" are the same as given in the preceding paragraph, the equation for centrifugal force is:

Fc = mvo2/r = ma

Gravity is the acceleration that balances the "a" in the equation, so:

mvo2/r = mg       and if we take "m" from both sides:

vo2/r = g       This can be re-written as:

(2)     vo2 = rg

If we divide equation (1) with equation (2):
vi2/vo2 = 2rg/rg = 2       then the equation can be re-written as:

(3)     vi = (2vo2)1/2

The sun's mass is approximately 333,000 times the mass of the earth. The approximate average orbital velocity of the earth is 18.5169 miles/second. The approximate average distance of the earth from the sun is 93,000,000 miles. Using these figures, the gravity program I devised using the older equations (which do not include orbital velocity) provides an answer for "vi" of 26.12849 miles/second. Equation (3) shown above provides an answer of 26.18682 miles/second. The input figures are approximations, so the two answers should be slightly different from one another. Of the two, the answer provided by equation (3) should be the most correct if we were to be able to find our precise orbital velocity at any given moment. Perhaps this will be possible in the future if someone has not already discovered a means to do so.


Another Derivation of Nether Velocity from Orbital Velocity
and its implications.
This derivation was added to the website on May 3, 2013.
Copyright (C) 2013 by Lew Paxton Price
 
The equation for gravity and escape velocity is

vi = (2rg)1/2   in which vi is incoming nether velocity, g is gravity, and r is the radius from the attracting center of mass to the point being examined.   Squaring both sides gives us

vi2 = 2rg       or

g = vi2/2r
 

The equation for centrifugal force is

Fc = mvo2/r

in which Fc is centrifugal force, vo is orbital velocity, and and r is the radius from a point within the gravity funnel.

Fc = ma       so as applied to centrifugal force

ma = mvo2/r       If we remove m from both sides

a = vo2/r

a must equal g for an orbiting body to maintain its orbit, so

a = g     or

g = a     Substituting, we have

vi2/2r = vo2/r     Removing r from both sides, the equation is

vi2/2 = vo2     Multiplying both sides by 2 gives us

vi2 = 2vo2     Takng the square root of both sides gives us equation (3) above

(3)     vi = (2vo2)1/2

Which is the same as

(4)   vi = (21/2)vo

 
This equation implies a 45 degree right triangle in which the sides are vo and the hypotenuse is vi.   We know that one side is vo and that the hypotenuse is vi.   The other side must be the opposing centrifugal force.

Vectors taken from an orbiting body at a point in its orbit would show

vo at its place tangent to the orbital curve,

vo as the component of centrifugal force pulling outward, and

vi as the component of gravity that is at 45 degrees to the two that are each equal to vo.

The kinetic energy of an object moving at vo is (1/2)mvo2/r
Since kinetic energy is F(d/2) in which d is the distance that the force is applied, and F = ma, its acceleration is vo2/r.

The acceleration we call centrifugal force is also vo2/r.

Gravity is also an acceleration and is vi2/2r.
We have found that vi is (21/2)vo.   So

vi2 = vo2 + vo2     is the same as

[(21/2)vo]2 = vo2 + vo2     which means

2(vo2) = 2(vo2)

The foregoing may seem to be overkill, but it shows

(1) that energy used by gravity to counter the energy used to keep a body in orbit is the same as the energy of the body in orbit,

(2) explains why the incoming velocity of nether at a point must be greater than the velocity of a body in orbit, and

(3) shows that the kinetic energy of the orbiting body is equal to the energy used to pull the body outward against gravity.

Looking at it another way, gravity must pull the orbiting body inward while also pulling it back enough to make the orbit curve in a nearly circular manner.   It has two vectors to overcome rather than one.   Those vectors are equal for a circular orbit but not for an elliptical orbit.



   
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[The following has been taken from my book, The Oldest Magic, which was created from magazine articles published in 1989, 1991, and 1992, with the addition of several other chapters published in 1995.]
 

The Role of Music in the Ancient World

Due in part to the complexities of the language of music and, probably, to the difficulties of translating ancient Chinese into English, much has been erroneously written about the Chinese musical scales and related subjects.   This fact is evident because much of what has been written is mathematically impossible and could not have been the case.   Also, there are conflicting accounts so that, when musical math is used, the correct account can be distinguished from that which is incorrect.   As best I can judge, the following is how things were.

As far back as we can go into Chinese prehistory, the pentatonic scale has reigned supreme.   Indeed, that is the case with all the cultures of which I am familiar because the human ear prefers this scale, and, even today, our music is basically pentatonic.   Our best music, the songs that have stood the test of time, is pentatonic.

The old Chinese pentatonic scale eventually evolved into five basic notes called Kung, Shang, Chiao, Chih, and Yu.   The first note, Kung, was from a musical pipe (what we call a panpipe) called the Huang Chung, meaning Yellow Bell, and is the note upon which the entire scale is based.   Supposedly, the Huang Chung, the official five note scale, and what we call the circle of fifths, were established during the reign of Huang Ti (2697 BCE). [In Ancient India, the earth element was represented by a yellow square.   It is possible that this had something to do with why the Chinese pipe had the name Huang Chung.]

It was upon music that the entire civilization of China was believed to rest.   Good music and the proper scale meant a prospering China, and the entire musical scale rested upon the Huang Chung which was set by a bamboo pipe of a particular theoretical length.

The Huang Chung established the first note, Kung.   The next note was set by a pipe two-thirds the length of the first.   The third note was set by a pipe two-thirds the length of the second.   The fourth note was set by a pipe two-thirds the length of the third, and so forth for a total of thirteen pipes, each being two-thirds of the length of the pipe before it.   This system of lengths was theoretical because each actual pipe is slightly shorter than the quarter wavelength of the note upon which it is based.   However, the Chinese knew this and tried to take it into account.

This means that each pipe produced a sound with a wavelength two-thirds as long as the wavelength of the sound of the pipe before it.   Two-thirds is the ratio required to make a note that is the fifth note up, on our modern diatonic scale, from the reference note with which we begin, so we call this note the fifth.   If we continue to go up in pitch by shortening each succeeding pipe to two-thirds of the one before it, then each succeeding pipe creates a fifth note up from the note of the pipe preceding it.   When thirteen pipes are used in this fashion, the thirteenth is almost an exact same pitch as the one with which we began but is eight octaves higher [each time one goes up one octave, the frequency is doubled).   This thirteen-note sequence is called today, the circle of fifths.   Very probably, the piano keyboard was based upon this same circle because the circle of fifths covers seven full octaves and eight octaves of the reference note just as does our piano if we use "C" as the reference.

The first five notes of this system were originally based upon a Huang Chung of about nine inches in length which would have produced approximately the note we call F sharp (F#) today.   Progressing upward on the circle of fifths, we have the following:
Kung (F#), Chih (C#), Shang (G#), Yu (D#), and Chiao (A#)

This is the Chinese pentatonic scale shown as the first five notes in the circle of fifths.   If we rearrange them to be consecutive notes within an octave, we have the following:
Kung (F#), Shang (G#), Chiao (A#), Chih (C#), and Yu (D#).

This pattern of five notes corresponds to the black notes on the piano keyboard.   And the intervals between them are not all the same.   There are larger gaps on the keyboard between A# and C#, and between D# and F#.   The location of the gaps that are larger within the sequence of notes gives the notes a distinctive "tune" or melody when they are played in ascending or descending order.   This tune is called a mode.   On a scale of five notes with larger gaps occurring at certain points, there are five possible modes - one for each note with which it is possible to begin.

The black notes on the piano keyboard constitute only a portion of the scale we use, and, in a sense, this was also true in ancient China.   Because, within a circle of fifths there are every one of the same twelve notes (pitches) that we use today.   Given as a circle of fifths, and beginning with Kung, we have: F#, C#, G#, D#, A#, F, C, G, D, A, E, and B - which is what we call the transposition pattern.

These twelve notes were needed to allow melodies to be played in different keys - and it ancient China this was much more important than it is today.   The ancient Chinese had a zodiac of twelve signs just as we do.   Their first sign (the one we call Aries) began just after the vernal equinox (just as does Aries).   To this sign they attributed the note Kung (F#).   A half-step up from Kung was "G", attributed to their second sign (Taurus in our language).   And, of course, G# was attributed to their third sign.   Each sign was known only by a number and a note.   It was not given an animal name until much later, and each sign was like one of our months.   Indeed, our months are simply misgotten travesties of what was once a very good calendar based upon equinoxes, solstices, and signs.

The note (pitch, actually, as well be seen later) attributed to each sign (month) was the key in which all music was to be played during that time - for this was the key most prevalent in the world for this duration and changing it would not be in the prescribed order of things.   A change in key for that time could presage a change in thinking and that could eventually foment a revolution - and a revolution or change of any kind might adversely affect the emperor.   Ergo, each emperor took great pains to choose the correct length for the Huang Chung and to see that the correct key was was played during each sign (month).   The signs were based upon the position of our sun in the heavens and the zodiac was seasonal just as were Western zodiacs.

As in most cultures, after a period of time with the pentatonic scale, the two large gaps in it were noticed.   To cure the gaps below Kung and below Chih, two other notes were added to the standard scale.   These notes were called pein-Kung and pien-Chih, meaning below Kung and below Chih.   The Chinese now had a diatonic scale officially.   However, the pentatonic scale continued to be used almost exclusively.   The old circle of fifths had given the Chinese the diatonic and the chromatic scales long before, but the diatonic scale was recognized only as the scale with notes other than Kung, Shang, Chiao, Chih, and Yu, and the chromatic scale was used only to change keys for melodies written in the pentatonic scale.   [Humans must have optimism, and each month the practice of playing music in a slightly higher key promoted optimism - just as the zodiac of increasingly higher keys does through the year until once more arriving at the beginning key based upon Kung.   This was like a helix where one moves forward with each revolution.   In truth, this is the way things are as the sun moves around the galaxy with its north pole pointed in its direction of motion.]   It is interesting to note that when the diatonic scale was introduced, the notes other than the official five were not presented as part of a diatonic scale.   Instead, two notes were added to the pentatonic scale.

The Chou Dynasty (1027 - 256 BCE) is known as the classical age of China, during which there was a renaissance in prose, poetry, history, philosophy, agriculture, flood control, military techniques, diplomacy, astrology, arts, and music.   The Chou came from the northwest and were keepers of herds and flocks.   It is likely that they brought with them much of the knowledge we think of as Western from Europe and mid-Eurasia.   In any case, during the Chou Dynasty, the diatonic scale that was known in the West became part of China by means of the addition of the two new notes.   The new seven-note system in China allowed two more modes in which music could be written and China now had seven modes just as Old Sumer had prior to 2600 BCE...
 

There is a very old tradition that has come down to us in the form of our Western zodiac and musical scale.   No one knows exactly where it began.   However, the oldest civilization to have had similar things was Old Sumer on the shores of the Persian Gulf where the Tigris and Euphrates Rivers empty into the sea.   Very likely it was people from the island of Dilmun (now known as Bahrein) who began to settle there prior to 4000 BCE.   They called themselves Sumerians (properly pronounced "Shoomerian, but due to a problem in translation now called "Soomerian").   It was from them that we received the sexagesimal numerical system, based upon the number sixty, the key to our time units of hours, minutes, and seconds.   Also from them we received our circle of 360 degrees and our musical scale.   There were other highly advanced cultures in the Middle East such as the Egyptian, the Assyrian, and the Indian, but none so old as that of Sumer.

In later times, the Babylonians developed the most advanced civilization we have on record.   The written record does not go back to the beginning of Old Sumer or to Dilmun, but there is reason to believe that in many ways, the older culture was superior to that of Babylon.   Though not as advanced as ours today in such things as gadgetry and certain kinds of technical information, the Babylonians of the Chaldean Renaissance were far more advanced in other ways which, I believe, were far more important - and these people built their civilization upon the knowledge of Old Sumer.

The old zodiac has come down to us from the height of civilization in the Middle East.   This zodiac has undoubtedly been altered a bit here and there by other civilizations.   Also, much has been lost due to lootings and burnings.   So we don't know all we would like to know about how it was used.   We know, however, that like the old Chinese zodiac, there were notes attributed to signs.   We know also that there were colors attributed to signs.

We can hear about ten octaves of sound and see about one octave of light.   Technically speaking, an octave is a range of frequencies between a reference frequency such as a note and another frequency that is either double the reference frequency (when going "up") or half the reference frequency (when going "down").   Because going up one octave is doubling, each succeeding octave has a bandwidth (range of frequencies) that is as large as the sum of all the bandwidths of the preceding octaves (analogous to each time our population doubles).   This means that the octave or visible light, which is 41 octaves above the middle octave of our sound range, is an octave with a bandwidth many times greater than the bandwidth of all ten octaves of our hearing range.   So the human eye has a rather wide range even though it sees but one octave.

Each color can be likened to a note in a higher octave than those of our sound range.   If we double the frequency for middle C just 41 times, we have the frequency for green.   If we double the frequency for E just 41 times, we have the frequency for violet.   In like fashion, all the notes are colors as well according to the following.

C is green, C# is blue-green, D is blue, D# is blue-violet, E is violet, F is red-violet, F# is red, G is red-orange, G# is orange, A is yellow-orange, A# is yellow, and B is yellow-green.

When an object radiates it own color, the frequency at which its electrons are vibrating in their outer orbits is the same as the color that comes from the object.   This is additive color.   We can add it to other colors to make still more colors or shades.   For instance, blue light and orange light will look white when in the correct proportions because they are color complements.   Likewise with red and green, yellow and violet, or any two colors that are diametrically opposed on the color wheel.

When an object reflects a color as does a leaf in the sun, its outer-orbit electrons are absorbing red and reflecting what is left over which is green.   So a leaf is really red - not green.   This is called subtractive color because the red is subtracted from the white light of the sun, leaving green as the apparent color of the leaf.

In the ancient Middle East the sign we call Aries was given the color red.   In China, this sign was given the note F# which is the same as red but in a much lower octave.   In the ancient Middle East, the sign we call Leo was given the color yellow.   In China, this sign was given the note we call A# which is the same as yellow but in a much lower octave.   The same is true of the other signs.   The color attributions of the Middle East were the same as the note attributions of the Chinese.   This is not only a little bit odd for coincidence, but indicates that someone back then could tell which color was which note.   This may seem astounding to one who is unfamiliar with the problem.   After all, higher octaves of sound can be determined with shorter panpipes.   But color wavelengths are so short that they are measured in angstroms and one angstrom is just a one hundred millionth of a centimeter - and that is small.

The color attributions of the Middle East were accompanied by notes also.   But these were notes of the color complements rather than those of the colors themselves, which indicates that the people of the time were aware of the principles of subtractive color and/or balancing with complements.

In both the Middle East and China the notes, which were considered synonymous with the signs, were given a gender.   The same system of gender was used in both areas.   The signs/notes/colors that we call fire and air were male.   Those we call water and earth were female.   There may have been other uses for this assigning of genders, but one of them was probably aiding in predicting the sex of an unborn child, which is done with the sign genders even today.

Thor Heyerdahl has done much to improve our knowledge of ancient global navigation and, from his research, it is now understood that reed vessels with up to three masts were used to carry goods to all parts of the world in times predating the Sumerians on the Tigris-Euphrates delta.   In fact, it may have been a reed vessel or a fleet of them that deposited the first Sumerians on that delta.   In any case, the sea was one avenue that was used to spread the word of the wise throughout the ancient world.

In Old Sumer, religion and science were the same, and groups of missionaries were sent to various parts of the globe to teach those who were willing to learn.   Imhotep or Hermes Trismegistus, the legendary vizier of Zoser in ancient Egypt, was probably a missionary from Sumer.   He was responsible for the sudden rise of advanced civilization in Egypt.   The records of the "Indians" of Central and South America indicate that a similar legendary figure was the key to their sudden advancement in the fields of math and science.   There are other indications, not so specific, indicating that the old sciences spread from Sumer to the Far East as well, accounting for the similarities in the musical scales of China and the Middle East.

The land connecting China with Europe probably accounts for the similarities between types of flutes found in these places.   Although the sea route from India may have played a part as well.   There are mountain ranges separating China from the rest of Eurasia, and mountain tribes of nomads may have formed the link between the Chinese and other cultures.   Between the Tien Shan and Altai Mountains is a lower area through which most of the traffic would have come.   This may have been the route taken by the Chou late in the second millenium BCE...
 

In any circle of fifths based upon the two-thirds ratio, the thirteenth pipe or string does not turn out to be the perfect note-maker for the octave.   It will be a bit sharper than it should be as long as the two-thirds ratio was rigidly kept throughout the circle.   Multiples of pipes going up higher and higher will not help.   In fact, in theory, they will only compound and magnify the error.   However, due to slight variations in the inner dimensions of pipes, and possibly, due to very slight errors in hearing each pipe, the nature of the thirteenth pipe can vary.   It may have been this variation that encouraged Ching Fang in 45 BCE to add more notes (called lu) to the official scale.   Very likely, the legendary magic of the number sixty may have influenced him as well because sixty was the total number of notes that the new scale had.   At least this is what most authorities would consider to be correct.

It seems more logical to assume a different reasoning for going to sixty in lieu of twelve.   I believe it was done because someone heard that it had been done in the Middle East.   The people of Sumer were very good at pure math.   Very likely they had discovered the secret of what we call the tempered scale.   This secret would have allowed them to go to a system of sixty notes to the octave.   Why should they want so many notes?   Probably because they were doing something very similar to the Chinese who were using music to augment the seasonal influences by officially changing keys at certain times.   But the Chinese did not have the secret of the tempered scale and would have failed miserably in developing a workable sixty-lu system.   This frustration, coupled with the desire to have a different note for each degree in the zodiac, led to a 360-lu system started by Ch'ien Yueh-Chih in about 438 AD.

The next eleven and half centuries must have been terribly frustrating for Chinese musicians because the 360-lu system would have been much less accurate that even the sixty-lu system due to the nature of simple mathematics.   The problem was resolved in 1596 AD by Prince Chu Tsai-Yu who established the tempered scale in China.   They had finally discovered the secret about one hundred years before the Europeans.   The Prince also restored the twelve-note scale to the relief of everyone concerned.

The Huang Chung had changed somewhat from time to time because each emperor's desire to keep the status quo dictated that his staff constantly check to be sure it was correct - and there was plenty of chance to err.   When the Huang Chung was about nine inches long, it had a tone approximately equal to our modern F#.   The last Huang Chung had a frequency of about 601.5 cycles per second (*Hertz) according to some, and this is about D#.   On the Chinese calendar, F# was the first sign after the vernal equinox.   So D# would have been the first sign after the winter solstice.   Again, we show the two starting points for the year, that began or were earliest known in the Middle East.

*[Cycles per second is now called Hertz, seemingly as a means of confusing most of us, but supposedly to honor a particular man by that name.]

There seems to have been an attitudinal schism between the East and the Middle East that has come down to us today as between East and West, because most of our Western tradition began in the Middle East.   The notes assigned to the Chinese calendar were for going along with the vibrations of the zodiac.   When the sun was energizing a certain sign, the Chinese music was played to augment or magnify its effects.   However, the Middle Eastern system seems to be designed to use the complementing note each month to balance the effects of the activated sign.   This same tendency is here today.   The traditional Oriental way is to accept and to go along with authority.   The Western way is usually not the same.   There is something to be said for both systems.
 

The Nether Flow Constant

The preceding section on ancient civilizations was provided to make the point that their people were very particular about music and the role it played in their societies.   This point must be understood to fully appreciate what follows.

Suppose that we were to relate instantaneous nether velocity with gravity - both as found a the earth's surface.   Let:

M = nether Mass
ve= nether velocity at the earth's surface
K = constant for earth
ge = gravity at earth's surface.

The equation would be:   Mve = Kge.

If we solve this equation for the value of K, we have what I have called the nether flow constant.   At sea level on earth, we can give M, nether Mass, a value of one and use it as a standard, allowing us to use the simpler equation:

ve = Kge.

When we use M at sea level as a standard, K can be used at places other than earth's surface, but only to find Mv when g is known.   With this in mind, let us play with K for a bit.

1.   ve = Kge       ve = distance/time       ge = distance/time2
2.   From dimensional analysis, K must be a measure of time.
3.   ve = (2rege)1/2       This is the equation for nether velocity.
4.   Kge = (2rege)1/2       Equivalencies of ve.
5.   (Kge)2 = 2rege       Squaring both sides of the equation.
6.   K2ge2 = 2rege       Removing the parentheses.
7.   K2 = (2re/ge)       Dividing both sides by ge2.
8.   K = (2re/ge)1/2       Taking the square root of both sides.

We should solve for K using the values for the earth's radius and gravity as found at the poles because at any other points, there is centrifugal force from the earth's rotation, which causes re to increase and ge to decrease.   At the poles, re is 3,950.19 miles, and ge is 32.2577759 feet per second2.   Solving for K at the poles, we discover that it is almost equal to 256pi times the square root of two.   In fact, if we use 256pi times the square root of two, and then solve for the radius of the earth, re = 3,951.67 miles which differs only .0375% from the re at the poles (3,950.19 miles).

This apparent coincidence would not have made any impression on me except that 256 times the square root of two is the frequency for the ancient F sharp (F#) of China, their Huang Chung, upon which their whole society was based.   This causes me to wonder what they knew, in this regard, that we do not.

By manipulating the equations we know, we can find various equivalencies for K.   The unit for K is the second (a time unit).

K = 2re/ve,   the time it would take for nether moving at escape velocity to pass through the earth.

K = (2re/ge)1/2

K = ve/ge

K = (21/2)(256pi) (this is approximate)

Looking at K = ve/ge, we realize that it can be best be used at all points within the earth's gravity funnel as K = Mv/g where M, v, and g all vary from point to point while the ratio (or time interval K), Mv/g, remains the same.   What we are actually seeing in this equation is the ratio of the velocity of incoming nether Mass per unit of gravitational acceleration.   So in this sense, K is the planetary signature, unique to Earth and the same throughout its gravity funnel.

Why is K apparently the same as pi multiplied by the frequency for ancient F# ?   As a coincidence this is astronomically unlikely.   There are 2pi radians in the circumference of a circle.   So 2pi constitutes the fundamental time unit for one wavelength to occur if we are speaking of vibration.   The fact that K appears to signify that a vibration is evident leads us to believe that K = (21/2)(256pi) is a natural phenomenon.   However, using everything I know, I cannot find a means of attributing this to a natural phenomenon.

Is it possible that the ancient Chinese knew about K and adjusted their Huang Chung to be 256pi multiplied by the square root of two?   If so,
(1) they would have known the value of pi and used it as code meaning "cycle" or "cycles" (this was not very surprising as other ancient cultures could do this),
(2) they would have probably understood C as a base of 256 (again, this is not too surprising as this seems to have been the case in the Middle East),
(3) they would have known of the tempered scale very early (perhaps before the secret was lost) in which roots of numbers used the base of 256, and
(4) they would have had a means of knowing the value of K.

This is just one more reason for us to wonder what came before us in our pre-history that might have been carried forward and taught by the ancient Sumerian teams of scientific missionaries.
 

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