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Russian Precession of the Planets or What was Plato Writing About? by V.L.Pakhomov
PART TWO
The Precession Law of the Planets System
"The chief aim of all investigations of the external
world
Now we shall proceed to the analysis of galactic latitude of the North Pole of the planets (see tab. 1). As we are interested only in the slope angle of the rotation axis of the planets to the galactic plane, for Pluto it is possible to change the sign of the galactic latitude, that is, to use the South Pole latitude (all the more that Pluto has a retrograde rotation). In the following table, planets are ordered on the increase of the slope angle of their rotation axis to the galactic plane.
Table 3
These data are shown on the following diagram.
Figure 8
To find the analytical formula for this dependence we shall present the same data on a logarithmic scale.
Figure 9
As you can see, two straight segments of this diagram [1, 3] and [4, 10] allow the approximation by the function b = a·p^{k}.
If we shift the origin of the coordinates to the point (4, 0) then the initial diagram will look so.
Figure 10
Here two segments are easily visible. Please note: Plato also takes two sequences of numbers. Using the method of least squares we get the values of parameters a and p.
b_{k} = a·p^{k }, k = 0, 1,... ,6 where: a = a_{1} = 20.5839, p = p_{1} = 1.0673
for the group of seven planets (fig. 10).
For the group of four planets (k = 3, 2, 1, 0) there will be the same formula, but with the values of parameters
a = a_{2} = 24.006, p = p_{2} = 1.876
The parameter p_{2} is close to 2, as well as was specified by Plato when he wrote about the geometrical progression (2, 4, 8) with a factor 2.
Approximation on the first three planets (k = 3, 2, 1) gives the parameter p_{2'} = 2.109.
On the following diagram, the obtained approximation is shown.
Figure 11
Here the source data are drawn in blue. The average relative error is equal to 2.43%.
Taking into account the data of ancient science explained by Plato, and that this dependence remain same today, it is possible to state that during the precession of the planets, the values of parameters p_{1} and p_{2} remain constant.
Parameters p_{1} and p_{2} can be named the harmonic constants of the precession. This law allows us to calculate the parameters of the precession of all planets and the Sun using the known data about the precession of one planet (for example, the Earth).
Further, I shall present information which leads to a deeper understanding of parameters p_{1} and p_{2}.
Diatonic Solar System
"...the people of their island [Laputa]
The theory that planets at their rotation around of the Earth make some sounds differing from each other depending on their size, velocity, and distances, was accepted by the ancient Greeks.
The diatonic scale is of ancient origin, but the particular tuning incorporated into modern just intonation is due to Ptolemy (Claudius Ptolemaeus, Greek astronomer, ca. 100  ca. 170). He gave it as one of a dozen or so possible tunings for the diatonic scale (calling it the "syntonic diatonic"). There are three intervals in the diatonic scale: ^{9}/_{8} (1.125), ^{10}/_{9} (1.111), and ^{16}/_{15} (1.067). The first two are called whole steps and the third a half step (or semitone).
As you can see, the parameter p_{1} = 1.0673, obtained in the chapter "The Precession Law of the Planets System", really coincides with the coefficient (interval) of the diatonic musical scale (semitone)!
The intervals mentioned by Plato (which determines the structure of planetary spheres), in music are named: ^{3}/_{2}  a quint, ^{4}/_{3}  a quart, ^{9}/_{8}  a second. And the fraction ^{256}/_{243} = 1.0535... is the basis of construction of the modern musical scale — the equally tempered scale, created only after 2000 years [2]. Modern music uses an equal tempered scale with the coefficient ^{12}√2 = 1.0595... The appropriate sound interval is named a semitone. Equal temperament makes the ratio between each semitone a constant. I wrote about it in the book [3]. In 1722, J. S. Bach, having learned of the equaltempered 12 note per octave scale described in 1691 by the German scientist and musician Andreas Werckmeister (16451706), wrote book 1 of The Well Tempered Clavier. It was the first experience of application of the equal tempered scale in the history of music.
Thus, if we accept that Mars corresponds to the note 'do' of the 1st octave, then on the modern musical scale, the planets can be arranged so:
Figure 12
All of these sound simultaneously as one chord. The accidental symbol — # (sharp), means on a halftone above. For example, #do is read 'do sharp', etc. Figure 12 is the musical approximation of the diagram of the galactic latitude of the North Pole of planets (see fig. 8). Still further, the approximations shown in figures 11 and 12 are completely equivalent in their numerical expression.
Taking into account the abovestated, it is possible to tell that all 10 "planets" take an interval in 4 octaves on a musical scale.
Thus, Pythagorean "harmony" and "the music of the spheres" express the real law of celestial mechanics.
What force caused the planets to precess according to the musical scale, and organized the solar system as a welltuned instrument?
To be continued (see Part Three)
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