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It was Plato who said that geometry is the knowledge of the eternally existent. In the preceding article an attempt was made to simplify and consolidate diverse geometrical perspectives from the past and bring forward from them something that could serve as the basis for a modern phenomenology of weather observations. We saw how Kepler, using the circumference of the earth as a fundamental akin to the string on a great monochord or musical instrument, constructed a scale of harmonic ratios. It is significant that Kepler wished to do this in order, initially, to study the weather. His later contributions to astronomy and celestial mechanics grew out of a need to get more accurate data for his weather experiments. This need drove him into his collaboration with Tycho Brahe and his subsequent discovery of laws governing the orbits of the planets. The fruits of Kepler's work is now present in the world as incredibly accurate ephemerides done by computers. These ephemerides give modern researchers an advantage of which Kepler would be envious. Using such refined astronomical data as we have available today it is possible to search for the eternally existent geometries of planetary movements with an unprecedented degree of accuracy. This motion in arc data, when coupled with global synoptic weather maps can provide an extremely rich field or phase space in which harmonic nodes of time and space can be phenomenologically studied. This is the method through which the experiments and case studies in the planetary flux model were undertaken.
Simple, repetitious observations of weather phenomena recorded against a backdrop of planetary movements in an ephemeris slowly reveals that the two realms of weather and celestial motion have a common root in periodicity. This is nowhere more apparent than in the phenomena surrounding eclipses. "Drought patterns and eclipse rhythms on the West Coast":http://docweather.com/. By observing eclipses again and again it is possible to see that on the day of an eclipse it is often possible to find high and low pressure areas lining up up with each other in significant geometric patterns. Most often they line up along the eclipse line. The eclipse line is a line that can be drawn from the position in longitude of an eclipse to a point 180° across the earth.
The eclipse line itself appears to behave exactly like a string as waves of high and low pressure alternate along the eclipse line like the nodes and oscillations in the sounding string. Aside from the awesome physics of wave trains in the atmosphere, the significance of an observation like this lies in the potential for correlating the appearance of such a pattern with a specific time window. However, over the years, it has become evident that the eclipse line does not actually fade away once the eclipse has passed. Kepler himself suggests that the eclipse influence remains in the atmosphere for a number of days. With highly accurate ephemerides and daily synoptic maps of the whole northern hemisphere it has been possible to observe the influences of an eclipse line lasting in the atmosphere until the next eclipse six months later shifts the line to a new position in longitude. These observations came were supported by watching planets transit the eclipse point (especially the moon) and seeing the eclipse line on that day behaving as if there were another eclipse. In actuality, the area around the eclipse line exhibits enhanced vortexial potentials whenever any planet crosses the eclipse point.
Setting up an experiment to test these eclipse point transits is a good way to develop a long range weather eye and will convince even the most skeptical observer of the effectiveness of eclipse point transits. These effects can be seen particularly in lows that are crossing the eclipse line as a planet is transiting the eclipse point. To be most effective both the lunar and solar eclipses should be put on a geodetic projection chart "flux chart basics":http://docweather.com/ and followed daily looking for the effects of transiting planets. Working this way it is possible to give rise to many questions about harmonic intervals and just what it is about the relationship between the Earth, the Sun and the moon which has such a profound effect upon the atmosphere. To try and shed some light upon these questions it is fruitful to return once again to the monochord and the study of ratio. Kepler, in his work Harmonica Mundi, makes great use of what he calls a semi-diameter measurement of the earth.This measurement was seen as a kind of tuning fork for all of the other spatial relationships in the earth's cosmic and terrestrial environment. Today we would call the semi-diameter the radius of the earth. In the ancient world division as a mathematical process was akin to sexual union and new relationships between the part and the whole could be produced by means of division. A ratio in and of itself is a process of division. In trying to understand harmony it is very useful to divide one number into another. The ratio 1:2 can be written 1/2 i.e. one divided by two. To produce a tone in a string (a new birth) the string is divided in certain ratios and the intervals which sound produce music. In like manner we could divide the radius of the earth as a part of the whole circumference by the radius of the moon to see if there is a significant ratio.
Earth radius 3,960 mi
Moon radius 1,080 mi
3,960:1,080 = 3.6
In all of these numbers the number nine is a significant integer. The earth radius (3,960 mi) when divided into the solar radius (432,000 mi) is 109.090909. Again, the number nine shows up in a significant way. Through the quality of the number nine we can see that there is an inner harmonic working between these three radii of the Sun, Moon and Earth.
But what is it about the number nine that makes these relationships significant? Aren't there many things in the world that have the integer nine as part of their mathtmatical values? In order to look more deeply into the nine"ness" of these relationships it would be useful to look at the harmonic element of numbers rather than the computational element. The following diagram recapitulates the material presented in the first essay.
8th Octave,compression and new fundamental-polarity
7th distillation 144°
6th turning 120°
5th expansion 90°
4th compression 72°
3rd turning 45°
2nd expansion 36°
In this diagram we see the numbers of the intervals in the octave as forces of motion and flow rather than as computational integers. We see that there are two compression points where a fundamental is established. One is at the beginning of the sequence (the fundamental). And the other is at the end of the sequence (the octave). In between is the compressive motion of the fourth as a "little octave." These relationships are illustrated in the following diagram.
Figure 2 shows the spatial qualities of the intervals up to the octave. The firs pitch at unity expands strongly into the second, then shortens and turns inward into the third as a compression into the fourth shortens the string length further into a very small interval. The fifth expands like the second, the sixth turns inward like the third, the seventh maintains a similar interval space with the sixth in a kind of distillation process, with only a slight diminishing of string length. Then a surge into the octave gives rise to a new fundamental. If we were to try to classify meteorological phenomena as one of these intervals we could say that the octave gives a sense of duration coupled with newness. The fundamental to the fourth represents a strong compression. The fifth begins a new expansion towards the octave. At the sounding of the octave, the gravity and compression of the fourth is transformed and balanced by the levity of the octave into a quality of duration coupled with the newness of a new fundamental at a higher level of vibration.
These qualities present in the lower octave become carried up into the higher intervals. Here they are transformed but still bear the mark of their inverval in the lower octave.
15 (8th) Highest Compression
14 (7th) distillation
13 (6th) turning
12 (5th) Expansion above Octave
11 (4th) Compression above Octave
10 (3rd) Turning
9 (2nd) Higher expansion
Above the octave the 9th above the fundamental has the quality of a second above the fundamental. And yet it is different, it is an expansion above the octave, a higher more fluid expansion now in the realm of air and warmth. The 10th above the fundamental is a new third, the 11th is a higher fourth with another compressive pulse, the 12th is again a higher second, expanding away from the compression of the 11th. . This links the 9th and the 12th harmonically. The 13th and 14th repeat the 6th and 7th but above the octave, and finally the 15th above the fundamental gets to the second octave yet this second octave is above the first by a whole process of development. We could call the action of this harmonic in the atmosphere a force of radical duration. In the planetary flux model, the number 15 plays into the struggle between drought cycles and floods by allowing high pressure cells to maintain their positions (duration) against all movements of the jet stream. In Doc Weather, 15th above the fundamental or the second octave figures prominently in patterns of drought.
To these consonant intervals we can also add the dissonance of the tritone or 41/2 above the fundamental which is represented by the angular aspect of 108°. This number figures prominently in situations where low pressure dominates in tropical latitudes. The other interval which is very useful for foul weather prediction is the minor sixth, that represents an angle of arc of 135°. The angle 135° has strong implications for low pressure events near the polar regions. This is a multiple of 9x15. Both the expansive force of the 9 and the compressive force of the 15 are present in this interval. 135° is an angular aspect with very clear effects on disturbing weather patterns along with the angular aspect of 108°. This is because they both represent dissonant elements in an otherwise harmonious sequence of pitches.
We now need to turn our attention to the actual concept of harmony and dissonance in order to develop a perception of the idea of the harmonic mean. The harmonic mean allows us to find other angular aspects that influence the eclipse points. The harmonic mean is any number found between two other numbers that represents the ideal harmony between them. If we were to take any two numbers and try to find the harmonic mean between them we would need to use the formula 2ab/a+b. If for instance we took 9 and 18 as two numbers in a series and we wanted to find the number between them which would be a perfect expression of their harmonic relationship, following our formula we would get; 12.
Twelve is the perfect harmonic interval between nine and eighteen. If we wished to find the next rational i.e. perfectly divisible harmonic we would have to go to 9:45 =15. That is the next number in which a whole number is the harmonic mean number would be 45. So between 9 and 18 the rational or whole harmonic is the number 12. Searching for the next whole number or rational harmonic leads us to the relationship between 9 and 45 (9:45). The harmonic mean for these two numbers is 15. The next rational harmonic is 9:72=16.
Searching for the next rational harmonic is a bit of a surprise. We are looking for the rational number that has 17 as the harmonic to 9. If we took 9:180=17.142857 we don't get a rational number. Searching farther along we can use 9:360=17.560975, and still get an irrational number as the harmonic mean. Searching farther still we can explode the harmonic relationships to 9 and end up with irrational numbers related to 17. For instance, 9:100,000=17.99838, 9:900,009=17.99982. We seem to have reached an abyss. This means that for 9 there are only a few rational harmonies in the universe, 18,45, 72. Every number between 2 and 9 will exhibit the same pattern of a few harmonic rational intervals before a leap into infinite or irrational (no ratios) numbers. The following is a list of those numbers and their harmonics.
From this we can see that the simple harmonics arising from the numbers in the first octave quickly reach infinity. Above the first octave, however, more harmonics can be found. Instead of shooting randomly throughout the universe of number for harmonic intervals above the octave, it would be a good starting point to use, as Kepler did, the circumference of the earth as a wholeness or fundamental. Since we are looking for harmonic degrees of angular aspect between planets we can use 360 as our fundamental. The following chart shows the rational harmonics to 360 present from 2 to 180 using the formula 2ab/a+b. It should be noted that between 2 and 24 there are no natural harmonics to 360. All of the relationships are irrational numbers. This is why we start this list at 24:360. We are looking for rational or natural numbers as a base to the harmonic series the numbers chosen,24,30,40,45 etc, are the next natural harmonics in the series. No numbers were passed over to make this list.
The figure shows the Kepler scale. This is a diagram of the angular aspects drawn from an eclipse point in the fundmental out through a half circle that represents the circumference of the earth at the equator. In this sequence of lines we can see that the Kepler scale illustrated in the figure is well represented in natural harmonics to 360. The intervals of the second (45°), third (72° ), fifth (120° ) and the sixth (144° ) and the octave are rational harmonics to 360. Pitches not included in this sequence but are found in Kepler's scale are the fourth, and the F# and minor 6th. The rational relationships of harmony for 90 (4th), 108 (F#) and 135 (minor 6th) are found by building ratios with other significant parts of 360. The fourth can be found as a result of a beautiful little harmonic roundelay illustrated below.
2:180=36 (the harmonic mean between 2 and 180 is 36). Take that product as the first term.
36:180=60 (the harmonic mean between 36 and 180 is 60) Take that product as the first term.
60:180=90 (the harmonic mean between 60 and 180 is 90 )
From this we can see the integrated forces of the fourth (compression) expressed as ratios involved with the half of the whole, i.e. the number 2 is the number of the division of unity. This is placed in ratio to half of the degrees in a circle(180). This yields a tenth part of the whole (36). This tenth part (36) set in ratio against the half of the whole (180) yields a third of the half (60) or a sixth of the whole. This part (60) set in harmonic ratio to the half of the whole (180) is harmonic to half (90) of the half (180) of the whole (360). This is a thoroughly integrated series of ratios giving the distinct impression of high integration and sustainability (duration/compression).
For the F# or fourth and a half above the fundamental (108° of arc) there is a ratio linking 72, and 216
The harmonic of 108 is not found as a whole step but is contained as a rational harmonic in the series of numbers that are multiples of 72°. This angle is generated by the pentagon; 5x72°=360°. This angle of 72° contains the secrets of the last of the Kepler scale intervals the minor 6th, or 135°. We saw how in the last example the 108 was harmonic between 72 and 216. We can recall that 108 was a number which arose when we divided the radius of the Earth into the radius of the Sun. 108 is also the number of miles in the radius of the Moon. If we look at the harmonic ratio between 108 and 360 we do not find a rational harmonic, neither is 108 harmonic to 216. However the rational harmonic between 108 and 180 is 135. i35 is also the harmonic mean between 90 and 270.
In summary we have designated 9 as the number that is characterized by expansion and fluid change. Fifteen, as a representation of the octave above the octave, is characterized as duration and resistance to change. The 135 then is a harmonic between change and duration. In astrological systems, 135° is the angle of agitation. In the planetary flux model, the 135° angular aspects arise when polarities between flow and resistance become accentuated to the point of agitation.
With the 216 and the 270 we are extending beyond the 180 point in the circle. We can use 108 as the first term for a new harmonic relationship that extends beyond the 180° point.
144 is also present harmonically between 90 and 360.
From this we see that 144 is also a significant harmonic angle with numerous connections within the octave.
Here is a chart that summarizes all of the previous elements of harmonies in this article.
Kepler harmonics and angular aspects
Pitch ratio angle multiple
Fundamental 1:1 360° 1X360
2nd 7:8 45° 3X15
mi3rd 5:6 60° 4X15
maj 3rd 4:5 72° 2X36
4th 3:4 90° 5X18, 6X15
tritone 7:10 108° 3X36
5th 2:3 120° 8X15
mi6th 5:8 135° 9X15
6th 3:5 144° 4X36
7th 7:12 150° 10X15
octave 1:2 180° 10X18
A final nod towards the mystery of the seventh, the distiller and perfecter among the angles can be obtained by finding the mean between 155 and 360.
This places 155 close to the 150° of the seventh above the fundamental in the Kepler scale. The seventh has shown little influence on climate patterns but it is a very enigmatic interval that bears further research.
The 135° angles, when projected onto the earth, are found in the very disturbed zones of the polar front. See "projective fields in the planetary flux model":http://docweather.com/
From this brief description we have seen that two primary angles of arc that are highly generative of weather polarities, are the 9° and 15° angles. We also have seen in the Kepler scale a series of harmonics which we can utilize in constructing a grid of lines and significant curves known in the planetary flux model as jet curves. "jet curves":http//docweather.com/ These lines can be projected over the earth for the purpose of tracking the onset of storms and the growth and decay of blocking ridges. The arising and decaying of these weather features is melodic and can be experienced through the new approach to the music of the spheres.