Post by glaucus on Mar 17, 2022 5:09:25 GMT
I have been investigating the Harmonics of the Golden Aspects that Dr. Theodor Landscheidt introduced for astrological use.
"The golden section divisions within cycles formed by the rotating earth may be considered a set of astrological aspects. The complete set emerges when we superimpose the two schematic diagrams in Figures 19 and 23, related to minor and major of the golden section, and compare all of those angles we find on the right and on the left of the origin 0°. Imagine that you are standing at the rising point R, or 0°, of the diurnal circle and are looking over to the setting point at 180°. Then the superimposed golden section divisions on your right form the set 21.25°, 42.49°, 47.51°, 68.75°, 111.25°, 132.49°, 137.51° and 158.75°. The angles 338.75°, 317.51°, 312.49° and so forth, on your left repeat the set on your right when subtracted from 360°. If we extend the fractal beyond the semicircle and include the quarter circle, the golden section operation generates the additional angles 34.38°, 55.62°, 124.38° and 145.62°. It is not an arbitrary procedure to divide cycles in halves and quarters. Observation shows that spectral peaks can appear at twice and four times the driving frequency, or at half or a quarter of it (Burroughs, 1992). Statistical tests indicate that the twelve golden aspects in the complete set are reliable."
Golden Aspects in their degrees and minutes are
21°15', 34°22', 42°30', 47°31', 55°37', 68°45', 111°15',
124°22 , 132°29', 137°30', 145°37', 158°45'
Dr. Landscheidt introduced the Geocentric Planetary Nodes in Astrology, and he mentions 85° as a Golden Aspect which any astrologer would recognize is the square (90°) with an orb of 5 degrees. He included Golden Aspects as aspects to work with the Geocentric Planetary Nodes.
David Cochrane identifies the 137°30' as The Golden Mean Aspect and the 85° as The Golden Mean of the Golden Mean Aspect.
David Cochrane's astrology program Sirius lists the following Golden Aspects that Dr. Lanscheidt devised as
maximum orb of 1 degree
He lists the 85° as 84°59'
He doesn't include the 47°31', 124°22',
137°30' is viewed by astrologers as the 3/8th Harmonic Aspect Sesquiquadrate (145°) with an orb of 2 degrees, 30 minutes. If using harmonic proportionate orb based on a 1st Harmonic Aspect Conjunction (0°) having a maximum orb of 15 degrees, the Sesquiquadrate has a maximum orb of 1 degree, 53 minutes.
84°59' is viewed as the 1/4th Harmonic Aspect Square (90°) with an orb of 5 degrees, 1 minute by mainstream astrologers. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the square has a maximum orb of 3 degrees, 45 minutes. Most astrologers use a maximum orb of well over 5 degrees. Some astrologers even use a maximum of 10 degrees. If using wide orbs like that, you run into an aspect of one of the first 12 harmonics, and that is the 3/11th Harmonic aspect Triundecile (98°10'54'') which has a harmonic proportionate orb of 1 degree, 21 minutes.
124°22' is viewed by astrologers as the 1/3rd harmonic aspect Trine (120°) with an orb of 4 degrees, 22 minutes by mainstream astrologers. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the trine has a maximum orb of 5 degrees. Most astrologers use a maximum orb well over 5 degrees. Some astrologers even use a maximum orb of 10 degrees.
132°29' is viewed as the Sesquiquadrate with an orb of 2 degrees, 31 minutes by mainstream astrologers. It is the 7/19th Harmonic Aspect with an orb of 8 minutes.
145°37' is viewed as the 2/5th Harmonic aspect Biquintile (144°) with an orb of 1 degree, 37 minutes by mainstream astrologers. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the Biquintile has a maximum orb of 3 degrees. Even though both the 1/5th Harmonic Aspect Quintile (72°) and Biquintile are of the same harmonic, some astrologers consider only the Quintile or give smaller maximum orb for the Biquintile like it's the lesser aspect like the 1/10th Harmonic Aspect Decile (36°) which is also known as Semiquintile.
158°45' is viewed as the 4/9th Harmonic Aspect Quadranovile (160°) with an orb of 1 degree, 15 minutes. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the Quadranovile has a maximum orb of 1 degree, 40 minutes.
47°31' is not close to any of the first 12 harmonics. It is the 2/15th harmonic aspect Biquindecile (48°) with an orb of 29 minutes. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the Biquindecile has a maximum orb of 1 degree.
I can see why highly harmonically oriented astrologer David Cochrane included 137°30' and 84°59' as Golden Aspects and didn't include 47°31', 124°22',132°29',145°37',158°45' as Golden Aspects.
He also included 52°31' as a Golden Aspect which can be viewed as the 1/7th Harmonic aspect Septile (51°25'42'') with an orb of 1 degree, 6 minutes. If using harmonic proportionate orb based on Conjunction, the Septile has a maximum orb of 2 degrees, 8 minutes. 52°31 is definitely a Septile, but this aspect and the 84°59' aspect are strongly related to the Golden Mean.
137.5° x 0.618 = 84.975°
137.5° x 0.382 = 52.525°
The Golden Mean is strongly connected to the Fibonacci numbers. The Golden Mean Aspect is multiple harmonic aspects with Fibonacci numbers as both the numerator and denominator that Sirius harmonic grid show. It's THE Fibonacci Aspect.
5/13th Harmonic Aspect
8/21st Harmonic Aspect
13/34th Harmonic Aspect
21/55th Harmonic Aspect
34/89th Harmonic Aspect
55/144th Harmonic Aspect
89/233rd Harmonic Aspect
Harmonic Grid is limited to the 360th Harmonic.
The Golden Mean is also
144/377th Harmonic Aspect
233/610th Harmonic Aspect
377/987th Harmonic Aspect
The following are the harmonic grids for Moon-Uranus Golden Aspects here in Tacoma, Washington.
Moon-Uranus Golden Mean (137°30'28'')
March 18, 2022
89/233 at 96% (Fibonacci harmonic)
55/144 at 92% (Fibonacci harmonic)
123/322 at 83%
other Fibonacci harmonics: 5/13, 8/21, 13/34, 21/55, 34/89
The Golden Mean Aspect is THE Fibonacci Aspect.
Moon-Uranus Golden Mean 137°30' (rounded off)
March 18, 2022 3:57:49 AM
55/144 at 100%
131/343 at 83%
My view is that the Golden Mean's primary harmonic aspect is 55/144.
Moon-Uranus Golden Mean of Golden Mean 84°59 '
March 28, 2022 12:17:06 AM
72/305 at 99%(shares with the Golden Mean)
55/233 at 91% (shares Fibonacci harmonic with the Golden Mean)
other shared Fibonacci harmonics :13/55, 21/89
other shared harmonics: 29/123, 47/199
Moon-Uranus Golden Mean of Golden Mean 85° (rounded off)
March 28, 2022 12:15:27 AM
17/72 at 100%
It being primarily a 72nd Harmonic and Golden Mean Aspect being primarily a 144th Harmonic Aspect. 72°is the Quintile and 144° Biquintile which are the 1st and 2nd 5th Harmonic Aspects.
March 30, 2022 9:09:33 AM
48/329 at 99%
41/281 at 94%
shared Fibonacci harmonics: 5/34, 8/55, 13/89, 34/233
other shared harmonics: 29/199, 47/322
Moon-Uranus 52°31 (rounded off)
March 30, 2022 9:09:55 AM
43/329 at 94%
41/281 at 87%
April 1, 2022 6:13:48 PM
17/288 at 100%
18/305 at 92% (shares with the Golden Mean)
16/271 at 91%
19/322 at 84% (shares harmonic with the Golden Mean)
other shared harmonics: 15/254
March 31, 2022 5:55:43 AM
17/178 - at 97%
19/199 - at 93% (shares harmonic with the Golden Mean)
Fibonacci harmonics shared with the Golden Mean: 2/21
other harmonics shared with the Golden Mean: 32/335
March 31, 2022 3:09:55 AM
36/305 at 99%
17/144 at 92% (shares Fibonacci Harmonic with the Golden Mean)
19/161 at 92%
other harmonics shared with the Golden Mean: 9/76, 25/212
March 30, 2022 9:09:55 AM
26/197 at 96%
45/341 at 95%
shared Fibonacci harmonics: 19/144
March 30, 2022 3:38:41 AM
36/233 at 99% (shares Fibonacci harmonic with the Golden Mean)
53/343 at 90% (shares harmonic with the Golden Mean
55/356 at 89%
other shared Fibonacci harmonics: 2/13
other shared harmonics: 19/123, 53/343
March 29, 2022 4:30:02 AM
55/288 at 93%
shared Fibonacci harmonics: 4/21, 17/89
shared harmonics: 38/199, 59/309, 64/335
March 16, 2022 4:16:47 AM
72/233 at 98% (shares Fibonacci harmonic with the Golden Mean)
89/288 at at 92%
106/343 at 82% (shared harmonic with the Golden Mean
shared Fibonacci harmonics: 4/13, 17/55, 72/233
other shared harmonics: 38/123, 93/301, 106/343
March 17, 2022 AM 4:23:22 AM
123/356 at 95%
104/301 at 89% (shares with the Golden Mean)
shared Fibonacci harmonics: 19/55
other shared harmonics: 104/301
March 17, 2022 7:00:58 PM
99/269 at 99%
shared Fibonacci harmonics: 53/144
March 18, 2022 6:20:15 PM
36/89 at 99% (shares a Fibonacci harmonic with the Golden Mean)
other shared harmonics: 125/309
March 19, 2022 5:18:15 PM
127/288 at 100%
71/161 at 92%
shared Fibonacci harmonics: 15/34
In his book Harmonics, John Addey wrote:
The number 13 is also interesting. It represents the one in the midst of the twelve and so suggests the idea of spiritual (?) lordship. Keith Critchlow in his valuable study of Order in Space points out that twelve spheres of equal size will exactly fit round a central sphere of the same size so that all are exactly touching their neighbors. The aspects in the the thirteenth series are as follows:
27°42' 55°23 110°46 138°28 166°09 (orb about 1°)
Using his recommended 1 degree orb, the following Golden Aspects are also 13th Harmonic Aspects:
55°37 is the 2/13th Harmonic aspect 55°23 with an orb of 14 minutes
111°15' is the 4/13th Harmonic aspect 110°46 with an orb of 29 minutes
137°30 (The Golden Mean Aspect) is the 5/13th Harmonic aspect 138°28 with an orb of 58 minutes.
13 is one of the Fibonacci numbers, and so the 13th Harmonic is a Fibonacci Harmonic. Therefore, they are Fibonacci Harmonic Aspects.
The Golden Aspects that I believe are most important are:
137°30', 85°, and 52°31
132°29 is just as significant as any of them, but I don't consider it to be a Golden Aspect. The aspect is very close to the Euler's aspect 132°26'12'' which is created by dividing 360° by the Euler's Number which is 2.71828. For that reason alone, I'd leave out 132°29 as a Golden Aspect and focus on it as just the Euler's Aspect. It is the 71/193rd Harmonic Aspect. I'd use an orb of 4 minutes, 36 minutes which is the harmonic proportionate orb for the 193rd Harmonic.
The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 which can be characterized in many ways. It is the base of the natural logarithms. It is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest.
It is also the unique positive number a such that the graph of the function y = ax has a slope of 1 at x = 0.
The (natural) exponential function f(x) = ex is the unique function f that equals its own derivative and satisfies the equation f(0) = 1; hence one can also define e as f(1). The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are various other characterizations.
e is sometimes called Euler's number (not to be confused with Euler's constant \gamma), after the Swiss mathematician Leonhard Euler, or Napier's constant. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.
The number e is of great importance in mathematics,[page needed] alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity, and play important and recurring roles across mathematics. Like the constant π, e is irrational (that is, it cannot be represented as a ratio of integers) and transcendental (that is, it is not a root of any non-zero polynomial with rational coefficients).[
On the Cosmological Significance of Euler’s Number Hartmut Muller ¨
The paper derives and exemplifies the stabilizing significance of Euler’s number in particle physics, biophysics, geophysics, astrophysics and cosmology.
The paper discusses Euler’s number stabilizing the solar system, biological rhythms, the atom, and the universe
The maximum orb that I recommend for the Golden Aspects is 6 minutes which is the harmonic proportionate orb for the 144th Harmonic. 1 degree orb is way too wide for the Golden Aspects. They'll be of totally different harmonics and operate on quite different frequencies. 'Major aspects' and the major 'minor aspects' may work differently because of having different frequencies in connection to different harmonics.
John Addey noted that the advantage of having a definite principle upon which to judge orbs is that it enables one to move with confidence into the sphere of micro-aspects which result from the division of the circle by numbers above 20. I definitely find it useful for the micro-aspects. I am highly geometrically oriented in how I do Astrology and consider more than just the ecliptic longitude aspects, and so I use smaller maximum orbs than mainstream Astrologers. Revisiting Theodor Landscheidt's work has gotten me interested in nontraditional aspects including especially the micro-aspects. He was a modern Kepler. Johannes Kepler was the pioneer of Harmonic Astrology, and he introduced minor aspects in Astrology. David Cochrane named one of his astrology programs after Kepler which is a program that I used before I upgraded to Sirius. I happen to have asteroid Urania (Muse of Astronomy) trine Kepler-Pluto conjunction within all aspects within 1 degree orb.