Post by glaucus on May 6, 2022 20:13:22 GMT
Fibonacci Harmonic Aspects
This is a post about Fibonacci Harmonic Aspects which are Aspects with Fibonacci Number as the denominator if you look at Harmonic Aspects as Fractions of a Circle.
I listed the 1st Harmonic Aspect, 2nd Harmonic Aspect, 3rd Harmonic Aspects, 5th Harmonic Aspects, 13th Harmonic Aspects, 21st Harmonic Aspects, 34th Harmonic Aspects, 55th Harmonic Aspects, 89th Harmonic Aspects, 144th Harmonic Aspects, and 233rd Harmonic Aspects.
In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omit the initial terms and start the sequence from 1 and 1 or from 1 and 2. Starting from 0 and 1, the next few values in the sequence are:[1]
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
The Fibonacci numbers were first described in Indian mathematics,[2][3][4] as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.[5]
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a pine cone's bracts.
Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181
Fibonacci numbers are strongly connected to the Golden Mean which is 1.618 represented by by the Greek letter phi, Φ.
Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
en.wikipedia.org/wiki/Fibonacci_number
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with {\displaystyle a>b>0,}
{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {=:}\ \varphi }
where the Greek letter phi (\varphi or \phi) represents the golden ratio. It is an irrational number that is a solution to the quadratic equation {\displaystyle x^{2}-x-1=0,} with a value of[2][1]
{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=}1.618033988749....(OEIS: A001622)
The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[3][4] Other names include extreme and mean ratio,[5] medial section, divine proportion (Latin: proportio divina),[6] divine section (Latin: sectio divina), golden proportion, golden cut,[7] and golden number.[8][9][10]
Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[11] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
en.wikipedia.org/wiki/Golden_ratio
21 divided by 13 = 1.615384615384615
34 divided by 21 = 1.619047619047619
55 divided by 34 = 1.617647058823529
89 divided by 55 = 1.618181818181818
144 divided by 89 = 1.617977528089888
The following is from David Cochrane:
Cosmic cybernetics is a highly simplified form of astrological analysis which uses an extremely narrow segment of astrological ideas. A fundamental assumption of cosmic cybernetics, as with harmonic astrology as proposed by John Addey, is that the angular distance between planetary positions, or the projected position of planets onto the ecliptic plane, produces in some as yet undiscovered way a wave form. Each planet acts as the node or crest of a wave function and there is an effect on life on our planet from these undiscovered waves.
Based on harmonic theory we expect that waves of higher harmonics will have proportionately smaller wave sizes so that all harmonics occur approximately equal often by chance. Even if harmonics do not occur equally often by chance due to the complex motion of planetary positions, based on harmonic theory the orb would still be based on these proportional wave sizes.
• To establish the proportional wave sizes, the orb for the first harmonic is divided by the harmonic number.
I do not believe that higher harmonics are weaker than lower harmonics, at least up to the 180th harmonic. I believe that higher harmonics are more internal and less visible externally but are not weaker. As an example of how aspects in a harmonic chart work, a trine in the 5th harmonic chart is a 15th harmonic aspect (5 x 3). In many cases the harmonic may reduce to a lower harmonic.
www.astrosoftware.com/harmonicfirst32.pdf
A very small orb of an aspect is not just the indicator of an aspect that is very strong. It is also an indicator that the aspect is also of higher octave harmonics. Example: a square within 6 minute orb is not just the 1/4 aspect, but it is also the 2/8 aspect, the 4/16 aspect, the 8/32 aspect, the 16/64 aspect, the 32/128 aspect. It is also the 9/36 aspect, the 18/72 aspect and the 36/144 aspect.
I listed all aspects of Harmonics of Fibonacci Numbers up to the 233rd.
I stopped at 233rd because I will be going over 360th. David Cochrane doesn't have the view that Harmonics beyond 360 work, and I am in agreement with him.
Fibonacci Harmonic Aspects
1st Harmonic Aspect (Conjunction in 1st Harmonic Chart)
1 is a Fibonacci Prime Number
1st Harmonic Aspect is a Fibonacci Harmonic Aspect
Maximum Orb - 15°
1 - 360° aka waning Conjunction with waxing degree being 0°
The following aspects have proportionate orbs based on 15 degree orb for Conjunction
2nd Harmonic Aspect (Conjunctions in 2nd Harmonic Chart)
2 is a Fibonacci Prime Number
Divisors are 1, 2
1st Harmonic aspect and 2nd Harmonic aspect are Fibonacci Harmonic aspects
Maximum Orb - 7° 30'
1/2 - 180°
2/2 - 360° aka waning Conjunction with waxing degree being 0°
3rd Harmonic Aspect (Conjunctions in 3rd Harmonic Chart)
3 is a Fibonacci Prime Number
Divisors - 1, 3
1st Harmonic aspect and 3rd Harmonic aspect are Fibonacci Harmonic Aspects
Maximum Orb - 5°
1/3 - 120° aka Trine
2/3 - 60° aka Sextile
3/3 - 360° aka waning Conjunction with waxing degree being 0°
5th Harmonic Aspects (Conjunctions in 5th Harmonic Chart)
5 is a Fibonacci Prime Number
Divisors - 1, 5
1st Harmonic and 5th Harmonic are Fibonacci Harmonics
1/5 - 72° aka Quintile
2/5 - 144° aka BiQuintile
5/5 - 360° which is waning degree of Conjunction with waxing degree being 0°
8th Harmonic Aspects (Conjunctions in 8th Harmonic Chart)
8 is a Fibonacci Number
Factorization is 2 to the 3rd Power
Divisors - 1, 2, 4, 8
1st Harmonic aspect, 2nd Harmonic aspect, 4th Harmonic aspects, and 8th Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 3° 45'
1/8 - 45° aka SemiSquare
2/8 - 90° aka Square
3/8 - 135° aka SesquiSquare
4/8 - 180° aka Opposition
8/8 - 360° aka waning Conjunction with waxing degree being 0°
13th Harmonic Aspects (Conjunctions in 13th Harmonic Chart)
13 is a Fibonacci Prime Number
Divisors - 1, 13
1st Harmonic aspect and 13th Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 1° 09'
1/13 - 27° 41' 31
2/13 - 55° 23′ 02′′
3/13 - 83° 04′ 33′′
4/13 - 110° 46′ 4′′
5/13 - 138° 27′ 35′′
6/13 - 166° 09′ 06′′
13/13 - 360° aka waning Conjunction with waxing degree being 0°
21st Harmonics (Conjunctions in 21st Harmonic Chart)
21 is a Fibonacci Number
Factorization - 3 x 7
Divisors - 1, 3, 7, 21
1st Harmonic aspect, 3rd Harmonic aspects, 7th Harmonic aspects, and 21st Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 0° 43'
1/21 - 17° 08′ 35′′
2/21 - 34° 17′ 10′′
3/21 - 51°25' 42'' aka Septile
4/21 - 68° 34′ 20′′
5/21 - 85° 42' 55''
6/21 - 102° 51' 25'' aka BiSeptile
7/21 - 120° aka Trine
8/21 - 137° 08' 40"
9/21 - 154° 17' 08'' aka TriSeptile
10/21 - 171° 25' 50''
21/21 - 360° aka waning Conjunction with waxing degree being 0°
34th Harmonic Aspects (Conjunctions in 34th Harmonic Chart)
34 is a Fibonacci Number
Factorization - 2 × 17
Divisors - 1, 2, 17, 34
1st Harmonic aspect, 2nd Harmonic aspect, 17th Harmonic aspects, and 34th Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 0° 26'
1/34 - 10° 35' 17
2/34 - 21° 10′ 34′′
3/34 - 31° 45′ 51′′
4/34 - 42° 21′ 08′′
5/34 - 52° 56′ 25′′
6/34 - 63° 31′ 42′′
7/34 - 74° 06′ 59′′
8/34 - 84° 42′ 16′′
9/34 - 95° 17′ 33′′
10/34 - 105° 52′ 50′′
11/34 - 116° 28′ 07′′
12/34 - 127° 03′ 24′′
13/34 - 137° 38′ 41
14/34 - 148° 13′ 58′′
15/34 - 158° 49′ 15′′
16/34 - 169° 24′ 32′′
17/34 - 180° aka Opposition
34/34 - 360° aka waning Conjunction with waxing degree being 0°
55th Harmonic Aspects (Conjunctions in 55th Harmonic Chart)
55 is a Fibonacci Number
Factorization - 5 × 11
Divisors - 1, 5, 11, 55
1st Harmonic aspect, 5th Harmonic aspects, 11th Harmonic aspects, and 55th Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 0° 16′
1/55 - 6° 32' 42"
2/55 - 13° 05′ 24′′
3/55 - 19° 38′ 06′′
4/55 - 26° 10′ 48′′
5/55 - 32°43'37'' aka Undecile
6/55 - 39° 16′ 12"
7/55 - 45° 48′ 54′′
8/55 - 52° 21′ 36′′
9/55 - 58° 54′ 18′′
10/55 - 65°27'16'' aka BiUndecile
11/55 - 72° aka Quintile
12/55 - 78° 32′ 24′′
13/55 - 85° 05′ 06′′
14/55 - 91° 37′ 48′′
15/55 - 98°10'54'' aka TriUndecile
16/55 - 104° 43′ 12′′
17/55 - 111° 15′ 54′′
18/55 - 117° 48′ 36′′
19/55 - 124° 21′ 18′′
20/55 - 130°54'32'' aka QuadriUndecile
21/55 - 137° 26′ 42′′
22/55 - 144° BiQuintile
23/55 - 150° 32′ 06′′
24/55 - 157° 04′ 48′′
25/55 - 163° 38' 10'' aka QuinqueUndecile
26/55 - 170° 10′ 12′′
27/55 - 176° 42′ 54′′
55/55 - 360° aka waning Conjunction with waxing degree being 0°
89th Harmonic Aspects (Conjunctions in 88th Harmonic Chart)
89 is a Fibonacci Prime Number
Divisors - 1, 89
1st Harmonic aspect and 8th Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 0° 10′
1/89 - 4° 02' 42"
2/89 - 8° 05′ 24′′
3/89 - 12° 08′ 06′′
4/89 - 16° 10′ 48′′
5/89 - 20° 13′ 30′′
6/89 - 24° 16′ 12′′
7/89 - 28° 18′ 54′′
8/89 - 32° 21′ 36′′
9/89 - 36° 24′ 18′′
10/89 - 40° 27′
11/89 - 44° 29′ 42′′
12/89 - 48° 32′ 24′′
13/89 - 52° 35′ 06′′
14/89 - 56° 37′ 48′′
15/89 - 60° 40′ 30′′
16/89 - 64° 43′ 12′′
17/89 - 68° 45′ 54′′
18/89 - 72° 48′ 36′′
19/89 - 76° 51′ 18′′
20/89 - 80° 54′
21/89 - 84° 56′ 42′′
22/89 - 88° 59′ 24′′
23/89 - 93° 02′ 06′′
24/89 - 97° 04′ 48′′
25/89 - 101° 07′ 30′′
26/89 - 105° 10′ 12′′
27/89 - 109° 12′ 54′′
28/89 - 113° 15′ 36′′
29/89 - 117° 18′ 18′′
30/89 - 121° 21′
31/89 - 125° 23′ 42′′
32/89 - 129° 26′ 24′′
33/89 - 133° 29′ 06′′
34/89 - 137° 31′ 48′′
35/89 - 141° 34′ 30′′
36/89 - 145° 37′ 12′′
37/89 - 149° 39′ 54′′
38/89 - 153° 42′ 36′′
39/89 - 157° 45′ 18′′
40/89 - 161° 48′
41/89 - 165° 50′ 42′′
42/89 - 169° 53′ 24′′
43/89 - 173° 56′ 06′′
44/89 - 177° 58′ 48′′
89/89 - 360° aka waning Conjunction with waxing degree being 0°
144th Harmonic Aspects (Conjunctions in 44th Harmonic Chart)
144 is a Fibonacci Number
Square of 12
Factorization - 2 to the 4th Power × 3 to the 2nd Power
Divisors - 1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 36, 48, 72, 144
1st Harmonic aspect, 2nd Harmonic aspect, 3rd Harmonic aspects, 4th Harmonic aspect, 4th Harmonic aspects, 6th Harmonic aspects, 8th Harmonic aspects, 12th Harmonic aspects, 16th Harmonic aspects, 18th Harmonic aspects, 24th Harmonic aspects, 36th Harmonic aspects, 48th Harmonic aspects, 72nd Harmonic aspects and 144th Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 0° 06′
1/144 - 2° 30'
2/144 - 5° aka 1/72
3/144 - 7° 30' aka 1/48
4/144 - 10° aka 1/36
5/144 - 12° 30' aka 2/48
6/144 - 15° aka 1/24 and Viginquartile
7/144 - 17° 30'
8/144 - 20° aka SemiNovile
9/144 - 22° 30' aka 1/16
10/144 - 25° aka 5/72
11/144 - 27° 30'
12/144 - 30° aka SemiSextile
13/144 - 32° 30'
14/144 - 35° aka 7/72
15/144 - 37° 30' aka 5/48
16/144 - 40° aka Novile
17/144 - 42° 30'
18/144 - 45° aka SemiSquare
19/144 - 47° 30'
20/144 - 50° aka 5/36
21/144 - 52° 30' aka 7/48
22/144 - 55° aka 11/72
23/144 - 57° 30'
24/144 - 60° aka Sextile
25/144 - 62° 30'
26/144 - 65° aka 13/72
27/144 - 67° 30' aka 3/16
28/144 - 70° aka 7/36
29/144 - 72° 30'
30/144 - 75° aka 5/24 aka Squile
31/144 - 77° 30'
32/144 - 80° aka BiNovile
33/144 - 82° 30' aka 11/48
34/144 - 85° aka 17/72 and Golden Mean of Golden Mean
35/144 - 87° 30'
36/144 - 90° aka Square
37/144 - 92° 30'
38/144 - 95° aka 19/72
39/144 - 97° 30' aka 13/48
40/144 - 100° aka 5/18
41/144 - 102° 30'
42/144 - 105° aka 7/24 and Squine
43/144 - 107° 30'
44/144 - 110° aka 11/36
45/144 - 112° 30' aka 5/16
46/144 - 115° aka 23/72
47/144 - 117° 30'
48/144 - 120° aka Trine
49/144 - 122° 30'
50/144 - 125° aka 25/72
51/144 - 127°30' aka 17/48
52/144 - 130° aka 13/36
53/144 - 132° 30'
54/144 - 135° aka SesquiSquare
55/144 - 137° 30' aka Golden Mean
56/144 - 140° aka 7/18
57/144 - 142° 30' aka 19/48
58/144 - 145° aka 29/72
59/144 - 147° 30'
60/144 - 150° aka Quincunx
61/144 - 152° 30'
62/144 - 155° aka 31/72
63/144 - 157° 30' aka 7/16
64/144 - 160° aka QuadriNovile
65/144 - 162° 30'
66/144 - 165° aka 11/24 and Johndro
67/144 - 167° 30'
68/144 - 170° aka 17/36
69/144 - 172° 30' aka 23/48
70/144 - 175° aka 35/72
71/144 - 177° 30'
72/144 - 180° aka Opposition
144/144 - 360° aka waning Conjunction with waxing degree being 0°
233rd Harmonic Aspects (Conjunctions in 233rd Harmonic Chart)
233 is a Fibonacci Prime Number
Divisors - 1, 233
1st Harmonic aspect and 233rd Harmonic aspects are Fibonacci Harmonic aspects
Maximum Orb - 0° 04′
1/233 - 1° 32' 42"
2/233 - 3° 05′ 24′′
3/233 - 4° 38′ 06′′
4/233 - 6° 10′ 48′′
5/233 - 7° 43′ 30′′
6/233 - 9° 16′ 12′′
7/233 - 10° 48′ 54′′
8/233 - 12° 21′ 36′′
9/233 - 13° 54′ 18′′
10/233 - 15° 27′
11/233 - 16° 59′ 42′′
12/233 - 18° 32′ 24′′
13/233 - 20° 05′ 06′′
14/233 - 21° 37′ 48′′
15/233 - 23° 10′ 30′′
16/233 - 24° 43′ 12′′
17/233 - 26° 15′ 54′′
18/233 - 27° 48′ 36′′
19/233 - 29° 21′ 18′′
20/233 - 30° 54′
21/233 - 32° 26′ 42"
22/233 - 33° 59′ 24′′
23/233 - 35° 32′ 06′′
24/233 - 37° 04′ 48′′
25/233 - 38° 37′ 30′′
26/233 - 40° 10′ 12′′
27/233 - 41° 42′ 54′′
28/233 - 43° 15′ 36′′
29/233 - 44° 48′ 18′′
30/233 - 46° 21′
31/233 - 47° 53′ 42′′
32/233 - 49° 26′ 24′′
33/233 - 50° 59′ 6′′
34/233 - 52° 31′ 48′′
35/233 - 54° 04′ 30′′
36/233 - 55° 37′ 12′′
37/233 - 57° 09′ 54′′
38/233 - 58° 42′ 36′′
39/233 - 60° 15′ 18′′
40/233 - 61° 48′
41/233 - 63° 20′ 42′′
42/233 - 64° 53′ 24′′
43/233 - 66° 26′ 06′′
44/233 - 67° 58′ 48′′
45/233 - 69° 31′ 30′′
46/233 - 71° 04′ 12′′
47/233 - 72° 36′ 54′′
48/233 - 74° 09′ 36′′
49/233 - 75° 42′ 18′′
50/233 - 77° 15′
51/233 - 78° 47′ 42′′
52/233 - 80° 20′ 24′′
53/233 - 81° 53′ 06′′
54/233 - 83° 25′ 48′′
55/233 - 84° 58′ 30′′
56/233 - 86° 31′ 12′′
57/233 - 88° 03′ 54′′
58/233 - 89° 36′ 36′′
59/233 - 91° 09′ 18′′
60/233 - 92° 42′
61/233 - 94° 14′ 42′′
62/233 - 95° 47′ 24′′
63/233 - 97° 20′ 06′′
64/233 - 98° 52′ 48′′
65/233 - 100° 25′ 30′′
66/233 - 101° 58′ 12′′
67/233 - 103° 30′ 54′′
68/233 - 105° 03′ 36′′
69/233 - 106° 36′ 18′′
70/233 - 108° 09′
71/233 - 109° 41′ 42′′
72/233 - 111° 14′ 24′′
73/233 - 112° 47′ 06′′
74/233 - 114° 19′ 48′′
75/233 - 115° 52′ 30′′
76/233 - 117° 25′ 12′′
77/233 - 118° 57′ 54′′
78/233 - 120° 30′ 36′′
79/233 - 122° 03′ 18′′
80/233 - 123° 36′
81/233 - 125° 08′ 42′′
82/233 - 126° 41′ 24′′
83/233 - 128° 14′ 06′′
84/233 - 129° 46′ 48′′
85/233 - 131° 19′ 30′′
86/233 - 132° 52′ 12′′
87/233 - 134° 24′ 54′′
88/233 - 135° 57′ 36′′
89/233 - 137° 30′ 18′′
90/233 - 139° 03′
91/233 - 140° 35′ 42′′
92/233 - 142° 08′ 24′′
93/233 - 143° 41′ 06′′
94/233 - 145° 13′ 48′′
95/233 - 146° 46′ 30′′
96/233 - 148° 19′ 12′′
97/233 - 149° 51′ 54′′
98/233 - 151° 24′ 36′′
99/233 - 152° 57′ 18′′
100/233 - 154° 30′
101/233 - 156° 02′ 42′′
102/233 - 157° 35′ 24′′
103/233 - 159° 08′ 06′′
104/233 - 160° 40′ 48′′
105/233 - 162° 13′ 30′′
106/233 - 163° 46′ 12′′
107/233 - 165° 18′ 54′′
108/233 - 166° 51′ 36′′
109/233 - 168° 24′ 18′′
110/233 - 169° 57′
111/233 - 171° 29′ 42′′
112/233 - 173° 02′ 24′′
113/233 - 174° 35′ 6′′
114/233 - 176° 07′ 48′′
115/233 - 177° 40′ 30′′
116/233 - 179° 13′ 12′′
233/233 - 360° aka waning Conjunction with waxing degree being 0°
All of the regular aspects used in Mainstream Astrology are 144th Harmonic aspects if they are within an orb of 6 minutes
12/144 - 30° aka SemiSextile
24/144 - 60° aka Sextile
36/144 - 90° aka Square
48/144 - 120° aka Trine
60/144 - 150° aka Quincunx
72/144 - 180° aka Opposition
144/144 - 360° which is waning degree of Conjunction with waxing degree of 0°
additional aspects regularly used in Cosmobiology and Uranian Astrology are 144th aspects if they are within an orb of 6 minutes
9/144 - 22° 30' aka 1/16 (used only in Uranian Astrology)
18/144 - 45° aka SemiSquare
27/144 - 67° 30' aka 3/16 (used only in Uranian Astrology)
54/144 - 135° aka SesquiSquare
63/144 - 157° 30' aka 7/16 (used only in Uranian Astrology)
In his article The Golden Section: A Cosmic Principle, Dr. Landscheidt discussed Golden Aspects
"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."
plasmaresources.com/ozwx/landscheidt/pdf/TheGoldenSection_ACosmicPrinciple.pdf
Some of the aspects that Dr. Theodor Landscheidt listed are near 144th Harmonic aspects
He was using a 1 degree orb for them.
34.38° - 14/144 - 35° aka 7/72
42.49° - 17/144 - 42°30'
47.51° - 19/144 - 47°30'
55.62° - 22/144 - 55° aka 11/72
124.38° - 50/144 - 125° aka 25/72
132.49° - 53/144 - 132°30'
137.51° - 55/144 - 137°30' aka Golden Mean
145.62° - 58/144 - 145° aka 29/72
360° divided by 1.618 = 222.4969097651422° which rounds off to 222.5°
360° - 222.5° = 137.5° which is close to Dr. Landscheidt's Golden Aspect 137.51°
137.5° is The Golden Angle
137.5° divided by 1.618 = 84.9814585908529° which rounds off to 85°
137.5° × .618 = 84.975 which rounds off to 85°
137.5° × .382 = 52.525 which rounds off to 52.53
137.5° divided by 2 = 68.75 which is one of Dr. Landscheidt's Golden Aspects
137.5° divided by 4 = 34.374 which is close to Dr. Landscheidt's Golden Aspect 34.38°
360° - 137.5° = 222.5°
222.5° divided by 2 = 111.25° which is one of Dr. Landscheidt's Golden Aspects
222.5° divided by 4 = 55.625° which is close to one of Dr. Landscheidt's Golden Aspect 55.62°
85° is one of the aspects that Dr. Landscheidt mentioned in his article on Geocentric Planetary Nodes and is The Golden Mean of Golden Mean Aspect. He included Golden Aspects as aspects that he used with Geocentric Planetary Nodes. He used a maximum of 1 degree with them.
2 objects in Golden Mean of Golden Mean aspect in Golden Mean to a 3rd object form a 144th Harmonic Isosceles Triangle.
85° divided by 2 = 42.5° which is close to Dr. Landscheidt's Golden Aspect 42.49°
85° divided by 4 = 21.25° which is one of the Dr. Landscheidt's Golden Aspects 21.25°
One of Dr. Landscheidt's Golden Aspects 132.49° is very close to the angle of the Euler's number.
The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828
360° divided by 2.71828 = 132.4366879055874°
132°26'12'' is the Euler's angle.
divergentastrology.blogspot.com/2022/03/the-eulers-aspect.html
One of Theodor Landscheidt's Golden Aspects 47.51° is very close to the Silver Mean-Pi Product 47° 29' 24''
Silver Mean is 2.4142135623
Pi is 3.14159
2.4142135623 × 3.14159 = 7.584469185186057 rounded off to 7.58
360° divided by 7.58 = 47.49 aka 47° 29' 24"
In mathematics, two quantities are in the silver ratio (or silver mean)[1][2] if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like.
en.wikipedia.org/wiki/Silver_ratio
The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22/7 are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.
en.wikipedia.org/wiki/Pi
Blog Posts about Golden Aspects
divergentastrology.blogspot.com/2022/02/the-orb-of-golden-mean-aspect-based-on.html
divergentastrology.blogspot.com/2022/02/golden-aspects-and-harmonics.html
divergentastrology.blogspot.com/2022/03/the-golden-mean-aspect-and-its.html
divergentastrology.blogspot.com/2022/03/golden-aspects-and-their-harmonics.html