
Image credit: beforeitsnews.com
The aim here is to show how the synodic periods and orbits of these three planets align with the so-called Grand Synod, a period of about 4628 years which has 27 Uranus-Neptune conjunctions and almost 233 Jupiter-Saturn conjunctions. Its half-period is sometimes referred to as the Hallstatt cycle (2314 years +/- a variable margin).
1. U-N ‘long period’
1420 Uranus-Neptune conjunctions = 1477 Neptune orbits
(for calculations, see Footnote)
1477 – 1420 = 57
Uranus-Neptune 360 degrees return is 1420/57 U-N = 24.91228 U-N long period = 4270.119 years
2. GS : U-N ratio
Grand Synod = 27 U-N = 4627.967 years (= ~233 Jupiter-Saturn conjunctions)
27 / 24.91228 = 1.0838028
1.0838028 * 12 = 13.005633
Therefore the ratio of 4627.967:4270.119 is almost exactly 13:12 (> 99.956% true)
3. Orbital data
Turning to the orbit periods nearest to the Grand Synod:
28 Neptune = 4614.157y
55 Uranus = 4620.927y
(Data: https://ssd.jpl.nasa.gov/?planet_phys_par )
4. Factor of 12
These periods fall slightly short of the 27 U-N Grand Synod (~4628 years).
However, multiplying by 12 and adding one orbit to each, gives:
28*12,+1 (337) Neptune = 55534.67y
55*12,+1 (661) Uranus = 55535.14y
27*12 (661 – 337) U-N = 55535.61y
Now the numbers match to within a year +/- 55535 years.
Also, the period is 12 Grand Synods (12*4628 = 55536y), or 13 U-N ‘long’ periods.
5. Pluto data
Pluto’s orbit period is 247.92065 years.
55535 / 247.92065y = 224.003
So 224 Pluto orbits also equate to 12 Grand Synods.

Therefore, a U-N-P synodic chart can be created for that period of time.
6. Neptune:Pluto orbits
Neptune has one more orbit in the period than an exact 3:2 ratio with Pluto – a planetary resonance.
224 P = 112*2
337 N = 112*3, +1
113 N-P = 112, +1
7. Phi factor
Uranus and Neptune both have one more orbit than this ratio:
660:336 = (55*12):(21*16)
55/21 = Phi²
12/16 = 3/4
Therefore the U:N ratio is almost (3/4 of Phi²):1
The U-N-P chart should repeat every 12 Grand synods i.e. every 55,535 years or so.
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Footnote
360 / Neptune orbit (164.79132) = 2.184581
2.184581 * U-N conjunction (171.40619) = 374.4507
374.4507 – 360 = 14.4507
Obtain nearest multiple of 360 degrees:
1420 * 14.4507 = 20519.9994
20520 / 360 = 57
1420 + 57 = 1477
1420 U-N = 1477 Neptune orbits
1420 + 1477 = 2897 Uranus orbits
Update (Dec. 2020): the number of occurrences of the Uranus-Neptune conjunction precession in the period is 337 – 324 = 13. Therefore the ratio of that period to the Grand Synod (GS) is 13:12, because the GS is 27 U-N and 324 = 27*12.