Lunar cycles – more about the 1:81 ratio

Expanding on the ‘square year’ post, we’ll generate numbers of all the lunar month variants (calculations further down) and analyse the results. It will show that the nodal and apsidal groups must both sum to a multiple of 81 in the period of the 9865 year chart. Summaries included for the time-poor!

From the lunar chart:
8750 full moon cycles + 1115 apsidal cycles = 9865
9865-year lunar chart

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From the Chandler Wobble post:
8326 Chandler Wobbles (CW) + 1539 CW cycles = 9865

8750 – 8326 = 1539 – 1115 = 424

The significance of 424 is that it’s 4/5ths of the 530 lunar nodal cycles (LNC). That tells us that in every 5 LNC the difference between numbers of full moon cycles and Chandler Wobbles is 4, as is the difference between numbers of apsidal cycles and CW cycles. We already saw in the first post in the series that 5 LNC consisted of 13*81 evections in latitude.

Turning to the de Rop cycles, 1645 (see chart) minus 1539 CW cycles = 106, which is 1/5th of 530 LNC. So in every 5 LNC the difference between numbers of de Rop cycles and Chandler Wobble cycles is 1, and between de Rop cycles and apsidal cycles it’s 5 (one per LNC).

Summary: the differences of 1,4 and 5 show how these cycles fit together every 5 nodal cycles.

The QBO and lunar year link

Based on a 2:1 ratio with the Chandler Wobble, QBO in the period of the chart should be 8326/2 = 4163. The lunar year occurs 36.3 times per 35.3 tropical years:
4163 QBO – 530 LNC = 3633
9865 / 3633 = 2.7153867
2.7153867 * 13 = 35.300027
This points back to the calculation in the first post in the 1:81 series. The lunar year is 13 tropical months.

Calculations

Recalling the published numbers for the square year: 385 draconic years = 4519 lunar synodic months (SM), we add 385 for draconic months = 4904 and subtract 385 for evections in latitude = 4134. Then multiply by 27 to get the full period of 27*385 = 10395 DY:
27*4519 = 122013 SM
27*4904 = 132408 DM
27*4134 = 111618 ELAT
The sum of these is 81*4519.
DM – SM = SM – ELAT = 10395 DY.

That’s the nodal side of the picture, now we want the apsidal equivalent. This can be derived from the full moon cycle (FMC), which gives us (almost): 251 SM = 269 anomalistic months (AM), a difference of 18 FMC. Working back to evections in longitude: 251-18 = 233 ELONG. Then:
251*18 = 4518 SM, only one less than the square year number in almost 18*18 = 324 FMC.
269*18 = 4842 AM
233*18 = 4194 ELONG.
Multiplying by 27:
4518*27 = 121986 SM in 8748 FMC.
4842*27 = 130734 AM
4194*27 = 113238 ELONG
To match the total SM in 10395 DY, we need to add 27. The other two figures need to add 27*2 between them, but as they’re not the same period, the shorter one (DM) gets 2 more (so 29) and the longer one (ELONG) 2 less, so 25:
121986+27 = 122013 SM (as before)
130734+29 = 130763 AM
113238+25 = 113263 ELON

Now the sum of each of the two sets (nodal and apsidal) of three numbers is 4519*81 (both include SM) . The point of all this is to check the differences between these numbers, compared to the lunar chart.

10395 draconic years : see nodal summary above.
8750 full moon cycles: SM minus ELON, AM minus SM
1645 de Rop cycles: ELON minus ELAT, DM minus AM

The other numbers are linked to the tropical months, which = 131878 TM (122013 SM + 9865 TY).
1115 apsidal cycles: TM minus AM
530 nodal cycles: DM minus TM

Summary: the month numbers confirm the figures in the chart. The Chandler Wobble isn’t calculated from month numbers but fits between the de Rop cycle and the full moon cycle in a straightforward pattern, as shown earlier in this post.

Footnote:
The de Rop cycle has a beat period with the tropical year:
5.9969604 TY *1 / (5.9969604 TY -1) = 1.2001216 TY
9865 TY / 1.2001216 = 8220
(This period is referred to by Sidorenkov here as a 1.2 year cycle. He sees it as a variant of the Chandler Wobble. here)
1645 de Rop cycles + 8220 ‘1.2 year’ cycles = 9865.
* * *
Links to all the 1:81 ratio posts here.

Source: https://tallbloke.wordpress.com/2025/11/17/lunar-cycles-more-about-the-181-ratio/