A novel compound epicyclic gear system I can't seem to understand

[Quantum of Solace]

TL;DR Summary

This moon-phase wheel using an epicyclic gear train driven by the hour hand is supposed to have a gear ratio of 118.122449:1 but I can't make the numbers work. What am I doing wrong?

This watch uses a novel geartrain fixed to a lunar-phase display wheel which is supposed to rotate once per synodic(lunar) month. More precisely, it should rotate once every 29.530589 days. The geartrain is described in detail and the explanation claims it achieves an accuracy of once every 29.5306122449 days.

In the website here: https://www.ochsundjunior.swiss/watches/moon-phase/ It is described as follows:

"An epicyclic gear train driven by a central finger bonded directly to the hour pipe turns the lunar disk beneath the dial counter-clockwise. The central finger engages with a wheel bearing 12 teeth, whose pinion with 14 teeth meshes with a wheel bearing 18 teeth, whose pinion with 14 teeth meshes with the fixed recessed ring gear with its 109 teeth machined into the underside of the dial"

There is a diagram linked in the description.

So I decided to verify the gearing and came up with the following:
The total R were it fixed, is 12/1, 18/14, 109,14. i.e. 23544/196, or 5886/49, or 120.211449.
But if it's a planetary gear system with a fixed annular ring, the hour wheel driving and the carrier being the output, I believe I'm supposed to multiply the ratio by a factor of 1-(1/R) which would give me 119.122449. (notably reduced because the direction of the planetary carrier is reversed by its internal geartrain). This was my intuition and further "confirmed" by looking up the formulas in places such as this:

https://roymech.org/Useful_Tables/Drive/Epi_cyclic_gears.html

So the moon wheel would rotate once every 119.122449 rotations of the driving wheel. Since it revolves once every twelve hours this would imply a period of 59.56122449 days. OK, it's more like two cycles than just one, but also it's inaccurate by quite a large factor and different from the assertion in the linked page. I even asked Grok and it came up with exactly the same answer as me, not as the assertion on the manufacturers page.

Trying to find my mistake, I accidentally repeated the factor reduction on this new gear ratio... i.e. 119.122449 * (1 - 1 / 119.122449).
Behold.. 118.122449. Which would give a period of 59.06122449 days, 2x a lunar cycle of 29.53061224. Exactly what is stated by the manufacturer.

Firstly... I can't see why I accidentally got the "right" figure by applying the factor recursively twice. That goes against my intuition, the formulas I could find, and, well... Grok agreed with me but only once, the rest of the AIs made a complete hash of it.

I can't find an example of a similar planetary system with a compound geartrain attached to the carrier, interacting only with itself, other than the single input/output to and from the sun and annular rings. Is there some maths I'm missing? How does this figure just drop out by doing a factor I don't think I was supposed to do?

I'm going to overlook the glaring "mistake" in the linked page that says the wheel revolves once per month with these gearings. Clearly they'll be gearing up 2:1 at some point to achieve that, but the main point is the accuracy of that period or any simple multiple thereof.

Thanks for any and all explanation or pointers to texts that might help.

The folly of calculating a moonphase accurate enough to keep synch for over three thousand years on a mechanically wound wristwatch is not lost on me, but these things are pretty interesting for more permanent fixtures and something I've been looking into for a while. This is the first time I've seen an epicyclic system put to use and it seems to provide more flexibility to achieve the desired ratios.

 

[Quantum of Solace]

Found it. Sorry about my haphazard approach, but the result was the same. The inconsistencies appear to be linked to misleading information.

The correct answer with the given information is that the ratio between the moon wheel carrier C and the hour wheel S is:
G_CS = (K N_S N_B) / (N_R N_A + K N_S N_B)

Where N_S is the number of teeth on the Sun wheel (hour wheel) S
N_A is the number of teeth on the gear A interfacing with the sun wheel
N_B is the number of teeth on the gear B interfacing with the outer ring, R
N_R is the number of teeth on the outer ring, R
K is the gear ratio implemented internally to the carrier, between the gears A and B.

In this case:
N_S = 1
N_R = 109
N_A = 12
N_B = 14
N_Ai = 14 (14 teeth on the internal gearing for sprocket fixed to A
N_Bi = 18 (18 teeth on the internal gearing for sprocket fixed to B
so:
K = - 14/18 or -7/9

G_CS = ((-7/9) * 1 * 14) / (109 * 12 + (-7/9) * 1 * 14) or -49/5837.

So the moon wheel rotates once for every 5837/49 rotations of the hour wheel, or 119.122449 rotations of 12 hours. So 1429.469388 hours, or 59.56122449 days.

This is the answer I got.

Now, if you simply increase the number of teeth(fingers) on the hour wheel to 2, the calculation works out at:

G_CS = ((-7/9) * 2 * 14) / (109 * 12 + (-7/9) * 2 * 14) = -49/2894.

Working in reverse and disregarding the direction again, 2894/49 is 59.06122449 hour wheel rotations. So, 708.7346939 hours, or 29.53061224 days. Exactly what I was looking for.

So, reapplying my "factor" twice was purely coincidental with the missing information, the actual number (2) of input teeth needed. But that would fix it. Problem solved. Hope this helps someone someday.