Moon Phase Lunation Gear Train for Watch

How do you design a simple gear train to allow a watch or clock to show the phases of the moon accurately? Well, a lunar month is an average of 29.5 days long. It's kind of hard to make a gear with 29.5 teeth--specifically the ".5" tooth. You can't really have half a tooth on a gear.

So, double the 29.5 x 2 = 59 tooth gear. You paint two little moons on the gear and have a little pointy lever poke at the gear once per day. One of the moons will slowly go beneath a cutout mask in the face of the watch clock. The next month will have the other moon slowly emerging.

Unfortunately lunar months are actually 29 days, 12 hours, 44 minutes and 2.8 seconds long and they basically plus or minus up to 7 hours over the course of 9 year cycles. A 59 tooth gear will by off by one day every two years!

What gets us closer to the 29 days, 12 hours, 44 minutes and 2.8 seconds = (29.53059xxxxx). Or if you want to make it easier twice that which would be 59.061 days for two lunar months?

You want to have a set of gear and pinion (smaller gears) teeth that are whole numbers.

35 x 40 x 71 divided by 9 x 11 x 17 = 59.061. These six gears give an accuracy of losing a day every 6661.3 years. You and your ancestors will never have to adjust this watch.

You attach this set of gears to an hour-hand gear. The gears are paired on 3 shafts like so: 9/35 and 11/40 and 17/71. 

On the 71 tooth gear you paint a nice little moon.

Above all these gears you put an stationary egg-shaped mask that covers more and more of the little painted moon as the gear turns day after day, eventually turning the full moon into a crescent and then it disappears--only to reappear on the other side as a growing crescent then into a full moon!

Sound complicated? Nope! I made a working model out of paper. The clock face is white with a cut out and a non-moving black mask. 

The 71 tooth gear is orange with a white moon on it.

As the orange gear spins you see less and less of the little moon because of the stationary black mask. Eventually the moon turns into a sliver of a crescent. As you can guess, when it keeps revolving it will pass the top of the black mask it will start growing again.

I haven't gotten my hands on a rotary table dividing index head for my milling machine (yet) to make the physical gears out of brass; but all this seems to work on my crummy little paper and cardboard mock ups. 

Who would need a watch that displays the phases of the moon accurately to within 7 hours over the course of over 6000 years? Hmm...I dunno, a werewolf?

Can you make a lunar watch even more accurate? Yes: but you need gears with over 800+ teeth--and that would be immensely difficult to make in my basement. Also the gears would be the size of hulahoops.

Some Serbian college professors and grad students came up with sets of real number rationalization permutations for these gears...and I found an error in their math! They had a 2 to the 3rd power instead of 2nd power. But, it turns out when you factor it all out that's what gets rid of having to draw a second little moon on the 71 tooth gear. https://www.researchgate.net/publication/265859964_SYNTHESIS_SOLID_MODELING_AND_WORKING_SIMULATION_OF_MOON_PHASE_CLOCK_MECHANISM 

#0 - N9 - ratio: 1:1 - RPM: 30

#1 - N9 - ratio: 1:1 - RPM: 30

#2 - N35 - ratio: 3.89:1 - RPM: 7.71

#3 - N11 - ratio: 3.89:1 - RPM: 7.71

#4 - N40 - ratio: 14.14:1 - RPM: 2.12

#5 - N17 - ratio: 14.14:1 - RPM: 2.12

#6 - N71 - ratio: 59.06:1 - RPM: 0.51

I might find some kid to build this out of Legos for  me as a test.