Moon phases
Mume moon goes through eight different phases and each phase is measured in full days. Moonphases change from one to another exactly at mume time 14:17. Why? Because of reasons… I'm guessing it's just a random time during a day when moon is barely visible.

New Moon (waxing) and (waning) - 120 ticks (5 mume days)
Quarter Moon (waxing) - 72 ticks (3 mume days)
Half Moon (waxing) - 72 ticks (3 mume days)
Three-Quarter Moon (waxing) - 72 ticks (3 mume days)
Full Moon (waxing) and (waning) - 24 ticks (1 mume day)
Three Quarter Moon (waning) - 72 ticks (3 mume days)
Half Moon (waning) - 72 ticks (3 mume days)
Quarter Moon (waning) - 72 ticks (3 mume days)
The full cycle takes exactly 24 mume days and the cycle starts at 1st of Afteryule from the middle of New Moon. It will take 12 mume days from there to get to full moon and then the cycle will repeat itself 15 times (12 months*30 days/24 = 15) until another new years eve. Based on that 24 day cycle, you can calculate the times when the full moon will be up - 12th of Afteryule, 6th of Solmath, 30th of Solmath, 24th of Rethe and so on.

When the moon cycle starts can differ based on last time restart which is done by the MUME management periodically. -Elestir

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Moonrise and moonfall
Moonfalls are fixed and will repeat after every 25 mume hours. Moonrises will be either 12 hours or 13 hours before the moonfall and they depend on, what I assume to be, a bug.

Between New Moon (waning) and Three Quarter Moon (waxing), the moon will be up for 12 hours.

At the start of 3/4 moon (waxing) the moon will rise twice. Once on the normal time and then again an hour later, meaning that after 3/4 waxing moon, the moon will be up for 13 hours and down for 12 hours. This interval will change again at the New moon (waning), but since the moon is almost never visible at that time, it's hard to verify when it changes back.

Moonfall also drifts 1 hour around every full moon, aparently. This was a presumption made early, but turned out to be true. The extra hour seems to be tied to the moon rising twice at the beginning of 3/4 moon.

First moonfall is 1st of Afteryule 18:00 on year 2936. Taking the 1 hour drift under consideration, it's possible to calculate the exact times of moonrise, moondown.

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Visibility
New moon and quarter moon might not be visible during the day, even if the moon should be up. They're usually visible during dawn and dusk. Half moon, three quarter moon and full moon are visible during the day.