I wanted to give my planet a moon that spins every SIX days and waxes-wanes every THIRTY days.
The bad news: Luna always shows the same face to Terra because of something called tidal locking explained here. And not just Luna: "All twenty known moons in the Solar System that are large enough to be round are tidally locked with their primaries, because they orbit very closely and tidal force increases rapidly (as a cubic function) with decreasing distance" A small and irregularly-shaped moon like Phobos and Deimos could spin, but that's not pretty and the world should be pretty.
The workaround: I'm not sure that 'tidal locking' means 'no rotation'. I thiiink (and here I am asking clarification from someone who knows the science better than I) that the forces that create tidal locking are a strong pull between the orbiter and the orbitee, and that doesn't necessarily mean stillness, it can mean a steady rhythm. I got my current feeble understanding mostly from this page which talks about how Mercury is locked to Sol in spin-orbit resonance ("This was the first calculation that showed that locking in fractional ratio is actually possible.")
Why I wanted to do this: Terra has months (lunar), years (solar), and weeks, which are based on nothing celestial. Many Terran cultures have had a time-division of 7±2 days. I thought wouldn't it be cool if we have celestial weeks. So then we need three celestial events, to track months, years and weeks. I was going to add two moons (one with a weeklong cycle and one with a monthlong cycle), but "looking up at the two alien moons" feels too much like Astounding Science Fiction, feels too alien. Another issue is two moons + Sun has religious implications, rather than a Moon-Goddess + Sun-God you'd need a triad. Solution: if the moon changes faces once a week and waxes once a month, you track weeks and months with one celestial body. Then I thought it was an exciting bonus that the 'fractional locking' thing is a real scientific justification for a nice tidy perennial calendar; there are scientific reasons to expect nice ratios like 5:1 (rather than something messy like 4.418941:1)
So the question in a nutshell is: would this work? The moon is big and close, like our Luna is. That implies that it is tidally locked (according to my understanding). But is it possible that it's tidally locked in a 5-spins-per-month rhythm rather than a no-spin rhythm?
Possibly relevant: The world in question was engineered by alien scientists far more advanced than the people who live on it. Extraterrestrial creationism. Like Slartibartfast. If the orbits and periods seem too good to be true, suspiciously tidy, that is actually good; the people in the story will notice their world is suspiciously well-designed.
Background research:
https://arxiv.org/pdf/1710.01405.pdf talks about how "Planets in an eccentric orbit can be captured in other spin-orbit resonance states (p, ratio of orbital period to spin period) other than p = 1.0." so if it can happen with planets surely it can happen with moons? The paper also discusses eccentricity a lot, which I haven't considered enough yet.
https://ui.adsabs.harvard.edu/abs/2017CeMDA.129..509B/abstract – mre about the locking of planets