A synodic month — new moon to new moon — is 29.53059 days. Twelve of them make about 354 days, eleven short of a solar year. Every lunisolar culture is an argument about those eleven days: add a thirteenth month, or let festivals wander.
The Gregorian months are fossils of lunar counting that no longer follow the Moon. January has 31 days because Rome would not let July and August be shorter than the emperors who named them.
Stand outside at night and the month is still honest. The phase is a clock that does not need a committee.