Astronomers count five different months, and none of them fits the calendar
From one full moon to the next takes about 29.53 days, yet the Moon completes an orbit relative to the stars in just 27.32. Astronomers distinguish five kinds of month, most first recognised by Babylonian sky-watchers. None divides neatly into a year, which is why calendars have wrestled with the Moon for millennia.
The words month and Moon share a root, and tally sticks suggest Paleolithic people already counted days by lunar phases. The familiar cycle is the synodic month, averaging 29.53059 days from one new moon to the next, though because the lunar orbit is eccentric an individual month can run up to seven hours longer or shorter. The sidereal month, measured against distant stars, is about 27.32166 days.
The gap comes from Earth's own motion. While the Moon circles Earth, the pair are travelling around the Sun in the same direction, so the Moon needs roughly 2.2 extra days to line up with the Sun again. The tropical month, timed by the equinox point, is a whisker shorter than the sidereal because of precession. The anomalistic month, perigee to perigee, runs about 27.55 days because the closest point of the orbit creeps forward once every nine years. The draconic month, about 27.21 days, is shortest, since the orbit's nodes, where eclipses can occur, drift backward once every 18.6 years.
Calendars must reconcile all this with the solar year, which is 10 or 11 days longer than twelve lunations. A crude lunar calendar alternates full 30-day and hollow 29-day months. Lunisolar systems either add a few extra days each year, as the ancient Egyptian calendar did, or slip in a thirteenth leap month every two or three years, as the Chinese calendar does. The Metonic cycle uses the fact that 235 lunations almost equal 19 years; it underlies the Hebrew calendar, the reckoning of Easter and the calendar built into the Antikythera Mechanism about 21 centuries ago, though it drifts a day every two centuries.
The Gregorian calendar abandoned lunar months altogether, using twelve months of 28 to 31 days. Its names betray Roman roots: September to December come from the Latin for seven to ten, because they were once the seventh to tenth months.
Source: Month
