The Good News is that We Have Only One Moon
This is neither a treatise on astronomy nor a history of calendars, but if you don't know what a lunar calendar is, you should probably read the first few paragraphs. On the other hand, if you already know the difference between a synodic month and a sidereal month, you already know more than is good for you!
First, all calendars are systems of reckoning time over extended periods. They are all based on the movement of the moon or the perceived motion of the sun (or possibly both), and they let cultures organize political, social and religious affairs, and life in general. The common western calendar in use today is the Gregorian solar calendar, which measures the passage of the earth around the sun to give us a year of a bit more than 365 days (with the addition of a "leap" day added every so often) divided into 12 months. A lunar year, on the other hand, is a year of 12 synodic months (that is, from the appearance of either the dark “new moon” or else the first crescent to the next new moon or first crescent), a time of approximately 29 and one-half days, giving us a synodic lunar year of about 354 and one-third days. Notice the uses of "a bit more," “approximately,” and “about”. The complexity of calendars stems from the fact that the natural periods of day, month and year are not commensurate with one another: that is, the month is not a simple fraction of the year and the day is not a simple fraction of the month or the year. Thus, to keep calendars from “slipping”, they all have to, at some point, fudge a little bit and put in “leap” days or even months. Our western solar calendar has had two major revisions when they got unbearably out of sync: the Julian (at time of Julius Caesar) and the Gregorian (for Pope Gregory XIII in 1582). Thus we have months with 30 or 31 days and one month with 28 or 29 (in a leap year), with increasingly accurate astronomy reminding us that it will have to...
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