Editing Metonic cycle (section)

==Mathematical basis==

The Metonic cycle is the most accurate cycle of time (in a timespan of less than 100 years) for synchronizing the [[tropical year]] and the lunar month ([[synodic month]]), when the method of synchronizing is the [[intercalation (timekeeping)|intercalation]] of a thirteenth lunar month  in a calendar year from time to time.{{sfnp|Richards|1998|pages=94–96}} The traditional lunar year of 12 synodic months is about 354 days, approximately eleven days short of the solar year. Thus, every 2 to 3 years there is a discrepancy of 22 to 33 days, or a full synodic month. For example, if the winter solstice and the new moon coincide, it takes 19 tropical years for the coincidence to recur. The mathematical logic is this:
 
*A tropical year lasts 365.2422 days.{{sfnp|glossary|2022|loc=s.v. year, tropical}} 
*:a span of 19 tropical years (365.2422 × 19) lasts 6,939.602 days 
That duration is almost the same as 235 synodic months:
*A synodic month lasts 29.53059 days.{{sfnp|Richards|2013|page=587}}
*:a span of 235 synodic months (29.53059 × 235) lasts 6,939.689 days 
Thus the algorithm is correct to 0.087 days (2 hours, 5 minutes and 16 seconds).

For a [[lunisolar calendar]] to 'catch up' to this discrepancy and thus maintain seasonal consistency, seven [[Intercalation (timekeeping)|intercalary months]] are added (one at a time), at intervals of every 2–3 years during the course of 19 solar years. Thus twelve of those years have 12 lunar months and seven have 13 months. Of these months, 125 were full (30 days) and 110 were deficient (29 days), to give a total of 6940 days.{{sfn|Richards|1998|p=95}}