Lunisolar calendar

A lunisolar calendar is a calendar that combines monthly lunar cycles with the solar year. As with all calendars which divide the year into months, there is an additional requirement that the year have a whole number of months (Moon cycles). The majority of years have twelve months but every second or third year is an embolismic year, which adds a thirteenth intercalary, embolismic, or leap month.

Lunisolar calendars are found in many cultures. In contrast to purely lunar calendars such as the Islamic calendar, lunisolar calendars have additional intercalation rules that reset them periodically into a rough agreement with the solar year and thus with the seasons.

Reconciling lunar and solar cycles

Determining leap months

A tropical year is approximately 365.2422 days long and a synodic month is approximately 29.5306 days long, so a tropical year is approximately 365.2422 / 29.5306 ≈ 12.36826 months long. Because 0.36826 is between 1⁄3 and 1⁄2, a typical year of 12 months needs to be supplemented with one intercalary or leap month every 2 to 3 years. More precisely, 0.36826 is quite close to 7⁄19 (about 0.3684211): several lunisolar calendars have 7 leap months in every cycle of 19 years (called a 'Metonic cycle'). The Babylonians applied the 19-year cycle in the late sixth century BCE.

Intercalation of leap months is frequently controlled by the "epact", which is the difference between the lunar and solar years (approximately 11 days). The classic Metonic cycle can be reproduced by assigning an initial epact value of 1 to the last year of the cycle and incrementing by 11 each year.