Lunisolar Calendars - Determining Leap Months

Determining Leap Months

To determine when an embolismic month needs to be inserted, some calendars rely on direct observations of the state of vegetation, while others compare the ecliptic longitude of the sun and the phase of the moon. The Hawaiians observe the movement of specific stars and insert months accordingly.

On the other hand, in arithmetical lunisolar calendars, an integral number of months is fitted into some integral number of years by a fixed rule. To construct such a calendar (in principle), the average length of the tropical year is divided by the average length of the synodic month, which gives the number of average synodic months in a tropical year as:

12.368266......

Continued fractions of this decimal value give optimal approximations for this value. So in the list below, after the number of synodic months listed in the numerator, approximately an integer number of tropical years as listed in the denominator have been completed:

Note however that in none of the arithmetic calendars is the average year length exactly equal to a true tropical year. Different calendars have different average year lengths and different average month lengths, so the discrepancy between the calendar months and moon is not equal to the values given above.

The 8-year cycle (99 synodic months, including 99−8×12 = 3 embolismic months) was used in the ancient Athenian calendar. The 8-year cycle was also used in early third-century Easter calculations (or old Computus) in Rome and Alexandria.

The 19-year cycle (235 synodic months, including 235−19×12 = 7 embolismic months) is the classic Metonic cycle, which is used in most arithmetical lunisolar calendars. It is a combination of the 8- and 11-year period, and whenever the error of the 19-year approximation accumulates to 1⁄₁₉ of a mean month, a cycle can be truncated to 11 years (skipping 8 years including 3 embolismic months), after which 19-year cycles can resume. Meton's cycle had an integer number of days, although Metonic cycle often means its use without an integer number of days. It was adapted to a mean year of 365.25 days by means of the 4×19 year Callipic cycle (used in the Easter calculations of the Julian calendar).

Rome used an 84-year cycle for Easter calculations from the late third century until 457. Early Christians in Britain and Ireland also used an 84-year cycle until the Synod of Whitby in 664. The 84-year cycle is equivalent to a Callipic 4×19-year cycle (including 4×7 embolismic months) plus an 8-year cycle (including 3 embolismic months) and so has a total of 1039 months (including 31 embolismic months). This gives an average of 12.3690476... months per year. One cycle was 30681 days, which is about 1.28 days short of 1039 synodic months, 0.66 days more than 84 tropical years, and 0.53 days short of 84 sidereal years.

The next approximation (arising from continued fractions) after the Metonic cycle (such as a 334-year cycle) is very sensitive to the values one adopts for the lunation (synodic month) and the year, especially the year. There are different possible definitions of the year so other approximations may be more accurate for specific purposes. For example a 353-year cycle including 130 embolismic months for a total of 4366 months (12.36827195...) is more accurate for a northern hemisphere spring equinox year, whereas a 611-year cycle including 225 embolismic months for a total of 7557 months (12.36824877...) has good accuracy for a northern hemisphere summer solstice year, and a 160-year cycle including 59 embolismic months for a total of 1979 months (12.36875) has good accuracy for a sidereal year (approx 12.3687462856 synodic months).