Lunisolar calendar

A lunisolar calendar is a calendar in many cultures whose date indicates both the moon phase and the time of the solar year. If the solar year is defined as a tropical year, then a lunisolar calendar will give an indication of the season; if it is taken as a sidereal year, then the calendar will predict the constellation near which the full moon may occur. Usually there is an additional requirement that the year have a whole number of months. In this case ordinary years consist of twelve months but every second or third year is an embolismic year, which adds a thirteenth intercalary, embolismic, or leap month.

Examples

The Hebrew, Jain, Buddhist, Hindu, Kurdish and Bengali calendars, as well as the traditional Chinese, Tibetan, Japanese, Vietnamese, Mongolian and Korean calendars (in the East Asian cultural sphere), plus the ancient Hellenic, Coligny, and Babylonian calendars are all lunisolar. Also, some of the ancient pre-Islamic calendars in South Arabia followed a lunisolar system.^[1] The Chinese, Coligny and Hebrew^[2] lunisolar calendars track more or less the tropical year whereas the Buddhist and Hindu lunisolar calendars track the sidereal year. Therefore, the first three give an idea of the seasons whereas the last two give an idea of the position among the constellations of the full moon. The Tibetan calendar was influenced by both the Chinese and Hindu calendars. The Germanic peoples also used a lunisolar calendar before their conversion to Christianity.

The Islamic calendar is lunar, but not a lunisolar calendar because its date is not related to the sun. The civil versions of the Julian and Gregorian calendars are solar, because their dates do not indicate the moon phase — however, both the Gregorian and Julian calendars include undated lunar calendars that allow them to calculate the Christian celebration of Easter, so both are lunisolar calendars in that respect.

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 ([12; 2, 1, 2, 1, 1, 17, ...]) 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:

12 /	1 =	12	= [12]	(error =	−0.368266... synodic months/year)
25 /	2 =	12.5	= [12; 2]	(error =	0.131734... synodic months/year)
37 /	3 =	12.333333...	= [12; 2, 1]	(error =	−0.034933... synodic months/year)
99 /	8 =	12.375	= [12; 2, 1, 2]	(error =	0.006734... synodic months/year)
136 /	11 =	12.363636...	= [12; 2, 1, 2, 1]	(error =	−0.004630... synodic months/year)
235 /	19 =	12.368421...	= [12; 2, 1, 2, 1, 1]	(error =	0.000155... synodic months/year)
4131 /	334 =	12.368263...	= [12; 2, 1, 2, 1, 1, 17]	(error =	−0.000003... synodic months/year)

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 the octaeteris used in the ancient Athenian calendar. The 8-year cycle was also used in early third-century Easter calculations (or old Computus) in Rome .

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 Callippic cycle (used in the Easter calculations of the Julian calendar).

Rome used an 84-year cycle for Easter calculations from the third century until 457. The native British Christians continued its use as late as 768, when Bishop Elfodd of Bangor finally persuaded them to adopt the improved calendars introduced by St Augustine's mission. The 84-year cycle is equivalent to a Callippic 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).

Calculating a leap month

A rough idea of the frequency of the intercalary or leap month in all lunisolar calendars can be obtained by the following calculation, using approximate lengths of months and years in days:

Year: 365.25, Month: 29.53
365.25/(12 × 29.53) = 1.0307
1/0.0307 = 32.57 common months between leap months
32.57/12 = 2.7 common years between leap years

A representative sequence of common and leap years is ccLccLcLccLccLccLcL, which is the classic nineteen-year Metonic cycle. The Buddhist and Hebrew calendars restrict the leap month to a single month of the year; the number of common months between leap months is, therefore, usually 36, but occasionally only 24 months. Because the Chinese and Hindu lunisolar calendars allow the leap month to occur after or before (respectively) any month but use the true motion of the sun, their leap months do not usually occur within a couple of months of perihelion, when the apparent speed of the sun along the ecliptic is fastest (now about 3 January). This increases the usual number of common months between leap months to roughly 34 months when a doublet of common years occurs, while reducing the number to about 29 months when only a common singleton occurs.