Lunisolar calendar

Encyclopedia

A

in many culture

s 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 which case most years have twelve months but every second or third year has thirteen.

, Buddhist

, Hellenic, Hindu lunisolar

, Burmese, Tibetan

, Chinese

, Vietnamese, Mongolian

, and Korean

calendars are all lunisolar, as was the Japanese calendar

until 1873, the pre-Islamic calendar, the first century Gaulish Coligny calendar

, and the Babylonian calendar

. The Chinese, Coligny and

Hebrew 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 Calendar

s 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.

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:

Continued fraction

s 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 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

The 19-year cycle (235 synodic months, including 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

Rome used an 84-year cycle for Easter calculation

s 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 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 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 has good accuracy for a sidereal year.

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.

began their count of lunar months from the arrival of spawning chinook salmon

(in Gregorian calendar October), and counted 10 months, leaving an uncounted period until the next chinook salmon run.

**lunisolar calendar**is a calendarCalendar

A calendar is a system of organizing days for social, religious, commercial, or administrative purposes. This is done by giving names to periods of time, typically days, weeks, months, and years. The name given to each day is known as a date. Periods in a calendar are usually, though not...

in many culture

Culture

Culture is a term that has many different inter-related meanings. For example, in 1952, Alfred Kroeber and Clyde Kluckhohn compiled a list of 164 definitions of "culture" in Culture: A Critical Review of Concepts and Definitions...

s whose date indicates both the moon phase and the time of the solar year

Year

A year is the orbital period of the Earth moving around the Sun. For an observer on Earth, this corresponds to the period it takes the Sun to complete one course throughout the zodiac along the ecliptic....

. If the solar year is defined as a tropical year

Tropical year

A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

then a lunisolar calendar will give an indication of the season

Season

A season is a division of the year, marked by changes in weather, ecology, and hours of daylight.Seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earth's axis relative to the plane of revolution...

; if it is taken as a sidereal year

Sidereal year

A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...

then the calendar will predict the constellation

Constellation

In modern astronomy, a constellation is an internationally defined area of the celestial sphere. These areas are grouped around asterisms, patterns formed by prominent stars within apparent proximity to one another on Earth's night sky....

near which the full moon

Full moon

Full moon lunar phase that occurs when the Moon is on the opposite side of the Earth from the Sun. More precisely, a full moon occurs when the geocentric apparent longitudes of the Sun and Moon differ by 180 degrees; the Moon is then in opposition with the Sun.Lunar eclipses can only occur at...

may occur. Usually there is an additional requirement that the year have a whole number of months, in which case most years have twelve months but every second or third year has thirteen.

## Examples

The HebrewHebrew calendar

The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...

, Buddhist

Buddhist calendar

The Buddhist calendar is used on mainland Southeast Asia in the countries of Thailand, Cambodia, Laos, Burma and Sri Lanka in several related forms. It is a lunisolar calendar having months that are alternately 29 and 30 days, with an intercalated day and a 30-day month added at regular intervals...

, Hellenic, Hindu lunisolar

Hindu calendar

The hindu calendar used in ancient times has undergone many changes in the process of regionalization, and today there are several regional Indian calendars, as well as an Indian national calendar. Nepali calendar, Bengali calendar, Malayalam calendar, Tamil calendar, Telugu calendar, Kannada...

, Burmese, Tibetan

Tibetan calendar

The Tibetan calendar is a lunisolar calendar, that is, the Tibetan year is composed of either 12 or 13 lunar months, each beginning and ending with a new moon. A thirteenth month is added every two or three years, so that an average Tibetan year is equal to the solar year.The Tibetan New Year...

, Chinese

Chinese calendar

The Chinese calendar is a lunisolar calendar, incorporating elements of a lunar calendar with those of a solar calendar. It is not exclusive to China, but followed by many other Asian cultures as well...

, Vietnamese, Mongolian

Mongolian calendar

The Mongol calendar is a lunisolar calendar based on system developed in 1747 by monk Ishbaljir . The Mongol year is composed of either 12 or 13 lunar months, each beginning and ending with a new moon...

, and Korean

Korean calendar

The traditional Korean calendar is a lunisolar calendar, like the traditional calendars of other East Asian countries. Dates are calculated from Korea's meridian, and observances and festivals are based in Korean culture....

calendars are all lunisolar, as was the Japanese calendar

Japanese calendar

On January 1, 1873, Japan adopted the Gregorian calendar. Before 1873, the Chinese style lunisolar calendar had been in use since 7th century. Japanese eras are still in use.-System:...

until 1873, the pre-Islamic calendar, the first century Gaulish Coligny calendar

Coligny calendar

The Gaulish Coligny calendar was found in Coligny, Ain, France near Lyon in 1897, along with the head of a bronze statue of a youthful male figure. It is a lunisolar calendar...

, and the Babylonian calendar

Babylonian calendar

The Babylonian calendar was a lunisolar calendar with years consisting of 12 lunar months, each beginning when a new crescent moon was first sighted low on the western horizon at sunset, plus an intercalary month inserted as needed by decree. The calendar is based on a Sumerian precedecessor...

. The Chinese, Coligny and

Hebrew lunisolar calendars track more or less the tropical year

Tropical year

A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

whereas the Buddhist and Hindu lunisolar calendars track the sidereal year

Sidereal year

A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...

. 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

Germanic calendar

The Germanic calendars were the regional calendars used amongst the early Germanic peoples, prior to the adoption of the Julian calendar in the Early Middle Ages....

before their conversion to Christianity.

The Islamic calendar

Islamic calendar

The Hijri calendar , also known as the Muslim calendar or Islamic calendar , is a lunar calendar consisting of 12 lunar months in a year of 354 or 355 days. It is used to date events in many Muslim countries , and used by Muslims everywhere to determine the proper day on which to celebrate Islamic...

is lunar

Lunar calendar

A lunar calendar is a calendar that is based on cycles of the lunar phase. A common purely lunar calendar is the Islamic calendar or Hijri calendar. A feature of the Islamic calendar is that a year is always 12 months, so the months are not linked with the seasons and drift each solar year by 11 to...

, but not a lunisolar calendar because its date is not related to the sun. The civil versions of the Julian

Julian calendar

The Julian calendar began in 45 BC as a reform of the Roman calendar by Julius Caesar. It was chosen after consultation with the astronomer Sosigenes of Alexandria and was probably designed to approximate the tropical year .The Julian calendar has a regular year of 365 days divided into 12 months...

and Gregorian Calendar

Gregorian calendar

The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...

s are solar

Solar calendar

A solar calendar is a calendar whose dates indicate the position of the earth on its revolution around the sun .-Tropical solar calendars:...

, 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

Computus

Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....

the Christian celebration of Easter

Easter

Easter is the central feast in the Christian liturgical year. According to the Canonical gospels, Jesus rose from the dead on the third day after his crucifixion. His resurrection is celebrated on Easter Day or Easter Sunday...

, 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

Tropical year

A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

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 fraction

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

s 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:

12 / | 1 = | 12 | (error = | −0.368266... synodic months/year) |

25 / | 2 = | 12.5 | (error = | 0.131734... synodic months/year) |

37 / | 3 = | 12.333333... | (error = | −0.034933... synodic months/year) |

99 / | 8 = | 12.375 | (error = | 0.006734... synodic months/year) |

136 / | 11 = | 12.363636... | (error = | −0.004630... synodic months/year) |

235 / | 19 = | 12.368421... | (error = | 0.000155... synodic months/year) |

4131 / | 334 = | 12.368263... | (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 3 embolismic months) was used in the ancient Athenian calendar. The 8-year cycle was also used in early third-century Easter calculations

Computus

Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....

(or old

*Computus*) in Rome and Alexandria.The 19-year cycle (235 synodic months, including 7 embolismic months) is the classic Metonic cycle

Metonic cycle

In astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the solar year and the synodic month...

, 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 calculation

Computus

Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....

s from the late third century until 457. Early Christians in Britain and Ireland also used an 84-year cycle until the Synod of Whitby

Synod of Whitby

The Synod of Whitby was a seventh century Northumbriansynod where King Oswiu of Northumbria ruled that his kingdom would calculate Easter and observe the monastic tonsure according to the customs of Rome, rather than the customs practised by Iona and its satellite institutions...

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 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 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 has good accuracy for a sidereal year.

## 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 − 1 = 1.7 common years between leap years

A representative sequence of common and leap years is ccLccLcLccLccLccLcL, which is the classic nineteen-year Metonic cycle

Metonic cycle

In astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the solar year and the synodic month...

. 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

Sun

The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, their leap months do not usually occur within a couple of months of perihelion, when the apparent speed of the sun along the ecliptic

Ecliptic

The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

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.

## Lunisolar calendars with uncounted time

An alternative way of dealing with the fact that a solar year does not contain an integer number of months is by including uncounted time in the year that does not belong to any month. Some Coast Salish peoples used a calendar of this kind. For instance, the ChehalisSts'Ailes

The Sts'ailes are a First Nations people in the Lower Mainland of the Canadian province of British Columbia. They are a Halqemeylem-speaking people but are distinct historically and politically from the surrounding Sto:lo peoples...

began their count of lunar months from the arrival of spawning chinook salmon

Chinook salmon

The Chinook salmon, Oncorhynchus tshawytscha, is the largest species in the pacific salmon family. Other commonly used names for the species include King salmon, Quinnat salmon, Spring salmon and Tyee salmon...

(in Gregorian calendar October), and counted 10 months, leaving an uncounted period until the next chinook salmon run.

## External links

- Lunisolar calendar year 2011 (by Serge Bièvre)
- Lunisolar Calendar
- Calendar studies
- Acano by Professor Jose Barrios Garcia
- Acano: a lunar calendar from the Canary Islands