Intercalation or embolism in timekeeping is the insertion of a

leap day A leap year (also known as an intercalary year or bissextile year) is a calendar year that contains an additional day (or, in the case of a lunisolar calendar, a month) compared to a common year. The 366th day (or 13th month) is added to kee ...

week A week is a unit of time equal to seven days. It is the standard time period used for short cycles of days in most parts of the world. The days are often used to indicate common work days and rest days, as well as days of worship. Weeks are ofte ...

, or month into some

calendar year A calendar year begins on the New Year's Day of the given calendar system and ends on the day before the following New Year's Day, and thus consists of a whole number of days. The Gregorian calendar year, which is in use as civil calendar in ...

s to make the calendar follow the seasons or moon phases.

Lunisolar calendar A lunisolar calendar is a calendar in many cultures, 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 mont ...

s may require intercalations of days or months.

Solar calendars

The solar or

tropical year A tropical year or solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky – as viewed from the Earth or another celestial body of the Solar System – thus completing a full cycle of astronom ...

does not have a whole number of days (it is about 365.24 days), but a

must have a whole number of days. The most common way to reconcile the two is to vary the number of days in the calendar year. In solar calendars, this is done by adding an extra day ("leap day" or "intercalary day") to a

common year A common year is a calendar year with 365 days, as distinguished from a ''leap year'', which has 366 days. More generally, a common year is one without Intercalation (timekeeping), intercalation. The Gregorian calendar, used by the majority of ...

of 365 days, about once every four years, creating a

leap year A leap year (also known as an intercalary year or bissextile year) is a calendar year that contains an additional day (or, in the case of a lunisolar calendar, a month) compared to a common year. The 366th day (or 13th month) is added to keep t ...

that has 366 days ( Julian, Gregorian and

Indian national calendar The Indian national calendar, also called the Shaka calendar or Śaka calendar, is a solar calendar that is used alongside the Gregorian calendar by ''The Gazette of India'', in news broadcasts by All India Radio, and in calendars and officia ...

s). The

Decree of Canopus The Decree of Canopus is a trilingual inscription in three scripts, which dates from the Ptolemaic period of ancient Egypt. It was written in three writing systems: Egyptian hieroglyphs, demotic, and koine Greek, on several ancient Egyptian memor ...

, issued by the pharaoh

Ptolemy III Euergetes Ptolemy III Euergetes (, "Ptolemy the Euergetes, Benefactor"; c. 280 – November/December 222 BC) was the third pharaoh of the Ptolemaic dynasty in Egypt from 246 to 222 BC. The Ptolemaic Kingdom reached the height of its military and economic ...

Ancient Egypt Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...

in 239 BC, decreed a solar leap day system; an Egyptian leap year was not adopted until 25 BC, when the Roman Emperor

Augustus Gaius Julius Caesar Augustus (born Gaius Octavius; 23 September 63 BC – 19 August AD 14), also known as Octavian (), was the founder of the Roman Empire, who reigned as the first Roman emperor from 27 BC until his death in A ...

instituted a reformed

Alexandrian calendar The Coptic calendar, also called the Alexandrian calendar, is a liturgical calendar used by the farming populace in Egypt and used by the Coptic Orthodox and Coptic Catholic churches. It was used for fiscal purposes in Egypt until the adoption ...

. In the

Julian calendar The Julian calendar is a solar calendar of 365 days in every year with an additional leap day every fourth year (without exception). The Julian calendar is still used as a religious calendar in parts of the Eastern Orthodox Church and in parts ...

, as well as in the

Gregorian calendar The Gregorian calendar is the calendar used in most parts of the world. It went into effect in October 1582 following the papal bull issued by Pope Gregory XIII, which introduced it as a modification of, and replacement for, the Julian cale ...

, which improved upon it, intercalation is done by adding an extra day to February in each leap year. In the Julian calendar this was done every four years. In the Gregorian, years divisible by 100 but not 400 were exempted in order to improve accuracy. Thus, 2000 was a leap year; 1700, 1800, and 1900 were not. Epagomenal days are days within a solar calendar that are outside any regular month. Usually five epagomenal days are included within every year (

Egyptian ''Egyptian'' describes something of, from, or related to Egypt. Egyptian or Egyptians may refer to: Nations and ethnic groups * Egyptians, a national group in North Africa ** Egyptian culture, a complex and stable culture with thousands of year ...

, Coptic,

Ethiopian Ethiopians are the native inhabitants of Ethiopia, as well as the global diaspora of Ethiopia. Ethiopians constitute several component ethnic groups, many of which are closely related to ethnic groups in neighboring Eritrea and other parts of ...

, Mayan Haab' and French Republican Calendars), but a sixth epagomenal day is intercalated every four years in some (Coptic, Ethiopian and French Republican calendars). The

Solar Hijri calendar The Solar Hijri calendar is the official calendar of Iran. It is a solar calendar, based on the Earth's orbit around the Sun. Each year begins on the day of the March equinox and has years of 365 or 366 days. It is sometimes also called the S ...

, used in Iran, is based on solar calculations and is similar to the Gregorian calendar in its structure, and hence the intercalation, with the exception that its

epoch In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured. The moment of epoch is usually decided b ...

is the

Hijrah The Hijrah, () also Hegira (from Medieval Latin), was the journey the prophets and messengers in Islam, Islamic prophet Muhammad and his followers took from Mecca to Medina. The year in which the Hijrah took place is also identified as the e ...

. The Bahá'í calendar includes enough epagomenal days (usually 4 or 5) before the last month (, ''ʿalāʾ'') to ensure that the following year starts on the

March equinox The March equinox or northward equinox is the equinox on the Earth when the subsolar point appears to leave the Southern Hemisphere and cross the celestial equator, heading northward as seen from Earth. The March equinox is known as the ver ...

. These are known as the

Ayyám-i-Há Ayyám-i-Há is a period of intercalary days in the Baháʼí calendar, when Baháʼís celebrate the Festival of Ayyám-i-Há. The four or five days of this period are inserted between the last two months of the calendar (Mulk and ʻAláʼ). ...

Lunisolar calendars

The solar year does not have a whole number of lunar months (it is about 365/29.5 = 12.37

lunation In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Euro ...

s), so a lunisolar calendar must have a variable number of months per year. Regular years have 12 months, but embolismic years insert a 13th "intercalary" or "leap" month or "embolismic" month every second or third year. Whether to insert an intercalary month in a given year may be determined using regular cycles such as the 19-year

Metonic cycle The Metonic cycle or enneadecaeteris (from , from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The recurrence is not perfect, and by precise obser ...

(

Hebrew calendar The Hebrew calendar (), also called the Jewish calendar, is a lunisolar calendar used today for Jewish religious observance and as an official calendar of Israel. It determines the dates of Jewish holidays and other rituals, such as '' yahrze ...

and in the determination of Easter) or using calculations of lunar phases ( Hindu lunisolar and

Chinese calendar The traditional Chinese calendar, dating back to the Han dynasty, is a lunisolar calendar that blends solar, lunar, and other cycles for social and agricultural purposes. While modern China primarily uses the Gregorian calendar for officia ...

s). The

Buddhist calendar The Buddhist calendar is a set of lunisolar calendars primarily used in Tibet, Cambodia, Laos, Myanmar, Bangladesh, India, Sri Lanka, Thailand and Vietnam as well as in Malaysia and Singapore and by Chinese populations for religious or o ...

adds both an intercalary day and month on a usually regular cycle.

Lunar calendars

In principle, lunar calendars do not employ intercalation because they do not seek to synchronise with the seasons, and the motion of the moon is astronomically predictable. But religious lunar calendars rely on actual observation. The

Lunar Hijri calendar The Hijri calendar (), also known in English as the Islamic calendar, is a lunar calendar consisting of 12 lunar months in a year of 354 or 355 days. It is used to determine the proper days of Islamic holidays and rituals, such as the an ...

, the purely lunar calendar observed by most of Islam, depends on actual observation of the first crescent of the moon and thus has no intercalation. Each month still has either 29 or 30 days, but due to the variable method of observations employed, there is usually no discernible order in the sequencing of 29- or 30-day month lengths. Traditionally, the first day of each month is the day (beginning at sunset) of the first sighting of the ''hilal'' (crescent moon) shortly after sunset. If the ''hilal'' is not observed immediately after the 29th day of a month (either because clouds block its view or because the western sky is still too bright when the moon sets), then the day that begins at that sunset is the 30th. The tabular Islamic calendar, used in Iran, has 12 lunar months that usually alternate between 30 and 29 days every year, but an intercalary day is added to the last month of the year 12 times in a 33-year cycle. Some historians also linked the pre-Islamic practice of

Nasi' Nasiʾ (, ''an-Nasīʾ'', "postponement"), also Romanized Nasii, or Nasie, was an aspect of the pre-Islamic Arabian calendar, mentioned in the Quran in the context of the "four forbidden months". In pre-Islamic Arabia, the decision of "postponem ...

to intercalation.

Leap seconds

The

International Earth Rotation and Reference Systems Service The International Earth Rotation and Reference Systems Service (IERS), formerly the International Earth Rotation Service, is the body responsible for maintaining global time and reference frame standards, notably through its Earth Orientation P ...

can insert or remove

leap second A leap second is a one-second adjustment that is occasionally applied to Coordinated Universal Time (UTC), to accommodate the difference between precise time (International Atomic Time (TAI), as measured by atomic clocks) and imprecise solar tim ...

s from the last day of any month (June and December are preferred). These are sometimes described as intercalary.leap second by Merriam-Webster OnLine
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Other uses

ISO 8601 ISO 8601 is an international standard covering the worldwide exchange and communication of date and time-related data. It is maintained by the International Organization for Standardization (ISO) and was first published in 1988, with updates in ...

includes a

specification A specification often refers to a set of documented requirements to be satisfied by a material, design, product, or service. A specification is often a type of technical standard. There are different types of technical or engineering specificati ...

for a 52/53-week year. Any year that has 53 Thursdays has 53 weeks; this extra week may be regarded as intercalary. The ''xiuhpōhualli'' (year count) system of the

Aztec calendar The Aztec or Mexica calendar is the calendar, calendrical system used by the Aztecs as well as other Pre-Columbian era, Pre-Columbian indigenous peoples of Mexico, peoples of central Mexico. It is one of the Mesoamerican calendars, sharing the bas ...

had five intercalary days after the eighteenth and final month, the ''

nēmontēmi In the Aztec (Mexica) culture, the Nahuatl word refers to a period of five intercalary days inserted between the 360 days labeled with numbers and day-names in the main part of the Aztec seasonal calendar. Their location was roughly around ...

'', in which the people fasted and reflected on the past year.

References