Intercalation or embolism in timekeeping is the insertion of a

leap day February 29, also known as leap day or leap year day, is a date added to leap years. A leap day is added in various solar calendars (calendars based on the Earth's revolution around the Sun), including the Gregorian calendar standard in mo ...

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 Generally speaking, 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. A year can also be measured by starting on any o ...

s to make the

calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A date is the designation of a single and specific day within such a system. A calendar is also a physi ...

follow the seasons or moon phases.

Lunisolar calendar A lunisolar calendar is a calendar in many cultures, combining lunar calendars and solar calendars. The date of Lunisolar calendars therefore indicates both the Moon phase and the time of the solar year, that is the position of the Sun in the Ea ...

s may require intercalations of both days and 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 of a celestial body of the Solar System such as the Earth, completing a full cycle of seasons; for example, the time fro ...

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 to a

common year A common year is a calendar year with 365 days, as distinguished from a leap year, which has 366. More generally, a common year is one without intercalation. The Gregorian calendar (like the earlier Julian calendar) employs both common years ...

of 365 days, an extra day ("leap day" or "intercalary day") about every four years, causing 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) added to keep the calendar year synchronized with the astronomical year or s ...

to have 366 days ( Julian, Gregorian and

Indian national calendar The Indian national calendar, sometimes called the Saka 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 official communications i ...

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 Greek, on several ancient Egyptian memorial sto ...

, which was issued by the pharaoh

Ptolemy III Euergetes , predecessor = Ptolemy II , successor = Ptolemy IV , nebty = ''ḳn nḏtj-nṯrw jnb-mnḫ-n-tꜢmrj'Qen nedjtinetjeru inebmenekhentamery''The brave one who has protected the gods, a potent wall for The Beloved Land , nebty_hiero ...

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

Augustus Caesar Augustus (born Gaius Octavius; 23 September 63 BC – 19 August AD 14), also known as Octavian, was the first Roman emperor; he reigned from 27 BC until his death in AD 14. He is known for being the founder of the Roman Pri ...

successfully instituted a reformed Alexandrian calendar. In the

Julian calendar The Julian calendar, proposed by Roman consul Julius Caesar in 46 BC, was a reform of the Roman calendar. It took effect on , by edict. It was designed with the aid of Greek mathematicians and astronomers such as Sosigenes of Alexandr ...

, as well as in the

Gregorian calendar The Gregorian calendar is the calendar used in most parts of the world. It was introduced in October 1582 by Pope Gregory XIII as a modification of, and replacement for, the Julian calendar. The principal change was to space leap years dif ...

, 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 years of ...

Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern Egypt but also in Sudan and Libya * Coptic language, a Northern Afro-Asiatic language spoken in Egypt until at least the 17th century * Coptic alphabet ...

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 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 verna ...

. 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 in a year. Regular years have 12 months, but embolismic years insert a 13th "intercalary" or "leap" or "embolismic" month every second or third year (see blue moon). 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 grc, ἐννεακαιδεκαετηρίς, 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 recu ...

(

Hebrew calendar The Hebrew calendar ( he, הַלּוּחַ הָעִבְרִי, translit=HaLuah HaIvri), also called the Jewish calendar, is a lunisolar calendar used today for Jewish religious observance, and as an official calendar of the state of Israel. I ...

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

Chinese calendar The traditional Chinese calendar (also known as the Agricultural Calendar ��曆; 农历; ''Nónglì''; 'farming calendar' Former Calendar ��曆; 旧历; ''Jiùlì'' Traditional Calendar ��曆; 老历; ''Lǎolì'', is a lunisolar calendar ...

s). The

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

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

Lunar calendars

The

tabular Islamic calendar The Tabular Islamic calendar ( ar, التقويم الهجري المجدول, altaqwim alhijriu almujadwal) is a rule-based variation of the Islamic calendar. It has the same numbering of years and months, but the months are determined by arith ...

usually has 12 lunar months that alternate between 30 and 29 days every year, but an intercalary day is added to the last month of the year 12 times within a 33-year cycle. Some historians also linked the pre-Islamic practice of

Nasi' Nasiʾ ( ar, ٱلنَّسِيء, ''an-Nasīʾ'', "postponement"), also spelled 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 de ...

to intercalation. The

Solar Hijri calendar The Solar calendar ( fa, گاه‌شماری هجری خورشیدی, Gâhšomâri-ye Xoršidi; ps, لمريز لېږدیز کلیز, lamrez legdez kalhandara; ku, ڕۆژژمێری کۆچیی ھەتاوی, Salnameya Koçberiyê) is a solar ...

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

Hijrah The Hijrah or Hijra () was the journey of the Islamic prophet Muhammad and his followers from Mecca to Medina. The year in which the Hijrah took place is also identified as the epoch of the Lunar Hijri and Solar Hijri calendars; its date eq ...

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 Pa ...

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 observe ...

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 Geneva-based International Organization for Standardization (ISO) and was first published in 1988, wi ...

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 had five intercalary days after the eighteenth and final month, the '' nēmontēmi'', in which the people reflect on the past year and do fasting.

References