Intercalation (timekeeping)

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Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. [1] Lunisolar calendars may require intercalations of both days and months.


Solar calendars

The solar or tropical year does not have a whole number of days (it is about 365.24 days), but a calendar year 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 of 365 days, an extra day ("leap day" or "intercalary day") about every four years, causing a leap year to have 366 days (Julian, Gregorian and Indian national calendars).

The Decree of Canopus, which was issued by the pharaoh Ptolemy III Euergetes 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 successfully instituted a reformed Alexandrian calendar.

In the Julian calendar, as well as in the Gregorian calendar, 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 [2] days are days within a solar calendar that are outside any regular month. Usually five epagomenal days are included within every year (Egyptian, Coptic, Ethiopian, 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 (علاء, ʿalāʾ) to ensure that the following year starts on the March equinox. These are known as the Ayyám-i-Há.

Lunisolar calendars

The solar year does not have a whole number of lunar months (it is about 365/29.5 = 12.37 lunations), 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 (Hebrew calendar and in the determination of Easter) or using calculations of lunar phases (Hindu lunisolar and Chinese calendars). The Buddhist calendar adds both an intercalary day and month on a usually regular cycle.

Lunar calendars

The tabular Islamic calendar 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 11 times within a 30-year cycle. Some historians also linked the pre-Islamic practice of Nasi' to intercalation.

The Solar Hijri calendar are 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 Hegira. [3]

Leap seconds

The International Earth Rotation and Reference Systems Service can insert or remove leap seconds from the last day of any month (June and December are preferred). These are sometimes described as intercalary. [4]

Other uses

ISO 8601 includes a specification 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.

See also

Related Research Articles

Calendar System for organizing the days of year

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, specific day within such a system. A calendar is also a physical record of such a system. A calendar can also mean a list of planned events, such as a court calendar or a partly or fully chronological list of documents, such as a calendar of wills.

Islamic calendar Lunar calendar used by Muslims to determine religious observances

The Hijri calendar, also known as the Lunar Hijri calendar and as the Islamic, Muslim or Arabic 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 annual period of fasting and the proper time for the Hajj. In almost all countries where the predominant religion is Islam, the civil calendar is the Gregorian calendar, with Syriac month-names used in the Levant and Mesopotamia. Notable exceptions to this rule are Iran and Afghanistan, which use the Solar Hijri calendar. Rents, wages and similar regular commitments are generally paid by the civil calendar.

The Julian calendar, proposed by Julius Caesar in AUC 708, was a reform of the Roman calendar. It took effect on 1 January AUC 709 , by edict. It was designed with the aid of Greek mathematicians and astronomers such as Sosigenes of Alexandria.

A leap year is a calendar year that contains an additional day added to keep the calendar year synchronized with the astronomical year or seasonal year. Because astronomical events and seasons do not repeat in a whole number of days, calendars that have a constant number of days in each year will unavoidably drift over time with respect to the event that the year is supposed to track, such as seasons. By inserting an additional day or month into some years, the drift between a civilization's dating system and the physical properties of the solar system can be corrected. A year that is not a leap year is a common year.

Lunar calendar Type of calendar

A lunar calendar is a calendar based on the monthly cycles of the Moon's phases, in contrast to solar calendars, whose annual cycles are based only directly on the solar year. The most commonly used calendar, the Gregorian calendar, is a solar calendar system that originally evolved out of a lunar calendar system. A purely lunar calendar is also distinguished from a lunisolar calendar, whose lunar months are brought into alignment with the solar year through some process of intercalation. The details of when months begin varies from calendar to calendar, with some using new, full, or crescent moons and others employing detailed calculations.

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. As with all calendars which divide the year into months 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.

A month is a unit of time, used with calendars, which is approximately as long as a natural period related to the motion of the Moon; month and Moon are cognates. The traditional concept arose with the cycle of Moon phases; such months (lunations) are synodic months and last approximately 29.53 days. From excavated tally sticks, researchers have deduced that people counted days in relation to the Moon's phases as early as the Paleolithic age. Synodic months, based on the Moon's orbital period with respect to the Earth-Sun line, are still the basis of many calendars today, and are used to divide the year.

Metonic cycle

The Metonic cycle or enneadecaeteris is a period of approximately 19 years after which the phases of the moon recur at the same time of the year. The recurrence is not perfect, and by precise observation the Metonic cycle defined as 235 synodic lunar months is just 1 hour, 27 minutes and 33 seconds longer than 19 tropical years. Meton of Athens, in the 5th century BC, judged the cycle to be a whole number of days, 6,940. Using these whole numbers facilitates the construction of a luni-solar calendar.

Roman calendar Calendar used by the Roman Kingdom and Roman Republic

The Roman calendar was the calendar used by the Roman kingdom and republic. The term often includes the Julian calendar established by the reforms of the dictator Julius Caesar and emperor Augustus in the late 1st century BC and sometimes includes any system dated by inclusive counting towards months' kalends, nones, and ides in the Roman manner. The term usually excludes the Alexandrian calendar of Roman Egypt, which continued the unique months of that land's former calendar; the Byzantine calendar of the later Roman Empire, which usually dated the Roman months in the simple count of the ancient Greek calendars; and the Gregorian calendar, which refined the Julian system to bring it into still closer alignment with the tropical year.

Coptic calendar Egyptian liturgical calendar

The Coptic calendar, also called the Alexandrian calendar, is a liturgical calendar used by the Coptic Orthodox Church and also used by the farming populace in Egypt. This calendar is based on the ancient Egyptian calendar. To avoid the calendar creep of the latter, a reform of the ancient Egyptian calendar was introduced at the time of Ptolemy III which consisted of the intercalation of a sixth epagomenal day every fourth year. However, this reform was opposed by the Egyptian priests, and the reform was not adopted until 25 BC, when the Roman Emperor Augustus imposed the Decree upon Egypt as its official calendar. To distinguish it from the Ancient Egyptian calendar, which remained in use by some astronomers until medieval times, this reformed calendar is known as the Coptic or Alexandrian calendar. Its years and months coincide with those of the Ethiopian calendar but have different numbers and names.

A solar calendar is a calendar whose dates indicate the season or almost equivalently the apparent position of the Sun relative to the stars. The Gregorian calendar, widely accepted as a standard in the world, is an example of a solar calendar. The main other type of calendar is a lunar calendar, whose months correspond to cycles of Moon phases. The months of the Gregorian calendar do not correspond to cycles of the Moon phase.

Egyptian calendar Calendar used in ancient Egypt before 22 BC

The ancient Egyptian calendar – a civil calendar – was a solar calendar with a 365-day year. The year consisted of three seasons of 120 days each, plus an intercalary month of five epagomenal days treated as outside of the year proper. Each season was divided into four months of 30 days. These twelve months were initially numbered within each season but came to also be known by the names of their principal festivals. Each month was divided into three 10-day periods known as decans or decades. It has been suggested that during the Nineteenth Dynasty and the Twentieth Dynasty the last two days of each decan were usually treated as a kind of weekend for the royal craftsmen, with royal artisans free from work.

The epact, used to be described by medieval computists as the age of a phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, the epact is reckoned as the age of the ecclesiastical moon on 1 January. Its principal use is in determining the date of Easter by computistical methods. It varies from year to year, because of the difference between the solar year of 365–366 days and the lunar year of 354–355 days.

The history of calendars, that is, of people creating and using methods for keeping track of days and larger divisions of time, covers a practice with ancient roots.

Calendar reform or calendrical reform is any significant revision of a calendar system. The term sometimes is used instead for a proposal to switch to a different calendar design.

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 predecessor preserved in the Umma calendar of Shulgi.

The Ethiopian calendar is the principal calendar used in Ethiopia and also serves as the liturgical year for Christians in Ethiopia and Eritrea belonging to the Orthodox Tewahedo Churches, Eastern Catholic Churches, and Eastern Protestant Christian P'ent'ay Churches. The Ethiopian calendar is a solar calendar that has more in common with the Coptic calendar of the Coptic Orthodox Church of Alexandria and Coptic Catholic Church, but like the Julian calendar, it adds a leap day every four years without exception, and begins the year on August 29 or August 30 in the Julian calendar. A gap of seven to eight years between the Ethiopian and Gregorian calendars results from an alternate calculation in determining the date of the Annunciation.

The Buddhist calendar is a set of lunisolar calendars primarily used in mainland Southeast Asian countries of Cambodia, Laos, Myanmar and Thailand as well as in Sri Lanka and Chinese populations of Malaysia and Singapore for religious or official occasions. While the calendars share a common lineage, they also have minor but important variations such as intercalation schedules, month names and numbering, use of cycles, etc. In Thailand, the name Buddhist Era is a year numbering system shared by the traditional Thai lunisolar calendar and by the Thai solar calendar.

The intercalary month or epagomenal days of the ancient Egyptian, Coptic, and Ethiopian calendars are a period of five days in common years and six days in leap years in addition to those calendars' 12 standard months, sometimes reckoned as their thirteenth month. They originated as a periodic measure to ensure that the heliacal rising of Sirius would occur in the 12th month of the Egyptian lunar calendar but became a regular feature of the civil calendar and its descendants. Coptic and Ethiopian leap days occur in the year preceding Julian and Gregorian leap years.


  1. Chisholm, Hugh, ed. (1911). "Intercalary"  . Encyclopædia Britannica (11th ed.). Cambridge University Press. p. 683.
  2. From ἐπαγόμενος, epagomenos (present participle passive of ἐπάγειν, epagein "to bring in") + -al
  3. "Hijri-Shamsi Calendar". Al Islam. Retrieved June 14, 2014.
  4. leap second by Merriam-Webster OnLine