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

Solar calendars[link]

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 BC, 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 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 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 four epagomenal days, with a fifth intercalary day matching the pattern of intercalation in the Gregorian calendar.

An alternative method of solar-calendar intercalation, which does not vary the number of days in the year, works by varying the lengths of certain days. The proposed Long-Sabbath Perennial Calendar extends to 36 hours the lengths of each of a pair of Saturdays and Sundays every year, and an additional pair in leap years. With this method, seventy-two hours (three 24-hour periods) are covered by only two calendar dates, resulting in a constant 364-day calendar year.

Lunisolar calendars[link]

The solar year does not have a whole number of lunar months (it is about 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 "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 (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 calendar[link]

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.

Baha'i Calendar[link]

See Ayyam-i-Ha

Leap seconds[link]

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,^[1] although the word intercalation strictly applies only to periods of 24 hours or longer.

Other uses[link]

ISO 8601 includes a specification for a 52-week year. Any year that has 53 Thursdays has 53 weeks; this extra week may be regarded as intercalary.

See also[link]

• Bahá'í calendar and Ayyám-i-Há
• Celtic calendar
• Chinese calendar
• Coptic calendar
• Egyptian calendar
• Hebrew calendar
• Hindu calendar
• Islamic calendar
• Iranian calendar
• Thai lunar calendar
• Bengali calendar
• Ethiopian calendar

References[link]

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