Intercalation (timekeeping)

Intercalationorembolismin timekeeping is the insertion of aleap day,week,or month into somecalendar yearsto make the calendar follow the seasons or moon phases.Lunisolar calendarsmay require intercalations of days or months.

Solar calendars

Further information:Intercalary month (Egypt)

The solar ortropical yeardoes not have a whole number of days (it is about 365.24 days), but acalendar yearmust 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 acommon yearof 365 days, about once every four years, creating aleap yearthat has 366 days (Julian,GregorianandIndian national calendars).

TheDecree of Canopus,issued by the pharaohPtolemy III EuergetesofAncient Egyptin 239 BC, decreed a solar leap day system; an Egyptian leap year was not adopted until 25 BC, when the Roman EmperorAugustusinstituted a reformedAlexandrian calendar.

In theJulian calendar,as well as in theGregorian 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.

Epagomenaldays are days within a solar calendar that are outside any regular month. Usually five epagomenal days are included within every year (Egyptian,Coptic,Ethiopian,MayanHaab'andFrench Republican Calendars), but a sixth epagomenal day is intercalated every four years in some (Coptic, Ethiopian and French Republican calendars).

TheSolar Hijri calendar,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 itsepochtheHijrah.

TheBahá'í calendarincludes enough epagomenal days (usually 4 or 5) before the last month (علاء,ʿalāʾ) to ensure that the following year starts on theMarch equinox.These are known as theAyyám-i-Há.

Lunisolar calendars

The solar year does not have a whole number of lunar months (it is about 365/29.5 = 12.37lunations), 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-yearMetonic cycle(Hebrew calendarand in thedetermination of Easter) or using calculations of lunar phases (Hindu lunisolarandChinese calendars). TheBuddhist calendaradds both an intercalary day and month on a usually regular cycle.

Lunar calendars

Further information:Nasi'andIslamic calendar § Astronomical considerations

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.

TheLunar Hijri calendar,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 thehilal(crescent moon) shortly after sunset. If thehilalis 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.

Thetabular 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 ofNasi'to intercalation.

Leap seconds

TheInternational Earth Rotation and Reference Systems Servicecan insert or removeleap secondsfrom the last day of any month (June and December are preferred). These are sometimes described as intercalary.

Other uses

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

Thexiuhpōhualli(year count) system of theAztec calendarhad five intercalary days after the eighteenth and final month, thenēmontēmi,in which the people fasted and reflected on the past year.

See also

• Lunisolar calendar
• Egyptian,Coptic,andEthiopian calendars
• Iranian calendar
• Islamic calendar
• Mandaean calendar
• Celtic calendar
• Thai lunar calendar
• Bengali calendar
• Igbo calendar
• World Calendar
• Intercalated Games