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. 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, 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 about every four years, causing a leap year to have 366 days. 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 days are days within a solar calendar that are outside any regular month. Usually five epagomenal days are included within every year, but a sixth epagomenal day is intercalated every four years in some. The Bahá'í calendar includes enough epagomenal days before the last month 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, 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. Whether to insert an intercalary month in a given year may be determined using regular cycles such as the 19-year Metonic cycle or using calculations of lunar phases. The Buddhist calendar adds both an intercalary day and month on a usually regular cycle.
Islamic 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 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 Hegira.
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 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.
