Intercalation (timekeeping) in the context of "Leap years"

A lunisolar calendar is a calendar in many cultures, that combines monthly lunar cycles with the solar year. As with all calendars which divide the year into months, there is an additional requirement that the year have a whole number of months (Moon cycles). The majority of years have twelve months but every second or third year is an embolismic year, which adds a thirteenth intercalary, embolismic, or leap month.

In contrast to purely lunar calendars such as the Islamic calendar, lunisolar calendars have additional intercalation rules that reset them periodically into a rough agreement with the solar year and thus with the seasons.

The French Republican calendar (French: calendrier républicain français), also commonly called the French Revolutionary calendar (calendrier révolutionnaire français), was a calendar created and implemented during the French Revolution and used by the French government for about 12 years from late 1793 to 1805, and for 18 days by the Paris Commune in 1871, meant to replace the Gregorian calendar. The calendar consisted of twelve 30-day months, each divided into three 10-day cycles similar to weeks, plus five or six intercalary days at the end to fill out the balance of a solar year. It was designed in part to remove all religious and royalist influences from the calendar, and it was part of a larger attempt at dechristianisation and decimalisation in France (which also included decimal time of day, decimalisation of currency, and metrication). It was used in government records in France and other areas under French rule, including Belgium, Luxembourg, and parts of the Netherlands, Germany, Switzerland, Malta, and Italy.

A lunar calendar is a calendar based on the monthly cycles of the Moon's phases (synodic months, lunations), in contrast to solar calendars, whose annual cycles are based on the solar year, and lunisolar calendars, whose lunar months are brought into alignment with the solar year through some process of intercalation – such as by insertion of a leap month. The most widely observed lunar calendar is the Islamic calendar. The details of when months begin vary from calendar to calendar, with some using new, full, or crescent moons and others employing detailed calculations.

Since each lunation is approximately 29+1⁄2 days, it is common for the months of a lunar calendar to alternate between 29 and 30 days. Since the period of 12 such lunations, a lunar year, is 354 days, 8 hours, 48 minutes, 34 seconds (354.36707 days), lunar calendars are 11 to 12 days shorter than the solar year. In lunar calendars, which do not make use of lunisolar calendars' intercalation, the lunar months cycle through all the seasons of a solar year over the course of a 33–34 lunar-year cycle (see, e.g., list of Islamic years).

The Ethiopian calendar (Amharic: ዐውደ ወር; Ge'ez: ዐውደ ወርኅ; Tigrinya: ዓዉደ ኣዋርሕ), or Geʽez calendar (Geʽez: ዐውደ ወርኅ; Tigrinya: ዓዉደ ኣዋርሕ, Amharic: የኢትዮጲያ ዘመን አቆጣጠር) is the official state civil calendar of Ethiopia and serves as an unofficial customary cultural calendar in Eritrea, and among Ethiopians and Eritreans in the diaspora. It is also an ecclesiastical calendar for Ethiopian Christians and Eritrean Christians belonging to the Orthodox Tewahedo Churches (Ethiopian Orthodox Tewahedo Church and Eritrean Orthodox Tewahedo Church), Eastern Catholic Churches (Eritrean Catholic Church and Ethiopian Catholic Church), and Protestant Christian P'ent'ay (Ethiopian-Eritrean Evangelical) Churches. The Ethiopian calendar is a solar calendar that has much 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 11 or 12th of September in the Gregorian calendar (from 1900 to 2099). Its epoch (first day of first year) equates to 29 August 8 AD. The neighbouring Coptic calendar is very similar to the Ethiopian calendar, except that it has a different epoch (29 August, 284 AD) and different names for the days of the week and months of the year.

The Ethiopian calendar has twelve months, all thirty days long, and five or six epagomenal days, which form a thirteenth month. The Ethiopian months begin on the same days as those of the Coptic calendar, but their names are in Geʽez. A sixth epagomenal day is added every four years, without exception, on 29 August of the Julian calendar, six months before the corresponding Julian leap day. Thus, the first day of the Ethiopian calendar year, 1 Mäskäräm, for years between 1900 and 2099 (inclusive), is usually 11 September (Gregorian). It falls on 12 September in years before the Gregorian leap year, however.

A solar term (or jiéqì, simplified Chinese: 节气; traditional Chinese: 節氣) is any of twenty-four moments in traditional Chinese lunisolar calendars that matches a particular astronomical event and signifies some natural phenomenon. The points are spaced 15° apart along the ecliptic and are used by lunisolar calendars to stay synchronized with the seasons, which is crucial for agrarian societies. The solar terms are also used to calculate intercalary months; which month is repeated depends on the position of the sun at the time.

According to the Book of Documents, the first determined term was Dongzhi (Winter Solstice) by Dan, the Duke of Zhou, while he was trying to locate the geological center of the Western Zhou dynasty, by measuring the length of the sun's shadow on an ancient type of sundial called tǔguī [ja] (土圭). Then four terms of seasons were set, which were soon evolved as eight terms; not until the Taichu Calendar of 104 BC were all twenty-four solar terms officially included in the Chinese calendar.

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) compared to a common year. The 366th day (or 13th month) is added to keep the calendar year synchronised with the astronomical year or seasonal year. Since astronomical events and seasons do not repeat in a whole number of days, calendars having a constant number of days each year will unavoidably drift over time with respect to the event that the year is supposed to track, such as seasons. By inserting ("intercalating") an additional day—a leap day—or month—a leap month—into some years, the drift between a civilisation's dating system and the physical properties of the Solar System can be corrected.

An astronomical year lasts slightly less than 365⁠1/4⁠ days. The historic Julian calendar has three common years of 365 days followed by a leap year of 366 days, by extending February to 29 days rather than the common 28. The Gregorian calendar, the world's most widely used civil calendar, makes a further adjustment for the small error in the Julian algorithm; this extra leap day occurs in each year that is a multiple of 4, except for years evenly divisible by 100 but not by 400. Thus 1600, 2000 and 2400 are leap years, but not 1700, 1800, 1900, 2100, 2200, and 2300.

The Hebrew calendar (Hebrew: הַלּוּחַ הָעִבְרִי, romanized: HaLuakh ha'Ivri, IPA: [ha‿ˈluaχ ha‿ʔivˈʁi]), also called the Jewish calendar, is a lunisolar calendar used today for Jewish religious observance and as an official calendar of Israel. It determines the dates of Jewish holidays and other rituals, such as yahrzeits and the schedule of public Torah readings. In Israel, it is used for religious purposes, provides a time frame for agriculture, and is an official calendar for civil holidays alongside the Gregorian calendar.

Like other lunisolar calendars, the Hebrew calendar consists of months of 29 or 30 days which begin and end at approximately the time of the new moon. As 12 such months comprise a total of just 354 days, an extra lunar month is added every 2 or 3 years so that the long-term average year length closely approximates the actual length of the solar year.