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A lunar calendar is a calendar that is based on cycles of the lunar phase. Because there are about twelve lunations (synodic months) in a solar year, this period (354.37 days) is sometimes referred to as a lunar year.
A common purely lunar calendar is the Islamic calendar or Hijri Qamari calendar. A feature of the Islamic calendar is that a year is always 12 months, so the months are not linked with the seasons and drift each solar year by 11 to 12 days. It comes back to the position it had in relation to the solar year approximately every 33 Islamic years. It is used mainly for religious purposes, but in Saudi Arabia it is the official calendar. In other systems, a lunar calendar may include extra months added that synchronize it with the solar calendar.
Most lunar calendars are in fact lunisolar calendars. That is, months reflect the lunar cycle, but then intercalary months (e.g. "second Adar" in the Hebrew calendar) are added to bring the calendar year into synchronisation with the solar year. Some examples are the Chinese and Hindu calendars, and most calendar systems used in antiquity.
All these calendars have a variable number of months in a year. The reason for this is that a year is not evenly divisible by an exact number of lunations, so without the addition of intercalary months the seasons would drift each year. This results in a thirteen-month year every two or three years.
Some lunar calendars are calibrated by annual natural events which are affected by lunar cycles as well as the solar cycle. An example of this is the lunar calendar of the Banks Islands which includes three months in which the edible palolo worm mass on the beaches. These events occur at the last quarter of the lunar month, as the reproductive cycle of the palolos is synchronised with the moon.
Even though the Gregorian calendar is in common and legal use, lunar calendars serve to determine traditional holidays in parts of the world such as India, China, Korea, Japan, Vietnam and Nepal. Some examples include Ramadan, Diwali, Chinese New Year/Tết (Vietnamese New Year), Mid-Autumn Festival/Chuseok and Nepal Sambat.
Lunar calendars differ as to which day is the first day of the month. For some lunar calendars, such as the Chinese calendar, the first day of a month is the day when an astronomical new moon occurs in a particular time zone. For others, such as some Hindu calendars, each month begins on the day after the full moon or the new moon. Others were based in the past on the first sighting of a lunar crescent, such as the Hebrew calendar.
The length of a month orbit/cycle is difficult to predict and varies from its average value. Because observations are subject to uncertainty and weather conditions, and astronomical methods are highly complex, there have been attempts to create fixed arithmetical rules.
The average length of the synodic month is 29.530589 days. This requires the length of a month to be alternately 29 and 30 days (termed respectively hollow and full). The distribution of hollow and full months can be determined using continued fractions, and examining successive approximations for the length of the month in terms of fractions of a day. In the list below, after the number of days listed in the numerator, an integer number of months as listed in the denominator have been completed:
These fractions can be used in the construction of lunar calendars, or in combination with a solar calendar to produce a lunisolar calendar. The 49-month cycle was proposed as the basis of an alternative Easter computation by Isaac Newton around 1700. The tabular Islamic calendar's 360-month cycle is equivalent to 24×15 months minus a correction of one day.