A lunar calendar is a calendar that is based on cycles of the moon phase. The only widely used purely lunar calendar is the Islamic calendar or Hijri calendar, whose year always consists of 12 lunar months. A feature of a purely lunar year, on the Islamic calendar model, is that the calendar ceases to be linked to the seasons, and drifts each year by 11 days (or 12 days in case of leap year), and comes back to the position it had in relation to the solar year approximately every 33 Islamic years. It is used predominantly for religious purposes. In Saudi Arabia it is also used for commercial purposes.
Most lunar calendars, except the Hijri, are in fact lunisolar calendars. That is, months are kept on a lunar cycle, but then intercalary months are added to bring the lunar cycles into synchronisation with the solar year.
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.
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.
For some lunar calendars, such as the Chinese calendar, the first day of a month is the day when an eclipse moon appears in a particular time zone.
Many other lunar calendars are based on the first sighting of a lunar crescent.
The average length of the synodic month is 29.530589 days. This means the length of a month is 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:
29 / 1 (error: 1 day after about 2 months)
30 / 1 (error: 1 day after about 2 months)
59 / 2 (error: 1 day after about 33 months)
443 / 15 (error: 1 day after about 30 years)
502 / 17 (error: 1 day after about 70 years)
1447 / 49 (error: 1 day after about 3 millennia)25101 / 850 (error: dependent on change of synodic month value}
The "Edwardian" (probably Edward II, late 13th or early 14th century) ballad of Robin Hood for example has "How many merry months be in the year? / There are thirteen, I say ...", amended by a Tudor editor to "...There are but twelve, I say....". Robert Graves in the introductions to Greek Myths comments on this with "Thirteen, the number of the sun's death-month, has never lost its evil reputation among the superstitious."
Even in the late 20th century, British financial institutions were still administering lunar mortgages, requiring an annual adjustment.