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Most calendars are based on the solar year. Solar years have the disadvantage of not being easily observable. Many years of observations are required to fix them with any significant degree of accuracy. On the other hand, the phases of the Moon -- and the first visibility after the new moon in particular -- are very easy and quick to observe. Therefore, the first calendars defined a lunar year, usually consisting of 12 synodic months. A synodic month is the interval from one new moon to the next and lasts 29.530588 days. This is equivalent to 29 days, 12 hours, 44 minutes and 2.9 seconds. Since for practical reasons a month should contain an integer number of days, most calendars alternated between months of 29 and 30 days, respectively. A year made out of six months of each type has 354 days and is thus too short by 0.3672 days as compared with a true lunar year. Therefore lunar calendars have to insert one leap day about every third year to keep in step with the moon phases. A pure lunar calendar is not synchronous with the seasons and after 16 years will put the winter in the summer and vice versa. Over a period of 32 years it will cycle through a complete year.
A luni-solar year is the attempt to combine the phases of the moon and the seasons into one calendar. This is possible if leap months are inserted. Several schemes were used in history. The best known solution was found by the Greek Meton in the year 432 BC but apparently was known to other cultures before. The Metonic cycle encompasses a total of 235 months of which 125 are full (i.e. they have 30 days) and 110 are `hollow' (having 29 days). The months are combined into 12 normal years with 12 months each and 7 leap years with 13 months each. The cycle covers 6940 days whereas 225 synodic months sum up to 6939.688 days and 19 tropical years to 6939.602 days. The difference in motion between Sun and Moon amounts to only 0.0866 days so that eclipses repeat in the Metonic cycle with high accuracy.
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