The Metonic cycle or Enneadecaeteris in astronomy and calendar studies is a particular approximate common multiple of the tropical year and the synodic (lunar) month. The Greek astronomer Meton of Athens observed that a period of 19 tropical years is almost exactly equal to 235 synodic months, and rounded to full days counts 6940 days. The difference between the two periods (of 19 tropical years and 235 synodic months) is only 2 hours.
Taking a year to be 1/19th of this 6940day cycle gives a year length of 365 + 1/4 + 1/76 days (the unrounded cycle is much more accurate), which is slightly more than 12 synodic months. To keep the 12month lunar year in pace with the solar year, an intercalary 13th month would have to be added on seven occasions during the nineteenyear period. Meton introduced a formula for intercalation in circa 432 BC.
The cycle's most significant contemporary use is to help in flight planning (trajectory calculations and launch window analysis) for lunar spacecraft missions as well as serving as the basis for the Hebrew calendar's 19 year cycle. Another use is in computus, the calculation of the date of the Christian feast of Easter.
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19 tropical years differ from 235 synodic months by about 2 hours. The Metonic cycle's error is one full day every 219 years, or 12.4 parts per million.
This cycle is an approximation of reality. The period of the Moon's orbit around the Earth and the Earth's orbit around the Sun (ignoring also exact definition of the year) are independent and have no known physical resonance. Examples of a real harmonic lock would be Mercury, with its 3:2 spinorbit resonance or other orbital resonance.
A lunar year of 12 synodic months is about 354 days on average, 11 days short of the 365day solar year. Therefore, in a lunisolar calendar, every 3 years or so there is a difference of more than a full lunar month between the lunar and solar years, and an extra (embolismic) month should be inserted (intercalation). The Athenians appear not to have had a regular means of intercalating a 13th month; instead, the question of when to add a month was decided by an official.
Traditionally (in the ancient Attic and Babylonian lunisolar calendars, as well as in the Hebrew calendar), the years 3, 6, 8, 11, 14, 17, and 19 are the long (13month) years of the Metonic cycle. This cycle can be used to predict eclipses, forms the basis of the Greek and Hebrew calendars, and is used in the computation of the date of Easter each year.
The Chaldean astronomer Kidinnu (4th century BC) knew of the 19year cycle, but the Babylonians may have learned of it earlier. They measured the moon's motion against the stars, so the 235:19 relation may originally have referred to sidereal years, instead of tropical years as it has been used in various calendars.
The Bahá'í calendar, established in the middle of the 19th century, is also based on cycles of 19 years.
The Metonic cycle incorporates two less accurate subcycles, for which 8 years = 99 lunations (an Octaeteris) to within 1.5 days, i.e. an error of one day in 5 years; and 11 years = 136 lunations within 1.5 days, i.e. an error of one day in 7.3 years. Adding the 11 year cycle to 17 or 18 Metonic cycles creates the more accurate cycles of 334 years in 4131 lunations and 353 years in 4366 lunations (see lunisolar calendar).
Meton of Athens approximated the cycle to a whole number (6940) of days, obtained by 125 long months of 30 days and 110 short months of 29 days. In the following century Callippus developed the Callippic cycle of four 19year periods for a 76year cycle with a mean year of exactly 365.25 days.
The 19year cycle is also close (to somewhat more than half a day) to 255 draconic months, so it is also an eclipse cycle, which lasts only for about 4 or 5 recurrences of eclipses.

