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The 'Metonic cycle' or 'Enneadecaeteris' in astronomy and calendar studies is a particular approximate common multiple of the year (specifically, the seasonal ''i.e.'' tropical year) and the synodic month. Nineteen 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.
:19 tropical years = 6939.602 days
:235 synodic months = 6939.688 days
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 spin-orbit resonance or other orbital resonance.
A year of 12 synodic or lunar months is about 354 days on average, 11 days short of the 365 day solar year. Therefore in a lunisolar calendar every 3 years or so there is a difference of more than a full 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. The Greek astronomer Meton of Athens introduced a formula for intercalation in 432 BC. It was his observation that a period of 19 solar years is almost exactly 235 lunar months, and rounded to full days counts 6940 days. The error is only 2 hours between 19 years and 235 months. This rounded day count defines the length of the year to be 365 + 1/4 + 1/76 days (the unrounded cycle is much more accurate). Seven of the 19 years would have an intercalated month. Traditionally (in the ancient Attic and Babylonian calendars, as well as in the Hebrew calendar), the years: 3, 6, 8, 11, 14, 17, and 19, are the long (13-month) 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. In Antiquity the Metonic cycle was sometimes called the Great year.
The Chaldean astronomer Kidinnu (4th century BC) knew of the 19-year 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; however, ancient astronomers did not make a clear distinction between sidereal and tropical years before Hipparchus discovered precession of the equinoxes c. 130 BC.
It is possible that Homer knew about the cycle some centuries before Meton. In the Odyssey, after Odysseus leaves Ithaca, he returns to secretly meet Penelope at the exact moment when one Metonic cycle has passed. ^{[1] [2] [3]}
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. The Metonic cycle itself is a subcycle of the next more correct approximation: 334 years = 4131 lunations to within 41 minutes, ''i.e.'' an error of one day in 11598 years, if the length of the lunation and the year would not change over such a long period (which they do). The 334-year cycle has a mean year that is slightly too short. An even more accurate approximation is a 353-year cycle with 4366 lunations.
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. This was a more accurate approximation, obtained by taking one day away from every fourth of Meton's cycles, so creating a 76-year cycle with a mean year of exactly 365.25 days. Later the solar Julian calendar was designed to have a year of this length by using leap days.
The 19-year 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.
The Bahá'í calendar, established in the middle of the 19th century, is also based on cycles of 19 years.

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References

1. Gilbert Murray. ''The Rise of the Greek Epic''. Oxford, (1907)
2. Joseph Campbell. ''The Masks of God: Occidental Mythology'', vol III (1964)
3. J.F. del Giorgio. ''The Oldest Europeans''. A.J. Place (2006)

See also

★ Antikythera mechanism

★ Baha'i calendar

★ Saros cycle