Metonic cycle

Depiction of the 19 years of the Metonic cycle as a wheel, with the Julian date of the Easter New Moon, from a 9th-century computistic manuscript made in St. Emmeram’s Abbey (Clm 14456, fol. 71r)

For example, by the 19-year Metonic cycle, the full moon repeats on or near Christmas day between 1711 and 2300.^[1]^[2] A small horizontal libration is visible comparing their appearances. A red color shows full moons that are also lunar eclipses.

The Metonic cycle or enneadecaeteris (from Ancient Greek: ἐννεακαιδεκαετηρίς (enneakaidekaeteris), “nineteen”) is a period of approximately 19 years after which the phases of the moon recur on the same day of the year. The recurrence is not perfect, and by precise observation the Metonic cycle is defined as 235 synodic lunar months, a period which is just 1h27m33s longer than 19 tropical years. Using these integer numbers facilitates the construction of a luni-solar calendar.

A tropical year is longer than 12 lunar months and shorter than 13 of them. The arithmetical equation
12×12 + 7×13 = 235
allows it to be seen that a combination of 12 ‘shorter’ (12 months) years and 7 ‘longer’ (13 months) years will be equal to 19=12+7 solar years.

Application in traditional calendars

Traditionally, for the Babylonian and Hebrew lunisolar calendars, the years 3, 6, 8, 11, 14, 17, and 19 are the long (13-month) years of the Metonic cycle. This cycle forms the basis of the Greek and Hebrew calendars, and is used for the computation of the date of Easter each year.

The Babylonians applied the 19-year cycle since the late sixth century BC.^[3] As they measured the moon’s motion against the stars, the 235:19 relationship may originally have referred to sidereal years, instead of tropical years as it has been used for various calendars.

According to Livy, the king of Rome Numa Pompilius (753–673 BC) inserted intercalary months in such a way that “in the twentieth year the days should fall in with the same position of the sun from which they had started.”^[4] As “the twentieth year” takes place nineteen years after “the first year”, this seems to indicate that the Metonic cycle was applied to Numa’s calendar.

Diodorus Siculus reports that Apollo is said to have visited the Hyperboreans once every 19 years.^[5]

The Metonic cycle has been implemented in the Antikythera mechanism which offers unexpected evidence for the popularity of the calendar based on it.^[6] Meton of Athens approximated the cycle to a whole number (6,940) of days, obtained by 125 long months of 30 days and 110 short months of 29 days. During the next century, Callippus developed the Callippic cycle of four 19-year periods for a 76-year cycle with a mean year of exactly 365.25 days.

The (19-year) Metonic cycle is a lunisolar cycle, as is the (76-year) Callippic cycle.^[7] An important example of an application of the Metonic cycle in the Julian calendar is the 19-year lunar cycle insofar as provided with a Metonic structure.^[8] Around AD 260 the Alexandrian computist Anatolius, who became bishop of Laodicea in AD 268, was the first to construct a version of this efficient computistical instrument for determining the date of Easter Sunday.^[9] However, it was some later, somewhat different, version of the Metonic 19-year lunar cycle which ultimately, as the basic structure of Dionysius Exiguus’ and also of Bede’s Easter table, would prevail throughout Christendom for a long time,^[10] at least until in the year 1582, when the Julian calendar was replaced with the Gregorian calendar.

The Runic calendar is a perpetual calendar based on the 19-year-long Metonic cycle. It is also known as a Rune staff or Runic Almanac. This calendar does not rely on knowledge of the duration of the tropical year or of the occurrence of leap years. It is set at the beginning of each year by observing the first full moon after the winter solstice. The oldest one known, and the only one from the Middle Ages, is the Nyköping staff, which is believed to date from the 13th century.

The Bahá’í calendar, established during the middle of the 19th century, is also based on cycles of 19 years.

In China, the traditional Chinese calendar used the Metonic cycle ever since the first known ancient China calendar. The cycle was continually used until the 5th century when it was replaced by a more accurate cycle.^[11]

Mathematical basis

The importance of the tropical year for agriculture came to be realized much later than the adoption of lunar months for time keeping. However, it was recognized that the two cannot be easily coordinated over a short time span, so longer intervals were considered and the Metonic cycle was discovered as rather good, but not perfect, schema. The currently accepted values are:

235 synodic months (lunar phases) = 6,939.688 days (Metonic period by definition).

19 tropical years = 6,939.602 days

The difference is 0.086 days for a cycle which mean that after a dozen returns there will be a full day of delay between the astronomical data and calculations. The error is actually one day every 219 years, or 12.4 parts per million. However, the Metonic cycle turned out to be very close to other periods:

254 sidereal months (lunar orbits) = 6,939.702 days

255 draconic months (lunar nodes) = 6,939.1161 days.

20.021 eclipse years (40 eclipse seasons)

Being close (to somewhat more than half a day) to 255 draconic months, the Metonic cycle is also an eclipse cycle, which lasts only for about 4 or 5 recurrences of eclipses. The Octon is ¹⁄₅ of a Metonic cycle (47 synodic months, 3.8 years), and it recurs about 20 to 25 cycles.

This cycle seems to be a coincidence. The periods of the Moon’s orbit around the Earth and the Earth’s orbit around the Sun are believed to be independent, and not to have any known physical resonance. An example of a non-coincidental cycle is the orbit of Mercury, with its 3:2 spin-orbit resonance.

A lunar year of 12 synodic months is about 354 days, approximately 11 days short of the “365-day” solar year. Therefore, for a lunisolar calendar, every 2 to 3 years there is a difference of more than a full lunar month between the lunar and solar years, and an extra (embolismic) month needs to be inserted (intercalation). The Athenians initially seem 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. Meton’s discovery made it possible to propose a regular intercalation scheme. The Babylonians seem to have introduced this scheme around 500 BC, thus well before Meton.

Further details

The Metonic cycle is related to two less accurate subcycles:

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.

By combining appropriate numbers of 11-year and 19-year periods, it is possible to generate ever more accurate cycles. For example, simple arithmetic shows that:

687 tropical years = 250,921.39 days;
8,497 lunations = 250,921.41 days.

This gives an error of only about half an hour in 687 years (2.5 seconds a year), although this is subject to secular variation in the length of the tropical year and the lunation.

At the time of Meton, axial precession had not yet been discovered, and he could not distinguish between sidereal years (currently: 365.256363 days) and tropical years (currently: 365.242190 days). Most calendars, like the commonly used Gregorian calendar, are based on the tropical year and maintain the seasons at the same calendar times each year.

Name	type	group	introduction	usage	comments
Vikram samwat	Lunisolar	Indian^{[citation needed]}	Ancient India
Egyptian calendar	fixed (365 days)	Egyptian	Bronze Age	Middle Kingdom	The year is based on the heliacal rising of Sirius (Sothis) and divided into the three seasons of akhet (Inundation), peret (Growth) and shemu (Harvest). The heliacal rising of Sothis returned to the same point in the calendar every 1,460 years (a period called the Sothic cycle).^[1]
Umma calendar	lunisolar	Mesopotamian	Bronze Age	Sumer/Mesopotamia	Recorded in Neo-Sumerian records (21st century BC), presumably based on older (Ur III) sources.
Pentecontad calendar	solar	Mesopotamian	Bronze Age	Amorites	A Bronze Age calendar in which the year is divided into seven periods of fifty days, with an annual supplement of fifteen or sixteen days for synchronisation with the solar year.
Four Seasons and Eight Nodes (四時曆)	solar	Chinese	Bronze Age(?)	China	The years is divided into four seasons, and each season is divided into a festival(四立) and three months. The start and middle of each season is the key node of the year.
Gezer Calendar	lunar	Mesopotamian	1000 BC	Israel/Canaan	The years are divided into monthly or bi-monthly periods and attributes to each a duty such as harvest, planting, or tending specific crops.
Roman calendar	lunisolar	Roman	713 BC	Roman Republic	Based on the reforms introduced by Numa Pompilius in c. 713 BC.
Six Ancient Calendars (古六曆)	lunisolar	Chinese	Iron Age	China	Six classical (Zhou era) calendars: Huangdi (黃帝曆), Zhuanxu (顓頊曆), Xia (夏曆), Yin (殷曆), Zhou’s calendar (周曆) and Lu (魯曆).
Nisg̱a’a	seasonal / lunisolar	Indigenous North America	^{[citation needed]}	Nisg̱a’a	The Nisga’a calendar revolves around harvesting of foods and goods used. The original year followed the various moons throughout the year.
Haida	lunar	Indigenous North America	^{[citation needed]}	Haida	The Haida calendar is a lunar calendar broken into two seasons (winter and summer) of six months each with an occasional thirteenth month between seasons.
Inuit	seasonal	Indigenous North America	^{[citation needed]}	Inuit	The Inuit calendar is based on between six and eight seasons as solar and lunar timekeeping methods do not work in the polar regions.
Haab’	fixed (365 days)	Pre-Columbian (Maya)	1st millennium BC^{[citation needed]}	Maya
Tzolk’in	fixed (260 days)	Pre-Columbian (Maya)	1st millennium BC^{[citation needed]}	Maya
Xiuhpohualli	fixed (365 days)	Pre-Columbian (Aztec)	^{[citation needed]}	Aztecs
Tonalpohualli	fixed (260 days)	Pre-Columbian (Aztec)	^{[citation needed]}	Aztecs
Attic calendar	lunisolar (354/ 384 days)	Hellenic	6th century BC	Classical Athens	The year begins with the new moon after the summer solstice. It was introduced by the astronomer Meton in 432 BC. Reconstructed by Academy of Episteme.
Old Persian calendar	lunisolar(?)	Iranian	4th century BC(?)	Persian Empire	Based on earlier Babylonian/Mesopotamian models
Seleucid calendar	lunisolar	Hellenic/Babylonian	4th century BC	Seleucid Empire	Combination of the Babylonian calendar, ancient Macedonian (Hellenic) month names and the Seleucid era.
Genesis Calendar (太初曆)	lunisolar	Chinese	Han dynasty	China	Introduced the “month without mid-climate is intercalary” rule; based on a solar year of 365³⁸⁵⁄₁₅₃₉ days and a lunar month of 29⁴³⁄₈₁ days (19 years=235 months=6939⁶¹⁄₈₁ days).
Ptolemaic calendar	lunisolar	Egyptian	238 BC	Ptolemaic Egypt	The Canopic reform of 238 BC introduced the leap year every fourth year later adopted in the Julian calendar. The reform eventually went into effect with the introduction of the “Alexandrian calendar” (or Julian calendar) by Augustus in 26/25 BC, which included a 6th epagomenal day for the first time in 22 BC.
Julian calendar	solar	Roman	45 BC	Western World	Revision of the Roman Republican calendar, in use in the Roman Empire and the Christian Middle Ages, and remains in use as liturgical calendar of Eastern Orthodox Churches.
Coptic calendar	solar	Egyptian	1st century^{[citation needed]}	Coptic Orthodox Church	Based on both the Ptolemaic calendar and the Julian calendar
Ethiopian calendar	solar	Egyptian	1st century^{[citation needed]}	Ethiopia, Ethiopian Christians	Eritrea, Eritrean Christians	the calendar associated with Ethiopian and Eritrean Churchs, based on the Coptic calendar
Berber calendar	solar	Julian	In Roman times	North Africa	Julian calendar used for agricultural work.
Qumran calendrical texts	fixed (364 days)	—	c. 1st century^{[citation needed]}	Second Temple Judaism	Description of a division of the year into 364 days, also mentioned in the pseudepigraphical Book of Enoch (the “Enoch calendar“).
Gaulish calendar	lunisolar	Gauls/Celts (no longer in use)	Iron Age	Gauls/Celts	Early calendars used by Celtic peoples prior to the introduction of the Julian calendar, reconstruction mostly based on the Coligny calendar (2nd century), which may be partially influenced by the Julian calendar.
Zoroastrian calendar	fixed (365 days)	Iranian	3rd century	Sassanid Persia	Based on both the Old Persian and Seleucid (Hellenic) calendars. Introduced in AD 226, reformed in AD 272, and again several times in the 5th to 7th centuries.
Chinese Calendar, Dàmíng origin (大明曆)	lunisolar	Chinese	510	China	Created by Zu Chongzhi, most accurate calendar in the world at its invention
Japanese calendar	lunisolar	Chinese-derived	6th century	Japan	Umbrella term for calendars historically and currently used in Japan, in the 6th century derived from the Chinese calendar
Chinese Calendar, Wùyín origin(戊寅元曆)	lunisolar	Chinese	619	China	First Chinese calendar to use the true moon motion
Islamic calendar	lunar	Muslim	632	Islam	Based on the observational lunisolar calendars used in Pre-Islamic Arabia. Remains in use for religious purposes in the Islamic world.
Pyu calendar	lunisolar	Hindu/Buddhist-derived	640^{[dubious– discuss]}	mainland Southeast Asia	Traditional calendar of Southeast Asia, in use until the 19th century. Traditionally said to originate in 640 (the calendar era) in Sri Ksetra Kingdom, one of the Burmese Pyu city-states.
Nepal Sambat	lunar	Buddhist/ Hindu	9th century	Nepal	A lunar Buddhist calendar traditional to Nepal, recognition in Nepal in 2008.
Byzantine calendar	solar	Julian	988	Ecumenical Patriarchate of Constantinople	Julian calendar with Anno Mundi era in use c. 691 to 1728.
Armenian calendar	fixed (365 days)	Iranian	medieval^{[citation needed]}	medieval Armenia	Calendar used in medieval Armenia and as liturgical calendar of the Armenian Apostolic Church. Derived from the Zoroastrian (or related medieval Iranian calendars such as the Sogdian/Choresmian ones^[2]). It uses the era AD 552. In modern Armenian nationalism, an alternative era of 2492 BC is sometimes used.
Bulgar calendar	solar	Bulgarian	Bronze Age	Volga Bulgaria	A reconstruction based on a short 15th-century transcript in Church Slavonic called Nominalia of the Bulgarian Khans, which contains 10 pairs of calendar terms.
Florentine calendar	solar	Julian	medieval	Republic of Florence	Variant of the Julian calendar in use in medieval Florence
Pisan calendar	solar	Julian	medieval	Republic of Pisa	Variant of the Julian calendar in use in medieval Pisa
Tamil calendar	solar	Hindu	Ancient	Tamil Nadu	The Hindu calendar used in Tamil Nadu
Nepali calendar	solar	Hindu/ Buddhist	medieval^{[clarification needed]}	Nepal	One of the Hindu calendars
Bengali calendar	lunisolar	Bengali	medieval^{[clarification needed]}	Bengal	Revised in 1987.
Thai lunar calendar	lunisolar	Hindu/Buddhist^{[clarification needed]}	medieval^{[clarification needed]}	Thailand	A Buddhist calendar
Pawukon calendar	fixed (210 days)	Hindu	^{[citation needed]}	Bali
Old Icelandic calendar	solar		10th century	medieval Iceland	partly inspired by the Julian calendar and partly by older Germanic calendar traditions. Leap week calendar based on a year of 364 days.
Jalali calendar	solar	Iranian	1079	Seljuk Sultanate	A calendar reform commissioned by Sultan Jalal al-Din Malik Shah I
Hebrew calendar	lunisolar	Babylonian/Seleucid-derived	11th/12th century	Judaism	recorded by Maimonides in the Mishneh Torah, resulting from various reforms and traditions developing since Late Antiquity. The Anno Mundi era gradually replaced the Seleucid era in Rabbinical literature in the 11th century.
Tibetan calendar	lunisolar	Buddhist/Chinese-derived	13th century	Tibet	The Kalacakra, a Buddhist calendar introduced in 13th-century Tibet
Seasonal Instruction (授时曆)	solar	Chinese	1281	China	Based on a solar year of 365.2425 (equal to the Gregorian year)
Runic calendar	solar	Julian	13th century	Sweden	A written representation of the Metonic cycle used in medieval and early modern Sweden, allowing to calculate the dates of the full moons relative to the Julian date. The introduction of the Gregorian calendar in Sweden in 1753 rendered the runic calendars unusable.
Six Imperial Calendars (ß)	solar	Chinese	Ming dynasty	China	In use 1368-1644
Incan calendar	lunisolar	Pre-Columbian	15th century	Inca Empire
Muisca calendar	lunisolar	Pre-Columbian	15th century	Muisca	Complex lunisolar calendar with three different years, composed of months divided into thirty days. After the Spanish conquest of the Muisca Confederation in present-day central Colombia in 1537 first replaced by the European Julianand as of 1582 the Gregorian calendar.
Chula Sakarat	lunisolar	Burmese	16th century	Southeast Asia
Gregorian calendar	solar	Julian-derived	1582	worldwide	Introduced as a reform of the Julian calendar in the Roman Catholic church, since the 20th century in de facto use worldwide.
Javanese calendar	lunar	Islamic influenced	1633	Java	Based on the Hindu calendar using the Saka era (78 CE), but changed to the lunar year following the Islamic calendar.
Seasonal Constitution (时宪历)	solar	Chinese	1645	China	First Chinese Calendar to use the true motion of the sun.
Swedish calendar	solar	Julian-derived	1700	Sweden	Part of the controversy surrounding the adoption of the Gregorian calendar, in use 1700–1712.
Astronomical year numbering	solar	Julian-derived	1740	astronomy	A mixture of Julian and Gregorian calendar, giving dates before 1582 in the Julian calendar, and dates after 1582 in the Gregorian calendar, counting 1 BC as year zero, and negative year numbers for 2 BC and earlier.
French Republican Calendar	solar	Gregorian	1793	First French Republic	In use in revolutionary France 1793 to 1805.
Pancronometer	solar	Gregorian	1745	—	Universal Georgian Calendar proposed by Hugh Jones
Rumi calendar	solar	Julian	1839	Ottoman Empire	Julian calendar using the Hijri era introduced in the Ottoman Empire.
Positivist calendar	solar	Gregorian	1849	—	solar calendar with 13 months of 28 days.
Badí‘ calendar	solar	Baháʼí	1873	Baháʼí	Uses a year of 19 months of 19 days each and a 1844 era. Also known as the “Baháʼí Calendar” or the “Wondrous Calendar”.
Thai solar calendar	solar	Gregorian	1888	Thailand	The Gregorian calendar but using the Buddhist Era (543 BC)
Invariable Calendar	solar	Gregorian	1900	—	Gregorian calendar with four 91-day quarters of 13 weeks
International Fixed Calendar	solar	Gregorian	1902	—	A “perpetual calendar” with a year of 13 months of 28 days each.
Minguo calendar	solar	Gregorian	1912	Republic of China	Months and days use the Gregorian calendar, introduced in China in 1912.
Revised Julian calendar	solar	Julian-derived	1923	some Orthodox churches	currently synchronized with the Gregorian calendar, but different leap rule and cycle (900 years), also called Meletian calendar or Milanković calendar, after Serbian scientist Milutin Milanković who developed it.
Solar Hijri calendar	solar	Iranian/Islamic	1925	Iran, Afghanistan	New Year is the day of the astronomical vernal equinox. The calendar as introduced in 1925 revived Iranian month names but counted the years of the Hijri era. The era was changed in 1976 to 559 BC (reign of Cyrus the Great), but was reverted to the Hijri era after the Iranian Revolution.
Era Fascista	solar	Gregorian	1926	Italy	Epoch is 29 October 1922; in use from 1926–1943
Soviet calendar	solar	Gregorian	1929	Soviet Union	Gregorian calendar with 5- and 6-day weeks, used during 1929 to 1940.
World Calendar	solar	Gregorian	1930	—	Perpetual calendar with 1–2 off-week days, preferred and almost adopted by the United Nations in 1950s
Pax Calendar	solar	Gregorian	1930	—	Leap week calendar
Pataphysical calendar	solar	Gregorian	1949	—	Absurdist variant of the Gregorian calendar by Alfred Jarry.
Indian national calendar	solar	Gregorian-derived	1957	Republic of India	Gregorian calendar with months based in traditional Hindu calendars and numbering years based on the Saka era(AD 78).
Assyrian calendar	lunar	Babylonian	1950s	Assyrianism	Lunar calendar with an “Assyrian era” of 4750 BC, introduced in Assyrian nationalism in the 1950s
Discordian calendar	solar	Gregorian	1963	Discordianism	Calendar invented in the context of the absurdist or parody religion of Discordianism, Gregorian calendar variant with a year consisting of five 73-day seasons.
World Season Calendar	solar	Gregorian	1973	—	Divides the year into four seasons.
Dreamspell	lunar/solar galactic	Mayan	1990	esotericism	13 months of 28 days each, synchronized with the Maya 260-day Tzolkin, calibrated to the Chilam Balam timing systems
Tranquility Calendar	solar	Gregorian	1989	—	Modification of the International Fixed Calendar
Holocene calendar	solar	Gregorian	1993	—	The Gregorian calendar with the era shifted by 10,000 years.
Juche era calendar	solar	Gregorian	1997	North Korea	Gregorian calendar with the era 1912 (birth of Kim Il-sung)
Nanakshahi calendar	solar	Gregorian-derived	1998	Sikhism	Gregorian calendar with months based in traditional Hindu calendars and numbering years based on the era 1469.
Symmetry454	solar	Gregorian	2004	—	Leap week calendar with 4:5:4 weeks per month
Hanke-Henry Permanent Calendar	solar	Gregorian	2004	—	Leap week calendar with 30:30:31 days per month, revised in 2011 and 2016
Igbo calendar	lunar	Indigenous West African	2009	Igbo people	Proposal^[3] based in Igbo tradition dating back to 13th century, 13 lunar months of 28 days divided into seven 4-day periods, plus leap days.

Tradition	culture	comments
Germanic calendar	Germanic	Medieval records of Germanic names of lunar months later equated with the Julian months.
Berber calendar	Berber	reconstructed medieval Berber-language names of the Julian months used in pre-Islamic (Roman era) North Africa
Lithuanian calendar	Lithuania	Lithuanian names for the Gregorian months and days of the week, officially recognized in 1918.
Rapa Nui calendar	Easter Islands	Thirteen names of lunar months recorded in the 19th century.
Xhosa calendar	Xhosa people	^{[clarification needed]}
Turkmen	Turkmenistan	Turkmen names officially adopted in 2002 following Ruhnama by president-for-life Saparmurat Niyazov.
Hellenic calendars	Hellenistic Greece	A great variety of regional month names in Ancient Greece, mostly attested in the 2nd century BC.
Slavic calendar	Slavic	Local month names in various Slavic countries, based on weather patterns and conditions, and agricultural activities that take place in each respective month.
Romanian calendar	Romania	Traditional names for the twelve months of the Gregorian calendar, which are usually used by the Romanian Orthodox Church.

Tradition	week length	comments
Bali	various
Korea	5 days	^{[citation needed]}
Java	5 days	^{[citation needed]}
Discordian	5 days
Akan	6 days	A traditional “six-day week” which combined with the Gregorian seven-day week gave rise to a 42-day cycle.
Ancient Rome	8 days	The Roman nundinal cycle.
Burmese	8 days
Celtic	8 days	reconstructed.^[4]^[5]
Baltic	9 days	Linguistic reconstruction^{[citation needed]}; the Gediminas Sceptre indicated that a week lasted for nine days during King Gediminas’ reign.
Chinese	10 days
Egyptian Calendar	10 days	The 10-day period was known as decans or decades
French Republican Calendar	10 days
Aztecs	13 days	Trecena, division of the Tonalpohualli 260-day period

r u clocked up?

Colonial era

Metonic cycle

Contents

Application in traditional calendars

Mathematical basis

Further details

See also

Callippic cycle

List of calendars

Contents

List of calendars

Variant month names

Non-standard weeks

Calendaring and timekeeping standards

Non-Earth or fictional

See also

List of astrological traditions, types, and systems

Contents

Current traditions

Historical traditions

Recent Western developments

Esoteric systems of astrology

See also