AboutCalendars.txt

---------------
BUILDING BLOCKS
---------------

All timekeeping is, when you get right down to it, applied astronomy.

Prehistoric humans noticed alternating periods of light and darkness, and gave them names meaning "day" and "night" respectively. In Sumer and China, the day was considered to end at sunset; in Egypt, Iran, and Mexico, the day began at sunrise. This subtle difference in consideration became important when humans gathered into societies and record-keeping became important. When this happened, it made sense to consider a day and a night as a single unit, which was also called a day. As a result, in Sumer, a calendar day lasted from sunset to sunset, while in Egypt, it lasted from sunrise to sunrise. The Romans instead defined the day at starting at midnight, when the sun is at its nadir; this has the useful effect that one goes to bed on one day and wakes up the next day[0][16].

Early subdivisions of the day reflect the early concept of day and night as fundamentally different. Sumerians and Egyptians both divided the period of sunset to sunrise into 12 equal parts, and also divided the period from sunrise to sunset into 12 parts, identified by the rising of particular stars [1][2]; this is the basis for the modern hour. As you can imagine, the length of the daytime and nighttime hours were usually unequal, and the lengths of the respective hours would change across the course of Earth's orbit around the sun. This changed with the invention of mechanical clocks in the late Middle Ages; because a mechanical clock cannot compensate for change in the length of daylight and nighttime, an hour was now fixed as 1/24 of a day/night cycle [3][4]. In an echo of having seperate sets of 12 hours each for daytime and nighttime, it is common in everyday speech to refer to the 12 hours between midnight and noon as a.m. (ante meridian, Latin for before noon), while those between noon and midnight are called p.m. (post meridian, after noon), with a.m. and p.m. hours each running from 1 to 12. In contexts where clarity is important, it is more common to refer to hours by a 24-hour scheme, running from starting at 00 at midnight, continuing on through 12 at noon until 24 is reached, at which point the count resets to 00.

Accurate timekeeping entails subdividing the hour. Semitic peoples divide the hour in 1080 chalakim (singular: chelek), which is important when calculating the exact start of the month [5]. The Sumerian preference for the number 60 led to Indo-European peoples dividing the hour into 60 minutes; as mechanical clocks became more accurate, the minute itself was divided into 60 seconds [3].

Even defining a day throws up some unexpected complications. In ancient times, people quite reasonably defined the length of a day as the time between successive noons or midnights; in modern times, we know this is the time over which Earth returns to the same orientation relative to the sun. This averages to 24 hours, or 86400 seconds. However, due to orbital mechanics, the actual length of the solar day can vary by up to 29 seconds [6]. Astronomers commonly use the sidereal day, the time taken for Earth to rotate once on its axis, as measured relative to the fixed stars. This time lasts 86164.1 seconds, or 23 hours 56 minutes 4.1 seconds. [7]

But days add up quickly, and in a time when most people can't count beyond 100, some scheme to group days together is necessary. The phases of the moon provide a handy guide, as the moon passes through its phases at a regular interval (but not quite regular; it decreases by about 25 microseconds each time) [5]. This period was known as a month, and in Sumer and China, each month began on the first sighting of the waxing crescent; in modern times, this is widely replaced with the more precise astronomical definition of the moment when the centre of the moon crosses the ecliptic longitude of the centre of the sun. The Akkadians (Assyrians and Chaldaeans) adopted the Sumerian practice, and transmitted it to the Hebrews and Arabs [8][9][10][11]; the veneration of Chinese culture in East Asia meant the system was also adopted in Cambodia, Thailand, Laos, Vietnam, Korea, and Japan.

But as with days, defining a month can be tricky. The period between solar conjunctions is referred to as the synodic month, which averages to 29 days 12 hours 44 minutes 2.8 seconds. There is, of course, a sidereal month, the time taken for the moon to return to the same position relative to the fixed stars; this lasts 27 days 7 hours 43 minutes 11.6 seconds, the discrepancy with the synodic month due to the fact that Earth is moving around the sun and thus from the perspective of someone on Earth, the moon needs a little over 2 more days to catch up to the sun. The tropical month is the time between ascendant crossings of the ecliptic, which comes out to 27 days 7 hours 43 minutes 4.7 seconds due to orbital wobble. The draconic month is the time between crossings of the point where the plane of the moon's orbit intersects the celestial equator, which is 27 days 5 hours 5 minutes 35.8 seconds. Finally, there is the anomalistic month, which is the time between perigees; it is 27 days 13 hours 18 minutes 33.2 seconds long. [15]

The other grouping of days is the time taken for Earth to orbit the sun, known as a year. Sumerians, Egyptians, Chinese, and Mayans figured out the length of the year by counting the number of days between vernal equinoxes and between summer solstices, which comes out to 365 days. In Sumer and China, it was noticed that this was 11 days longer than 12 full months, and so a standard year was defined as a cycle of 12 months. However, this would result in the months slipping out of sync with the equinoxes, and so some years would have 13 months to catch up [11]. In Egypt, people held the sun in greater reverence than the moon, and so the civil year was simply defined as 365 days, with no attempt to keep months in sync with the moon.

As ever, defining a year can be tricky. The sidereal year is the time it takes for Earth to orbit the sun once, relative to the fixed stars; this lasts 365 days 6 hours 9 minutes 9.76 seconds [14]. There are two possible meanings of tropical year. The more common , also called the equinoctial year, is the time between two moments of northward equinox, the moment when the centre of the north ascendant sun crosses the point where the celestial equator intersects the ecliptic. Because Earth's axis is preceding slowly, this period lastsis 365 days 5 hours 48 minutes 45 seconds [12]. Alternatively, there is the time taken for the sun to return to the same position in the sky, which is 365 days 5 hours 49 minutes 1 second [13].

[0]  https://en.wikipedia.org/wiki/History_of_calendars
[1]  https://en.wikipedia.org/wiki/Decan
[2]  https://www.nist.gov/pml/time-and-frequency-division/popular-links/walk-through-time/walk-through-time-early-clocks
[3]  http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=191
[4]  https://www.nist.gov/pml/time-and-frequency-division/popular-links/walk-through-time/walk-through-time-revolution
[5]  http://individual.utoronto.ca/kalendis/hebrew/drift.htm
[6]  https://en.wikipedia.org/wiki/Solar_time
[7]  https://earthsky.org/astronomy-essentials/what-is-sidereal-time
[8]  http://world.clndr.org/history/ancient-sumerian
[9]  https://www.livescience.com/44964-why-60-minutes-in-an-hour.html
[10] https://www.ancient-origins.net/history/sumerians-looked-heavens-they-invented-system-time-and-we-still-use-it-today-007341
[11] https://www.livius.org/articles/concept/calendar-babylonian/
[12] https://web.archive.org/web/20190119121207/http://asa.usno.navy.mil/SecM/Glossary.html
[13] http://aramis.obspm.fr/~heydari/divers/ir-cal-eng.html
[14] http://hpiers.obspm.fr/eop-pc/models/constants.html
[15] https://en.wikipedia.org/wiki/Lunar_month
[16] http://www.beaglesoft.com/timehistoryroman.htm

------------------------
CHRONOLOGICAL JULIAN DAY
------------------------

The Julian day (not to be confused with the Julian calendar) is a system which assigns a unique number to each date. Strictly speaking, each Julian day begins at noon UTC, because they scheme was originally developed by and for astronomers. Specifically, Ludwig Ideler and Sir John Herschel simplified the Julian Period to just a count of days with the same epoch, in which day 0 well before any recorded history (1 January 4713 BC by the Julian calendar; 24 November 4714 BC by the Gregorian calendar; 3 Azar 5335 BH by the Solar Hijri calendar). Astronomers and computer programmers find Julian days useful due to often having to do maths with dates, and most every calendar converter you will find (including this one) uses Julian days as an intermediary to convert between other calendars.

This converter uses the consecutive Julian days rather than the standard version. Each consecutive Julian day starts at midnight rather than noon; as such, day 0 in the canonical Julian day scheme is day (-1) in the consecutive Julian day scheme.

https://en.wikipedia.org/wiki/Julian_day
https://www.hermetic.ch/cal_stud/jdn.htm
https://www.aa.quae.nl/en/reken/juliaansedag.html

---------------
JULIAN CALENDAR
---------------

The Julian calendar was designed by Sosigenes of Alexandria, building on the work of Eudoxus of Cnidus, and introduced as Julius Caesar in 44 BC as a replacement for the old Roman calendar. This was necessary because the Roman calendar was notoriously inaccurate, irregular, and subject to political manipulation to the point that when Caesar became dictator, nobody could agree on what the date actually was. To this end, Caesar declared that the year he took power would last 445 days, which brought the calendar back in sync with the solstices and equinoxes, and the Julian calendar began the next year.

The Julian calendar consists of 12 months of 30 or 31 days each, except for February, which has 28 days in normal years and 29 in leap years. Leap years occur every 4th year, without exception. The average year is thus 365 days 6 hours long, which is 11 minutes 15 seconds longer than the mean tropical year; as a consequence, the calendar loses 1 day every 128 years. Days begin at midnight. It is a completely solar calendar that makes no effort to maintain synchronisation with the phases of the moons.

The modern calendar era is the nominal birth year of Jesus Christ (though hisotrians and Bible scholars ge\
nerally agree that Jesus was more likely born four years earlier), placing the epoch at Julian Day 1721423.

Today, it has been almost completely supplanted by the Gregorian calendar, though Orthodox churches continue to use it to determine the dates of religious festivals and observances.

The Anno Domini dating system simply means that each year AD refers to the nth year of Jesus Christ, while each year BC is the nth year before Jesus was born. As such, both are ordinal rather than cardinal, and there is no year 0.

https://en.wikipedia.org/wiki/Julian_calendar

------------------
GREGORIAN CALENDAR
------------------

The Gregorian calendar was designed by Aloysius Lilius and introduced in 1582 AD by Pope Gregory XIII in 1582 AD. As early as the 8th century AD, the innacuracies of the Julian calendar were known, and by the 16th century AD, the drift of Easter relative to the equinoxes had become unacceptably severe. Modified the Julian calendar such that years that are integer multiples of 100 would NOT be leap years, except for those that are also integer multiples of 400. The overall result is a cycle of 400 years in which 97 are leap years. The average year length is thus 365 days, 5 hours, 49 minutes, and 12 seconds, which is just 27 seconds longer than the mean equinoctial year. This means that the Gregorian calendar accumulates loses just 1 day in 3200 years. When the second-to-second effects of gravity and Earth's variable speed are taken into account, it may only lose one day in 7700 years, by which point the slowing of Earth's rotation will become an issue.

Lilius and Gregory calibrated the calendar so that the solstices and equinoxes of 1582 would match those of the First Council of Nicaea in 325. Because the Julian calendar had alrady lost two days by then, the epoch of the Gregorian calendar is Julian Day 1721425, two days after the start of the Julian calendar. Days begin at midnight. It is a purely solar calendar which makes no effort to maintain synchronisation with the phases of the moon.

It is sometimes said that the Gregorian calendar cannot truly be used for events before 1582, when the calendar was introduced. This is false, because a calendar is simply a scheme to assign every day a unique identifier. As such, it is perfectly possible to extend the calendar back to before it was introduced. This converter uses the standard practice of no year 0, so 1 BC is followed by 1 AD.

When the Gregorian calendar was introduced, the day after 4 October became 15 October to correct for calendar drift. The new calendar was adopted readily in Catholic countries, but Orthodox and Protestant countries at first refused, not trusting anything from the Vatican. Britain famously did not adopt it until 1752, at which point they had to skip 11 rather than jsut 10 days, and it remained customary for decades after to refer to dates in both the Julian and Gregorian calendars.

Sweden formally adopted the Gregorian calendar in 1700, but the king decided to gradually transition by leaving out leap years from 1700 to 1740. However, due to poor management, leap years were accidentally inserted in 1704 and 1708, and then Sweden got into a war with Russia, so the plan was abandoned and Sweden reverted to the Julian calendar, with the result that February 1712 had 30 days. The Gregorian calendar was finally properly introduced in Swden in 1753, with a single 11-day jump.

https://en.wikipedia.org/wiki/Gregorian_calendar
https://en.wikipedia.org/wiki/Adoption_of_the_Gregorian_calendar
https://courses.lumenlearning.com/astronomy/chapter/the-calendar/
https://scienceworld.wolfram.com/astronomy/GregorianCalendar.html
https://www.fourmilab.ch/documents/calendar/IG_Latin.html

---------------
COPTIC CALENDAR
---------------

The Coptic calendar is a descendent of the Ancient Egyptian calendar, which incorporates the Julian calendar's rule of a leap year every four years. Days begin at midnight. The epoch is Julian Day 1825028, which corresponds to 29 August 284 by the Julian calendar, the Year of Martyrs or the Diocletian era, named for the severe persecutions of Christians begun by Emperor Diocletian.

The calendar consists of 12 months of 30 days plus five extra days (six in leap years). Days begin at midnight. It is a purely solar calendar that makes no effort to maintain synchronisation with the phases of the moon. The calendar is not used for civil purposes today, being only a liturgical calendar for the Coptic Orthodox Church.

https://en.wikipedia.org/wiki/Coptic_calendar
http://www.copticchurch.net/easter.html
https://web.archive.org/web/20060928041231/http://www.copticheritage.org/parameters/copticheritage/calendar/The_Coptic_Calendar.pdf

------------------
ETHIOPIAN CALENDAR
------------------

The calendar used in Ethiopia and Eritrea is also a descendent of the ancient Egpytian calendar, and like the Julian calendar, it includes a leap year every four years without exception; as in the Coptic calendar, leap day is added to the extra days at the end of the year.

The Ethiopian calendar, like the Gregorian, is dated from the birth of Jesus Christ. However, it uses a later estimate of his birth, and so the calendar epoch is 1724222, corresponding to 29 August 8 AD by the Gregorian calendar. Days begin at midnight. It is an entirely solar calendar, and makes no effort to maintain synchronisation of the months with the phases of the moon.

https://en.wikipedia.org/wiki/Ethiopian_calendar
https://www.timeanddate.com/calendar/ethiopia-calendar.html
https://theculturetrip.com/africa/ethiopia/articles/why-is-the-ethiopian-calendar-7-years-behind/

-------------------------------
ANCIENT EGYPTIAN CIVIL CALENDAR
-------------------------------

The Egyptians developed the first known completely solar calendar. While there is evidence that the very early Egyptian civilisation had a lunar calendar, the civil calendar never attempted to maintain synchronisation with the phases of the moon.

From the earliest times, Egyptian civilisation has been dependent on the regular flooding of the Nile. It was noticed in antiquity that the Nile would always flood right after Sirius returned to the sky for its heliacal rising, and so this even was celebrated as the start of the year. While modern Egpytians follow Islam an Christianity, the date is still celebrated.

By simply counting the days between the heliacal risings of Sirius, the Egyptians measured the year as 365 days, and a civil calendar of this length was introduced by Pharaoh Shepsekaf consisting of three seasons of four months, each with 30 days, plus five extra days consecrated to the children of Nuit, the goddess of the sky, and Geb, the earth god. Given the prominence of the sun and the year marker occurring just before sunrise, the day begins at sunrise; as such, any Egyptian date on this calendar should actually refer to the previous day if the time falls between midnight and sunrise.

The mythology holds that the year was originally 360 days long, but when Nuit became pregnant, she was cursed by Ra, the sun god to never give birth in any month. Thoth, the god of wisdom and thus under whose domain fall legal loopholes, was sympathetic to Nuit and gambled with Khonsu, the god of the moon, to win enough light to create five extra days. Because these five days were not part of any month, Nuit was free to give birth to Osiris, Isis, Seth, Nephthys, and Horus the Elder. The Rosetta Stone records a decree by Pharaoh Ptolemy III to introduce a leap year mechanism which would fix New Year's Day to the heliacal rising of Sirius; however, this was opposed by the priesthood on the grounds that only gods can create new days.

The Egyptians did not have a continuous count of years, instead describing each year by how long the current pharaoh had been in power. For the era, I went with 4724 BC based on an article in Egpyt Today. the veracity of Egypt Today is questionable at best, but it's at last something. It is known that Egyptian New Year in 632 AD fell on 16 July (Julian calendar), and so I chose Julian Day 160550 as the epoch to make the dates line up.

https://en.wikipedia.org/wiki/Egyptian_calendar
http://www.copticchurch.net/easter.html
https://web.archive.org/web/20060928041231/http://www.copticheritage.org/parameters/copticheritage/calendar/The_Coptic_Calen\
dar.pdf
https://www.egypttoday.com/Article/4/22184/September-11-marks-the-beginning-of-a-new-Egyptian-year
https://www.iranchamber.com/calendar/articles/old_iranian_calendars1.php

-----------------
ARMENIAN CALENDAR
-----------------

The Old Armenian calendar is derived from the Zoroastrian calendar, whcih in turn is another derivative of the ancient Egyptian calendar. It was in use in Armenia until the 11th century AD, when it was replaced by the Julian calendar.

Folklore relates how it was established by Hayk, the legendary founder of Armenia, to commemorate his victory of General Belos of Babylon; consensus among historians is that it was introduced by king Artaxias I (188 BC - 161 BC). The calendar epoch is 1922867 (Gregorian: 13 May 552 AD; Solar Hijri: 22 Tir 70 BH), when the Great Armenian Era was declared.

https://en.wikipedia.org/wiki/Armenian_calendar
http://haytomar.com/calendar.php
http://www.rahamasha.net/uploads/2/3/2/8/2328777/armenian_calendar.pdf
http://www.armeniapedia.org/wiki/Armenian_calendar

------------------------
TABULAR ISLAMIC CALENDAR
------------------------

The tabular Islamic calendar is a version of the Solar Hijri calendar based entirely on arithmetical calculations of rational numbers. The year consists of 12 months of alternating 30 and 29 days. In addition, an extra day is added to Dhu al-Hijjah according to a 30 year leap cycle in which 2, 5, 7, 10, 13, 16, 18, 21, 24, 26 and 29 are leap years. Until 1441 AH, an alternative algorithmic calendar, the Umm al-Qura calendar, was the official civil calendar of Saudi Arabia; however, I have not been able to find any good descriptions of the algorithms used for that one. In 1437, the Arabian government switched to a homebrew solar calendar for purposes of paying civil servants, and since 1439 have used the Gregorian calendar.

The epoch of this calendar is Julian Day 1948439, the first day of Muharram in the year that the prophet Muhammad and his acolytes moved from Mecca to Medina and established the first ummah.

https://en.wikipedia.org/wiki/Islamic_calendar
https://en.wikipedia.org/wiki/Tabular_Islamic_calendar
http://www.webexhibits.org/calendars/calendar-islamic.html

---------------
JALALI CALENDAR
---------------

The Jalali calendar is a purely solar calendar designed by Omar Khayyam in 457 AH (1079 AD) at the behest of Grand Vizier Abu Ali Hasan ibn Ali Tusi of the Seljuq Empire, and promulgated by Sultan Malik Shah I (born Jalāl al-Dawla), after whom it is named. This calendar operates on a 33-year cycle in which the fifth year and every fourth year there after is a leap year, leap day being added to Esfand. The average year in this calendar lasts 365 days 5 hours 49 minutes 5.5 seconds, which is 20.35 seconds longer than the mean equinoctial year. As such, the algorithmic calendar accumulates 1 day of error every 4247 years. New Year's Day is timed to fall on the day of the northward equinox.

Khayyam's original formulation was entirely observational, each month beginning with the transition of the sun into the relevent portion of the Zodiac; as such, every month could have between 29 and 32 days depending on the state of every particle in the universe. Forward planning thus required a carefully-determined ephemeris to be worked out in advance; for simplicity, this was abandoned in favour of the purely algorithmic calendar used here.

The Jalali calendar was set aside around 638 AH (1259 AD) and replaced with a duodecimal system in which each year as assigned an animal based on the Chinese zodiac, introduced by Mongol conquerors. This remained the official calendar of Iran until 1911, when the Jalali calendar was restored.

http://aramis.obspm.fr/~heydari/divers/ir-cal-eng.html
https://en.wikipedia.org/wiki/Jalali_calendar
http://www.iranicaonline.org/articles/calendars

--------------------
SOLAR HIJRI CALENDAR
--------------------

The Solar Hijri calendar is the civil calendar used in Iran and Afghanistan. As its name suggests, it is a solar calendar that makes no effort to maintain consistency with the phases of the moon. As with the Lunar Hijri calendar, its era is the flight of the prophet Muhammad from Mecca to Medina. Days begin at midnight.

The modern Solar Hijri calendar is observational, with Nowruz falling the instant the centre of the north ascendant sun crosses the celestial equator; New Year's Day is then the day whose midnight falls closes to the moment of vernal equinox as observed in Tehran. 

http://aramis.obspm.fr/~heydari/divers/ir-cal-eng.html

------------------------
AHMAD BIRASHK'S CALENDAR
------------------------

Ahmad Birashk came up with a new calendar in 1993, building on the work of Zabih Beruz, published in 1952. Birashk proposed a baroque algorithmic calendar which he claimed was more accurate; it does better match the tropical year, but Khayyam's calendar is still a better match for the equinoctial year.

Birashk's calendar is based on the same 33-year leap year cycle as Khayyam's, but also includes a 29-year cycle, in which the last four years are omitted, and a 37-year cycle, which adds an additional four years with a leap year in year 37. These are then arranged in 21 cycles of one 29-year cycle and three 33-year cycles (128) years, followed by a cycle of one 29-year cycle, two 33-year cycles, and one 37-year cycle, for a total cycle length of a dizzying 2820 years. This results in an average year length of 365 days 5 hours 48 minutes 45.952 seconds, just 0.023 second shorter than the mean tropical day; as such, it gains just one day in 3.8 million years, by which point the accuracy will have been cancelled out by Earth's slowing rotation.

http://aramis.obspm.fr/~heydari/divers/ir-cal-eng.html

------------------------
MODERN ASSYRIAN CALENDAR
------------------------

The Assyrian calendar included in this converter is not the one used by the ancient Assyrian empires. It is an entirely solar calendar used by the modern Assyrian people, a Christian minority mostly living in Syria. Days begin at midnight, and New Year's Day falls on 1 April by the Gregorian calendar. It consists of six months of 31 days followed by five months of 30 days and a final month of 29 days, which has 30 days in leap years. Days begin at midnight.

The epoch is Julian Day (-13387), which corresponds to 1 April 4750 BC and 12 Farvardin 5371 BH. In Assyrian mythology, this is considered to be the year of the Great Flood, the story on which the Biblical Flood narrative was based. References to this calendar tend to get caught up in Assyrian nationalism, so be skeptical when reading the sources.

http://www.nineveh.com/Assyrian%20Calendar.html
https://en.wikipedia.org/wiki/Assyrian_calendar
https://web.archive.org/web/20100728001440/http://www.assyriatimes.com/engine/modules/news/article.php?storyid=3410

-----------------------
OLD BABYLONIAN CALENDAR
-----------------------

The Babylonians inherited their calendar from the Sumerians. It is a lunisolar calendar, which attempts to simultaneously keep the months synchronised with the phases of the moon while also ensuring that New Year's Day falls around the vernal equinox. In the earliest days, it was a purely observational calendar; each month began with the first sighting of the waxing crescent moon, and an intercalary month was inserted whenever the priests thought it was about the right time. As such, days begin at sunset; because this converter midnight-indexes all days for the sake of simplicity, each date is a day early if the time is between sunset and midnight.

The lengths of the Babylonian months were not fixed; instead, each month referred to the duration of the actual lunar cycle. Civil days begin at sunset, and the month begins with the first sighting of the crescent moon at sunset. Getthing the exact time involves going down a rabbit hole of celestial mechanics and atmospheric refraction, so I have simply declared that the first crescent is visible the day after the New Moon; I have furthermore simplified sunset as always happening at 18:00, which is reasonably accurate around the northward equinox.

The Chaldean Nation website claims that 2019 AD corresponds to 7319 PD in the Babylonian calendar. Even more than Egypt Today and my sources for the Assyrian calendar, Chaldean Nation is suspect, and there is a lot of Chaldean nationalism going on, but it's the only source I could find for an era.

https://chaldeannation.com/blog/2019/03/31/akitu-7319-chaldean-new-year/
https://en.wikipedia.org/wiki/Babylonian_calendar
https://www.friesian.com/calendar.htm

--------------------------
FIXED BABYLONIAN CALENDAR
--------------------------

When Nabû-nāṣir, also called Nabonasser, came to power in Babylon in 747 BC, he implemented a policy of meticulous astronomical and chronological record keeping. As a result, by 424 BC, Babylonian astronomers had worked out that 235 lunar months was almost exactly equal to 19 solar years, and thus concluded that seven intercalary months should be added to each cycle on 19 years. This pattern, known as the Metonic Cycle after Meton of Athens (who learned about it from the Babylonians) is used today in the Jewish calendar.

Days begin at sunset, and the month begins with the first sighting of the waxing crescent moon at sunset; as with the Old Babylonian Calendar, I amm assuming that the sun sets at 18:00 each day and that the first crescent is visible the day after teh New Moon. The Metonic Cycle as used by the Babylonians is indexed to 747 BC, the year of Nabû-nāṣir's accession; as such, this date is taken as the calendar era, as was done by Greek astronomers working with Babylonian records.

theos-sphragis.info/hebrew_babylonian_intercalation.html
https://en.wikipedia.org/wiki/Babylonian_calendar
https://www.livius.org/articles/concept/calendar-babylonian/

----------------------
JEWISH HEBREW CALENDAR
----------------------

And now we come to the Jewish Hebrew calendar, the most fiendishly baroque calendar currently in use. This is the liturgical calendar used by Jews all over the world and it also one of two official civil calendars used in Israel, the other being the Gregorian calendar. Looking around the Internet, I get the impression that most Israelis use the Gregorian calendar almost exclusively, only checking the Jewish calendar to see when certain religious festivals fall. All national holidays in Israel are fixed to the Jewish calendar and government correspondence is supposed to be issued in both date formats, though until computers became widely available it was common to ignore the Hebrew date.

The year begins on the first of Tishrei, around the southward equinox, with the festival of Rosh Hashanah. Confusingly, Tishrei is ususally listed as the seventh month, with Nisan as the first; this is most likely due to Babylonian influence, since all the month names are derived from those of the Babylonian calendar. The 1st of Nisan is also considered a New Year's Day, most likely due again to Babylonian influence; this day is considered the ecclesiastical new year, as opposed to 1 Tishrei, the civil new year (and the day on which the year increments).

To confuse matters further, the 1st of Elul is the start of the year when calculating the payment of tithes, and the 15th of Shevat is identified as the New Year of Trees (Tu BiShvat). In ancient times, this had to do with calculating when it was acceptable to eat fruit from a given tree, and when that fruit could be tithed; in modern times, it is an environmental awareness day, often marked by planting trees. To keep things simple, this converter only considers the civil year, and so Tishrei is considered the first month of the year.

The Jewish calendar is lunisolar, like the Babylonian. In ancient times, months began with the first observation of the new moon, known as the molad. In the late 41st century AM, Hillel II ben Jehudah worked out a purely mathematical calendar that would maintain alignment with the phases of the moon. This gradually caught on among the scattered Jews as a way to maintain their traditions even in the face of persecution; indeed, Jews even managed to make calendars and carry on their traditions on the correct dates while in concentration camps.

The calendar epoch is Julian Day 347997 (Gregorian: 7 September 3761 BC; Solar Hijri: 16 Shahrivar 4382 BH), which Moses ben Maimon (AKA Maimonides, Rambam) in 4938 AM (1178 AD, 557 AH) determined to be the date of the Genesis creation account... sort of. Adam is traditionally held to have been created on Rosh Hashanah, but because there were five days of creation preceding Adam, this was technically Rosh Hashanah of the second year. Maimonides, assuming that time before Creation was still meaningful, extended this backwards to the molad of Tishrei that would have existed had God gotten around to creating the moon yet; because this occurs in before Creation, it is termed the molad tohu, or new moon of chaos (which is also a good name for a heavy metal band).

The basic Jewish year consists of 12 months of alternating 30 and 29 days. In a leap year, Adar is renamed to Veadar, Adar Sheini, Adar Beth, or Adar II, and a 30-day month, called Adar I, Adar Aleph, Adar Rishon, or simply Adar, is inserted before it. Thus a year by default has 354 days in a normal year and 384 in a leap year, but in Judaism, things are rarely simple.

Days begin at sunset, so a given date in this converter actually refers to the next day if viewed between sunset and midnight; for purposes of calculation, we can fortunately assume that every sunset occurs at exactly 18:00. An important unit of time when calculating the start of the month is the chelek (plural chalakim), equal to 1/1080 hours, 1/18 minutes, or 10/3 seconds.

Hillel II determined the mean lunation to be 29 days 12 hours 793 chalakim. This was correct at the time, but we now know that due to orbital mechanics, the mean lunation time is shrinking by about 25 microseconds a month, and it is now about 0.6 seconds shorter than it was when this was first worked out. Still, this difference has not yet become significant.

To prevent Yom Kippur from falling on a Friday or Sunday, and to keep Hoshanah Rabbah from falling on a Saturday, while keeping the length of the year more or less consistent, Rosh Hashanah will often be postponed by a day or two. This happens if the molad of Tishrei falls on a "Tuesday" after 03:11 and 20 seconds am when it's not a leap year, or on a "Monday" in the year after a leap year after 09:32 and 43 seconds am, or in any year if the molad of Tishrei falls after noon. After accounting for all that, if Rosh Hashanah would fall on a "Sunday", "Wednesday", or "Friday", it falls the next day. To ensure the correct form of the calendar is used, Rosh Hashanah postponements must be worked out at least a year in advance.

The upshot of all that is that a year may be a day shorter than the basic year, in which case it is termed deficient and a day is removed from Marcheshvan; it may also be a day longer, in which case it is called abundant and a day is added to Kislev. It also means that the exact cycle of year types will only repeat every 689,472 years!

Having to maintain this calendar shows just how protective Jews are of their culture and traditions. Karaite Jews in Israel have reverted to the ancient practice of going outside and looking at the moon, and they also add a leap month based not on calculations but on the ripening of barley.

This calendar only loses 1 day in 13800 years against the moon, but is less accurate with regard to the sun, gaining 1 day every 216 years. Irv Bromberg, a professor of medicine at the University of Toronto and calendar enthusiast, has proposed a revised (or rectified as he calls it) Hebrew calendar, based on a 353-year leap year cycle and incorporating a progressively shorter molad interval; unfortunately, I was unable to accurately translate his scheme or his algorithms into Python. Perhaps someone smarter than me can give it a try.

http://www.webexhibits.org/calendars/calendar-jewish.html
https://www.individual.utoronto.ca/kalendis/hebrew/molad.htm
https://www.individual.utoronto.ca/kalendis/hebrew/drift.htm
https://www.individual.utoronto.ca/kalendis/lunar/index.htm
https://www.individual.utoronto.ca/kalendis/hebrew/postpone.htm
https://www.individual.utoronto.ca/kalendis/hebrew/rect.htm
https://www.tondering.dk/claus/cal/julperiod.php/hebrew.php
https://www.tondering.dk/claus/cal/hebrew.php
https://en.wikipedia.org/wiki/Hebrew_calendar
http://www.jewfaq.org/calendar.htm
http://www.jewfaq.org/calendr2.htm

-------------------------
SAMARITAN HEBREW CALENDAR
-------------------------

The Samaritans are a small Israeli community numbering 818 as of 4 May 2020. They mostly live around Mount Gezirim and Holon. They consider themselves to uphold the true religion of Moses, and modern Jews to be heretics. They have their own calendar, which is similar but not identical to the Jewish calendar; based on The Samaritan Update, they also appear to mostly use the Gregorian calendar in day to day life. The exact nature of the calendar and how it is calculated is a secret known only to the high priest and his family. At some point in the late 20th century, high priest Avraham ben Pinchas developed a computer algorithm to calculate the calendar six months in advance, using a scheme which the high priest verifies every six month by astronomical observations. Samaritan sources claim this calendar is accurate to one second in one million years; unfortunately, the algorithm and its rules are a secret, and so this converter can only estimate them. However, some information can be gleaned.

The Samaritan calendar is based on the mythological date of the entry of the Israelites under Joshua into Canaan. Samaritan year 3658 AY corresponds to 2020 AD and 5780 AM. The Samaritans use the same 19-year leap year cycle as the Jews, but starting approximately 2000 years later, which results in the two calendars' leap year schedules being out of phase. The Samaritans celebrate Rosh Hashanah on 1 Nisan rather than 1 Tishrei, likely influenced by the Assyrian occupation of Israel. They also do not apply any postponement rules; if Yom Kippur should fall on a Friday or Sunday, they just plough through two consecutive days of Sabbath restrictions. They also do not appear to use willow branches in their version of Sukkoth ceremonies, obviating the need to preent Rosh Hashanah from falling on a Saturday.

https://www.israelite-samaritans.com/history/
https://www.israelite-samaritans.com/religion/passover/
https://www.israelite-samaritans.com/religion/sukkah/
https://www.israelite-samaritans.com/religion/sukkah/
https://www.israelite-samaritans.com/religion/calendar/
http://shomron0.tripod.com/articles/samaritancalendar.pdf
http://thesamaritanupdate.com/
https://www.the-samaritans.net/
https://www.the-samaritans.net/the-festival-of-sukkoth-tabernacles/

----------------
KURDISH CALENDAR
----------------

The Kurdish calendar is, naturally, the calendar of the Kurds. It is commonly used by Iraqi and Iranian Kurds, in Iraq due to their status as a protected minority and in Iran because the Kurdish calendar is functionally the same as the Iranian calendar, but its reference era is the Battle of Nineveh in 1233 BH; as such, the calendar epoch is Julian Day 1497976. The Battle of Nineveh marked the end of Assyrian dominance of the Middle East and the ascendance of the Median Empire, from which the Kurds are descended. The calendar is little used in Turkey or Syria due to its association with Kurdish nationalism.

https://en.wikipedia.org/wiki/Kurdish_calendar
https://web.archive.org/web/20080222053000/http://www.kurdistanica.com/english/culture/ncharacters/calendar/calendar.html

---------------
TRUE JULIAN DAY
---------------

The true Julian day is the Julian day used by astronomers. Unlike other days, the day increments at noon, and so is 0.5 days ahead of the chronological Julian day. (Strictly speaking, the Julian day is the time at Greenwich Observatory, but for the sake of a converter this can be ignored).

https://en.wikipedia.org/wiki/Julian_day
https://www.hermetic.ch/cal_stud/jdn.htm
https://www.aa.quae.nl/en/reken/juliaansedag.html

------------------
REDUCED JULIAN DAY
------------------

The reduced Julian Day was introduced by the Smithsonian Astrophysical Observatory in 1957 for use with satellite tracking. Because the full Julian day was too big to fit in the memory of the computers used at the time, the SAO subtracted 2400000 from the Julian day, yielding an epoch of noon on 16 November 1858.

https://www.slac.stanford.edu/~rkj/crazytime.txt

-------------------
MODIFIED JULIAN DAY
-------------------

The modified Julian day is a variant of the reduced Julian day, which drops the half day for an epoch of midnight on 17 November 1858; in effect, it is a midnight-indexed version of the reduced Julian calendar. The modified Julian date was the base value used in Digital's popular operating systems TOPS-10, TOPS-20, and VMS.

https://www.slac.stanford.edu/~rkj/crazytime.txt

--------------------
TRUNCATED JULIAN DAY
--------------------

The truncated Julian day was introduced by NASA's Goddard Space Flight Center in 1979 to fit a count of days into 14 bits; this is accomplished by subtracting 2440000.5 from the true Julian day, giving it an epoch of midnight on 12 May 1968.

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800007830.pdf
https://archive.org/stream/nasa_techdoc_19890006406/19890006406_djvu.txt

-----------------
DUBLIN JULIAN DAY
-----------------

The Dublin Julian day was determined by the International Astronomical Union at their 1955 meeting in Dublin. It is based on the epoch used by Simon Newcomb in his Tables of the Motion of the Earth on its Axis and Around the Sun, which is 31 December 1899. This is the date format used internally by LibreOffice.

https://en.wikipedia.org/wiki/Julian_day

---------------
CNES JULIAN DAY
---------------

The CNES Julian day is, as you can probably guess, the day format used by the Centre national d'études spatiales. Its epoch is midnight on 1 January 1950, the "present" epoch in the Before Present dating system.

http://www.mlb.co.jp/linux/science/tptime/doc/index-5.html

----------------
CCSDS JULIAN DAY
----------------

The CCSDS Julian day is used by the Consultative Committe for Space Data Systems, an international space agency forum. Its epoch is midnight on 1 January 1958

http://www.mlb.co.jp/linux/science/tptime/doc/index-5.html

--------------
LOP JULIAN DAY
--------------

The LOP Julian day is the day format used by Laboratoire d'Oceanographique Physique. Its epoch is midnight on 1 January 1992.

http://www.mlb.co.jp/linux/science/tptime/doc/index-5.html

----------
LILIAN DAY
----------

The Lilian Day is a count of the number of days since midnight on 15 October 1582, the day that the Gregorian calendar was introduced; it is named after Aloysius Lilius, the inventor of the calendar. Unlike other integral day counts, the Lilian Day is 1-indexed, not 0-indexed. It was introduced by Bruce G. Ohms at IBM in 1986, and is the internal date format used in IBM's modern mainframes.

https://en.wikipedia.org/wiki/Lilian_date

--------
RATA DIE
--------

Rata Die is a count of days since the start of the Gregorian calendar; like the Lilian Day, rata die is 1-indexed, so day 1 corresponds to 1 January 1. It is the internal date format used in the Rexx programming language as well as some other calendar converters.

https://en.wikipedia.org/wiki/Rata_Die

---------
UNIX TIME
---------

Unix time is the internal time format used in Unix and Unix-like operating systems. It is a count of the number of seconds since 1 January 1970. Unix treats each day as containing exactly 86400 seconds, and thus does not account for the variation in Earth's rotation; leap seconds are accounted for by having the preceding 'second' last twice as long.

Unix stores time as a signed integer. In the 1970s, this integer was 32 bits, which will max out and reset to 0 at 03:14:08 on 19 January 2038. Fortunately, most Unix computers have switched to a 64-bit integer, which will last beyond the heat death of the universe.

https://en.wikipedia.org/wiki/Unix_time
https://www.mail-archive.com/leapsecs@rom.usno.navy.mil/msg00109.html
https://en.wikipedia.org/wiki/Year_2038_problem
https://www.scienceabc.com/innovation/what-is-the-2038-problem.htm
https://fossbytes.com/year-2038-problem-linux-unix/

---------------------
CONSECUTIVE SOL COUNT
---------------------

The consecutive sol count is an integer count of sols (Martian days), beginning on the sol corresponding to 29 December 1873. Each sol lasts 24 hours 39 minutes 35.244 seconds.

https://en.wikipedia.org/wiki/Julian_day
https://en.wikipedia.org/wiki/Sol_(day_on_Mars)
https://jtauber.github.io/mars-clock/

--------
VMS TIME
--------

VMS time is the time standard used in Digitals VMS operating system, developed for their VAX family of minicomputers, and their TOPS-10 and TOPS-20 mainframe operating systems. Time is counted in 100ns tics since the start of the Modified Julian Day epoch.

https://www.slac.stanford.edu/~rkj/crazytime.txt

----------------
AMAZIGH CALENDAR
----------------

The Amazigh calendar is a variant of the Julian calendar used by the Amazigh people (also called the Berbers). In times past, it was the standard civil calendar used by the Amazigh, because the Lunar Hijri calendar was unsuitable for agricultural purposes. Today, Amazigh people live mostly in Morocco, Algeria, and Mali, which use the Gregorian calendar; however, the Amazigh continue to celebrate the first day of their calendar, which is known as Yenneyar after the month.

Unlike the Julian calendar, leap day is added to the 12th rather than the 2nd month of the year. The epoch is Julian day 1374435, which puts it in 950 BC / 1572 BH. This era was chosen as the approximate ascension of Sheshonq I, the first Amazigh pharaoh of Egypt.

https://en.wikipedia.org/wiki/Berber_calendar
https://web.archive.org/web/20060716215321/http://tamurth.net/article.php3?id_article=625
https://www.thenational.ae/world/africa/happy-2968-berber-new-year-becomes-holiday-in-algeria-1.69486
http://world.clndr.org/calendars/Berber-calendar
https://mastersfdl.hypotheses.org/443]
http://www.tamazgha.fr/La-fete-de-Yennayer-pratiques-et-presages,1841.html
https://www.best-country.com/en/africa/algeria/new_year

------------------------------
OTTOMAN FISCAL (RUMI) CALENDAR
------------------------------

The Rumi calendar is a version of the Julian calendar dated from the Hijra. It was used in Turkey for tax calculations from 1677 AD (1055 AH), and adopted as the civil calendar in 1840 AD (1218 AH). It was replaced by the Gregorian calendar in 1926 AD (1304 AH) as part of Mustafa Kemal Atatürk's reforms.

Dating this calendar is tricky. Because for most of its history, it was ony a fiscal calendar, and the Lunar Hijri calendar was the civil calendar of Turkey. As such, the year number of the Rumi calendar is the same as the corresponding Lunar Hijri year, but because the Lunar Hijri calendr is only 354 or 355 days long, one will occasionally find a civil year that starts in one fiscal year and ends before the end of that same fiscal year. To compensate for this, the Ottoman government would occasionally skip a year to bring the fiscal calendar back into synchronisatin with the civil calendar; this happened approximately (but not consistently) every 33 years until 1256, when skipping was abolished and the Rumi calendar was made the civil calendar in Turkey. This situation lasted until 1342 (1926 AD), when the newly-constituted Republic of Turkey switched to the Gregorian calendar

https://en.wikipedia.org/wiki/Rumi_calendar
https://www.takvim.com/takvim_donusum.php
https://www.jstor.org/stable/23060497?read-now=1&seq=11#page_scan_tab_contents

--------------------------
REVISED GREGORIAN CALENDAR
--------------------------

William Herschel proposed a revision to the Gregorian calendar by which years that are integer multiples of 4000 would not be leap years, though this was never officially adopted. The mean calendar year in the revised Gregorian calendar is 365 day 5 hours 48 minutes 50.4 seconds, a mere 5.4 seconds longer than the mean equinoctial year and thus accurate to 1 day in 19500 years, assuming Earth's period of rotation does not change.

http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-94926&I=649&M=tdm p 29
https://tondering.dk/claus/cal/gregorian.php

---------------
PARKER CALENDAR
---------------

The Parker calendar is an alternative revison to the Gregorian calendar suggested by stand-up mathematician Matt Parker when describing improved leap year algorithms. In this version, a year is NOT a leap year if, when divided by 10000, it yields a remainder of 2800, 5600, or 8400. This works out to a mean calendar year of 365 days 5 hours 48 minutes 46.08 seconds, just 1.08 seconds longer than the current mean equinoctial year; this adds up to a loss of just one day in 91,743 years.

As you can tell from the name, the Parker calendar is almost, but not quite, a perfect match for the astronomical year. Celestial mechanics mean that any algorithmic calendar will have some error, but fortunately Matt did independently derive the Goucher-Parker calendar.

https://www.youtube.com/watch?v=qkt_wmRKYNQ

-----------------------
GOUCHER-PARKER CALENDAR
-----------------------

The Goucher-Parker calendar was independently worked out, by different approaches, by Matt Parker and mathematician Adam P. Goucher. It dispenses with the 100- and 400-year rules of the Gregorian calendar in favour of omitting leap years in years that are integer multiples of 128. Matt describes this as rolling back to the Julian calendar, and so I have chosen to date this calendar from the Julian rather than the Gregorian epoch..

The mean Goucher-Parker calendar year is less than 1 second longer than the mean equinoctial year, meaning it only loses 1 day in 625,000 years, if we ignore the chaotic effects of the gravity and momentum of everything else in the solar system.

https://www.youtube.com/watch?v=qkt_wmRKYNQ
https://cp4space.wordpress.com/2012/09/12/lunisolar-calendars/
https://www.theguardian.com/science/2011/feb/28/leap-year-alex-bellos

-----------------------
SERBIAN CHURCH CALENDAR
-----------------------

The Serbian church calendar was designed by Maksim Trpković in 1900. At the time, the Julian calendar was the civil calendar in Serbia; the Gregorian calendar had not been adopted because Serbia was (and is) majority Eastern Orthodox, and until the early 19th century AD had been part of Ottoman Turkey. Since the Julian calendar was by the 15 days off, there was considerable interest in a better calendar, preferably one not linked to the Catholic Church, an Trpković stepped up to the task.

Trpković's calendar adopts the Gregorian calendar's 100-year rule, but replaces the 400-year rule with a rule whereby a year which yields a remainder of 0 or 400 when divided by 900 is a leap year. Like the revised Julian calendar devised by Milutin Milanković, Trpković's calendar is accurate to 1 day in 28,800 years. Trpković's calendar has an advantage over Milanković's in that the astronomical northward equinox will fall more often on 21 March; however, Milanković's calendar spreads the leap years out more evenly and is better aligned to the Gregorian calendar, which is the de facto standard. Because Trpković said to only skip 13 days rather than 15 and wanted to keep the equinoxes aligned with those of 325 AD, the year in which the Council of Nicaea met, it shares a calendar epoch with the Gregorian.

Trpković's calendar was endorsed by the Serbian government and the Serbian Orthodox Church, and was put forward at the Pan-Orthodox Congress as a candidate to replace the old Julian calendar. The calendar met with considerable approval, but the Congress ultimately opted for Milanković's submission.

http://publications.aob.rs/75/pdf/301-306.pdf

-----------------------
REVISED JULIAN CALENDAR
-----------------------

The revised Julian calendar was developed by Milutin Milanković in 1923 AD, building on the work of Maksim Trpković. It implements the Gregorian 100-year rule, but instead of the 400-year rule, adds a rule that any year which leaves a remainder of 200 or 600 when divided by 900 is a leap year. Due to Milanković, like Gregory, seeking to preserve the date of the northward equinox of the Council of Nicaea, the revised Julian calendar has the same epoch as the Gregorian calendar and not the original Julian.

the Orthodox churches had refused to accept the Gregorian calendar on the grounds that it came from the pope of Rome, and as such the majority-Orthodox countries had stuck with the Julian calendar. By the 20th century AD, it had become drifted drastically out of sync with the seasons, and in 1902 AD Ecumenical Patriarch Joakim III wrote to the other patriarchs and bishops to invite proposals on calendar reform. Patriarch Miletios IV convened a Pan-Orthodox Congress in Istanbul, then under Entente occupation, in May 1923 AD. Trpković's calendar was supported by the Serbian delegation, but ultimately Milanković persuaded the council to adopt his proposal instead, and this change was promulgated by Miletios IV and King George II of Greece. The revised Julian calendar was never used for civil purposes, all the Eastern Orthodox-majority countries having already adopted the Gregorian calendar as their civil calendar by this time; the revised Julian calendar was thus only ever used as a liturgical calendar.

However, things did not proceed smoothly. Miletios IV was a controversial figure in the Eastern Orthodox Church, and the patriarchs of Alexandria, Antioch, Jerusalem, and Russia did not attend. The new calendar was adopted by the Orthodox Churches of Constantinople, Greece, Serbia, Romania, and Cyprus, but initially no others. Since then, the revised calendar has been adopted by the Patriarchates of Alexandria and Antioch, but not Jerusalem. It has also been adopted by the Churches of Bulgaria, Poland, Albania, Estonia, Czechai and Slovakia, and America (except Alaska); these are collectively referred to as the 'new calendarists'. The Finnish Orthodox Church uses the Gregorian calendar for all purposes, and the rest of the Eastern Orthodox churches continue to date their festivals by the old Julian calendar. A few churches use both calendars for different purposes, and all except Finland use the original Julian calendar to calculate the date of Easter.

The average calendar year length is 365 days 5 hours 48 minutes 48 seconds, just 3 seconds longer than the mean equinoctial year, which works out to a loss of one day in 28,800 years; the revised Julian calendar is the most accurate algorithmic calendar ever implemented.

https://en.wikipedia.org/wiki/Revised_Julian_calendar
http://images.astronet.ru/pubd/2008/09/28/0001230709/145-147.pdf
https://www.holy-trinity.org/modern/calen3.html
http://www.byzcath.org/forums/ubbthreads.php/topics/309500/Which_Churches_follow_what_cal
http://publications.aob.rs/75/pdf/301-306.pdf

--------------
WORLD CALENDAR
--------------

The World calendar was designed by Elisabeth Achelis in 1930 AD, one of many proposed after the League of Nations called for proposed calendar reforms in 1923.

The World calendar has all the same months as the Gregorian calendar, but changes the number of days such that they year divides into four quarters of three months, each having 31, 30, and 30 days. An additional day, Worldsday, comes after 30 December and falls outside of the week; thus, in every year, Saturday 30 December is followed by Worldsday, which in turn is followed by Sunday 1 January. Leap years follow the same cycle as the Gregorian calendar; Leap Day is inserted between 30 June and 1 July, and like Worldsday it falls outside the week. Achelis proposed that Worldsday and Leap Day be international holidays. Every year starts on a Sunday, and Friday the 13th falls four times a year. Leap years follow the same rules as the Gregorian calendar, giving the World calendar exactly the same accuracy.

The World Calendar Association point to the benefits of having months and quarters of consistent length making it easier for businesses to compare performance over fractions of a year. The fact that every day of the year always falls on the same day of the week makes it easier to schedule events in advance, and also has ecological benefits as there is no need to buy a new wall calendar every year.

The World calendar faced considerable opposition from Jews with the support of Christians, spearheaded by Rabbi Joseph Herman Hertz. Hertz pointed out that because Jews and Christians [and also Muslims] worship on a seven-day cycle, having a day fall outside the week would wreak havoc on religious observances; people would be forced to either worship on the wrong day of the week, or of shifting the time of the weekends, which cancels out the whole point of the World calendar. Achelis argued that people could view Worldsday and Leap Day as double Sabbaths, but this doesn't really change the fact that such a double Sabbath would still cover two solar days, plus the Jewish New Year algorithm has checks specifically to prevent double Sabbaths as two consecutive days of Sabbath restrictions is considered to be overly harsh. Moreover, because Muslims worship on Friday and Worldsday and Leap Day falls between Saturday and Sunday, they would be unable to use the holidays as a double day of rest.

Faced with this, the to-then strong support for the calendar in America faltered as Congress decided not to endorse the reform. In the end, the calendar was not implemented, but the World Calendar Foundation, founded by Achelis, continues to advocate for it.

https://en.wikipedia.org/wiki/World_Calendar
http://myweb.ecu.edu/mccartyr/achelis.html
http://myweb.ecu.edu/mccartyr/world-calendar.html
http://www.theworldcalendar.org/TWCandDescription.pdf
http://www.theworldcalendar.org/CalendarMathProblemSolution100206.pdf

----------------------------
INTERNATIONAL FIXED CALENDAR
----------------------------

The International Fixed Calendar (henceforth IFC) was designed at the turn of the 20th century AD by Moses Cotsworth, a railway advisor with the North Eastern Railway who grew frustrated with trying to compare monthly incomes and expenditures. Cotsworth shopped the new calendar around various companies in America and the British Empire, but people at the time were more concerned with the approaching World War I at the time.

The IFC divides the year into 13 months of 28 days, making each one exactly four weeks long; the extra month, called Sol, is inserted between June and July. As in the World calendar, each year has one extra day, Yearsday, after December 28 which falls outside the week. Leap Day falls between June and Sol, and is also outside of the week. Leap years follow the same rules as the Gregorian calendar, giving the IFC exactly the same accuracy. Cotsworth 

The IFC has many of the same advantages as the World Calendar. Every day of the year falls on the same day of the week, which makes scheduling easier. Every month is of perfectly consistent length, and is an integer number of weeks, which makes month-to-month comparisons easier for businesses; however, quarterly comparisons don't line up well with monthly ones, because 13 is a prime number and so any even subdivision of the year will contain a non-integer number of months. It also means that workers who are paid on a monthly basis have a far more reliable, regular income.

The IFC was presented alongside the World calendar and others at the League of Nations' 1928 AD calendar reform call for proposals, and it failed for the same reasons as the World calendar, since Yearsday and Leap Day falling outside the year wreak havoc with traditional religious worship cycles. However, George Eastman was an enthusiastic fan, and his Kodak company used the IFC internally from 1928 until 1992.

https://en.wikipedia.org/wiki/International_Fixed_Calendar
https://www.jta.org/2012/02/24/default/calendar-reform-and-joseph-herman-hertz
http://myweb.ecu.edu/mccartyr/eastman.html
http://myweb.ecu.edu/mccartyr/cotsworth.html
https://www.rd.com/article/international-fixed-calendar/
https://www.unz.com/print/Outlook-1927sep28-00109?__cf_chl_jschl_tk__=69f7afabc2eb77ff8dd0af23f69afa34b2155314-1594419113-0-AecdfOlrq79eBgNEnZ8l09Qsb4M0yN_jiA69aqUPiaDEINHpDW0exKuMkKgfAXsmofpRfzTsfSyQSt4IQMURLAYlDbvvgHMzXI3JPcxOcbO_483Tr3kSbDeawEO6Snh5S3TjgVf9nSlOgF8WoUVv_8PONWbAgvFPb6TA9z7UnKQ9egYgeeyYIbO6St90r0IJzI-12-ScaJtX3i_fCVTHAuO82QigVePEeKI7EYQhZ4mZfSo804Yt_YetSadLqCbwSqPohCBUddQQAayVt21Uqm3ixSpg-AwZnqVwrPYgllRb_RM3VS_HPmLJ6GGMPgEL6Q
https://www.bloomberg.com/news/articles/2014-12-11/the-death-and-life-of-the-13-month-calendar

------------
PAX CALENDAR
------------

The Pax calendar is yet another proposed reform presented to the League of Nations in 1930 AD, designed by James Colligan. Like the World calendar and IFC, the Pax calendar aims to be perennial. Unlike them, it contains an integer number of weeks in each year; as such, each leap year adds an extra week rather than just a day. All the other months contain exactly four weeks (28 days), with an extra month, Columbus, falling between November and December. In leap years, the extra week forms a bonus mini-month, falling between Columbus and December. Leap years fall in any year whose last two digits are 00, 99, or an integer multiple of 6, unless the year is an integer multiple of 400. This all means that 400 years in the Pax calendar have the same number of days as 400 years in the Gregorian calendar, giving it exactly the same accuracy.

The Pax calendar brings in all the advantages of the World calendar and the IFC while avoiding the religious problems, at the cost of the dates of the solstices and equionxes moving about the year. It seems to have failed mostly because people were concerned with the first rumblings of World War II.

https://en.wikipedia.org/wiki/Pax_Calendar
http://myweb.ecu.edu/mccartyr/colligan.html

---------------
GORMAN CALENDAR
---------------

The Gorman calendar is a variation of the IFC designed by comedian Dave Gorman out of things which annoy him about the Gregorian calendar. The year starts in March so that those months whose names contain numerican components are in the right position, and each month is exactly four weeks long. July and August are renamed to Quintilis and Sextilis because naming a month after oneself is arrogant, and a new month, Gormanuary, is added afer February to round out the year. Between 28 Gormanuary and 1 March, there is a single day which falls outside the week, just called Intermission; in leap years, Intermission is two days long. Leap years follow the Gregorian rules, and so the Gorman calendar is exactly as accurate as the Gregorian.

https://www.youtube.com/watch?v=EcMTHr3TqA0

--------
PAX 2020
--------

The Pax 2020 calendar is completely unrelated to the 1920s Pax calendar. Pax 2020 was designed by a 9-year-old child who was being homeschooled during the 2020 AD Covid-19 lockdown. After noticing a subtle ambiguity in a maths problem, he read up on calendars on Wikipedia and realised he could design his own.

Pax 2020 consists of 13 months. 12 of these have 28 days (exactly four weeks), and the last has 29 days in normal years or 30 in leap years; the creator suggests these last two days be holidays.

https://bryanmmathers.com/?mailpoet_router&endpoint=view_in_browser&action=view&data=WzMsImJhYzE3YTBiOWYzYyIsMSwiNGEwNGEzIiwzLDFd
https://bryanmmathers.com/pax-calendar/


-------------------
POSITIVIST CALENDAR
-------------------

The Positivist calendar was designed by Auguste Comte in 1849, which he intended as a break with both Julian and Gregorian traditions. The Positivist calendar is the forebear of the various other perennial calendars included in this converter; it consists of thirteen months of exactly four weeks each, plus one extra day, the Festival of all the Dead. In leap years, this day is followed by the Festival of Holy Women; both year-end festivals fall outside the week. Unlike other perennial calendrs, the Positivist calendar starts each year on Monday.

In the Positivist calendar, Comte imbues his humanist goals, naming each month after a significant philosopher, writer, or political reformer, all originating from a variety of countries to show an inernational spirit. Alas, Comte's 19th-century biases are clearly visible; 12 of the months are named after white European men, and one after a Jewish man. There are no women at all, nor are there any Africans, Pacific Islanders, aboriginal people of Australia or the Americas, nor any Asians except for Moses.

The calendar was never adopted anywhere, at least partly because months named after people came across as silly at the time. Plus, as we have seen, religious people weren't about to give up their seven-day cycle of worship.

https://en.wikipedia.org/wiki/Positivist_calendar
http://positivists.org/calendar.html
https://gallica.bnf.fr/ark:/12148/bpt6k6108866f/f359.image
http://myweb.ecu.edu/mccartyr/pos-cal.html
http://webspace.ship.edu/cgboer/comtescalendar.html
http://myweb.ecu.edu/mccartyr/comte.html

-------------------------------
ASTRONOMICAL GREGORIAN CALENDAR
-------------------------------

The astronomical Gregorian calendar is a variant of the Gregorian calendar which includes a year 0. In short, each year n BC in the normal Gregorian calendar is year -(n + 1) in the astronomical Gregorian calendar.

------------
NEX CALENDAR
------------

The Nex calendar is yet another proposed perennial calendar. In the Nex calendar, each year begins on Monday. The months have the same lengths as in the Gregorian calendar, except that February has 29 days each year and December had 30, except in leap years when December has 31 days. The 30th and 31st of December are both Sunday, and the last two days of August are both Saturday.

http://nexcalendar.org/

-----------------
HOLOCENE CALENDAR
-----------------

The Holocene calendar was proposed by geologist Cesare Emiliani in 1993 AD (11993 HE, 1371 AH). It is a variant of the astronomical Gregorian calendar, differing only in its epoch. The Holocene calendar begins on Julian day (-1931366), which corresponds to 1 January 10001 BC, the approximate start of the Holocene geologic epoch. Humans first started to transition from hunting and gathering to agriculture in the early Holocene, and so this calendar allows all of human history to have a positive number. The calendar is closer to cultural neutrality than is the Gregorian. Any relatively recent date can be converted into the Holocene calendar simply by prepending a 1.

https://en.wikipedia.org/wiki/Holocene_calendar
https://www.youtube.com/watch?v=czgOWmtGVGs
https://holocenecalendar.org/
https://www.easytimeline.org/
https://www.youtube.com/watch?v=aDnoTk5mOQc

---
ADA
---

ADA, an abbreviation for After the Development of Agriculture, is another dating system based on the origin of farming. ADA was proposed by feminist archaeologist Merlin Stone and begins in 8001 BC (2000 HE), the approximate date which Stone believes that women invented agriculture. This dating system never caught on, possibly at least in part due to many of Stone's ideas about prehistory being questionable at best. Still, a calendar is a calendar.

https://en.wikipedia.org/wiki/After_the_Development_of_Agriculture

---------------------------
FRENCH REPUBLICAN CALENDARS
---------------------------

The French Republican (or Revolutionary) calendar was introduced in France on 2 January 1793 AD (13 Dey 1171 AH; 13 Nivôse 1 AR) following a period of debate which only fully ended on 24 October 1793 AD (3 Aban 1172 AH; 3 Brumaire 2 AR). The calendar was part of a wide-ranging series of radical reconstitutions of society intended to make a clean break with the ancien régime; other reforms included the introduction of the metric system and decimal currency, which have lasted until today. The Revolution also brought a decimal time system, in which each day was divided into 10 "hours", each of which was subdivided into 100 "minutes" consisting of 100 "seconds" each. Unlike the other reforms, decimal time was unpopular, and was abandoned after less than a year and a half.

The calendar committee sought a more regular calendar than the Gregorian, and so decided that each year would consist of 12 months of 30 days each, grouped into 10-day décades (not to be confused with the English word 'decade'). Décadi, the tenth day of the décade, was a national day of rest, and the fifth day, Quintidi, was a half day. Each month was given a Latin- or Greek-derived name referring to the dominant weather in Metropolitan France at that time of year. Due to the presence of poets on the calendar committe, the three months in each season rhyme. To make up the length of the year, a five-day period (six in leap years) of natinal holidays was appended to the year, referred to as the complementary days or the Sans-culottides in honour of the working-class people who made the Revolution.

If this seems familiar, you are correct. The French Republican calendar ended up basically the same as the ancient Egyptian civil calendar, but with a leap year mechanism. However, where the Egyptian calendar was intended to start around the northern solstice when Sirius returns to the sky, the French Republican calendar began on the southward equinox, possibly inspired by the Jalali calendar. Its epoch is Julian Day 2375839, corresponding to 1 Vendémiaire Ans du Révolution; the era is 1792 AD, the year the Revolution began. While the calendar was in use, it was customary to write the year in Roman numerals; I have ignored this rule for the sake of readability.

The determination of leap years presented a problem which was never resolved while the calendar was in use. The initial proclamation is itself confused; it declares that each year will begin on the autumnal (ie southward) equinox as observed from Paris Observatory, and also that leap day would occur every four years. As such, an in order to ensure the calendar would be more accurate than at least the Julian, there was some debate as to hwo leap years should be calculated. Initially, the observational rule was adopted, though this was unsatisfactory because determining the start of the year required consulting astronomers, and errors could accrue if the equinox occurred very close to midnight. Tying it to the southward equinox also make the calendar a harder sell abroad, where the New Year's Day tended to fall around the southern solstice or northern equinox. (It also necessarily privileges Paris time above all others). Gilbert Romme, who served on the committee which originally designed the calendar, proposed an algorithmic solution: the Republican calendar would use the same rules as the Gregorian and also implement William Herschel's 4000-year rule. Romme lost his head soon after this proposal and it was never formally implemented.

Both these rules are implemented in this converter. The algorithmic French Republican calendar, naturally, uses Romme's proposal; as with the revised Gregorian calendar, it loses one day in 19500 years. The astronomical French Republican calendar uses the length of the mean equinoctial year (365 days 5 hours 48 minutes 45 seconds) to determine when each New Year's Day happens. Earth's angular velocity is gradually being transferred to the moon, which results in the length of the solar day increasing by about 2ms per century. I have not taken this change into account, and so the algorithm used in this converter will gain one day every 4.32*10^9 years.

As part of an ongoing policy of Dechristianisation, businesses were only permitted to close and religious services allowed to be held only on Décadi in order to mess up the seven-day cycle of religious observance (there were also restrictions on which day of the décade fish could be sold so as to make it harder for Christians to refrain from poultry and red meat on Friday). In addition, individual days were associated with crops, farm animals, farm equipment, and minerals rather than Catholic saints. The décade system was tremendously unpopular, because one day of rest in 10 is objectively worse than one in seven; as such, the seven-day week was reintroduced in 3 AR (1795 AD); at the same time, the Gregorian calendar was restored to co-official status to facilitate international relations.

The Republican calendar was formally abandoned by Napoleon on 1 January 1806 AD (11 Nivôse 14 AR), though it was briefly revived in 1871 AD (79 AR) by the Paris Commune. As such, France, one of the first countries to adopt the Gregorian calendar, was also the first to abandon it, and is so far the only country to abandon and then readopt it.

https://www.fourmilab.ch/documents/calendar/fr_decret.html
https://en.wikipedia.org/wiki/French_Republican_calendar
http://www.webexhibits.org/calendars/calendar-french.html
https://www.cooksinfo.com/french-revolutionary-calendar
http://www.windhorst.org/calendar/
https://www.tondering.dk/claus/cal/french.php
https://www.napoleon.org/en/history-of-the-two-empires/the-republican-calendar/
https://www.worldslastchance.com/ecourses/lessons/changing-weeks-hiding-sabbath-ecourse/18/french-republican-calendar.html
https://en.wikipedia.org/wiki/Sansculottides
https://web.archive.org/web/20070618041235/http://personal.telefonica.terra.es/web/jlmartinmas/repcalendar/calendar.html

--------------------
SOLAR HIJRI CALENDAR
--------------------

The Solar Hijri calendar is a refinement of the Jalali calendar, designed by partially going back to Omar Khayyam's original formulation. Like the French Republican calendar, the Solar Hijri calendar determines the new year, known as Nowruz, entirely by observation. The beginning of the year is based on the observation of the northward equinox; new year's day, known as Nowruz, is the day whose midnight in Iran Standard Time falls closest to the moment of equinox; to put it another way, if the equinox happens before noon, Nowruz is the same day, while if it falls after noon, Nowruz is the following day. Culturally, the moment of equinox is considered to be the real start of the year, as that is when everybody celebrates the transition. As the calendar is completely observation-based, it remains perfectly synchronised with the seasons. To keep things manageable, this converter assumes constant values for the year and the day; in fact, both are gradaully getting longer, and this should be taken into account if you are forecasting 4.32 billion years into the future.

The Solar Hijri calendar was first adopted in 1301 AH (1922 AD), replacing the animal-based duodecimal calendar introduced by the Mongols. It was later adopted as the official calendar of Iran in 1304 AH (1922 AD), replacing the Jalali calendar.

https://iranicaonline.org/articles/calendars
https://en.wikipedia.org/wiki/Solar_Hijri_calendar
http://aramis.obspm.fr/~heydari/divers/ir-cal-eng.html

----------------
THELLID CALENDAR
----------------

The Thellid calendar was designed by Warren Mars as a liturgical calendar for the Church of Infinite Dimensions, a rather quirky secular alternative to religion which he founded; fortunately, he designed it to be perfectly usable as a purely civil calendar.

The Thellid calendar incorporates ideas from the Holocene, Positivist, and French Republican calendars; Mars claims he was unaware of the Positivist calendar when he came up with the Thellid, and I have no reason to doubt this. Each year consists of 13 months of 28 days, plus one extra day, called Old Year's Day. Old Year's Day always falls on the day of the southern solstice as measured from Greenwich Obervatory; in leap years, Leap Day is inserted before Old Year's Day. As with the Positivist, World, Gorman, and International Fixed calendars, Leap Day and Old Year's Day are not part of any month, and fall outside of the week; as such, every day of the year always falls on the same day of the week.

To keep things as secular as possible, the Thellid calendar uses the Holocene era, though its epoch is Julian Day (-1931005), five days before the Holocene calendar epoch due to different definitions of New Year's Day. Each month and weekday is given a distinctive name; weekdays are all two syllables long, while months get three. The names of the weeks and the months, as well as the name 'Thellid', were all concocted by Mars himself; they are not derived from any root words and are thus culturally neutral (though there is a clear echo of the Roman alphabet in the naming of the months).

http://infinitedimensions.org/devices/thellid_calendar/thellid_calendar.htm

------------------------------
LUNAR HIJRI (ISLAMIC) CALENDAR
------------------------------

The Lunar Hijri calendar is a slightly modified version of the pre-Islamic Arab calendar. Most notably, the prophet Muhammad ended the practice of intercalary leap months, resulting in an entirely lunar calendar that makes no effort to maintain synchronisation with the sun.

Each month of the religious calendar begins with the first sighting of the waxing crescent moon, and days thus begin at sunset; as such, any date give by this converter actually refers to the next day if the time is between sunset and midnight. Years are numbered AH, for Anno Hejirae; this is the year that the prohpet Muhammad moved from Mecca to Medina and founded the first Muslim community. For simplicity, I have taken the month to be exactly 29 days 12 hours 44 minutes 3.3 seconds long, based on Hebrew calculations; this will become inaccurate over millions of years as angular momentum is transferred from Earth to the moon.

The version used in this converter takes an epoch of Julian Day 1948437 and 1807/4320 of a day; this was derived by first taking the date and time of the New Moon of Muharram 1442 AH and then subtracting 1441 * the length of the astronomical lunar year. The date and time of the new moon is the instant in Saudi Arabian Standard Time when the New Moon is visible anywhere in the world. This is the convention used by many Muslim communities for convenience and to allow for Muslims across the world to celebrate religious observances at the same time. Strictly speaking, however, the month is supposed to begin on the night of the first local sighting of the New Moon, which can mean that the Islamic months begin on different nights depending on time zones; this reliance on direct observations causes problems when it's cloudy.

As such, the dates given in this converter might not exactly match those decreed by Muslim authorities; they can also differ from those of the tabular calendar. In the modern age, this difference can be up to two days; over centuries, the extra 44 minutes 3.3 seconds add up, resulting in a greater divergence millennia in the past and future.

https://en.wikipedia.org/wiki/Islamic_calendar
https://en.wikipedia.org/wiki/Tabular_Islamic_calendar
http://www.webexhibits.org/calendars/calendar-islamic.html
https://icfresno.org/2019/06/eid-prayer-at-iccf-2/

-------------------------
PRE-ISLAMIC ARAB CALENDAR
-------------------------

The nature of the calendar used by the Arabs before the introduction of Islam is little-known and controversial. Some evidence exists that two seperate calendars were in use, one entirely lunar, the other lunisolar. If this is the case, the purely lunar calendar was basically identical to the modern liturgical Muslim calendar, and so has already been implemented.

Tentative reconstructions of the pagan Arab lunisolar calendar point to one quite similar to the Hebrew calendars. An extra month, Nasiʾ, was intercalated according to the Babylonian algorithm which Rabbi Hillel II had introduced to the Jews in 4118 AM (358 AD, 273 BH lunar, 269 BH solar). Some historians have suggested that the intercalation algorithm followed a simpler scheme in which Nasiʾ was added every two, three, or four years, or that nine out of every 24 years were intercalated; however, the most plausible lines of evidence point to the Arabs copying the Jewish scheme.

Other historians question whether Nasiʾ was in fact an extra month, pointing out that the word just means 'postponement', and as such may have simply referred to postponing the sacredness of one month to the next without actually adding any months. Even if this was the case at some point, the evidence most strongly indicates that by 200 BH, the Arabs were indeed adding months to the year.

While other lunisolar calendars add months at a fixed point in the year, the Arab calendar appears to have cycled which month it would follow; if Nasiʾ followed Muharram one year, it would follow Safar the next, and so on. This is all rather complicated, and as in other cultures, it fell to an individual to declare when Nasiʾ would be added. In the case of the Arab calendar, this was al-Qalammas, a position apparently named after the first person to hold the office, who was always of the Qinanah tribe.

A variety of calendar eras were in use at different times. It appears that, after Islam became dominant, the Wathanists adopted the Hijra era, and so this converter dates years Anno Hegirae. 

Muhammad abolished the practice of Nasiʾ on the grounds that adding months screws with what God has ordained; it has also been suggested that this was done to weaken the influence of the Qinanah. Wathanists, naturally, continued the practice.

Muhammad finally got around to performing the Hajj in Dhu al-Hijjah 10 AH, stating that at this time, the Muslim and pagan calendars had gotten back in sync. Thereafter, he banned non-Muslims from performing Hajj. Based on this, I have set the epoch as the new moon of Julian Day 1948319 so that Dhu al-Hijjah 10 AH will start and end on the same days in the Lunar Hijri and Arab calendars.

https://en.wikipedia.org/wiki/Pre-Islamic_Arabian_calendar
http://irigs.iiu.edu.pk:64447/gsdl/collect/islamics/import/v37i34.pdf
https://en.wikipedia.org/wiki/Nasi%27
http://etheses.dur.ac.uk/978/1/978.pdf?EThOS%20(BL)
http://www.eretzyisroel.org/~jkatz/The%20Islamic%20Jewish%20Calendar.pdf

-------------
IGBO CALENDAR
-------------

The Igbo calendar is the traditional calendar used by the Igbo people. While Nigeria uses the Gregorian calendar, the Igbo continue to use their own traditional calendar in parallel to determine days of rest and market days.

A year in the Igbo calendar contains only 364 days. These are divided into 13 months, each of which contains seven izu of four days each. People work and farm three days of the izu, setting aside the fourth as a day of rest. Eke is the standard, but each community is free to designate any day as an alternative rest day. Since Nigeria is quite close to the Equator, the weather doesn't change much with the seasons, so presumably the Igbo didn't feel any particular need to keep pace with the solstices and equinoxes.

The count of years begins in 1003 AD, but I couldn't find any reference as to the origin of this era. The epoch is thus the first of Mbụ this year, which is Julian Day 2087628.

https://en.wikipedia.org/wiki/Igbo_calendar
https://www.thenigerianvoice.com/nvnews/85709/1/day-massob-took-over-nri-kingdom.html
https://igbocalendar.com/igbo-calendar/view-igbo-calendar-for-this-year/
http://www.ekwenche.org/igbo-calendar-2020-2021.htm

--------------
ROMAN CALENDAR
--------------

The Romans were bad at calendars.

The legend goes that the earliest Roman calendar was designed by Romulus himself, consisting of four months of 31 days and six of 30, for a total of 304; the remaining 61 or so days were just assigned to an indeterminate winter period. Numa Pompilius later assigned the winter days to two new months, Ianuarius and Februarius.

In fact, the evidence is that the Romans originally used some sort of lunar calendar, which later evolved into a calendar of fixed-length months. Presumably this at one point consisted of months with alternating lengths of 30 and 29 days; however, the Romans thought that even numbers were unlucky, and so it was restructured into four months having 31 days, seven having 29, and Februarius having 28. This gives us a 355-day calendar which does not track the sun or the moon.

In order to somewhat stay in sync with the sun, a leap month, Mercedonius, was sometimes added, and this is where things get complicated. Wikipedia claims that Mercedonius fell between 23 and 24 Februarius, but this isn't really accurate, because the Romans didn't think of days in terms of "the nth day of the month". Rather, they referred to days in relation to the kalends, nones, and ides, days which fell at fixed points in the month (these at one time corresponded to the days of the new moon, first quarter moon, and full moon, but the calendar used in the Roman Republic had long since stopped corresponding to the lunar cycle). Thus, for example, the festival of Carmentalia, which we would think of as falling on 11 January, was considered by the Romans to fall on the fifth day before the Nones of Ianuarius. So in reality, in leap years, Februarius was shortened, and the days between Mercedonius and Martius of the next year were just considered to be a group of days before the kalends of Martius.

It is known that leap years in the Roman calendar could be either 377 or 378 days long, and there is some controversy as to whether it was Februarius or Mercedonius that got the extra day in a 378-day year. Bennett cites the work of historian A Michels as well as the writings of Livy, where it is stated that Mercedonius sometimes fell between Terminalia and Regifugium, and sometimes the day after Regifugium. Thus, it appears that the length of Mercedonius was fixed and the length of Februarius was variable in leap years; this is the convention I have used in this converter.

Nominally, years were supposed to progress in a four-year cycle of 355, 377, 355, and 378 days, for an average of 366.25 days; as such, the calendar would lose 5 days in 16 years. But leap years weren't assigned by any rule. Instead, the task of deciding when to declare a leap year was given to the pontifex maximus, who would routinely declare a leap year when he supported the consuls, and fail to declare one when he did not (or when there were more pressing matters). This is also probably why it was actually more common to declare a year of 378 than of 377 years. Furthermore, since the declaration of when a leap year fell had to be proclaimed across the empire, people in far-off provinces would not know for some time what kind of year year it was, which had implications for tax collection.

As you can see, the Roman calendar was hopelessly unfit for purpose, which is why fixing it was one of the planks of Julius Caesar's dictatorship. Caesar wisely hired Greek astronomers and mathematicians to design a sensible calendar, most notably Sosigenes of Alexandria. Caesar declared the year 708 AUC (46 BC, 667 BH) would be 445 days long, and the new calendar took effect from the start of the next year, 708 AUC (45 BC, 666 BH).

For the calendar era, I am using Ab Urbe Condita (From the Founding of the City [of Rome]) convention, which dates from the legendary foundation of Rome in 753 BC (1374 BH). This dating system was occasionally used by Roman writers, but was pretty rare; it was mostly used by Romanophiliac European writers of the Middle Ages and Renaissance, who make it look more common than it really was when they sometimes insert it into translations of authentic Roman writings. Romans mostly referred to the year by who the consuls were, or by the year of the current king or emperor.

I have also decided to use the convention that the calendar begins in Martius rather than Ianuarius. This is the original form of the calendar, with New Year's Day shifting to Ianuarius at some point apparently in the early Republican era, or possibly during the middle Kingdom era. Because Sosigenes and Caesar calibrated the Julian calendar to restore sanity to the dates of the solstices and equinoxes, I have chosen as the epoch Julian Day 1446449, which corresponds to 1 March 753 BC by the Julian calendar.

As you can tell, this converter will not convert an actual Roman date to the correct date in any other calendar, because the Roman calendar was unstable and inconsistent, and constantly micromanaged by the government. Rather, it will give the dates corresponding what would have been the date in the Roman calendar had it been implemented sensibly and consistently.

https://en.wikipedia.org/wiki/Roman_calendar
https://archive.org/stream/historyrome00dickgoog#page/n242/mode/2up
https://en.wikipedia.org/wiki/Mercedonius
http://elfinspell.com/ClassicalTexts/Maude/Censorinus/DeDieNatale-Part2.html
https://en.wikipedia.org/wiki/Ab_urbe_condita
https://books.google.ie/books?id=krcp3GU2MssC&pg=PA14&redir_esc=y#v=onepage&q&f=false
http://www.instonebrewer.com/TyndaleSites/Egypt/ptolemies/chron/roman/chron_rom_cal_fr.htm
http://www.webexhibits.org/calendars/calendar-roman.html
http://penelope.uchicago.edu/~grout/encyclopaedia_romana/calendar/romancalendar.html

-------------------
MACEDONIAN CALENDAR
-------------------

A variety of calendars were in use in Ancient Greece; in fact, each city-state had its own calendar. All were luni-solar and the months were supposed to start on the first sighting of the new moon, but that's where the similarities end. Different city-states tied the start of the year to each of the equinoxes and sunsets, and days were variously reckoned to begin at sunset, sunrise, midnight, and even noon. Some employed the Metonic cycle to decide when to add leap months, while others just added extra months whenever it looked about right. City governments would sometimes insert extra days into a given month, causing it to drift out of sync with the moon; this habit forms the plot of Aristophanes' satirical play Clouds. To make things more irritating, the Greeks did not use calendar eras, instead referring to each year by who the archon was at the time; this has been causing headaches for historians since ancient times. A scheme occasionally used was to date events by the Olympiad, but this could be ambiguous due to each city-state reckoning the lengths of years differently.

While most Greek calendars are fragmentary and lack a reasonable era, the Macedonian calendar is quite well known. Each year began on the first new moon following the southward equinox. Intercalary months would normally be added in spring, but could happen at the end of the year (or really whenever the king wanted). It appears that the Macedonians at some point adoped the Metonic cycle to assign leap years; for this converter I have chosen to go with the observational scheme used in the older version of the calendar. For the era, I have somewhat arbitrarily chosen the year Alexander the Great ascended the throne of Macedon; this puts the epoch at Julian Day 1598617.

https://en.wikipedia.org/wiki/Ancient_Greek_calendars
https://en.wikipedia.org/wiki/Ancient_Macedonian_calendar
https://books.google.ie/books?id=x2AD3M77TgMC&pg=PA166&lpg=PA166&dq=Macedonian+epigraphy&source=bl&ots=MOvxl5luuB&sig=StnUhRUO0D5aowvC3muiUfjbyOw&hl=en&ei=lPoxSu_9O4LAsAbK8NXNCQ&sa=X&oi=book_result&ct=result&redir_esc=y#v=onepage&q=Macedonian%20epigraphy&f=false
http://www.ancientgreecefacts.com/greek-calendar/
http://www.polysyllabic.com/?q=calhistory/earlier/greek
https://www.nancybiska.com/ancient-macedonia-and-its-calendars/#
https://www.researchgate.net/publication/292592727_THE_MACEDONIAN_CALENDAR_IN_MACEDONIA

-----------------
SELEUCID CALENDAR
-----------------

The Seleucid Calendar can be considered a variant of either the Macedonian or the fixed Babylonian calendar. It was used in the Seleucid Empire and dates from the second conquest of Babylon by Seleucus I Nicator in 312 BC, giving it an epoch of Julian Day 1607743. Like the Macedonian calendar, it begins on the new moon following the southward equinox; as in the fixed Babylonian calendar, intercalary months are assigned as per the Metonic cycle rather than by deciding the weather is about right. I have designed this converter so that the addition of intercalary months in the Seleucid and fixed Babylonian calendars will sync up; indeed, Wikipedia says that farmers in the Seleucid Empire used the Babylonian calendar with Macedonian month names, and as such the commoners began the year six or seven months before the nobility, at the northward equinox.

https://en.wikipedia.org/wiki/Seleucid_era
http://theos-sphragis.info/hebrew_babylonian_intercalation.html
https://www.nancybiska.com/ancient-macedonia-and-its-calendars/#

----------
ANNO LUCIS
----------

Anno Lucis is one of several dating systems used internally by Freemasons. Anno Lucis, meaning 'In the Year of Light', dates events from (approximately) the Biblical creation account. Freemasons do this by adding 4,000 to the current Gregorian date.

https://en.wikipedia.org/wiki/Anno_Lucis
https://www.masonic-lodge-of-education.com/anno-lucis.html
https://freemasonrymatters.co.uk/knowledge/anno-lucis/
https://pagrandlodge.org/shedding-light-on-anno-lucis/
https://grandlodgeofiowa.org/docs/Masonic_History/AnnoLucis.pdf

----------------
ANNO INVENTIONIS
----------------

Anno Inventionis (Year of Discovery) is another Freemason calendar, this one dating from the construction of the Second Temple in Jerusalem in 530 BC and is used by the Royal Arch Masons.

https://pagrandlodge.org/shedding-light-on-anno-lucis/
http://www.masonicdictionary.com/calendar.html

-----------------------
MESOAMERICAN LONG COUNT
-----------------------

The Long Count is a system used by various Central American civilisations to date historical events. It is commonly called the Maya calendar, though there is evidence it was introduced by the Olmecs; it was also used by the Tarascans, Toltect, Mixtecs, Zempaolans, and Aztecs. It was not used in daily life; agriculture and religion were scheduled according to the Haab' and the Tzolki'in respectively, which will be added in a future update.

The Long Count is based on powers of 20, with one exception; it does not make any serious effort to keep in sync with the sun or the moon (but Maya inscriptions do usually include the lunar phase). 20 k'ins (days) make a uinal, and 18 uinals make a tun; a tun is thus 360 days, and this exception to the rule of 20 is most likely to make the tun approximate a solar year. 20 tuns make a k'atun, and 20 k'atuns make a b'ak'tun.

Mayan inscriptions generally only go up to the b'ak'tun; since a single b'ak'tun lasts about 1,400 years, there was little need to go any higher. However, inscriptions at Palenque do indeed advance to the next place, which historians call a piktun. There is evidence of even bigger groups, termed kalabtun, k'inchiltun, and alautun, but they are poorly attested and so are not included in this converter. It is unclear whether a piktun was 20 b'ak'tun long or just 13. In my own research, I came across people arguing both ways; to me (and I stress I am no sort of historian), it looks like a piktun was indeed 20 b'ak'tuns long, and the idea it was only 13 derives from the fact that the Mayans considered 13 to be an auspicious number. The transition from one b'ak'tun to the next was believed to be the end of an era and a time of great change, and so was marked by important rituals and ceremonies, and so the beginning of b'ak'tun 13 would be particularly well-marked. This significance attached to b'ak'tun 13 is why people believed the Long Count would end on 21 December 2012, and with it the world, despite actual Mayans consistently saying that it was simply the end of an era.

The epoch is Julian Day 584282. Uniquely among traditional calendars, all positions in the Long Count start at 0 rather tha 1, which makes conversion algorithms much simpler.

https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar
https://en.wikipedia.org/wiki/Maya_calendar
https://en.wikipedia.org/wiki/Baktun
https://en.wikipedia.org/wiki/Aztec_calendar
https://en.wikipedia.org/wiki/Mesoamerican_calendars
http://www.maya-portal.net/calendar/long_count
https://mayan-calendar.com/ancient_longcount.html
https://www.ancientpages.com/2016/08/11/on-this-day-in-history-mesoamerican-long-count-calendar-begins-on-august-11-3114-b-c/
https://maya.nmai.si.edu/calendar/maya-calendar-converter
https://www.timeanddate.com/calendar/mayan.html
http://coyote-wind.com/studios/MayaDate/about.html
https://mayanpeninsula.com/mayan-calendar/
https://www.nationalgeographic.com/news/2012/12/121207-maya-truly-did-not-predict-doomsday-apocalypse/

------------------
GEORGIAN CALENDARS
------------------

The Georgian calendar was proposed in 1745 AD/1123 AH by Rev. Hugh Jones of Maryland, under the pseudonym Hirossa Ap-Iccim. It is named in honour of King George II of Britain, and has nothing to do with Georgia, which uses the Gregorian calendar. It appears to be the first stab at designing a perennial calendar, and thus is ancestral to the Positivist, World, Gorman, and International Fixed Calendars.

Jones actually designed two different calendars, both of which he called the Georgian. The original was published in the July 1745 issue of The Gentleman's Magazine, in which Jones set out a number of problems he had with both the Julian and Gregorian calendars, as well as how to fix them. The year is divided into 13 months of exactly 4 weeks each, and each month always begins on a Sunday. Jones proposed that the months simply be referred to by number, but if people wanted to retain individual names, the 13th should be named Georgy aftr King George, in the same way July is named after Julius Caesar. The last day of a common year falls outside the week, and is the day on which Christmas is celebrated; Jones chose this scheme because a celebration as important as Christmas deserved a unique day.

Leap years fall every four years, except in years which are integer multiples of 132; leap day, which may be called Olympiad (as it is in this converter), British Lustrum, or National Day, comes the day after Christmas and also falls outside the week. Jones calculated that this would be accurate to one day in 10,000 years, based on a measurement of the equinoctial year as being 365 days 5 hours 48 minutes 47 seconds. Based on current measurements, which give the year as 2 seconds shorter, this scheme is accurate to about 20.34 seconds each year, which means it actually gains about 1 day in 4,247 years; this is still better than the Gregorian calendar, and comparable to the Jalali.

The calendar is intended to begin on the southern solstice, with the epoch falling in 4 BC so as to coincide with the actual birth of Jesus. Year 1750 in this calendar is intended to begin on 10 December 1746 by the Julian calendar, which means its epoch is Julian Day 1720319.

In 1753, Jones published The Pancronometer, or Univesal Georgian Calendar, a pamphlet in which he set out a new version of the Georgian calendar. The structure is the same, but this version's epoch is Julian Day 2362414, corresponding to 10 December 1755 in the Julian calendar; thus, like the French Republican calendar, it would represent a clean break and the start of a new era. In this version, the months are all named after Christian saints.

http://myweb.ecu.edu/mccartyr/hirossa.html

--------------
JUCHE CALENDAR
--------------

The Juche calendar is another variant of the Gregorian calendar, used exclusively in North Korea. Its epoch is 1 January 1912 AD, the year Kim Il-sung was born. It was implemented in 1997, three years after Kim's death, and prior year numbers retroactively changed. All official North Korean records and correspondence use Juche, but usually include the Gregorian year as well for convenience.

For years prior to Juche 1, North Korean records simply use the Gregorian date; for date ranges which cross the Juche epoch, Gregorian dates are used exclusively. But that's boring, so I decided to implement it as if the Juche calendar was proleptic. (It will still give the correct date correspondence for any legitimate Juche date).

http://world.clndr.org/calendars/Juche-calendar
https://koryogroup.com/blog/what-year-is-it-in-north-korea-understanding-the-juche-calendar
https://www.uritours.com/blog/north-korea-juche-calendar/
https://www.youngpioneertours.com/understanding-juche-calendar/
http://wafflesatnoon.com/north-korea-calendar/
https://en.wikipedia.org/wiki/North_Korean_calendar

--------------
INCA CALENDARS
--------------

The reckoning of time in Tiwantisuyu, also called the Inca Empire, is obscure and likely lost due to fact that the Andeans never developed writing . What they did have were khipus, devices which recorded data in the form of knots and colours in string, sometimes with cartouches attached to provide additional information. Alas, the knowledge of how to read teh khipus is itself lost due to suppression by the Spanish, and information found on the Internet is vague and contradictory. In attempting to work out the nature of the calendars used in Tiwantisuyu, I am relying primarily on Laurencich-Minelli, Laura & Magli, Giulio. (2008): A calendar Quipu of the early 17th century and its relationship with the Inca astronomy. Laurencich-Minelli and Magli analyse the Pachaquipu and correlate it with the Miccinelli documents to tease out the nature of the Inca calendars.

The Incas appear to have used a purely solar calendar for agricultural purposes. Like the ancient Egyptian calendar, it consisted of twelve 30-day months plus five extra days. However, since the agricultural calendar was not the civil calendar, the Incas of Tiwantisuyu were under no obligation to keep the year to a particular length; based on the observation towers around Cuzco, it appears that the first day of the agricultural year was determined by astronomical observations. But which observations? Some sources suggest it began with the heliacal rising of the Pleiades, similar to how the Egyptian agricultural year began with the heliacal rising of Sirius. Other sources point to the summer solstice as the starting point, but Laurencich-Minelli and Magli argue that this was most likely an attempt by the Anonymous Chronicler to impose European calendrical ideas on the Tiwantisuyu system, either to make it easier for his king to understand or to facilitate the adoption of Christianity in Tiwantiduyu (or both) by putting Capac Raymi, the festival of the summer solstice, nest to the Julian New Year's Day. Based on the significance of Inti Raymi, the festival of the winter solstice, in the civil calendar, I suspect the agricultural calendar began on the northern solstice, and have used this convention in this converter.

The civil year was a lunisolar calendar with some similarities to the old Babylonian calendar. Laurencich-Minelli and Magli argue that the Incas did not use the leap month convention seen in Middle Eastern calendars, but instead added extra days as needed to make up 365/6 in a calendar year; however, this gets hopelessly difficult to keep track of given that the civil year officially began on the last new moon before the northern solstice. It is possible that the Pachaquipu was intended to be used as both a lunar and solar calendar, depending on how one read it. In any case, I have decided to use the leap month convention which Laurencich-Minelli and Magli argue against on the grounds that it is simpler; if anyone can work out a clearer or more accurate picture of the Inca calendar, please let me know. Evidence from the Pachaquipu is that each month began on the sighting of the first crescent of the waxing moon. Some sources claim that the Incas also measured time by sidereal months; while sidereal months were certainly recorded, there is no evidence in the Pachaquipu that they factored into dating.

As to the era, if the Incas had a single epoch like the Olmecs did, we don't know about it. It is entirely possible they simply went by how long the current sapa inca was in power, or perhaps they gave each year a name. But we need an epoch, and so I have chosen 1438 AD/817 AH as the era. This is the year Pachacútec officially ascended as sapa inca of the Kingdom of Cuzco, and it was under his reign that the kingdom conquered the surrounding lands to become the Empire of Tiwantisuyu, or the Inca Empire as the Spanish called it. More relevant to our purposes, he is traditionally held to have introduced Inti Raymi, to which the civil and agricultural years were anchored. English Wikipedia, citing a random web page, says this happened in 1412 AD/808 AH, which would be quite the feat considering he wasn't even born until 1418 AD/814 AH. Spanish Wikipedia places the first Inti Raymi in the 1430s AD, which is just about possible given that Pachacútec ascended the throne in 1438 AD. As such, the northern solstice of 1438 AD is our best bet.

Sources (some of these are in Spanish):

https://www.researchgate.net/publication/1906435_A_calendar_Quipu_of_the_early_17th_century_and_its_relationship_with_the_Inca_astronomy
http://www.machupicchu-inca.com/inca-calendar.html
http://www.gogeometry.com/incas/inca_calendar.htm
https://www.kuodatravel.com/before-christmas-the-andean-celebrated-capac-raymi/
https://globalvoices.org/2012/01/16/peru-capac-raymi-the-andean-christmas/
https://www.worldhistory.biz/ancient-history/56365-inca-calendars.html
https://www.britannica.com/topic/Inca-calendar
http://amsacta.unibo.it/2350/7/Cap2.pdf
http://cuzcoeats.com/days-of-the-andean-week/
https://www.combonimissionaries.co.uk/index.php/2020/06/17/andean-calendar-the-celebration-of-the-sun/
https://www.history.com/topics/south-america/inca
https://en.wikipedia.org/wiki/Inca_Empire
https://www.ducksters.com/history/inca/science_and_technology.php
https://ancient-inca-empire.weebly.com/religious-festivals.html
https://en.wikipedia.org/wiki/Inti_Raymi
https://es.wikipedia.org/wiki/Inti_Raymi
http://www.discover-peru.org/inti-raymi/
https://www.classzone.com/books/en_espanol_shared/ML1/ML_1_Inti_Raymi/ml_1_inti_raymi.html
https://www.cachilife.com/inti-raymi-peru-festival/
https://es.wikipedia.org/wiki/Calendario_incaico
https://es.wikipedia.org/wiki/Cápac_Raymi
https://pueblosoriginarios.com/sur/andina/inca/calendario.html
https://www.boletomachupicchu.com/astronomia-y-calendario-inca/
https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/calendars-south-american-calendars
https://en.wikipedia.org/wiki/Pachacuti
https://es.wikipedia.org/wiki/Pachacútec
https://www.cachilife.com/inti-raymi-peru-festival/

----------------
BAHÁ'Í CALENDARS
----------------

The Bádí, or Báhá'í, calendar is the liturgical calendar of the Bahá'í faith. It combines aspects various traditional Iranian calendars.

It was first described by the Báb, foundational prophet of Bábism and Báhá'í, in the Book of Divine Names and the Persian Bayán, around 1223 AH. The Báb described a calendar consisting of 19 "months" of 19 days each, plus 4 or 5 extra days, but did not describe where the extra days should be placed. In Kitáb-i-Aqdas, published in 1252 AH, Bahá'ulláh clarifies this, stating that the extra days go between the 18th and 19th months. Bahá'ulláh also specified that the year begins on the day the sun enters Aries, which Bahá'ís understand as referring to the northward equinox.

The 19×19 structure appears to be derived from Islam, in which the number 19 is considered significant. Likewise, the day officially begins as sunset, as in the Lunar Hijri calendar. The year beginning on the northward equinox continues the earlier Zoroastrian tradition.

The era is, naturally, the Báhá'í era (BE). 1 BE is the year of publication of the Book of Divine Names, corresponding to 1223 AH (1844 AD). For 140 years or so after the foundation of Bahá'í, there was discrepancy between practice in different countries. In Asia, practitioners were mostly converts from Islam, and commonly observed festivals according the phases of the moon. In Europe and the Americas, they were mostly former Christians, and celebrated Naw-Ruz on the 21st of March. In 171 BE (1393 AH), the Universal House of Justice ruled that the calendar would henceforth begin on the day of the northward equinox according to Iranian Standard Time, bringing worldwise practices into harmony; they also clarified that practitioners would celebrated Naw-Ruz on the day of the equinox according to their local timezones.

An important break with Islam is that Bahá'í allows for intercalation. The year sometimes has an extra day to keep it aligned with the equinoxes. Furthermore, the births of the Báb and Bahá'ulláh are celebrated on the 1st and 2nd days respectively of the 8th new moon after the northward equinox (where new moon appears to refer to the first visible crescent rather than the dark moon).

The sidereal Bahá'í calendar is one invented by me, based on an alternative interpretation of Bahá'ulláh's intruction that Nowruz fall on the First Point of Aries. In the sidereal calanedar, Nowruz falls on the day that the sun enters conjunction with 4 Arietis, the equinoxmost star in Aries.

Suurces:
* https://en.wikipedia.org/wiki/Bah%C3%A1%CA%BC%C3%AD_calendar
* https://www.bahaiblog.net/bahai-calendar/
* http://www.bahai-ottawa.org/significant_bahai_dates_178_be_2021_2022.pdf
* https://bahaipedia.org/Calendar
* https://www.bahai.ca/holy-days/
* https://bahai-library.com/taylor_novelty_badi_calendar
* https://bahai-library.com/mihrshahi_wondrous_new_day
* https://reference.bahai.org/en/t/b/KA/ka-43.html#gr2
* https://universalhouseofjustice.bahai.org/activities-bahai-community/20140710_001
* https://bahai-library.com/nakhjavani_ninth_cycle
* https://web.archive.org/web/20061002101423/http://www.bcca.org/bahaivision/docs/today.html
* https://web.archive.org/web/20151222225356/http://www.religiouslife.emory.edu/documents/Baha_i%20Holy%20Days%2050%20year%20calendar.pdf
* https://bahai-library.com/velasco_mihrshahi_badi_calendar
* https://en.wikipedia.org/wiki/Bah%C3%A1%CA%BC%C3%AD_Holy_Days
* https://bahaisofketchikan.org/the-bahai-calendar/
* https://oceanoflights.org/file/012_ORD_The_Baha'i_Year_en.pdf
* https://bahaisofketchikan.org/wp-content/uploads/2021/08/20141211_UHJ_50-Yr_Table_of_Dates_English.pdf

-------------------------
IRANIAN NATIONAL CALENDAR
-------------------------

The Shahanshahi, or Imperial, calendar is a variant of the Solar Hijri calendar used briefly in Iran from 1354 to 1358. Shah Pahlavi and the Parliament decided to court nationalist opinion by changing the era to 1181 BH, the year that Cyrus the great ascended the throne; as such, the year 1354 instantly became 2535. When the Islamic Revolution happened, the Ayatollah Khamenei reverted Iran to counting years from the Hijra to emphasis that Iran was a Muslim country.

Sources:
* https://iranicaonline.org/articles/calendars

-----------------
CHINESE CALENDARS
-----------------

A variety of calendars have been used in China throughout history, which have influenced the calendars of the surrounding countries where they were not adopted wholesale.

A feature peculiar to the Chinese calendars is the solar term, or jiéqì, which is key to determining dates. Solar terms are a generalisaton of the concepts of solstices and equinoxes, and divides the ecliptic into 24 equal sections of 15º, anchored on the southern solstice (known as Dōngzhì). 12 of the solar terms are termed major solar terms (zhōngqì), four of which begin on the solstices and equinoxes; the other 12 are known as minor solar terms. Minor and major solar terms alternate through the year.

There is evidence that the earliest Chinese calendars were purely solar, beginning on either the date of the southern solstice or the first of Lìchūn. The modern calendar consists of 12 lunar months, making the Chinese calendar an oddity in having gone from purely solar to lunisolar when the general trend is to do the opposite.

Days in the Chinese calendars start at midnight. Each solar term begins on the day when the sun crosses that part of the ecliptic, regardless of when in the day that happens. For example, if the sun crosses 330º at 23:59 on 19 February, then Yǔshuı̌ begins on 19 February. Likewise, each month begins on the day of the new moon, regardless of when in the day the new moon actually happens. Rather than maintain seperate calendars for every single timezone, overseas Chinese consider the months and solar terms to begin on the dates they do according to Chinese standard time, a convention also used by this converter.

The calendar is designed so that the southern solstice will always fall on a day during Shíyīyuè, the eleventh month of the year. The leap month can fall at any point in the year; if there are 12 complete lunations between the end of one Shíyīyuè and the start of the next, the first of these lunations to begin after the start of one major solar term and finish before the beginning of the next major solar term is the leap month. The leap month shares its number with that of the month preceding it, which allows the southern solstice to still fall in the eleventh month. In most cases, the leap month also shares it sname with the preceding month, differentiated by the addition of "Rùn" (for "extra") prior to the month name. The exception is if the leap month follows the 12th month; in this case, the leap month, being the final month of the year, is called "Làyuè", while the 12th month is called "Shíèryuè" ("12th moon").

Older versions of the calendar assigned leap months according to the Metonic cycle, whereby there are seven leap years according to a fixed 19-year cycle. It also used mean values for the positions of the sun and the moon, which meant that the leap month could fall anywhere in the year with equal probability. In the early 17th century, Jesuit missionaries introduced new astronomical and mathematical tools, most notably logarithms, which made calculations much easier. Since then, months and solar terms have been determined by observation, which meant that to make this converter work, I had to write a Fortran module to do some serious number crunching.

For most of history, the Chinese calendar did not make use of any for of calendar era, instead simply referring to years by the reigning emperor. In the early 20th century, Liu Shipei took inspiration from the Western practices of dating events based on Jesus and Muhammad, and proposed dating events from the reign of Emperor /Huangdi, or the Yellow Emperor, the legendary first Emperor of China, as part of a scheme to promote Han nationalism and oppose the Manchu Qing social order. Other proposals for a Chinese era include basing the year on Confucious, due to his profound influence on Chinese culture; Emperor Qin Shi Huang, for first uniting China and indeed introducing the very idea of Chinsa; the Gonghe Regency, as this is the earliest point in Chinese history for which a definitive date can be assigned; and Emperor Yao, considered to be the model of the ideal monarch. King Fuxi, the mythological first king of China, has also been proposed, but reliable information on him is so scarce it's not even certain whether he really existed.

Sources:
* https://en.wikipedia.org/wiki/Chinese_calendar
* https://en.wikipedia.org/wiki/Chinese_New_Year
* https://en.wikipedia.org/wiki/Solar_term
* https://www.chinesecalendaronline.com
* https://chinese-year.com
* https://chinesenewyear.net/
* https://www.almanac.com/content/chinese-new-year-chinese-zodiac
* https://www.almanac.com/fact/chinese-new-year-the-new-year-is-holiday
* http://www.chinesenewyears.info/chinese-new-year-calendar.php
* https://www.travelchinaguide.com/intro/1985-chinese-zodiac-ox-year.htm
* http://www.cma.gov.cn/en2014/20150311/20170119/201701/t20170119_388239.html
* https://www.chineseastrologyonline.com/chinesezodiac/
* http://en.nongli.info/years/index.php?year=1985
* https://www.teasetc.com/education/chinese-lunar-new-year-calendar-and-zodiac-animals
* https://www.hko.gov.hk/en/gts/time/calendar/pdf/files/1985e.pdf
* https://www.hko.gov.hk/en/gts/time/24solarterms.htm
* https://www.friesian.com/chinacal.htm
* https://www.weather.gov/media/ind/seasons.pdf
* https://www.calendar-12.com/moon_phases/1985
* https://www.mastertsai.com/ChineseCalendar/The_Mathematics_of_the_Chinese_Calendar.pdf
* https://www2.ihp.sinica.edu.tw/file/1097kaBVwNb.pdf

---------------------------------------------
CHINESE QUARTER REMAINDER CALENDAR (SIFEN LI)
---------------------------------------------

The Quarter Remainder calendar, Sifen li, or Later Han Qarter Calendar (四分曆), was introduced during the reign of Emperor Zhang of the Eastern Han dynasty; it should not be confused with the earlier Warring States Quarter Calendars. It defines the solar year as being exactly 365 + 1/4 days long, and the synodic month as 29 + 499/940 days; this results in 19 calendar years containing precisely 235 months. Leap months are added as needed to ensure that the southern solstice always falls in the 11th month. Since this calendar predates the introduction of solar terms, I am inserting the leap month immediately before the 11th month each time.

Sources:
* https://en.wikipedia.org/wiki/Chinese_calendar
* https://zh.wikipedia.org/wiki/%E5%BE%8C%E6%BC%A2%E5%9B%9B%E5%88%86%E6%9B%86
* http://www.webexhibits.org/calendars/calendar-chinese.html
* https://chinesehsc.org/chinese_calenders.shtml
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English)

-------------
ZHOU CALENDAR
-------------

The Zhou calendar is the oldest attested lunisolar calendar used in China, dating to the early Western Zhou dynasty. It is a good deal more straightforward than the modern Chinese calendar; indeed, its structure is basically the same as the Inca civil calendar. Under the Zhou calendar, each year begins with the last new moon on or before the winter solstice, and leap years put the extra month right at the end. For the Zhou epoch, I have chosen the new moon of December 428 BC, which corresponds to the year the calendar was introduced. The new moon and the southern solstice both fell on the same day that year, and both around local noon; this corresponds to year 39 of the 76-year Callipic cycle (year 38 if one starts counting at 0). I have also chosen a timezone of UTC + 7 hr 14 min 52.5 sec, based on when local midnight falls in Fenghao, the ancient capital of Zhou.

Sources:
* https://en.wikipedia.org/wiki/Chinese_calendar
* https://en.wikipedia.org/wiki/Western_Zhou
* http://en.chinaculture.org/library/2008-02/01/content_26219.htm
* http://www.webexhibits.org/calendars/calendar-chinese.html
* https://chinesehsc.org/chinese_calenders.shtml

----------------
ZHUANXU CALENDAR
----------------

The Zhuanxu calendar was introduced in 366 BC by Qin Xian II as an assertion of Qin's independence from Zhou during the Warring States period. It has the same pattern of months and years as the Zhou calendar, but starts in Guìyuè rather than Júyuè, the 10th month in the modern Chinese calendar. I have chosen 384 BC, the start of Xian's reign, as the epoch.

Sources:
* https://zh.wikipedia.org/wiki/%E9%A1%93%E9%A0%8A%E6%9B%86

------------
XIA CALENDAR
------------

The Xia calendar was named for the calendar believed to have been used in the legendary Xia dynasty, and is modelled on descriptions found in Chinese classics. However, barring a surprising coincidence, it is *not* that calendar. Rather, the Xia calendar was introduced in 422 BC by one of Zhou's various rival states. The year begins two or three lunations after the new moon on or preceding the say on which the southern solstice falls, as in the modern calendar. Unlike the modern calendar, the leap month is always placed at the end of the year, because this is before the introduction of solar terms.

Sources:
* https://zh.wikipedia.org/wiki/%E9%A1%93%E9%A0%8A%E6%9B%86
* https://zh.wikipedia.org/wiki/%E5%A4%8F%E6%9B%86_(%E5%8F%A4%E5%85%AD%E6%9B%86)

--------------
SHANG CALENDAR
--------------

The Shang calendar, named after the legendary state that preceded the Zhou, was another calendar that arose during the Warring States Era. There is very little information available on this calendar, other than that the first month of the year was the one in which the southern solstice fell. Based on Chinese Wikipedia, it appears to have first been implemented eight years after the Zhuanxu.

Sources:
* https://zh.wikipedia.org/wiki/%E6%88%98%E5%9B%BD%E5%9B%9B%E5%88%86%E5%8E%86

-----------
LU CALENDAR
-----------

The Lu calendar was another quarter-remainder calendar from the Warring States Era which started the year on the month of the southern solstice. It was established by Lu, apparently around the same time as the Shang calendar. Since there it little information available, I have chosen as its epoch the first year of the reign of Ning, Duke Dao of Lu.

Sources:
* https://zh.wikipedia.org/wiki/%E6%88%98%E5%9B%BD%E5%9B%9B%E5%88%86%E5%8E%86

------------
YIN CALENDAR
------------

The Yin calendar is the last of the Six Ancient Calendars, introduced in the Song state during the Warring States Era. Unlike the others, it starts the year on the new moon following the southern solstice. Like the others, it defines the tropical year as exactly 365.25 days long, and has precisely 235 months in 19 tropical years.

Sources:
* https://zh.wikipedia.org/wiki/%E6%88%98%E5%9B%BD%E5%9B%9B%E5%88%86%E5%8E%86

---------------
TAICHU CALENDAR
---------------

The Taichu, or Great Inception, calendar was introduced in the 36th year of Emperor Wu of the Han Dynasty, and set the structure for all future Chinese calendars. It is also the oldest Chinese calendar for which full descriptions are available.

The Taichu calendar takes the basic structure of the Xia calendar, in which there are 235 month in 19 years, but defines the tropical year as 365 + 385/1539 days and the lunar month as 29 + 43/81 days. This is actually slightly less accurate to the seasons than the Warring States calendars; the numbers seem to have been chosen because 1539 == 81 * 19.

The most significant feature of the Taichu calendar is the introduction of the solar terms, which are key to fixing the time of the leap month. Furthermore, the leap month is for the first time defined as the first lunation betweeen two southern solstices during which the sun does not pass into a new major solar term.

Every subsequent calendar would be a refinement of the Taichuli.

Sources:
* https://zh.wikipedia.org/wiki/%E5%A4%AA%E5%88%9D%E6%9B%86
* http://www.chinastyle.cn/chinese-zodiac-calendar/tai-chu-calendar.htm
* http://chinaknowledge.de/History/Terms/calendar.html
* https://link.springer.com/content/pdf/10.1007%2F978-1-4614-6141-8_225.pdf

----------------
SANTONG CALENDAR
----------------

The Santong, or Triple Concordance, calendar is an adjustment to the Taichu calendar, introduced by Liu Xin. This system establishes that the southern solstice coincided with the new moon 143,126 years before the calendar was introduced... which also means they coincided in the 37th year of Emperor Wu. Needless to say, this is inaccurate, but the basic principle would continue to be used for centuries after until better methods of predicting the exact positions of the sun and the moon were discovered.

Sources:
* https://zh.wikipedia.org/wiki/%E4%B8%89%E7%B5%B1%E6%9B%86
* https://link.springer.com/content/pdf/10.1007%2F978-1-4614-6141-8_225.pdf

------------------------------
HAN QUARTER REMAINDER CALENDAR
------------------------------

In the 10th year of Emperor Zhang, Han China reverted to the Quarter Remainder system of the Warring States era. In this version, compiled by Li Fan, the tropical year is 365 + 1/4 days long, and the synodic month is fixed at 29 + 499/940 days. As in the Xia calendar, the southern solstice falls in the 11th month. As in the Triple Concordance calendar, solar terms are a key part of the calendar; each solar term lasts exactly 1/24 of a tropical year, and the leap month is the first month of the year which does not contain the beginning of a major solar term.

Sources:
* https://zh.wikipedia.org/wiki/%E5%BE%8C%E6%BC%A2%E5%9B%9B%E5%88%86%E6%9B%86
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English) chapter 8

------------------
QIANXIANG CALENDAR
------------------

The Qianxiang calendar was designed by Liu Hong towards the end of the Han dynasty, and was the official calendar of Sun Wu during the Three Kingdoms Era. It was notably accurate in predicting the phase of the moon and the dates of eclises; indeed, it was so accurate it was adopted as the official calendar of the Han Empire as soon as Liu published it.

The Qianliang calendar defines the tropical year as 365 + 189/589 days (365 days 7 hours 42 minutes 4.278 seconds), and the synodic month as 29 + 1117/1457 days (29 days 17 hours 54 minutes 28.14Z seconds); this is about 2 hours 7 minutes 43 seconds longer than the mean tropical year, and 4 hours 49 minutes 29 seconds longer than the mean synodic month, and the fractions preserve the equivalence of 19 tropical years to 235 synodic months yet is still less accurate than the Quarter Remainder calendar. The Qian Lian Li is notably the first ever work to make use of the argument of perigee for calculating the new moon.

Sources:
* https://zh.wikipedia.org/wiki/%E4%B9%BE%E8%B1%A1%E6%9B%86
* https://en.wikipedia.org/wiki/Liu_Hong_(astronomer)

----------------
JINGCHU CALENDAR
----------------

The Luminous Inception, or Jingchu, calendar was introduced by Cao Rui, the second king of Cao Wei, during the Three Kingdoms period. The introduction of the calendar marked the beginning of Rui's third era, Jingchu, for which the calendar is named.

It defines the tropical year as 365 + 455/1843 days, the synodic month as 29 + 2419/4559 days, and the anomalistic month as 27 + 2528/4559 days. It is thus somewhat more accurate than the Qianxiang calendar, but still less accurate than the Quarter Remainder calendar. Still, it remained in use for a remarkably long time - after the reunification of China, it was adopted as the official calendar by the Sima Jin, Liu Song, and Toba Wei. It was in the Eastern Jin that the calendar lasted the longest, persisting until the 8th year of Emperor Wu of Liang, for a total of 274 years (that's a long time for a Chinese calendar).

Sources:
* https://zh.wikipedia.org/wiki/%E6%99%AF%E5%88%9D%E6%9B%86
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English) chapter 9

--------------
SANJI CALENDAR
--------------

The Sanji calendar was introduced in Yao Qin, one of the Sixteen Kingdoms in existence after the fragmentation of Sima Jin, by King Wuzhao at the very start of his reign. It was the first algorithmic calendar to be more accurate than the Quarter Remainder calendar - it defined the tropical year to be 365 + 605/2451 days long, and the synodic month as 29 + 3217/6063 days, an arrangement that maintains the convention of exactly 235 months in 19 years. Thus, the calculated year is about 6 min 18 sec, and the calculated month is less than 1.5 seconds longer than the actual mean synodic month.

Sources:
* https://zh.wikipedia.org/wiki/%E4%B8%89%E7%B4%80%E6%9B%86

----------------
YUANJIA CALENDAR
----------------

The Yuanjia calendar was developed by He Chengtian and introduced in the 22nd year of Emperor Wen of the Liu Song dynasty. It refines the Sanji calendar with more accurate approximations of the tropical year and synodic month, which still maintain the 19:235 ratio. The tropical year is defined as 365 + 75/304 days, which is about 6 minutes 29 seconds longer than the mean tropical year. The synodic month is defined as 29 + 399/752 days, less than half a second longer than the mean synodic month. Its use persisted into the Southern Liang dynasty, until it was replaced by the Daming calendar during the reign of Emperor Wu.

Sources:
* https://zh.wikipedia.org/wiki/%E5%85%83%E5%98%89%E6%9B%86

----------------
XUANSHI CALENDAR
----------------

The Xuanshi calendar (also called the Yuanshi, but this can also refer to a later calendar introduced during the Yuan dynasty) was introduced at the very beginning of the Xuanshi era, the second era of King Juqu Mengxun of Bei Liang, and was also used in Tuoba Wei. It is most notable for breaking with the 19:235 year:month ratio; instead, 600 years in this calendar comprise 7421 months. The synodic month is defined as 29 + 47251/89052 days, which is a little less than 1 second longer than the mean synodic month. The tropical year is defined as 365 + 1759/7200 days, almost exactly 5 minutes longer than the mean tropical year. (Chinese Wikipedia gives the tropical year as 365 + 17589/72000 days, but the maths do not work with this value. I suspect it is a typo, especially since Martzloff gives the correct answer in his book).

Sources:
* https://zh.wikipedia.org/wiki/%E7%8E%84%E5%A7%8B%E6%9B%86
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendix F

---------------
DAMING CALENDAR
---------------

The Daming calendar was designed by Zu Chongzhi and presented to Emperor Xiaowu of Liu Song during the fifth year of the Daming era (462 AD, 162 BH). However, it was not implemented during his lifetime, only coming into use during the 8th year of Emperor Wu of Liang.

The Daming calendar defines the synodic month as 29 + 2090/3939 days, which is just 0.107 seconds longer than the true mean synodic month. The tropical year is defined as 365 + 9589/39491 days, less than 2 minutes longer than the actual tropical year; this works out to 144 leap months in 391 years. Zu furthermore defined the draconic month as 27 + 5598/26377 days, a mere 0.9 seconds less than the true mean value.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendix F
* https://zh.wikipedia.org/wiki/%E5%A4%A7%E6%98%8E%E6%9B%86_(%E7%A5%96%E6%B2%96%E4%B9%8B)
* http://www.herongyang.com/1000124_Zu_Chongzhi_and_the_Chinese_Calendar_Reform_of_462_AD.html

-------------------
ZHENGGUANG CALENDAR
-------------------

The Zhengguang calendar was worked out by a council of scholars appointed by Emperor Xuanwu of Tuoba Wei at the very end of his reign, when errors in the Xuanshi calendar had started to become apparent. It is also known as the Tortoise calendar after the Shengui (Divine Tortoise) era, when it was first presented to Emperor Xiaoming. It was finally implemented halfway through the Zhengguang era, hence the name.

The Zhengguang calendar defines the tropical year as 365 + 1477/6060 days, which is about 1 minute 13 seconds longer than the mean tropical year. The synodic month is defined as 29 + 39769/74952 days, just 0.4 seconds longer than the actual mean synodic month.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices F and G
* https://zh.wikipedia.org/wiki/%E6%AD%A3%E5%85%89%E6%9B%86

---------------
XINGHE CALENDAR
---------------

The Xinghe calendar was compiled by Lin Yexing towards the Dong Wei dynasty and introduced in the first year of Xinghe, the second era of King Heliuhun (AKA Gao Huan, Xianwu). It defines the tropical year as 365 + 4117/19860 days, which is slightly under 2 minutes longer than the mean tropical year. The synodic month is defined as 29 + 100647/208530 days, approximately 1.25 seconds longer than the modern mean synodic month. It is thus slightly less accurate than the Zhengguan calendar. This results in 6951 months in 562 years.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices D, F, and G
* https://zh.wikipedia.org/wiki/%E8%88%88%E5%92%8C%E6%9B%86

----------------
TIANBAO CALENDAR
----------------

The Tianbao calendar was designed by Song Jing and introduced by Emperor Wenxuan at the very start of his reign. The tropical year is defined as 365 + 5787/23660 days, about 3 minutes 10 seconds longer than the mean tropical year. The synodic month is defines as 29 + 155272/292635 days, just 0.8 seconds longer than the actual mean synodic month. This worked out to 8361 months in 676 years. It was thus better than the Xinghe calendar, but still less accurate than the Zhengguang calendar.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices D, F, and G
* https://zh.wikipedia.org/wiki/%E5%A4%A9%E4%BF%9D%E6%9B%86_(%E4%B8%AD%E5%9C%8B)

---------------
TIANHE CALENDAR
---------------

The Tianhe calendar was designed by Zhen Luan and introduced at the beginning of Tianhe, the second era of Emperor Wu of Bei Zhou. It adopts the Zhengguang calendar's convention of 4836 months in 391 years, but redefines the lengths of the tropical year and synodic month. The tropical year is defined as 365 + 5731/23460 days, less than 40 seconds longer than the modern mean tropical year (Martzloff's book has a typo; on p. 362, it gives the numerator of the fractional part as 5761, which is incorrect). The synodic month is defined as 29 + 153991/290160 days, a little over 10 seconds longer than the modern mean synodic month. The Tianhe calendar thus trades solar for lunar accuracy.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices D, and G
* https://zh.wikipedia.org/wiki/%E5%A4%A9%E5%92%8C%E6%9B%86

----------------
DAXIANG CALENDAR
----------------

The Daxiang, or Bingyin Yuan, calendar was designed by Ma Xiang and introduced at the beginning of the reign of Emperor Jing of Bei Zhou; it continued to be used into the early Sui dynasty. The tropical year is defined as 365 + 3167/12992 days, which is about 1 minute 16 seconds longer than the modern mean tropical year. The synodic month is defined as 29 + 28422/53563 days, about 2.5 seconds longer than the modern mean synodic month. There are 5541 months in 448 years.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices D, F, and G
* https://zh.wikipedia.org/wiki/%E5%A4%A7%E8%B1%A1%E6%9B%86

-----------------
KAIHUANG CALENDAR
-----------------

After the Sui dynasty reunited China, Emperor Wen decided that a new calendar was in order and commissioned Zhang Bin to design one. Zhang's new calendar was introduced in the fourth year of the Kaihuang era, which was Emperor Wen's first era.

The tropical year is defined as 365 + 25063/102960 days, about 46 seconds longer than the modern mean tropical year. The month is defined as 29 + 96529/181920 days, less than 2 seconds longer than the modern mean synodic month. There are 5306 months in 429 years.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices D, F, and G
* https://zh.wikipedia.org/wiki/%E9%96%8B%E7%9A%87%E6%9B%86

-------------
DAYE CALENDAR
-------------

The Daye calendar was developed by Zhang Hexuan towards the end of the Kaihuang era, when he noticed some errors in the Kaihuang calendar; however, it was not introduced until early in the reign of Emperor Yang, hence the name Daye calendar, after Yang's era name.

The tropical year is defined as 365 + 10363/42640 days, just 13 seconds longer than the actual mean tropical year when measured from one souther solstice to the next. The synodic month was defined as 29 + 607/1144 days, less than 0.4 seconds longer than the modern mean synodic month. There are 5071 months in 410 years. It is the most accurate calendar I am aware of that uses mean values - indeed, it matches the period from one solstice to another better than the Gregorian calendar matches the period from one equinox to the next!

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices D, F, and G
* https://zh.wikipedia.org/wiki/%E5%A4%A7%E6%A5%AD%E6%9B%86

--------------
WUYIN CALENDAR
--------------

The Wuyin, or Wuyin Yuan, calendar, is the last hisotrical Chinese calendar to use exclusively mean values. Designed by Fu Renjun and Cui Shan, it was introduced early in the Tang dynasty, in the second year of Emperor Gaozu.

There are 249 leap years in 676 year. The tropical year is defined as 365 + 2315/9464 days, about 2 and a half minutes longer than the modern tropical year. The synodic month was defined as 29 + 6901/13006 days, just under a second longer than the mean synodic month.

While it was a step back from the Daye calendar, further calendar reforms in the Tang dynasty would make the calendar ever more accurate as the variable speed of the sun and the moon across the zodiac were taken into account to generate incredibly precise almanacs.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), Appendices D, F, and G
* https://zh.wikipedia.org/wiki/%E6%88%8A%E5%AF%85%E5%85%83%E6%9B%86

------------------
XUANMMING CALENDAR
------------------

When Emperor Gaozong took the throne, he commissioned renowned scholar Li Chunfeng to revise the calendar yet again. Li's Linde calendar adapted Liu Zhou's proposed but never implemented Huangji li, formally introducing corrections to account for the sun's variable speed across the zodiac. It also employed the anomalistic month, first described in the Jingchu calendar, to fix the exact position of the moon relative to the ecliptic. It lasted for 66 years, but by the reign of Emperor Xuanzong, discrepancies between predictions and observations had become apparent. Around this time, the illustrious Buddhist monk Yi Xing, armed with Indian astronomical knowledge, produced the even more accurate Dayan calendar. His methods were slightly refined towards the end of Emperor Suzong's reign to produce the Baoying calendar. This was followed by the Zhengyuan (not to be confused with the Zhengguang) and Guanxiang calendars.

Unfortunately, this is where I hit a brick wall in my research. All the real information about the Linde and Dayan calendars is written in Classical Chinese and has not been translated into English, Irish, French, or Spanish, and due to the involved calculations, I was unable to derive the required figures mathematically. As such, I had no choice but to move past them and go straight to the Xuanming calendar, which succeeded the Guanxiang and has possibly the most intricate calculations of any algorithmic calendar.

The Xuanming calendar was implemented in the third year of Emperor Muzong. It defines the tropical year as 365 + 2055/8400 days, about 3 minutes 14 seconds longer than the mean tropical year. The synodic month is defined as 29 + 4457/8400 days, less than half a second longer than the modern mean synodic month. Finally, the anomalistic month is defined as 27 + 4658/8400 days, about 2 1/4 seconds shorter than the modern mean synodic month.

Based on the pioneering work of Liu Zhou, the Xuanming calendar applies asmall correction to the length of each solar term so they will begin at a point closer to astronomical reality. Taoist principles of balance are visible in the algorithm - the true positions of the the solstices are assumed to be identical to their mean positions. Furthermore, a set of values are subtracted from the length of each solar term between the southern solstice and the northward equinox, and then those same values are added to each solar term between the northward equinox and the northern solstice but in reverse order. The same numbers are then added in the original order to each solar term between the northern solstice and southward equinox, and finally subtracted in reverse order to each solar term between the southward equinox and the southern solstice. This might not sound like much, but when you plot it out, or even program an algorithm, the whole thing visually and mathematically echoes the taijitu.

The lunar adjustments are much more complex. The phase of the moon has two modifications, one solar and one lunar. The solar correction applies a modification based on the time between the start of the current solar term and the mean lunar phase, while the lunar correction uses the time from the start of the last anomalistic month to the lunar phase. Oh, but these aren't just simple addition and subtraction, oh no. The solar and lunar corrections are defined by complex, multi-stage algorithms that even have special cases for when it#s the 7nth day of the anomalistic month. Like I said, complex and intricate. Uniquely among calendars derived from the work of Liu Zhou, the Xuanming calendar splits the anomalistic month into two 'phases', with the exact modification dependent on both the phase of the anomalistic month and the time from the start of the anomalist month to the phase of the moon in question.

Sources:
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), chapters 5 and 10.

-------------------------------
VIETNAMESE TRADITIONAL CALENDAR
-------------------------------

For most of their history, Vietnam has imported calendars from China; the Gregorian calendar was also introduced in the 19th century by French conquerors. Today, the Gregorian calendar is used for official purposes, but the traditional calendar is still used for everyday activities and determining the dates of some national holidays and observances.

Prior to 1967 AD, the Vietnamese calendar was identical to the Chinese one. In 1967, North Vietnam switched from UTC+8, the same as China, to UTC+7 to better match local astronomy. This infamously led to the Tet offensive happening a day earlier than the South Vietnamese forces were expecting; because the new moon happened very close to midnight in 1968, it ended up falling a day earlier in North Vietnam, hence the South's lack of readiness.

For the epoch, I have chosen the reign of the mythical Kinh Dương Vương, traditionally said to be the first king of Văn Lang, the first Vietnamese nation. According to legend, Kinh Dương Vương united the tribes in Xích Quỷ in 2879 BC.

Sources:
* https://ling-app.com/vi/vietnamese-calendar/
* https://www.vietnamonline.com/az/vietnam-lunar-calendar.html
* https://www.vietnam-tour.biz/blog/traditional-calendar-vietnamese-people/
* https://informatik.uni-leipzig.de/~duc/amlich/calrules_en.html
* https://informatik.uni-leipzig.de/~duc/amlich/vncal_engl.html

----------------------------------
DANGUN (KOREAN LUNISOLAR) CALENDAR
----------------------------------

Like Vietnam, Korea has always imported calendars from China (and then exported them to Japan). The Dangun calendar is the Chinese lunisolar calendar, but calibrated to Korean Standard Time. Korea has used the Gregorian calendar since 1896, with North Korea switching to the Juche calendar for years after 1997 AD; however, the lunisolar calendar continues to be used to set national holidays and observances.

For the epoch, I have chosen 2333 BC, the traditional year in which Korea was founded by the mythical King Dangun.

Sources:
* https://en.wikipedia.org/wiki/Korean_calendar
* http://www.lifeinkorea.com/Calendar/holidays.cfm
* https://books.google.ie/books?id=H4CsWDEi52IC&pg=PA91&redir_esc=y#v=onepage&q&f=false
* https://ui.adsabs.harvard.edu/abs/2017PKAS...32..407P/abstract


---------------------
KOREAN ROYAL CALENDAR
---------------------

Korea adopted the Gregorian calendar for official purposes in 1896 AD. For 16 years following World War II, years were officially numbered from the mythical date of the foundation of Gojoseon, the first Korean kingdom, by Dangun, a system which had been used unofficially at a small scale for some years beforehand. And darn it, I do love me an alternate calendar, so it's going in.

Sources:
* https://en.wikipedia.org/wiki/Korean_calendar

---------------------------
JAPANESE LUNISOLAR CALENDAR
---------------------------

All surviving Japanese calendrical records describe calendars as having been imported from Korea, which in turn got them from China. The Nihonkoki, the second-oldest chronicle of Japanese history, describes the court of Emperor Kinmei inviting a scholar from the Korean kingdom of Baekje to Japan to design a calendar; it further states that the calendar, a translation of the Daye calendar, was implemented in the 12th year of Empress Suiko (604 AD). Other calendars followed; indeed, standard practice was to import a stack of almanacs from China or Korea every year and use those. Most notably, the Xuanming calendar was introduced in 862 AD and saw continued use for 823 years, at which point discrepancies between the Xuanming li's algorithms and astronomical observations had become too great for the calendar to be useful. To this end, the first native Japanese calendar was designed in the second year of Jokyo (1685 AD), which corresponds to the 21st year of Emperor Reigen. Unfortunately, I have been unable to find any good information about this calendar. This was itself revised a number of times before the adoption of the Gregorian calendar in 1872, in the fourth year of the Meiji era.

The Japanese lunisolar calendar is not a real calendar; rather, it is a calendar I designed, which takes the rules of the Chinese calendar and adapts them to the Japanese meridian, and thus represents a system for calculating the dates of traditional Japanese festivals if Japan had adopted the same policy towards calendars as Korea and Vietnam did. Due to how timezones work, the day and month in the Japanese lunisolar calendar are identical to those of the Korean lunisolar calendar. For the era, I have chosen 660 BC, the date that Emperor Jimmu is said to have arrived from Korea and founded Japan.

Sources:
* https://www.ndl.go.jp/koyomi/e/history/index.html
* https://www.ndl.go.jp/koyomi/e/history/calendar.html
* https://www.ndl.go.jp/koyomi/e/quiz/index.html
* https://en.wikipedia.org/wiki/Japanese_calendar
* Astronomy and Calendars: The Other Chinese Mathematics 104 BC - AD 1644; Jean-Claude Martzloff, Springer, 2009 (French), 2016 (English), p. 277 (ch. 10)
* https://books.google.ie/books?id=nSNLAAAAIAAJ&pg=PA160&dq=&redir_esc=y#v=onepage&q&f=false
* https://books.google.ie/books?id=9sawofv6lJsC&pg=PA46&dq=&redir_esc=y#v=onepage&q&f=false
* https://www.nippon.com/en/japan-glances/jg00122/the-japanese-calendar.html
* https://smilenihongo.com/months-in-japanese

--------------------------
JAPANESE IMPERIAL CALENDAR
--------------------------

The Koki, or Japanese imperial calendar, is a variant of the Gregorian calendar first attested in 1872 AD (Koki 2532), which dates events from the arrival of the mythical Emperor Jimmu in Japan from Korea. It was in popular use during the Meji Contitution, and WWII-era Japanese military equipment was often identified by the IJC year number. This is why the Mitsubishi A6M Carrier Fighter was known as the Zero - since it entered service in 1940 AD, which was Koki 2600, it was given the designation Type 00.

This calendar is still occasionally used by Shinto priests, but is no longer officially used in Japan. Instead, years are officially dated by the reign of the current emperor†, with the Gregorian calendar used for everyday purposes.

†Very strictly speaking, years are dated according to the imperial era. Since 1868 AD (Koki 2528), the rule has been that the era only changes upon the ascension of the new emperor; prior to the Meiji era, the emperor could declare a new era whenever he wanted.

Sources:
* https://en.wikipedia.org/wiki/Japanese_imperial_year
* https://www.nippon.com/en/japan-glances/jg00122/the-japanese-calendar.html
* https://www.sljfaq.org/afaq/number-dates.html
* https://wattention.com/understanding-japanese-eras/
* https://notesofnomads.com/japanese-date-system/
* https://smilenihongo.com/months-in-japanese

------------------------------
MONGOLIAN TRADITIONAL CALENDAR
------------------------------

The Mongolian traditional calendar was designed by Ishbaljir, a Buddhist monk, during the Qing dynasty. It is directly modelled on the Chinese lunisolar calendar; the primary difference is that the Mongolian month ends on the new moon, and so each month starts a day later than the corresponding month in China. Like in China, the Gregorian calendar (referred on Mongolian as 'argyn toolol', or chronology of method) is used for everyday purposes. The traditional calendar (referred to as 'bilgiin toolol', or chronology of wisdom) is used for the setting of traditional festivals and certain national holidays and observances.

For the epoch, there is only one choice. I went with the ascension of Temujin Borjigin to the position of Genghis Khan.

Sources:
* https://en.wikipedia.org/wiki/Mongolian_calendar
* http://www.tsalo.fi/Mongolian%20Calendar.html
* https://publicholidays.asia/mongolia/lunar-new-year/
* https://www.travelbuddies.info/mongolian-lunar-new-year-tsagaan-sar/
* https://www.selenatravel.com/mongolia-lunar-new-year
* https://www.officeholidays.com/holidays/mongolia/tsagaan-sar
* https://panoramicjourneys.com/experience/the-festivals-of-mongolia
* https://linguapedia.info/en/phrases/seasons-months-in-mongolian.html
* https://danielroy.tripod.com/cgi-bin/alternate/mongolia/time-reckoning.html
* https://www.babelstone.co.uk/Blog/2013/03/todays-date-in.html

---------------
QADIMI CALENDAR
---------------

The Qadimi ("ancient", "traditional") calendar is the liturgical calendar traditionally used by Iranian Zoroastrians. It is derived from the Vihezakik calendar, when Iranian Zoroastrians stopped performing intercalations afte 385 AH due to their community being in chaos.

Structurally, it is a slight revision of the Young Avestan calendar. A year consists of 365 days with no intercalation at all. Each year consists of 12 months of 30 days plus five extra days, known as the Gathas, placed at the very end of the year.

Three calendar eras are in common use. Before the Muslim conquest, Iranians usually dated events according to how long the current shah had been in power; the Yazdegerdi era continues the count from the reign of Yazdegerd III, the last Zoroastrian shah of Iran, and its epoch is Julian Day 1952062.

Anno Zoroastres dates events from the notional date of Zarathustra's birth, placing its epoch on Julian Day 1579397. This era arose during the Seleucid dynasty; the Zoroastrian priesthood did not care for the Greek ruling class dating things according to the Era of Alexander, and sought to counter this by establishing the Era of Zarathustra. Based on Babylonian records, they determined that Cyrus the Great conquered Babylon when Zarathustra was 30 years old, which would place his birth at 30 years before Alexander.

The final era is the Zoroastrian Religious Era, proposed by the Zarathushtrian Society of Northern California. In this scheme, Zarathstra's mission coincided with the beginning of the Age of Aries, and so the epoch should be the northward equinox of 2360 BH, or Julian Day 1086712. However, I have chosen Julian Day 1086647 in order to keep the months consistent.

* https://en.wikipedia.org/wiki/Zoroastrian_calendar
* https://iranicaonline.org/articles/calendars
* https://www.iranchamber.com/calendar/articles/old_iranian_calendars1.php
* https://www.jstor.org/stable/4299877
* https://jihs.ut.ac.ir/article_36130_4d9c41f00283f0368d75be229e4d3af1.pdf
* http://www.avesta.org/zcal.html
* https://authenticgathazoroastrianism.org/2021/08/18/1268/
* https://zoroastrian.org/articles/The%20Precise%20Iranian%20Calendar.htm
* http://www.heritageinstitute.com/zoroastrianism/calendar/index.htm
* https://www.iranchamber.com/calendar/articles/calendar_systems_origins.php
* https://www.iranchamber.com/calendar/articles/iranian_months.php
* https://www.iranchamber.com/calendar/articles/iranian_months.php
* http://www.zcserv.com/
* https://parsikhabar.net/education/jamshedi-navruz-a-detailed-explanation-of-the-calendars/3954/
* http://www.avesta.org/antia/Enigmatic_Response_of_Three_Dasturjis_to_the_United_Zarthoshti_Calendar.pdf
* http://www3.sympatico.ca/zoroastrian/cal.html

-----------------
SHENSHAI CALENDAR
-----------------

The Shenshai, or Shahanshi ("imperial", "royalist") calendar is the one followed by some Indian Zoroastrians. It differs from the Qadimi calendar by being one month behind, due to the Indian Zoroastrians keeping up intercalation for one more cycle than those in Iran. The discrepancy was noticed by Jamasp Peshotan Velati in 1099 AH (1642 SE), and the majority of Indian Zoroastrians decided to adopt the Iranian reckoning on the basis that it came from the Zoroastrian homeland. A minority decided to maintain their own tradition, and this calendar became known as the Shahanshai or Shenshai calendar, while the Iranian calendar was known as the Qadimi.

Calendar eras are the same as the Qadimi calendar.

* https://en.wikipedia.org/wiki/Zoroastrian_calendar
* https://iranicaonline.org/articles/calendarslendar
* https://www.iranchamber.com/calendar/articles/old_iranian_calendars1.php
* https://www.jstor.org/stable/4299877
* https://jihs.ut.ac.ir/article_36130_4d9c41f00283f0368d75be229e4d3af1.pdf
* http://www.avesta.org/zcal.html
* https://authenticgathazoroastrianism.org/2021/08/18/1268/
* https://www.ndtv.com/india-news/parsi-new-year-2021-navroz-date-history-significance-and-food-2511333
* http://www.heritageinstitute.com/zoroastrianism/calendar/index.htm
* https://www.iranchamber.com/calendar/articles/calendar_systems_origins.php
* https://www.iranchamber.com/calendar/articles/iranian_months.php
* https://www.iranchamber.com/calendar/articles/iranian_months.php
* http://www.zcserv.com/
* https://parsikhabar.net/education/jamshedi-navruz-a-detailed-explanation-of-the-calendars/3954/
* http://www.avesta.org/antia/Enigmatic_Response_of_Three_Dasturjis_to_the_United_Zarthoshti_Calendar.pdf
* http://www3.sympatico.ca/zoroastrian/cal.html

----------------------
YOUNG AVESTAN CALENDAR
----------------------

The Egyptian calendar was introduced to Iran following the conquest of Egypt by Cambyses. The Zoroastrian priests considered the Egyptian calendar much simpler than their own (too had it was less accurate), and the Young Avestan calendar replaced the Old Avestan calendar as the official calendar of Iran around 1063 BH.

The Young Avestan calendar consists of 12 months of 30 days each plus five Gathas between Aban and Azar. This arises because the Iranians copied the Egyptian calendar exactly, but because at that point the 1st of Choak fell around the northward equinoox, the fourth month of the Egyptian calendar thus became the first month of the Avestan calendar, with both calendars retaining the same says for the start of the month and the position of the extra days.

Calendar eras are the same as for the Qadimi and Shenshai calendars.

* https://iranicaonline.org/articles/calendars
* https://www.iranchamber.com/calendar/articles/old_iranian_calendars1.php
* https://www.jstor.org/stable/4299877
* https://zoroastrian.org/articles/The%20Precise%20Iranian%20Calendar.htm
* https://www.iranchamber.com/calendar/articles/iranian_months.php
* https://www.iranchamber.com/calendar/articles/iranian_months.php
* https://parsikhabar.net/education/jamshedi-navruz-a-detailed-explanation-of-the-calendars/3954/
* http://www.avesta.org/antia/Enigmatic_Response_of_Three_Dasturjis_to_the_United_Zarthoshti_Calendar.pdf
* http://www3.sympatico.ca/zoroastrian/cal.html
* BI Marshak, The Historico-Cultural Significance of the Sogdian Calendar, Iran, Vol. 30 (1992), pp. 145-154

--------------------
SHAHANSHAHI CALENDAR
--------------------

The Shahanshahi ("Imperial") calendar is a former fixed solar calendar used by Zoroastrians in Iran and India, and was the liturgical calendar of Iran before the Arab conquest. It consists of 12 months of 30 days each plus five extra days; additionally, an extra month is added every 120 years to keep the start of the year in sync with the northward equinox. As such, its accuracy is equal to that of the Julian calendar. The Shahanshahi calendar is a slight modification of the Vihezakik calendar, implemented when Yazdegerd the Sinner moved the Gathas to the end of the year.

Al-Biruni suggests that there were in fact two variants of the Shahanshahi calendar - one with a leap month every 120 years, and another with a leap month every 116 years, which would sync up well with the sidereal year. However, Antia points out that al-Biruni describes the extra month as falling every 120 years and every 116 years in the same sentence, and he seems to be the sole source for this claim. As such, the majority of scholars believe he was simply mistaken.

Eras are the same as the Qadimi, Shenshai, and Young Avestan calendars.

Soures:
* https://www.iranchamber.com/calendar/articles/old_iranian_calendars1.php
* https://en.wikipedia.org/wiki/Zoroastrian_calendar
* http://www.avesta.org/zcal.html
* http://www.avesta.org/antia/Enigmatic_Response_of_Three_Dasturjis_to_the_United_Zarthoshti_Calendar.pdf

--------------------
OLD AVESTAN CALENDAR
--------------------

The Old Avestan calendar is one of the two earliest known calendars to have been used in Iran; along with the Elamite calendar, it predates even Zoroastrianism. A regular year consists of 12 months of 30 days each; since anybody can tell this is noticeably shorter than the solar year, an extra month is added every six years, and an additional intercalary month is added every 120 years to account for the extra fraction of a day. This means that a normal year is 360 days long while a leap year is 390 days, and the double-leap year reaches a whopping 420 days.

In the absence of any better options, I chose the same eras as the various Zoroastrian calendars. Taqizadeh places the switch from the Old Avestan to the Young Avestan (basically Egyptian) calendar in 441 BC (1063 BH), during the Achaemenid dynasty. I expect that the first day of the Young Avestan calendar in this year would have been the same as the first day of the Old Avestan calendar, and so I have adjusted the epochs by 10 days to make sure that the calendars coincide on Julian Day 1560392.

Sources:
* https://www.iranchamber.com/calendar/articles/old_iranian_calendars1.php
* https://iranicaonline.org/articles/calendars#prettyPhoto[content]/0/
* https://iranicaonline.org/articles/calendars

--------------
FASLI CALENDAR
--------------

The Fasli ("seasonal") calendar is the most modern Zoroastrian calendar. Like the Jalali and Solar Hijri calendars, the year begins on the northward equinox. Days begin at sunrise. Eras are the same as for other Zoroastrian calendars.

The Fasli calendar is standard among modern Zoroastrians, and since it is observational, Nowruz syncs up perfectly with the Solar Hijri calendar. However, it is not without controversy. In particular, prominent Indian Zoroastrians have objected to it, on the grounds that intercalating a full day is forbidden by the Bundahishn, and that the Fasli calendar is an attempt to force the Muslim Jalali calender on Zoroastrians. The second objection is clearly false, since the Jalali and Solar Hijri calendars have completely different monthly structures; furthermore, Antia makes a strong case that the Fasli reckoning was known, and indeed practiced, in ancient times.

Sources:
* https://en.wikipedia.org/wiki/Zoroastrian_calendar
* http://www.avesta.org/zcal.html
* https://zoroastrian.org/articles/The%20Precise%20Iranian%20Calendar.htm
* https://theguibordcenter.org/zoroastrian-holy-days-and-observances/
* https://zoroastrian.org/articles/nowruz.htm
* http://www.heritageinstitute.com/zoroastrianism/calendar/index.htm
* https://www.iranchamber.com/calendar/articles/calendar_systems_origins.php
* https://www.iranchamber.com/culture/articles/norooz_iranian_new_year.php
* https://www.iranchamber.com/culture/articles/norooz_zarathushtrian_new_year.php
* http://www.zcserv.com/
* https://parsikhabar.net/education/jamshedi-navruz-a-detailed-explanation-of-the-calendars/3954/
* http://www.avesta.org/antia/Enigmatic_Response_of_Three_Dasturjis_to_the_United_Zarthoshti_Calendar.pdf
* https://zoroastrians.net/wp-content/uploads/2015/03/jamshedi-navruz-on-march-21st.pdf

----------------
SOGDIAN CALENDAR
----------------

The Sogdian calendar is a derivative of the fixed Zoroastrian calendar. As its name suggests, it was used in Sogdia a pre-Islamic Iranic kingdom in central Asia. Due to the Iranian influence on Sogdia, as well as the fact that Sogdia was part of Iran during the Achaemenid period, the Sogdians adopted one form of the Zoroastrian calendar. It has the same structure as the Egyptian, Qadimi, Shenshai, and Armenian calendars, consisting of 12 months of 30 days plus five extra days. In fact, the Sogdian calendar is actually identical to the Armenian, save that the Armenian calendar has a canonical epoch, which the Sogdian does not (and also the Armenian month names are, obviously, in Armenian); the Sogdian and Armenian calendars' New Years are both five days after that of the Qadimi.

The most commonly-used epoch among modern scholars is the Yazdegerdi era, which I have gone with here. However, for various reasons, the Sogdian year 1 is one year before the Zoroastrian year 1.

Sources:
* BI Marshak, The Historico-Cultural Significance of the Sogdian Calendar, Iran, ol. 30 (1992), pp. 145-154
* http://www.rahamasha.net/uploads/2/3/2/8/2328777/the_structure_of_the_later_sogdian_calenda1_1.pdf
* http://www.rahamasha.net/uploads/2/3/2/8/2328777/scf_m_147_2.pdf

------------------------
INDIAN NATIONAL CALENDAR
------------------------

The Indian national calendar is a solar calendar used, unsurprisingly, in India, where it is co-official with the Gregorian calendar. Years are counted from the Saka era, which in the case of India begins on 22 March 78 AD (2 Farvardin 544 BH). In a normal year, the first month, Chaitra, has 30 days, the next five months have 31 days, and the last six have 30; this gives it an overall structure reminiscent of the Solar Hijri calendar. In leap years, an extra day is added to Chaitra. The Indian national calendar is by design synchronised with the Gregorian calendar so that New Year's Day falls on 22 March in normal years and 21 March in leap years.

The calendar was designed by a team led by Meghnad Saha and introduced in 1879 SE (1957 AD, 1336 AH); Saha's team also drew up guidelines for determining the dates of traditional and religious observances. In designing the calendar, the team sought to inntegrate elements of the >30 calendars then in use by various groups across India while also keeping pace with the international standard Gregorian. It is the calendar by which the government operates, though regional holidays and observances are still largely set by the various traditional calendars.

Sources:
* https://en.wikipedia.org/wiki/Indian_national_calendar
* https://www.fourmilab.ch/documents/calendar/
* https://www.india.gov.in/calendar?date=2022-04
* https://knowindia.india.gov.in/national-identity-elements/national-calendar.php
* https://eclipse.gsfc.nasa.gov/SEhelp/calendars.html
* https://web.archive.org/web/20180417203122/http://www.math.nus.edu.sg/aslaksen/projects/lcl.pdf, p. 32

-----------------
MANDAEAN CALENDAR
-----------------

The Mandaean calendar is yet another derivative of the traditional Zoroastrian calendar, and thus it ultimately dates back to the Egyptian calendar. The year consists of 12 months of 30 days, plus five extra days between the eighth and ninth months, with no leap years. The day traditionall began at sunset, but nowadays starts at sunrise. The epooch is a little under 480,000 years in the past, and dates from the date of Adam's creation in Mandaean tradition.

Sources:
* https://en.wikipedia.org/wiki/Mandaean_calendar

---------------
SOTHIC CALENDAR
---------------

The Sothic calendar was at no point used in ancient Egypt. It is a calendar I invented, based on Egyptian customs.

The Egyptian civil year, as already described, was 365 days long, no more, no less. Egyptians were aware of the discrepancy between calendar and climate, but they refused to change it because only the gods can create new days. However, this was only really of interest to the priesthood and the royal household. Regular people lived their lives by an agricultural calendar of three seasons, in which akhet, the first season, began on whatever day the Nile flooded. This, naturally, could vary considerably depending on climate, even multiple months in extreme cases.

The flooding ususally happened quite soon after the heliacal rising of Sirius, which in Pharaohnic times roughly coincided with the summer solstice in Egypt and Ethiopia. Around this time, a series on monsoons in the Ethiopian highlands swell the various Nile tributaries, bringing about a long period of flooding all across the river system. Since Sirius is associated with Isis, ancient Egyptians interpreted the rainfall as her weeping for Osiris.

The heliacal rising of Sirius was faithfully recoded in ancient Egpytian annals, since it provided such a reliable indicator of when the Nile was about to flood. Indeed, this was even regionally accurate; due to celestial mechanics, Sirius becomes visible just above the horizon about a day earlier with each degree of latitude one goes south, and so the rising would be observed several days earlier in Aswan than at the delta, which would also flood several days earlier.

This, however, presents a challenge in attempting to actually implement the Sothic calendar, since one must choose a particular latitude on which to base calculations. I went with the Great Pyramid of Giza, because what's more Egyptian than the Great Pyramid? As to when Sirius first becomes visible, Tetley calculates it to be when Sirius is 2º above the horizon while the sun is at least 5.5º below. As it happens, this separation recurs almost exactly every 365 days 6 hours 1 minute, and so the Julian calendar is considerably more accurate to the Sothic calendar than to the equinoxes. Since I was unable to implement an accurate algorithm to determine the altitudes of celestial bodies, I have instead defined the heliacal rising as the first day when Sirius rises at least 30 minutes before the sun.

Sources:
* M. Christine Tetley, THE RECONSTRUCTED CHRONOLOGY OF THE EGYPTIAN KINGS, 2014, BW Tetley, chapter 3. Available at https://www.egyptchronology.com/pdf-for-download/
* https://earthsky.org/brightest-stars/sirius-the-brightest-star/
* https://en.wikipedia.org/wiki/Egyptian_calendar
* https://en.wikipedia.org/wiki/Flooding_of_the_Nile
* https://www.space.com/21702-sirius-brightest-star.html
* https://www.space.com/39638-sirius-dog-star-visible-this-week.html
* https://web.archive.org/web/20131225071503/https://www.lavia.org/english/archivo/egyptiancalendaren.html

-----------------------
MADHYAMA SOLAR CALENDAR
-----------------------

The Madhyama, or mean, Hindu solar calendar originates with the Vedic culture, in the distant past before there were even kingdoms in India. Before about the 12th century AD/5th century AH, it was a purely arithmetical calendar based on mean values of the sidereal year and rāśi. The year is defined as being 365 + 149/576 days; a rāśi, or sidereal "month", is exactly 1/12 of this, and is the approximate time it takes for the sun to traverse 1/12 of the zodiac. These values are given by the great mathematician and astronomer Aryabhata I in the Ãryabhaṭīya. One will quickly spot parallels with the zhongqi of the Chinese calendar, and the tropical months of the Jalali and Solar Hijri calendars.

Days begin at sunrise, which for the purpose of calculations is considered to be precisely halfway between midnight and noon. A day falls within the rāśi which is current at sunrise; thus, if the calculated beginning of the rāśi falls even one truti after calculated sunrise, then the rāśi in question begins the next day.

Three epochs are in common use. The first is the start of the Kali Yuga, the fourth age of the world in the great Yuga Cycle; it is calculated to have begun at sunrise on 18 Februarty 3102 BC in the Julian calendar. The others come from recorded history. The Vikram Samvat dates from Emperor Vikramaditya's defeat of the Shakas. The other, the Shaka era, is also used in the Indian National Calendar, dates from the ascension of King Chashtana as chief of the Western Satraps.

The value of the mean year is approximately 3 minutes longer than the actual length of the sidereal year, which is an impressive accomplishment for naked-eye astronomy. Over time, however, this discrepancy adds up, and so the calendar would have been occasionally recalibrated by astronomical observations.

Sources:
* Edward M. Reingold and Nachum Dershowitz, *Calendrical Calculations: The Ultimate Edition*, Cambridge University Press, 2018, chapter 10.
* https://en.wikipedia.org/wiki/Hindu_calendar

--------------
TAMIL CALENDAR
--------------

The Tamil calendar is the traditional calendar used by the Tamil people, primarily in Tamil Nadu, Paducherry, Sri Lanka, and Malaysia. It is a sidereal, purely solar calendar; the Calendar Reform Committee Report includes two Tamil-language panchangams that are described as lunisolar, but from my research, these are more likely to be solar calendars which include religious celebrations tied to the phases of the moon. (As an analogy, Christian celebrations and observances related to Jesus' death and resurrection are also synched with the phase of the moon, but that doesn't mean the Gregorian calendar is lunisolar).

Days begin at sunrise, but the reckoning of months is a bit unusual. If the sun enters a constellation after sunset, the corresponding month officially begins the following day; otherwise, it begins on the same day, even if the sun enters the constellation well after sunrise.

There is no consensus as to which epoch should be used, and indeed Tamils in Tamil Nadu often just go by the year's position in the 60-year cycle, while using the Gregorian calendar for everyday purposes. Among those who do use an era when working in the Tamil calendar, the Śaka era seems to be the most popular, so that is the one I went with.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report
* The Calendrical System in Tamil Nadu, CJ Fuller, The Journal of the Royal Asiatic Society of Great Britain and Ireland, № 1 (1980), pp 52–63
* https://www.prokerala.com/general/calendar/tamilcalendar.php
* http://world.clndr.org/calendars/Tamil-calendar
* https://tamil.calendarin.com/tamil-daily-calendar
* https://www.tamildailycalendar.com/tamil_monthly_calendar.php?msg=Tamil%20Monthly%20Calendar%202022
* http://packolkata.gov.in/INDIAN_CALADAR_PAC.pdf

-----------------
BENGALI CALENDARS
-----------------

The Bengali calendar is the traditional calendar of West Bengal and Assam. The 1954 Calendar Reform Committee report describes five variants; two of these are slightly different wordings of calculations based on the Sūrya Siddhānta, and three are the modern method with very minor discrepancies that will be ironed out by observation.

The Bengali calendar is the quintessential Indian solar calendar. Days begin at sunrise, the year begins with the sun's entry into Meṣa, and each month lasts as long as it takes the sun to cross one sign of the zodiac. Its introduction is variously attributed to either King Shashanka or Emperor Akbar, but some form of it most likely predates both. Its era is the Bengali Era, which seems to date from the accession of Shashanka to the throne of Gauda.

The calendar made headlines in 1428 BE (1943 SE, 2021 AD) when the state legislative assembly of Assam announced that it would be made coöfficial with the Gregorian and Indian National calendars; this was widely (and incorrectly) reported as the local calendar becoming the *only* official calendar. The actual result was rather less dramatic; all official Assamese government communications carry the Assamese date alongside the Indian and Gregorian dates.

Sources:
* https://www.nextias.com/ca/current-affairs/21-10-2021/bhaskarabda-calendar
* https://www.adda247.com/upsc-exam/bhaskarabda-calendar/
* https://web.archive.org/web/20230104023923/https://testbook.com/current-affairs/bhaskarabda-calendar-to-be-added-to-official-assam-calendar/
* https://www.drishtiias.com/daily-updates/daily-news-analysis/bhaskarabda-a-luni-solar-calendar
* https://iasscore.in/current-affairs/prelims/bhaskarabda-a-luni-solar-calendar
* https://www.pratidintime.com/latest-assam-news-breaking-news-assam/bhaskarabda-calendar-official-calendar
* https://www.vishvasnews.com/english/viral/fact-check-bhaskaravda-will-not-be-used-as-an-official-calendar-in-assam-but-as-a-part-of-the-official-calendar-viral-posts-are-misleading-2/
* https://www.pratidintime.com/latest-assam-news-breaking-news-assam/bhaskarabda-calendar-official-calendar
* https://www.thehindu.com/news/national/other-states/bhaskarabda-to-be-added-to-official-assam-calendar/article37068273.ece
* https://www.sentinelassam.com/north-east-india-news/assam-news/bhaskarabda-luni-solar-calendar-to-be-used-in-official-calendar-assam-cm-himanta-biswa-sarma-559383
* https://www.iastoppers.com/articles/bhaskarabda-to-be-added-to-official-assam-calendar
* https://www.adda247.com/upsc-exam/bhaskarabda-calendar/
* https://web.archive.org/web/20211019093335/https://www.thehindu.com/news/national/other-states/bhaskarabda-to-be-added-to-official-assam-calendar/article37068273.ece
* https://assam.gov.in/sites/default/files/dms/notificationholidaylist2022.pdf
* https://www.assamgovernmentjob.com/assam-govt-holiday-list-2024-gad-assam/
* https://en.wikipedia.org/wiki/Bengali_calendars
* https://www.cuttingthechai.com/2019/04/8125/pohela-boishakh-a-brief-history-of-the-bengali-calendar/
* https://bengalicalendar.com/#
* https://www.prokerala.com/calendar/bengalicalendar.html
* https://www.fluentin3months.com/bengali-calendar/
* https://www.prokerala.com/calendar/bengalicalendar-2022.html
* https://www.prokerala.com/calendar/bengalicalendar-2021.html
* https://www.theworldcalendar.org/bengali-calendar/
* https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%99%E0%A7%8D%E0%A6%97%E0%A6%BE%E0%A6%AC%E0%A7%8D%E0%A6%A6#%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%B7%E0%A6%AA%E0%A6%9E%E0%A7%8D%E0%A6%9C%E0%A6%BF
* https://en.wikibooks.org/wiki/Bengali/Months
* https://www.fluentin3months.com/bengali-calendar/
* https://netkibaten.com/bengali-month-name/
* http://world.clndr.org/calendars/bengali-calendar/
* http://packolkata.gov.in/INDIAN_CALADAR_PAC.pdf
* https://drive.google.com/file/d/183xBV5EgCJsNmiv3bsVd9PUXbZiOkDs1/view?pli=1
* https://ganpatisevak.in/assamese-calendar-pdf-assam-govt-festivals-holidays-list/
* https://www.getbengal.com/details/a-few-bengali-calendar-basics-as-we-begin-1429
* https://www.opindia.com/2020/04/bengali-new-year-noboborsho-akbar-calendar-shashanka-deva-dynasty/
* https://beautifulbengal.com/bengali-calendar.html
* https://www.tbsnews.net/thoughts/saga-bangla-calendar-403202
* https://tanmoy.tripod.com/bengcal.html
* https://www.newindianexpress.com/nation/2023/aug/23/kolkata-diary-state-decides-to-adopt-bengali-lunar-calender-2607988.html
* https://www.cuttingthechai.com/2019/04/8125/pohela-boishakh-a-brief-history-of-the-bengali-calendar/
* https://www.wb.gov.in/pdf/Holiday-2021.pdf

--------------------
BANGLADESHI CALENDAR
--------------------

A radical redesign of the Bengali calendar was designed by a committe at Bangla Academy, led by Muhammad Shahidullah, and first presented in late 1372 BE (1887 SE, 1344 AH, early 1966 AD). The new calendar would exactly synchronise with the Gregorian, such that Pohela Bôisakh (Bengali New Year) would always fall on 14 April. The first five months of the year have 31 days, the next six have 30, and the last month has 30 in normal years and 31 in leap years. The new calendar was officially adopted in Bangladesh in 1393 BE and made coöfficial with the Gregorian calendar; it was also considered by rejected in West Bengal, because the traditional calendar is too deeply intwined with local culture and Hindu tradition.

The calendar was slightly revised in 1426 BE (1398 AH, 1941 SE, 2019 AH), resulting in a calendar that more closely resembles the Solar Hijri calendar. In the new Bangladeshi calendar, the first six months of the year have 31 days each, the next five have 30, and the last month has 29 days in normal years and 30 in leap years. This and the Gregorian calendar are the official calendars in use in Bangladesh today.

Sources:
* https://www.tbsnews.net/thoughts/saga-bangla-calendar-403202
* https://www.daily-bangladesh.com/english/national/30081
* https://en.wikipedia.org/wiki/Bangladeshi_calendar
* https://en.wikipedia.org/wiki/Bengali_calendars
* https://en.wikipedia.org/wiki/Pohela_Boishakh
* https://books.google.ie/books?id=QVOFAAAAQBAJ&pg=PA114&redir_esc=y#v=onepage&q&f=false
* http://en.banglapedia.org/index.php?title=Bangabda
* http://en.banglapedia.org/index.php?title=Panjika

---------------------------
MADHYAMA LUNISOLAR CALENDAR
---------------------------

The lunisolar calendar arose in India around the same time as the solar calendar, and developed in parallel with it. Given the number of similarities, I suspect the Indian lunisolar calendar was influential on the Chinese calendar.

As in the solar calendar, a sidereal year is defined as 365 + 149/576 days, and a rāśi as 1/12 of this. One synodic month is defined as 1577917500/53433336 days, or 29 + 2362563/4452778 days; this is only about ½ second longer than the modern value. Each month takes the name of the rāśi during which it begins, and the first day of each calendar month is the day of the first sunrise after the calculated instant of the new moon. (Some traditions begin the month the day after the full moon, but it is more common to start the month on the day after the new moon). If an entire month passes inside of a single rāśi, it is considered a leap (or "adhik") month, and shares its name with the regular month that follows; this system is almost identical to that of the Chinese calendar, except that the Chinese calendar names the leap month after the previous regular month, and zhongqi are determined by tropical rather than sidereal measurements.

The "days" in the lunisolar calendar are not actual days; rather, they are tithi (sometimes translated as "lunar day", not to be confused with days on the moon), defined as 1/30 of a synodic month, or the time it takes for the ecliptic angle between the sun and the moon to change by 12º. Tithi are counted from the first visible crescent, and the day is given the number of the tithi which is current at sunrise. Because a calendar month can be up to 30 days long and a tithi is slightly shorter than a calendar day, an entire tithi will occasionally pass between one sunrise and the next; in this case, that number is skipped.

Eras are the same as in the solar calendar, though the lunisolar year starts a month earlier than the solar year.

Sources:
* Edward M. Reingold and Nachum Dershowitz, *Calendrical Calculations: The Ultimate Edition*, Cambridge University Press, 2018, chapter 10.
* https://en.wikipedia.org/wiki/Hindu_calendar

----------------------------------
TRADITIONAL INDIAN SOLAR CALENDARS
----------------------------------

The more recent Hindu solar calendars are based on algorithms written in the Sūrya Siddhānta, published in around 400 SE (c 500 AD, 100 BH).

The modern solar calendar is a sidereal calendar, in which a year is defined as the calculated time it takes the sun to completely traverse the zodiac. Each solar month, or raśi, lasts for as long as the calculated time for the sun to traverse 1/12 of the zodiac, or 30° of sky; the names of the raśi correspond to the constellations, but only roughly correspond to the actual locations of the constellations. The beginning of the year is the point on the zodiac corresponding to the northward equinox when the Sūrya Siddhānta was first published.

The Sūrya Siddhānta describes a geocentric model of the solar system, in which the sun, moon, and planets all orbit Earth. As they do so, they trace out epicycles, which accounts for the variable speeds and occasional retrogression of celestial bodies. Of note, the epicycles of the sun, moon, and planets progress according to the anomalistic year and not the sidereal (or even tropical) year; as such, this makes it a very basic [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) to represent the ellipses that the planets actually move in.

Based on certain technical terms' resemblence to Greek works (specifically, कॅड्नअ is a loanword from the Greek χενδρος) the model appears to have originally been learned from the Greeks; while the Greeks believed the epicyclic model was a real description of reality, the Indians assumed it was merely a mathematical technique that gave the right answer, and that the planets really did sometimes speed up, slow down, and occasionally move backwards. The Sūrya Siddhānta's algorithms employ those epicycles in elaborate and complex schemes to correct for the true positions of the sun and the moon; for most of history, Indian astronomers used lookup tables derived from the algorithms because the actual process is incredibly long and fiddly, but since this is exactly why computers were invented, calconv uses the actual algorithms.

Notably, the Surya Siddhanta contains the earliest known descriptions of the jya, koti-jya, and utkrama-jya functions, which are the ancestors of the sine, cosine, and versine functions. Jyas are important in the calculations presented, and so a description is in order. The jya gives the length of a line drawn from one point of an arc perpendicular to a radius intersecting another point on that same arc. This is hard to describe with just words, but when you see it drawn geometrically, it is immediately obvious that if an angle θ subtends an arc AB of a circle with radius r, then jya(AB) == r×sin(θ). The Sūrya Siddhānta gives the jyas for various angles in increments of 1/24 of a right angle on a circle of radius 3438 units, with interpolation used for intermediate angles. Since jya is a function of the length of an arc, I have opted to define it such that any jya(x) will always return a positive value; hence, the sine and related functions used in this implementation will also always return a value between 0 and 1, unlike the modern understanding of sines, which can have a value down to (-1).

Extant editions descibe the precession of the equinoxes, but as Burgess notes, the awkward placement and description of the phenomenon suggests it was not present in the earliest versions, and was instead interpolated in two parts at later dates, possibly in order to incorporate Roman astronomical knowledge. The calendar is sidereal, and as such the precession of the equinoxes is only used to determine the sun's true declination, which is used to determine the times of sunrise and sunset. A careful reading of the description of the precession shows that the Indians imagined it a librational system; in other words, rather than the equinoxes gradually moving backward through the zodiac, the zodiac osicallates about the equinoxes such that, over the course of a kalpa, the tropical and sidereal years work out the same. It also advises the astronomer to use a sundial to determine the days of the equinoxes and the direction of precession, and indeed the language suggests that at one point, it described a constant rate and direction of precession rather than a libration. Nonetheless, I have gone with a librational algorithm, because that is what the siddhānta describes.

Days begin at sunrise. The mean sidereal year lasts 365+279457/1080000 days, and a mean month is 1/12 of that. Month start on the day that the sun rises in their corresponding raśi.

Sources:
* Surya Siddhanta, chapters 1–3; see:
** https://archive.org/details/SuryaSiddhantaTranslation/mode/2up
** https://archive.org/details/suryasiddhantaandsiddhantasiromanibimalaprasadsidhhnatasarasvatidanvirgoswamyiskon_202003_23_r
** https://archive.org/details/suryasiddhantaenglishtranslationlancelotwilkinsonbapudevasastri1861_727_C
* https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81

DO NOT SEE Edward M. Reingold and Nachum Dershowitz, *Calendrical Calculations: The Ultimate Edition*, Cambridge University Press, 2018, chapter 20. Because Reingold and Dershowitz are idiots with no idea what they're talking about and don't understand what they're reading.

--------------------------------------
TRADITIONAL INDIAN LUNISOLAR CALENDARS
--------------------------------------

The traditional Indian lunisolar calendars are, like the solar calendars, based on the values and algorithms described in the Sūrya Siddhānta. These calendars assume a mean synodic month of 29+7087771/13358334 days, which is approximately ½ second longer than the modern measured mean synodic month; a sidereal month of 27+4644439/14438334 days, approximately 1.1 second longer than the modern measured mean sidereal month; and an anomalistic month of 27+31759129/57265137 days, about 4.165 seconds longer than the modern measured mean anomalistic month.

Days begin at sunrise. The first day of the month is the day when the sun rises when the moon is new; to put it another way, months end of the day of the new moon. The month is determined by the position of the sun in the zodiac at sunrise on the first day of the month; thus, if the sun is in Mīna when it rises, the month is Chaitra; if the sun rises in Meṣa, it's Vaisākha, and so on. If two lunar months begin while the sun is in the same sign, the second is the "official" month; the first is a leap month, and is prefixed with "Adhik", a word meaning "empty". In some cases, it is possible for the sun to cross an entire sign within a single lunar month; in this case, the lunar month corresponding to that sign is skipped. Such a rare event is compensated with a leap month elsewhen in the year.

This is the traditional calendar used by the Telugu minorities in Tamil Nadu (who use the Śaka era), Delhi, and Karnataka (where dates are counted as to the Vikram Samvat), as well as Kolhapur; it is also used by some in Maharashtra, though the observational calendar is far more common there.

Sources:
* Sūrya Siddhānta, chapters 1–3; see:
** https://archive.org/details/SuryaSiddhantaTranslation/mode/2up
** https://archive.org/details/suryasiddhantaandsiddhantasiromanibimalaprasadsidhhnatasarasvatidanvirgoswamyiskon_202003_23_r
** https://archive.org/details/suryasiddhantaenglishtranslationlancelotwilkinsonbapudevasastri1861_727_C
* https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81
* Calendar Reform Committee Report, 1954: https://archive.org/details/calendar_reform_comittee_report

DO NOT SEE Edward M. Reingold and Nachum Dershowitz, *Calendrical Calculations: The Ultimate Edition*, Cambridge University Press, 2018, chapter 20. Because Reingold and Dershowitz are idiots with no idea what they're talking about and don't understand what they're reading.

-----------------
TRIPURI CALENDARS
-----------------

The sidereal Tripuri calendar is the traditional calendar of Tripura. It is a very slight variant of the Bengali calendar; its era begins three years earlier, and it uses the Hindi rather than Bengali month names. It was commonly used up until the end of British rule in India; since the accession of Tripura to the Republic of India, the Gregorian and Bengali calendars have been in standard use in Tripura. More recently, a revival of the Tripuri calendar has been promoted by Tripura nationalists.

In researching it, I found scattered references to a Tripuri calendar linked to the southern solstice. I was not able to find anything concrete, and so this appears to be an idea some Tripurans have that has no published definition. As such, the tropical Tripuri calendar here is one of my own design. Its epoch is consecutive Julian Day 1936544, corresponding to the southern solstice preceding the epoch of the sidereal calendar. Each month lasts as long as it takes the sun to cross 30° of the ecliptic, and each month begins on the day of the sunrise immediately at or following the sun's entry into that segment of the ecliptic.

Sources:
* https://en.wikipedia.org/wiki/Tripuri_calendar
* https://thenortheasttoday.com/states/tripura/tripura-5-facts-on-the-history-behind-tring-tripuri-new/cid2552853.htm
* https://www.telegraphindia.com/india/tripura-tribal-front-rewrites-calendar/cid/937480
* http://www.tripura.org.in/truthoftringe1.htm
* http://www.packolkata.gov.in/INDIAN_CALADAR_PAC.pdf

------------------------------------
OBSERVATIONAL INDIAN SOLAR CALENDARS
------------------------------------

The observational Indian calendars are based on the same ideas as the traditional calendars, but they use modern observational data, ephemerides, astronomical values, and forecasting techniques.

The solar calendar, like the traditional calendar, is sidereal. Nominally, the start point of the sun's journey across the zodiac is the point opposite Spica; however, the sun is normally opposite Spica on 12 April (Gregorian), while Indian calendar makers put the first day of the solar calendar on the 14th, give or take a day. The Calendar Reform Committee, in proposing a switch from sidereal to tropical years for the timing of Hindu festivals, reccomended that the beginning of the tropical year for religious purposes be set at the point when the sun's ecliptic longitude is 23°15'. In 1955, this occurred when the sun was 0.6749121716696571° past counter-Spica (by my calculations), and so I have taken this point on the zodiac as the zero point.

Days begin at sunrise. In the solar calendars, months begin on the day that the sun rises in the corresponding zodiac sign. This is the traditional calendar of Punjab, where the Vikram Samvat era is used.

In the lunisolar calendars, months begin on the day of the first sunrise after the new moon. Months are named according to the star sign in which said sunrise falls. If there are two new moons while the moon is in the same star sign, the first of these is designated as a leap month, and its name is preceded by "Adhik". Due to orbital mechanics, it is possible for the sun to cross an entire zodiac sign in between two new moons; in such a case, the corresponding month is skipped, and the difference is made up by a leap month elsewhere in the year.

Sources:
* Sūrya Siddhānta, ch 1–3 (translated by Ebeneezer Burgess)
* 1954 Report of the Indian Calendar Reform Committee
* https://www.indiatoday.in/information/story/mesha-sankranti-2023-date-time-significance-and-all-you-need-to-know-2359839-2023-04-14
* https://www.jagranjosh.com/general-knowledge/solar-new-year-or-mesha-sankranti-1681443910-1
* https://blog.devdarshanapp.com/festivals/mesha-sankranti/
* https://www.prokerala.com/festivals/tula-sankranti.html
* https://www.mpanchang.com/vrats/sankranti/?date=2023
* https://en.wikipedia.org/wiki/Mesha_Sankranti
* https://en.wikipedia.org/wiki/Indian_New_Year%27s_days
* https://planetcalc.com/9166/
* https://timesofindia.indiatimes.com/religion/festivals/mesha-sankranti-2021-date-time-and-significance/articleshow/82060121.cms
* https://www.ganeshaspeaks.com/festival-calendars/information/mesha-sankranti/
* https://www.news18.com/news/lifestyle/mesha-sankranti-2022-rituals-importance-and-all-you-need-to-know-about-the-solar-new-year-4977763.html
* https://www.prokerala.com/festivals/mesha-sankranti.html

---------------------
MUGHAL FAṢLĪ CALENDAR
---------------------

Not to be confused with the Zoroastrian Faṣlī calendar, the Mughal Faṣlī calendar was introduced in India by Emperor Akbar in order to try and make the traditional Indian lunisolar calendar conform to the Lunar Hijri calendar, for the same reason the Solar Hijri was introduced in Iran. This most noticeable change was changing the calendar era to count from Muhammad's flight to Medina. For this converter, I have also set days to begin at sunset in accordance with Islamic practice.

Sources:
* https://en.wikipedia.org/wiki/Fasli_calendar
* https://www.britannica.com/topic/Fasli-era

-------------
ODIA CALENDAR
-------------

The Odia calendar is the version of the traditional Indian solar calendar used in Odisha, Punjab, and Haryana. Like other Indian calendars, it uses the sidereal year; unlike others Indian calendars, it reckons dates by the Chinese method. Days run from midnight to midnight, and the day of an astronomical event is the day that the event occurs, regardless of when in the day it falls. The epoch is one year earlier than the Bengali epoch.

For determining the dates of religious festivals, it uses the purnimanta system, in which lunar months run from full moon to full moon.

Sources:
* Leow Choon Lian, Indian Calendars, National University of Singapore: https://web.archive.org/web/20180417203122/http://www.math.nus.edu.sg/aslaksen/projects/lcl.pdf
* GM Ballabh, Indian Calendars, Kerala Positional Astronomy Centre
* https://en.wikipedia.org/wiki/Odia_calendar
* https://www.odiacalendar.com/
* https://jantrajyotisha.com/odia-calendar/
* http://shreekhetra.com/oriyacalender.html
* https://incredibleodisha.in/odia-months-name-list-seasons-rutus/
* https://nijuktiodisha.in/odia-calender
* https://jagannath.co.in/jagannath-odia-calendar/
* https://www.odiacalendar.com/odia-calendar-2020/
* https://web.archive.org/web/20110419195447/http://www.odishanewstoday.com/features/354-odisha-celebrates-maha-vishuba-sankranti-with-fervor-.html
* https://www.odiacalendar.com/blog/odia-months-name/

----------------------------------------
OBSERVATIONAL INDIAN LUNISOLAR CALENDARS
----------------------------------------

The modern Indian lunisolar calendar is based on the same principles as the traditional calendars, but uses modern ephemerides, observational data, and computational methods to calculate the date.

Days begin at sunrise, and the month begins on the first day following the new moon; in other words, when the sun rises after the moon passes between Earth and the sun. The name of the month is determined by the sign of the zodiac in which the sun rises, beginning with Chaitra in Mīna (Pisces), followed by Vaisākha in Meṣa (Aries), and so on. If two months begin within the same star sign, the second one is the official month; the first is given the prefix "Adhik" and is considered a leap month. Occasionally, no new moon will fall within a sign; in that case, the corresponding month is skipped, and is called kshaya. A kshaya month will always be balanced out by an adhik month elsewhen in the year.

Three eras are in common use: Kali Yuga, Śaka, and Vikram Samvat, of which the Śaka era is the most common. The observational lunisolar calendar, dating as to the Śaka era, is the calendar used for traditional purposes in Andhra Pradesh, Uttar Pradesh, Manipur, Karnataka, and most of Maharashtra; it is also used to an extent in Tamil Nadu to calculate the dates of religious observances. The 1954 Calendar Reform Committee Report mentions that two panchangas, published by Siddhanta Panchangam and Sri Saptarishi Panchang, report using the tropical rather than sidereal year; however, this appears to be because the respondants misread the survey question, as their other answers correspond to a sidereal reference frame.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report

------------------------------
TROPICAL INDIAN SOLAR CALENDAR
------------------------------

The tropical Indian solar calendar was proposed by the Calendar Reform Committee in their 1954 report. Intended to facilitate as seamless as possible a transition from the use of the sidereal year to the tropical year for religious and cultural festivals, this calendar begins on the day that the sun hits an ecliptic longitude of 23°15', which was the sun's ecliptic longitude as of Meṣa Saṃkrānti in 1878 SE (1956 AD); each month lasts as long as it takes the sun to cross 30° of the ecliptic, reminiscent of how months are determined in the Jalali calendar. This was intended as a temporary measure to facilitate prevent solar and lunar festivals from drifting further from their seasons, and the Committee intended that, at a future date, the calendar might be reformed again to restore traditional festivals to their original seasons. However, traditional calendar makers rejected this proposal and kept on determining dates of festivals by sidereal markers.

While civil days in India today run from midnight to midnight, the Calendar Reform Committee reccomended that the religious tropical calendar continue to reckon days from sunrise to sunrise, and I have implemented that convention. Years are counted per the Śaka Era.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report

----------------------------------
TROPICAL INDIAN LUNISOLAR CALENDAR
----------------------------------

The tropical Indian lunisolar calendar was proposed by the Calendar Reform Committee in their 1954 AD report as a replacement for existing traditional calendars. The goal was to shift the calculation of the dates of Hindu and Buddhist religious observances from a sidereal to a tropical basis, and thus prevent the dates of traditional festivals from drifting any further away from their intended seasons. It uses the tropical solar calendar (described above) as its basis, and from there uses the phases of the moon to determine the months. It was intended as a temporary measure, with a future reform moving the dates of festivals back to their original seasons; however, panchanga makers considered even the switch to a tropical system to be too much, and so the new calendar was never officially adopted.

Days begin at sunrise. Years are numbered from the Śaka Era.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report

----------------------------
MALAYAM (KERALITE) CALENDARS
----------------------------

The Indian Calendar Reform Committee received desriptions of five Keralite calendars in response to their survey, but as far as I can tell, these only represent three distinct calendars. All are purely solar calendars, apparently descended from the Tamil solar calendar. In contrast to most Indian calendars, and to every solar calendar I have come across, days in the Keralite calendars begin at sunset rather than sunrise or midnight.

The Keralite calendars are also distinguished by not starting the year upon the sun's entry into Meṣa. The Uthara Malayala Panchangam uses modern observations and techniques to calculate the position of the sun, and the year begins with the sun's entry into the sign of Kanni (Kanyā in Hindi; Greek Virgo). It also has different names for the months, some (but not all) of which are clearly related to the Hindi names. The Jogakshemam Panchangam and Jyothir Deepika, meanwhile, begin the year with the sun's entry into Siṃha; the difference is that the Jogakshemam Panchangam calculates dates and times according to the Sūrya Siddhānta, while the Jyothir Deepika uses modern methods.

The Jyolsyabharanam and Vidyabhivardhini Kanakajoobili Prasasti uses modern methods and seems to be the same as the Jyothir Deepika. Its descriptions in the Committee Report includes very slight differences in the length of the sidereal year and amount of precession as of the northward equinox of 1954 AD, but because both calendars rely on modern methods, such differences will be ironed out by observation.

The Nava Bharathi Panchangam takes an altogether different tack. From what I understand, rather than casting an entire calendar, it uses the methods of the Sūrya Siddhānta to determine the dates of traditional festivals in the Gregorian calendar.

The Keralite calendars use a very unusual method for determining the month. Rather than taking the sun's position at sunrise, sunset, midnight, or even noon as important, the critical time is the aparahna, the instant ⅗ between sunrise and sunset (which Dershowitz and Reingold misunderstand as meaning 13:12; this is true only on the days of the equinoxes).

The Keralite calendars use the Kollam era, which begins in 3927 Kali Yuga (748 Śaka Era, 881 Vikram Samvat, 826 Anno Domini, 205 Anno Hejirae). The origin of this era is obscure; the most plausible explanation I found is that it comemorates the construction of the temple to Shiva in Kandiyoor by king Cheraman Perumal.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report
* https://www.mpanchang.com/calendar/malayalm-festivals/
* https://kerala.me/language/malayalam-calendar
* http://vaikhari.org/kollavarsham.html
* https://www.keralatourism.org/malayalam-calendar
* http://world.clndr.org/calendars/Malayalam-calendar
* http://packolkata.gov.in/INDIAN_CALADAR_PAC.pdf
* https://www.narit.or.th/files/JAHH/2018JAHHvol21/2018JAHH...21..134G.pdf
* Kollam Era, KV Sarna, Indian Journal of the History of Science 31(1), 1996

----------------
JALGAON CALENDAR
----------------

The Jalgaon calendar is a variant of the modern Indian lunisolar calendar, used in Jalgaon. In Jalgaon, each month falls a month early, such that Chaitra in Jalgaon corresponds to Phālguna elsewhere, Vaisākha corresponds to Chaitra, and so on. It uses the Śaka era.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report

-----------------
MANIPURI CALENDAR
-----------------

The Manipuri calendar is the local calendar of Manipur. It is a minor variation on the observational Indian lunisolar calendar, with its own era starting in 710 SE. Days begin at sunrise and month names are in Manipuri.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report

-----------------
GUJARATI CALENDAR
-----------------

In Gujarat, a variant of the observational Indian lunisolar calendar is used in which the year begins with Kārtika, in the constellation of Tulā (Libra); once upon a time, this would have corresponded to the first new moon after the southward equinox. The era is Vikram Samvat.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report
* The Calendars of India, Vinod K. Mishra: https://web3.arxiv.org/pdf/1007.0062.pdf
* Indian Calendars, Leow Choon Lian: https://telibrary.com/wp-content/uploads/2022/08/Indian-Calender-Article.pdf

--------------
KUTCH CALENDAR
--------------

The Kutch calendar is a local variant of the observational Indian lunisolar calendar, used in Kutch and Saurashtra. The year begins with Āṣāḍha, the first moon in Mithuna (Gemini), which in times past would have corresponded to the northern solstice. The era is Vikram Samvat.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report
* The Calendars of India, Vinod K. Mishra: https://web3.arxiv.org/pdf/1007.0062.pdf
* Indian Calendars, Leow Choon Lian: https://telibrary.com/wp-content/uploads/2022/08/Indian-Calender-Article.pdf

--------------------------------------------------
TRADITIONAL INDIAN LUNISOLAR CALENDAR (PURNIMANTA)
--------------------------------------------------

All traditional Indian lunisolar calendars described thus far use the amanta scheme; each month ends on the day in which the new moon falls. In the purnimanta scheme, months instead end of the day when the full moon falls. Festivals and observances fall on the same day regardless of whether the amanta or purnimanta scheme is used; a festival which falls on the first day of an amanta month will be celebrated on the same day by people who use purnimanta months, but the purnimanta people will consider that day to be in the middle of the month.

As with the amanta scheme, in the purnimanta scheme, the name of the month is determined by the zodiacal position of the sun at the first sunrise after the new moon. The month begins on the day of the first sunrise following the full moon immediately prior to that full moon; as such, the month can begin up a fortnight before the sun enters the corresponding star sign. As far as I can tell, leap months are determined by the same method as in the amanta scheme.

According to the Calendar Reform Committee Report, a version based on the Sūrya Siddhānta is used in parts of Uttar Pradesh, and years are numbered according to Vikram Samvat.

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report
* The Calendars of India, Vinod K. Mishra: https://web3.arxiv.org/pdf/1007.0062.pdf
* Indian Calendars, Leow Choon Lian: https://telibrary.com/wp-content/uploads/2022/08/Indian-Calender-Article.pdf
* Indian Calendars, GM Ballabh: http://www.packolkata.gov.in/INDIAN_CALADAR_PAC.pdf

----------------------------------------------------
OBSERVATIONAL INDIAN LUNISOLAR CALENDAR (PURNIMANTA)
----------------------------------------------------

The Calendar Reform Committee Report describes a version of the purnimanta scheme as being standard in Uttar Pradesh and Himachal Pradesh. Mishra, Lian, and Ballabh each describe it as also being standard in Jammu and Kashmir, Rajasthan, Madhya Pradesh, and Bihar; they also agree that states and communities whose traditional calendars are purely solar also use the purnimanta scheme to determine the dates of lunar festivals. As with the traditional version, Vikram Samvat seems to be the standard era for lunisolar calendars using the 

Sources:
* 1954 Report of the Calendar Reform Committee: https://archive.org/details/calendar_reform_comittee_report
* The Calendars of India, Vinod K. Mishra: https://web3.arxiv.org/pdf/1007.0062.pdf
* Indian Calendars, Leow Choon Lian: https://telibrary.com/wp-content/uploads/2022/08/Indian-Calender-Article.pdf
* Indian Calendars, GM Ballabh:	http://www.packolkata.gov.in/INDIAN_CALADAR_PAC.pdf

--------------------
NANAKSHAHI CALENDARS
--------------------

The Nanakshahi calendars are a two calendars used by Sikhs for the timing of religious rituals, festivals, and observances.

Prior to 2060 VS (2003 AD), most Sikhs used one of the Indian lunisolar calendars to schedule events. The first Sikh-specific calendar was designed by Pal Singh Purewal in 2056 VS; he discarded the use of the stars and the phases of the moon for timekeeping in favour of a purely solar calendar, synchronised with the Gregorian calendar. New Year's Day always falls on 14 March. The first five months of the year have 31 days and the rest have 30, with leap day being added to the last day. The era is 1526 VS (1469 AD), the birth of Guru Nanak Dev. It is usually just referred to as the Nanakshahi calendar, but is sometimes called the Moon (original) Nanakshahi calendar when ambiguity is possible.

The Mool Nanakshahi calendar was officially adopted by all Sikh governing bodies in 535 NE (2060 VS, 2003 AD), but did not immediately filter through to every practitioner. In 542 NE (2067 VS, 2010 AD), a group of conservatives expressed opposition on the grounds that Nanak himself used the traditional Indian lunisolar calendar, and they further objected that the Mool Nanakshahi calendar is Christian in origin. This faction's argument that Sikhs should date their festivals by traditional Indian lunisolar methods was rejected by Sikh governing bodies; nonetheless, I have included it here on the grounds that it is used by a small number of practitioners. It is simply the observational Indian lunisolar calendar, but dating years from the birth of Guru Nanak Dev.

Sources:
* http://www.purewal.biz/Gurbani_and_Nanakshahi_Calendar.pdf
* https://www.bbc.co.uk/religion/religions/sikhism/customs/nanakshahi.shtml
* https://www.hindustantimes.com/punjab/ht-explainer-know-about-the-controversial-nanakshahi-calendar/story-UeKm0yTcDtat1FCRIel86K.html
 http://www.purewal.biz/RepLamba.pdf

--------------
JAIN CALENDARS
--------------

Jains use traditional Indian lunisolar calendars to set the dates of festivals and observances. Vikram Samvat is a common era, but there are also attestations of an era dating from the year that Mahavira achieved Nirvana. This year is given as -470 VS (527 BC) in the Śvetāmbara tradition, and as -605 VS (662 BC) in the Digimbaras tradition. Both versions are included here.

Sources:
* https://jainpedia.org/themes/practices/festivals/jain-calendar/

---------------
VEDĀṄGA JYOTIṢA
---------------

The Vedāṅga Jyotiṣa is the oldest known Indian astronomical text; based on its descriptions of various phenomena, TS Kuppanna Sastry estimates it was composed around 1450 BS (1370 BC).

The Vedāṅga Jyotiṣa describes a 5-year yuga as containing 1830 tropical days and 62 synodic months; this equates to 366 days per year, but Sastry argues, quite reasonably, that rulers and astronomers back then knew full well that a sidereal and a tropical year contained about 365 days. Rather, this yuga was only intended to give a rough relation between units of time which could be used for scheduling secular events, while religious events proceded according to direct observations, which would be periodically used to correct the secular forecasts.

The tropical year begins on the day of the heliacal rising of Śraviṣṭhā (Alpha and Delta Delphini), with the first month of the year most likely being the next visible crescent. Since the conjunction of the sun with Śraviṣṭhā is stated to fall in the month of Māgha, it follows that Phālguna is the first month of the year.

Leap months are stated to be inserted every 2½ years, either in the middle of the year or at the end; the leap month is always named Adhikmasa. Since the calendar was frequently recalibrated to match observations, I have opted to determine leap years according to when new moons occur in relation to the heliacal rising of Śraviṣṭhā, with the position of Adhikmasa being determined by the year's position in a five-year cycle.

Sources:
* https://web.archive.org/web/20110501084408/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005abd_s1.pdf
* https://archive.org/details/VedangaJyotisha/page/n17/mode/2up

---------------------------------
NEPALI LUNISOLAR (NEWAR) CALENDAR
---------------------------------

Indian calendars have been used in Nepal for centuries, of which the Vikram Samvat has long been popular.

The lunisolar calendar included here is the one primarily used by the Newar minority, though it known throughout Nepal, and its advocates are very clear that it is for all Nepalis, not just the Newar.

The structure is basically the same as the Gujarati calendar, but I have adjusted it to base its calculations on the coördinates of Kathmandu rather than Ujjain. Days begin at sunrise, and months begin on the day which begins with the sunrise following the new moon.

The solar epoch, known as Nepal Sambat, is consecutive Julian Day 2042401. This corresponds to the date that legendary merchant Sankhadhar Sākhwā paid off the debts on everybody in Nepal.

Sources:
* https://en.wikipedia.org/wiki/Nepal_Sambat
* https://web.archive.org/web/20120706163359/http://www.gorkhapatra.org.np/gopa.detail.php?article_id=25573&cat_id=10
* https://web.archive.org/web/20100102032418/http://www.myrepublica.com/portal/index.php?action=news_details&news_id=10891
* https://web.archive.org/web/20150610231101/http://www.ekantipur.com/the-kathmandu-post/2012/01/28/free-the-words/the-sands-of-time/230946.html
* https://web.archive.org/web/20140808053114/http://www.gorkhapatra.org.np/gopa.detail.php?article_id=9026&cat_id=29
* https://web.archive.org/web/20111027091055/http://www.myrepublica.com/portal/index.php?action=news_details&news_id=37568
* https://web.archive.org/web/20150610193828/http://www.ekantipur.com/2011/10/27/top-story/mha-puja-today-nepal-sambat-1132-being-observed/342845.html
* http://world.clndr.org/calendars/nepal-calendar/
* https://nepalipatro.com.np/blog/en/nepal-sambat/
* https://web.archive.org/web/20201004205943/https://medium.com/@hyoojala/overalls-of-nepal-sambat-3f63e514c984
* https://web.archive.org/web/20240204234745/https://old.risingnepaldaily.com/nation/lmc-brings-nepal-sambat-into-official-use
* https://www.ashesh.com.np/nepali-calendar/

---------------------
NEPALI SOLAR CALENDAR
---------------------

The Nepali solar calendar was introduced in 1141 NS (2077 BS, 2020 AD). It is an algorithmic calendar, structured so that the year always begins on 20 October in the Gregorian calendar.

Month names are the same as in the lunisolar calendar. The last six months of the year always have 31 days, the first five have 30, and the sixth has 29 in normal years and 30 in leap years.

Sources:
* https://web.archive.org/web/20201004205943/https://medium.com/@hyoojala/overalls-of-nepal-sambat-3f63e514c984
* https://web.archive.org/web/20230324213834/https://old.risingnepaldaily.com/main-news/nepal-sambat-gets-solar-calendar-for-the-first-time
* https://web.archive.org/web/20220815104429/https://old.risingnepaldaily.com/nation/nepal-sambat-calendar-completes-one-year
* http://nepalsambat.org/solar-calendar/