This page explains modern calendars. Related pages deal with:
The following calendars are described on this page:
The calendar in official use today in most Western countries, and also in the Christian faith, is the Gregorian calendar.
Days in the Gregorian calendar start at midnight. The Gregorian calendar is a solar calendar and tries run in step with the beginning of spring in the northern hemisphere (related to the Christian feast of Easter [Steel]) without keeping track of the phases of the Moon. The Gregorian calendar year consists of 12 months that each contain between 28 and 31 days, inclusive), as is shown in the following table.
Most years consist of 365 days, but some years (the leap years) have 366. The average length of the calendar year is 365 97/400 = 365.2425 days, which is about 11 seconds longer than the average time from one spring to the next, and about 11 days longer than 12 synodical months. In the Gregorian calendar, 400 years are exactly equal to 146,097 days, and after such a period the sequence of ordinary and leap years repeats itself. The first day of the year, 1 January, falls just after the middle of the winter half od the year in the Northern hemisphere. The first year of the calendar is tied to the birth of Jesus (from the Christian faith).
The rules for leap years are as follows:
Following these rules, the years 1700, 1800, and 1900 weren't leap years, but 1600 and 2000 were.
The Gregorian calendar is a modified version of the Julian calendar. Modification was necessary because the average length of the Julian calendar year, 365 1/4 day, was about 11 minutes too long compared to the March equinox year so that the vernal equinox (the beginning of spring in the northern hemisphere) moved backwards through the calendar year by about one day per 131 years. When the Julian calendar was introduced, in −44, the vernal equinox fell around 25 March. At the time of the Council of Nicaea in 325 the vernal equinox had shifted to 21 March, and the rules for determining the date of Easter were based on that date. Around 1500 the vernal equinox had shifted further back to about 11 March. For this and other reasons, Pope Gregory XIII proclaimed the new calendar rules that define what we now call the Gregorian calendar.
To make the calendar year follow the return of the vernal equinox more closely, the last two leap year rules listed above were promulgated. To get the vernal equinox back to 21 March, the Pope decreed that the day following 4 October 1582 be called 15 October 1582, thereby skipping 10 days of the calendar. Of course, this does not mean that 10 days were taken from everybody's life, but only that something changed in what the days were called.
The Gregorian calendar reform was adopted quickly in the Catholic regions of the world (of which the Pope was the religious leader), but there was a lot of resistance to the reform in Protestant areas. Many Protestant areas started using the Gregorian calendar around 1700. Great Britain and its possessions followed in 1752, and Russia in 1918.
It is clear that the gradual acceptance of the Gregorian calendar causes no end of problems in accurately dating dates from the period. A date in 1612, for example, is probably on the Gregorian calendar if it occurs on a letter sent from a Catholic country, but probably on the Julian calendar if it is on a letter sent from a Protestant country.
Dating is made even more difficult because not everybody celebrated the beginning of a new year on January 1st. That January 1st is now considered to be the first day of the year is a historical accident. In the past, other dates have been used for New Year's Day, such as 25 March and 25 December, but eventually 1 January was the most popular.
If I'm not mistaken, then in Christianity the 25th of March is associated with the conception of Jezus, the 25th of December with the birth of Jezus, and the 1st of January with the circumcision of Jezus. In pre-Christian times, 25 March was the ascending equinox (in the northern hemisphere the beginning of the summer half of the year), 25 December the southern solstice (the darkest day in the north, the middle of the winter half of the year, the midwinter feast), and 1 January was the New Year of the Romans (at least since Julius Caesar in 45 B.C.).
In the Middle Ages, each civil or religious authority could determine when New Year would be celebrated in their jurisdiction. If you thought that the conception of Jezus or the beginning of the summer half of the year was the most important, then you chose 25 March. If you thought the birth of Jezus or the midwinter feast (Yule) was the most important, then you picked 25 December. If you thought the circumcision of Jezus was the defining moment, or if you preferred to have the first day of the year at the beginning of a month, or if you wanted to keep to the tradition of the Romans, then you used 1 January.
So, if one city celebrated New Year on 25 March but another city on 1 January, then the same day could be called 1 March 1504 in the one city but 1 March 1505 in the other one. There were also differences in the year counts that were used. For the Julian calendar, for example, the eras of the Foundation of Rome (−752) and the start of the reign of Emperor Diocletian (284) were used before the era of Christ (invented in 532) became popular.
February can only have five of any particular day of the week in a leap year, because in an ordinary year February has exactly four weeks, so then every weekday occurs exactly four times in February. In a leap year, the five-of-a-kind day of the week occurs on both the first day and the last day of February (the latter is also the leap day or bissextile day). The year 2004 is a leap year and the first day of February in 2004 is a Sunday, so February has five Sundays in 2004.
This situation repeats itself every 28 years, forward until the end of the 21st century and backward until the beginning of the 20th century, so there were and will be five Sundays in February of 1920, 1948, 2004, 2032, 2060, and 2088. There are also months of February with five Sundays before the 20th century and after the 21st century, but not in the same arithmetical progression, because the years 1900 and 2100 are not leap years.
Each other month of the calendar has two or three weekdays of which there are five that month, every year.
ISO standard 8601 defines a calendar that assigns a week number to each week of the year. The week always begins with Monday and the first Thursday of a year is always in week number 1. For the Gregorian calendar, this means that the week that includes 4 January is always week number 1.
In this calendar, all years in which 1 January is a Thursday have 53 weeks, and leap years in which 1 January is a Wednesday do so as well. Other years have 52 weeks.
The explanation for this is as follows:
If 1 January falls on a Monday, Tuesday, Wednesday, or Thursday, then it is in week number 1. If 1 January falls on a Friday, Saturday, or Sunday, then it is in the week preceding week number 1, so it is week 52 or 53 of the preceding year. For convenience, I refer to the week preceding week 1 as week 0.
An ordinary year contains 52 weeks plus one day. If New Year's Day in such a year falls on a Monday, Tuesday, or Wednesday (week 1), then the next New Year's Day falls on a Tuesday, Wednesday, or Thursday (week 1), so the just concluded year then contains 52 weeks (as far as week numbers are concerned). If New Year's Day falls on a Thursday (week 1), then the next New Year's Day falls on a Friday (week 0), so then the year contained 53 weeks. If New Year's Day falls on a Friday or Saturday (week 0), then the next New Year's Day is on a Saturday or Sunday (week 0), so then the just concluded year contained 52 weeks. If New Year's Day is on a Sunday (the last day of week 0), then the next New Year's Day falls on a Monday (the first day of week 1), but there is an extra week boundary between those two (from Sunday to Monday), so the year yet contained 52 weeks.
A leap year contains 52 weeks plus 2 days. If New Year's Day in such a year falls on a Monday or Tuesday, then such a year has 52 weeks. If New Year's Day falls on a Wednesday or Thursday (week 1), then the next New Year's Day is a Friday or Saturday (week 0), so the just concluded year contained 53 weeks. If New Year's Day falls on a Friday, then the next New Year's Day is on a Sunday (week 0), so then the year has 52 weeks. If New Year's Day falls on a Saturday or Sunday (week 0), then the next one is on a Monday or Tuesday (week 1), but there's an extra week boundary in between, so the year yet contained only 52 weeks.
The next table shows how New Year's Day is distributed across the days of the week during a period of 400 Gregorian years. After such a period the distribution of leap years repeats itself, and that period is also equal to a whole number of weeks (namely 20,871) so the next period of 400 years starts with the same day of the week as the previous one.
For example: During a period of 400 years, there are 15 leap years that have 1 January fall on a Sunday. The years that contain 53 weeks are marked with an *. There are 71 of those, and that fits a total of 71*53 + 329*52 = 20,871 weeks.
After a year with 53 weeks, the next year with 53 weeks is 5, 6, or 7 years later. Usually (43 out of 71 times) this is 6 years later, quite often (27 of 71 times) 5 years later, and once per 400 years it is 7 years later.
The first years since 2000 that contain 53 weeks are 2004, 2009, 2015, 2020, 2026, and 2032. The years 2296 and 2303 also contain 53 weeks, and that is the next time that there are 7 years between two years with 53 weeks.
We saw that there was much variation in the precise details of the calendars that have been used in Western Europe, even though they're all called Julian or Gregorian. For dating of historical events (for example, for history books or astronomical tables) it is useful to have an unambiguous calendar. For this unambiguous calendar, one uses the rules of the Gregorian calendar for dates from 15 October 1582, and the rules of the Julian calendar for dates up to and including 4 October 1582, with the era of Christ, even for days from before that era or the Julian calendar were even invented. The application of the rules of a calendar to days when those rules weren't in use (or even invented yet) is called proleptic. So, we can talk about 1 March of the year −300 in the Julian (proleptic) calendar, even though the Julian calendar hadn't been invented yet at that time, and the beginning of the era of Christ was still 300 year in the future.
Outside of astronomy, it is common to designate years before the epoch as "before Christ" (or BC), and years after the epoch as "ante diem" (or AD), such that the first year is AD 1, and the years immediately preceding that year are 1 BC, 2 BC, and so on. With this year count, which we may refer to as the historical year count, one must be extra careful when calculating the length of a period that includes the epoch. For example, the number of years between 1 March of year 3 BC and 1 March of year AD 2 is not 2 + 3 = 5, but one less: one year from 3 BC to 2 BC, another one to 1 BC, one more to AD 1, and the last one to AD 2. Ordinary arithmetic falters because the historical year count does not include a year 0. The most popular Christian date for the creation of the world is 23 October 4004 BC, and some people celebrated the 6000th anniversary of that date in 1996, presumably because 4004 + 1996 = 6000, but because of the absence of a year 0 there were only 5999 years between 4004 BC and AD 1996.
In astronomy there are quite a few calculations with years, and for
those it is much more convenient to use an unbroken sequence of
numbers for the years, including negative year numbers, and including
a year 0. For years after the epoch, the astronomical and historical
year counts coincide: the "astronomical" year 2001 or +2001 is
equivalent to the "historical" year AD 2001. The years immediately
preceding year +1 are years 0, −1, −2, and so on. To figure the
astromical year number that is equivalent to a particular historical
BC year number, subtract one from the BC year number and then put a
minus sign in front of it. The year 3 BC corresponds to the year −2,
and the number of years between that year and the year +2 is simply
+2 − (−2) = 2 + 2 = 4.
For identifying leap years, the astronomical year numbers are more convenient than the historical year numbers, too. According to the rules of the Julian proleptic calendar the year AD 8 was a leap year, because that year number is a multiple of 4. The calendar has a leap year each 4 years, so if we repeatedly count back 4 years from a leap year, then we find more leap years. In the historical year count, you then find leap years AD 4, 1 BC, 5 BC, 9 BC, and so on. The BC leap year numbers are therefore not multiples of 4. With the astronomical year count, however, those leap years are referred to as +4, 0, −4, −8, and so on, so then leap years before the epoch also have year numbers that are multiples of 4.
Julius Caesar introduced the Julian Calendar and with it the bissextile day in February. The first year of that calendar was the year −44, but that year was not considered to be a leap year, although it matches the rule that says that year numbers evenly divisible by four indicate leap years. That rule was not applied correctly for some time, which meant that leap years were observed more frequently than intended. This was noticed around the year −7, and then the leap year rule was suspended for a while until the excess number of leap years from the recent past had been balanced. From about the year 8 the calendar ran as intended, with leap years for year numbers evenly divisible by four.
I suspect that the counting error came from a cultural difference between the Romans and the advisor (a Sosigenes from Egypt) who recommended the leap year rule to Caesar. If you asked a Roman to point out "each fourth year" starting from year I, then that Roman would first count I - II - III - IV and end up at year IV = 4, and then count IV - V - VI - VII and end up at year VII = 7, and then VII - VIII - IX - X and end up at year X = 10. The difference between the successive last years was 3 and not 4, because they counted the last year of the previous period also as the first year of the next period. We would have called that "every third year", and I think Caesar's adviser did, too. That adviser surely explained to Caesar what was meant by "every fourth year" in the leap year rule, but Caesar was murdered long before the first leap year according to the new rule began, and his successor apparently had not gotten the same explanation.
It is not entirely clear which years in that first period of the new calendar were counted as leap years, because the Romans misapplied the rule and no documents have been handed down to us that say which years they had taken to be leap years. However, when they became aware of their mistake, then they suspended the leap year rule for a while until the calendar was back in line with what was originally intended. The Romans themselves did know which years out of the previous 40 years they had taken to be leap years, so they knew also how many bissextile days they needed to skip to get back in line.
For that reason we can assume that the number of leap years between
the years −44 and +7 is the number that fits the rule that every year
number divisible by four indicates a leap year, with as only exception
that the year −44 itself was not a leap year. That means that year 8
was the 13th leap year. After that there was a leap year every 4th
year, which means 25 leap years per century. The number
N of leap years in the Julian Calendar up to and
j ≥ 8) is equal to
N = ⌊j/4⌋ + 11
The year 100 was the 36th leap year (
N = ⌊100⁄4⌋ + 11 = 25 + 11
= 36). The year 1000 was the 261st leap year (
⌊1000⁄4⌋ + 11 = 250 + 11 = 261).
The now most widely accepted proposal for which years were regarded as leap years by the Romans in the first period of the Julian calendar is that of Scaliger from 1583 (see http://en.wikipedia.org/wiki/Julian_calendar), who regarded the following years as leap years between the years −44 and +7: −41, −38, −35, −32, −29, −26, −23, −20, −17, −14, −11, −8. After that, the year +8 was the first leap year.
From 1582 (in different years at different locations) the Julian
Calendar was replaced by the Gregorian Calendar, which has leap years
slightly less frequently. Until 1582 there had been (in the Julian
N = ⌊1582⁄4⌋ + 11 = 395 + 11 = 406 leap
years. Since that time (in the Gregorian Calendar) years with year
numbers evenly divisible by 100 but not by 400 are no longer leap
years. The years 1600 and 2000 were leap years (just like for the
rules of the Julian Calendar), but the years 1700, 1800, and 1900 were
N of leap years since the introduction of
the Julian Calendar up to and including the year
the Gregorian Calendar (
j ≥ 1582) is equal to
N = ⌊j/4⌋ − ⌊j/100⌋ + ⌊j/400⌋ + 23
In some countries the Gregorian Calendar was introduced much later than in the year 1582, but the difference between the leap year rules of the Julian and Gregorian Calendars was taken into account then, so that a country that switched later ran in step from then on with a country that had switched already in 1582.
The year 1600 was the 411th leap year (for countries that had switched to the Gregorian Calendar, and also for countries that still used the Julian Calendar then ― the year 1700 was the first year for which the leap year rules of the two calendars said different things about whether that year was a leap year). Since that time the Gregorian Calendar has only 97 leap years per 400 years, instead of 100. The year 2000 was the 508th leap year. The first couple of leap years from the year 2000 are:
At the beginning of the 20th century some Eastern Orthodox Churches wanted to switch from the Julian calendar to a more accurate calendar, but preferred not to use the Gregorian calendar of their religious competitors the Catholic Church. Milutin Milanković (1879-1958) invented a calendar change for these Churches that yields a calendar that follows the average of the seasons more accurately than the Gregorian Calendar does.
This Milanković Calendar (also called the Revised Julian Calendar) was equal to the Gregorian Calendar, except in the determination of which years are leap years. Years of which the number is divisible by 4 but not by 100 are leap years, just like in the Gregorian Calendar, but years of which the number is divisible by 100 are leap years only if the number leaves a remainder of 200 or 600 when it is divided by 900 (while in the Gregorian Calendar the year number must be evenly divisible by 400). With this, the average length of a year in the Orthodox Calendar is 365 109/450 days.
This new calendar was adopted by a synod of Orthodox Church leaders in Istanbul in 1923. http://www.friesian.com/russia.htm says that the calendar was used sporadically in the 1920s and not after that, but http://en.wikipedia.org/wiki/Revised_Julian_calendar mentions many Orthodox churches that use the calendar and does not say anything about the calendar being abandoned.
The Islamic calendar is used officially in countries where Islam is the state religion (especially in the Middle East), and in the Islamic faith. The Islamic calendar is a lunar calendar, and tries to follow the phases of the Moon without regards to the seasons.
For religious use, an Islamic month start with the first appearance of the crescent Moon in the evening sky after a New Moon. Because the beginning of the month depends on observations it cannot be accurately predicted. It may also happen that Muslims in different countries or cities observe the new Moon first on different days, and so put the start of the month on different days. Those differences are ordinarily at most one day, but can cause some problems when making appointments. Therefore, there is also an administrative calendar that is based on fixed rules (and can thus be predicted arbitrarily far into the future) but yet follows the phases of the Moon pretty closely. The difference between the religious and administrative Islamic calendars is usually at most one day.
An Islamic calendar year always has 12 months. In the administrative calendar, the months alternate between 30 and 29 days, and the last month can get an extra day (and so have 30 rather than 29 days), as is shown by the following table
|3||Rabī`a Ⅰ (ربيع الأوّل)||30|
|4||Rabī`a Ⅱ (ربيع الثاني)||29|
|5||Jumādā Ⅰ (جمادى الأولى)||30|
|6||Jumādā Ⅱ (جمادى الثانية)||29|
|11||Dhu al-Qa`dah (ذو القعدة)||30|
|12||Dhu al-Ḥijjah (ذو الحجّة)||29/30|
With this, a calendar year has 354 or 355 days. In the administrative calendar, a month lasts on average 29 191/360 days, and the year 354 11/30 days. With this, the calendar month is about 3 seconds short of the average synodical month. After 30 years = 10631 days the administrative calendar repeats itself (as far as the order of leap years is concerned).
Because the Islamic calendar year is about 11 days shorter than the tropical year, the beginning of the Islamic year moves through all seasons in about 34 years' time, and the difference between the Gregorian year number and the Islamic year number (now about 580) decreases by one about every 34 years. The beginning of the Islamic year 1420 is around the third day before the Gregorian year 2009. The administrative Islamic and the Gregorian calendar will "intersect" around the 5th month of the year 20,874: the 5th month of the year 20,874 in the Gregorian calendar will start on the same day as the 5th month of the year 20,874 in the Islamic administrative calendar.
The epoch of the Islamic calendar is sunset of 15 July 622 on the Julian proleptic calendar, and is tied to the migration of Muhammad from Mecca to Medina.
The next table shows at what dates in the Gregorian calendar the Islamic New Year and the beginning of the Islamic month of Ramaḍān fall in the Islamic years 1424 through 1430, according to the administrative calendar.
The Jewish calendar is used in Israel and in the Jewish faith. The Jewish calendar is a lunisolar calendar: the days start at sunset, months start with New Moon, and the year follows the seasons. The calendar months have 29 or 30 days, and the calendar year 12 or 13 months. The months of the Jewish calendar and their lengths are listed in the following table.
|2||Iyar (אִיָּר / אייר)||29|
|3||Sivan (סִיוָן / סיוון)||30|
|8||Ḥeshvan (מַרְחֶשְׁוָן / מרחשוון)||29/30|
|9||Kislev (כִּסְלֵו / כסליו)||29/30|
|-/12||Adar Ⅰ (אֲדָר א׳)||0/30|
|12/13||Adar / Adar Ⅱ (אֲדָר / אֲדָר ב)||29|
The Bible names Nisan as the first month, but mentions 1 Tishi as New Year's Day. Leap (bissextile) days can be inserted at the end of Ḥeshvan and Kislev, and an embolismic month Adar Ⅱ can be inserted before Adar, which is then called Adar Ⅰ. With this, there are six possible lengths of the calendar year:
|year length||type name|
|353||deficient ordinary year|
|354||regular ordinary year|
|355||complete ordinary year|
|383||deficient leap year|
|384||regular leap year|
|385||complete leap year|
The calendar month on average lasts 29 13753/25920 days, and the calendar year 365 617/2500 days. The calendar year is about 6 minutes longer than a tropical year, and the calendar month about 0.3 seconds longer than a synodical month. The beginning of the Jewish calendar year is around the beginning of autumn in the northern hemisphere, between 3 September and 12 October on the Gregorian calendar.
The epoch of the Jewish calendar is at sunset of 6 October −3760 on the Julian proleptic calendar, and is tied to the (assumed) creation of the world.
The Julian day count is a calendar that knows no months or years, but merely counts the days since the epoch (Julian Day 0), which was at noon of 1 January −4712 on the Julian proleptic calendar. This calendar is useful for astronomical calculations, and also as way station for the conversion of a date from one calendar to another calendar. Also, the day of the week is easy to calculate from the number of the Julian day that starts on the desired date: take the Julian day number and determine the remainder after dividing by 7. A remainder of 0 indicates the day is a Monday; 1 means Tuesday, and so on.
The following table shows some correspondences between modern calendars, the Egyptian calendar, the Maya Long Count, and the Julian day count. The correspondences were calculated for noon (12:00) of the dates. For the Jewish calendar, Tishri is month 1. For the Egyptian calendar, the 5 days after the 12 regular months are counted as a single month 13. Dates are given in the order year number, month number, day number.
At noon on 1 January 2001, JD (= Julian day) 2,451,911 began. The Julian Day Number Calculation Page explains how you can calculate the Julian Day Number for dates in the Gregorian calendar.
The traditional Chinese Calendar is a lunisolar calendar. Calendar days begin at midnight, calendar months begin on the day of New Moon, and calendar years follow the seasons. A calendar year has 12 or 13 calendar months, depending on how many New Moons there are between two successive southern solstices. The months get numbers 1 through 12. A leap month gets the same number as the preceding month, but is marked as a "leap month" (闰月 rùnyuè). Not all leap months have the same number.
Besides the calendar months (lunar months) there are also solar months. Those solar months are needed to determine the numbering of months in a leap year. The solar months are tied to the position of the Sun along the ecliptic and correspond to the (Western) astrological signs of the zodiac: Each solar month begins when the ecliptical longitude of the Sun is a multiple of 30°.
The rules for determining the month numbers are:
Chinese Calendar years have a name from a set of 60 names that repeat themselves (干支 gānzhī). The first year from the first cycle began on the day that corresponds to 8 March −2636 in the Julian proleptic calendar (CJD 758326). The name of a year is formed from a "celestial stem" (天干 tiāngān) from a list of 10 and a "terrestrial branch" (地支 dìzhī) from a list of 12. The (untranslatable) heavenly stems are:
The terrestrial branches are:
The celestial stem and the terrestrial branch both shift for each year, so after year jǐ-chǒu (stem 6, branch 2) follows year gēng-yín (stem 7, branch 3). The first year from the cycle of 60 is jiǎ-zǐ (stem 1, branch 1).
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Last updated: 2013-01-22