First draft: October, 2018 Last major update: April, 2019

China officially adopted the Gregorian calendar since 1912. However, Chinese people still use the traditional Chinese calendar (農曆) today. Many Chinese festivals and some public holidays are still based on the Chinese calendar. It is therefore useful to understand how the Chinese calendar works. This page intends to give only a brief introduction to the Chinese calendar. For a more detail and comprehensive introduction to the subject, I suggest Helmer Aslaksen's wonderful article "Mathematics of the Chinese calendar"^{[Aslaksen10]}. This is the best English reference that I know of. Another nice article is Section 15.8 in the book *Explanatory Supplement to the Astronomical Almanac*^{[Richards13]}. The best Chinese reference I know of is the book 历书百问百答 (*Calendars: 100 Questions and Answers*)^{[Tang86]} by Táng Hànliáng (唐汉良).

**Contents**

- Introduction
- Important Concepts
- Rules for the Modern Chinese Calendar
- Examples of Computing the Chinese Calendar
- Why Do the Rules Work?
- New Standards 2017 (GB/T 33661-2017)

Calendars were created to keep track of time. There are three cycles occurring in nature that were important to people in the ancient time. They are the cycle of day and night, the cycle of moon phases (lunar cycle) and the cycle of seasons (solar cycle). The cycle of day and night can be characterized by the average value of a solar day, which is the average time between two successive solar noons. The lunar cycle can be characterized by the average time between two successive new moons, or the synodic month. The cycle of seasons can be characterized by the tropical year, which is *approximately* the average time between two successive winter solstices (or between two successive vernal equinoxes)^{[fn1]}. The concept of a day is directly related to the cycle of day and night. The concept of a month is (at least initially) related to the lunar cycle, and the concept of a year is related to the cycle of seasons.

One synodic month has 29.5306 days and one tropical year has 365.2422 days. None of these three cycles are commensurate with each other (i.e. none of the ratios of any pair of the cycles is a rational number). The challenge is to derive a scheme to arrange days into months and years to keep track of the other two cycles. One approach is to ignore the lunar cycle and only keep track of the cycle of seasons. This is called a *solar calendar* because its arrangement of days and months in a year depends only on the position of the Sun on the ecliptic. Gregorian calendar is an example of a solar calendar. Another approach is to ignore the cycle of seasons and only keep track of moon phases. The resulting calendar is called a *lunar calendar*. Islamic calendar is an example of a lunar calendar. The third approach is to keep track of all of the other two cycles. The resulting calendar is called a *luni-solar* calendar. The Chinese calendar is an example of a luni-solar calendar.

In Chinese calendar, the first day of a month is determined by the day of the lunar conjunction (new moon). Since the synodic month is 29.5306 days, a Chinese month can have 29 days (short month) or 30 days (long month), depending on the number of days between the dates of two successive new moons. One tropical year has 365.2422 days. Therefore, there are on average 12.37 Chinese months in a tropical year. A Chinese year normally has 12 months. To keep the calendar in sync with seasons, an extra month has to be inserted to the year about every 3 years, in which case there are 13 months in the year. This extra month is called a leap (intercalary) month.

The challenge is to derive a scheme to insert a leap month so that a Chinese year is close to the tropical year on average. Various schemes were used over the history of China. There were five significant reforms in the Chinese calendar. The last one occurred in 1645. Here I only talk about the rules applied to the modern Chinese calendar. Several important concepts need to be explained first before describing the rules.

To fully understand how the modern Chinese calendar works, we need to first understand how we measure time today.

Universal Time (UT) is a time standard based on Earth's rotation with respect to the Sun. There are several versions of universal time. The most commonly used are UT1 and the Coordinated Universal Time (UTC). UT1 is based on the orientation of a reference meridian^{[fn2]} on Earth's surface relative to distant stars, and then use a simple transformation to obtain a time standard in accord with the mean solar time. The problem is that Earth's rotation is not uniform, which makes UT1 inconvenient for many applications. As a result, the Coordinated Universal Time (UTC) is introduced to approximate UT1.

UTC is defined by two components: International Atomic Time (TAI) and UT1. TAI is a weighted average of the time kept by over 400 atomic clocks in over 50 national laboratories worldwide. Each second of in a TAI is a constant. UTC is defined to be TAI plus an integer number of seconds. The duration of one second in UTC is therefore exactly equal to one TAI second. Leap seconds are added to ensure that the difference between UTC and UT1 is smaller than 0.9 seconds. Leap seconds are usually inserted to the last minute on June 30 at 23:59:59 UTC or on December 31 UTC at 23:59:59 UTC of a year. In principle, a leap second can be positive or negative. The minute with a positive leap second has 61 seconds and the minute with a negative leap second has only 59 seconds. So far only positive leap seconds appear. As of today (October 2018), a total of 27 leap seconds have been inserted since the system of adjustment was implemented in 1972. The most recent leap second occurred on December 31, 2016 at 23:59:60 UTC. Thus, time in UTC is uniform, except for occasional "glitches" when leap seconds are added. These "glitches" cause some problems and there have been discussions on the abolishment of leap seconds since 2005.

Civil time is related to UTC by a UTC offset. A UTC offset is a multiple of 15 minutes, and the majority of offsets are in whole hours. As far as the Chinese calendar is concerned, the most important civil time is the China standard time, which is UTC+08:00, meaning that it is 8 hours ahead of UTC. This is the local time for the meridians of 120° East.

In Chinese calendar, years are counted in sexagenary cycle. Months are simply indicated by numbers 1, 2, 3, ..., 12. A leap month is indicated by the same number as the previous month, but a "leap" is added before the number. Days in a month are also simply indicated by 1, 2, 3, ..., 30. On my calendar page, dates in the Chinese calendar are indicated by MM-DD, where MM represents the month number and DD represents the day number in the month. Dates in a leap month are represented by leap MM-DD. For example, 11-15 means the 15th day of month 11. It should be noted that this MM-DD convention is not used everywhere else.

The lunar conjunction, or new moon, is the moment when the Moon and the Sun are in the same direction. The above description is too vague by modern standard. In modern astronomy, lunar conjunction (new moon) is defined as the time when the apparent geocentric longitude of the Moon λ_{M} is equal to the apparent geocentric longitude of the Sun λ_{S}. Three other related concepts are the first quarter, full moon (or lunar opposition) and third quarter. First quarter is defined as the time when λ_{M} - λ_{S} = 90° (modulo 360°); full moon is defined as the time when λ_{M} - λ_{S} = 180° (modulo 360°); third quarter is defined as the time when λ_{M} - λ_{S} = 270° (modulo 360°).

The day on which a new moon occurs is the day of the new moon. Note that in China days are measured from midnight to midnight, so 0:00:00 and 23:59:59.999 are considered to be on the same day. Also, times are measured in the China standard time (UTC+08:00). For example, a new moon occurred at UTC 17:36 on December 5, 2010. In China standard time, this new moon occurred at 01:36 on December 6, 2010. So the date of this new Moon was December 6, 2010.

One of the rules of the Chinese calendar is that the first day of a month must occur on the day of a new moon. Therefore, December 6, 2010 was the first day of a Chinese month. It was in fact the first day of month 11 (see below). The next new moon occurred on January 4, 2011 at 17:03 (UTC+8). So January 4, 2011 was the first day of a month. It was month 12. Since there were 29 days between December 6, 2010 and January 4, 2011, month 11 had 29 days. It was a short month.

As explained in the 24 solar term page, ancient Chinese used the 24 solar terms to keep track of the Sun's position on the ecliptic. The 24 solar terms are grouped into 12 major solar terms and 12 minor solar terms. The major solar terms can be defined as the times when the apparent geocentric longitude of the Sun reaches integral multiples of 30°. The major solar terms are labelled by Z followed by a number on the 24 solar term page.

Winter solstice (Z11) is one of the major solar terms. It plays an important role in the Chinese calendar. As explained in the sexagenary cycle page, the zǐ month is defined as the month containing the winter solstice. Since the yín month is designated as the first month of a year, the zǐ month corresponds to month 11. Thus, month 11 always contains the winter solstice. For example, in 2010 the winter solstice occurred on December 22 at 07:38 (UTC+8). We mentioned above that a new moon occurred on December 6, 2010 and the following new moon was on January 4, 2011. The month starting on December 6 contained the winter solstice, so it was month 11.

As mentioned in the solar term page, the period between two successive winter solstices is called a *suì*. Suì can also be referred to a period starting on the first day of a zǐ month (month 11) and ending on the day before the first day of the next zǐ month. Simple calculations show that the number of months between the two "month 11"s (counting the first month 11 but not the second month 11) can be 12 or 13. If there are 12 months, I call the suì as a *regular suì*. If there are 13 months, I call the suì as a *leap suì*, using the terminology in Aslasken's article^{Aslaksen10}. One of the rules in the Chinese calendar is that in a leap suì, one of the 13 months is a leap month.

In a regular suì, the 11 months following month 11 are assigned the numbers 12, 1, 2, ..., 10 and we are done in designating the months in a regular suì. In a regular suì, the Chinese new year is the first day of the second month after month 11, which is the day of the second new moon after the day of winter solstice.

Since there are 12 major solar terms in a suì and there are 13 months in a leap suì, it follows that at least one month in a leap suì does not contain a major solar term. Here is an important rule: in a leap suì, the first month (after month 11) that does not contain a major solar term is a leap month. In a leap suì, the 11 *regular* months (i.e. non-leap months) following month 11 are assigned the numbers 12, 1, 2, ..., 10. The leap month is assigned the same number as its preceding month. In a leap suì, the Chinese new year is the first day of the second regular month after month 11.

Most of the rules have already been mentioned as the important concepts are explained. It is time to gather them together. The following is a summary of the rules.

- The instants of lunar conjunctions and solar terms are calculated for meridians of 120° East.
- The days are measured from midnight to midnight.
- The first day of a month is the day in which a conjunction of the Moon (new moon) falls.
- The winter solstice (Z11) always falls in month 11.
- If a suì contains 13 complete months, one of them is a leap month. This leap month is the first in the suì that contains no major solar term.
- The second month (not counting the leap month) after month 11 is the first month of a year.
- Years are counted in sexagenary year cycles.
- Months are assigned numbers 1 to 12; a leap month is assigned the same number as its predecessor but with the word "leap" added before the number.
- Days in a month are assigned numbers 1 to 29 or 30. Sexagenary days can also be used to indicate dates.

Except for Rules 1 and 5, all of the rest of the rules are passed down from ancient time. Rules 4 and 6 are the statement "the month containing the winter solstice is the zǐ month, and the yín month is designated as the first month" expressed in a language without referencing the branch name of a month (see *Jiàn* and the "Three Standards"). Rule 5, which was revised from an older rule, was created around the calendar reform in 1645, and Rule 1 was added in 1928.

Today Rule 1 simply means that times are expressed in UTC+8. However, UTC was invented in 1960. There were several changes in UTC until it was finalized in 1972. On my calendar page, times are given in UT1+8 before 1972 and UTC+8 in and after 1972. This has almost no effect because times are given to the nearest minute on the calendar page and the difference between UT1 and UTC was at most a few seconds between 1960 and 1972.

Rule 1 was only adopted since 1929. Before 1929, times were based on the Beijing meridian (116°25' East), which is about 14 minutes earlier than the times based on the meridian of 120°E. If a solar term or new moon occurred close to the midnight, the date could be off by one day. For example, the new moon associated with the Chinese new year in 1916 occurred on February 4 at 00:05 (UT1+8), but the Chinese new year was celebrated on February 3 because the new moon was on February 3 at 23:51 according to the local time of the Beijing meridian.

It is easy to derive an algorithm to calculate the Chinese calendar based on these rules. For simplicity, I use the symbol N_{Y} to represent the Chinese year whose New Year day is closest to Jan. 1 in the Western year Y. The following shows the steps of determining each month in the suì S_{Y} from month 11 in N_{Y-1} to the day before month 11 in N_{Y}

- Calculate the dates of the winter solstices W
_{Y-1}and W_{Y}in Gregorian years Y-1 and Y. - Calculate the dates of the first lunar conjunction M
_{-1}that occurs on or before W_{Y-1}, and the first lunar conjunction M_{11}that occurs on or before W_{Y}. - Calculate the number of complete lunations L
_{Y}between M_{-1}and M_{11}. This is the integer closest to the number

(M_{11}- M_{-1})/29.53. - If L
_{Y}= 12, there is no leap month in the suì S_{Y}. Calculate the dates of the 11 lunar conjunctions between M_{-1}and M_{11}. Label them in chronological order as M_{0}, M_{1}, M_{2}, ..., M_{10}. - If L
_{Y}= 13, there is a leap month in the suì S_{Y}. Calculate the dates of 11 major solar terms between W_{Y-1}and W_{Y}, and the dates of the 12 lunar conjunctions between M_{-1}and M_{11}. Determine the first month after M_{-1}that does not contain a major solar term. This is the leap month. Label the dates of the 11 lunar conjunctions after M_{-1}that are*not*associated with the leap month in chronological order as M_{0}, M_{1}, M_{2}, ..., M_{10}. Label the date of the lunar conjunction associated with the leap month as M^{*}_{i}, where i is the month number of the previous month.

- If L
- M
_{-1}is the first day of month 11 of in the Chinese year N_{Y-1}; M_{0}is the first day of month 12 in N_{Y-1}; M_{1}is the Chinese New Year of N_{Y}; M_{2}, M_{3}, ..., and M_{11}are the first days of month 2, month 3, ..., and month 11 in N_{Y}. If there is a leap month, the first day of the leap month is M^{*}_{i}.

These steps determine the months from month 11 in N_{Y-1} to month 10 in N_{Y}. To determine the rest of the months in N_{Y}, one needs to go through the above steps for the suì S_{Y+1}.

In May 2017, the Chinese government issued a document (labelled GB/T 33661-2017) drafted by astronomers at the Purple Mountain Observatory (PMO) on new standards for computing the Chinese calendar^{PMO17}. The reason for issuing this document is probably related to the following history.

In the past, the book *Wànniánlì* (萬年曆) was quite popular in China. *Wànniánlì* literally means *ten-thousand-year calendar*. However, the Chinese character *wàn* (萬) (literally means ten thousand) is often used to mean "many". So a better translation of *Wànniánlì* is *Calendar Covering Many Years*. According to the book *New Edition of Wànniánlì*^{PMO86} edited by the PMO and the book *Pocket Edition of 100-Year Chinese Calendar*^{Liu93}, *Wànniánlì* was edited by astronomers in the Imperial Astronomical Bureau (欽天監) of the Qīng dynasty. The first *Wànniánlì* contained Chinese calendar covering years from 1624 to 1835. In 1787, a new edition of the book was published, extending the years to 1935. From then on, every time a new emperor ascended to the throne, a new edition of *Wànniánlì* would be published, in which calendar up to about 200 years into the future would be calculated. The last *Wànniánlì* before the end of the Qīng dynasty was published in 1910, covering years up to 2108.

The Chinese government at that time, the Qīng government, issued the *Shíxiàn Calendar* every year, providing the official calendar for the upcoming year. *Shíxiàn Calendar* was also calculated by astronomers in the Imperial Astronomical Bureau. Sometimes they would modify the calendar data in *Wànniánlì*. For example, in the *Shíxiàn Calendar* for the year N_{1841}, the first day of month 12 was one day earlier than that listed in the *Wànniánlì* published in 1799; in the *Shíxiàn Calendar* for the year N_{1856}, the first day of month 11 was one day later than that listed in the *Wànniánlì* published in 1824; in the *Shíxiàn Calendar* for the year N_{1890}, the first day of month 7 was one day later than that listed in the *Wànniánlì* published in 1862. These are just a few examples to show that *Wànniánlì* was not officially used even in the Qīng dynasty.

In 1912, the Republic of China was established and adopted the Gregorian calendar as the official calendar. However, Chinese people still used the Chinese calendar and *Wànniánlì* was still popular. The *Republic of China Calendar* published by the Beiyang government of the Republic of China still listed the dates of the Chinese calendar in addition to the Gregorian calendar. In 1929, the Nationalist government tried to ban the use of the traditional Chinese calendar. The *Kuómín Calendar* published by the government no longer listed the dates of the Chinese calendar. However, Chinese people were used to the traditional calendar and many traditional customs were based on the Chinese calendar. The ban was not successful and was lifted in 1934.

Since there was no longer official version of the Chinese calendar, two different versions of Chinese calendar sometimes appeared and caused confusion. For example, there were two different versions of Chinese calendar appearing in southern China in 1978. One version was based on the calculation by the PMO. The other version, which was imported from Hong Kong, was based on the *Wànniánlì* published in 1910. According to the PMO version, the lunar conjunction associated with month 8 occurred at 0:09 (UTC+8) on Sep. 3, 1978. The old calendar, in which apparent solar time was calculated for the Beijing meridian, put the lunar conjunction on Sep. 2, 1978 a few minutes before midnight. This meant that there was one day difference between the two versions of the calendar on the first day of month 8. The 15th day of month 8 is called the mid-autumn festival, a harvest festival celebrated by the Chinese. As a result, people in Hong Kong and mainland China celebrated the mid-autumn festival on different days in 1978. From then on, the Chinese calendars published in Hong Kong have been based on the PMO calculation.

In May 2017, the Chinese government issued the document GB/T 33661-2017 to reconcile traditional calendrical practices with modern astronomical concepts. The rules for the Chinese calendar are the same as the ones stated above, but the astronomical concepts relevant to the calendar calculation are defined very precisely using modern terminologies following the IERS conventions. It also requires that calculations of the times of the 24 solar terms and lunar conjunctions must be accurate to about one second, excluding the unpublished leap seconds at the time of calculation. The document also requires that calendar data from 1912 to 2017 should refer to calendars published by the PMO. The Purple Mountain Observatory is also responsible for providing the official, most up-to-date calendar data for the upcoming year on a yearly basis. Thus, Chinese calendar has an official version again.

It is clearly a good thing to set unified standards for all Chinese calendar calculations to prevent confusion. However, the requirement of the one-second accuracy in the calculation of times of lunar conjunctions and 24 solar terms is baffling to me, because such accuracy is rarely, if ever, needed for calendar calculations. Let's take a closer look at this accuracy requirement.

What does it take to achieve a one second time accuracy? The 24 solar terms are defined as the times when the apparent longitude of the Sun reaches integer multiples of 15°. The mean longitude of the Sun increases by 360° in a tropical year (365.2422 days). This means that in one second, the Sun moves, on average, by 360°/(365.2422×86400) = 0.04" (0.04 arcseconds). To calculate the times of the 24 solar terms to within one second, we must calculate the position of the Sun to accuracy better than 0.04". For the lunar conjunction, a similar calculation shows that the accuracy requirement for the position of the Moon is about 360°/(29.5306×86400) = 0.5". No ephemerides in the world could achieve this level of accuracy until perhaps in the late 1970s. In addition, a precise definition of the ecliptic and equinoxes is needed to meet the accuracy requirement, which is the reason why the document has to specify precisely how the relevant astronomical terms are defined.

High accuracy is only required for the situations in which lunar conjunctions or solar terms occurring very close to the midnight. However, those situations will occur decades from now and even in those situations an accuracy of one second does not help. Note that the one-second accuracy does not account for the unknown leap seconds that will be added to the UTC, as specified by the GB/T 33661-2017 document. Times for lunar conjunctions and 24 solar terms can be computed very accurately in *barycentric dynamical time* (TDB), which is the time used in modern ephemerides. TDB is a uniform time scale defined by general relativity. To convert TDB to UTC, TDB can be first transformed to the terrestrial time (TT) using a transformation formula in general relativity. However, the difference between TDB and TT remains less than two milliseconds over several millennia and can be ignored in calendar calculations. TT is related to TAI described above by a constant offset: TT = TAI + 32.184 seconds. As mentioned above, UTC and TAI differ by an integral number of seconds. Hence we need to know the number of leap seconds that will be added to UTC in order to convert TDB to UTC. However, Earth's rotation is irregular and it is very difficult, if not impossible, to predict precisely the number of leap seconds that will be added to UTC decades ahead. This means that even though times in TDB can be calculated to within one second, their accuracy in UTC is still dominated by the uncertainty of the amount of leap seconds that will be added to UTC. Hence the accuracy of times in UTC+8 decreases with time even if we can compute the times in TDB with an accuracy of one second. The best approach to handle the situations in which lunar conjunctions or solar terms occurring very close to the midnight is to acknowledge that it is currently impossible to determine their exact dates, which is what this website does.

The situation in which lunar conjunctions and solar terms occurring near the midnight obviously also happened in the past. In the past, the definitions of lunar conjunction and solar terms were not as accurate as today's, and the methods used to compute their times were also less accurate. In addition, times based on the 120° East meridians were not imposed until 1929. Thus, there are inconsistencies in some of the dates of lunar conjunctions and solar terms between the calculations in the past and calculations using modern methods, which may result in discrepancies in the calculated Chinese calendar. In these situations, we should use the calendar issued by the government at that time. On this website, all printed times are based on calculations using the modern method, but the Chinese calendar (before ~2000) is based on the actual calendar issuded by the Chinese government. As for the 24 solar terms, the data based on the modern calculation and old calculations are both listed on my calendar page for years before 1733. The dates of the solar terms based on the old calculations are labelled as "calendrical solar terms", meaning solar terms calculated based on the astronomical system used at that time. After 1733, the dates of the calendrical solar terms were mostly the same as the dates calculated using the modern method and therefore they are omitted, except when there were discrepancies.

- [Aslaksen10] Aslaksen, Helmer, "Mathematics of the Chinese calendar" (PDF) (July 17, 2010), Department of Mathematics, National University of Singapore (now at University of Oslo).
- [fn1] Tropical year is now defined as the time needed for the Sun's mean longitude to increase by 360°. Its value is currently 365.2422 days. The average time between two successive winter (December) solstices is currently 365.2427 days. The average time between two successive vernal (March) equinoxes is currently 365.2424 days. They are close but not exactly the same. This is caused by the precession of the equinoxes and the fact that Earth's orbit around the Sun is slightly eccentric, as explained in this article.
- [fn2] In the past, the reference meridian was the prime meridian passing through the Royal Observatory at Greenwich. The current reference meridian used to define UT1 is the Terrestrial Intermediate Origin (TIO), which is about 100 meters east of the Greenwich meridian.
- [Liu93] Liú, Bǎolín (劉寶琳), 《100年袖珍干支月曆》 (
*Pocket Edition of 100-Year Chinese Calendar*), 商務印書館(香港) (The Commercial Press, Hong Kong), 1993. - [PMO17] 《农历的编算和颁行》("Calculation and promulgation of the Chinese calendar"), revised version (June 28, 2017), issued jointly by General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China and Standardization Administration of the People's Republic of China, drafted by Purple Mountain Observatory. PDF version of the draft can be downloaded here.
- [PMO86] 《新编万年历（修定本）》 (
*New Edition of Wànniánlì*, revised edition), edited by 中国科学院紫金山天文台 (Purple Mountain Observatory, CAS), 科学普及出版社 (Popular Science Press), 1986. - [Richards13] Richards, E.G., in Section 15.8 of
*Explanatory Supplement to the Astronomical Almanac*, ed. by S.E. Urban and P.K. Seidelmann, third edition, University Science Books, Mill Valley, California (2013). A PDF version of Chapter 15 of the book is available online at the bottom of the U.S. Naval Observatory calendar page (direct link). - [Tang86] Táng, Hànliáng (唐汉良), 历书百问百答 (
*Calendars: 100 Questions and Answers*), 江苏科学技术出版社 (Jiangsu Science & Technology press), 1986.