# The Mathematics of the Chinese Calendar

Chinese New Year is the main holiday of the year for more than one quarter of the world's population; very few people, however, know how to compute its date. For many years I kept asking people about the rules for the Chinese calendar, but I wasn't able to find anybody who could help me. Many of the people who were knowledgeable about science felt that the traditional Chinese calendar was backwards and superstitious, while people who cared about Chinese culture usually lacked the scientific knowledge to understand how the calendar worked. In the end I gave up and decided that I had to figure it out for myself.

 Chinese astronomers determining the summer solstice

## Paper on the Mathematics of the Chinese Calendar

I have a written a long paper on The Mathematics of the Chinese Calendar. It gives you all the details. This web page is just an introduction to the topic. I have also written a shorter introduction called When is Chinese New Year?. This paper won a fourth prize in the Fifth Annual Boeing Writing Contest, which is organized by the Griffith Observatory. The article appeared in the Griffith Observer, 66 (2002), no. 2 (February), 1-17. I have also written a paper on Fake Leap Months in the Chinese Calendar: From the Jesuits to 2033.

I give a lot of public lectures on calendar topics and here are lecture notes on Heavenly Mathematics: The Mathematics of the Chinese, Indian, Islamic and Gregorian Calendars, The Mathematics of the Public Holidays of Singapore, The Mathematics of the Chinese Calendar. You may want to only download the first file. The two other are subsets of the first.

The main focus on my paper is the study of leap months in the Chinese calendar. In the early 1990s, Chinese astronomers discovered that there was an error in the Chinese calendar for 2033. The traditional calendar claimed that the leap month would follow the 7th month, while in fact it comes after the 11th month. It is very unusual that the 11th month has a leap month, in fact it hasn't happened since the calendar reform in 1645 (before 1645, all months had the same probability for having a leap month). But many Chinese astronomers still claim that there will never be a leap month after the 12th and 1st month. I have found that there will be a leap month after the 1st month in 2262 (in fact, it should have happened in 1651, but they got the calculations wrong) and there will be a leap month after the 12th month in 3358. Since the Chinese calendar is an astronomical calendar, predictions require delicate astronomical calculations, so my computations for 3358 should probably be taken with a grain of salt. I also discuss other mathematical issues related to the Chinese calendar.

## Reading and Writing Chinese Characters and Pinyin on the Web Using Unicode

If you can't read the Chinese characters or the pinyin on this page, please go to my page on Reading and Writing Chinese Characters and Pinyin on the Web Using Unicode.

 Astronomical instruments in the Imperial Observatory in Beijing made by the Jesuit missionary Ferdinand Verbiest, 1670

## The Date of Chinese New Year

The mathematics behind the date of Chinese New Year is explained in full detail in my paper The Mathematics of the Chinese Calendar or the shorter introduction When is Chinese New Year?, but I will give two quick rules of thumb here.

One rule of thumb is that Chinese New Year should be the new Moon closest to the beginning of spring (立春, lìchūn). This rule is correct most of the time, but it can fail if Lìchūn falls close to halfway between two new Moons. It failed in 1985 and will fail again in 2015. Since Lìchūn falls around February 4, this helps explain why Chinese New Year will always fall between January 21 and February 21. It also helps explain why Chinese New Year is called the spring festival. If you have a Western calendar that indicates the phases of the Moon, this will give you an approximation of the date of Chinese New Year. But notice that the Chinese calendar uses the time of new Moon in China.

As explained above, Chinese New Year will always fall between January 21 and February 21. The tropical (or solar) year is about 365.25 days, while a synodic (or lunar) month is about 29.5 days. Hence a lunar year consisting of 12 months will be about 12 x 29.5 = 354 days. So a lunar year is about 11 days shorter than a solar year.

The second rule of thumb is therefore that most of the time Chinese New Year will fall 11 (or sometimes 10 or 12) days earlier than the previous year, but if that would take us outside of the Chinese New Year range of January 21 to February 21, we must add a leap month, so Chinese New Year jumps 19 (or sometimes 18) days later. If this rule takes you close to January 21, you can end up being one month wrong, otherwise you will be at most one day off.

 year date of CNY change in date of next CNY 1999 2000 2001 2002 2003 2004 2005 2006 2007 Feb 16 Feb 5 Jan 24 Feb 12 Feb 1 Jan 22 Feb 9 Jan 29 Feb 18 11 12 19 11 10 18 11 20

## The Sexagenary Cycle

An important aspect of the Chinese calendar is the sexagenary cycle (干支, gān zhī). This is a combination of the 10 heavenly stems (天干, tiān gān), and the 12 earthly branches (地支, dì zhī).

Stems 天干 tiān gān Element Branches 地支 dì zhī Animal
1 jiǎ Wood 1 Rat
2 Wood 2 chǒu Ox
3 bǐng Fire 3 yín Tiger
4 dīng Fire 4 mǎo Rabbit
5 Earth 5 chén Dragon
6 Earth 6 Snake
7 gēng Metal 7 Horse
8 xīn Metal 8 wèi Goat
9 rén Water 9 shēn Monkey
10 guǐ Water 10 yǒu Chicken
11 Dog
12 hài Pig

To explain how this cycle works, let us denote both the stems and the branches by their numbers. We denote 1 by (1,1) or (甲,子), 2 by (2,2) or (乙,丑) and so on up to (10,10) or (癸,酉). But now we have run out of stems, so we denote 11 by (1, 11) or (甲,戌) and 12 by (2, 12) or (乙,亥). Now we have run out of branches, too, so 13 becomes (3, 1) or (丙,子). We continue in this way through 6 cycles of stems and 5 cycles of branches up to 60, which is (10, 12) or (癸,亥). The next number is then (1,1) or (甲,子), which starts a new sexagenary cycle.

## Why is 2000 a Golden Dragon Year?

This cycle is used for keeping track of years, months, days and (double) hours in Chinese astrology. Your date and time of birth is determined by the eight characters (八字) formed by the pair of cyclical characters, or pillar, (﻿柱, zhù), for the year, month, day and hour. The 60-day cycle has been used for keeping track of days since ancient times and go back to at least the 13th century BCE during the Shāng Dynasty (商朝, 1600--1046 BCE). The 60-month cycle is also old. The 60-year cycle was introduced during the Hàn Dynasty ((汉朝 [漢朝]) and is related to the orbital period of Jupiter. In modern times, the year cycle is the only one in common use. The branches are often associated with the sequence of 12 animals: rat, ox, tiger, rabbit, dragon, snake, horse, sheep, monkey, rooster, dog, and pig. It is not clear when the branches were associated with the 12 animals, but it seems to have taken place around the time of the Táng Dynasty.

Notice that each branch, or animal, occurs five times in each 60-year cycle. An animal corresponding to an odd number, will meet the stems that correspond to the odd numbers. Year 2000 is the 17th year in the current cycle, so it corresponds to (7,5) (17 = 10 + 7 = 12 + 5) or (庚, 辰). So we see that it is a metal dragon year, or a golden dragon.

Determining the stem corresponding to a month is easy. The 11th month has branch 1, the 12th month has branch 2, the first month has branch 3 and so on. So the only problem is to keep track of the stem. There are two things to notice here. First of all, this system ignores leap months. The month pillar of a leap month is the same as the month pillar of the previous months! Secondly, why does the first branch correspond to the 11th month and not the first month?

In fact, both of these paradoxes are easy to explain. Since two months can have at most 60 days, the day pillar will still separate two different days. In a sense you can think of a month and its following leap month just as a one long month. And why does the first branch correspond to the 11th month? Because the 11th month contains the winter solstice, which is fundamental to Chinese astronomy!

The hour cycle is similar to the month cycle. The first branch corresponds to the double hour from 11 p.m. to 1 a.m., and so on. Again, we only need to worry about the stem.

According to Ho Peng Yoke, the year of birth was considered the most important in astrology until the Táng Dynasty, when the month of birth assumed greater importance. Since the Míng Dynasty, the day of birth has become the most important in Chinese eight characters astrology.

The cycle of 12 branches is probably related to the 12 months, while the cycle of 10 stems is probably related to the ancient Chinese 10-day ``week'', xún (旬). The seven-day week was probably introduced not earlier than the Sòng Dynasty (宋朝, 960-1279).

## Which Year is it in the Chinese Calendar?

Because of this web page, I get a lot of e-mail about the Chinese calendar. I once got an e-mail from a greeting cards company who needed to know which year 2000 would be in the Chinese calendar. The answer is that the Chinese do not have a continuous year count. They started counting from one again with each new emperor. However, some scholars tried to reconstruct ancient Chinese chronology by adding up years of reigns, much the same way some westerners in the past tried to reconstruct Biblical chronology. Some claim that the calendar was invented by the Yellow Emperor,Huángdì (黄帝)), in 2637 BCE in the 61st year of his reign. However, others prefer to start the count with the first year of his reign in 2697 BCE. Since these years are 60 years apart, it follows that 1984 was the first year of either the 78th or 79th 60-year cycle. Using this as a starting point, Chinese New Year in 2000 marks the beginning of the Chinese year 4637 or 4697. To give you an example of the level of confusion on this point, in Chapter 3 of Volume III of the translation of the Shoo King (Shūjīng, 书经) by James Legge, he refers to the current year, 1863, as being in the 76th cycle, implying a starting point of 2697 BCE. However, the book has an appendix on Chinese astronomy, written by John Chalmers, where the starting point is taken to be 2637 BCE! Chalmers actually writes 2636 BCE, but that really mean -2636, using the astronomical year count, where 1 BCE is year 0, 2 BCE is -1, etc. This is fairly typical of the level of confusion about the continuous year count in the Chinese calendar, and simply illustrates the fact that the continuous year count is not an integral part of the Chinese calendar, but rather an afterthought. While there isolated incidents of Chinese scholars who have used it, it only gained popularity with the Jesuit missionaries. Most of the people who use it are Westerners who refuse to believe that it is possible to have a ``civilized'' society without a linear, continuous year count. That's why I told the greeting cards company to stick with calling it the year of the Dragon!

To add to the confusion, some authors use an epoch of 2698 BCE. I believe this because they want to use a year 0 as the starting point, rather than counting 2697 BCE as year 1, or that they assume that the Yellow Emperor started his year with the Winter solstice of 2698 BCE. In particular, this system was used by Sun Yat-sen (孫逸仙, Sūn Yìxiān or 孫中山, Sūn Zhōngshān, 1866--1925). He and other political activists wanted to use a republican and “modern” year numbering system. This system actually won some acceptance in the overseas Chinese community, and is for example used occasionally in San Francisco's Chinatown. (At least around the time of Chinese New Year!)

However, let me stress again that using an epoch is not the traditional way of counting years in Chinese history. The traditional way was to use emperor's era name (年号 [年號], nían hào) together with the 60-year cycle. In the past, the emperor would often change his era name during his reign, but by the time of the Míng and Qíng dynasties, the emperors would use the same era name for their whole reign. This system worked well most of the time, but the Kāngxī Emperor (康熙) ruled more than 60 years. He ruled from February 7, 1661 to December 20, 1722. Since Chinese New Year fell on January 30 in 1661, the first year of his reign started on February 18, 1662, and the last year of his reign ended on February 4, 1723. Since both 1662 and 1722 are rényín years, the term Kāngxī rényín (康熙壬寅) is ambiguous. However, this is the only such problem in Chinese history. His grandson, the Qiánlóng Emperor (乾隆) ruled from October 18, 1735, to February 8, 1796. The first year of his rule started on February 12, 1736, but he chose to retire on February 8, 1796, as a filial act in order not to reign longer than his grandfather, the illustrious Kāngxī Emperor. Despite his retirement, however, he retained ultimate power until his death in 1799.

It is well known that the 60-year cycle was introduced during the Hàn Dynasty, so it came as something of a surprise when scholars realized that the 60-day cycle had been in use in the Shāng Dynasty (商朝, 1600--1046 BCE). This shows that the two systems are independent, and there is no point looking for an ancient origin with a (甲,子) day in a (甲,子) month in a (甲,子) year in either 2637 BCE or 2697 BCE. I should also point out, that while Chinese chronology is fairly reliable going back to 841 BCE, and oracle bones with date inscription go back to the 13th century BCE, modern scholars consider the Yellow Emperor to be a mythological figure. So this whole discussion of ancient dates is just a curiosity.

## Software and Calendar Conversion

The best source for information about calendrical calculations is the book Calendrical Calculations by Nachum Dershowitz and Edward M. Reingold. If you need a calendar conversion program, you can either go to their Calendar Applet or get the program Chinese Calendrics from Hermetic Systems: Calendars, Encryption, Astronomy, Prime Numbers and More.

Two undergraduate students at the NUS, KUAN Shau Hong and TENG Keat Huat have written a UROPS (Undergraduate Research Opportunities Programme in Science) report on The Chinese Calendar of the Later Han Period. They have also written a program for doing computations involving the Sìfēn lì (四分历) calendar. Both the DOS executable and the C source code are available. Their project is very interesting, and shows that during the later Han, the no zhōngqi rule was not just a rule of thumb, but the actual rule used for determining leap months. We are writing up a paper on this.

I have developed a Mathematica package, ChineseCalendar.m (version 2.0, 2011 June 3) that I use for Chinese calendrical computations. It uses the code from the second edition of the book Calendrical Calculations by Nachum Dershowitz and Edward M. Reingold. Their Lisp functions were translated into the Mathematica package Calendrica by Robert C. McNally. Please note that this package is only available from the book. If you don't have the book, you can use V1 of the Calendrica package, which is freely available (I have a version that is updated for Mathematica V8) and version 1.07 of the my package, ChineseCalendardV1.m.

I have created some notebooks that illustrate some of the computations I have done. (If you don't have Mathematica, you can download a copy of the MathReader.) ChineseCalendar.nb (or ChineseCalendarV1.nb if you use the V1 code) demonstrates the commands. LeapMonths.nb lists the leap months between 1645 and 3944, and ChineseNewYear1000.nb lists the date of Chinese New Year between 1645 and 2644,

## Heavenly Mathematics

I'm teaching a General Education Module called Heavenly Mathematics & Cultural Astronomy.

## Astronomical Java Applets and Animations

Together with Tey Meng Khoon and Frederick H. Willeboordse of CITA, I have developed several interactive Java applets that I hope will help you understand the motion of the Earth and the Sun.

We have also developed several interactive applets to explain What Does the Waxing or Waning Moon Look Like in Different Parts of the World?

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I have a separate page about the student projects I have supervised. The following have been related to the Chinese calendar.

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Back to Helmer Aslaksen's page on Calendars in Singapore.

Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

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