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|
The Mathematics of the Chinese Calendar
The Mathematics of the Chinese Calendar
Adam
Schall (汤若望 [湯若望], Tāng Rùowàng, 1591-1666), Imperial Astronomer
in Beijing. Designed the current Chinese calendar
Why This Page?
Paper on the Mathematics
of the Chinese Calendar
Reading and Writing
Chinese Characters and Pinyin on the Web Using Unicode
The Date of Chinese New Year
The Sexagenary Cycle
Why is 2000 a Golden Dragon Year?
Which Year is it in the
Chinese Calendar?
Software and Calendar
Conversion
Heavenly Mathematics
Astronomical Java Applets
and Animations
Public Lectures
Student Projects
Links
Back to Helmer
Aslaksen's page on Calendars in Singapore
Back to Helmer
Aslaksen's home page
Why This Page?
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, February 2002, vol. 66, no. 2. 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.
To view PDF files, you need to download the free Adobe
Acrobat Reader.
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
1999
2000
2001
2002
2003
2004
2005
2006
2007
date of CNY
Feb 16
Feb 5
Jan 24
Feb 12
Feb 1
Jan 22
Feb 9
Jan 29
Feb 18
change in date of next CNY
11
12
19
11
10
18
11
20
Equatorial
(left) and ecliptic (right) armillary spheres in the Imperial
Observatory in Beijing made by the Jesuit missionary Ferdinand
Verbiest, 1670
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
甲
jiǎ
Wood
1
子
zǐ
Rat
2
乙
yǐ
Wood
2
丑
chǒu
Ox
3
丙
bǐng
Fire
3
寅
yín
Tiger
4
丁
dīng
Fire
4
卯
mǎo
Rabbit
5
戊
wù
Earth
5
辰
chén
Dragon
6
己
jǐ
Earth
6
巳
sì
Snake
7
庚
gēng
Metal
7
午
wǔ
Horse
8
辛
xīn
Metal
8
未
wèi
Goat
9
壬
rén
Water
9
申
shēn
Monkey
10
癸
guǐ
Water
10
酉
yǒu
Chicken
11
戌
xū
Dog
12
亥
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.
Dengfeng
Observatory, designed by Guo Shoujing in 1276
Armillary
spheres at the Purple Mountain Observatory in Nanjing designed
by Guo Shoujing
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.
Chinese
sundials from the Adler Planetarium
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
1.06, Oct. 23 1999) that I use for Chinese calendrical computations.
It uses the code of Nachum Dershowitz and Edward M. Reingold from
their book Calendrical Calculations. Their Lisp functions were translated into the Mathematica package Calendrica by Robert C. McNally. Please note that this is the code from the first edition, which
is freely available. The code from the new edition will only be available
with the book.
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 (version 1.06, Oct. 23 1999) 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?
Public
Lectures
I have a separate page about my public
lectures on topics in astronomy and mathematics.
Student Projects
I have a separate page about the student
projects I have supervised. The following have been related to the Chinese
calendar.
Undergraduate Research Opportunities Programme in Science
(UROPS) Projects. These students are second or third year students.
KUAN Shau Hong and TENG Keat Huat, The
Chinese Calendar of the Later Han Period (abstract), 1999.
NG Say Tiong, Calendars,
Interpolation, Gnomons and Armillary Spheres in the Work of Guo
Shoujing (1231-1314) (abstract), 2000.
ZHANG Jieping, Strings
of Long Months and Short Months in the Chinese Calendar (abstract).
Term projects for my General Education Module Heavenly
Mathematics & Cultural Astronomy. I don't really supervise these.
Zu
Chongzhi and the Chinese Calendar Reform of 462 AD by Yang Shao-yun, Lim Nyek Fong, Elayne Ong Sze Jie and Desmond Teo
Chih Leang.
The
Chinese Calendar by Henky Jamiko Gunawan, Hng Wee Kwang, Lim Chee Kian, Seah Nam Shien,
Wong Chee Yeh.
Honours Projects. Honours year is an additional year for
our better students.
Rachel LEE Tang Hwee, Calendars
in Singapore, 2000.
Ministry of Education's Gifted Education Programme's Mentorship
Programme Projects. This is a programme for gifted secondary (US:
Junior high school) students.
AKMAL bin Abd. Rahman and AMIT Jain, Raffles Institution,
Astronomy and Calendars, 1995.
Science Research Programme (SRP) Projects. This is a programme
for gifted Junior College (US: High school) students.
Veronica CHIN Hei Ting, Raffles Junior College, The
Mathematics of the Chinese Calendar, 1999.
Links
I have a separate page of links.
Back to Helmer
Aslaksen's page on Calendars in Singapore.
Helmer
Aslaksen
Department
of Mathematics
National
University of Singapore
aslaksen@math.nus.edu.sg
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