Sexagenary cycle

 ( Chinese Calendars ) 

The sexagenary cycle, also known as the gānzhī (干支) or stems-and-branches, is a cycle of sixty terms, each corresponding to one year, thus amounting to a total of sixty years every cycle, historically used for recording time in China and the rest of the East Asian cultural sphere, as well as in Southeast Asia.For non-mathematical readers, a simple explanation for the number "60" in the 60-year cycle of the Lunar calendar can be found in . It appears as a means of recording days in the first Chinese written texts, the of the late second millennium BC Shang dynasty. Its use to record years began around the middle of the 3rd century BC. The cycle and its variations have been an important part of the traditional calendrical systems in Chinese-influenced Asian states and territories, particularly those of Japan, Korean calendar, and Vietnam, with the old Chinese system still in use in Taiwan, and in Mainland China.For example, the annual CCTV New Year's Gala gala has continued to announce the sexagenary term of the upcoming year (庚子, gengzi for 2020). In India, the Ahom people (descendants of the Dai people of Yunnan who migrated to Assam in the 13th century) also used the sexagenary cycle known as Lak-Ni."...the Ahom reckoned time by means of a sexagenary cycle""..migration from Mong Mao in Yunnan into Mungdunshunkhām..."

This traditional method of numbering days and years no longer has any significant role in modern Chinese time-keeping or the official calendar. However, the sexagenary cycle is used in the names of many historical events, such as the Chinese Xinhai Revolution, the Japanese Boshin War, the Korean Imjin War and the Vietnamese Famine of Ất Dậu, Tet Offensive. It also continues to have a role in contemporary Chinese astrology and fortune telling. There are some parallels in this with Tamil calendar.

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Overview Each term in the sexagenary cycle consists of two Chinese characters, the first being one of the ten Heavenly Stems of the Shang-era week and the second being one of the twelve Earthly Branches representing the years of Jupiter's orbital cycle. The first term jiǎzǐ () combines the first heavenly stem with the first earthly branch. The second term yǐchǒu () combines the second stem with the second branch. This pattern continues until both cycles conclude simultaneously with guǐhài (), after which it begins again at jiǎzǐ. This termination at ten and twelve's least common multiple leaves half of the combinations—such as jiǎchǒu (甲丑)—unused; this is traditionally explained by reference to pairing the stems and branches according to their yin and yang properties.

This combination of two sub-cycles to generate a larger cycle and its use to record time have parallels in other calendrical systems, notably the Akan calendar.For the Akan calendar, see .

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History The sexagenary cycle is attested as a method of recording days from the earliest written records in China, records of divination on oracle bones, beginning .

Jean-Claude Martzloff (2025). 9789401143011, Springer. ISBN 9789401143011

Almost every oracle bone inscription includes a date in this format. This use of the cycle for days is attested throughout the Zhou dynasty and remained common into the Han dynasty for all documentary purposes that required dates specified to the day.

Almost all the dates in the Spring and Autumn Annals, a chronological list of events from 722 to 481 BC, use this system in combination with regnal years and months () to record dates. Eclipses recorded in the Annals demonstrate that continuity in the sexagenary day-count was unbroken from that period onwards. It is likely that this unbroken continuity went back still further to the first appearance of the sexagenary cycle during the Shang period.

The use of the sexagenary cycle for recording years is much more recent. The earliest discovered documents showing this usage are among the silk manuscripts recovered from Mawangdui, sealed in 168 BC. In one of these documents, a sexagenary grid diagram is annotated in three places to mark notable events. For example, the first year of the reign of Qin Shi Huang (秦始皇), 246 BC, is noted on the diagram next to the position of the 60-cycle term yǐmǎo (乙卯, 52 of 60), corresponding to that year. Use of the cycle to record years became widespread for administrative time-keeping during the Western Han dynasty (202 BC – 8 AD). The count of years has continued uninterrupted ever since: the year 1984 began the present cycle (a 甲子— jiǎzǐ year), and 2044 will begin another. Note that in China the new year, when the sexagenary count increments, is not January 1, but rather the lunar new year of the traditional Chinese calendar. For example, the jichou 己丑 year (coinciding roughly with 2009) began on January 26, 2009. (However, for astrology, the year begins with the first solar term "Lìchūn" (立春), which occurs near February 4.)

In Japan, according to Nihon shoki, the calendar was transmitted to Japan in 553. But it was not until the Empress Suiko era that the calendar was used for politics. The year 604, when the Japanese officially adopted the Chinese calendar, was the first year of the cycle.

The Korean (환갑; 還甲 hwangap) and Japanese tradition (還暦 kanreki) of celebrating the 60th birthday (literally 'return of calendar') reflects the influence of the sexagenary cycle as a count of years.

The Tibetan calendar also counts years using a 60-year cycle based on 12 animals and 5 elements, but while the first year of the Chinese cycle is always jiǎzǐ (the year of the Wood Rat), the first year of the Tibetan cycle is dīngmǎo (丁卯; year 4 on the Chinese cycle, year of the Fire Rabbit).

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Heavenly Stems

甲	yang	wood
乙	yin
丙	yang	fire
丁	yin
戊	yang	earth
己	yin
庚	yang	metal
辛	yin
壬	yang	water
癸	yin

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Earthly Branches

Rat (italic=no 𤝞)	23:00–01:00
Water buffalo (italic=no 𤛠)	01:00–03:00
Tiger (italic=no /italic=no 𧲫)	03:00–05:00
Cat (italic=no 猫)	05:00–07:00
Dragon (italic=no 龍)	07:00–09:00
Snake (italic=no 𧋻)	09:00–11:00
Horse (italic=no 馭)	11:00–13:00
Goat (italic=no 羝)	13:00–15:00
Monkey (italic=no 𤠳)	15:00–17:00
Rooster (italic=no 𪂮)	17:00–19:00
Dog (italic=no 㹥)	19:00–21:00
Pig (italic=no 𤞼/italic=no 㺧)	21:00–23:00

*The names of several animals can be translated into English in several different ways. The Vietnamese Earthly Branches use cat instead of Rabbit.

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Sexagenary years

1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043

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Conversion between cyclic years and Western years As mentioned above, the cycle first started to be used for indicating years during the Han dynasty, but it also can be used to indicate earlier years retroactively. Since it repeats, by itself it cannot specify a year without some other information, but it is frequently used with the Chinese era name (; ) to specify a year. The year starts with the new year of whoever is using the calendar. In China, the cyclic year normally changes on the Chinese Lunar New Year. In Japan until recently it was the Japanese lunar new year, which was sometimes different from the Chinese; now it is January 1. So when calculating the cyclic year of a date in the Gregorian year, one has to consider what their "new year" is. Hence, the following calculation deals with the Chinese dates after the Lunar New Year in that Gregorian year; to find the corresponding sexagenary year in the dates before the Lunar New Year would require the Gregorian year to be decreased

As for example, the year 2697 BC (or −2696, using the astronomical year count), traditionally the first year of the reign of the legendary Yellow Emperor, was the first year (甲子; jiǎzǐ) of a cycle. 2700 years later in 4 AD, the duration equivalent to 45 60-year cycles, was also the starting year of a 60-year cycle. Similarly 1980 years later, 1984 was the start of a new cycle.

Thus, to find out the Gregorian year's equivalent in the sexagenary cycle use the appropriate method below.

1. For any year number greater than 4 AD, the equivalent sexagenary year can be found by subtracting 3 from the Gregorian year, dividing by 60 and taking the remainder. See example below.
2. For any year before 1 AD, the equivalent sexagenary year can be found by adding 2 to the Gregorian year number (in BC), dividing it by 60, and subtracting the remainder from 60.
3. 1 AD, 2 AD and 3 AD correspond respectively to the 58th, 59th and 60th years of the sexagenary cycle.
4. The formula for years AD is and for years BC is .

The result will produce a number between 0 and 59, corresponding to the year order in the cycle; if the remainder is 0, it corresponds to the 60th year of a cycle. Thus, using the first method, the equivalent sexagenary year for 2012 AD is the 29th year (壬辰; rénchén), as (2012–3) modulo operation 60 = 29 (i.e., the remainder of (2012–3) divided by 60 is 29). Using the second, the equivalent sexagenary year for 221 BC is the 17th year (庚辰; gēngchén), as 60- (221+2) = 17 (i.e., 60 minus the remainder of (221+2) divided by 60 is 17).

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Examples Step-by-step example to determine the sign for 1967:

1. 1967 – 3 = 1964 ("subtracting 3 from the Gregorian year")
2. 1964 ÷ 60 = 32 ("divide by 60 and discard any fraction")
3. 1964 – (60 × 32) = 44 ("taking the remainder")
4. Show one of the Sexagenary Cycle tables (the following section), look for 44 in the first column (No) and obtain Fire Goat (丁未; dīngwèi).

Step-by-step example to determine the cyclic year of first year of the reign of Qin Shi Huang (246 BC):

1. 246 + 2 = 248 ("adding 2 to the Gregorian year number (in BC)")
2. 248 ÷ 60 = 4 ("divide by 60 and discard any fraction")
3. 248 – (60 × 4) = 8 ("taking the remainder")
4. 60 – 8 = 52 ("subtract the remainder from 60")
5. Show one of the Sexagenary Cycle table (the following section), look for 52 in the first column (No) and obtain Wood Rabbit (乙卯; yǐmǎo).

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Shorter equivalent method Start from the AD year (1967), take directly the remainder mod 60, and look into column AD of the table "Sexagenary years" (just above).

• 1967 = 60 × 32 + 47.

Remainder is therefore 47 and the AD column says 'Fire Goat' as it should be.

For a BC year: discard the minus sign, take the remainder of the year mod 60 and look into column BC. Applied to year -246, this gives:

• 246 = 60 × 4 + 6. Remainder is therefore 6 and the BC column of table "Sexagenary years" (just above) gives 'Wood Rabbit'.

When doing these conversions, year 246 BC cannot be treated as −246 AD due to the lack of a year 0 in the Gregorian AD/BC system.

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Recent cycles

+ Recent years (in the Gregorian calendar) and their corresponding years in the cycles !rowspan=2	No. ! Heavenly stem !! Earthly branch !!colspan=4	New Year Day
子 Rat	02 Feb 1984
丑 Ox	21 Jan 1985
寅 Tiger	09 Feb 1986
卯 Rabbit	30 Jan 1987
辰 Dragon	18 Feb 1988
巳 Snake	06 Feb 1989
午 Horse	27 Jan 1990
未 Goat	15 Feb 1991
申 Monkey	04 Feb 1992
酉 Rooster	23 Jan 1993
戌 Dog	11 Feb 1994
亥 Pig	31 Jan 1995
子 Rat	19 Feb 1996
丑 Ox	07 Feb 1997
寅 Tiger	28 Jan 1998
卯 Rabbit	16 Feb 1999
辰 Dragon	05 Feb 2000
巳 Snake	24 Jan 2001
午 Horse	12 Feb 2002
未 Goat	01 Feb 2003
申 Monkey	22 Jan 2004
酉 Rooster	09 Feb 2005
戌 Dog	30 Jan 2006
亥 Pig	18 Feb 2007
子 Rat	07 Feb 2008
丑 Ox	26 Jan 2009
寅 Tiger	14 Feb 2010
卯 Rabbit	03 Feb 2011
辰 Dragon	23 Jan 2012
巳 Snake	10 Feb 2013
午 Horse	31 Jan 2014
未 Goat	19 Feb 2015
申 Monkey	08 Feb 2016
酉 Rooster	28 Jan 2017
戌 Dog	16 Feb 2018
亥 Pig	05 Feb 2019
子 Rat	25 Jan 2020
丑 Ox	12 Feb 2021
寅 Tiger	01 Feb 2022
卯 Rabbit	22 Jan 2023
辰 Dragon	10 Feb 2024
巳 Snake	29 Jan 2025
午 Horse	17 Feb 2026
未 Goat	06 Feb 2027
申 Monkey	26 Jan 2028
酉 Rooster	13 Feb 2029
戌 Dog	03 Feb 2030
亥 Pig	23 Jan 2031
子 Rat	11 Feb 2032
丑 Ox	31 Jan 2033
寅 Tiger	19 Feb 2034
卯 Rabbit	08 Feb 2035
辰 Dragon	28 Jan 2036
巳 Snake	15 Feb 2037
午 Horse	04 Feb 2038
未 Goat	24 Jan 2039
申 Monkey	12 Feb 2040
酉 Rooster	01 Feb 2041
戌 Dog	22 Jan 2042
亥 Pig	10 Feb 2043

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Sexagenary months The branches are used marginally to indicate months. Despite there being twelve branches and twelve months in a year, the earliest use of branches to indicate a twelve-fold division of a year was in the 2nd century BC. They were coordinated with the orientations of the Big Dipper, (建子月: jiànzǐyuè, 建丑月: jiànchǒuyuè, etc.). There are two systems of placing these months, the lunar one and the solar one.

One system follows the ordinary Chinese calendar and connects the names of the months directly to the central solar term (; zhōngqì). The jiànzǐyuè ((建)子月) is the month containing the winter solstice (i.e. the 冬至 Dōngzhì) zhōngqì. The jiànchǒuyuè ((建)) is the month of the following zhōngqì, which is Dàhán (大寒), while the jiànyínyuè ((建)寅月) is that of the Yǔshuǐ (雨水) zhōngqì, etc. Intercalary months have the same branch as the preceding month. In the other system (節月; jiéyuè) the "month" lasts for the period of two solar terms (two 氣策 qìcì). The zǐyuè (子月) is the period starting with Dàxuě (大雪), i.e. the solar term before the winter solstice. The chǒuyuè (丑月) starts with Xiaohan (小寒), the term before Dàhán (大寒), while the yínyuè (寅月) starts with Lichun (立春), the term before Yǔshuǐ (雨水), etc. Thus in the solar system a month starts anywhere from about 15 days before to 15 days after its lunar counterpart.

The branch names are not usual month names; the main use of the branches for months is astrological. However, the names are sometimes used to indicate historically which (lunar) month was the first month of the year in ancient times. For example, since the Han dynasty, the first month has been jiànyínyuè, but earlier the first month was jiànzǐyuè (during the Zhou dynasty) or jiànchǒuyuè (traditionally during the Shang dynasty) as well.

For astrological purposes stems are also necessary, and the months are named using the sexagenary cycle following a five-year cycle starting in a jiǎ (甲; 1st) or jǐ (; 6th) year. The first month of the jiǎ or jǐ year is a bǐngyín (丙寅; 3rd) month, the next one is a dīngmǎo (丁卯; 4th) month, etc., and the last month of the year is a dīngchǒu (丁丑, 14th) month. The next year will start with a wùyín (戊寅; 15th) month, etc. following the cycle. The 5th year will end with a yǐchǒu (乙丑; 2nd) month. The following month, the start of a jǐ or jiǎ year, will hence again be a bǐngyín (3rd) month again. The beginning and end of the (solar) months in the table below are the approximate dates of current solar terms; they vary slightly from year to year depending on the leap days of the Gregorian calendar.

Jiayin / 甲寅月
Month of Mao (卯月)	Jingzhe – Qingming||Chunfen / 春分||March 6||April 5||Dingmao / 丁卯月||Jimao / 己卯月||Xinmao / 辛卯月||Guimao / 癸卯月|Yimao / 乙卯月
Bingchen / 丙辰月
Dingsi / 丁巳月
Wuwu / 戊午月
Jiwei / 己未月
Gengshen / 庚申月
Xinyou / 辛酉月
Renxu / 壬戌月
Guihai / 癸亥月
Jiazi / 甲子月
Yichou / 乙丑月

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Sexagenary days

+ Table for sexagenary days
24
25
26
27
28
29
30
31
20
21
Dates with yellow background indicate they are for this year.																	22
																	23

• N for the year: (5y + y/4) mod 10, y = 0–39 (stem); (5y + y/4) mod 12, y = 0–15 (branch)
• N for the Gregorian century: (4c + c/4 + 2) mod 10 (stem); (8c + c/4 + 2) mod 12 (branch), c ≥ 15
• N for the Julian century: 5c mod 10, c = 0–1 (stem); 9c mod 12, c = 0–3 (branch)

The table above allows one to find the stem & branch for any given date. For both the stem and the branch, find the N for the row for the century, year, month, and day, then add them together. If the sum for the stems' N is above 10, subtract 10 until the result is between 1 and 10. If the sum for the branches' N is above 12, subtract 12 until the result is between 1 and 12.

For any date before October 15, 1582, use the Julian calendar column to find the row for that century's N. For dates after October 15, 1582, use the Gregorian century column to find the century's N. When looking at dates in January and February of leap years, use the bold & italic Feb and Jan.

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Examples

• Step-by-step example to determine the stem-branch for October 1, 1949.
• Stem
• (day stem N + month stem N + year stem N + century stem N) = number of stem. If over 10, subtract 10 until within 1 – 10.
• Day 1: N = 1,
• Month of October: N = 1,
• Year 49: N = 7,
• 49 isn't on the table, so we'll have to Modulo operation 49 by 40. This gives us year 9, which we can follow to find the N for that row.
• Century 19: N = 2.
• (1 + 1 + 7 + 2) = 11. This is more than 10, so we'll subtract 10 to bring it between 1 and 10.
• 11 – 10 = 1,
• Stem = 1, .
• Branch
• (day branch N + month branch N + year branch N + century branch N)= number of branch. If over 12, subtract 12 until within 1 – 12.
• Day 1: N = 1,
• Month of October: N = 5,
• Year 49: N = 5,
• Again, 49 is not in the table for year. Modding 49 by 16 gives us 1, which we can look up to find the N of that row.
• Century 19: N = 2.
• (1 + 5 + 5 + 2) = 13. Since 13 is more than 12, we'll subtract 12 to bring it between 1 and 12.
• 13 – 12 = 1,
• Branch = 1, .
• Stem-branch = 1, 1 (, 1 in sexagenary cycle = 32 – 5 + 33 + 1 – 60).

• Stem-branch for December 31, 1592
• Stem = (day stem N + month stem N + year stem N + century stem N)
• Day 31: N = 1; month of December: N = 2; year 92 (92 mod 40 = 12): N = 3; century 15: N = 5.
• (1 + 2 + 3 + 5) = 11; 11 – 10 = 1.
• Stem = 1, .
• Branch = (day branch N + month branch N + year branch N + century branch N)
• Day 31: N = 7; month of December: N = 6; year 92 (92 mod 16 = 12): N = 3; century 15: N = 5.
• (7 + 6 + 3 + 5) = 21; 21 – 12 = 9.
• Branch = 9,
• Stem-branch = 1, 9 (, 21 in cycle = – 42 – 2 + 34 + 31 = 21)
• Stem-branch for August 4, 1338
• Stem = 8,
• Day 4: N = 4; month of August: N = 0; year 38: N = 9; century 13 (13 mod 2 = 1): N = 5.
• (4 + 0 + 9 + 5) = 18; 18 – 10 = 8.
• Branch = 12,
• Day 4: N = 4; month of August: N = 4; year 38 (38 mod 16 = 6): N = 7; century 13 (13 mod 4 = 1): N = 9.
• (4 + 4 + 7 + 9) = 24; 24 – 12 = 12
• Stem-branch = 8, 12 (, 48 in cycle = 4 + 8 + 32 + 4)
• Stem-branch for May 25, 105 BC (−104).
• Stem = 7,
• Day 25: N = 5; month of May: N = 8; year −4 (−4 mod 40 = 36): N = 9; century −1 (−1 mod 2 = 1): N = 5.
• (5 + 8 + 9 + 5) = 27; 27 – 10 = 17; 17 – 10 = 7.
• Branch = 3,
• Day 25: N = 1; month of May: N = 8; year −4 (−4 mod 16 = 12): N = 3; century −1 (−1 mod 4 = 3): N = 3.
• (1 + 8 + 3 + 3) = 15; 15 – 12 = 3.
• Stem-branch = 7, 3 (, 27 in cycle = – 6 + 8 + 0 + 25)
• Alternately, instead of doing both century and year, one can exclude the century and simply use −104 as the year for both the stem and the branch to get the same result.

Algorithm for mental calculation

$SB\; =\; (y\; +\; c\; +\; m\; +\; day)\; \backslash bmod\; 60$

$S\; =\; SB\; \backslash bmod\; 10,\; B\; =\; SB\; \backslash bmod\; 12$

$y\; =\; (year(\backslash bmod\; 400)\; \backslash bmod\; 80\; (\backslash bmod\; 12)\; \backslash times\; 5\; +\; \backslash left\backslash lfloor\backslash frac\{year(\backslash bmod\; 400)\; \backslash bmod\; 80\}\{4\}\backslash right\backslash rfloor)\; \backslash bmod\; 60$

$c\; =\; \backslash left\backslash lfloor\backslash frac\{year\}\{400\}\backslash right\backslash rfloor\; -\; \backslash left\backslash lfloor\backslash frac\{year\}\{100\}\backslash right\backslash rfloor\; +\; 10$ for Gregorian calendar and $c\; =\; 8$ for Julian calendar.

$m\; =(month\; +\; 1)\backslash bmod\; 2\backslash times\; 30+\backslash left\backslash lfloor\{0.6\backslash times\; (month\; +\; 1)\; -\; 3\}\backslash right\backslash rfloor\; -\; i$
$i\; =\; 5$ for Jan or Feb in a common year and $i\; =\; 6$ in a leap year.

• Stem-branch for February 22, 720 BC (−719).

y = 5 x (720–719) + 1/4 = 5

c = 8

m = 30 + 0.6 – 5 = 31

d = 22

SB = 5 + 8 + 31 + 22 – 60 = 6

S = B = 6, 己巳

• Stem-branch for November 1, 211 BC (−210).

y = 5 x (240–210) + 30/4 = 5 x 6 + 7 = 37

c = 8

m = 0 + 0.6 = 4

d = 1

SB = 37 + 8 + 4 + 1 = 50

S = 0, B = 2, 癸丑

• Stem-branch for February 18, 1912.

y = 5 x (1912–1920) + -8/4 + 60 = 18

c = 4 – 19 + 10 = -5

m = 30 + 0.6 – 6 = 30

d = 18

SB = 18 – 5 + 30 + 18 – 60 = 1

S = B = 1, 甲子

• Stem-branch for October 1, 1949.

y = 5 x (1949–1920) + 29/4 = 5 x 5 + 7 = 32

c = -5

m = 30 + 0.6 = 33

d = 1

SB = 32 – 5 + 33 + 1 – 60 = 1

S = B = 1, 甲子

+Look up table for sexagenary days	Centuries
Years of the century
										10 20 30
										癸
82
86
90
94
98
83
87
91
95
99
干支纪日速查表
96
99
97
98
Years of the century
													Jan May
Centuries

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Sexagenary hours

+ Table for sexagenary hours (5-day cycle)

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See also

• Doumu (斗母元君)
• Tai Sui (太歲)
• Chinese calendar
• Chinese era name
• Lunisolar calendar
• Japanese invasions of Korea (1592–1598), known in Korean as the "Imjin War", after the imjin (Yang Water Dragon) year 1592.
• Koshien Stadium (Japan), named after the kōshi (Yang Wood Rat) year 1924. One of the last examples of general usage of the cycle in Japan.
• Tet Offensive – Vietnamese name of the event, "Tết Mậu Thân Event", named after the mậuthân (Yang Earth Monkey) year 1968.
• Xinhai Revolution (China), named after the xinhai (Yin Metal Pig) year 1911
  (2025). 9789004433243, Brill. ISBN 9789004433243
• Samvatsara

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Citations

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Sources

• Marc Kalinowski (2025). 9789004160637, Brill. ISBN 9789004160637
• Adam Smith (2025). 9781842179871, Oxbow. ISBN 9781842179871

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External links

• Ganzhi.io An Open Source application and implementation of Gan & Zhi as well as Jeiqi

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