:Sexagenary cycle

{{Sexagenary cycle}}

File:Changchun-Temple-Jiazi-Dian-a-row-of-Tai-Sui-0340.jpg deities responsible for individual years of the sexagenary cycle]]

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 duodecennial orbital cycle. The first term jiǎzǐ ({{lang|zh|{{linktext|甲子}}}}) combines the first heavenly stem with the first earthly branch. The second term yǐchǒu ({{lang|zh|{{linktext|乙丑}}}}) combines the second stem with the second branch. This pattern continues until both cycles conclude simultaneously with guǐhài ({{lang|zh|{{linktext|癸亥}}}}), 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 ({{lang|zh|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 {{harv|Bartle|1978|}}.

File:Heji 37986 Ganzhi table.jpg

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 {{Circa|1100 BC}}.{{Cite book |editor-last=Selin|editor-first=Helaine| editor-link= Helaine Selin |chapter-url=https://books.google.com/books?id=DbsqBgAAQBAJ&pg=PA390|title=Mathematics Across Cultures: The History of Non-Western Mathematics|date=2012|publisher=Springer |isbn=978-94-011-4301-1|page=390|language=en| chapter=Chinese Mathematical Astronomy | last= Martzloff | first= Jean-Claude | doi= 10.1007/978-94-011-4301-1_18 | orig-date= 2000 | series= Science Across Cultures: The History of Non-Western Science, vol. 2 | edition= illustrated}} 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 period 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 (lunations) 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.{{sfn|Smith|2011|pages=24, 26–27}}

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 tomb 3, 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 ({{lang|und-Hani|秦始皇}}), 246 BC, is noted on the diagram next to the position of the 60-cycle term yǐmǎo ({{lang|und-Hani|乙卯}}, 52 of 60), corresponding to that year.{{sfn|Kalinowski|2007|page=145, fig. 3}}{{Sfn|Smith|2011|page=29}} 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:{{sfn|Smith|2011|page=28}} the year 1984 began the present cycle (a {{lang|und-Hani|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 {{lang|und-Hani|己丑}} year (coinciding roughly with 2009) began on January 26, 2009. (However, for astrology, the year begins with the first solar term "Lìchūn" ({{lang|und-Hani|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 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.{{cite web |publisher=National Diet Library |url=http://www.ndl.go.jp/koyomi/e/history/02_index1.html |title=Calendar History; the Source |archive-url=https://web.archive.org/web/20130106054946/http://www.ndl.go.jp/koyomi/e/history/02_index1.html |archive-date=January 6, 2013 |access-date=January 1, 2013}}

The Korean ({{lang|ko|환갑; 還甲}} hwangap) and Japanese tradition ({{lang|ja|還暦}} kanreki) of celebrating the 60th birthday (literally 'return of calendar') reflects the influence of the sexagenary cycle as a count of years.{{cite encyclopedia |encyclopedia=Encyclopedia of Shinto |url=http://eos.kokugakuin.ac.jp/modules/xwords/entry.php?entryID=1040 |title=Kanreki |access-date=January 1, 2013}}

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 ({{lang|und-Hani|丁卯}}; year 4 on the Chinese cycle, year of the Fire Rabbit).{{cite book |last=Chattopadhyaya |first=Alaka |date=1996 | orig-date =1967 |chapter-url=https://books.google.com/books?id=H8Ssa5r7MZkC&pg=PA566 |title=Atīśa and Tibet: Life and Works of Dipamkara Srijnana in relation to the history and religion of Tibet |pages=566–568|publisher=Motilal Banarsidass |isbn=9788120809284 | chapter= On the Tibetan sexagenary cycle | author2= R.N. Bhattacharya }}

{{main|Heavenly Stems}}

{{main|Earthly Branches}}

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

File:China 24 cardinal directions.svg]]

File:sexagenary cycle years spirals.svg

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 ({{zhi|年号, 年號}}; {{tlit|zh|niánhào}}) to specify a year.{{cite web |url=http://www.math.nus.edu.sg/aslaksen/calendar/chinese.shtml |title=Mathematics of the Chinese calendar |last=Aslaksen |first=Helmer |date=July 17, 2010 |format=PDF |language=en |website=www.math.nus.edu.sg/aslaksen |publisher=Department of Maths, National University of Singapore |access-date=December 12, 2011 |archive-url=https://web.archive.org/web/20060424100637/http://www.math.nus.edu.sg/aslaksen/calendar/chinese.shtml |archive-date=April 24, 2006 |url-status=dead |df=mdy-all }} 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 {{nowrap|by 1.}}

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 {{math|(year – 3) mod 60}} and for years BC is {{math|(year + 2) mod 60}}.

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) mod 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) mod 60] = 17 (i.e., 60 minus the remainder of (221+2) divided by 60 is 17).

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).

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.

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, ({{lang|und-Hani|建子月}}: jiànzǐyuè, {{lang|und-Hani|建丑月}}: jiànchǒuyuè, etc.).{{sfn|Smith|2011|pages=28, 29 fn2}}{{cite dictionary |dictionary=Kōjien |publisher=Iwanami Shoten |place=Tokyo |script-title=ja:建す}} There are two systems of placing these months, the lunar one and the solar one.

One system follows the ordinary Chinese lunar calendar and connects the names of the months directly to the central solar term ({{zh|labels=no|中氣}}; zhōngqì). The jiànzǐyuè (({{lang|und-Hani|建}}){{lang|und-Hani|子月}}) is the month containing the winter solstice (i.e. the {{lang|und-Hani|冬至}} Dōngzhì) zhōngqì. The jiànchǒuyuè (({{lang|und-Hani|建}}){{zh|labels=no|丑月}}) is the month of the following zhōngqì, which is Dàhán ({{lang|und-Hani|大寒}}), while the jiànyínyuè (({{lang|und-Hani|建}}){{lang|und-Hani|寅月}}) is that of the Yǔshuǐ ({{lang|und-Hani|雨水}}) zhōngqì, etc. Intercalary months have the same branch as the preceding month.{{cite book |script-title=zh:新唐書 |title=Xīn Tángshū |trans-title=New Book of Tang|chapter=本紀第六 肅宗 代宗 (Records part 6)| date= 1060 | author1= Ouyang Xiu | author2= Song Qi | author1-link= Ouyang Xiu | author2-link= Song Qi |quote={{zh|labels=no|二年……，九月壬寅，大赦，去「乾元大圣光天文武孝感」号，去「上元」号，称元年，以十一月为岁首，月以斗所建辰为名。赐文武官阶、勋、爵，版授侍老官，先授者叙进之。停四京号。
　　元年建子月癸巳，曹州刺史常休明及史朝义将薛崿战，败之。己亥，朝圣皇天帝于西内。丙午，卫伯玉及史朝义战于永宁，败之。己酉，朝献于太清宫。庚戌，朝享于太庙及元献皇后庙。建丑月辛亥，有事于南郊。己未，来瑱及史朝义战于汝州，败之。乙亥，侯希逸及朝义将李怀仙战于范阳，败之。宝应元年建寅月甲申，追册靖德太子琮为皇帝，妃窦氏为皇后。乙酉，葬王公妃主遇害者。丙戌，盗发敬陵、惠陵。甲辰，李光弼克许州。吐蕃请和。戊申，史朝义陷营州。建卯月辛亥，大赦。赐文武官阶、爵。五品以上清望及郎官、御史荐流人有行业情可矜者。停贡鹰、鹞、狗、豹。以京兆府为上都，河南府为东都，凤翔府为西都，江陵府为南都，太原府为北都。壬子，羌、浑、奴剌寇梁州。癸丑，河东军乱，杀其节度使邓景山，都知兵马使辛云京自称节度使。乙丑，河中军乱，杀李国贞及其节度使荔非元礼。戊辰，淮西节度使王仲升及史朝义将谢钦让战于申州，败绩。庚午，敦子仪知朔方、河中、北庭、潞仪泽沁节度行营，兴平、定国军兵马副元帅。壬申，鄜州刺史成公意及党项战，败之。建辰月壬午，大赦，官吏听纳赃免罪，左降官及流人罚镇效力者还之。甲午，奴剌寇梁州。戊申，萧华罢。户部侍郎元载同中书门下平章事。建巳月庚戌，史朝义寇泽州，刺史李抱玉败之。壬子，楚州献定国宝玉十有三。甲寅，圣皇天帝崩。乙丑，皇太子监国。大赦，改元年为宝应元年，复以正月为岁首，建巳月为四月。丙寅，闲厩使李辅国、飞龙厩副使程元振迁皇后于别殿，杀越王系、兗王亻闲。是夜，皇帝崩于长生殿，年五十二。查《壽星萬年曆》，
唐肅宗之元年
冬至所在月（761.12）：初一壬午大雪，十三癸巳，十七冬至，十九己亥，廿五丙午，廿八己酉，廿九庚戌
大寒所在月（762.02）：初一辛亥，初三小寒，初九己未，十八大寒，廿五乙亥
雨水所在月（762.03）：初一辛巳，初三立春，初四甲申，初五乙酉，初六丙戌，十八雨水，廿四甲辰，廿八戊申
春分所在月（762.3）：初一辛亥，初四驚蜇，初二壬子，初三癸丑，十五乙丑，十八戊辰，十九春分，二十庚午，廿一壬申，
穀雨所在月（762.4）：初一庚辰，初三壬午，初五清明，十五甲午，二十穀雨，廿九戊申
小滿所在月（762.5）：初一庚戌，初三壬子，初五甲寅立夏，初五乙丑，十六丙寅。
大寒所在月初一辛亥，已稱建丑月，初三才小寒
春分所在月初一辛亥，已稱建卯月，初四才驚蜇
穀雨所在月初三壬午，已稱建辰月，初五才清明
小滿所在月初一庚戌、初三壬子，已稱建巳月，初五才立夏
由此可見，唐代地支紀月自朔日始，非自節氣始。}}}}

In the other system ({{lang|und-Hani|節月}}; jiéyuè) the "month" lasts for the period of two solar terms (two {{lang|und-Hani|氣策}} qìcì). The zǐyuè ({{lang|und-Hani|子月}}) is the period starting with Dàxuě ({{lang|und-Hani|大雪}}), i.e. the solar term before the winter solstice. The chǒuyuè ({{lang|und-Hani|丑月}}) starts with Xiǎohán ({{lang|und-Hani|小寒}}), the term before Dàhán ({{lang|und-Hani|大寒}}), while the yínyuè ({{lang|und-Hani|寅月}}) starts with Lìchūn ({{lang|und-Hani|立春}}), the term before Yǔshuǐ ({{lang|und-Hani|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.{{citation |script-title=ja:三正 |dictionary=Kōjien |publisher=Iwanami Shoten |place=Tokyo}}

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 ({{lang|und-Hani|甲}}; 1st) or jǐ ({{zh|labels=no|己}}; 6th) year. The first month of the jiǎ or jǐ year is a bǐngyín ({{lang|und-Hani|丙寅}}; 3rd) month, the next one is a dīngmǎo ({{lang|und-Hani|丁卯}}; 4th) month, etc., and the last month of the year is a dīngchǒu ({{lang|und-Hani|丁丑}}, 14th) month. The next year will start with a wùyín ({{lang|und-Hani|戊寅}}; 15th) month, etc. following the cycle. The 5th year will end with a yǐchǒu ({{lang|und-Hani|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.

|-

| 01 || 11 || 21 || 31 || Sep || Oct || colspan="2" | || {{tooltip|2=leap year|04}} || 06 || 25 || 27 || || colspan="2" | || 21 || || {{yes|01}} || || colspan="4"| ||colspan="6"| || 14 || colspan="4" | || 01 || 13 || 25

|-

| 02 || 12 || 22 || || Nov || Dec || colspan="2" | || {{tooltip|2=leap year|08}} || 10 || 29 || 31 || || || 19 || colspan="2" | || {{yes|02}} || || {{tooltip|2=1600 is a leap year|16}} || 19 || colspan="2" | ||colspan="2"| || {{maybe|}}05 ||colspan="4"| || {{tooltip|2=In a leap year|Feb}} || Apr || colspan="2" | || 02 || 14 || 26

|-

| 03 || 13 || 23 || || colspan="4"| || {{tooltip|2=leap year|12}} || 14 || 33 || 35 || || colspan="4" | || {{yes|03}} || 03 || colspan="2" | || 22 || || || 03 ||colspan="3"| || {{tooltip|2=leap year|12}} || || Feb || Jun || colspan="2" | || 03 || 15 || 27

|-

| 04 || 14 || 24 || || colspan="4"| || {{tooltip|2=leap year|16}} || 18 || 37 || 39 || || 17 ||colspan="2" | || {{tooltip|2=2400 is a leap year|24}} || {{yes|04}} || || colspan="4"| ||colspan="4"| || 10 ||colspan="2"| || colspan="2" | || Aug || || 04 || 16 || 28

|-

| 05 || 15 || 25 || || colspan="4"| || 01 || 03 || {{tooltip|2=leap year|20}} || 22 || style="text-align:center| 01 || colspan="2" | || 22 || 15 || {{yes|05}} || || 15 || colspan="3"| || 01 ||colspan="6"| || colspan="3" | || Oct || 05 || 17 || 29

|-

| 06 || 16 || 26 || || colspan="4"| || 05 || 07 || {{tooltip|2=leap year|24}} || 26 || || colspan="4"| || {{yes|06}} || 02 || ||18 || colspan="2" | || colspan="3"| || {{tooltip|2=leap year|08}}||colspan="2"| || 15 || colspan="3" | || Dec || 06 || 18 || 30

|-

| 07 || 17 || 27 || || || Mar || || {{tooltip|2=In a leap year|Jan}} || 09 || 11 || {{tooltip|2=leap year|28}} || 30 || || || {{maybe|}}{{tooltip|2=2000 is a leap year|20}} || colspan="2" | || {{yes|07}} || || colspan="2" | || 21 || ||colspan="2"| || 06 ||colspan="4"| || {{tooltip|2=In a leap year|Jan}} || Mar || colspan="2" | || 07 || 19 || 31

|-

| 08 || 18 || 28 || || Jan || Apr || May || {{tooltip|2=In a leap year|Feb}} || 13 || 15 || {{tooltip|2=leap year|32}} || 34 || || 18 || colspan="3" | || {{yes|08}} || || colspan="3" | || {{tooltip|2=2400 is a leap year|24}} ||colspan="5"| || 13 || || Jan || May || colspan="2" | || 08 || 20

|rowspan="4"|

|-

| 09 || 19 || 29 || || Feb || Jun || Jul || || 17 || 19 || {{tooltip|2=leap year|36}}|| 38 || || colspan="2" | || 23 || || {{yes|09}} || 01 || colspan="4"| || || {{tooltip|2=leap year|04}} ||colspan="2"| || 11 ||colspan="2"| || colspan="2" | || Jul || || 09 || 21

|-

|rowspan="2" colspan="17"| Dates with yellow background indicate they are for this year.

| {{yes|10}} || ||17 || colspan="3" | || 02 ||colspan="6"| || colspan="4"| || 10 || 22

|-

| {{yes|11}} || || colspan="2" | || {{maybe|}}{{tooltip|2=2000 is a leap year|20}} || 23 ||colspan="3"| || 09 ||colspan="3"| || colspan="2" | || Sep || || 11 || 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 century 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.

• 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 mod 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, {{zh| labels=no|c=甲}}.
• 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, {{zh| labels=no|c=子}}.
• Stem-branch = 1, 1 ({{zh| labels=no|c=甲子}}, 1 in sexagenary cycle = 32 – 5 + 33 + 1 – 60).

{{collapse top |title=More detailed examples}}

• 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, {{zh| labels=no|c=甲}}.
• 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, {{zh| labels=no|c=申}}
• Stem-branch = 1, 9 ({{zh| labels=no|c=甲申}}, 21 in cycle = – 42 – 2 + 34 + 31 = 21)
• Stem-branch for August 4, 1338
• Stem = 8, {{zh|labels=no|c=辛}}
• 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, {{zh|labels=no|c=亥}}
• 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 ({{zh|labels=no|c=辛亥}}, 48 in cycle = 4 + 8 + 32 + 4)
• Stem-branch for May 25, 105 BC (−104).
• Stem = 7, {{zh|labels=no|c=庚}}
• 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, {{zh|labels=no|c=寅}}
• 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 ({{zh|labels=no|c=庚寅}}, 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 x 15 – 3] – 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 x 12 – 3] = 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 x 15 – 3] – 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 x 11 -3] = 33

:d = 1

:SB = 32 – 5 + 33 + 1 – 60 = 1

:S = B = 1, 甲子

{{collapse bottom}}

|-

!Yi or Geng day
({{lang|und-Hani|乙/庚}})

|13 {{lang|und-Hani|丙子}}||14 {{lang|und-Hani|丁丑}}||15 {{lang|und-Hani|戊寅}}||16 {{lang|und-Hani|己卯}}||17 {{lang|und-Hani|庚辰}}||18 {{lang|und-Hani|辛巳}}||19 {{lang|und-Hani|壬午}}||20 {{lang|und-Hani|癸未}}||21 {{lang|und-Hani|甲申}}||22 {{lang|und-Hani|乙酉}}||23 {{lang|und-Hani|丙戌}}||24 {{lang|und-Hani|丁亥}}

|-

!Bing or Xin day
({{lang|und-Hani|丙/辛}})

|25 {{lang|und-Hani|戊子}}||26 {{lang|und-Hani|己丑}}||27 {{lang|und-Hani|庚寅}}||28 {{lang|und-Hani|辛卯}}||29 {{lang|und-Hani|壬辰}}||30 {{lang|und-Hani|癸巳}}||31 {{lang|und-Hani|甲午}}||32 {{lang|und-Hani|乙未}}||33 {{lang|und-Hani|丙申}}||34 {{lang|und-Hani|丁酉}}||35 {{lang|und-Hani|戊戌}}||36 {{lang|und-Hani|己亥}}

|-

!Ding or Ren day
({{lang|und-Hani|丁/壬}})

|37 {{lang|und-Hani|庚子}}||38 {{lang|und-Hani|辛丑}}||39 {{lang|und-Hani|壬寅}}||40 {{lang|und-Hani|癸卯}}||41 {{lang|und-Hani|甲辰}}||42 {{lang|und-Hani|乙巳}}||43 {{lang|und-Hani|丙午}}||44 {{lang|und-Hani|丁未}}||45 {{lang|und-Hani|戊申}}||46 {{lang|und-Hani|己酉}}||47 {{lang|und-Hani|庚戌}}||48 {{lang|und-Hani|辛亥}}

|-

!Wu or Gui day
({{lang|und-Hani|戊/癸}})

|49 {{lang|und-Hani|壬子}}||50 {{lang|und-Hani|癸丑}}||51 {{lang|und-Hani|甲寅}}||52 {{lang|und-Hani|乙卯}}||53 {{lang|und-Hani|丙辰}}||54 {{lang|und-Hani|丁巳}}||55 {{lang|und-Hani|戊午}}||56 {{lang|und-Hani|己未}}||57 {{lang|und-Hani|庚申}}||58 {{lang|und-Hani|辛酉}}||59 {{lang|und-Hani|壬戌}}||60 {{lang|und-Hani|癸亥}}

|}

• 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{{Cite book |chapter-url=https://books.google.com/books?id=0fP_DwAAQBAJ&pg=PA640|title=Conflicting Memories: Tibetan History under Mao Retold|date=2020|publisher=Brill|isbn=978-90-04-43324-3|page=640|language=en | doi= 10.1163/9789004433243 | chapter= Glossary of Chinese Terms | series =Series: Inner Asia Book Series, vol. 12| editor1= Robert Barnett | editor2= Benno Weiner| editor3= Françoise Robin }}
• Samvatsara

{{Reflist|1}}

{{refbegin}}

• {{cite journal | volume = 48 | issue = 1 | pages = 80–84 | last = Bartle | first = P. F. W. | title = Forty days: the Akan calendar | journal = Africa: Journal of the International African Institute | year = 1978 | doi = 10.2307/1158712 | jstor = 1158712 | s2cid = 143751434 }}
• {{cite book | publisher = Brill | isbn = 978-90-04-16063-7 | pages = 137–168 | editor = Francesca Bray | last = Kalinowski | first = Marc | title = Graphics and text in the production of technical knowledge in China : the warp and the weft | chapter = Time, space and orientation: figurative representations of the sexagenary cycle in ancient and medieval China | location = Leiden |year = 2007 }}
• {{cite book | publisher = Oxbow | isbn = 978-1-84217-987-1 | pages = 1–37 | editor-first = John |editor-last = Steele | last = Smith | first = Adam | title = Calendars and Years II: Astronomy and time in the ancient and medieval world |chapter = The Chinese sexagenary cycle and the ritual origins of the calendar |location = Oxford |year = 2011 |chapter-url = https://academiccommons.columbia.edu/catalog/ac:128930 | doi = 10.7916/D8891CDX}}
• {{citation |last=Kapoor |first=Ramesh|date=2021

|title=Fixing the chronology in Tai-Ahom chronicles by using astronomical references

|journal=Journal of Astronomical History and Heritage|doi=10.3724/SP.J.1440-2807.2021.03.07 |volume=24 |issue=3 |pages=665–687 |url=https://www.researchgate.net/publication/354689630 |language=en}}

• {{cite book |last=Phukan Baruah|first=B. N.N.|date=1964|title= Ahom lexicon |publisher=The government of Assam in the department of historical and antiquarian studies |url=https://books.google.com/books?id=MfqkboCUdzwC}}

{{refend}}

• {{cite web|url=http://www.hko.gov.hk/gts/time/stemsandbranches.htm|title=Heavenly Stems and Earthly Branches|website=Hong Kong Observatory|language=en|access-date=2018-11-04|url-status=live|archive-url=https://web.archive.org/web/20181104043156/http://www.hko.gov.hk/gts/time/stemsandbranches.htm|archive-date=2018-11-04}}
• [https://ganzhi.io/ganzhi Ganzhi.io] An Open Source application and implementation of Gan & Zhi as well as Jeiqi

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{{chronology}}

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Category:Chinese calendars

Category:Chinese astrology

Category:Units of time

Category:Sexagenary cycle