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2014-02-03T05:17:05Z

24.128.97.194: /* Hydraulics */

In [[mathematics]], a [[system of linear equations]] or a [[system of polynomial equations]] is considered '''underdetermined''' if there are fewer equations than unknowns (in contrast to an [[overdetermined system]], where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting. Each [[Variable (mathematics)|unknown]] can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom.

Therefore the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint removing a degree of freedom. The '''underdetermined''' case, by contrast, occurs when the system has been underconstrained—that is, when the unknowns outnumber the equations.

== Solutions of undetermined systems ==

An underdetermined linear system has either no solution or infinitely many solutions.

For example
:<math>\begin{align}
x+y+z&=1\\
x+y+z&=0
\end{align}</math>
is an underdetermined system without any solution; any system of equations having no solution is said to be [[System of linear equations#Consistency|inconsistent]]. On the other hand, the system

:<math>\begin{align}
x+y+z&=1\\
x+y+2z&=3
\end{align}</math>

is consistent and has an infinitude of solutions, such as (''x'', ''y'', ''z'') = (1, −2 ,2), (2 , −3 ,2), and (3, −4 ,2).

More specifically, according to the [[Rouché–Capelli theorem]], any system of linear equations (underdetermined or otherwise) is inconsistent if the [[rank (linear algebra)|rank]] of the [[augmented matrix]] is greater than the rank of the [[coefficient matrix]]. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution; since in an underdetermined system this rank is necessarily less than the number of unknowns, there are indeed an infinitude of solutions, with the general solution having ''k'' free parameters where ''k'' is the difference between the number of variables and the rank.

There are [[algorithm]]s to decide if an underdetermined system has solutions, and if it has any, to express all solutions as linear functions of ''k'' of the variables (same ''k'' as above). The simplest one is [[Gaussian elimination]]. See [[System of linear equations]] for more details.

== Underdetermined polynomial systems ==
The main property of linear underdetermined systems, of having either no solution or infinitely many, extends to [[systems of polynomial equations]] in the following way.

A system of polynomial equations which has fewer equations than unknowns is said to be '''underdetermined'''. It has either infinitely many complex solutions (or, more generally, solutions in an [[algebraically closed field]]) or is inconsistent. It is inconsistent if and only if {{nowrap|1=0 = 1}} is a linear combination (with polynomial coefficients) of the equations (this is [[Hilbert's Nullstellensatz]]). If an underdetermined system of ''t'' equations in ''n'' variables (''t'' < ''n'') has solutions, then the set of all complex solutions is an [[algebraic set]] of [[dimension of an algebraic variety|dimension]] at least {{nowrap|''n'' - ''t''}}. If the underdetermined system is chosen at random the dimension is equal to {{nowrap|''n'' - ''t''}} with probability one.

==Underdetermined systems with other constraints and in optimization problems==

In general, an underdetermined system of linear equations has an infinite number of solutions, if any. However, in [[mathematical optimization|optimization problems]] that are subject to linear equality constraints, only one of the solutions is relevant, namely the one giving the highest or lowest value of an objective function.

Some problems specify that one or more of the variables are constrained to take on integer values. An integer constraint leads to [[integer programming]] and [[Diophantine equations]] problems, which may have only a finite number of solutions.

Another kind of constraint, which appears in [[coding theory]], especially in [[error correcting codes]] and [[signal processing]] (for example [[compressed sensing]]), consists in an upper bound on the number of variables which may be different from zero. In error correcting codes, this bound corresponds to the maximal number of errors that may be corrected simultaneously.

==See also==

*[[Overdetermined system]]
*[[Regularization (mathematics)]]

==References==

<references/>

[[Category:Linear algebra]]
[[Category:Equations]]

Chinese calendar

2014-02-02T01:35:02Z

24.128.167.38: /* Continuous year numbering system */

[[File:The Chinese calendar of Jiawunian.png|thumb|right|A single page of the Chinese calendar of Jiǎwǔnián, which contains the Tibetan and Islamic date]]
'''Chinese calendar''' may refer to any of the official and civil calendars used in [[China]] and some neighbouring countries in different periods of history; however, the phrase is generally synonymous with ''Han calendar''.

The official calendar in China today is the [[Gregorian calendar]], which is a [[solar calendar]]. It is used for public and business affairs.

The civil calendar in much of China is the '''Han calendar''', which is a [[lunisolar calendar]]. It is used for selecting the day of a wedding or funeral, for opening a venture, or a relocation. A similar calendar is used in [[Japan]], [[Korea]], and [[Vietnam]] for these purposes. Muslims living in [[Xinjiang]], [[Ningxia]] and other parts of northern China use the [[Islamic calendar]], which is a mean moon [[lunar calendar]], as their civil calendar. The civil calendar for [[Tibet]] is the Tibetan calendar, which is a lunisolar calendar. The civil calendar for [[Miao people|Miao]] is the Miao calendar, which is a solar calendar. The civil calendar for [[Dai people|Tai]] is the Tai calendar, which is a lunisolar calendar.

In China, some public holidays relate to the Gregorian calendar, such as [[International Workers' Day|Labor Day]] and [[National Day of the People's Republic of China|National Day]] while others relate to the Chinese Calendar, such as [[Chinese New Year]], [[Duanwu Festival]], and the [[Mid-Autumn Festival]]. In specified provinces(Autonomous~) of China, some extra public holidays related to Islamic calendar or [[Tibetan calendar]], such as [[Eid ul-Fitr|Islamic New Year]] and [[Eid al-Adha|the Major Festival]] in Ningxia and Xinjiang, [[Losar|Tibetan New Year]] and [[Sho Dun Festival|Summer Assembly]] in Tibet.

The Han calendar is a lunisolar calendar, which indicates both the moon phases and the solar terms. In Han calendar, a year usually begins on the second dark moon after the winter solstice but occasionally on the third dark moon after the winter solstice.

The year from January 31, 2014 to February 18, 2015 is a ''Wǔnián'' or ''Mǎnián'' (Year of the [[Horse (zodiac)|Horse]]).
[[File:Chinesecalendar.JPG|thumb|An example of a Gregorian calendar and Chinese lunisolar calendar below it]][[File:Chinese Zodiac Gregorian calendar.jpg|thumb|A combination calendar, with Gregorian system below and a [[Chinese zodiac]] chart above.]]

==Early Chinese calendars==
It is found on the [[oracle bone]]s of the [[Shang Dynasty]] (late 2nd millennium BC), which seem to describe a lunisolar year of 12 months, with a possible [[Intercalation (timekeeping)|intercalary]] 13th, or even 14th, added empirically to prevent calendar ''drift''. The [[Sexagenary cycle]] for recording days was already in use. Tradition holds that, in that era, the year began on the first new moon after the winter solstice.

Early [[Eastern Zhou]] texts, such as the ''[[Spring and Autumn Annals]]'', provide better understanding of the calendars used in the [[Zhou dynasty]]. One year usually had 12 months, which were alternately 29 and 30 days long (with an additional day added from time to time, to catch up with "drifts" between the calendar and the actual moon cycle), and intercalary months were added in an arbitrary fashion at the end of the year.

These arbitrary rules on day and month intercalation caused the calendars of each state to be slightly different, at times. Thus, texts like the Annals will often state whether the calendar they use (the calendar of Lu) is in phase with the ''Royal calendar'' (used by the Zhou kings).

Although tradition holds that in the Zhou, the year began on the new moon which preceded the [[winter solstice]], the ''[[Spring and Autumn Annals]]'' seem to indicate that (in Lu at least) the Yin calendar (the calendar used in [[Shang dynasty]], with years beginning on the first new moon after the [[winter solstice]]) was in use until the middle of the 7th century, and that the beginning of the year was shifted back one month around 650 BC.

By the beginning of the [[Warring States]], progress in astronomy and mathematics allowed the creation of calculated calendars (where intercalary months and days are set by a rule, and not arbitrarily). The ''sìfēn'' 四分 (quarter remainder) calendar, which began about 484 BC, was the first calculated Chinese calendar, so named because it used a solar year of 365¼ days (the same as the 1st-century BC [[Julian Calendar]] of [[Ancient Rome|Rome]]), along with a 19-year (235-month) Rule Cycle ''zhang'' 章, known in the West as the [[Metonic cycle]].<ref name="deng 67">Deng, Yingke. (2005). ''Ancient Chinese Inventions''. Translated by Wang Pingxing. Beijing: China Intercontinental Press ({{lang|zh-s|五洲传播出版社}}). ISBN 7-5085-0837-8. Page 67.</ref> The year began on the new moon preceding the winter solstice, and intercalary months were inserted at the end of the year.

In 256 BC, as the last Zhou king ceded his territory to Qin, a new calendar (the Qin calendar) began to be used. It followed the same principles as the Sifen calendar, except that the year began at Shíyuèshuò (十月朔, the closest new moon of the winter beginning). The Qin calendar was used during the [[Qin dynasty]], and in the beginning of the Western [[Han dynasty]]. According to the ''[[Han Shu]]'' 21a, 973, for the moment of unification the Middle kingdoms had 6 different calendars: those of the mythological progenitors [[Yellow Emperor]] (黄帝曆) and [[Zhuanxu]] (顓頊曆); of the dynasties [[Xia Dynasty|Xia]] (夏曆), [[Yin Dynasty|Yin]] (殷曆), and [[Zhou Dynasty|Zhou]] (周曆), and of the Zhou Dynasty state of [[Lu (state)|Lu]] (鲁曆). Of those, the second was taken to substitute the rest. The Han imperial library is said to contain 82 volumes of descriptions of all those systems (''Han Shu'' 30, 1765-6), now mostly lost.<ref name="Cullen 27">Cullen, Atronomy and Methematics in Ancient China. Cambridge, 1996.</ref>

The two oldest printed Chinese calendars are dated 877 and 882; they were found at the Buddhist pilgrimage site of [[Dunhuang]]; Patricia Ebrey writes that it is no surprise that some of the earliest printed items were calendars, since the Chinese found it necessary to calculate and mark which days were auspicious and which were not.<ref>Needham, Joseph. (1986). ''Science and Civilization in China: Volume 5, Chemistry and Chemical Technology, Part 6, Missiles and Sieges''. Cambridge University Press., reprinted Taipei: Caves Books, Ltd.(1986). Page 151.</ref><ref>Ebrey, Patricia Buckley (1999). ''The Cambridge Illustrated History of China''. Cambridge: Cambridge University Press. ISBN 0-521-66991-X (paperback). Pages 124–125.</ref>

==Ancient Han calendar==
[[Emperor Wu of Han China|Emperor Wu]] of the Western [[Han dynasty]] introduced reforms that have governed Han calendar ever since. His Tàichū ({{lang|zh|太初}}, "Grand Inception") calendar of 104 BC had a year with the [[winter solstice]] in the 11th month and was the first to use the system of 24 [[solar term]]s to determine intercalary months, in which calendar months (a month of 29 or 30 whole days) during which the sun does not pass a principal term (that is, remained within the same sign of the [[zodiac]] throughout) are designated as intercalary. The solar year of the Taichu calendar was defined as {{nowrap|365 <math>\tfrac{385}{1539}</math>}} days and the lunar month as {{nowrap|29 <math>\tfrac{43}{81}</math>}} days. Because the sun's mean motion was used to calculate the solar terms until 1645, this intercalary month was equally likely to occur after any month of the year. The conjunction of the sun and moon (the astronomical new moon) was calculated using the mean motions of both the sun and moon.

*Nineteen year cycle
It was realized in the early age of Han Calendar, that 19 solar years have 235 months. So, as 235=12*19+7, there are 7 intercalary months in 19 years.

When the 1 tropical year was calculated as <math>365\tfrac{1}{4}</math> days, 1 synodic month was accordingly <math>29\tfrac{499}{940}</math> days.

Also, 76 years = 365*76+76/4 = 27759 days, and likewise 940 months = 29*940+499 = 27759 days.
In other words, 76 years = 940 months, meaning that 76 years is a cycle, or in other words the sun and moon both return to their original positions after 76 years.

The 19-year-cycle was used in the "Six Early Calendars" ([[:zh:古六历|古六历]], Gǔlìulì). And a 600 year cycle with 221 intercalary months is used in the Yuánshǐ Calendar in the Liang(N) dynasty (401－439, established in Liangzhou). And then a 391 year cycle with 144 intercalary months was used in the Dàmíng Calendar from the Ninth year of Tianjian (天监九年, 510) in the Liang Dynasty (502-557, established in Jiankang). After the Délíng Calendar of the Tang Dynasty (618-907, established in Chang'an), the calendar was made with the true new moon, the cycle could no longer be used to determine the intercalary months.

However the Gregorian date may be estimated with a 19-year cycle. The error is a day, or about a month.
===Name of ancient Han calendar===
The authorities give a name for each version of the Han calendar, such as: ''Tàichū calendar''(太初历, the genesis calendar),''Dàmíng calendar''(大明历, the legend calendar), ''Huángjí calendar''(皇极历, the ultimate calendar), ''Shòushí calendar''(授时历, the time signal calendar), etc.

There're about 100 versions, so there're about 100 names for Han calendar. But, the civilians called it as ''Huánglì''(皇历, the imperial calendar) generally.

==Modern Han Calendar==

===True sun and moon===
Though the fact of the irregularity of the lunar orbit was known in the 1st century BC, the starts of the months were calculated using the mean motions of both the sun and moon until 619, the second year of the [[Tang dynasty]], when chronologists began to use true motions modeled using two offset opposing [[parabola]]s (with small linear and cubic components). Unfortunately, the parabolas did not meet smoothly at the mean motion, but met with a discontinuity or jump.

With the introduction of European astronomy into China via the [[Society of Jesus|Jesuits]], the motions of both the sun and moon began to be calculated with [[trigonometric function|sinusoids]] in the 1645 [[Chongzhen calendar]] ({{lang|zh-t|時憲書}}, Book of the Conformity of Time) of the [[Qing dynasty]], made by the Jesuits [[Johann Adam Schall von Bell|Adam Schall]] and [[Giacomo Rho]]. The true motion of the sun was now used to calculate the [[#Jieqi|jiéqì]], which caused the intercalary month to often occur after the second through the ninth months, but rarely after the tenth through first months. A few autumn-winter periods have two or three calendar months in which the sun stays within one sign during the month (i.e. months that would normally be treated as intercalary months), interspersed with one or two calendar months in which the sun enters two signs of the zodiac during the month, something that was impossible using mean sun motion.

===Standard time===
Before 1929, the traditional calendar was calculated by the Central Observatory (formerly the Imperial Observatory) in Beijing using Beijing local time at a longitude of 116°25'E {{nowrap|(UTC+7:45:40)}}. From 1929 to 1949 it was calculated by the Institute of Astronomy in Nanjing and since 1949 by the [[Purple Mountain Observatory]] outside of Nanjing using Chinese standard time at a longitude of 120°E (UTC+8). This shifted the midnight marking the beginning of each day in both the traditional and Gregorian calendars by plus 14 minutes 20 seconds. This shift meant that any dark moon which formerly occurred just before midnight Beijing local time now occurred just after midnight Chinese standard time, causing the first day of a lunar month to occur one day later. However, unlike the official tables, most public calendars relied on the old ''Wannian Shu'' (Long-term (lit. "10,000-year") Calendar; simplified Chinese: 万年书; traditional Chinese: 萬年書) last published in 1910 using Beijing time until they were forced to adopt the official traditional calendar using Chinese standard time when the two disagreed. In 1953 public calendars placed the dark moon and the first day of a lunar month on August 9, whereas the official traditional calendar placed it on August 10, which caused public calendars in most of the People's Republic of China to use the official tables and standard time after 1953. In 1978 the dates were respectively September 2 and 3, causing public calendars in the Hong Kong and Canton areas to do the same after 1978. In 1989 the dates were August 1 and 2, which caused Taiwan to do the same after 1989.<ref>F. Richard Stephenson and Liu Baolin, "A brief contemporary history of the Chinese calendar" ''Orion'', Jahrgang 56, Nr. 287, 1998)</ref><ref>Helmer Aslaksen, [http://www.math.nus.edu.sg/aslaksen/calendar/cal.pdf The mathematics of the Chinese calendar] pages 18 & 28.</ref>

===Name of modern Han Calendar===
The first version of Modern Han calendar is called as ''Shíxiàn Calendar''(时宪历, the time constitution), which is issued at the second year of ''Shùnzhì''(顺治二年, 1645). But, authority called it as ''Hànlì''(汉历, the calendar of Han) when they issue a document for general use, such as ''Imperial Hànlì collection by Kāngxī''(《康熙御制汉历大全》)

After 1912, the Han calendar became unofficial calendar. So no authority gave a name for it. People called Han calendar as ''Jìulì''(旧历, the former calendar), ''Lǎolì''(老历, the traditional calendar), ''Yīnlì''(阴历, the lunar calendar), etc.

In the public media, the Han calendar is always called as ''Nónglì'' (农历, the rural calendar).

In addition, some people appeal to called it as ''Xiàlì''(夏历, the calendar of Xià) for some reason.

==Structure of Han calendar==

===Day (''Rì'', 日)===
A day in the Chinese calendar runs from midnight to midnight, as in the Gregorian calendar. Currently, midnight is based on Chinese Standard Time, the mean solar time at longitude 120° east (equivalent to [[UTC+08]]).

====Subdivisions of a day====
In modern Chinese, the day is divided according to the Western hour-minute-second system, but the older standards are still used in some instances.
{| class="wikitable" style="float: right;"
|+Time table of ''shí'' and ''kè''(''am'' = ''ante mid-'', ''rm'' = ''right at mid-'', ''pm'' = ''post mid-'')
|-
!
!Range of!!''1 kè am''!!''2 kè am''!!''3 kè am''!!''4 kè am''!!''kè rm''!!''1 kè pm''!!''2 kè pm''!!''3 kè pm''!!''4 kè pm''
|-
!''Mǎoshí''
|05:00-07:00||05:02:24||05:16:48||05:31:12||05:45:36||06:00:00||06:14:24||06:28:48||06:43:12||06:57:36
|-
!''Chénshí''
|07:00-09:00||07:12:00||07:26:24||07:40:48||07:55:12|| ||08:09:36||08:24:00||08:38:24||08:52:48
|-
!''Sìshí''
|09:00-11:00||09:07:12||09:21:36||09:36:00||09:50:24|| ||10:04:48||10:19:12||10:33:36||10:48:00
|-
!''Wǔshí''
|11:00-13:00||11:02:24||11:16:48||11:31:12||11:45:36||12:00:00||12:14:24||12:28:48||12:43:12||12:57:36
|-
!''Wèishí''
|13:00-15:00||13:12:00||13:26:24||13:40:48||13:55:12|| ||14:09:36||14:24:00||14:38:24||14:52:48
|-
!''Shēnshí''
|15:00-17:00||15:07:12||15:21:36||15:36:00||15:50:24|| ||16:04:48||16:19:12||16:33:36||16:48:00
|-
!''Yǒushí''
|17:00-19:00||17:02:24||17:16:48||17:31:12||17:45:36||18:00:00||18:14:24||18:28:48||18:43:12||18:57:36
|-
!''Xūshí''
|19:00-21:00||19:12:00||19:26:24||19:40:48||19:55:12|| ||20:09:36||20:24:00||20:38:24||20:52:48
|-
!''Hàishí''
|21:00-23:00||21:07:12||21:21:36||21:36:00||21:50:24|| ||22:04:48||22:19:12||22:33:36||22:48:00
|-
!''Zǐshí''
|23:00-01:00||23:02:24||23:16:48||23:31:12||23:45:36||00:00:00||00:14:24||00:28:48||00:43:12||00:57:36
|-
!''Chǒushí''
|01:00-03:00||01:12:00||01:26:24||01:40:48||01:55:12|| ||02:09:36||02:24:00||02:38:24||02:52:48
|-
!''Yínshí''
|03:00-05:00||03:07:12||03:21:36||03:36:00||03:50:24|| ||04:04:48||04:19:12||04:33:36||04:48:00
|}
*''shí''-''kè'' system
A day is divided into 12 ''shí'' (时/時, dual hour). The ''shí'' are named according to the [[Earthly Branches]].

The halfway point of each ''shí'' is ''zhèng'' (正, mid-), for example, midday is called ''zhèngwǔshí'' (正午时/正午時, mid-''wǔshí''). The first hour of each ''shí'' is called as ''chū'' (初, ante mid-); and the second hour of each ''shí'' is called as ''zhèng'' (正, post mid-).

For the purposes of calculating the calendar, a day starts at midnight (''zhèngzǐshí'', 正子时/正子時), but people tend to regard a day as starting at from dawn (during ''yínshí'').

A day is divided into 100 units as well. The boundaries of theses units are called as ''[[Ke (unit)|kè]]'' (刻, scale). The units are be called as ''[[Ke (unit)|kè]]'' (刻, centiday) too, although it's a vague statement.

''Shí'' is equivalent to 2 hours or 120 minutes. And ''kè'' (centiday) is equivalent to 14.4 minutes. So, each ''shí'' can't be divided into several whole ''kè'' (centidays), although efforts had been made to redefine the ''kè''. Currently, the word ''kè'' is used to denote a quarter of an hour.

The ''kè'' is always joined to ''shí'', for example, in ''Chapter 6, Volume 1, The Illustrated Wings of the Classified Canon'' (类经图翼, 第一卷, 第六章）

At the winter solstice, which is the 11th primary solar term, the sun moves to the 4th degree of the ''Jī-mansion''. The sunrise is 1 ''kè am'' of ''Chénshí'', and the sunset is 4 ''kè pm'' of ''Shēnshí''. so the day is 41 ''kès'', and the night is 59 ''kès'' (冬至十一月中，日躔箕四度，出辰初初刻，入申正四刻，昼长四十一刻，夜长五十九刻。)

*''gēng''-''diǎn'' system

In another subdivision system, there're 10 ''gēng'' (更, alternation) in a day. The 5 ''gēng'' in the night are named according to Heavenly Stems (''Jiǎ'', ''Yǐ'', ''Bǐng'', ''Dīng'', ''Wù'') at first, and use ordinal name (''Yīgēng'', ''Èrgēng'', ''Sāngēng'', ''Sìgēng'', ''Wǔgēng'') later. The names of 5 ''gēng'' in the day are ''Zhāo'' (朝, morning), ''Yú'' (禺, forenoon), ''Zhōng'' (中, noon), ''Bū'' (晡, afternoon), ''Xī'' (夕, evening).

There was a night watch system (called ''dǎgēng'' 打更, referring to the striking of gongs or clappers) corresponding to the 5 ''gēng'' in the night, so that the time between two watches is a deciday.

There are five ''diǎn'' (点/點, check points) between two night watches. The time between two points is also called ''diǎn'' (点/點, one sixtieth day), and it's 24 minutes.

The ''diǎn'' (point) is always joined to ''gēng'' (alternation, night watch), for example, in ''Section 4, Military Records, History of the Yuán Dynasty'' (元史·兵志四)

The curfew rule: after ''yīgēng sāndiǎn''(20:24) when the bell tone stopped, walking is forbidden; after ''wǔgēng sāndiǎn'' (06:00) when the bell tone rang, walking is allowed. (其夜禁之法，一更三点，钟声绝，禁人行；五更三点，钟声动，听人行。”)
''Yīgēng'' 19:12; ''Yīgēng yīdiǎn'' 19:36; ''Yīgēng èrdiǎn'' 20:00; ''Yīgēng sāndiǎn'' 20:24; ''Yīgēng sìdiǎn'' 20:48; ''Yīgēng wǔdiǎn'' 21:12
''Èrgēng'' 21:36; ''Èrgēng yīdiǎn'' 22:00; ''Èrgēng èrdiǎn'' 22:24; ''Èrgēng sāndiǎn'' 22:48; ''Èrgēng sìdiǎn'' 23:12; ''Èrgēng wǔdiǎn'' 23:48
''Sāngēng'' 00:00; ''Sāngēng yīdiǎn'' 00:24; ''Sāngēng èrdiǎn'' 00:48; ''Sāngēng sāndiǎn'' 01:12; ''Sāngēng sìdiǎn'' 01:36; ''Sāngēng wǔdiǎn'' 02:00
''Sìgēng'' 02:24; ''Sìgēng yīdiǎn'' 02:48; ''Sìgēng èrdiǎn'' 03:12; ''Sìgēng sāndiǎn'' 03:36; ''Sìgēng sìdiǎn'' 04:00; ''Sìgēng wǔdiǎn'' 04:24
''Wǔgēng'' 04:48; ''Wǔgēng yīdiǎn'' 05:12; ''Wǔgēng èrdiǎn'' 05:36; ''Wǔgēng sāndiǎn'' 06:00; ''Wǔgēng sìdiǎn'' 06:24; ''Wǔgēng wǔdiǎn'' 06:48
''Zhāoshí'' 07:12; ''Zhāoshí yīdiǎn'' 07:36; ''Zhāoshí èrdiǎn'' 08:00; ''Zhāoshí sāndiǎn'' 08:24; ''Zhāoshí sìdiǎn'' 08:48; ''Zhāoshí wǔdiǎn'' 09:12
''Yúshí'' 09:36; ''Yúshí yīdiǎn'' 10:00; ''Yúshí èrdiǎn'' 10:24; ''Yúshí sāndiǎn'' 10:48; ''Yúshí sìdiǎn'' 11:12; ''Yúshí wǔdiǎn'' 11:36
''Zhōngwǔ'' 12:00; ''Zhōngwǔ yīdiǎn'' 12:24; ''Zhōngwǔ èrdiǎn'' 12:48; ''Zhōngwǔ sāndiǎn'' 13:12; ''Zhōngwǔ sìdiǎn'' 13:36; ''Zhōngwǔ wǔdiǎn'' 14:00
''Būshí'' 14:24; ''Būshí yīdiǎn'' 14:48; ''Būshí èrdiǎn'' 15:12; ''Būshí sāndiǎn'' 15:36; ''Būshí sìdiǎn'' 16:00; ''Būshí wǔdiǎn'' 16:24
''Xīshí'' 16:48; ''Xīshí yīdiǎn'' 17:12; ''Xīshí èrdiǎn'' 17:36; ''Xīshí sāndiǎn'' 18:00; ''Xīshí sìdiǎn'' 18:24; ''Xīshí wǔdiǎn'' 18:48
The time between two points are divided into 100 ''fēns'' (分,cents).
1 ''shí'' = 120 minutes; 1 ''kè'' = 14.4 minutes; 1 ''diǎn'' = 24 minutes; 1 ''fēn'' = 1.44 seconds
1 ''shí'' = 8⅓ ''kès'' = 500 ''fēns''; 1 ''kè'' = 60 ''fēns''; 1 ''diǎn'' = 100 ''fēns'';

===Week===
Days are grouped within several kinds of weeks.
*[[Names of the days of the week|Luminaries week]] (''Xīngqī/Qīyào'', 星期/七曜)

The days are grouped within a 7-days week, which is called a Luminary week (''Xīngqī/Qī-yào'', 星期/七曜). The name of the weekdays are Sun-day (''Rìyào'', 日曜), Moon-day (''Yuèyào'', 月曜), Mars-day (''Huǒyào'', 火曜), Mercury-day (''Shuǐyào'', 水曜), Jupiter-day (''Mùyào'', 木曜), Venus-day (''Jīnyào'', 金曜), and Saturn-day (''Tǔyào'', 土曜).

In modern China, the names are identified by ordinal numbers, such as: ''Xīngqīyī'' (星期一, First-day), ''Xīngqīèr'' (星期二, Second-day), ''Xīngqīsān'' (星期三, Third-day), ''Xīngqīsì'' (星期四, Fourth-day), ''Xīngqīwǔ'' (星期五, Fifth-day), ''Xīngqīlìu'' (星期六, Sixth-day). The exception is Sunday, which is known as ''Xīngqīrì'' (星期日, Sunday).

31 January 2013 is ''Xīngqīsì'' (Thursday).

Each 4 weeks are grouped within a 28-days week. the week days of a 28-days week are marked with [[Twenty-eight mansions|''Èrshíbāxìu'']] (二十八宿值日). For example, 31 January 2014 is ''Níujīnníu'' (牛金牛).
''Jiǎomùjiāo''角木蛟 | ''Kàngjīnlóng''亢金龙 | ''Dǐtǔhé'' 氐土貉 | ''Fángrìtù'' 房日兔 | ''Xīnyuèhú'' 心月狐 | ''Wěihuǒhǔ'' 尾火虎 | ''Jīshuǐbào'' 箕水豹
''Dǒumùxiè'' 斗木獬 | ''Níujīnníu'' 牛金牛 | ''Nǚtǔfú'' 女土蝠 | ''Xūrìshǔ''虚日鼠 | ''Wēiyuèyàn''危月燕 | ''Shìhuǒzhū''室火猪 | ''Bìshuǐyǔ'' 壁水貐
''Kuímùláng''奎木狼 | ''Lóujīngǒu'' 娄金狗 | ''Wèitǔzhì'' 胃土雉 | ''Ángrìjī'' 昴日鸡 | ''Bìyuèwū'' 毕月乌 | ''Zīhuǒhóu''觜火猴 | ''Shēnshuǐyuán''参水猿
''Jǐngmùhān'' 井木犴 | ''Guǐjīnyáng''鬼金羊 | ''Lǐutǔzhāng''柳土獐 | ''Xīngrìmǎ'' 星日马 | ''Zhāngyuèlù'' 张月鹿 | ''Yìhuǒshé''翼火蛇 | ''Zhěnshuǐyǐn'' 轸水蚓

*[[Heavenly Stems]] week

The days are grouped within a 10-days week, and is called Heavenly stems. The names of the weekdays are ''Jiǎrì'', ''Yǐrì'', ''Bǐngrì'', ''Dīngrì'', ''Wùrì'', ''Jǐrì'', ''Gēngrì'', ''Xīnrì'', ''Rénrì'', and ''Guìrì''.

31 January 2013 is ''Dīngrì''.

Some tradition holiday is established according to Heavenly Stems week. Such as the Spring/Autumn Assembly (''Chūnshè/Qīushè'', 春社/秋社) is the fifth ''Wùrì'' after the Vernal Begins/Autumn Begins.

* [[Earthly Branches]] week

The days are grouped within a 12-days week, which are called [[Earthly Branches]]. The names of the weekdays are ''Zǐrì'', ''Chǒurì'', ''Yínrì'', ''Mǎorì'', ''Chénrì'', ''Sìrì'', ''Wǔrì'', ''Wèirì'', ''Shēnrì'', ''Yǒurì'', ''Xūrì'', and ''Hàirì''.

31 January 2013 is ''Yǒurì''.

Some traditional holidays are established according to [[Earthly Branches]] week at first. Such as the ''Shàngsì'' Festival (上巳节) is the first ''Sìrì'' in ''Sānyuè'', and the ''Duānwǔ'' Festival (端午节) is the first ''Wǔrì'' in ''Wǔyuè'' at first. These festivals are moved to the fixed calendar later. The ''Shàngsì'' Festival is fixed to ''Sānyuè'' 3rd, and the ''Duānwǔ'' Festival is fixed to ''Wǔyuè'' 5th.

*[[Stem-branches]] week
{| class="wikitable" style="float: right;"
|-align="right"
|''Jiǎzǐ''||''Yǐchǒu''||''Bǐngyín''||''Dīngmǎo''||''Wùchén''||''Jǐsì''||''Gēngwǔ''||''Xīnwèi''||''Rénshēn''||''Guìyǒu''||''Jiǎxū''||''Yǐhài''
|-align="right"
|''Bǐngzǐ''||''Dīngchǒu''||''Wùyín''||''Jǐmǎo''||''Gēngchén''||''Xīnsì''||''Rénwǔ''||''Guìwèi''||''Jiǎshēn''||''Yǐyǒu''||''Bǐngxū''||''Dīnghài''
|-align="right"
|''Wùzǐ''||''Jǐchǒu''||''Gēngyín''||''Xīnmǎo''||''Rénchén''||''Guìsì''||''Jiǎwǔ''||''Yǐwèi''||''Bǐngshēn''||''Dīngyǒu''||''Wùxū''||''Jǐhài''
|-align="right"
|''Gēngzǐ''||''Xīnchǒu''||''Rényín''||''Guìmǎo''||''Jiǎchén''||''Yǐsì''||''Bǐngwǔ''||''Dīngwèi''||''Wùshēn''||''Jǐyǒu''||''Gēngxū''||''Xīnhài''
|-align="right"
|''Rénzǐ''||''Guìchǒu''||''Jiǎyín''||''Yǐmǎo''||''Bǐngchén''||''Dīngsì''||''Wùwǔ''||''Jǐwèi''||''Gēngshēn''||''Xīnyǒu''||''Rénxū''||''Guìhài''
|}

The Heavenly stems and [[Earthly Branches]] run together and, when combined, make a 60-day week which is called stem-branches week.

31 January 2013 is ''Dīngyǒurì''.

The earliest evidence of stem-branches week was found on [[oracle bone]]s dated c. 1350 BC in the [[Shang Dynasty]]. The stem-branches week continues to this day, and can still be found on Chinese calendars today.

Although the stem-branches week cannot decided the actual date alone in historical events, it can locate the accurate date along with context and other statement about time, and the difference between versions of the calendar may be neglected. For this reason, the stem-branches week is always used to mark date in the annals. Such as:

''Chronicle of Pope Rén, History of Sung Dynasty'' (宋史·仁宗本纪)

In the ''Bǐngyínrì'' of vernal ''Zhēngyuè'' in the first year of ''Tiānshèng'', which is the first day of the month, changed the era name; ...(天聖元年春正月丙寅朔，改元；………)

In the ''Wùxūrì'' of ''Èryuè'', accepted the rule of ''Gusiluo''{{'s}} annual tribute; in ''Dīngsìrì'', established the portrait of ''Grand Chris'' and ''Grand Pope'' at ''Hóngqìng Palace'' of the ''Southern Capital'' (''Suiyang''); lunched the ''Monopolizing-Tea-Discount Law'' at the 13 tea plantations of Huáinán (二月戊戌，許唃厮啰歲一入貢；丁巳，奉安太祖、太宗禦容於南京鴻慶宫；行淮南十三山場貼射茶法)

In the ''Jiǎxūrì'' of ''Sānyuè'', established the portrait of ''Pope Zhēn'' at ''Yìngtiān Temple'' of the ''Western Capital'' (''Loyang''); ...; in the ''Xīnmǎorì'', the ''Imperial Astronomer'' present the ''Chóngtiān Calendar'' for the approval to issue; ...(三月甲戌，奉安真宗禦容于西京應天院；……；辛卯，司天监上崇天曆； ……)

===Lunar phase and lunar month (''Yuè'', 月)===
{{anchor|Month}}
[[Lunar phase]] refers to the shape of the illuminated (sunlit) portion of the Moon as seen by an observer on Earth. The lunar phases corresponds to the celestial longitude difference of the moon and sun. The celestial longitude difference is cyclic variation from 0° to 360°. The principal lunar phases are new moon (0° celestial longitude difference), first quarter moon (90° celestial longitude difference), full moon (180° celestial longitude difference) and last quarter moon (270° celestial longitude difference).

In Han calendar, A lunar month is corresponds to a variation cycle (0°-360°) of the celestial longitude difference. So, the month in Han calendar starts on the day of a new moon (i.e. the astronomical new moon, not the crescent new moon) and ends the day before the next one.

Synodic month(29.48 days) 2013-01-12 03:43:36 ~ 2013-02-10 15:20:06
+ New moon day 2013-01-12 00:00:00 ~ 2013-01-13 00:00:00
- Next new moon day 2013-02-10 00:00:00 ~ 2013-02-11 00:00:00
-----------------------------------------------------------------------------------
The month(29 days) 2013-01-12 00:00:00 ~ 2013-02-10 00:00:00

Synodic month(29.52 days) 2013-02-10 15:20:06 ~ 2013-03-12 03:51:00
+ New moon day 2013-02-10 00:00:00 ~ 2013-02-11 00:00:00
- Next new moon day 2013-03-12 00:00:00 ~ 2013-03-13 00:00:00
------------------------------------------------------------------------------------
The month(30 days) 2013-02-10 00:00:00 ~ 2013-03-12 00:00:00

*The days in the month

The days in the month are called with two characters, such as:
''Chūyī''(初一) ''Chūèr''(初二) ''Chūsān''(初三) ''Chūsì''(初四) ''Chūwǔ''(初五) ''Chūlìu''(初六) ''Chūqī''(初七) ''Chūbā''(初八) ''Chūjǐu''(初九) ''Chūshí''(初十)
''Shíyī''(十一) ''Shíèr''(十二) ''Shísān''(十三) ''Shísì''(十四) ''Shíwǔ''(十五) ''Shílìu''(十六) ''Shíqī''(十七) ''Shíbā''(十八) ''Shíjǐu''(十九) ''Èrshí''(二十)
''Niànyī''(廿一) ''Niànèr''(廿二) ''Niànsān''(廿三) ''Niànsì''(廿四) ''Niànwǔ''(廿五) ''Niànlìu''(廿六) ''Niànqī''(廿七) ''Niànbā''(廿八) ''Niànjǐu''(廿九) ''Sānshí''(三十)

In the late ''Qīng'' Dynasty, selected ''yùnmù'' (韵目, the representative character of the rhymes) are on behalf of the two date characters for telegram use. It was applied into the date in the Gregorian calendar later.

The ''yùnmù'' on behalf of date is used to mark the key historical events too. Such as:

''Yàndiàn'' (艳电/艷電) is a telegram issued by ''Chao-ming Chi-hsin Wang'' (''Wang Ching-wei'', VP of ''Kuomintang'') on December 29, 1938 and ''Yàn'' (艳/艷) is on behalf of 29th.

''Wénxī'' Fire (文夕大火) is a conflagration on the eve of 1938-11-12 (''Evening of 1938-11-12, Chángshā was going to be occupied by the enemy, municipality burnt and fielded the city''), ''Wén'' (文) is on behalf of ''12th'' and ''xī''(夕) is on behalf of ''evening''.

===Solar year (''suìcì'', 岁次/歲次) and solar month (''Xīngcì'', 星次)===
{| class="wikitable" style="float: right;"
|+ The ecliptic position of each solar term (SST = sectional solar term; PST = primary solar term)
|-
! colspan="2" | Solar Month
! Solar term !! Date !! Longitude !! Zodiac
|-
| rowspan="2" | ''Xīngjì'' (星纪)
| SST || '''GS''', Great Snow (''Dàxuě'', 大雪) || Dec 7 || 255°
| Sagitarius 240-270°
|-
| PST || '''WS''', Winter Solstice (''Dōngzhì'', 冬至) || Dec 22 || 270°
| rowspan="2" | Capricornus 270-300°
|-
| rowspan="2" | ''Xuánxiāo'' (玄枵)
| SST || '''SC''', Slight Cold (''Xiǎohán'', 小寒) || Jan 6 || 285°
|-
| PST || '''GC''', Great Cold (''Dàhán'', 大寒) || Jan 20 || 300°
| rowspan="2" | Aquarius 300-330°
|-
| rowspan="2" | ''Jūzī'' (娵訾)
| SST || '''VB''', Vernal Begins (''Lìchūn'', 立春) || Feb 4 || 315°
|-
| PST || '''RW''', Rain Water (''Yǔshuǐ'', 雨水)|| Feb 19 || 330°
| rowspan="2" | Pisces 330-360°
|-
| rowspan="2" | ''Jiànglóu'' (降娄)
| SST || '''IA''', Insects Awaken (''Jīnzhé'', 惊蛰/驚蟄) || Mar 5 || 345°
|-
| PST || '''VE''', Vernal Equinox (''Chūnfēn'', 春分) ||Mar 20 || 360°/0°
| rowspan="2" | Aries 0-30°
|-
| rowspan="2" | ''Dàliáng'' (大梁)
| SST || '''CB''', Clear and Bright (''Qīngmíng'', 清明)|| Apr 5|| 15°
|-
| PST || '''GR''', Grain Rain (''Gǔyǔ'', 谷雨/穀雨) || Apr 20 || 30°
| rowspan="2" | Taurus 30-60°
|-
| rowspan="2" | ''Shíshěn'' (实沈)
| SST || '''SB''', Summer Begins (''Lìxià'', 立夏)|| May 6 || 45°
|-
| PST || '''GF''', Grain Full (''Xiǎomǎn'', 小满/小滿) || May 21 || 60°
| rowspan="2" | Gemini 60-90°
|-
| rowspan="2" | ''Chúnshǒu'' (鹑首)
| SST || '''GE''', Grain in Ear (''Mángzhǒng'', 芒种/芒種) || Jun 6 || 75°
|-
| PST || '''SS''', Summer Solstice (''Xiàzhì'', 夏至) || Jun 21 || 90°
| rowspan="2" | Cancer 90-120°
|-
| rowspan="2" | ''Chúhuǒ'' (鹑火)
| SST || '''SH''', Slight Heat (''Xiǎoshǔ'', 小暑) || Jul 7 || 105°
|-
| PST || '''GH''', Great Heat (''Dàshǔ'', 大暑) || Jul 23 || 120°
| rowspan="2" | Leo 120-150°
|-
| rowspan="2" | ''Chúnwěi'' (鹑尾)
| SST || '''AB''', Autumn Begins (''Lìqīu'', 立秋) || Aug 7|| 135°
|-
| PST || '''LH''', Limit of Heat (''Chúshǔ'', 处暑/處暑) || Aug 23|| 150°
| rowspan="2" | Virgo 150-180°
|-
| rowspan="2" | ''Shòuxīng'' (寿星)
| SST || '''WD''', White Dew (''Báilù'', 白露)|| Sep 8 || 165°
|-
| PST || '''AE''', Autumnal Equinox (''Qīufēn'', 秋分) || Sep 23|| 180°
| rowspan="2" | Libra 180-210°
|-
| rowspan="2" | ''Dàhuǒ'' (大火)
| SST || '''CD''', Cold Dew (''Hánlù'', 寒露)||Oct 8 || 195°
|-
| PST || '''FD''', Frost's Descent (''Shuāngjiàng'', 霜降)||Oct 23 || 210°
| rowspan="2" | Scorpio 210-240°
|-
| rowspan="2" | ''Xīmù'' (析木)
| SST || '''WB''', Winter Begins (''Lìdōng'', 立冬)|| Nov 7 || 225°
|-
| PST || '''LS''', Light Snow (''Xiǎoxuě'', 小雪) || Nov 22|| 240°
| Sagitarius 240-270°
|}
The solar year (''suì'', 岁次/歲次) in Han calendar is the period between two winter solstices.

In the late [[Spring and Autumn Period]] (722–481 BC), the former ''Sìfēn'' calendar (古四分历) was established, and set the [[solar year|tropical year]] at 365.25 days, the same length as the [[Julian calendar]] which was introduced in 46 BC.<ref name="deng 67">Deng, Yinke. (2005). ''Ancient Chinese Inventions''. Translated by Wang Pingxing. Beijing: China Intercontinental Press. ISBN 7-5085-0837-8, pp. 67 - 69.</ref> The Taichu calendar (太初历) of 104 BC under [[Emperor Wu of Han]] rendered the solar year at roughly the same ( <math>365\tfrac{385}{1539}</math>).<ref name="deng 67"/>

Many other calendars were established between then and the Yuan Dynasty (1271–1368), including those established by [[Li Chunfeng]] (602–670) and [[Yi Xing]] (683–727).<ref name="deng 67"/> In 1281, the Yuan astronomer [[Guo Shoujing]] (1233–1316) fixed the calendar at 365.2425 days, the same as the [[Gregorian calendar]] established in 1582; this calendar, the Shoushi calendar (授時曆), would be used in China for the next 363 years.<ref name="deng 67"/><ref>Asiapac Editorial. (2004). ''Origins of Chinese Science and Technology''. Translated by Yang Liping and Y.N. Han. Singapore: Asiapac Books Pte. Ltd. ISBN 981-229-376-0, p. 132.</ref> Guo Shoujing established the new calendar with the aid of his own achievements in [[spherical trigonometry]], which he derived largely from the work of [[Shen Kuo]] (1031–1095) who [[History of trigonometry|established trigonometry in China]].<ref>Needham, Joseph. (1959). ''Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth''. Cambridge University Press., reprinted Taipei: Caves Books, Ltd.(1986), pp. 109–110.</ref><ref>Ho, Peng Yoke. (2000). ''Li, Qi, and Shu: An Introduction to Science and Civilization in China''. Mineola: Dover Publications. ISBN 0-486-41445-0. p. 105.</ref><ref>Restivo, Sal. (1992). ''Mathematics in Society and History: Sociological Inquiries''. Dordrecht: Kluwer Academic Publishers. ISBN 1-4020-0039-1. p. 32.</ref>

A solar year is divided into 12 solar months. The 12 solar months is decided by the 24 solar terms (''Jiéqì'', 节气/節氣).
*Solar terms (''Jiéqì'', 节气/節氣)
{{Main|Solar term}}
The 24 solar terms are 24 seasonal markers, which may assist farmers to decide when to plant or harvest crops. Each successive term corresponds to a point 15° further along the ecliptic; two terms together correspond quite closely to the Western zodiac divisions.

In ancient astronomy records, the solar position is defined with 28 mansions and 12 ''Xīngcì'' (星次). such as: In the winter solstice, the sun stays at the 5 degree of the ''Jī-mansion'' (冬至日, 日在箕五度).

The 28 mansions are 28 uneven divisions of ecliptic and 12 ''Xīngcì'' are 12 even divisions of ecliptic.

A couple of terms are associated into a solar month. The first term called as the sectional solar term (''Jiéqì'', 节气/節氣, node of climate) is the beginning of the solar term. The second term called as the primary solar term (''Zhōngqì'', 中气/中氣, center of the climate) is the center of the solar month.<ref>"Solar term" is the standard modern English term for the 24 節氣. "Primary terms" and "sectional terms" are the English translation given by Liu Baolin 刘宝琳, chief calendarist at the Purple Mountain Observatory 中国科学院紫金山天文台 in his paper, Liu Baolin and F. Richard Stephenson, "The Chinese calendar and its operational rules" published: ''Orion'', Jahrgang 56, Nr. 286, Juni 1998, 16.</ref>

Usually, we use the 12 branches to mark the 12 solar months. And the period from a primary solar term to the next is associated with a Zodiac.

The 12 sectional solar terms were used by Shěn Kuò (沈括, 1031－1095) in his 12-qì calendar (12氣曆).

*Month Names (''Yuèfèn'', 月份)
An [[astronomical year]] is approximately 365¼ days, a period between 12 and 13 lunar months. So, to keep the pace with the astronomical year in a long term, years with 12 lunar months and years with 13 lunar months are interleaved. Generally, in each 19 years, there're 7 years which contains 13 lunar months each and 12 years which contains 12 lunar months each.
{| class="wikitable" style="float: right;"
|+The general relation of the month mark, the primary solar term in the month, and the zodiac which the sun enters
! month name !! ''Zhēngyuè'' !! ''Èryuè'' !! ''Sānyuè'' !! ''Sìyuè'' !! ''Wǔyuè'' !! ''Lìuyuè'' !! ''Qīyuè'' !! ''Bāyuè'' !! ''Jǐuyuè'' !! ''Shíyuè''!! ''Shíyīyuè'' !! ''Làyuè'' !! ''Rùnyuè''
|-align="center"
| ordinal || 1st || 2nd || 3rd || 4th || 5th || 6th || 7th || 8th || 9th || 10th || 11th || 12th || leap over
|-align="center"
| PST || RW || VE || GR || GF || SS || GH || LH || AE || FD || LS || WS || GC || N/A
|-align="center"
| zodiac|| Pisces || Aries || Taurus || Gemini || Cancer || Leo || Virgo || Libra || Scorpios || Sagittarius || Capricorns || Aquarius || N/A
|}
In Han calendar, the month name are defined to keep the pace with the climatic variation generally. As the center of the climatic, the primary solar term decides the month mark(It means that the zodiac which the sun enters decides the month name). It means that the 12 lunar months fits the 12 solar months at maximum ratio.

In the other words, the month name is decided by the closest sectional primary of the first day.

For example, the table below show the information of 2014.
Month Date PST Date Zodiac Closest SST Month name(Ordinal)
A 2014-01-31 ~ 2014-02-28 RW: 2014-02-19 enter Pisces VB(+05) ''Zhēngyuè''(1st)
B 2014-03-01 ~ 2014-03-30 VE: 2014-03-21 enter Aries IA(+06) '' Èryuè''(2nd)
C 2014-03-31 ~ 2014-04-28 GR: 2014-04-20 enter Taurus CB(+06) '' Sānyuè''(3rd)
D 2014-04-29 ~ 2014-05-28 GF: 2014-05-21 enter Gemini SB(+07) '' Sìyuè''(4th)
E 2014-05-29 ~ 2014-06-26 SS: 2014-06-21 enter Cancer GE(+09) '' Wǔyuè''(5th)
F 2014-06-27 ~ 2014-07-26 GH: 2014-07-23 enter Leo LH(+11) '' Lìuyuè''(6th)
G 2014-07-27 ~ 2014-08-24 LH: 2014-08-23 enter Virgo AB(+12) '' Qīyuè''(7th)
H 2014-08-25 ~ 2014-09-23 AE: 2014-09-23 enter Libra WD(+15) '' Bāyuè''(8th)
I 2014-09-24 ~ 2014-10-23 FD: 2014-10-23 enter Scorpios CD(+15) '' Jǐuyuè''(9th)
J 2014-10-24 ~ 2014-11-21 N/A CD(-15) '' Rùnyuè''(leap over)
K 2014-11-22 ~ 2014-12-21 LS: 2014-11-22 enter Sagittarius WB(-14) '' Shíyuè''(10th)
L 2014-12-22 ~ 2015-01-19 WS: 2014-12-22 enter Capricorns GS(-13) ''Shíyīyuè''(11th)
M 2015-01-20 ~ 2015-02-18 GC: 2015-01-20 enter Aquarius LC(-14) '' Làyuè''(12th)

From ''Shíxiàn'' calendar which is release at the second year of ''Shùnzhì'' (''Shùnzhì èrnián'', 顺治二年/順治二年, 1645), the true sun position at ecliptic is used to define the solar term, and the length of the solar terms are uneven. In the winter, the length of the solar terms may be as low as 14.7 days. Thus, a month may contain two primary solar terms in winter. If there are months with 2 primary solar terms, exception occurred. There is about 0.4% of exception.

Thus, the formal month naming rules of Han calendar are:

1. The months with the Winter Solstice are the ''Shíyīyuè'' always.

2. The months except the intercalary month following ''Shíyīyuè'' are ''Làyuè'', ''Zhēngyuè'', ''Èryuè'', ''Sānyuè'', ''Sìyuè'', ''Wǔyuè'', ''Lìuyuè'', ''Qīyuè'', ''Bāyuè'', ''Jǐuyuè'', ''Shíyuè''.

3. If there are 11 months, there is no intercalary month.

3. If there are 12 months, the first month without a primary solar term is the intercalary month.

4. The intercalary month follows the names of the month before it.

For example, the table below shows the information of 2032-2035. The months A, M and Z are the months with winter solstice. So they are ''Shíyīyuè''. There are 11 months between A and M. So, there is no intercalary month in 2033. There are 12 months between M and Z, and N is the first month without primary term. So, N is the intercalary month. The other months are named accordingly.
Month Date PST Date Month name
A 2032-12-03 ~ 2032-12-31 WS: 2032-12-21 0 ''Shíyīyuè''(11th)
B 2033-01-01 ~ 2033-01-30 GC: 2032-01-20 1 '' Làyuè''(12th)
C 2033-01-31 ~ 2033-02-28 RW: 2032-02-18 2 '' Zhēngyuè''( 1st)
D 2033-03-01 ~ 2033-03-30 VE: 2032-03-20 3 '' Èryuè''( 2nd)
E 2033-03-31 ~ 2033-04-28 GR: 2032-04-20 4 '' Sānyuè''( 3rd)
F 2033-04-29 ~ 2033-05-27 GF: 2032-05-21 5 '' Sìyuè''( 4th)
G 2033-05-28 ~ 2033-06-26 SS: 2032-06-21 6 '' Wǔyuè''( 5th)
H 2033-06-27 ~ 2033-07-25 GH: 2032-07-22 7 '' Lìuyuè''( 6th)
I 2033-07-26 ~ 2033-08-24 LH: 2032-08-23 8 '' Qīyuè''( 7th)
J 2033-08-25 ~ 2033-09-22 N/A 9 '' Bāyuè''( 8th)
K 2033-09-23 ~ 2033-10-22 AE: 2032-09-23 10 '' Jǐuyuè''( 9th)
L 2033-10-23 ~ 2033-11-21 FD: 2032-10-23 11 '' Shíyuè''(10th)
M 2033-11-22 ~ 2033-12-21 LS: 11-22 / WS: 12-21 0 ''Shíyīyuè''(11th)
N 2033-12-22 ~ 2034-01-19 N/A (1st month w/o PT) 1 '' Rùnyuè''(11th, Intercalary)
O 2033-01-20 ~ 2034-02-18 GC: 01-20 / RW: 02-18 2 '' Làyuè''(12th)
P 2034-02-19 ~ 2034-03-19 N/A (2nd month w/o PT) 3 '' Zhēngyuè''( 1st)
Q 2034-03-20 ~ 2034-04-18 VE: 2034-03-20 4 '' Èryuè''( 2nd)
R 2034-04-19 ~ 2034-05-17 GR: 2034-04-20 5 '' Sānyuè''( 3rd)
S 2034-05-18 ~ 2034-06-15 GF: 2034-05-21 6 '' Sìyuè''( 4th)
T 2034-06-16 ~ 2034-07-15 SS: 2034-06-21 7 '' Wǔyuè''( 5th)
U 2034-07-16 ~ 2034-08-13 GH: 2034-07-23 8 '' Lìuyuè''( 6th)
V 2034-08-14 ~ 2034-09-12 LH: 2034-08-23 9 '' Qīyuè''( 7th)
W 2034-09-13 ~ 2034-10-11 AE: 2034-09-23 10 '' Bāyuè''( 8th)
X 2034-10-12 ~ 2034-11-10 FD: 2034-10-23 11 '' Jǐuyuè''( 9th)
Y 2034-11-10 ~ 2034-12-10 LS: 2034-11-22 12 '' Shíyuè''(10th)
Z 2034-12-11 ~ 2035-01-08 WS: 2034-12-22 0 ''Shíyīyuè''(11th)

===Year (''nián'', 年)===
A year starts on ''Zhēngyuè'' 1st, and always ends on the last day of ''Làyuè'' (if there is an intercalary month after ''Làyuè'', the year will end on the last day of ''Rùnyuè'').

There are 12 or 13 months in a year. If there are 12 months in year, there are 353, 354 or 355 days in this year. If there are 13 months in a year, there are 383, 384 or 385 days in this year. For example, the current year starts at 2012-1-23 and ends at 2013-02-09. There are 13 months or 384 days.

*Age recognition in China

In China, the age recognition for official use is base on the Gregorian calendar. But, for traditional use, it's based on Han calendar. From birthday to the end of the year, it's one year old. And, add one year old after each New Year Eve. Such as, if one's birthday is ''Làyuè'' 29th 2013, he is 2 years old at ''Zhēngyuè'' 1st 2014.

*Year name (''Niánfèn'', 年份)

The years are named with the era name (which is a name of several years) and ordinal number generally. But, the first year of each era is called with ''Yuánnián'' (元年).

For the eras ante ''Emperor Wǔ of Hàn Dynasty'', the regnal names are regard as the era names. such as ''Yǐngōng Yuánnián'' (隐公元年, The first year of ''Duke Yǐn of Lǔ State'', 722 BC).

113 BC, ''Emperor Wǔ of Hàn Dynasty'' issued the first era name, ''Jiànyuán'' (建元), and 140 BC is marked as ''Jiànyuán Yuánnián'' (建元元年,140 BC).

In China, the first official era name is ''Jiànyuán'' (建元), the last official era name is ''Xuāntǒng'' (宣统/宣統).

In Japan, the first era name is ''Taika'' (大化), the last era name for Han calendar is ''Keiō'' (慶應). The first era name for the Gregorian calendar is ''Meiji'' (明治). Current era name is ''Heisei'' (平成).

After ''Xuāntǒng Sānnián'', the republic authority adopted the Gregorian calendar and establish the next year as the first year. The country name (Mínguǒ, 民国) is regard as the era name. The time system is used in Taiwan and some overseas Chinese societies still.

Since 1949, in mainland China, the authority abolished the era name of the ROC, and specified the Chinese of Christ Era as ''Gōngyuán'' (公元, Common Era). So, the ''Gōngyuán'' is regard as the current era name. In some Chinese society, ''Xīyuán'' (西元, the Western Era) takes the place of ''Gōngyuán''

====Continuous year numbering system====
There's not a widely recognized "[[epoch (reference date)|epoch]]" for Han calendar. So, there is not a widely recognized continuous year numbering system.

*''Yuán-Huì-Yùn-Shì''

[[Shao Yong]] (邵雍, 1011–1077, Courtesy name: Yáofū, Posthumous title: Kāngjié, a philosopher, cosmologist, poet and historian who greatly influenced the development of Non-Confucianism in China.) introduced a timing system in his ''Huángjíjīngshì'' (皇极经世, ''The Ultimate which Manages the World'')

In his time system, 1 ''yuán'' (元, round), which contains 12'9600 years, is a lifecycle of the world. Each ''yuán'' (round) is divided into 12 ''huì'' (会, assembly), which are named with Earthly Branches.

The ''Hài'', ''Zǐ'', and ''Chǒu'' are the ''Tàigǔ'' (太古, Remote Ages),which is the opening of the world (天地之分) just as the winter. The ''Yín'', ''Mǎo'', and ''Chén'' are the ''Shànggǔ'' (上古, Early Ages), which is the evolution of the world (天地之化) just as the spring. The ''Sì'', ''Wǔ'', and ''Wèi'' are the ''Zhōnggǔ'' (中古, Middle Ages), which is the climax of the world (天地之关) just as the summer. The ''Shēn'', ''Yǒu'', and ''Xū'' are the ''Xiàgǔ'' (下古, Late Ages), which is the closing of the world (天地之合) just as the autumn.

Each ''huì'' (assembly) is divided into 30 ''yùn'' (运, run), and each ''yùn'' (run) is divided into 12 ''shì'' (世, generation). So, each ''shì''(generation) is equivalent to 30 years.

The ''Yuán-Huì-Yùn-Shì'' is corresponded with ''Nián-Yuè-Rì-Shí''. So the ''Yuán-Huì-Yùn-Shì'' is called as the ''major tend'' or the ''numbers of the heaven'', and the ''Nián-Yuè-Rì-Shí'' is called as the ''minor tend'' or the ''numbers of the earth''.

The ''major tend'' or the ''numbers of the heaven'' is far away from people seemingly, but the ''minor tend'' or the ''numbers of the earth'' is close to people. So the ''minor tend'' or the ''numbers of the earth'' is adapted by people for predicting destiny or fate. The numbers of ''Nián-Yuè-Rì-Shí'' is marked with stem-branches. So the ''minor tend'' of the ''numbers of the earth'' is show a form of ''Bāzì'', and the four terms are be called as [[Four Pillars|Four Pillars of Destiny]]

For example, the eight characters of the birth of Emperor Qiánlóng is ''Xīnmǎo-Dīngyǒu-Gēngwǔ-Bǐngzǐ'' (辛卯、丁酉、庚午、丙子).

Shào{{'s}} ''Huángjíjīngshì'' recorded the history with stem-branches cycle from the first year of the 180th run or 2149th generation (''ARGY 6-30-1-1'', 2577 BC) and marked the year with reign title from the ''Jiǎchénnián'' of the 2156th generation (''ARGY 6-30-8-11'', 2357 BC, 1st year of ''Tángyáo'', 唐尧元年).

According to this timing system, 2014-1-31 is ''ARG/YMD 7-12-10/1-1-1''.

*''Huángdì'' Era

In the 17th century, the Jesuits tried to determine what year should be considered the epoch of Han calendar. In his ''Sinicae historiae decas prima'' (first published in Munich in 1658), Martino Martini (1614–1661) dated the royal ascension of Huangdi to 2697 BC, but started the Chinese calendar with the reign of Fu Xi, which he claimed started in 2952 BC. Philippe Couplet's (1623–1693) ''Chronological table of Chinese monarchs'' (''Tabula chronologica monarchiae sinicae''; 1686) also gave the same date for the Yellow Emperor. The Jesuits' dates provoked great interest in Europe, where they were used for comparisons with Biblical chronology. Modern Chinese chronology has generally accepted Martini's dates, except that it usually places the reign of Huangdi in 2698 BC and omits Huangdi's predecessors Fu Xi and Shennong, who are considered "too legendary to include".

Starting in 1903, radical publications started using the projected date of birth of the Yellow Emperor as the first year of Han calendar. Different newspapers and magazines proposed different dates. ''Jiangsu'', for example, counted 1905 as year 4396 (use an epoch of 2491 BC), whereas the ''Minbao'' (明报, the organ of the Tongmenghui) reckoned 1905 as 4603 (use an epoch of 2698 BC). ''Liu Shipei'' (劉師培; 1884–1919) claimed that the 1900 international expedition sent by eight foreign powers to suppress the Boxer Uprising entered Beijing in the 4611th year of the Yellow Emperor. Liu's calendar started with the birth of the Yellow Emperor, which was determined to be 2711 BC.

At January 2, 1912, Sun Yat-sen declared that ''the Republic of China adopt the Gregorian calendar, and establish Shíyīyuè 13th 4609 of Huángdì Era as the new year's day of the first year of the Republic of China'' (中華民國改用陽曆，以黄帝紀元四千六百零九年十一月十三日爲中華民國元年元旦). Sun Yat-sen's choice, which implied an epoch of 2698 BC, was adopted by many [[overseas Chinese]] communities outside [[Southeast Asia]] such as [[Chinatown, San Francisco|San Francisco's Chinatown]].<ref>Aslaksen, p. 38.</ref>
''Sun Yat-sen'''s version
years before 2698 BC Not mentioned
years before 1 AD HE=2699-CE
years after 1 BC HE=2698+CE such as 1912+2698=4610, the year after spring festival of 1912 CE is 4610 HE

In the earlier 20th century, kinds of year marking system is released. Except Huángdì era, the following system are well-known,
Confucius Era (孔子紀年), Based on the birth or dead year of Confucius
Yáo's Era (帝堯紀年), Based on the regnal of Emperor Yao
Xià's Era (夏禹紀年), Based on the regnal of Yu of Xia
Qín's Era (秦統一紀年), Based on the year when Qin Unified China
Yuán's Era (亡国紀年), Based on the year when Chinese (Song Dynasty) lost China completely
Gònghé's Era (共和紀年), Based on the first year of Gònghé. From then, there's clear year mark system.

But in any case, no reference date can be recognized as an epoch of Han calendar without the official recognition.

*Calendar Era

The frame is established at ''Tàichū Yuánnián'' (太初元年, 105 BC - 103 BC), and the calendars with this frame is used all the way. And, the first winter solstice of ''Tàichū Yuánnián'' which is a winter solstice at Jiǎzǐrì with dark moon (朔旦冬至得甲子), is a node. In the historic book, for modern Chinese, the date after then is idiomatic and the date before then is unidiomatic (''Such as: the Zhēngyuè is not always the first month in spring; the years starts at Shíyuè, etc.''). So, the first winter solstice of ''Tàichū Yuánnián'' is a key reference date for Han calendar. It should be regarded as the epoch of Han calendar. And it's clear that it will be a neutral epoch.
the years before 105 BC, 105-''Gōngyuán'', such as: 2698 BC is -1594th year(1594th year before the Calendar)
the year of 105 BC, 105-''Gōngyuán'', 105 BC is 0th year
the years form 104 BC to 1 BC, 105-''Gōngyuán'', Such as: 87 BC is 18th year
the years after 1 BC, ''Gōngyuán''+104, such as: 2013 is 2117th year

===Cycle of years===
*10-year cycle ([[Heavenly stems]] cycle)

The years are grouped within a 10-year cycle which is called the 10 Celestial Stems (or called as 10 Heavenly Stems). The names of each year are: ''Jiǎnián'', ''Yǐnián'', ''Bǐngnián'', ''Dīngnián'', ''Wùnián'', ''Jǐnián'', ''Gēngnián'', ''Xīnnián'', ''Rénnián'' and ''Guìnián''.

The current year (2014-1-31~2015-2-18) is ''Jiǎnián''.

Each year corresponds to an element in ''Wǔxíng''. ''Jiǎnián'' and ''Yǐnián'' is ''Mùnián'' (木, Wood), ''Bǐngnián'' and ''Dīngnián'' is ''Huǒnián'' (火, Fire), ''Wùnián'' and ''Jǐnián'' is ''Tǔnián'' (土, Earth), ''Gēngnián'' and ''Xīnnián'' is ''Jīnnián'' (金, Metal), ''Rénnián'' and ''Guìnián'' is ''Shuǐnián'' (水, Water).

Therefore the current year is a ''Mùnián''.

*12-year cycle ([[Earthly Branches]] cycle)

The years are grouped within a 12-year cycle which is called the 12 [[Earthly Branches]]. The names of they years are: ''Zǐnián'', ''Chǒunián'', ''Yínnián'', ''Mǎonián'', ''Chénnián'', ''Sìnián'', ''Wǔnián'', ''Wèinián'', ''Shēnnián'', ''Yǒunián'', ''Xūnián'', and ''Hàinián''.

The current year is ''Wǔnián''.

Each cycle year is corresponds to an animal of the [[Chinese zodiac]]. ''Zǐnián'' to ''Shǔnián'' (鼠, [[Rat (zodiac)|Rat]]), ''Chǒunián'' to ''Níunián'' (牛, [[Ox (zodiac)|Ox]]), ''Yínnián'' to ''Hǔnián'' (虎, [[Tiger (zodiac)|Tiger]]), ''Mǎonián'' to ''Tùnián'' (兔, [[Rabbit (zodiac)|Rabbit]]), ''Chénnián'' to ''Lóngnián'' (龙, [[Dragon (zodiac)|Dragon]]), ''Sìnián'' to ''Shénián'' (蛇, [[Snake (zodiac)|Snake]]), ''Wǔnián'' to ''Mǎnián'' (马, [[Horse (zodiac)|Horse]]), ''Wèinián'' to ''Yángnián'' (羊, [[Goat (zodiac)|Goat]]), ''Shēnnián'' to ''Hóunián'' (猴, [[Monkey (zodiac)|Monkey]]), ''Yǒunián'' to ''Jīnián'' (鸡, [[Rooster (zodiac)|Rooster]]), ''Xūnián'' to ''Gǒunián'' (狗, [[Dog (zodiac)|Dog]]) and ''Hàinián'' to ''Zhūnián'' (猪, [[Pig (zodiac)|Pig]]).

The seal characters of the earthly branches show the figure of the animals. For example, ''Wǔ'' shows the figure of the horse face.

Therefore the current year is ''Mǎnián'' ([[Horse (zodiac)|Horse]]).

{{Main|Chinese zodiac}}

{| class="wikitable" style="float: right;"
|+ Stem-branch of recent years
|-align="center"
! ''Jiǎzǐ'' №[1]
! Stem, branch
! ''Gānzhī''({{lang|zh|干支}})
! Year of the...
! Continuous[2]
! Gregorian date
|-align="center"
|79:26||07,03||''Gēngyínnián'' ({{lang|zh|庚寅年}})||Metal [[Tiger (zodiac)|Tiger]]||4708||Feb 14, 2010-Feb 2, 2011
|-align="center"
|79:27||08,04||''Xīnmǎonián'' ({{lang|zh|辛卯年}})||Metal [[Rabbit (zodiac)|Rabbit]]||4709||Feb 3, 2011-Jan 22, 2012
|-align="center"
|79:28||09,05||''Rénchénnián'' ({{lang|zh|壬辰年}})||Water [[Dragon (zodiac)|Dragon]]||4710||Jan 23, 2012-Feb 9, 2013
|-align="center"
|79:29||10,06||''Guǐsìnián'' ({{lang|zh|癸巳年}})||Water [[Snake (zodiac)|Snake]]||4711||Feb 10, 2013-Jan 30, 2014
|-align="center"
|79:30||01,07||''Jiǎwǔnián'' ({{lang|zh|甲午年}})||Wood [[Horse (zodiac)|Horse]]||4712||Jan 31, 2014-Feb 18, 2015
|-align="center"
|79:31||02,08||''Yǐwèinián'' ({{lang|zh|乙未年}})||Wood [[Goat (zodiac)|Goat]]||4713||Feb 19, 2015-Feb 7, 2016
|-align="center"
|79:32||03,09||''Bǐngshēnnián'' ({{lang|zh|丙申年}})||Fire [[Monkey (zodiac)|Monkey]]||4714||Feb 8, 2016-Jan 27, 2017
|-align="center"
|79:33||04,10||''Dīngyǒunián'' ({{lang|zh|丁酉年}})||Fire [[Rooster (zodiac)|Rooster]]||4715||Jan 28, 2017-Feb 15, 2018
|-align="center"
|79:34||05,11||''Wùxūnián'' ({{lang|zh|戊戌年}})||Earth [[Dog (zodiac)|Dog]]||4716||Feb 16, 2018-Feb 4, 2019
|-align="center"
|79:35||06,12||''Jǐhàinián'' ({{lang|zh|己亥年}})||Earth [[Pig (zodiac)|Pig]]||4717||Feb 5, 2019-Jan 24, 2020
|}

*60-year cycle ([[Sexagenary cycle|Stem-branches cycle]])

The Heavenly stems cycle and [[Earthly Branches]] cycle runs together, and become a 60-year cycle which is called the stem-branches cycle.

The current year is ''Jiǎwǔnián'', also called as ''Mùmǎnián'' (Wood [[Horse (zodiac)|Horse]]).

Around the [[Han Dynasty]], the stem-branches cycle was introduced. In ''Míng Dynasty'' and ''Qīng Dynasty'', the stem-branches cycle was used along with the year mark in the Oration to ''Yellow Emperor''. Such as:

''Oration to Yellow Emperor at the first year of Wànlì''

''That, at the first year of Wànlì, which is Guìyǒunián, at Sìyuè started at Gēngxūrì, at 16th day which is Yǐchǒurì, the Emperor sent... Enjoy!'' (万历元年，岁次癸酉，四月庚戌朔，越十六日乙丑，皇帝遣……。尚飨！)

The table right shows the stem/branch year names, correspondences to the Western (Gregorian) calendar, and other related information for the current decade.<ref>The following link provides conversion of Chinese calendar dates to Western calendar dates: [http://www.sinica.edu.tw/~tdbproj/sinocal/luso.html]</ref> Alternatively, see [[Chinese calendar correspondence table|this larger table]] of the full 60-year cycle.

'''Notes'''

'''1''' Regard 2697 BC as 01:00.

'''2''' Regard 2698 BC as the first year.

===Holidays===
{{Main|Traditional Chinese holidays}}
The Chinese calendar year has nine main festivals, seven determined by the lunisolar calendar, and two derived from the solar agricultural calendar. (Farmers actually used a solar calendar, and its 24 terms, to determine when to plant crops, due to the inaccuracy of the lunisolar traditional calendar. However, the traditional calendar has also come to be known as the agricultural calendar.) The two special holidays are the [[Qingming Festival]] (about 15 days after the spring equinox) and the [[Dongzhi Festival|Winter Solstice Festival]], falling upon the respective solar terms, at ecliptic longitudes of 15° and 270°, respectively. As for all other calendrical calculations, the calculations use civil time in China, [[UTC+08:00]]

{| class="wikitable"
|-
! Date
! English
! Chinese
! Vietnamese
! Remarks
! 2008
! 2009
! 2010
! 2011
! 2012
! 2013
|-
| ''Zhēngyuè'' 1st
| [[Chinese New Year]]
| {{lang|zh|春節}} ''chūnjié''
| [[Nguyên Đán]] (元旦)
| Family gathering and festivities for 3–15 days
| 0207
| 0126
| 0214
| 0203
| 0123
| 0210
|-
| ''Zhēngyuè'' 15th
| [[Lantern Festival]]
| {{lang|zh|元宵節}} ''yuánxiāo''
| Thượng Nguyên (上元)
| Lanterns parading and Eating [[Yuanxiao]]
| 0221
| 0209
| 0228
| 0217
| 0206
| 0224
|-
| ''Wǔyuè'' 5th
| [[Dragon Boat Festival]]
| {{lang|zh|端午}} ''duānwǔ''
| Đoan Ngọ (端午)
| [[Dragon boat]] racing and [[zongzi]] eating
| 0608
| 0528
| 0616
| 0606
| 0623
| 0612
|-
| ''Qīyuè'' 7th
| [[Qi Xi|Night of Sevens]]
| {{lang|zh|七夕}} ''qīxī''
| Thất tịch
| For lovers, like Valentine's Day
| 0807
| 0826
| 0816
| 0806
| 0823
| 0813
|-
| ''Qīyuè'' 15th
| [[Ghost Festival]]
| {{lang|zh|中元節}} ''zhōngyuán''
| {{nowrap|Trung Nguyên}} (中元)
| Offer tributes and respect to the deceased
| 0815
| 0903
| 0824
| 0814
| 0831
| 0821
|-
| ''Bāyuè'' 15th
| [[Mid-Autumn Festival]]
| {{lang|zh|中秋}} ''Zhōngqiū''
| Trung Thu (中秋)
| Family gathering and [[moon cake]] eating
| 0914
| 1003
| 0922
| 0912
| 0930
| 0919
|-
| ''Jǐuyuè 9th''
| [[Double Ninth Festival]]
| {{lang|zh|重陽}} ''chóngyáng''
| Trùng Cửu (重九)
| Mountain climbing and flower shows
| 1007
| 1026
| 1016
| 1005
| 1023
| 1013
|-
| ''Shíyuè 15th''
| [[Xia Yuan Festival|''Xiàyuán Festival'']]
| {{lang|zh|下元節}} ''xiàyuán''
| Hạ Nguyên (下元)
| Pray for a peaceful year to the Water God
| 1112
| 1201
| 1120
| 1110
| 1128
| 1117
|-
| ''Làyuè'' 23rd or 24th
| [[Kitchen God]] Festival
| {{lang|zh|小年}} ''xiǎonián''
| Táo Quân (竈君)
| Worshiping the kitchen god with thanks
| 0131
| 0119
| 0207
| 0127
| 0117
| 0204
|-
| April 4 or 5
| [[Qingming Festival]]
| {{lang|zh|清明節}} ''qīngmíng''
| Thanh Minh (清明)
| Tomb sweeping
| 0404
| 0404
| 0405
| 0405
| 0404
| 0404
|-
| December 21 or 22
| [[Dongzhi Festival|Winter Solstice Festival]]
| {{lang|zh|冬至}} ''dōngzhì''
| Lễ hội Đông Chí
| Family gathering, eat Tong yuen
| 1221
| 1221
| 1222
| 1222
| 1221
| 1221
|}

==Tibetan calendar==
{{main|Tibetan calendar}}
* Solar day

A solar day is the time round the clock. In Tibetan calendar, a solar day is start at dawn, and end at the next.
* Lunar month and lunar day (moon phase)
A lunar month is the time between two identical moon phases (new moons or full moons). In Tibetan calendar, a lunar month is start at a full month, and end at the next.

A lunar month is divided into 30 lunar days. Each lunar day corresponds to a 12° variance of celestial longitude difference of the moon and sun.

A lunar month is divided into two half, too. The first half is called as the waning half, and the second half is called as the waxing half.

*Date
The 15 phases in the waning half is the 16th to 30th phase, and the 15 phases in the waxing half is the first to 15th phase. So, the ''i'' th phase corresponds to ''i×12°'' celestial longitude difference of the moon and sun.

The phase number decides the date. If there're two phases within a solar day, the number of the first phase number decides the date; and if there's no phase number within a solar day, the following phase number decides the date. For example,
the first phase is at the 2nd point of day A; Day A is 1st of the month
the second phase is at the 57th point of day A; 2nd of the month is vacancy
the third phase is at the 53rd point of day B; Day B is 3rd of the month
...
the 10th phase is at the 55th point of day C; Day C is 10th of the month
no phase in day D; Day D is 11th of the month
the 11th phase is at the 4th point of day E; Day E is repeated 11th of the month

The difference of the date rules between Tibetan and Chinese calendar is moon phase against solar day. And the difference leads the bias of date sometimes, but the bias is within 1.

*Solar year and solar month(zodiac)

A solar year is the length of time that the Sun takes to return to the same position in the cycle of seasons.

In Tibetan calendar, a solar year is divided into 12 solar months (zodiac) equally. If a lunar month entered a zodiac, the zodiac decides the month mark; if a lunar month didn't enter a zodiac, the month is an intercalary month. Such as:
month A entered the Taurus at 30th, Month A is the third month of the year
month B did not enter a zodiac, Month B is an intercalary month
month C entered the Gemini at 2nd, Month C is the fourth month of the year

The 12 solar months is corresponding to the 24 solar terms (mean solar terms) in Chinese calendar. So, the difference of the month rules between Tibetan and Chinese calendar is mean solar against true solar. And the difference leads the bias of month sequence sometimes, but the bias is within 1.

*Compare with Chinese calendar

Considering the date and month rule, the bias between Tibetan and Chinese calendar is no bias, a bias of a day, a bias of a month, or a bias of a month and a day. Such as:
''Feb 28, 1987'' is ''Dangpo 1st, Fire rabbit year'' or '' Èryuè 1st, Dīngmǎonián'' with a bias of a month
''Feb 18, 1988'' is ''Dangpo 1st, Earth dragon year'' or ''Zhēngyuè 2nd, Wùchénnián'' with a bias of a day
''Feb 27, 1990'' is ''Dangpo 1st, Iron horse year'' or ''Zhēngyuè 1st, Gēngwǔnián'' with a bias of a month and a day
''Feb 15, 1991'' is ''Dangpo 1st, Iron goat year'' or ''Zhēngyuè 1st, Xīngwèinián'' without bias

==Miao calendar==
The Miao calendar is a solar calendar. The Miao calendar is used from ten-thousands years ago, and retired at 33rd year of ''Guāngxù'' (光绪三十三年, 1907). The New Year's Day in Miao calendar is the winter solstice.

A year in Miao calendar contains 12 months, which are the unfixed month (动月), the 0th month (偏月), the 1st month (正月), the 2nd month, the 3rd month, the 4th month, the 5th month, the 6th month, the 7th month, the 8th month, the 9th month, the 10th month.

The length of the 0th, 2nd, 4th, 6th, 8th, 10th month is 30 days. The length of the 1st, 3rd, 5th, 7th, 9th month is 31 days.

The length of the unfixed month is decided by the winter solstice. If the winter solstice comes after dusk, the length of the unfixed month is 31 days, otherwise the length is 30 days.

*84 mansion-branches
The 28 mansions and the 12 earthly branches runs together, and works out 84 mansion-branches. The 84 mansion-branches are used to count the day and year continuously, just as 60 stem-branches do.

==Tai calendar==
The Tai calendar is a lunisolar calendar. In the Tai calendar, the length of the 1st, 3rd, 5th, 7th, 9th, 11th month (except the leap month) is 30 days, and the length of the 2nd, 4th, 6th, 8th, 10th, 12th month (except the leap month) is 29 days.

*Leap day and leap month

After several years, a leap month (30 days) is added after the 9th month in order to keep the tropical year. There are 7 leap months in each 19 years. After several years, a leap day is added to the end of the 8th month in order to keep the synodic month. The leap day is added to the year with leap month.

*Season

In the Tai calendar, a year is divided into 3 seasons: cold season, heat season and rainy season. so the Tai calendar is 3 months earlier then the Chinese calendar.

*New year day

In some Tai community (such as Sip Song Pan Na), the [[Water-Sprinkling Festival]] is regarded as the new year day.

==Gregorian reform==

The [[Gregorian calendar]] was adopted by the nascent [[Republic of China]] effective January 1, 1912 for official business, but the general populace continued to use the traditional calendar. The status of the Gregorian calendar was unclear between 1916 and 1921 while China was controlled by several competing [[warlord]]s each supported by foreign colonial powers. From about 1921 until 1928 warlords continued to fight over northern China, but the [[Kuomintang]] or Nationalist government controlled southern China and used the Gregorian calendar. After the Kuomintang reconstituted the Republic of China on October 10, 1928, the Gregorian calendar was officially adopted, effective January 1, 1929. The People's Republic of China has continued to use the Gregorian calendar since 1949.

==Relevance of the calendar today==
There have been calls for reform in recent years from experts in China, because of the increasing irrelevance of the Chinese calendar in modern life. They point to the example in Japan, where during the [[Meiji Restoration]] the nation adopted the [[Gregorian calendar]], and simply shifted all traditional festivities onto an equivalent date. However, the Chinese calendar remains important as an element of cultural tradition, and for certain cultural activities.

===Practical uses===
The original practical relevance of the lunisolar calendar for date marking has largely disappeared. First, the Gregorian calendar is much easier to compute and more in line with international standards. Its adoption for official purposes has meant that the traditional calendar is rarely used for date marking. This, in turn, means that it is more convenient to remember significant events such as birth dates by the Gregorian rather than the Chinese calendar.

Second, the 24 solar terms were important to farmers who would not be able to plan agricultural activities without foreknowledge of these terms. However, the 24 solar terms (including the [[solstice]]s and [[equinox]]es) are more predictable on the Gregorian calendar than the lunisolar calendar since they are based on the solar cycle. It is easier for the average Chinese farmer to organize their planting and harvesting with the Gregorian calendar.

However, one practical advantage of using a calendar where the months are lunar months is that the phases of the moon, and astronomical and tidal phenomena associated with them, such as spring and neap tides, fall on approximately the same day in each lunar month, and the times of high and low water and the tidal streams experienced in a certain location on a certain day of the lunar month are likely to be similar to those for the same place and lunar day in any month. For many years, therefore, mariners in East and South-East Asia have related their tidal observations to the Chinese calendar, so as to be able to provide quick, rule-of-thumb approximations of tides and tidal conditions from memory, based on the day of the Lunar month, without needing to refer to tide tables. Certain inshore passages on the China coast, for example, where there are strong tidal streams associated with spring tides, were regarded by mariners to be passable on certain days of the lunar month, and impassable on others.

===Cultural issues===
The Chinese calendar remains culturally essential today. For example, most of the traditional festivals, such as [[Chinese New Year]] and the [[Mid-Autumn Festival]], occur on new moons or full moons. The traditional Chinese calendar, as an element of traditional culture, has much cultural and nationalistic sentiment invested in it.

The calendar is still used in the more traditional Chinese households around the world to pick 'auspicious dates' for important events such as weddings, funerals, and business deals. A special calendar is used for this purpose, called ''Huang Li'' ({{zh|t={{linktext|皇|曆}}|s={{linktext|皇|历}}|p=huánglì|links=no}}), literally "Imperial Calendar", which contains auspicious activities, times, and directions for each day. The calendar follows the Gregorian dates but has the corresponding Chinese dates. Every date would have a comprehensive listing of astrological measurements and fortune elements.

==Influence==
Other traditional [[East Asia]]n calendars are very similar to if not identical to the Chinese calendar: the [[Korean calendar]] is identical; the [[Thai lunar calendar]] substitutes a big snake for the [[Dragon (zodiac)|Dragon]] and a little snake for the [[Snake (zodiac)|Snake]]; the [[Tết|Vietnamese calendar]] substitutes the cat for the [[Rabbit (zodiac)|Rabbit]] in the [[#Twelve animals|Chinese zodiac]]; the [[Tibetan calendar]] differs slightly in animal names, and the traditional [[Japanese calendar]] uses a different method of calculation, resulting in disagreements between the calendars in some years.
The 12 year cycle, with the animal names translated into the vernacular, was adopted by the [[Göktürks]] (its use there is first attested 584), and spread subsequently among many if not most Turkic peoples, as well as the [[Mongols]]. [[Bulgar calendar|A similar]] calendar seems to have been used by the [[Bulgars]], as attested in the [[Nominalia of the Bulgarian Khans]] and in some other documents. The main differences between the Bulgar and the Chinese calendar are the different calculating system, the [[Tiger (zodiac)|Tiger]] has been replaced with a wolf, and the [[Dragon (zodiac)|Dragon]] and [[Monkey (zodiac)|Monkey]] with an unknown animal. Also, the Bulgar calendar is a solar one.<ref>[http://www.euro2001.net/issues/3_1997/7BR97F11.htm Перипетиите на календара, проф. Никола Николов]</ref>

===Chinese-Uighur calendar===
In 1258, when both North China and the [[Islamic world]] were part of the [[Mongol Empire]], [[Hulagu Khan]] established an observatory in [[Maragheh]] for the astronomer [[Nasir al-Din al-Tusi]] at which a few Chinese astronomers were present, resulting in the Chinese-Uighur calendar that al-Tusi describes in his ''[[Zij-i Ilkhani]]''.<ref>Benno van Dalen, E.S. Kennedy, Mustafa K. Saiyid, "The Chinese-Uighur Calendar in Tusi's Zij-i Ilkhani", ''Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften'' 11 (1997) 111–151.</ref> The 12 year cycle, including Turkic/Mongolian translations of the animal names (known as ''sanawat-e turki'' سنوات ترکی,) remained in use for chronology, historiography, and bureaucratic purposes in the Persian and Turkic speaking world from Asia Minor to India and Mongolia throughout the Medieval and Early Modern periods. In Iran it remained common in agricultural records and tax assessments until [[Iranian calendar#Modern calendar (Solar Hejri)|a 1925 law deprecated its use]].

==See also==
* [[Culture of China]]
* [[East Asian age reckoning]]
* [[Sexagenary cycle]]
* [[Dates in Chinese]]
* [[Public holidays in China]]

==References==
{{Reflist|30em}}

==External links==

===Calendars===
*[http://www.chinesetools.eu/tools/chinesecalendar Chinese months]
*[http://chinesecalendar.orados.com/ Chinese day with auspicious times]
*[http://www.hko.gov.hk/gts/time/conversion.htm Gregorian-Lunar calendar years] (1900–2100)
*[http://china-calendar.com/en_calendar/ Chinese calendar and holidays]
*[http://baike.baidu.com/view/2239159.htm Miao calendar]

===Calendar conversion===
*[http://www.mandarintools.com/calendar.html Western-Chinese calendar converter] (since 1912)
*[http://sinocal.sinica.edu.tw 2000-year Chinese-Western calendar converter] From 1 AD to 2100 AD. Useful for historical studies. Put the western year 年 month 月day 日in the bottom row and click on 執行.

===Rules===
*[http://www.math.nus.edu.sg/aslaksen/calendar/chinese.html Rules for the Chinese Calendar]
*[http://www.hermetic.ch/cal_stud/chinese_cal.htm The Structure of the Chinese Calendar]

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[[Category:Chinese culture]]
[[Category:Lunisolar calendars]]
[[Category:Specific calendars]]