The Chinese Calendar and Background Information
This chapter provides you:
 Astronomical background information for all calendar systems.
 Introduction to the Gregorian calendar system.
 Introduction to the Chinese calendar system.
Astronomical Bases of Calendars
There are three principal astronomical cycles that can be used
by a calendar:
 Day  Based on the rotation of the earth on its axis.
 Year  Based on the revolution of the earth around the sun.
 Month  Based on the revolution of the moon around the earth.
The year, also called the tropical year, is defined as the mean interval
between vernal equinoxes; corresponding to the cycle of the seasons.
The following expression, based on the orbital elements of Laskar (1986),
is used for calculating the length of the tropical year:
365.2421896698  0.00000615359 T  7.29E10 T^2 + 2.64E10 T^3 [days],
where T = (JD  2451545.0)/36525 and JD is the Julian day number.
However, the interval from a particular vernal equinox to the next
may vary from this mean by several minutes.
The month, also called the synodic month, is defined as the mean interval
between conjunctions of the moon and sun, corresponding to the cycle of lunar phases.
The following expression for the synodic month is based on the lunar theory
of ChaprontTouze' and Chapront (1988):
29.5305888531 + 0.00000021621 T  3.64E10 T^2 [days],
where T = (JD  2451545.0)/36525 and JD is the Julian day number.
Any particular phase cycle may vary from the mean by up to seven hours.
In the preceding formulas, T is measured in Julian centuries of
Terrestrial Dynamical Time (TDT), which is independent of the variable
rotation of the Earth. Thus, the lengths of the tropical year and synodic
month are here defined in days of 86400 seconds of International Atomic Time (TAI).
Julian day number seems to be defined as 4713. But, I am not sure if this is correct.
Depending which cycles are used, calendars are divided into three groups:
 Solar calendar  Designed to be synchronized with the tropical year.
For example, Gregorian calendar is a solar calendar.
 Lunar calendar  Designed to be synchronized with the synodic month.
For example, Islamic calendar is a lunar calendar.
 Lunisolar calendar  Designed to be synchronized with both tropical year
and synodic month. For example, Hebrew and Chinese calendars are lunisolar
calendars.
The Gregorian Calendar
The Gregorian calendar is a solar calendar. It follows the tropical year with
two types calendar years:
 Common year  365 days.
 Leap year  366 days.
All years are common years with exceptions defined by the following
leap year rules:
 1. If the year number is divisible by 4, it is a leap year.
 2. But if the year number is divisible by 100, rule 1 is not used,
so it is a common year.
 3. But if the year number is divisible by 400, rule 2 is not used,
so it is a leap year.
Now let's see how closely the Gregorian years are synchronized to the tropical
years.
Based on leap year rule 1, 4 Gregorian calendar years are about 0.03124
day more than 4 tropical years.
4 Gregorian years: 3 * 365 + 366 = 1461 days
4 tropical years: 4 * 365.2421896698 = 1460.9687586792 days
Based on leap year rule 2, 100 Gregorian calendar years are about 0.21897
day less than 100 tropical years.
100 Gregorian years: 76 * 365 + 24 * 366 = 36524 days
100 tropical years: 100 * 365.2421896698 = 36524.21896698 days
Based on leap year rule 3, 400 Gregorian calendar years are about 0.12413
day more than 400 tropical years.
400 Gregorian years: 305 * 365 + 95 * 366 = 146097 days
400 tropical years: 400 * 365.2421896698 = 146096.87586792 days
As you can see, there will be about one extra day in about
2500 Gregorian calendar years comparing 2500 tripocal years.
A Gregorian year is divided into 12 Gregorian months, named as
January, February, March, April, May, June, July, August, September,
October, November and December. The number of days
in each month varies as shown in the following tables for a common year:
Month 1 2 3 4 5 6 7 8 9 10 11 12
Days 31 28 31 30 31 30 31 31 30 31 30 31
For a leap year, the second month has 29 days.
Note that Gregorian month has no relation with the synodic month cycle.
Gregorian calendar also uses the week cycle, which is a 7day cycle, with
days names as Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday.
The first day of Gregorian year 1 is a Monday.
Since 400 Gregorian years have 146097 days, which is divisible by 7, the week cycle
and the year cycle meets every 400 years.
The Chinese Calendar
The Chinese calendar is a lunisolar calenday, which is based on the tropical year
and the synodic month.
The primary cycle in the Chinese calendar is the Chinese month, which is closely
synchronized to the synodic month. Since the synodic month has an average of
about 29.5305888531 days, the number of days in a Chinese month varies between
29 days and 30 days from month to month.
The first day of a Chinese month must be the new moon day, when the moon is completely
dark, that is the moon is in conjunction with the sun.
The secondary cycle in the Chinese calendar is the solar term system, which is
closely synchronized to the tropical year. The solar term system has
12 principal terms to indicate the sun's longitudes at every 30 degrees, with the
first principle term defined as the day when the sun's longitude at 330 degrees.
In addition to the 12 principal terms, 12 sectional terms are added to
break the sun's longitude into 24 15degree segments. The folowing table lists
the 24 solar terms and their approximate dates in Gregorian calendar.
Term Long. Greg. Dur. Ch. Name En. Name
S. 1 315 Feb. 4 Lichun Beginning of Spring
P. 1 330 Feb.19 29.8 Yushui Rain Water
S. 2 345 Mar. 6 Jingzhi Waking of Insects
P. 2 0 Mar.21 30.2 Chunfen Spring Equinox
S. 3 15 Apr. 5 Qingming Pure Brightness
P. 3 30 Apr.20 30.7 Guyu Grain Rain
S. 4 45 May 6 Lixia Beginning of Summer
P. 4 60 May 21 31.2 Xiaoman Grain Full
S. 5 75 Jun. 6 Mangzhong Grain in Ear
P. 5 90 Jun.22 31.4 Xiazhi Summer Solstice
S. 6 105 Jul. 7 Xiaoshu Slight Heat
P. 6 120 Jul.23 31.4 Dashu Great Heat
S. 7 135 Aug. 8 Liqiu Beginning of Autumn
P. 7 150 Aug.23 31.1 Chushu Limit of Heat
S. 8 165 Sep. 8 Bailu White Dew
P. 8 180 Sep.23 30.7 Qiufen Autumnal Equinox
S. 9 195 Oct. 8 Hanlu Cold Dew
P. 9 210 Oct.24 30.1 Shuangjiang Descent of Frost
S.10 225 Nov. 8 Lidong Beginning of Winter
P.10 240 Nov.22 29.7 Xiaoxue Slight Snow
S.11 255 Dec. 7 Daxue Great Snow
P.11 270 Dec.22 29.5 Dongzhi Winter Solstice
S.12 285 Jan. 6 Xiaohan Slight Cold
P.12 300 Jan.20 29.5 Dahan Great Cold
The third cycle in the Chinese calendar is the Chinese year, which is losely synchronized
to the tropical year. A common Chinese year has 12 Chinese months, with 353, 354, or 355 days.
which is about 11 days less than a tropical year. In order to bring the Chinese year cycle
in line with the tropical year cycle, a leap year is defined after about every 3 common
years. A leap year has 13 months, with the extra month called leap month.
To determine which year is a leap year and after which month to add the leap month,
the Chinese calendar uses the following leap year rules:
 1. The 11th principle term, winter solstice, must falls in month 11. A leap year is a year
that an extra month must be added to push month 11 to contain the 11th principle term.
 2. In a leap year, the first month after month 11 that contains no principle term
is the leap month. A leap month shares the same number and name of the preceding month.
Chinese years are counted in 60year cycles. Each year is named with a combination
of one of the 10 heavenly stems (tiangan) and one of the 12 earthly branches (dizhi)
sequentially, and repeatedly.
10 heavenly stems are mapped to 5 elements as Wood, Wood, Fire, Fire, Earth,
Earth, Metal, Metal, Water, and Water.
12 earthly branches are also mapped to 12 animals as Rat, Ox, Tiger, Rabbit,
Dragon, Snake, Horse, Goat, Monkey, Chicken, Dog, and Pig.
It is believed that the Chinese calendar has completed 78 cycles so far. This year, 1999 in
Gregorian calendar, is the 16th year in the 79th cycle, or year 4696. Note that January 1, 1999
was still in year 4695, because the Chinese year is about one month behind.
The Chinese calendar rules can be summarized as:
 1. New moon rule  The first day of the month is when the moon is completely dark.
This defines the Chinese month cycle.
 2. Solar term rule  24 solar terms divide the tropical year into 15degree longitude
segements. This links the Chinese calendar to the tropical year cycle.
 3. Winter solstice rule  The winter solstice must falls in month 11. This defines
leap years.
 4. Leap month rule  The first month after month 11 that contains no principle term is
a leap month in a leap year.
 5. Sexagesimal cycle rule  Chinese years are counted in 60year cycles.
 6. Calculation of rule 1 and 2 must be carried out for the meridian 120 degrees east of
Greenwich, roughly corresponds to the east coast of China.
