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 - 7.29E-10 T^2 + 2.64E-10 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 Chapront-Touze' and Chapront (1988): 29.5305888531 + 0.00000021621 T - 3.64E-10 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 7-day 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 15-degree 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 60-year 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 15-degree 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 60-year 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.