Chinese and Some Other Calendars

This year, the Chinese New Year’s Day falls on January 25, 2020. Please do not miss the TACAS CNY day celebration on Feb 1 all day at NC fairground in Raleigh. It is natural to ask why it is not Jan. 1, or at least why not the same day every year. To answer this question, I will go in depth how Chinese calendar works. Before then, let us analyze how calendar started and how some calendars work.

Ever since the beginning of civilization, no matter where they are on earth, people are influenced by the moon, mesmerized by moonlight, attracted by the moonlight night. So, the wax and wane of the moon cycle created our “month”. However, in an agricultural society like many early societies on earth, people’s lives depend strongly on the seasonal changes which were controlled by the Sun. Unfortunately, the sun cycle, a year, though very close but is not exactly 12 times of that of the moon cycle. Therefore, a twelve-month-to-a-year calendar could not be easily made. Different civilizations, thus, used various ways to create their calendars.

The most obvious and the most ancient way was to use the moon cycle alone. Hence, it is considered a “lunar” calendar. The Islamic calendar is a good example. The definition of a month is from one crescent moon (a new moon that can be observed by naked eyes) to another and that is very pleasant. There are 12 months to a year and the year is therefore 355 or 356 days long and the seasons shift every year by about ten days (i.e., a 10-day shift from the Gregorian year). The first year in the Islamic calendar was the year Muhammad emigrated from Mecca to Medina in 622AD. January 1, 2020 is the 6th day of the fifth month, Jumada Al-Awwal, of the Islamic year 1441. (As you notice, it is 1398 years after 622AD)

The Jewish calendar is also a lunar calendar. A month in the Jewish calendar is from one new moon to the next. To compensate the discrepancy of the difference of the length of a year and twelve times the length of the month, an extra (leap) month is added once a few years. We earthling are lucked out because a lunar month is about 29.5306 days and a solar year is about 365.2424 days (from one March Equinox to the next). That means that there are 7 extra months in 19 years off only by about two hours. So, in every 19 years the Jewish calendar has 7 “leap” years, the year with an intercalary, “leap,” month attached at the end. To make the addition evenly spread out, it is defined that there is one leap year every three years two times, then two years, then every three years three times, and then in two years (the 3, 3, 2, 3, 3, 3, 2 rule, with the numbers “3” and “2” distributed like the black notes on the piano keyboard). January 1, 2020 corresponds to the 4th day of the 10th month, Tevet, of year 5780.

Another way is to use the seasonal cycle alone to define a year. Gregorian calendar is a good example. The “month” is a borrowed concept of nearly a moon cycle aso that a year can be divided into 12 month. A complete seasonal cycle was defined by using Spring Equinox (Vernal Equinox) as the reference point. It is the point when the axis of Earth spin is perpendicular to the line drawn between the Earth and the Sun (indicated by the fact when the length of the day is the same as the night). As mentioned above, a year (a Tropical Year, 365.2424 days) is thus defined as the time from one Vernal Equinox to the next. (By the way, this is not the time to take for the earth to travel back to the same point on her orbit with respect to the fixed stars. That would be the Sidereal Year. Because the precession of the Earth’s spin, the lengths of time from the two definitions differ by about 20 minutes.) Since .2424 X 400 = 96.96, it is safe to add 97 leap days in four centuries. That is why we have the present rule of leap years (multiple of four, not on century years but on the four-century years). However, the distribution of numbers of days in various months is always a mystery to me.

The third way is, of course, to use both the Sun and the Moon. The Chinese calendar is the only one on Earth to do that, therefore a true lunisolar calendar. Let’s see how that is done. Presumably, Huangdi (the Yellow King, 2698 – 2599 BCD) first designed the calendar and started the 60 year cycle stem-branch (干支) counting (still continued counting, as explained later). There were, several modifications throughout the past five or six millenniums. For instance, winter solstice were supposed to be contained in the first month in Zhou dynasty but later it was supposed to be the middle of the winter and hence in the middle of the winter three (10, 11, 12) month, the 11th month. Now, let us describe the physical construct by using modern terms and concept we are familiar with like earth orbit, earth spin axis, etc.

Since the earth’s axis (of spinning) is not parallel to the axis of the earth orbit, the sun appears to move north in our summer and south in our winter. The ancient people did not know of the orbiting or spinning stuff but the Chinese astronomers knew exactly the position of the sun in the sky (using stars as background), the two solstices (the sun appeared to be northern and southern most), and the two equinoxes (same length for day and night). These four points form two perpendicular lines dividing the earth orbit in four quadrants as we know it now. The ancient Chinese added two more “markers” between each of these four basic markers by equally dividing the path of the sun in the sky.

Effectively, as we understand it now, they divided the earth’s orbit in twelve sectors by equal angles. Since the earth orbit is elliptical, and since the earth travels faster when it is near the sun, these twelve markers divide the year into twelve time slots but not exactly all equal. Nonetheless, there is usually one new moon occurring in between two of these markers. As we pointed out earlier, a month is less than a twelfth of a year, once a while (more precisely, seven times in 19 years), two new moons will fall between two markers. The month lying between these two neighboring markers is defined as a intercalary (leap) month. So, you see, the Chinese calendar is defined by the precise position of the sun as well as the moon and therefore the only true “lunisolar” calendar. It realigns the offset as soon as it occurs instead of waiting for the end of the second or the third year. Few more definitions are needed to define a calendar. A day is defined as the time from midnight to midnight at meridian 120 E (where Beijing is). The first day of a month is the day in which a new moon falls. The month in which the winter solstice (usually December 21 or 22 in the Gregorian calendar) falls is defined as the eleventh of the year.

Even though the Sun is furthest south on winter solstice, the weather usually lags and the coldest days usually come later. The above definition automatically sets the New Year’s Day in average 45 days after winter solstice, usually the coldest part of the year, suitable for a rest from farm work and an annual celebration. That is, under the condition there is no leap month for the eleventh or the twelfth month, CNYD should be the second new moon, at least 29.53 days and at most 59.06 days after winter solstice. This is indeed the case and you do find CNYD usually between Jan.22 and Feb 15 on the Gregorian calendar. Now, what is the chance to have a leap month for the eleventh or the twelfth month? Would you not estimate it to be (2 (for two month)/12) X( 7 / 19) = 7 / 114, about once every 17 years? In fact it is much less than that. Why? Well, as we all know, the earth orbit is an eclipse while the sun resides at one of the foci. Earth passes the periapsis on January 3, only a few days after winter solstice. That means the earth spends most of the winter on the part of orbit closest to the sun and therefore travels the orbit faster in the winter than in the summer. This is made clear by the fact that there are a bit more than 186 days from vernal equinox to autumnal equinox but only a little more than 178 days back through the winter. Of the twelve equal-angle markers we mentioned earlier, the sun will speed through each pair faster in the winter than in the summer. The three time periods through the four winter markers are: 29.55, 29.44, 29.59 days, very little chance to stick two new moons in any of the pairs. That is why it is extremely rare to have the third new moon after winter solstice as the CNYD. The last time it happened was in 1645 and the next time is in 2033. What is the latest date a CNYD will happen w.r.t. the Gregorian calendar? It is February 21 in the year of 2319AD, nearly 300 years from now.

The above defines the mechanism inside a year. How is it counted and when did the counting start? In general (independent from calendars), Chinese use the stem-branch way to assist counting. There are ten “haven-stems” (甲Jia, 乙 Yi, 丙 Bing, … up to ten of them) and twelve “earth-branches” (子zi, 丑 chou, 寅 yin, … up to 12, each associated with an animal). Run them in order and pair them to form a 60 cycle (甲子JiaZi, 乙丑 YiCHou, …up to 60). These sexagenary cycles has been used for counting years continually ever since Shan or Zhou Dynasty defining the first cycle the year HuangDi (2698 – 2599 BCE) invented calendar (2637 BCE) or the year before his birth (2697BCE which is 60 years before), now generally recognized. The current year, beginning on Jan.25, 2020, is the Year 庚子GungZi (36th year) in the 79th sexagenary cycle which started 1984 and therefore the 4775th year from the beginning of counting. It is the year of Rat because that is the animal associated with 子Zi.

Before adapting Gregorian calendar in 20th Century, practically all East Asian countries used Chinese calendar. That includes Korea, Vietnam, Tibet, Japan. In 1258, Yuan Dynasty, ruler of both China and Islamic world, set up an observatory in Maragheh resulting a Chinese-Uighur calendar that was or is in use in Persian and Turkish speaking world.

I got most of the above information from an article called “the Mathematics of Chinese Calendar” by Helmer Aslaksen at the Department of Mathematics, National University of Singapore. He used to have a website at and

However, I cannot find the above pages now. He is now in the University of Oslo.

You can find a Chinese calendar with an extremely detailed description plus references.