The Mathematics of the Chinese Calendar
By: Helmer Aslaksen
Why This Page?
Chinese New Year is the main holiday of the year for more than one quarter of the world's population; very few people, however, know how to compute its date. For many years I kept asking people about the rules for the Chinese calendar, but I wasn't able to find anybody who could help me. Many of the people who were knowledgeable about science felt that the traditional Chinese calendar was backwards and superstitious, while people who cared about Chinese culture usually lacked the scientific knowledge to understand how the calendar worked. In the end I gave up and decided that I had to figure it out for myself.
Paper on the Mathematics of the Chinese Calendar
I have a written a long paper on The Mathematics of the Chinese Calendar. It gives you all the details. This web page is just an introduction to the topic. I have also written a shorter introduction called When is Chinese New Year?. This paper won a fourth prize in the Fifth Annual Boeing Writing Contest, which is organized by the Griffith Observatory. The article appeared in the Griffith Observer, February 2002, vol. 66, no. 2. I have also written a paper on Fake Leap Months in the Chinese Calendar: From the Jesuits to 2033.
I give a lot of public lectures on calendar topics and here are lecture notes on Heavenly Mathematics: The Mathematics of the Chinese, Indian, Islamic and Gregorian Calendars, The Mathematics of the Public Holidays of Singapore, The Mathematics of the Chinese Calendar. You may want to only download the first file. The two other are subsets of the first.
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The main focus on my paper is the study of leap months in the Chinese calendar. In the early 1990s, Chinese astronomers discovered that there was an error in the Chinese calendar for 2033. The traditional calendar claimed that the leap month would follow the 7th month, while in fact it comes after the 11th month. It is very unusual that the 11th month has a leap month, in fact it hasn't happened since the calendar reform in 1645 (before 1645, all months had the same probability for having a leap month). But many Chinese astronomers still claim that there will never be a leap month after the 12th and 1st month. I have found that there will be a leap month after the 1st month in 2262 (in fact, it should have happened in 1651, but they got the calculations wrong) and there will be a leap month after the 12th month in 3358. Since the Chinese calendar is an astronomical calendar, predictions require delicate astronomical calculations, so my computations for 3358 should probably be taken with a grain of salt. I also discuss other mathematical issues related to the Chinese calendar.
The Date of Chinese New Year
The mathematics behind the date of Chinese New Year is explained in full detail in my paper The Mathematics of the Chinese Calendar or the shorter introduction When is Chinese New Year?, but I will give two quick rules of thumb here.
One rule of thumb is that Chinese New Year should be the new Moon closest to the beginning of spring (??, lìchun). This rule is correct most of the time, but it can fail if Lìchun falls close to halfway between two new Moons. It failed in 1985 and will fail again in 2015. Since Lìchun falls around February 4, this helps explain why Chinese New Year will always fall between January 21 and February 21. It also helps explain why Chinese New Year is called the spring festival. If you have a Western calendar that indicates the phases of the Moon, this will give you an approximation of the date of Chinese New Year. But notice that the Chinese calendar uses the time of new Moon in China.
As explained above, Chinese New Year will always fall between January 21 and February 21. The tropical (or solar) year is about 365.25 days, while a synodic (or lunar) month is about 29.5 days. Hence a lunar year consisting of 12 months will be about 12 x 29.5 = 354 days. So a lunar year is about 11 days shorter than a solar year.
The second rule of thumb is therefore that most of the time Chinese New Year will fall 11 (or sometimes 10 or 12) days earlier than the previous year, but if that would take us outside of the Chinese New Year range of January 21 to February 21, we must add a leap month, so Chinese New Year jumps 19 (or sometimes 18) days later. If this rule takes you close to January 21, you can end up being one month wrong, otherwise you will be at most one day off.
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