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The Mathematics of the Chinese Calendar

“When does the Chinese New Year begin this year?”, I often ask my wife, a native of China.

“Why, on Chinese New Year’s Day, of course,” she’s fond of answering.

OK, smarty-pants, next year I’ll have the upper hand, thanks to Helmer Aslaksen in the Department of Mathematics at the National University of Singapore. Dr. Aslaksen, apparently tired of the same types of comebacks from his Chinese wife, settled the score by writing an award-winning scholarly paper entitled “The Mathematics of the Chinese Calendar,” now available in PDF format.

He notes:

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….

Fascinating, but that won’t won’t make for a snappy comeback to the Mrs. I’ll try this instead:

One rule of thumb is that Chinese New Year should be the new Moon closest to (??, lýchun), the beginning of spring. This rule is correct most of the time, but it can fail if lýchun falls close to halfway between two new Moons. It failed in 1985 and will fail again in 2015. Since lýchun 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.

Need some more snappy comebacks in your own “When is Chinese New Year?” conversations? You’ll find a wealth of them on Dr. Aslaksen’s site.

September 2003

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