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<!DOCTYPE html><html lang="en-US"><meta name="keywords" content="Chinese Calendar Calculation, rules, examples, leap month, 2033 issue, fake leap month"><meta charset="UTF-8"><meta name="viewport" content="width=device-width,initial-scale=1"><head><title>Examples of Computing the Chinese Calendar</title><link rel="stylesheet" href="calendar_min.css"><style>body{counter-reset:subsec-counter}subsect::before{counter-increment:subsec-counter 1;content:counter(subsec-counter,lower-roman) ". "}</style><script src="header_min.js"></script><base href="https://ytliu0.github.io/ChineseCalendar/"><script async src="https://www.googletagmanager.com/gtag/js?id=G-SN0QJRDXXT"></script><script>function gtag(){dataLayer.push(arguments)}window.dataLayer=window.dataLayer||[],gtag("js",new Date),gtag("config","G-SN0QJRDXXT")</script></head><body onload='header(0,"","examples")'><div id="wrapper0"><div id="menu"></div><div id="language"></div><p><a href="rules.html#examples" style="background-color:pink">Back to the calendar rules page</a></p><br><h1>Examples of Computing the Chinese Calendar</h1><p>First draft: October, 2018</p><br><p>On this page I provide two examples of computing the Chinese calendar using the <a href="rules.html#rules">rules</a> stated on the rules page. I also analyze the special year 2033.</p><p>For simplicity, I label a Chinese year by N<sub>Y</sub>. It means the Chinese year whose New Year day is closest to Jan. 1 in Gregorian year Y. For example, N<sub>1984</sub> means the Chinese year Jiǎ zǐ (甲子) that begins on Feb. 2, 1984 and ends on Feb. 19, 1985.</p><p>It is also convenient to label the lunar conjunctions (new moons) by lunation numbers. A lunation number refers to the number of new moons counting from a starting point. It was invented by Ernest W. Brown in 1933. In Brown's system, lunation number 1 is the first new moon in 1923, which occurred on January 17, 1923 at 2:41 UT1. The Julian day number on Jan. 17, 1923 at noon UT is 2423437. Since the synodic month is 29.5306 days. The lunation number associated with a new moon is the integer closest to 1 + (JD - 2423437)/29.5306, where JD is the Julian day number at UT noon on the date of the new moon.</p><br><subsect id="ex1">Example 1: Determine all months in the Chinese year <span class="nowrap">N<sub>2017</sub></span>, which is the year Dīng yǒu (丁酉).</subsect><p>According to <i>Astronomical Phenomena for the Year 2016</i>, the winter solstice in 2016 was on Dec. 21 at 10:44 UTC, which was Dec. 21 at 18:44 (UTC+8). So <span class="nowrap">W<sub>2016</sub></span> = Dec. 21, 2016. According to <i>Astronomical Phenomena for the Year 2017</i>, the winter solstice in 2017 was on Dec. 21 at 16:28 UTC, which was Dec. 22 at 00:28 (UTC+8). So <span class="nowrap">W<sub>2017</sub></span> = Dec. 22, 2017. The times of all lunar conjunctions in 2016 and 2017 are also available in the books. Note that the new moons in the books are labelled by the lunation numbers. From the data in the books I find <span class="nowrap">M<sub>-1</sub></span> = Nov. 29, 2016 and <span class="nowrap">M<sub>11</sub></span> = Dec. 18, 2017, and so <span class="nowrap">M<sub>11</sub></span> - <span class="nowrap">M<sub>-1</sub></span> = 384. This means L = 13 and there was a leap month in the suì <span class="nowrap">S<sub>2017</sub></span>. Next we need the dates of all 12 major solar terms in 2017. Unfortunately, <i>Astronomical Phenomena for the Year 2017</i> only has times for the equinoxes and solstices, so we have to calculate them on our own. I have done the calculations and the times are available on the <a href="sunMoon.html?y=2017">Sun & Moon phenomena page</a>. Note that all times in the two books above are in UTC, whereas the times in the Sun & Moon page are in UTC+8. All times must be converted to UTC+8 for Chinese calendar calculation according to Rule 1.</p><p>The following table lists the dates of the major solar terms and relevant lunar conjunctions in chronological order. A lunar conjunction is labelled as L followed by the lunation number.</p><table><tr><th>Lunar Conjunction</th><th>Major Solar Term</th></tr><tr><td>L1162: Nov. 29, 2016</td><td></td></tr><tr><td></td><td>Z11: Dec. 21, 2016</td></tr><tr><td>L1163: Dec. 29, 2016</td><td></td></tr><tr><td></td><td>Z12: Jan. 20, 2017</td></tr><tr><td>L1164: Jan. 28, 2017</td><td></td></tr><tr><td></td><td>Z1: Feb. 18, 2017</td></tr><tr><td>L1165: Feb. 26, 2017</td><td></td></tr><tr><td></td><td>Z2: Mar. 20, 2017</td></tr><tr><td>L1166: Mar. 28, 2017</td><td></td></tr><tr><td></td><td>Z3: Apr. 20, 2017</td></tr><tr><td>L1167: Apr. 26, 2017</td><td></td></tr><tr><td></td><td>Z4: May 21, 2017</td></tr><tr><td>L1168: May 26, 2017</td><td></td></tr><tr><td></td><td>Z5: June 21, 2017</td></tr><tr><td>L1169: June 24, 2017</td><td></td></tr><tr><td></td><td>Z6: July 22, 2017</td></tr><tr><td>L1170: July 23, 2017</td><td></td></tr><tr><td>L1171: Aug. 22, 2017</td><td></td></tr><tr><td></td><td>Z7: Aug. 23, 2017</td></tr><tr><td>L1172: Sep. 20, 2017</td><td></td></tr><tr><td></td><td>Z8: Sep. 23, 2017</td></tr><tr><td>L1173: Oct. 20, 2017</td><td></td></tr><tr><td></td><td>Z9: Oct. 23, 2017</td></tr><tr><td>L1174: Nov. 18, 2017</td><td></td></tr><tr><td></td><td>Z10: Nov. 22, 2017</td></tr><tr><td>L1175: Dec. 18, 2017</td><td></td></tr><tr><td></td><td>Z11: Dec. 22, 2017</td></tr></table><p>Looking at the dates, we see that the month associated with the new moon on July 23, 2017 (lunation 1170) was the only month in the suì that does not contain a major solar term. It was a leap month. So we have <span class="nowrap">M<sub>0</sub></span> = Dec. 29, 2016, <span class="nowrap">M<sub>1</sub></span> = Jan. 28, 2017, ..., <span class="nowrap">M<sub>6</sub></span> = June 24, 2017, <span class="nowrap">M<sup>*</sup><sub>6</sub></span> = July 23, 2017, <span class="nowrap">M<sub>7</sub></span> = Aug. 22, 2017, ..., <span class="nowrap">M<sub>11</sub></span> = Dec. 18, 2017. The leap month was the month after month 6. To determine the rest of the month(s) in <span class="nowrap">N<sub>2017</sub></span>, we need to go through the steps again in principle. However, we can use a short cut here. We know that there can only be at most 13 months in a Chinese year. Therefore, there is only one month left in <span class="nowrap">N<sub>2017</sub></span> and it is month 12, which must be the month following month 11. The corresponding date of the lunar conjunction can be found in <i>Astronomical Phenomena for the Year 2018</i>. The following table summarizes the 13 months in N<sub>2017</sub>.</p><table><tr><th>Month</th><th>First day</th><th># of Days</th></tr><tr><td>1</td><td>Jan. 28, 2017</td><td>29</td></tr><tr><td>2</td><td>Feb. 26, 2017</td><td>30</td></tr><tr><td>3</td><td>Mar. 28, 2017</td><td>29</td></tr><tr><td>4</td><td>Apr. 26, 2017</td><td>30</td></tr><tr><td>5</td><td>May 26, 2017</td><td>29</td></tr><tr><td>6</td><td>June 24, 2017</td><td>29</td></tr><tr><td>leap 6</td><td>July 23, 2017</td><td>30</td></tr><tr><td>7</td><td>Aug. 22, 2017</td><td>29</td></tr><tr><td>8</td><td>Sep. 20, 2017</td><td>30</td></tr><tr><td>9</td><td>Oct. 20, 2017</td><td>29</td></tr><tr><td>10</td><td>Nov. 18, 2017</td><td>30</td></tr><tr><td>11</td><td>Dec. 18, 2017</td><td>30</td></tr><tr><td>12</td><td>Jan. 17, 2018</td><td>30</td></tr></table><p>The number of days in a month is computed by counting the number of days between the first day of the month and the first day of the following month. For example, the first day of month 4 was on Apr. 26 and the first day of month 5 was on May 26. This means that Apr. 26 was the first day of month 4; Apr. 27 was the second day of month 4; Apr. 28 was the third day of month 4; ...; May 25 was the 30th day of month 4; and May 26 was the first day of month 5. Thus, month 4 had 30 days.</p><br><subsect id="ex2">Example 2: Determine all months in the Chinese year <span class="nowrap">N<sub>2033</sub></span>, which is the year Guǐ chǒu (癸丑).</subsect><p>Here I am going to use my calculation for the times of lunar conjunctions and solar terms. They can be looked up on the <a href="sunMoon.html?y=2033">Sun & Moon page</a>. I find <span class="nowrap">W<sub>2032</sub></span> = Dec. 21, 2032; <span class="nowrap">W<sub>2033</sub></span> = Dec. 21, 2033; <span class="nowrap">M<sub>-1</sub></span> = Dec. 3, 2032; and <span class="nowrap">M<sub>11</sub></span> = Nov. 22, 2033. It follows that <span class="nowrap">M<sub>11</sub></span> - <span class="nowrap">M<sub>-1</sub></span> = 354 days and L = 12. Thus there is no leap month in the suì <span class="nowrap">S<sub>2033</sub></span> and <span class="nowrap">M<sub>1</sub></span> is the date of the second lunar conjunction after <span class="nowrap">M<sub>-1</sub></span>, which is Jan. 31, 2033. The first days of months 2-10 are the 9 lunar conjunctions following <span class="nowrap">M<sub>1</sub></span>, which can be looked up directly from the Sun & Moon page. The first day of month 11 is <span class="nowrap">M<sub>11</sub></span> = Nov. 22, 2033 determined above. There is no need to look at the major solar terms in the suì <span class="nowrap">S<sub>2033</sub></span> in this case because it is not a leap suì. However, there could be a leap month after month 11 in <span class="nowrap">N<sub>2033</sub></span>. So we need to determine if the suì <span class="nowrap">S<sub>2034</sub></span> is a leap suì.</p><p>The Sun & Moon page shows <span class="nowrap">W<sub>2034</sub></span> = Dec. 22, 2034 and the lunar conjunction on or before <span class="nowrap">W<sub>2034</sub></span> is <span class="nowrap">M<sub>23</sub></span> = Dec. 11, 2034. Here I use <span class="nowrap">M<sub>23</sub></span> to label the date of the lunar conjunction because it is associated with the 23rd <i>regular</i> month counting from the Chinese year <span class="nowrap">N<sub>2033</sub></span>. It follows that <span class="nowrap">M<sub>23</sub></span> - <span class="nowrap">M<sub>11</sub></span> = 384 days and so there are 13 months in the suì <span class="nowrap">S<sub>2034</sub></span>. One of the months is a leap month and it could be in the year N<sub>2033</sub>. We now need the dates of major solar terms and lunar conjunctions in the suì <span class="nowrap">S<sub>2034</sub></span> to determine which month is a leap month. The following table lists the first few dates starting from <span class="nowrap">M<sub>11</sub></span>.</p><table><tr><th>Lunar Conjunction</th><th>Major Solar Term</th></tr><tr><td>L1372: Nov. 22, 2033</td><td></td></tr><tr><td></td><td>Z11: Dec. 21, 2033</td></tr><tr><td>L1373: Dec. 22, 2033</td><td></td></tr><tr><td>L1374: Jan. 20, 2034</td><td>Z12: Jan. 20, 2034</td></tr><tr><td></td><td>Z1: Feb. 18, 2034</td></tr><tr><td>L1375: Feb. 19, 2034</td><td></td></tr></table><p>We see that the month associated with lunation 1373 is the first month after <span class="nowrap">M<sub>11</sub></span> that does not contain a major solar term, so it is a leap month. The previous month is month 11, so the leap month has a number 11. It is in the Chinese year N<sub>2033</sub>. The month following the leap month is month 12 of <span class="nowrap">N<sub>2033</sub></span> and the calculation of all months in <span class="nowrap">N<sub>2033</sub></span> is now complete. The following summarizes the result for the year <span class="nowrap">N<sub>2033</sub></span>.</p><table><tr><th>Month</th><th>First day</th><th># of Days</th></tr><tr><td>1</td><td>Jan. 31, 2033</td><td>29</td></tr><tr><td>2</td><td>Mar. 1, 2033</td><td>30</td></tr><tr><td>3</td><td>Mar. 31, 2033</td><td>29</td></tr><tr><td>4</td><td>Apr. 29, 2033</td><td>29</td></tr><tr><td>5</td><td>May 28, 2033</td><td>30</td></tr><tr><td>6</td><td>June 27, 2033</td><td>29</td></tr><tr><td>7</td><td>July 26, 2033</td><td>30</td></tr><tr><td>8</td><td>Aug. 25, 2033</td><td>29</td></tr><tr><td>9</td><td>Sep. 23, 2033</td><td>30</td></tr><tr><td>10</td><td>Oct. 23, 2033</td><td>30</td></tr><tr><td>11</td><td>Nov. 22, 2033</td><td>30</td></tr><tr><td>leap 11</td><td>Dec. 22, 2033</td><td>29</td></tr><tr><td>12</td><td>Jan. 20, 2034</td><td>30</td></tr></table><br><subsect id="exceptional">The Exceptional Year <span class="nowrap">N<sub>2033</sub></span></subsect><p>It has been pointed out by several people that <span class="nowrap">N<sub>2033</sub></span> is an exceptional year and some published Chinese calendars made mistakes in their calculations. However, we don't see anything special from the calculation shown in example 2. To understand why <span class="nowrap">N<sub>2033</sub></span> is special, I have to list all the relevant dates of lunar conjunctions and major solar terms, including those I skipped in the above calculation. The following table shows the dates.</p><table><tr><th>Lunar Conjunction</th><th>Major Solar Term</th><th>Date in the Chinese Calendar</th></tr><tr><td>L1360: Dec. 3, 2032</td><td></td><td>11-01, N<sub>2032</sub></td></tr><tr><td></td><td>Z11: Dec. 21, 2032</td><td>11-19, N<sub>2032</sub></td></tr><tr><td>L1361: Jan. 1, 2033</td><td></td><td>12-01, N<sub>2032</sub></td></tr><tr><td></td><td>Z12: Jan. 20, 2033</td><td>12-20, N<sub>2032</sub></td></tr><tr><td>L1362: Jan. 31, 2033</td><td></td><td>01-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z1: Feb. 18, 2033</td><td>01-19, N<sub>2033</sub></td></tr><tr><td>L1363: Mar. 1, 2033</td><td></td><td>02-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z2: Mar. 20, 2033</td><td>02-20, N<sub>2033</sub></td></tr><tr><td>L1364: Mar. 31, 2033</td><td></td><td>03-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z3: Apr. 20, 2033</td><td>03-21, N<sub>2033</sub></td></tr><tr><td>L1365: Apr. 29, 2033</td><td></td><td>04-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z4: May 21, 2033</td><td>04-23, N<sub>2033</sub></td></tr><tr><td>L1366: May 28, 2033</td><td></td><td>05-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z5: June 21, 2033</td><td>05-25, N<sub>2033</sub></td></tr><tr><td>L1367: June 27, 2033</td><td></td><td>06-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z6: July 22, 2033</td><td>06-26, N<sub>2033</sub></td></tr><tr><td>L1368: July 26, 2033</td><td></td><td>07-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z7: Aug. 23, 2033</td><td>07-29, N<sub>2033</sub></td></tr><tr><td>L1369: Aug. 25, 2033</td><td></td><td>08-01, N<sub>2033</sub></td></tr><tr><td>L1370: Sep. 23, 2033</td><td>Z8: Sep. 23, 2033</td><td>09-01, N<sub>2033</sub></td></tr><tr><td>L1371: Oct. 23, 2033</td><td>Z9: Oct. 23, 2033</td><td>10-01, N<sub>2033</sub></td></tr><tr><td>L1372: Nov. 22, 2033</td><td>Z10: Nov. 22, 2033</td><td>11-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z11: Dec. 21, 2033</td><td>11-30, N<sub>2033</sub></td></tr><tr><td>L1373: Dec. 22, 2033</td><td></td><td>leap 11-01, N<sub>2033</sub></td></tr><tr><td>L1374: Jan. 20, 2034</td><td>Z12: Jan. 20, 2034</td><td>12-01, N<sub>2033</sub></td></tr><tr><td></td><td>Z1: Feb. 18, 2034</td><td>12-30, N<sub>2033</sub></td></tr><tr><td>L1375: Feb. 19, 2034</td><td></td><td>01-01, N<sub>2034</sub></td></tr><tr><td>L1376: Mar. 20, 2034</td><td>Z2: Mar. 20, 2034</td><td>02-01, N<sub>2034</sub></td></tr><tr><td>L1377: Apr. 19, 2034</td><td></td><td>03-01, N<sub>2034</sub></td></tr><tr><td></td><td>Z3: Apr. 20, 2034</td><td>03-02, N<sub>2034</sub></td></tr><tr><td>L1378: May 18, 2034</td><td></td><td>04-01, N<sub>2034</sub></td></tr><tr><td></td><td>Z4: May 21, 2034</td><td>04-04, N<sub>2034</sub></td></tr></table><p>In the table above, dates in the Chinese calendar are indicated by MM-DD, <span class="nowrap">N<sub>Y</sub></span>. For example, 12-20, <span class="nowrap">N<sub>2032</sub></span> means the 20th day of month 12 in the year <span class="nowrap">N<sub>2032</sub></span>.</p><p>It can be seen from the table that the months associated with lunations 1369, 1373 and 1375 do not contain any major solar term but only one of them is a leap month. The mistake some people made in the calendar calculation arises from using the incorrect rule that "a month that does not contain a major solar term is a leap month". However, this can only be taken as a rule of thumb in the sense that it is true most of the time but can fail on rare occasions. It clearly doesn't work in <span class="nowrap">N<sub>2033</sub></span> and <span class="nowrap">N<sub>2034</sub></span>. People who performed the calendar calculation clearly realized that it was inappropriate to put three leap months there. Without understanding the details of how the Chinese calendar works, most people incorrectly assigned the leap month to the month associated with lunation 1369. A month that does not contain a major solar term but is not a leap month is called a <i>fake leap month</i> in <a href="http://www.math.nus.edu.sg/aslaksen/calendar/cal.pdf" target="_blank">Aslasksen's article</a>. A fake leap month also occurred in <span class="nowrap">N<sub>1984</sub></span>, but it occurred after the true leap month. So assigning the leap month to the first month without a major solar term worked in that case.</p><p>The rules of calendar calculation are robust and they can handle <span class="nowrap">N<sub>2033</sub></span> without any difficulty. Since there are 12 months in the suì <span class="nowrap">S<sub>2033</sub></span>, it is not a leap suì. It doesn't matter if there is a month with no major solar term within since there is no room for a leap month. That is why in example 2 I don't even bother to look up the dates of the major solar terms in the suì <span class="nowrap">S<sub>2033</sub></span>. The suì <span class="nowrap">S<sub>2034</sub></span> is a leap suì. In the calculation in example 2, once I find that the month after month 11 does not contain a major solar term I can immediately call it a leap month. Since there can only be one leap month in a suì, the rest of the months in the suì are determined and it doesn't matter if there are more months in the suì not containing a major solar term.</p><p>There is also a "rule" that says the month number in a regular month is determined by the major solar term contained in the month. For example, month 1 must contain Z1, month 2 must contain Z2 and so on. This is another rule of thumb that is true most of the time but can fail on rare occasions. We can only say for sure that month 11 must contain Z11 because it is imposed by Rule 4. This "rule" clearly fails when there are two major solar terms in a month, which happens twice in <span class="nowrap">N<sub>2033</sub></span>! This "rule" also fails in the case of fake leap months. As seen from the table above, in <span class="nowrap">N<sub>2033</sub></span>, month 8 does not contain Z8; month 9 does not contain Z9; month 10 does not contain Z10; month 1 in <span class="nowrap">N<sub>2034</sub></span> does not contain Z1.</p><p>These two "fake rules" were old rules before the 1645 calendar reform. As discussed in the <a href="solarTerms.html">solar term page</a>, the 24 solar terms before 1645 were defined using <i>píngqì</i> (based on the mean motion of the Sun) and a month could have at most one major solar term. There are 12 major solar terms in a suì. It is easy to show that each of the regular months in a suì must contain exactly one major solar term and there can only be one month in a leap suì that does not contain a major solar term. Hence any month that does not contain a major solar term must be a leap month and fake leap months cannot exist. Since month 11 must contain Z11 and there cannot be more than one major solar term in a regular month, it follows that month 12 must contain Z12, month 1 must contain Z1 and so on. As a result, the two rules were true if the major solar terms were defined using <i>píngqì</i>. They were no longer true when <i>píngqì</i> was abandoned after the 1645 calendar reform.</p><p>A fake leap month is clearly associated with a month with two major solar terms. In the new system, there can be two major solar terms in a month and a fake leap month appears. The old rules are no longer compatible with the new system. A new rule, i.e. Rule 5, was created to handle the new situation around 1645.</p><p>The calendar reform in 1645 was carried out by a Jesuit missionary called Johann Adam Schall von Bell. The appearance of two major solar terms in a month and fake leap months upset many intellectuals at the time. <a href="http://www.math.nus.edu.sg/aslaksen/calendar/cal.pdf" target="_blank">Aslasksen's article</a> points out that Chinese astronomy was not necessarily inferior to Western astronomy at that time. He thinks that none of the Jesuits was a better astronomer than Guō Shǒujìng, a famous Chinese astronomer in the 13th century. However, in the 17th century the Chinese officials responsible for calendar calculation in the imperial court did not understand their traditional calendar very well. Their positions had become hereditary. The Jesuits eventually gained the upper hand and won the debate.</p><br><p><a href="rules.html#examples" style="background-color:pink">Back to the calendar rules page</a></p></div><br><br></body></html>