Reconstruction of the Chunqiu Calendar

First draft: March, 2019

In ancient time, Chinese calendars were made based on astronomical observations. In the Spring and Autumn period (722 BCE – 481 BCE), people developed a simple method to predict the moon phases. Calendars began to be calculated in advance. China was divided into many states at that time and each state used its own calendar. Unfortunately, none of these calendars is preserved. Today we only have fragmented information about the calendar used by the Lu state from the chronicle Chunqiu (《春秋》) revised by Confucius. This calendar is called Chunqiu here.

Here I describe the Chunqiu calendar reconstructed using the method and table in Section 3.5 of the book Zhōng Guó Gǔ Dài Lì Fǎ (《中国古代历法》 or Ancient Chinese Calendars and Almanacs) written by Zhāng Péiyú (張培瑜), Chén Měidōng (陳美東), Bó Shùrén (薄樹人), and Hú Tiězhū (胡鐵珠), published by China Science Press (Beijing) in March 2008. Hereafter, I will refer to the book as Ancient Chinese Calendars and Almanacs. According to the preface, chapters in the book were written separately by the four authors. Chapter 3 was written by Zhāng Péiyú, a researcher in the Purple Mountain Observatory in Nanjing, China.

According to Zhāng's research, the lunar cycle adopted by the Chunqiu calendar was between 29.5306703 days and 29.5306755 days. For practical calculation, the number 30328/1027 days = (29 + 545/1027) days = 29.53067186 days may be used. This number is quite close to the lunar cycle at that time (29.530583 days) and is more accurate than (29 + 499/940) days adopted by the calendars in the Warring States period (480 BCE – 222 BCE). The accumulated error of the adopted lunar cycle would reach one day in 900 years. Chinese calendars before the 7th century CE used píngshuò for the calculation of lunar conjunction, which only takes into account the mean motions of the moon and Sun. If we know a particular conjunction time, the other conjunction times can be determined by simply adding integral multiples of the adopted lunar cycle.

For convenience, I use the notation N_y to denote the year of the Chunqiu calendar whose New Year day was closest to January 1 of the proleptic Julian calendar of year y. For example, N_-721 began on January 16, -721 (722 BCE) and ended on January 4, -720. According to Zhāng's reconstructed Chunqiu calendar, the predicted lunar conjunction time associated with the New Year day of N_-721 occurred on the first xīn sì (sexagenary) day after the winter solstice. The conjunction time was 268/1027 days from the midnight. Expressed in the Western calendar and the modern time system, the conjunction time was on January 16, -721 at 06:16. In ancient time, it was not known that different places had different local times. We can regard the time as being the local time in the state of Lu, whose capital was at the present-day city Qūfù in Shāndōng province in northeast China.

Decimals were not used in the calendar calculations in ancient China. The fractional part of a number was called xiǎoyú (小餘) and was usually expressed as a fraction. "Fractional day" may be used as an English translation of xiǎoyú. As a result, 268/1027 was the xiǎoyú of the conjunction time. The denominator 1027 is usually omitted and we simply refer the number 268 as the xiǎoyú. Xiǎoyú can be thought of as the time in the day expressed as a fraction of a day. Using today's 24-hour system, a xiǎoyú of 268/1027 is the same 6:15:46. According to the píngshuò rule, the next conjunction would occur (29 + 545/1027) days later, which was (268/1027 + 29 + 545/1027) days, or (29 + 813/1027) days from the midnight of the first conjunction day. Here the integer part was 29 and xiǎoyú was 813. The sexagenary day cycle was 29 days after xīn sì. The cycle of the heavenly stems is 10 and the cycle of earthly branches is 12. Since 29 = 30-1, the heavenly stem was one stem before xīn, which was gēng; mod(29, 12) = 5 and so the earthly branch was 5 branches after sì, which was xū. Here mod(X, Y) denotes the remainder of X divided by Y. The sexagenary day of the first day of month 2 was gēng xū. It is also easy to calculate the date in the proleptic Julian calendar: January (16+29) = February (16+29-31) = February 14.

In general, the lunar conjunction times predicted by the Chunqiu calendar are given by the following formula.

M_i = M₀ + i (29 + 545/1027) days     (1)

Here M₀ is a particular lunar conjunction time, which we will take as the conjunction time associated with the New Year day of N_-721, i is the number of accumulated months from M₀. Let A(y) be the accumulated months associated with the New Year day of N_y. Then the conjunction time associated with the New Year day is given by

Z(y) = M₀ + A(y) (29 + 545/1027) days    (2)

Once we know A(y), other months in N_y can be obtained by adding integer multiples of (29 + 545/1027) days. The accumulated months A(y) can be calculated by a recurrence relation. Suppose there was no leap month in N_y, then A(y+1) = A(y) + 12. If, on the other hand, there was a leap month in N_y, then A(y+1) = A(y) + 13. By definition, A(-721) = 0. Hence, if we know all the leap months in the Chunqiu calendar, we can easily reconstruct the calendar in any given year.

There was no fixed rule for inserting the leap months in the Chunqiu calendar. Instead, the insertion of leap months was determined by astronomical observations. Leap months in the Chunqiu calendar can only be inferred from the chronicle Chunqiu. Based on his research and studies of other scholars, Zhāng constructed a table of the New Year days in the Chunqiu calendar. The table was reproduced in Table 3-7 of Ancient Chinese Calendars and Almanacs. Table 3-7 lists the sexagenary days of the New Year days, the jiàn (i.e. the branch name, see sexagenary cycle page) of the first month, the xiǎoyú of the conjunction associated with the New Year day, the dates of winter solstices in proleptic Julian calendar, and also the Chunqiu calendar data reconstructed by Wáng Tāo (王韜), a scholar in the 19th century, for comparison. It is possible to determine if there was leap month from Table 3-7. The simplest method is to compare the xiǎoyús in two successive years.

As mentioned above, xiǎoyú is the conjunction time expressed as a fraction of a day from midnight, with the denominator 1027 omitted. If there was no leap month in N_y, the xiǎoyú of the new year conjunction K(y) is related to the xiǎoyú of the new year conjunction in N_y+1, K(y+1), by the following equation.
K(y+1) = mod(K(y) + 12×545, 1027) = mod(K(y) + 378, 1027)    (3)
If there was a leap month in N_y, then
K(y+1) = mod(K(y) + 13×545, 1027) = mod(K(y) - 104, 1027)    (4)
We can use these two equations to determine from Table 3-7 whether or not a given year had a leap month. The accumulated months A(y) can then be calculated from the recurrence relation mentioned above. The result is listed in the following table, which also lists the jiàn of the first month, the sexagenary day cycle of the New Year day, the date in the proleptic Julian calendar, and the xiǎoyú of the new year day conjunction.

Once A(y) is known, it is very easy to compute the Chunqiu calendar. The data in other columns are also easy to reproduce, except in some cases the jiàn of the first month. Here I demonstrate the calculation by an example for the year N_-649. From the table above, A(-649) = 890. It follows that the number of days between the new year day conjunction time in N_-649 and midnight of Jan. 16, 722 BCE was
[268/1027 + 890×(29 + 545/1027)] days= (26282 + 574/1027) days.
The integer part was 26282 and the xiǎoyú was 574, consistent with the xiǎoyú listed in the table. It is easy to calculate the sexagenary day cycle using modular arithmetic: the heavenly stems have a cycle of 10 days and the earthly branches have a cycle of 12 days. Simple calculation shows that 26282 days after xīn sì was guǐ wèi, which is also the same as the one listed in the table. The New Year day in the proleptic Julian calendar can be determined by first computing the number of days between Jan. 16, -721 and Jan. 16, -649. There are 72 years between -721 and -649. Years divisible by 4 have 366 days and others have 365 days. The leap years between -721 and -649 were -720, -716, ..., and -652, a total of 18 leap years. Hence the number of days between Jan. 16, -721 and Jan. 16, -649 was (72×365 + 18) = 26298 days = (26282 + 16) days. This means that the New Year day of N_-649 was 16 days before Jan. 16, -649, which was Dec. 31, -650 and agrees with the date listed in the table. As for the jiàn, the zǐ month is defined to be the month that contains the winter solstice (see sexagenary cycle page). The winter solstice in -650 can be looked up from the Sun & Moon phenomena page and it was on Dec. 28 in the proleptic Julian calendar^fn1. The New Year day was 3 days after the winter solstice. Thus the previous month was the zǐ month and the first month of N_-649 was one month after the zǐ month, so it was the chǒu month. This also agrees with the jiàn listed in the table.

Now that we know the New Year day of N_-649, the first days of the other months in the year can be calculated by adding (29 + 545/1027) days successively to the time of the new year day conjunction. There was no leap month in N_-649 according to the table. The first days of each of the 12 months in N_-649 are listed in the following table.

In the table, L means the month had 30 days (long month); S means the month had 29 days (short months). The number of days in a month is determined by the number of days between two successive lunar conjunctions, which can be calculated easily by the xiǎoyú. There is a simple relationship between the xiǎoyús of two successive months: the xiǎoyú of a month is equal to the xiǎoyú of the previous month plus 545, and then subtract 1027 if necessary. It is probably easier to understand the logic using today's 24-hour time system. The lunar cycle is 29 days 12 hours 44 minutes and 10 seconds. The time in the day of a conjunction can be obtained by adding 12 hours 44 minutes and 10 seconds from the previous conjunction time. If the number exceeds 24, subtract 24 from it. If the xiǎoyú of a month is smaller than 482, adding 545 to it will result in a number smaller than 1027. The number of days in the month will be 29 in this case and the month is a short month. If the xiǎoyú of the month is equal to or greater than 482, adding 545 to it will result in a number that is at least 1027 and the number has to be subtracted by 1027 to get the xiǎoyú of the following month. There will be 30 days in the month in this case. The conclusion is that a month with xiǎoyú less than 482 has 29 days and a month with xiǎoyú ≥482 has 30 days. It is also easy to show that if xiǎoyú exceeds 963, there will be two long months in a row. It is impossible to have two successive short months in the Chunqiu calendar. Since the synodic month is slightly greater than 29.5 days, it is in general impossible to have two successive short months for calendars based on the píngshuò algorithm.

I used the calculated A(y) to compute the sexagenary cycle of the New Year day and the jiàn of the first month in each year, and then compared with those listed in Table 3-7 in Ancient Chinese Calendars and Almanacs. I find agreement in most years, but there are discrepancies in 5 years. After further analysis, I find that the data in Table 3-7 for those 5 years are inconsistent with the xiǎoyús, which means that at least one of them is incorrect. My analysis suggests that the data should be corrected as shown below.

Let's look at each of them. The typo in N_-720 is easy to see. If there was no leap month in N_-721, the xiǎoyú of the new year day conjunction in N_-720 would be 646, which agrees with Table 3-7, but the sexagenary day would be yǐ hài. If there was a leap month in N_-721, the xiǎoyú would be 164, which disagrees with the value listed in Table 3-7, and the sexagenary day cycle would be yǐ sì. Therefore, the sexagenary day would not be jǐ hài no matter whether there was a leap month in N_-721. Clearly, jǐ hài is a typo.

Comparing the xiǎoyús between N_-682 and N_-681, I conclude that there was no leap month in N_-682 and the New Year day of N_-681 was Nov. 25, -682 in the proleptic Julian calendar. This was clearly before the winter solstice, which occurred on Dec. 28 in -682. Hence, it is impossible for the first month to be a chǒu month. There was 33 days before Nov. 25 and Dec. 28, which also makes it impossible for the jiàn to be zǐ. The first month was clearly a month before the zǐ month, making the jiàn to be hài. The sexagenary day of Nov. 25, -682 was jǐ wèi, which agrees with Table 3-7. The sexagenary day of the first day of the chǒu month was also jǐ wèi. However, there would be a serious problem if the jiàn ware really chǒu. In addition to the discrepancy of the xiǎoyú, N_-682 would have 14 months, which was impossible. Hence changing the jiàn from chǒu to hài is appropriate.

Having corrected the typo in N_-681, it is not difficult to see that the jiàn in N_-680 in Table 3-7 is also incorrect. It follows from the xiǎoyús in N_-681 and N_-680 that there was a leap month in N_-681 and the New Year day of N_-680 was Dec. 14, -681. The first month clearly contained the winter solstice and so was the zǐ month. The sexagenary day on Dec. 14, -681 was guǐ wèi, which agrees with Table 3-7. The sexagenary day of the conjunction day associated with the yín month was rén wǔ, in contradiction with Table 3-7. The xiǎoyú of the yín month conjunction was also different from the number listed in Table 3-7. Changing the jiàn from yín to zǐ removes the discrepancies.

Table 3-7 lists the sexagenary day of the New Year day of N_-654 as rén zǐ, and the conjunction xiǎoyú was 675; the xiǎoyú of the new year day conjunction of N_-653 was 26 = mod(675+378, 1027). It follows that there was no leap month in N_-654, and the number of days between the new year day conjunction of N_-653 and the midnight of the New Year day of N_-654 was
[675/1027 + 12×(29+545/1027)] days = (355 + 26/1027) days.
The integer part was 355 and the xiǎoyú was 26. It is easy to show, using modular arithmetic, that the sexagenary day 355 days after the day of rén zǐ is dīng wèi, which is one day after the day of bǐng wǔ. The bǐng wǔ listed in Table 3-7 is probably caused by a mistake in arithmetic calculation.

Table 3-7 lists the New Year day of N_-624 as the wù zǐ day in the zǐ month (December 25, 626 BCE) and the xiǎoyú was 551. If there was no leap month in N_-624, the New Year day of N_-623 would be the rén wǔ day in the zǐ month (December 13, 625 BCE) and the xiǎoyú would be 929. If there was a leap month in N_-624, the New Year day of N_-623 would be the rén zǐ day in the chǒu month (January 12, 624 BCE) and the xiǎoyú would be 447. It is therefore impossible for the xiǎoyú in N_-624 to be 614. Table 3-7 lists the New Year day of N_-623 as the rén wǔ day in the zǐ month, so the xiǎoyú should be 929.

Zhāng Péiyú has another book titled Sānqiān Wǔbǎiniǎn Lìrì Tiānxiàng (《三千五百年历日天象》 or 3500 Years of Calendars and Astronomical Phenomena), in which there are also calendar dates for the Chunqiu calendar. The New Year days of N_-721, N_-681 and N_-680 agree with my corrected dates. However, the New Year days of N_-675, N_-674, N_-600, N_-519 and N_-502 differ by one month compared to those in Table 3-7 of Ancient Chinese Calendars and Almanacs. I do not use the data of 3500 Years of Calendars and Astronomical Phenomena on this website in the Spring and Autumn period, as the reconstruction method is not stated and so it is difficult to assess the reliability of the data.^fn2

Scholars have not come to an agreement on the position of the leap months in the Chunqiu calendar. The 19th century scholar Wáng Tāo placed some of the leap months in the middle of the years in order to account for some of the dates recorded by the chronicle Chunqiu, which were otherwise irreconcilable with his reconstructed calendar. Zhāng points out that leap months are only mentioned twice in Chunqiu and they were all placed at the end of the years. He also doesn't think that there was any natural algorithm in the Chunqiu calendar to place a leap month in the middle of a year. Thus he thinks all leap months were placed at the end of the years. I adopt Zhāng's view and place all leap months at the end of the years on this website.

As there were no fixed rules for inserting leap months, the jiàns of the first months were not fixed either. Looking at the table above, we see that the jiàns wandered between hài (present-day month 10) and yín (present-day month 1), with chǒu (present-day month 12) occurring more often in the early years and zǐ (present-day month 11) occurring more often in the later years. Zhāng speculates that the Chunqiu calendar makers probably discovered the 19-year cycle (known as the Metonic cycle in ancient Greece) and the cycle of the tropical year in later years, and intentionally shifted the first month's jiàn from chǒu to zǐ. Sometime in the Warring States period (480 BCE – 222 BCE), the Lu state used the Lu calendar, one of the gǔliùlì (ancient six calendars) used in that period, in which a rule for inserting leap months was developed and the first month was fixed in the zǐ month.

Zhāng's reconstructed Chunqiu calendar is based on a lunar cycle and leap months deduced from the dates recorded in the chronicle Chunqiu. We can only say that it is a model constructed to fit the data. The lunar conjunction is computed by adding integer multiples of the lunar cycle to a constant, which is consistent with the píngshuò rule. However, using one lunar cycle for the entire period of 242 years implies that the rules of calendar did not change over 200+ years, which does not seem to be possible. However, using a simple model to fit the data is reasonable given the lack of information on the calendar evolution at that time. One way to assess the reliability of the reconstructed calendar is to examine how well the model fits the data.

According to Zhāng's description in Section 3.5 in Ancient Chinese Calendars and Almanacs, there are 393 pieces of calendrical dates mentioned in the chronicle Chunqiu. Most of the dates are consistent with the reconstructed calendar, but 45 of them are inconsistent. Of these 45, 11 of them can be resolved if leap months were inserted in the middle of the years. The current version of Chunqiu was passed down over many generations by hand copying the texts over 2000+ years. Mistakes are inevitable. It is also found that there are incompatible texts in Chunqiu. Of course, some discripancies may be explained by typos. However, it is difficult to decide whether a descrpancy is caused by typos or a true discrpancy between the reconstructed calendar and the texts. If we assume that all of the 45 mismatches are true descrpancies, the reconstructed calendar can explain about 89% of the calendrical dates mentioned in Chunqiu. Comparing the two reconstructed calendars between Zhāng and Wáng Tāo listed in Table 3-7 in Ancient Chinese Calendars and Almanacs, we see that sometimes their New Year days can differ by one month. This is caused by their different opinions on whether or not a leap month could appear in the middle of a year. Based on these pieces of information, I speculate it is very likely that there are true discrepancies between Zhāng's reconstructed calendar and the true calendar used at that time, but the amount of deviations may be small. The reconstructed calendar is of great value for historians.

In the example demonstrated above, time is measured from the midnight of the New Year day of N_-721. In modern calendar calculations, time is often measured from noon on January 1, 4713 BCE in the proleptic Julian calendar. This day-counting system is known as the Julian days. Here for convenience, the time origin is assumed to be the local time for the Lu state. The Julian day at midnight on the New Year day of N_-721 is 1457727.5. Therefore, the Julian day of any given time (measured from the midnight of the New Year day of N_-721) can be obtained by adding 1457727.5.

The advantage of using Julian days is that there are standard algorithms for converting Julian days to dates in the Julian/Gregorian calendar. Our sexagenary cycle page also lists formulae for converting Julian days to sexagenary days. However, using Julian days for the computation of the Chunqiu calendar alone is not simpler than the calculation demonstrated above. Dates in the Julian calendar are so regular that we can easily invent a conversion method ourselves without relying on the standard algorithms. On the other hand, Julian days are convenient if we want to compute gǔliùlì (ancient six calendars), and calendars used by the Qin and early Han dynasties in addition to the Chunqiu calendars. Each of these calendars have their own natural time origins. Using Julian days is just to unify the time origin at noon on Jan. 1, -4712. We can then do the calendar date conversion and sexagenary day calculation using a single procedure. This is very useful when writing a computer program for calendar calculations.

Here I introduce a method to calculate the Chunqiu calendar using Julian days. Equation (1) for the lunar conjunction time can be written in terms of Julian days by the following equation.
JD(M_i) = 1457727.5 + 268/1027 + i(29 + 545/1027)     (5)
Let's look at the example of computing the new year day conjunction of N_-649 again. From the table above, we have A(y) = 890. It follows from equation (5) that the Julian day of the conjunction was
JD = 1457727.5 + 268/1027 + 890× (29 + 545/1027) = 1484009.5 + 574/1027.
The Julian day at noon on the New Year day was 1484010 and the xiǎoyú of the conjunction was 574. Using a standard Julian day-Julian calendar conversion algorithm (e.g. algorithm by Richards), Julian day 1484010 corresponds to Dec. 31, -650 in the proleptic Julian calendar. According to the formulae in our sexagenary cycle page, the heavenly stem number of the New Year day was
1 + mod(1484010-1, 10) = 10, which means the heavenly stem of the day was guǐ; the earthly branch number was
1+ mod(1484010+1, 12) = 8 and the earthly branch was wèi. Thus the sexagenary cycle of the New Year day was guǐ wèi. All these results agree with the above calculation.

I use fractions in the above calculation. Calculations are now usually done on a computer and it is more complicated to write a program to handle fractions. Thus, using floating point numbers is easier even though the computational efficiency is slightly reduced. Modern computers are so fast that the slight decrease in efficiency is hardly noticeable. However, there is one thing we should be careful. When the xiǎoyú of a conjunction is exactly 0, the conjunction occurs exactly at midnight. The decimal part of the Julian day is exactly 0.5, but floating point roundoff error may give 0.49999999999 instead, resulting in an off-by-one-day error. This can be prevented by adding 0.0001 to the Julian day. Specifically, equation (1) is modified as follows.
JD(M_i) = 1457727.761054236 + i · 29.53067185978578      (6)
Since 0.0001 = 0.1027/1027, adding it to equation (5) results in the xiǎoyú changing from the range between 0 and 1026 to the range between 0.1027 and 1026.1027. Now the conjunction can never occur at midnight and so the floating point roundoff error will not produce a one-day error. Now that we are dealing with decimals, it is no longer necessary to calculate xiǎoyú, but we can still calculate it if we want:
xiǎoyú (omitting the denominator 1027) = floor(1027(JD + 0.5 - floor(JD + 0.5))     (7)
Here floor(x) denotes the largest integer smaller than or equal to x. JD + 0.5 - floor(JD + 0.5) is xiǎoyú expressed in decimals. Since 0.0001 is added to the Julian day, when the number is multiplied by 1027 it becomes an integer plus 0.1027. The floor() function then removes the decimals and the result is xiǎoyú with the denominator 1027 omitted. As mentioned, we no longer need to care about xiǎoyús when using decimals. The only reason to calculate xiǎoyús is to compare them with the values listed in Table 3-7 in Ancient Chinese Calendars and Almanacs for code validation.

Footnotes

• [fn1] The average number of days in the Julian calendar is 365.25 days, which is 0.0078 days longer than the tropical year (365.2422 days). As a result, the dates of the winter solstice gradually drifted earlier and earlier in the Julian calendar. The average number of days in a Gregorian year is 365.2425 days, much closer to the tropical year, so the dates of the winter solstice have not drifted much between now and the Gregorian calendar reform in 1582. Winter solstice occurs around Dec. 21 today. The Gregorian reform in 1582 skipped 10 days by specifying that the date following Oct 4, 1582 was Oct 15, 1582. Hence the date of the winter solstice before the calendar reform was around Dec. 11. There are 2232 years between 1582 and -650 and the winter solstice drifted by 0.0078×2232 days = 17.4 days. It follows that the winter solstice in -650 was around Dec. 28 in the proleptic Julian calendar.

• [fn2] The New Year days of N_-675 and N_-674 listed in the book are very dubious. If the data were correct, there would be 3 consecutive leap months occurring in N_-677, N_-676 and N_-675, and there would be no leap months in the next 6 years. I have not studied the Chunqiu calendar and cannot say for sure such unusual leap month pattern was impossible, but I find the data in Table 3-7 of Ancient Chinese Calendars and Almanacs to be more reasonable.