On the Periodicity of Chinese Sexagenary Indices
I have written a short note on the periodicity of Chinese sexagenary indices or bazi 八字, \(\mathbf{I} = (I_{\rm year}, I_{\rm month}, I_{\rm day}, I_{\rm hour})\), after reading a book by Hee, an accountant by training, on the same subject, many years ago. I was motivated primarily by a section of her book. In page 22, Hee wrote:
Gregorian dates with sexagenary indices \((1, 4, 48, I_{\rm hour})\)
It is clear from the table that the indices are repeated 60 years after 1984, namely on March 3, 2044. However, after 2044, the next identical indices appear only after 360 years in April 5, 2404 (As a rule of thumb, if \(n\) is large, the number of identical sexagenary indices in \(n\) years can be estimated by \(0.00829n \pm 1\)). Both of them clearly contradicts Hee's assertion of a 240-year cycle. It is not clear to me as to who is the originator of this theory, could this idea come from her mentor: Raymond Lo of Hong Kong?.
Another interesting observation is that some entries in the table share identical month and day with March 18, 1984. (i.e. March 18, 4 and March 18, 3964). They appear approximately 1980 years (or exactly 723,180 days) before and after March 18, 1984. This is not a coincidence.
Suppose \(f\) and \(f + g'\) are two successive fixed day numbers with identical month and day, then it can be shown that the interval \(g'\) is given by $$g' = 365\lambda_3 + 366\lambda_4$$ For \(g\) and \(g'\) to coincide, we need to solve the following Diophantine equation $$60(365\lambda_1 + \lambda_2) = 365\lambda_3 + 366\lambda_4$$ of which one of the solution is \( (\lambda_1, \lambda_2, \lambda_3, \lambda_4) = (33, 8, 1575, 402) \). The following additional constraints apply when solving the Diophantine equation: The ratio of \(\lambda_4 \) to \( \lambda_3 \) must be approximately 1 to 4, and that for large \(\lambda_1\), we have \(\pm \lambda_2 = \lfloor \pm\frac{1}{4}\lambda_1 \rfloor \).
. . . because of the numerous possible combinations, it takes 60 years for the same set of year pillars to repeat itself (by comparison, a set of month pillars repeats itself after just five years). Therefore, if you have a certain day and time, the set of four pillars will repeat itself in 60 years. However, since the same day may not appear in exactly the same month - and even if it is in the same month, the day may not be found in the same half month - it takes 240 years before the identical four pillars appear again. . .Basically, Hee's claims are as follows:
- When considered independently, the periodicity of the year-index, \(I_{\rm year}\), is 60 years, which is obviously correct.
- When considered independently, the periodicity of the month-index, \(I_{\rm month}\), is 5 years, which is also correct.
- In general, the complete set of sexagenary indices, \(\mathbf{I} = (I_{\rm year}, I_{\rm month}, I_{\rm day}, I_{\rm hour})\), will repeat itself after 240 years. This claim is incorrect and will be illustrated by a counterexample.
\(\lambda_1\) | \(\lambda_2\) | RDN | Gregorian date | Period |
−66 | −16 | −722,007 | March 18, −1976 | 夏 |
−62 | −15 | −634,347 | March 20, −1736 | 夏 |
−61 | −15 | −612,447 | March 6, −1676 | 夏 |
−59 | −14 | −568,587 | April 5, −1556 | 商 |
−58 | −14 | −546,687 | March 22, −1496 | 商 |
−55 | −13 | −480,927 | April 7, −1316 | 商 |
−54 | −13 | −459,027 | March 24, −1256 | 商・小辛 |
−53 | −13 | −437,127 | March 9, −1196 | 商・武丁 |
−50 | −12 | −371,367 | March 25, −1016 | 周・周康王 |
−49 | −12 | −349,467 | March 11, −956 | 周・周穆王 |
−48 | −12 | −327,567 | February 26, −896 | 周・周懿王 |
−47 | −11 | −305,607 | April 11, −836 | 周・周共和5年 |
−46 | −11 | −283,707 | March 27, −776 | 周・周幽王5年 |
−45 | −11 | −261,807 | March 12, −716 | 春秋・周桓王3年 |
−42 | −10 | −196,047 | March 29, −536 | 春秋・周景王8年 |
−41 | −10 | −174,147 | March 15, −476 | 春秋・周敬王43年 |
−40 | −10 | −152,247 | February 29, −416 | 戰國・周威烈王9年 |
−38 | −9 | −108,387 | March 31, −296 | 戰國・周赧王18年 |
−34 | −8 | −20,727 | April 2, −56 | 漢・五鳳0年 |
−33 | −8 | 1,173 | March 18, 4 | 漢・元始4年 |
−29 | −7 | 88,833 | March 20, 244 | 三國・正始5年 |
−28 | −7 | 110,733 | March 6, 304 | 晉・永安0年 |
−26 | −6 | 154,593 | April 5, 424 | 南北・景平2年 |
−25 | −6 | 176,493 | March 21, 484 | 南北・永明2年 |
−21 | −5 | 264,153 | March 24, 724 | 唐・開元12年 |
−20 | −5 | 286,053 | March 9, 784 | 唐・興元0年 |
−17 | −4 | 351,813 | March 25, 964 | 宋・乾德2年 |
−14 | −3 | 417,573 | April 11, 1144 | 宋・紹興14年 |
−13 | −3 | 439,473 | March 27, 1204 | 宋・嘉泰4年 |
−12 | −3 | 461,373 | March 12, 1264 | 宋・景定5年 |
−8 | −2 | 549,033 | March 15, 1504 | 明・弘治17年 |
−7 | −2 | 570,933 | February 29, 1564 | 明・嘉慶43年 |
−6 | −1 | 592,893 | April 14, 1624 | 明・天啟4年 |
−5 | −1 | 614,793 | March 30, 1684 | 清・康熙23年 |
−4 | −1 | 636,693 | March 16, 1744 | 清・乾隆9年 |
−1 | 0 | 702,453 | April 2, 1924 | 民初 |
0 | 0 | 724,353 | March 18, 1984 | |
1 | 0 | 746,253 | March 3, 2044 | |
7 | 2 | 877,773 | April 5, 2404 | |
8 | 2 | 899,673 | March 21, 2464 | |
9 | 2 | 921,573 | March 7, 2524 | |
12 | 3 | 987,333 | March 24, 2704 | |
13 | 3 | 1,009,233 | March 9, 2764 | |
14 | 3 | 1,031,133 | February 23, 2824 | |
15 | 4 | 1,053,093 | April 8, 2884 | |
16 | 4 | 1,074,993 | March 25, 2944 | |
20 | 5 | 1,162,653 | March 27, 3184 | |
21 | 5 | 1,184,553 | March 12, 3244 | |
22 | 5 | 1,206,453 | February 27, 3304 | |
28 | 7 | 1,337,973 | March 30, 3664 | |
29 | 7 | 1,359,873 | March 16, 3724 | |
33 | 8 | 1,447,533 | March 18, 3964 |
It is clear from the table that the indices are repeated 60 years after 1984, namely on March 3, 2044. However, after 2044, the next identical indices appear only after 360 years in April 5, 2404 (As a rule of thumb, if \(n\) is large, the number of identical sexagenary indices in \(n\) years can be estimated by \(0.00829n \pm 1\)). Both of them clearly contradicts Hee's assertion of a 240-year cycle. It is not clear to me as to who is the originator of this theory, could this idea come from her mentor: Raymond Lo of Hong Kong?.
Another interesting observation is that some entries in the table share identical month and day with March 18, 1984. (i.e. March 18, 4 and March 18, 3964). They appear approximately 1980 years (or exactly 723,180 days) before and after March 18, 1984. This is not a coincidence.
Suppose \(f\) and \(f + g'\) are two successive fixed day numbers with identical month and day, then it can be shown that the interval \(g'\) is given by $$g' = 365\lambda_3 + 366\lambda_4$$ For \(g\) and \(g'\) to coincide, we need to solve the following Diophantine equation $$60(365\lambda_1 + \lambda_2) = 365\lambda_3 + 366\lambda_4$$ of which one of the solution is \( (\lambda_1, \lambda_2, \lambda_3, \lambda_4) = (33, 8, 1575, 402) \). The following additional constraints apply when solving the Diophantine equation: The ratio of \(\lambda_4 \) to \( \lambda_3 \) must be approximately 1 to 4, and that for large \(\lambda_1\), we have \(\pm \lambda_2 = \lfloor \pm\frac{1}{4}\lambda_1 \rfloor \).