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:
. . . 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:
  1. When considered independently, the periodicity of the year-index, \(I_{\rm year}\), is 60 years, which is obviously correct.
  2. When considered independently, the periodicity of the month-index, \(I_{\rm month}\), is 5 years, which is also correct.
  3. 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.
It appeared in the paragraph quoted above that Hee believes that the interval between two successive sexagenary indices is regularly maintained at 240 years. I believe this assertion is flawed when I first read it and I believe I know enough math to check the validity of this claim. First of all, it is easy to formulate an expression for the interval \(g\) between two successive sexagenary indices. It is irregular and it is given by $$g = 60(365\lambda_1 + \lambda_2)$$ As a numerical example, the sexagenary indices associated with March 18, 1984 are given by \(\mathbf{I}_{\rm 18/Mar/1984} = (1, 4, 48, I_{\rm hour})\) = (甲子, 丁卯, 辛亥, \(I_{\rm hour}\)). Using calendrical algorithm by Reingold and Dershowitz, we can easily compute other dates with similar sexagenary indices, and they are tabulated in the table below.

Gregorian dates with sexagenary indices \((1, 4, 48, I_{\rm hour})\)
\(\lambda_1\) \(\lambda_2\) RDN Gregorian date Period
−66 −16 −722,007 March 18, −1976
−62 −15 −634,347 March 20, −1736
−61 −15 −612,447 March 6, −1676
−59 −14 −568,587 April 5, −1556
−58 −14 −546,687 March 22, −1496
−55 −13 −480,927 April 7, −1316
−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 \(\pm \lambda_2 = \lfloor \pm\frac{1}{4}\lambda_1 \rfloor \).

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