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. On 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, the 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, \(\mathbf{I}\), will repeat itself after 240 years.

The first two claims are obviously correct. But the last 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 $$\mathbf{I}_{\rm{1984-03-18}} = (1, 4, 48, I_{\rm hour}) = (甲子, 丁卯, 辛亥, I_{\rm hour})$$ 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	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 originate 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:

$$60(365\lambda_1 + \lambda_2) = 365\lambda_3 + 366\lambda_4$$

to 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 \(\pm \lambda_2 = \lfloor \pm\frac{1}{4}\lambda_1 \rfloor \).