The calendar of the sovereign hexagrams
The 12 hexagrams of the year's ebb and flow
Press play to walk the year month by month and see which line changes at each step.
Out of the 64 hexagrams, the Han tradition set 12 apart as bi gua (辟卦, “sovereign hexagrams”), also called xiao xi gua (消息卦, “of waning and waxing”), and assigned one to each lunar month. The association is attributed to Meng Xi and to the school of Jing Fang, the same one that ordered the eight palaces. Read in month order, the sovereigns draw the breathing of the year: yang enters from below one line per month until it fills the hexagram, and then yin empties it in the same order.
Its antipode is month 5: , its exact dui complement (every line inverted). Half a year apart, 6 lines of difference.
1. A closed Gray cycle. Each month differs from the next by a single line, and the return from month 10 (Kun, all yin) to month 11 (Fu, The Return) too: the sovereigns form a closed walk over the hypercube, a cycle of 12 edges out of the 64 states.
2. The lines change in order. The sequence of mutating lines is 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1: first yang rises filling from the bottom up for 6 months (from Fu to Qian), then yin empties in the same order (from Gou to Kun). It is the triangular wave of the number of yang lines: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0.
3. Opposite months are complements. Month m and month m+6 sit at the antipodes of the clock and are exact dui complements (their XOR is 63): Peace (Tài) in month 1 against Standstill (Pǐ) in month 7 is the classic example. It holds for all 6 pairs.
4. They are exactly the monotone ones. The 12 sovereigns are precisely the hexagrams where all the yang sits below all the yin (or the other way round): the “monotone” hexagrams of the cube. There are 12 of them, and these are they.
height = yang lines · the peak is Qian (month 4) and the trough is Kun (month 10)
the 12 marked on the binary ring 0–63, joined in month order: a star polygon that closes
The 12 sovereigns are not a loose set: they are exactly the first 6 generations of the palace of Qian and the 6 of Kun, and none appears in the other palaces of Jing Fang. It makes sense: filling the yang from the bottom up (the waxing sovereigns) is precisely changing lines 1 to 6 from Qian or from Kun, which is how the first generations of each palace are produced. The qualitative observation is in Yijing Dao; here it is proved and asserted by the suite.
Assigning a sovereign hexagram to each lunar month is documented tradition of the Han dynasty, tied to the calendars of Meng Xi (孟喜) and to the school of Jing Fang (京房), not a theorem. What are theorems, and are verified here and in the Python suite, are the four properties: the closed Gray cycle, the order of the lines 2, 3, 4, 5, 6, 1, the dui symmetry of opposite months and the identity with the 12 monotone hexagrams of the cube. The correspondence with the Gregorian months is approximate: the Chinese lunar year is lunisolar and its months do not coincide with those of the Western calendar.
Shao Yong-Leibniz convention: yang = 1, bottom line = most significant bit, Kun = 0, Qian = 63.
The 12 bi gua satisfy four properties: they form a closed Gray cycle, the changing line advances 2, 3, 4, 5, 6, 1, opposite months are dui complements, and they are exactly the 12 monotone hexagrams of Q6.
verificar_soberanosThe 12 bi gua (xiaoxi gua) were associated with the 12 lunar months in the Han tradition, linked to Meng Xi and to the gua qi system.
The 12 sovereign hexagrams (the monotone ones of Q6) are exactly the first six generations of the palace of Qian and the first six of Kun (six and six; zero in the other six palaces): a bridge between the sovereigns and Jing Fang's palaces.
verificar_soberanosThe qualitative observation is in Yijing Dao; here it is proved and asserted.
The types of claim and the full bibliography (APA) are in Foundations.