The calendar today is based on the Sun and its annual circuit around the sky against the background of the stars. The fundamental unit is the tropical year, the time from the summer solstice to the next summer solstice. Because one year is not exactly equal to an integral number of days, it is necessary to vary slightly the number of days in a year so that the calendar remains synchronized with the seasons over time. Thus every four years we add one extra day to the year. By contrast, the Chinese calendar was based on both the Sun and the Moon. A lunar-solar calendar was also used by the Babylonians and is fairly natural in any society that makes the month a fundamental unit to mark the passage of time. In ancient China the month was taken to begin on the day of conjunction of the Moon and the Sun. The last day of the month coincided with the last day that the crescent moon was observed in the eastern sky before sunrise. The lunar month as defined in this way actually varies slightly in value because the last visibility of the crescent moon is affected by such factors as the inclination of the ecliptic to the horizon and the latitude of the Moon with respect to the ecliptic. In addition, the length of the year measured as an integral number of days varies, holding to an average value that is known today to equal 365.2422 days.
In the Chinese lunar-solar calendar it was necessary to adjust the length of the month so that the calendar kept in step with the seasons. The fundamental problem was to find a cycle in which an integral multiple of years was equal to an integral multiple of months and to find other cycles that yielded both of these quantities in terms of an integral number of days. Another basic concept of the Chinese calendar was the original time, or epoch, from which all dates were computed. In any given dynastical system of calendrical astronomy the selection of the epoch was determined by political factors, astronomical considerations, and various numerological beliefs.
A fundamental cycle of the Chinese lunar-solar calendar was the equality 19 years = 235 months. The number 19 was regarded as significant because it was equal to 10 + 9, the yin and yang numbers from the I Ching. Another fundamental cycle was the equality 76 years = 940 months = 27,759 days. This is the smallest cycle that gives rise to a whole number of days, months, and years. We have 940 - (76 x 12) = 28, and so it follows that during a 76-year period, there will need to be 28 calendar years in which a 13th month is added to the standard 12-month year.
Up to the second century b.c., the traditional value used for the average length of the year was 365 1/4 days. In 100 b.c. the Han emperor Wu decided to institute a new calendar, or li. The most significant innovation was to change slightly the value adopted for the average length of the month. In the old system a month was equal, on average, to 29 499/940 days, while in the new system it was equal to 29 43/81 days. This change implied a change in the average length of the year, from 365 1/4 days to the value 365 385/1539. This change required various adjustments in the cycles making up the calendrical system.
The change in li under Emperor Wu illustrates how considerations of a numerological sort influenced thinking about the calendar. The basis of the change was a new value for the fractional part of the average month, namely 43/81. The number 81 in the denominator of this fraction was the square of nine, and nine was a special yang number in the mode of thinking that saw the cosmos as balanced between the cosmic forces of yin and yang. Eighty-one was also the volume capacity in conventional units of the standard pitch pipe, an instrument whose notes were believed to resonate with the cycles of the cosmos. The organic unity of the universe was expressed in resonances between the cycles of the calendar and the cycles found elsewhere in the affairs of man and nature.
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