Taienreki (大衍暦: also read as Daienreki) was a lunisolar calendar once used in China and Japan.
It was the calendar which a Chinese monk Yi Xing formulated by an imperial ordinance of Emperor Xuan Zong of Tang. Yi Xing and his group conducted a meridian determination covering the area from the southern Jiao-zhou up to the northern Tie-le, and also a large scale astronomical surveying over the whole China. The calendar had been used in China for 33 years from 729 to 761.
It was an extremely well-formulated calendar, which became a model for the later generations. A solar motion table was developed taking into account the unevenness of the solar motion, and the interpolation method using unevenly-spaced quadratic difference (不等間隔二次差補間法) was used as well. Also a concept of regional time difference was brought into the eclipse calculation (astronomy).
As a Japanese Calendar
KIBI no Makibi (a Japanese scholar and noble during the Nara period) brought it to Japan when he returned from China in 735. However, as there were not many people who were versed in astronomy in Japan in those days (according to the "Shoku Nihongi" [Chronicle of Japan Continued], paragraph of April 22, 730), careful preparations had to be made (according to the "Shoku Nihongi," paragraph of December 28, 757, concerning which "Ruiju Sandai Kyaku" (Assorted Regulations from Three Reigns) describes that an imperial decree was issued on the same day [December 28]) before it was implemented under the FUJIWARA no Nakamaro government.
It had been used in Japan for 98 years from 764 to 861. During the four years from 858, Taienreki was concurrently used with the Gokireki calendar for the preparation for a calendar reform, and the Senmyoreki calendar was introduced in 862.