Zhu was born in Qinyang, Henan Province to an aristocratic family, the sixth-generation descendant of the Hongxi Emperor, the fourth emperor of the Ming Dynasty. Zhu inherited the title the Prince of Zheng in 1593, but quickly resigned it to his cousin. On the emperor's order, he was granted a new princely title in 1606, the year he delivered a set of ten musicological treatises to the court, establishing his scholarly merit. His posthumous name was 鄭端靖世子 ("His Excellency The Dauphin of Zheng")
Zhu wrote on music theory and temperament (five treatises survive), music history (two treatises survive), dance and dance music (five treatises survive), and several other works. Three music theory works in particular are associated with the ideas of equal temperament, the「律學新說」 (" on the equal temperament ", 1584), 「律呂精義」("A clear explanation of that which concerns the equal temperament", 1595/96), and 「算學新說」(" Reflection on mathematics", 1603). His work has been described as "the crowning achievement of two millenia of acoustical experiment and research (Robinson 1962:224)" though it may be considered more justified to describe him as "one of the most important historians of his nation's music."
Zhu Zaiyu's equal temperament ratio in The Complete Works in Yuelu