Five tones and 12 pitches
The five tones and 12 pitches are terms in traditional Chinese music, which refer to the ancient Chinese method of tuning. The five tones – gong, shang, jue, zhi and yu – form the basic scales of ancient Chinese music. The term "five tones" first appeared in ancient Chinese literature during the Spring and Autumn period (770-476 B.C.), in works such as Zuo Zhuan (Commentary on the Spring and Autumn Annals), Rites of Zhou, and Guanzi. The ancient Chinese connected the five tones of Chinese scales with the five elements, or wuxing (as shown in the chart), resulting in a unique system of music theory.
The ancient Chinese proposed the 12 pitches tonal system, which used the "Method of dividing into three parts and taking away one part" to divide an octave into 12 different semitone pitches. In 1584, Zhu Zaiyu used a new mathematical method to calculate square roots using an extra-large abacus to derive a critical ratio equal to the 12th root of 2, which was instrumental in the development of the "12-tone equal temperament." Based on these findings, he designed and developed two types of tuning instruments. Zhu's ingenious method of arithmetic calculation of equal temperament was the first in the history of world temperament, which solved a long-standing problem in the field of music.
五音十二律
五音十二律是中國傳統樂律學名詞，是中國古代的定音方法。五音，即宮、商、角、徵、羽，是中國古代音樂的基本音階?！拔逡簟弊钤绯鲎源呵飼r期的典籍，如《左傳》《周禮》《管子》之中。中國古人把“五音”同“五行”相對應（見圖表），形成了獨特的音樂理論體系。
中國古人提出了“十二律”的定音方法，即用三分損益法將一個八度分為十二個不完全相同的半音的一種律制。1584年，朱載堉利用新的數學方法，用特大算盤進行開方計算，研究出十二平均律的關鍵數據——
，并以此為據，設計并制造出兩種定音樂器。朱載堉發現的“十二平均律”在世界律學史上屬首創，解決了音樂領域的千古學術難題。
