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
五音十二律是中国传统乐律学名词，是中国古代的定音方法。五音，即宫、商、角、徵、羽，是中国古代音乐的基本音阶。“五音”最早出自春秋时期的典籍，如《左传》《周礼》《管子》之中。中国古人把“五音”同“五行”相对应（见图表），形成了独特的音乐理论体系。
中国古人提出了“十二律”的定音方法，即用三分损益法将一个八度分为十二个不完全相同的半音的一种律制。1584年，朱载堉利用新的数学方法，用特大算盘进行开方计算，研究出十二平均律的关键数据——
，并以此为据，设计并制造出两种定音乐器。朱载堉发现的“十二平均律”在世界律学史上属首创，解决了音乐领域的千古学术难题。
