The twelve-tone equal temperament is a standard scale tuning in which an octave is divided into 12 equal semitones, with the interval value of any two adjacent semitones being the 12th root of 2. It was first proposed by Zhu Zaiyu (1536-1611),a musician of the Ming Dynasty. Zhu was the first scholar in the world to determine the twelve-tone equal temperament, and he not only created a new historical stage of Chinese temperament, but also set a milestone in the music history of the world. As early as the Zhou Dynasty, the ancient Chinese had discovered the twelve-tone temperament, but the temperament calculated by scholars over the ages in terms of equivalence could not enable modulation, concerning both key and mode changes. However, Zhu realized that the relationship between the pitches of the semitones was in fact an equiproportional relationship, and after clever calculations, he created a new method of dividing the temperament equally with geometric progression, which is exactly the same as the twelve-tone equal temperament of today, and has the advantage that it enables key and mode changes. Zhu also applied the principles of the twelve-tone equal temperament to musical instruments and designed and built some new instruments. The twelve-tone equal temperament is widely used in keyboard instruments all over the world and the piano is an instrument invented and tuned according to this principle.

In olden days, the Yellow Emperor commanded the court musician Lun to create a tuning system for music. Lun traveled westward beyond the Daxia mountains and onward to the north side of Mount Kunlun, where he took bamboo from the valley, selected a hollow bamboo with even thickness of the wall, cut a length from the middle part between two bamboo joints measuring three cun(寸) and nine fen (分), and designated the sound produced while blowing this bamboo pipe as Huangzhong Gong (黄钟宫), and named it Sheshao (舍少). After this, Lun made twelve bamboo pipes in turn which he took to the foot of Mount Kunlun to listen to the sound of the phoenixes, and used these to determine the temperament. The male phoenix’s song was divided into six tones, likewise that of the female phoenix. He then compared these with the sound of Huangzhong Gong, and found them both moderate and harmonious. (Master Lü’s Spring and Autumn Annals)

In general, instead of using the method of finding a pitch by mathematical calculation based on the proportional relationship between the length of the wind or string and the pitch of the sound, the Pythagorean theorem and the extraction of a cubic root are used to calculate the level of the temperament. This way, every eight or six temperament interval could be cycled infinitely, with the last and first temperament intervals being produced in succession, proving that the saying that temperament intervals cannot be cycled is wrong. This is probably a discovery that had never been made in the past two thousand years and more, and is a theory conceived in our own dynasty. (Zhu Zaiyu: On the Relationship Between Temperament, Astronomy and Calendars)