Equal Temperament

    Ancient Chinese had led the world in the studies of music and pitch. The earliest record of such studies was found in Guanzi, a book written during the Warring States period. The studies found that of the five tones - gong, shang, jue, zhi, yu ¨C on the ancient Chinese five-tone scale, the sound-wave frequencies of upper and lower notes formed simple mathematical ratios of 3:2 or 4:3,

    According to the circle-of-fifths principle, after 11 circles the sound-wave frequency of the last tone should be twice that of the first tone. But in reality that is not the case, which gave a great puzzle to ancient scholars who studied music and pitch and urged them to make further exploration. Ancient Chinese researchers had all along devoted their efforts to finding the absolute values of the 12 tones and the accurate ratios of upper and lower tones. Of the chimes of the Spring and Autumn Period unearthed in 1957 in Xinyang of Henan Province and those excavated in Suixian County of Hubei Province in 1978, the pitch and sound-wave frequency of each bell approach those determined in present-day equal temperament. But they were only improvements from the three-section method, and not accurate enough. From the Han, Wei, Jin, Sui and Tang dynasties to the Five Dynasties and the Song Dynasty, the search for accurate sound-wave frequencies continued, but the puzzle was not solved. Then a genius, Zhu Zaiyu, finally appeared to solve this problem.

    Zhu Zaiyu (1536-1611), a noted musician, mathematician and astronomer- calendarist of the Ming Dynasty, was born in Henei County of Huaiqing Fu (now Qinyang City of Henan Province). He was a prince of Zheng, and a grandson of the ninth generation of Zhu Yuanzhang, the first Ming emperor. His father was once persecuted and sent to prison, and the family fell into poverty. Despite his bad fortune, he devoted all of his energy to science, especially to the study of music and pitch. When the emperor decided to exonerate him and resume his royal family status, Zhu declined and continued his studies. By instinct and family training, Zhu developed a musical talent. In 1560 he finished his first work, Scores for the Se (Lute). In 1584 he completed the masterpiece, Luxue Xinshuo (New Explanation of the Theory of Pitch), in which he put forward the theory of equal temperament and method of calculation. This was a brilliant achievement in both Chinese and world history of music. Zhu's discovery came more than one hundred years earlier than his counterparts in the West.

    Zhu Zaiyu not only worked out the theory of equal temperament, but also put it into practice and created the world's first stringed instrument based on this theory - the xian zhun, which produces 12 accurate tones. Applying the extraction of square and cube roots to the study of musical tuning, Zhu discovered the difference of sound- wave frequencies of two neighboring tones to be the number 2 extracted for 12 times with an accuracy of more than 20 decimal places. The equal intervals ensure ideal musical modulation, and the revolutionary discovery has proved to be accurate and scientific. Zhu said, "Doing away with the three-section, the new method enables unending circulation of 12 tones in an orderly way, and this is an unprecedented achievement in the two thousand years of the history of music." Zhu's method has been tested by musicians in other countries, and his theory of equal temperament is now used throughout the world.

    Zhu Zaiyu achieved a great deal in his studies of mathematics, astronomy and music. Apart from musical tuning, his works also cover mathematics and astronomy.