Chapter 52 Section 3 Twelve Laws of Equality

Temperament is a science that studies the rules and rules of the ratio of high and low pronunciation of vocal bodies, and is a basic science in music.All dynasties in our country have attached great importance to temperament. As early as the Western Zhou Dynasty, there were "twelve temperaments", "pentatonic scales" and "seven-tone scales".To produce sounds of different frequencies, it is necessary to adjust the length of the pipe or string of the instrument.In ancient my country, strings were generally used for tuning.During the Spring and Autumn Period and the Warring States Period, the "three-point profit and loss method" was used to determine the length of the string.The so-called three-point profit and loss method is based on the length of a string that is customized as the pitch, divide it into three equal parts, and then add one point (benefit one, that is, multiply by 4/3) or subtract one point (loss one , namely multiplied by 2/3) to determine the corresponding chord lengths of other laws.Such a temperament is determined one by one until a tone that is exactly twice as high or half as low as the fundamental tone is obtained.However, the result obtained by the three-point profit and loss method, the tone and the fundamental tone can not be exactly a multiple relationship, but there is a certain error.At the same time, the frequency ratios of adjacent two laws are not completely equal.These defects were not resolved by Zhu Zaiyu until the Ming Dynasty.

Zhu Zaiyu (yuyu) (1536-1610 A.D.), courtesy name Boqin, nicknamed Qushanren, was a descendant of King Zheng Jing, the concubine son of Emperor Renzong of Ming Dynasty.Although he is the son of the royal family, he has devoted himself to the research of astronomy, mathematics and temperament for decades, and has made some achievements. The most important contribution is the creation of the "twelve equal temperament", that is, the use of geometric progression To determine the chord length of each law.After rigorous mathematical calculations, Zhu Zaiyu took the common ratio of each law as