Chapter 10 Section 6 Gongshangjiao Zhengyu and the three-point profit and loss method

The inscription on the chime bells of Zeng Houyi uses Chinese characters to name the various levels of the scale, among which five basic levels are recorded as "Gong", "Shang", "Jiao", "Zheng [zhi Zhi]", and "Yu".Correspondingly, use the modern scale roll name, the palace is "托(do)", the business is "唻(re)", the horn is "喜(mi)", the levy is "抓(sol)", and the feather is "拉(la)". )".These five tones are arranged together in order of height, which is a pentatonic scale commonly used in Chinese music throughout the ages. It is expressed in modern numbered notation, which is 1, 2, 3, 5, and 6.

In the pre-Qin period, the pentatonic scale was the basic scale of all kinds of music, and the pentatonic level was the basic sound level of various musical scales.Sometimes, in order to express special feelings, singers or musicians also add some change levels on the basis of the pentatonic level, thus forming a six-tone or seven-tone scale. In "Warring States Policy" there is a record about Jing Ke's singing of the "Voice of Bianzheng" before assassinating the King of Qin; in Liu Xiang's "New Preface" there is a record about someone singing "Yangchun Baixue" in Yingzhong (now Jiangling, Hubei) when the tune appeared The records of "Introducing merchants and engraving feathers, mixed with Liuzheng" describe the situation of singing changing levels in addition to the basic pentatonic levels.In the pre-Qin literature, the pentatonic level is called "Zhengsheng", and the changing sound level is called "Bian".The most common ways to change the sound level are to add "Bianzheng" between Jiao and Zheng and "Biangong" between Yu and Gong. Bianzheng and Biangong are collectively called "two changes".The second change and the pentatonic level are arranged together to form a seven-tone scale:

The definite calculation of the pentatonic scale in terms of melody was produced in China before the 3rd century BC.The pre-Qin document "Guanzi", which records the words and deeds of the statesman Guan Zhong (? - 645 BC) in the Spring and Autumn Period, specifically and clearly described the calculation method of the five levels of the pentatonic scale in the "Diyuan Pian": "Where the five tones are to be played, the first is to use one and three, and the four are divided to combine ninety-nine...to form a palace. Three parts are benefited by one, and there are eight in a hundred. It is enough to be a living shang. If there are three parts, return it to its place, so it is a feather. If there are three parts, get rid of its multiplication, so it is a raw horn."

This text can be calculated sentence by sentence to get five kinds of strings with different lengths, and when each string is pronounced with one sound, there are five tones: (1) "First master one and three, four openings to combine ninety-nine...to form a palace." The formula is: (1×3)=9×9=81, the relative ratio of the chord length obtained is 81, and it is determined as a palace. sound, the relative pitch is called do. (2) "One-third of the benefit is one, and there are eight out of a hundred, which is a sign." The formula is: 81×4/3=108, that is, adding 1/3 of the front chord length ratio 81 to get the chord length ratio number 108, pronounced as sign, and the relative pitch is called sol.

(3) "It is not without three points and multiplied by it, which is enough to be a living quotient." The formula is: 108×2/3=72, that is, reduce the 1/3 of the previous chord length ratio of 108, and get the chord length ratio of 72 , pronounced as quotient, relative pitch called re. (4) "There are three points and return to its place, so it is the feather." The formula is: 72×4/3=96, that is, add 1/3 of the front chord length ratio of 72, and get the chord length ratio of 96, Pronounced as Yu, the relative pitch is called la. (5) "There are three points and multiplied by it, so that it is a raw angle." The formula is: 96×2/3=64, that is, reduce the 1/3 of the front chord length ratio of 96, and get the chord length ratio of 64, which is pronounced as Angle, relative pitch called mi.

These five formulas and the resulting five tones are arranged in the following table from bass to treble: This calculation method for generating pentatonic scales was called "three-point profit and loss method" in ancient times. "Loss" means decrease; "gain" means increase.After the five tones are produced, if the "loss" and "benefit" continue, all 12 tones may be produced on the strings. "Lu's Spring and Autumn Annals - Yindu Pian" records this kind of production law that produces 12 tones. The "three-point profit and loss method" is the earliest known method of scientifically generating rhythm in ancient China.