Gong, Shang, Jiao, Zheng and Yu [1] are the names of five different tones in the pentatonic scale in China, which are similar to 1, 2, 3, 5 and 6 in the current notation. That is, the house is equal to 1(Do), the quotient is equal to 2(Re), the angle is equal to 3(Mi), the number is equal to 5(Sol), and the feather is equal to 6(La). The earliest name of "official business education" was discovered in the Spring and Autumn Period more than 2,600 years ago. In Guan Yuan Qu, there is a scientific method to obtain the five tones of Gong Shang, Jiao and Zheng Yin through mathematical operation, which is the famous "three-point gain and loss method" in China's music history.
If we set the house to 8 1, then we can get that the symbol, quotient, feather and angle are all integers according to the five-degree law, and they are 54, 72, 48 and 64 respectively.
If the profit and loss go on, then tone sandhi and sign sandhi are not integers, so the ancients only chose the five quotient angles of the integer part to mark the feathers.
Length/Bigong (8 1) quotient (72) angle (64) sign (54) feather (48)
Gong (81)19/881/643/227/16.
Quotient (72) 8/9 1 9/8 4/3 3/2
Angle (64) 64/818/9132/27 4/3
Sign (54) 2/3 3/4 27/32 1 9/8
Feather (48)16/27 2/3 3/4 9/81
Also, if we set the house to 729, then the changes of house and constellation are integers. See below for the length:
Gong 729
Signs (from the loss of a third of the palace) 486
Quotient (from the collection of three points and one point) 648
Feather (loss of one third from business) 432
Angle (from feather three points to gain one point) 576
Change the palace (from the perspective of losing a third) 384
Change the visa (from changing the palace to one third profit) 5 12
If it is changed to Qing angle, then the length of the palace must be 243*4=972, and the length of Qing angle is still 729. See below for the length:
Green pepper 729
Palace (from the three points of the Qing Dynasty) 972
Signs (from the loss of a third of the palace) 648
Quotient (from the collection of three points and one point) 864
Feather (loss of one third from business) 576
Angle (from feather three points to gain one point) 768
Change the palace (from the perspective of losing one third) 5 12
If leap is added, the length of the palace must be 16 times of 729, that is 1 1664.
See below for the length:
Leap 656 1
Clear angle (from leap three points to benefit one) 8748
Palace (three points from Qing Dynasty) 1 1664
Signs (from the loss of a third of the palace) 7776
Quotient (from the set of three points and one profit) 10368
Feather (one point less than one point in business) 69 12
Angle (from feather three points to benefit one) 92 16
Changing the palace (from the perspective of losing one third) 6 144
Change the visa (from changing the palace to one third profit) 8 192
The tone sandhi quotient, feathering and angle change of the other three are not detailed here.
The phonetic names they converted are c, #C/bD, d, #D/bE, e, f, #F/bG, g, #G/bA, a, #A/bB, b.
In addition, you can use the problem of "chickens and rabbits in the same cage" to explain why the palace will be eight degrees higher if you lose six times:
Assuming that 12 operations are all losses, then 7* 12=84 will eventually get a palace seven octaves higher, so it will lose six times, get six times, which is equivalent to six octaves lower, and finally just get a palace one octave higher;
In addition, it can be assumed that 12 operations are beneficial, so 5* 12=60 will eventually get a palace that is five octaves lower, so it must be damaged six times and benefited six times, which is equivalent to six octaves higher, and finally the same palace that is eight octaves higher can be obtained.
To sum up, only in the order of "profit and loss, profit and loss, profit and loss, profit and loss", can we get all the twelve tones of the same octave, the first six are input first and then repeated twice, and the last six are input first and then repeated twice.
Note: The above gains and losses are limited to the law of twelve averages. If the gains and losses are eliminated according to the five-degree rule, the last house that is one octave higher will last for 24 minutes, not half the length of the original house, but shorter, which will make the last house sound higher. These 24 minutes are the ancient sound difference, and this problem is "the problem that the yellow bell cannot be restored."
Zhu Zaiyu in the Ming Dynasty invented twelve tones, which not only returned to the palace at last, but also facilitated the mode adjustment, because the octave was divided into twelve identical semitones, all of which were 100.
Attachment: The so-called double negative second degree, negative first degree and double negative first degree are not defined in music theory.