Chapter 238 The Law of Twelve Equalities

So now the old man plays the piano and writes all day long, and also makes trouble for Su You.

The map of Yizhou Road has finally been completed. This is the first time anyone can see the complete Dujiangyan Irrigation District water conservancy project.

Zhao Pian looked at the grand map and kept asking Su You what the various legends meant.

Su You was very impatient because he was excitedly preparing to educate Old Man Zhao with a copper rod.

It's not to beat him, but because not long ago, the two had a musical disagreement.

Old man Zhao loves to play the piano. In fact, most scholar-bureaucrats and ladies in the Song Dynasty could play the piano, but Su You didn't understand music.

The boy still couldn't learn well, so Old Man Zhao scolded him as stupid as an ox.

Then Su You made irresponsible remarks and told Old Man Zhao that to him, sound is a sound wave, and music is a sound wave composed of multiple sounds superimposed.

There are two most common methods of superposition: one is called harmonic superposition, and the other is called beat superposition.

He also told the old man that each single tone is composed of the superposition of multiple harmonics, and the superposition of multiple single tones forms a beat sound. The single tones and beat tones from different instruments are intertwined to form harmony, and finally a series of

The harmonies make up wonderful music.

Then the old man and the young child started fighting each other again.

Old Man Zhao looked at Su You with disdain, and went from "Guoyu" to "Guanzi Diyuan Pian", and from "Guanzi" to "Lu's Spring and Autumn Rhythm Pian", and also analyzed the relevant content in "Huainanzi".

Crushing, crushing, crushing, finally summed up in three words: You! No! Understand!

This thing is called temperament number, which is the foundation of music. For the ancient Chinese, it was to use mathematical methods to find the five-tone twelve temperament.

This method is called the "three-point profit and loss method". Its basic principle is to use a section of sounding tube with an absolutely uniform diameter as the base number, and the musical scale obtained from this base number is the palace;

Then, divide the sound tube into three sections, discard one section and keep two sections. This is "three thirds lose one". The sound produced by the remaining two-thirds of the length of the sound tube is the pure sound of "Palace".

treble fifth sign;

Divide the levy tube into three equal parts, and add one part, which is four-thirds of the length of the levy tube. This is "three-thirds plus one", and thus the pure fourth-degree bass quotient of the levy tube is produced;

Shang Guan retains two thirds, "three thirds loses one", so Shang's pure fifth treble feather is obtained;

The quill "three-thirds equals one", that is, four-thirds of the length of the quill is the angle tube, and the angle tube emits a pure fourth-degree bass angle of the feather.

In this way, after having the basic sound "Gong", after two "three-thirds loss one" and two "three-thirds gain one", the five musical scales of Gong, Shang, Jiao, Zheng and Yu were generated.

Gong gives birth to Zheng, Zheng gives birth to Shang, Shang gives birth to Yu, and Yu gives birth to horns. Since "five and five are born together", when music experts talk about the five scales, they don't say "Gong, Shang, Jiao, Zheng, Yu".

, but it is said as "Gong, Zheng, Shang, Yu, Jiao".

According to the records of "Guanzi", Guan Zhong actually only gave birth to five sounds: Gong, Shang, Jiao, Zheng and Yu.

"Lu's Spring and Autumn Annals" continued the intergeneration of eleven more times on the basis of Guan Zhong's five tones, which completed the intergeneration of the twelve rhythms, including Huang Zhong, Lin Zhong, Taicu, Nanlu, Guxian, Yingzhong, Ruibin,

Da Lu, Yi Ze, Jia Zhong, Wu She, Zhong Lu.

These twelve laws were further distinguished and divided into six Yang laws and six Yin laws. The six Yang laws were called "Lv" and the six Yin laws were called "Lu".

This is the origin of the saying "Lv Lu Tiao Yang" in "The Thousand Character Classic".

The reason why intervals are important is that the ratio of the lengths of two homogeneous strings is equal to the inverse ratio of the frequency of the sound they emit. If the exact ratio between these lengths is known, the best musical standards can be formulated.

Unfortunately, there is a problem with the three-point profit and loss method. When the relationship reaches the eleventh time, that is, when it reaches the "Qinghuang Bell" after the twelve laws, the "Qinghuang Bell" cannot return to the original starting law, which leads to

This makes it impossible for the Twelve Rhythms to start over and over again, and it is impossible to achieve perfect cycles and modulations.

Because the twelve months are related to the order of the moon, the people who studied this issue in ancient China were generally mathematicians, astronomers, and people with in-depth research on the calendar, and then they studied this music issue.

Waves of geniuses have been studying for thousands of years, and many masters have emerged in the process. Until the Ming Dynasty, the ninth generation grandson of Ming Taizu, the musician prince in the Wanli period, and the sixth generation son of Zheng Fan, was known as the "Sage of Law"

Zhu Zaiyu, who developed the "new method of density", perfectly and accurately calculated the intervals of the uniform scale, which can be taken as the twelfth root of two, which is the twelve equal temperaments that guide modern music.

At the end of the 16th century, the missionary Matteo Ricci brought the Twelve Equal Laws from China to the West, and it was not until the 17th century that it began to be widely circulated on the European continent.

Since then, Chinese civilization has entered a period of decline, making the gap between the theoretical level of Chinese classical music and Western music ever wider.

The West obtained the golden law of twelve equal temperaments, and new harmony theories, natural modes and various related tonality emerged accordingly, and then developed new performance and composition techniques, as well as more complex harmony.

Vocal techniques, counterpoint techniques, etc., and of course the most important thing, staff music, were finally perfected and modern music theory emerged.

Twelve equal temperaments are easy to remember: two single tones with a frequency ratio of one to two, that is, the interval between two strings with a length ratio of two to one, are defined as a "pure octave". This pure octave

Dividing the degree into twelve segments according to the twelve equal temperaments can construct thirteen single tones, a perfect cycle.

As an engineering dog who has a deep knowledge of ancient cold knowledge, Su You often uses this example to educate the students in the guitar club in the university dormitory. Of course, all the students in the guitar club get a blank stare. Are you kidding me? Come on, come on.

I'll give you the guitar, want to listen to some music?

Anyway, at this point, all that's left is an interesting mathematical problem. The method known to Su You is to convert irrational numbers into continued fractions, and then calculate the approximate value of the twelfth root of two by truncating the value.

Accurate to six decimal places.

Su You seriously doubted that the Chinese mathematician and astronomer Zu Chongzhi had known this method for a long time. Because he raised the ratio between the tropical year and the synodic moon of previous calendarists from seven leaps in nineteen years to one hundred and fourteen in three hundred and ninety-one years.

The two ratios of fourteen leaps are completely coincidental with the different cutoff values ​​of continued fractions.

Similarly, pushing the value of π from 22 to 7 of the approximate ratio to 355 to 113 of the density is also a different cutoff value for the continued fraction.

However, Zu Dada’s mathematical works have been published since the Tang Dynasty. The textbooks for the Imperial Calculation Examination are too advanced and no one can understand them.

Then, it was miserably lost. This is the tragedy of using square words to express mathematics.

This is a digression. Su You wrote the calculated values ​​​​of the twelve equal laws in the letter and asked Ke Longli to make thirteen copper rods of different precise lengths according to the lengths written down.

The copper rod just arrived, so we won’t discuss the map today. Old man Zhao, please show your face quickly!

For people in the Song Dynasty, there is no dispute that cannot be resolved by gambling. If there is, then the two...well, it still depends on whether one party loses everything.

This was a big bet. Old Zhao bet on his own white turtle, while Su You bet on a pair of glasses and a set of "Southwestern Picture Chronicles" compiled by Long Changqi.

It's quite shameless, but you have to speak conscientiously. For the big shots in the DPRK and China, the value of White Turtle really cannot be compared with "Southwestern Picture Chronicles".

Everyone thought they were sure to win, and it was a good deal until Old Man Zhao took turns knocking thirteen copper sticks with a small copper stick.

Thirteen copper rods are exactly the same, connected at the bottom to form a beautiful curve.

A sweet voice sounded, but Old Man Zhao's face became increasingly pale.

Knocked again, knocked again, knocked again…

Old Man Zhao finally gave up struggling. The copper rod in his hand was shaking as hard as the beard on his face: "You...how did you do that..."

Su You was very proud: "This is an interesting mathematical problem, and mathematics is my strong point. If you give me the white turtle, I will give you glasses, give you "Southwestern Map", and then tell you this principle.

how are you?"

Old Man Zhao did not hesitate at all: "Deal!"

This is a huge achievement! Not a musical achievement, but a political achievement!

The Song Dynasty tried many times to restore and reconstruct ancient rituals and music, but the biggest problem was to fix the Huang Zhong and the rules.

This is really not an exaggeration. The Song Dynasty gathered the best politicians, historians, musicians, scientists, mathematicians, and astronomers at the time to solve this problem. The dispute over the right rules in Taichang Temple almost brought the court into chaos.

!

Zhao Pian continued to tremble with the small copper stick in his hand. He didn't know whether he should call the kid in front of him as stupid as a donkey or a genius. He said he was as stupid as a donkey because he had no idea of ​​the seriousness of the matter. He said he was a genius.
Chapter completed!