The five numbers are one, ten, hundred, thousand, and ten thousand. Ancient texts say: "Only after all things come into being do they have form; only after reproduction do they have quantity." Therefore, those who establish laws believe that numbers begin with the creation of the first unit, with the starting value of the Huangzhong pitch being one. Then, for every hour (time period), it is multiplied by three. After nine hours, by the time of the You hour, it reaches nineteen thousand six hundred eighty-three, completing five levels of magnitude, which serves as the basis for the law. Continuing to multiply by three until the end of the Hai period, a total of twelve hours, it reaches one hundred seventy-seven thousand one hundred forty-seven, completing the count of hours, which serves as the total value of the law. Dividing the total by the base value gives nine inches, which is the length of the Huangzhong pitch.
Thus, numbers arise from laws, and laws are fulfilled by numbers, allowing for the calculation of all things, summarizing and examining various phenomena. When calculating, bamboo strips are used, each measuring two fen in width and three inches in length, with each strip carved with three lines, forming a hexagon composed of 216 strips, symbolizing the Qian hexagram; each strip carved with four lines, forming a square composed of one hundred forty-four strips, symbolizing the Kun hexagram. Both the hexagon and the square have twelve sides, which is the largest number in the universe. Therefore, exploring profound and mysterious matters and tracing the origins of things all involve numbers. One, ten, hundred, thousand, ten thousand—this is the common source of numbers; laws, measurements, weights, balances, calendars, and rates—these are the different uses of numbers.
Thus, for objects with length, measuring with a ruler will not miss a fraction; for quantities, using containers will not miss a drop; for weights, using scales will not miss a bit; for sound clarity, using musical tuning will ensure the correct pitch; for the movements of the sun, moon, and stars, using calendars will not miss a second; for various interconnected matters, managing with ratios will not violate their fundamental laws. Therefore, hidden circumstances and subtle changes can all be quantified with numbers.
The so-called ratios consist of nine types: one is square field ratio, used to calculate the area and boundaries of fields; two is grain ratio, used to calculate the trading and exchange of grains; three is tax classification, used to categorize taxes of different levels; four is area and volume ratio, used to calculate area and volume; five is work efficiency ratio, used to calculate the progress and actual completion of projects; six is uniform distribution, used to calculate the transportation costs of different regions; seven is surplus proportion, used to calculate complex interrelated values; eight is equation, used to calculate complex positive and negative number problems; nine is sentence division, used to calculate the distance of height, depth, width, and length. All of these are calculated using multiplication for dispersion, division for concentration, and equivalent methods for connection; now there is a method to connect them. Therefore, the calculation method is all here.
It is said that in ancient times, the value of pi was roughly estimated as π=3, with the diameter of the circle being 1, so the circumference was 3; this method is quite unreliable! Starting from Liu Xin, Zhang Heng, Liu Hui, Wang Fan, and Pi Yanzong, these scholars gradually improved the algorithms and calculated different values of pi, but they have not been able to unify the standard.
It wasn't until the end of the Song Dynasty that there was a genius named Zu Chongzhi, who served as a historian in Southern Xuzhou and invented a super powerful calculation method. He set the diameter to one hundred million and calculated the precise value range of the circumference of a circle: the upper bound is three zhang one chi four fen five li nine hao two seconds seven hou, the lower bound is three zhang one chi four fen five li nine hao two seconds six hou; the true value of pi lies between these two bounds. He also determined two more convenient approximations: the dense rate, with a diameter of one hundred and thirteen and a circumference of three hundred and fifty-five; the approximate rate, with a diameter of seven and a circumference of twenty-two. Impressive, isn't it? He also studied high-level mathematical methods such as opening difference powers and opening difference bases, combined with the calculation of a perfect circle; his calculations were highly accurate, making him one of the leading figures in the field of mathematics at that time.
Zu Chongzhi wrote these achievements into a book called "Zhuishu." Unfortunately, at that time, the scholars could not understand the profound theories in his book; they thought it was too difficult, so this book was left on the shelf, and no one studied it, which is truly unfortunate!
It is said that the Yellow Emperor had Ling Lun cut bamboo, and the bamboo grew to three inches and nine tenths of an inch. When blown, it produced a sound known as the Huangzhong scale, referred to as "Han Shao." Then, twelve bamboo pipes were crafted to capture the calls of the phoenix, distinguishing the twelve pitches and the sounds of male and female phoenixes, thus determining the musical scales. The pitch of the sounds is mutually generated, starting with Huangzhong. The "Book of Songs" states: "When the leaves are in season, the moon is full, and the sun is at its zenith, the same scales are used to measure weights and measures." After his appointment, Yu the Great also employed sound to establish the scales and the body to determine measures. In the "Rites of Zhou," the crafting of musical instruments adheres to the twelve pitches as standards. Sima Qian noted in the "Book of Music": "Huangzhong is eight inches and seven tenths long, Taicu is seven inches and seven tenths and two hundredths long, Linzhong is five inches and seven tenths and three hundredths long, and Yingzhong is four inches and three tenths and two hundredths long." These three tones are the three starting tones of musical law and the foundation of the twelve pitches. Ban Gu and Sima Biao wrote in the "Record of Music": "Huangzhong is nine inches long and produces the most resonant sound; Taicu is eight inches long; Linzhong is six inches long; Yingzhong is four inches, seven tenths, and a little over four hundredths long, and has the clearest sound." Although the measurements show slight variations in Zheng Xuan's annotations in "Rites: Monthly Ordinances," Cai Yong's "Monthly Ordinance Commentary," and discussions by Du Kui, Xun Xu, and others, the measurements of the twelve pitches are largely consistent. The "History of Han" records that Jing Fang proposed a method of generating sounds at intervals of eight, starting from Huangzhong and ending at Zhonglv, thus completing the cycle of the twelve pitches. Zhonglv generates Huangzhong upward, which measures less than nine inches, referred to as "Zhi Shi," while downward it generates the sounds of decay. The pitches are mutually generated, ultimately reaching Nanlv, which introduces forty-eight additional pitches, bringing the total to sixty. Its operational pattern aligns with the movements of the stars, generating a sound that encompasses all things, emerging nine pitches later after the winter solstice. The division of sounds and the delayed tones gradually diminish to the clarity of Yingzhong.
During the Song and Yuan Dynasties, Tai Shi Qian Lezhi further developed the basis of the Jingfang Nanlu, creating three hundred musical pitches, eventually reaching Anyun, which measures just over four inches and four tenths. Originally, there were three hundred and sixty musical pitches. Each day corresponds to a bamboo tube used for sound, arranged in order of pitches such as Gong, Zhi, Xuan, and Yun. He Chengtian stated in his work "Legislation and Discussion": "The pitch of the sound is generated by each other, with a one-third gain or loss, which is the ancient simple method. Just as the ancient calendar divided the year into three hundred sixty-five days and four quarters, later generations modified the calendar, resulting in differences. Jingfang failed to grasp this concept and mistakenly established sixty musical pitches." He Chengtian then established a new pitch system, from which Huangzhong can be derived once more, with twelve Gong tones cycling, preserving both sound and rhythm. Huangzhong measures nine inches, Taicu measures eight inches and two fen, Linzhong measures six inches and one fen, and Yingzhong measures just over four inches and seven tenths. The gain generated from the middle Lu amounts to one hundred seventy-seven thousand one hundred and forty-seven, which corresponds to the number of twelve Chen. In the early period of the Liang Dynasty, there were no significant changes as the systems of the Jin, Song, and Qi Dynasties were adhered to. Later, Emperor Wu authored the book "Zhong Lu Wei," in which he discussed the advantages and disadvantages of the previous dynasty's system. The book likely stated the following:
According to the rules of the Lu system, Jing, Ma, Zheng, Cai, all the way to Rui Bin, are derived from the Da Lu; but Ban Gu, in the "Records of Rhythms and Calendars," after mentioning Rui Bin, proceeds to derive in reverse order. If following Ban Gu's explanation, the length of the Jia Zhong would be just over three inches and seven tenths. If the Lu pipes are too short, the sound of the Jia Zhong would become monotonous, the sound of the Zhong Lu would become even more discordant, and it would simply be out of tune. In mid-spring and early summer, when all things are growing, the climate is mild and gentle, which cannot accommodate overly short sounds. Considering both the sound and the actual conditions, Ban Gu's explanation is flawed. Zheng Xuan further explains the order of the Lu system using the Yin and Yang Hexagrams. If Yin and Yang are mutually derived, and only Yang Qi rises, how then do we explain the descent of Yin Qi? From the divination numbers, Qian Gua corresponds to Jia and Ren, deducing to the left; Kun Gua corresponds to Yi and Gui, deducing to the right, hence the reason for the rise and fall of Yin and Yang. The mutual derivation of Yin and Yang is fundamental, while the rise and fall of the six positions serve as metaphors and numerical representations. Zheng Xuan uses metaphors and numbers to explain the essence, giving it a veneer of logic, but the reasoning lacks conviction. He mentions the mutual generation of nine and six, but completely fails to explain how the twelve Lu Qi flow with each other, revealing his lack of thorough consideration, which is quite evident. Jing Fang's sixty Lu, calculated according to his method, are all accurate. However, the results derived from the Lu system are inconsistent; some yield five Lu while others yield six, marking the first inconsistency. Moreover, first deriving upwards, then inwards, then upwards once more, and finally dividing, presents the second inconsistency. Jing Fang is very knowledgeable in the principles of Yin and Yang; he surely has his reasons. If it's not due to deeply profound reasons that are difficult to grasp, then it is likely that later generations have not adequately studied his theories.
After careful comparison, I couldn't tell which one was correct. I used my spare time to try to deduce the principles, referenced ancient musical instruments and ancient jade bell tunings, and crafted a new ruler, precisely accurate. I made four musical instruments called "Tong." The string length of these four musical instruments is nine feet, with a height of one inch and two tenths at Mount Yue. The string length of the Huangzhong is nine feet, made of two hundred and seventy silk threads, and the number of strings and string lengths of the other twelve tunings are calculated according to the method of three-part gain and loss. I named them "Tong" according to their corresponding months, five elements, emperors, the beginning and end of musical notes, and the sequence of each tuning. The "Tong" musical instrument uses three strings to calculate lunar phases, and the results are all accurate. Using it to correct the ancient jade bell tunings is completely accurate.
I had people make twelve flutes to play various tones. These twelve flutes, played in the palace mode, have accurate and deviation-free tones. In the "Records of Shanqian," it is written: "The three bells in front of the palace were all cast by King Jing of Zhou and are called Wushe bells." I had the musicians play with the newly made Wushe-tuned flutes, but the tones were off. When played with the Yize-tuned flutes, it harmonized. We also checked the inscription on the bell outside the Duomen, confirming that it was tuned in Yize. The bell in the West Wing was moved to the east during the Tianjian era. When played with modern flutes, it is in the Nanlü mode. Upon checking its inscription, it was originally tuned to Taicu mode, which is two tones lower than modern flutes. I instructed Sisuanda, the Chief of the Imperial Music Bureau, to inspect it closely, and indeed there were marks of chiseling on both the inside and outside of the bell.
We asked around and found out that during the Taishi period of the Song Dynasty, Zhang Yong had chiseled this bell, removing a lot of copper, resulting in a lower tone. By studying the tuning of the bell, one can understand the reason behind it. After Emperor Wu of the Song Dynasty pacified the Central Plains, General Chen Qing presented three bells, one large and two small. The two bells in front of the Taiji Hall and the one outside the main gate are the ones mentioned. The inscription on the Xi bell reads "Qing Temple Bell," since there was no Qing Temple during the Qin Dynasty, indicating it must have been from the Zhou Dynasty. The inscription on another bell reads "Taicu Bell Zhi," which refers to the usage in the Lin Zhong Palace. Jing Fang developed the musical theory, seemingly based on this. There is no record of the Qin and Han dynasties in the inscriptions, only mentioning Yi Ze and Taicu, suggesting it does not date back to the Qin and Han periods. The ancients valued social hierarchy, so the names of servants are engraved on the bell, which further supports the idea that it is not a recent creation. Moreover, the accuracy of the musical scale is crucial to state affairs; the Five Tones and Six Musical Notes must be correct. Craftsmen are responsible for the musical scale, and Confucian scholars are responsible for the classics; over the years, these two areas have become disconnected and are no longer verifiable. Even if all the musical instruments are present, they cannot be utilized. The praises of the Zhou Dynasty and the music of the Han Dynasty celebrated their respective achievements; how can they be used on later emperors and kings, abusing their reputation? I will now share my thoughts in detail and order all officials to collaborate on this study, striving to find the correct answer.
Before any reforms to the music system could take place, the chaos of Hou Jing occurred. The system of the Chen Dynasty was also not reformed.
In the first year of the Western Wei Emperor Yuan, Emperor Zhou Wen took over the regency. He ordered the Shangshu Su Chao to carefully correct the musical scale. Su Chao at that time obtained a ruler from the Song Dynasty to determine the dimensions of various pipes. Just as he began, he encountered the abdication of Emperor Min, with political affairs now in the hands of the Prime Minister, and then the invasion of the Qi Kingdom, which ultimately remained unfinished. Later, when excavating the Tai Cang, ancient jade measuring vessels were discovered, and based on these, music scales and weighing instruments were made, but much of this knowledge was eventually lost.
During the early years of the Kaihuang era under Emperor Wen of the Sui Dynasty, the emperor ordered the Grand Minister of Rites, Niu Hong, to study and establish musical scales. Niu Hong widely solicited scholars to discuss methods for formulating these scales, but they were still unable to reach a conclusion. Later, Niu Hong happened to obtain twelve tuning pipes from the Chen Dynasty in the Pingjiang region (now Suzhou, Jiangsu), and he handed these pipes over to the emperor. The emperor then dispatched music experts, such as Mao Shuang, the magistrate of Shanyang County in Chenzhou, Cai Ziyuan, the Minister of Music, and Yu Puming, to create the "Lüpu" according to the solar terms. At that time, Mao Shuang was elderly and went to meet Emperor Wen of the Sui in plain clothes. The emperor appointed him as the governor of Huaizhou, but he declined the official position. Therefore, the emperor sent Zuxiaosun, a music official, to learn the principles of music from Mao Shuang. Niu Hong played the tuning pipes and established the musical scales.
Once the world was unified, musical instruments from various dynasties were gathered at the Music Bureau, and music scholars examined these instruments to set the standards for bell tones. New instruments were crafted to perform the fourteen pieces of "Huangxia." Emperor Wen of the Sui and the ministers in court listened together, and the emperor commented, "The sound of this music is grand, harmonious, and elegant, making listeners feel relaxed and invigorated."
However, nothing can come into being, take shape, or fade away without the Five Phases. Therefore, if the fire ruler is used to determine the Five Sounds, things will tend towards fire; if the metal ruler is used to determine the Five Sounds, there will be more military disasters; if the wood ruler is used to determine the Five Sounds, there will be more funerals; if the earth ruler is used to determine the Five Sounds, the world will fall into chaos; only by using the water ruler to determine the Five Sounds can the music be harmonious, and the world be peaceful. The Wei Dynasty, Northern Zhou Dynasty, and Northern Qi Dynasty were all greedy for the length of cloth, so they used the earth ruler to determine the musical tones. The music is now determined by the water ruler. The rulers in the Jiangdong region are shorter than the earth ruler but longer than the water ruler. The common people do not understand this. They see a ruler made of jade and call it a jade ruler; they see a ruler made of iron and call it an iron ruler. The emperor ordered the use of the "water ruler" to set the music and had all the gold and stone musical instruments from previous dynasties melted down to silence people's criticism.
In the fourth year of Ren Shou, Liu Zha wrote a letter to the Crown Prince discussing Zhang Zhou Xuan's calendar and also mentioned the tuning of musical instruments. His main point was that the essence of music lies in its pitch, which must be determined by standardized tuning tubes; if the pitch isn't standardized, achieving harmony is impossible. The standard tuning tubes should be based on the length of the Huang Zhong. However, previous calculation methods were inaccurate, and the length of the Huang Zhong had always been miscalculated. During the Han Dynasty, Jing Fang made a mess of it, resulting in sixty pitches; in the Song Dynasty, Qian Lezhi produced three hundred sixty pitches. Upon reviewing ancient texts, there was no mention of these claims. Forcing changes could lead to problems. This issue involves not just length errors, but also discrepancies in the thickness of the pipes compared to the calculations. Furthermore, ancient regulations regarding dimensions are also difficult to verify clearly. As a result, musical instruments can't be tuned properly, leading to chaotic measurements. Liu Zha recalibrated these data, hoping to resolve these issues. He calculated the actual length of the Huang Zhong pipe to be sixty-three, and then each tuning tube was three-tenths shorter than the previous one, with seven as one inch. According to this calculation, the Huang Zhong measures nine inches, the Tai Cu is eight inches and one-tenth four-hundredths, the Lin Zhong is six inches, and the Ying Zhong is four inches, two-tenths, eight-tenths, and four-hundredths. That year, Emperor Gaozu passed away, and Emperor Yang ascended the throne but he didn't have time to revise, leaving the matter unresolved, and Liu Zha's works were also lost.
In the second year of Daye, Emperor Yang of Sui ordered the adoption of the tuning tubes from the Liang Dynasty to remake bells, chimes, and other musical instruments, which were more in line with ancient standards than those of previous dynasties. However, the related systems and documents, along with the old tuning of Mao Shuang, were lost in the chaos of war in Jiangdu.
The "Book of Han" states: "The circumference of the Huang Zhong pipe is nine inches, the Lin Zhong is six-tenths, and the Tai Cu is eight-tenths." The "Book of Later Han" and Zheng Xuan both state: "The hollow tubes of the twelve pitches all have a diameter of three-tenths and a circumference of nine inches." Wang Anfeng of the Northern Wei Dynasty, following Ban Gu's "Book of Han," reported that the circumference of the hollow tube of the Lin Zhong is six-tenths and that of the Tai Cu is eight-tenths, but when these tuning tubes were used to play, the sound did not match the pitches of Huang Zhong, Shang, and Zhi. Only by changing the circumference of all hollow tubes to nine inches can they correspond with standard musical instruments.
After the Chen dynasty was pacified in the ninth year of the Kaihuang era, Niu Hong, Xin Yanzhi, Zheng Yi, He Tuo, and their contemporaries, based on the dimensions of ancient law instruments and the actual situation at that time, crafted the Huangzhong pipes, with a diameter of three fen (approximately 1.0 cm) and a length of nine cun (approximately 30.0 cm). Due to material loss during production, the pitch of the sound may vary; because the thickness and length of the pipes do not precisely conform to the standard dimensions, the number of grains of millet they can accommodate also varies. Here are the recorded measurements:
Before the Jin dynasty, the Huangzhong pipe could accommodate 888 grains of millet.
In the Liang dynasty, 828 grains of millet could be held by the Huangzhong pipe.
In the Liang dynasty, three variations of the Huangzhong pipes existed: one could hold 925 grains of millet, another could hold 910 grains of millet, and a third could hold 1,120 grains of millet.
According to the official measurements from the Han dynasty, the Huangzhong pipe could hold 939 grains of millet.
An antique silver piece inscribed with "Huangzhong" has a volume of 1,200 grains of millet.
In the Song dynasty, iron rulers were used for measurement, and there were two versions of the Huangzhong pipe's volume: one was 1,200 grains of millet, and the other was 1,047 grains of millet.
Going back to the early Northern Wei dynasty, the volume of the Huangzhong pipe measured with their ruler was 1,115 grains of millet.
During the Northern Zhou period, measured with a jade ruler, the volume of the Huangzhong pipe increased to 1,267 grains of millet.
In the later Northern Wei period, the volume of the Huangzhong pipe measured with a specific ruler rose to 1,555 grains of millet.
In the later Northern Wei period, the volume of the Huangzhong pipe measured with another ruler further increased to 1,819 grains of millet.
During the Eastern Wei dynasty, the volume of the Huangzhong pipe measured with their ruler reached an impressive 2,869 grains of millet.
Wan Baochang's water ruler indicated a volume of 1,320 grains of millet for the Huangzhong pipe.
Although the table ruler and iron ruler of the Liang dynasty had the same length and diameter, the volume of the Huangzhong pipe was different, some more and some less. This is because the artisans who crafted these rulers intentionally created uneven surfaces, leading to variations in volume.
Xin Dufang, a military officer in charge of the Tiancao under Emperor Shenwu of Northern Qi, was a very intelligent person. He could use a tube to measure temperature and predict the weather by looking at the clouds in the sky. Once, pointing to the sky, he said to someone, "The season of Meng Chun has begun!" When others checked the tube he used, they found that the ash in the tube had already shown a reaction. Every month, his weather predictions were remarkably accurate. He also made twenty-four wheel fans that he buried underground to measure the twenty-four solar terms. When a solar term arrived, only one fan would automatically rotate, while the others remained still, perfectly aligned with the ash's reaction in the tube, as if it had been predetermined, which was quite magical.
After defeating the Chen Kingdom in the ninth year of the Kaihuang reign, Emperor Yang Jian sent Mao Shuang, Cai Ziyuan, and Yu Puming to observe the solar terms. According to ancient methods, twelve wooden tables were placed in a room sealed in three layers. Each table corresponded to a solar term, with the corresponding tuning fork placed on each, then buried in the ground with a flat surface, filled with reed ash inside the table, and covered with a light silk over the opening of the tuning fork. When a solar term arrived each month, if it matched the frequency of the tuning fork, the reed ash would fly up and break through the silk, scattering outside. However, the ash's behavior varied in timing and amount. Some months, the solar term would be confirmed right at the start; in some cases, it might not react until the middle or late part of the month; in some cases, the ash would fly out and be finished in three to five days; in some months, only a small amount of ash would be released throughout the month.
Yang Jian found this quite strange and asked Niu Hong what this meant. Niu Hong replied, "If half of the ashes fly out, it indicates a mild season; if all of the ashes fly out, it indicates a strong season; if the ashes do not fly out, it indicates a weak season. A mild season corresponds to the stability of the nation's political climate, a strong season corresponds to the unruliness of the courtiers, and a weak season corresponds to the king's tyranny." Yang Jian countered, "If the courtiers are unruly and the king is tyrannical, the political climate is surely unstable, and this has nothing to do with the month. The reactions of the twelve months in a year are all different, so how can we say that a tyrant and unruly courtiers must be like this?" Niu Hong could not respond. So Yang Jian asked Mao Shuang and others to come up with a set of methods. Mao Shuang reviewed historical records and wrote an article called "Lu Pu." The article stated: Minister Mao Shuang verified that the Yellow Emperor sent Ling Lun to collect bamboo in the Yanshan Valley and created twelve musical tones beneath the phoenix's nest, marking the origin of numerology. The Yang tube represents the scale, while the Yin tube represents the pitch. Their tones can help predict the four seasons, and their numbers can be used to record all things. Yun Li was the first to create numerology, which likely serves as the foundation of the scale. Numbers such as one, ten, hundred, thousand, ten thousand, billion, trillion are derived from it, and measurements are also derived from it. Therefore, Yu Shi used the scale to coordinate sound, and Zou Yan modified it to determine the five elements. The colors of the calendar and clothing are also classified this way. The Xia Dynasty prioritized people, the Shang Dynasty focused on the earth, and the Zhou Dynasty focused on the heavens. Confucius said, "I have mastered the Xia Dynasty's calendar," implying that he had grasped the essence of numerology.
At the beginning of the Han Dynasty, Zhang Cang formulated laws and advocated the theory of "Five Victories," believing that water was the main element. In fact, this was due to government negligence during the Warring States period and the Qin Dynasty's destruction of academia. The principles of music and tuning gradually declined, and Zhang Cang only made some repairs without delving into deeper research. It wasn't until the time of Emperor Wu of Han that a specialized official position for music and tuning was established, appointing Li Yannian as the commander. Although Li Yannian was able to create some new tunes, he did not fully grasp the fundamental principles of music and tuning, so the sizes and tunings of musical instruments at that time were not established.
During the time of Emperor Yuan of Han, the emperor himself was knowledgeable in music and tuning, and the court musician Jing Fang was also proficient in music and tuning. Emperor Yuan had Wei Xuancheng and others seek advice from Jing Fang. Jing Fang explained his theory, stating, "I studied the theory of Jiao Yanshou and applied the principles of sixty pitches interacting and complementing each other. The upper notes give birth to the lower notes, all in groups of three and two; the lower notes give birth to the upper notes, all in groups of three and four. Yang notes give birth to Yin notes downwards, Yin notes give birth to Yang notes upwards, and eventually return to the fundamental notes of the palace mode; this is the law." Later, Liu Xin compiled this information into a memorial, detailing the development of music and tuning, gradually perfecting the theory of music and tuning. Ban Gu's "Book of Han, Records of Music and Calendars" mainly referenced Liu Xin's work, while Sima Biao's "Continuation of the Book of Han, Records of Music and Calendars" mainly referenced Jing Fang's theory.
During the Eastern Han Dynasty, the standards of music and tuning underwent slight changes. During the Wei Dynasty, Du Kui also established the Lu Lu, used to predict the weather, and it was said that even dust wouldn’t stir. During the Jin Dynasty, the Grand Master of the Palace, Xun Xu, obtained some ancient bronze pipes, compared them with the standards set by Du Kui, and found that the ancient bronze pipes were a quarter longer than those set by Du Kui, realizing that Du Kui's standards were flawed, and the issue was with his own methods. Therefore, drawing on the records from the "Rites of Zhou," Xun Xu recreated the ancient ruler to measure the size of the musical pipes, so that the musical instruments' music and tuning were accurate and flawless.
After the Jin Dynasty, musical theory gradually began to diverge. By the time of Emperor Wu of the Liang Dynasty, jade pipes were excavated from the Jizhong tombs. During the Cangwu period of the Southern Dynasties, someone transformed the jade pipes into horizontal wind instruments, though the key dimensions, such as length and thickness, of these instruments were still maintained. I previously studied at Qicheng, where I learned arithmetic from Zu Cheng and musical theory from He Chengtian, dedicating over thirty years to research, which gave me a solid understanding of musical theory. Later, I became the Minister of Music, overseeing music-related affairs, and I retrieved that jade pipe and the ruler used by the Grand Historian of the Southern Song Dynasty to present before the court. The court ordered craftsmen to make musical pipes according to the original design. Since then, the instruments could remain free of dust. During the Hou Jing rebellion, my brother Xi acquired these instruments from the Taiyue Office. Later, when Emperor Xuan of Chen was held hostage in Jingzhou, coinciding with Emperor Yuan of Liang's defeat, my brother died in the Northern Zhou. He was preparing to make a presentation to the court when Emperor Wu of Chen ascended the throne, deriving sixty musical notes from the twelve pipes, privately studying the patterns of climate change, and made several discoveries. By the time of the Tai Jian era of Chen, my brother became the Minister of Personnel and wanted to present to the court. Just then, Emperor Xuan passed away, and the later ruler ascended, appointing my brother as the Interior Minister in Yongjia, leaving these instruments at home to be passed down to his descendants. When the Chen Dynasty fell, these instruments were ultimately lost.
The twelve musical pipes used in the court are crafted according to the principles of yin and yang changes throughout the four seasons. In winter, the yang energy begins to grow, starting with the Huangzhong pitch pipe, and by summer, the yin energy begins to grow, ultimately ending with the Zhonglü pitch pipe, thus the energy of the entire year is encapsulated. "Yang descends to give birth to yin, starting with Huangzhong; yin ascends to give birth to yang, culminating with Zhonglü, and thus the energy of the year is completed here."
Starting from "Zhong Lu", we move up to "Zhi Shi", then down to "Qu Mie", and finally conclude with "Nan Shi." This way, all sixty solar terms are encompassed. In the eleventh month (Zhong Dong), the musical pitch used is Huang Zhong. Huang Zhong signifies the onset of Dongzhi, marking the initial emergence of yang energy. It measures nine inches, in accordance with heavenly principles. In the eleventh month, the Huang Zhong pitch should be used, as it can harmonize the six energies and perfect the nine virtues. "Huang Zhong, beginning at Dongzhi, the start of yang. When the energy arrives in the eleventh month, the Huang Zhong pitch should be used to nourish the six energies and harmonize the nine virtues."
From that time onwards, everyone used the musical pitches established by Jing Fang, with the tones of Gong and Zhi alternating in length each day. These twelve musical pitches each govern a certain range, which, when extrapolated, results in sixty. This can also be derived from the Bagua, turning into sixty-four. The principles of generation and restraint govern their transformations. For example, the Huang Zhong pitch, when deduced downwards, becomes Lin Zhong, representing the change from yang to yin. "What generates also transforms. Starting with the Huang Zhong pitch, Lin Zhong emerges below, with yang giving rise to yin, hence the change."
Complementary musical pitches influence one another; for instance, the Zhong Lu pitch affects the others, like a mother influencing her child. "That which combines also communicates. Like the Zhong Lu pitch, combining with the physical response, with the mother controlling the child." Therefore, these changing pitches will have different effects at different times, while those influencing each other will come into play successively in the same month. If there are timing inconsistencies, it indicates not the normal operation of the pitches, but rather the interaction of sub-pitches influenced by the mother pitch. "When there are early and late occurrences, it is not the normal energy of the correct pitch, but the interaction of sub-pitches, relying on the mother pitch for guidance."
During the Dali era, while Emperor Yang of Sui was in Jiangdu, all these musical pitches were lost. These pitches were lost in Jiangdu at the end of the Dali era.
Qian Lezhi of the Song Dynasty, after studying the musical theories of Jing Fang, created three hundred musical scales. Shen Zhongzhong, a scholar of the Liang Dynasty, said in his "Discussion on Laws": "The Book of Changes uses three hundred and sixty to represent the number of days in a year, which is the number of musical scales. The Huainanzi states: 'One scale produces five sounds, twelve scales produce sixty sounds, multiplied by six, you get three hundred and sixty sounds, corresponding to the number of days in a year. The number of scales reflects the laws of the movement of heaven and earth.' So these three hundred and sixty scales have been calculated in this way since ancient times." Shen Zhongzhong used the algorithm from the Huainanzi, applied Jing Fang's method of calculation, and obtained three hundred and sixty scales. Each month is based on a main scale, serving as a part. Using the number of scales in this part as the numerator, and the number of days contained in a solar term as the denominator, the numerator determines the denominator, and each scale corresponds to different days based on the number of days. Using it to distribute the seven tones, the scale corresponding to the winter solstice is as follows: Huangzhong as the tonic, Taicu as the supertonic, Linzhong as the mediant, Nanlv as the subdominant, Guxi as the dominant, Yingzhong as the submediant, Ruibin as the leading tone. This results in a harmonious arrangement of five sounds and seven tones. The scales corresponding to the following days are calculated accordingly. The scale corresponding to the day is considered the tonic, followed by the supertonic and other tones. Using it to examine sounds and airs, distinguish time order, and all things should follow the changes of solar terms. From Huangzhong to Zhuangjin, a total of one hundred and fifty scales were calculated using the method of subtracting one third; from Yixing to Yizhao, a total of two hundred and nine scales were calculated using the method of adding one third; only Anyun is the final scale and is not further calculated. The calculation of these scales is based on the actual length of Huangzhong, which is 177,147, using a ratio of three to nine, dividing by this basic length to obtain inches, minutes, and seconds, with the rest being discarded. This gives the length of each scale. Arranging these scales is done by ordering the tonic, mediant, and other sounds in the order of generation. Now I will briefly write about the Huangzhong section:
Huangzhong: the fundamental tone
Baoyu: the nurturing tone
Hanwei: the subtle tone
Dide: the tone of virtue
Guangyun: the tone of broad movement
Xiaji: the tone of lower harmony
Kezhong: the tone of overcoming the end
Zhishishi: the tone of holding the beginning
Woji: the tone of grasping the pivot
Tongsheng: the tone of connecting the sacred
Qiansheng: the tone of potential ascent
Yin Pu, Jing Sheng, Zi Meng, Guang Bei, Xian Heng, Nai Wen, Nai Sheng, Wei Yang, Fen Dong, Sheng Qi, Yun Fan Yu Yan (these are names associated with ancient Chinese music and philosophy).
Sheng Yin, Tun Jie, Kai Yuan, Zhi Wei, Ai Mei, Bu Jian, Xuan Zhong, Yu Zhu, Diao Feng
The right Huang Zhong consists of one section with thirty-four lü (musical pieces). (Each lü corresponds to thirty-one parts of thirty-four days)
Da Lü:
At the beginning, all things awaken, praising the vastness of heaven and earth; Kun Yuan (the earth) assists the times, correcting and harmonizing governance. The development of events has smooth times and complex times; only in the subtle can one see the signs. Letting go of daydreams helps one stay grounded. One must uphold justice and remain strong and unyielding, just as yin is attached to yang; one must recognize the deep and the developing to prosper and thrive. Recognizing when to adapt and weighing the pros and cons, saying less and doing more enables collaboration with others, embracing the light and passing it on.
This is the Da Lü Palace tune, with a total of twenty-seven lü (musical pieces). (Each lü corresponds to one day and one twenty-seventh of a day)
Next is the Tai Cu Palace tune:
Not recognizing one's own abilities while striving for success, like a small poisonous plant, which can gather strength, from beginning to end, from birth to death, like the performance of a gourd, starting from the initial angle (Shaoyang), with weak branches, the sound of Shang music is deep and mournful, supporting the weak and fostering development, as all things grow and flourish. Mountains and rivers cease, everything unfolds in accordance with the passage of time, leaping like a dragon, as Guo Mang, a divine figure, brings order, lush branches and leaves, flowers bloom, spreading and growing, justice prevails, and ranks are clearly defined. As spring plowing begins, we celebrate the harvest, hidden talents come to light, while all things eventually wither.
This is the Tai Cu Palace tune, with a total of thirty-four lü (musical pieces).
Then comes the Jia Zhong Palace tune:
Mutual cooperation, assisting each other, with Yin energy supporting Yang energy, wind coming from all directions, issuing decrees, all things flourishing, spring arriving, shaking the earth, riding on the gentle spring breeze, fragrant and full of vitality, sunny and beautiful, bringing joy with the spring breeze, everything flourishing freely, benevolent and dignified, growing eagerly towards the south, the rising sun in the east, morning arriving, everything growing, each with its own duties, pure and renewed.
This is the Jiazhong Gong Diao, with a total of twenty-seven verses.
Next is the Guxi Gong Diao:
Hope is brought by the south, offering sacrifices to the deities, only then can the divine majesty be manifested, carrying the power of the newborn, cleansing the old, changing old customs, displaying talents, propitious energy rising, beautiful auspicious clouds, opening up paths, following the seasons, day and night devoted to cultivation, hot summer, everything growing, branches tender, enforcing justice, all things equal, blooming flowers, removing weeds, abundant fruits, bright and expanding (prosperity of future generations), following the rules (upper birth nurturing), less choosing, following the rules, worshipping in red ceremonial attire, gaining fame in the court, maintaining chastity.
This is the Guxi Gong Diao, with a total of thirty-four verses.
Next is the Zhonglu Gong Diao:
The fire of the Vermilion Bird (southern constellation) brings good luck, the gentle spring breeze begins, birds and animals become active, all things in the south revive, spring arrives, guiding farmers to plant, orderly and methodical, the southern star guides respectfully, chasing each other, maintaining purity in the heart, red plants and trees shining brightly, submitting to the arrangement of fate, light shining everywhere, the weather is clear and warm at the time of Si (9-11 am), everything growing, prohibition on overfishing, desolate fields, chaste constellations, the celestial court bestows blessings upon the Zhou dynasty.
This is the Zhonglu Gong Diao, with a total of twenty-seven verses.
Lastly, the Ruibin Gong Diao:
Southern Affairs (The Final Rites of Jingfang), tranquility and peace, choosing the right moment, flowers in full bloom, full of vitality, latent power, tremendous changes, guests at ease, nostalgia for the homeland, sounds reaching afar, similar trajectories, calm sea, eliminating disasters, staying away from danger, strong and healthy, bright and noble, gazing into the distance, rising to the center, the phoenix soaring, the sun rising, seizing the opportunity, auspicious and smooth, Quail Fire (a constellation) appearing, orderly and serene, flames rising high, brilliant and resplendent.
This text is an ancient record of a series of rites, like chapter titles in some rituals or classics. "Right Ruibin: Twenty-Seven Rites" means that this part of Ruibin consists of a total of twenty-seven rites. Next, Linzhong, Yize, Nanlv, Wuxie, Yingzhong—these are all chapter names, followed by the number of rites contained in each. Let’s look at them one by one, translating these names and numbers into modern vernacular, as if reading an ancient catalog.
Alright, let’s begin.
- Linzhong: Humble Service, Cultivating Virtue, Walking the Path, Strong and Robust, Yin Rising, Free from Deceit, Eliminate Destruction, Splendor Vanishing, Friends Celebrating, Clouds Spreading, Equal Responsibilities Leading to Success, Open-Minded, Safe Passage, Virtue Equal, No Obstacles, Overflowing Etiquette, Deep Wisdom, Serious and Respectful, Pure and Diligent, Returning to Excellence, Beautiful Sound, Gentle Wind, Seasonal Changes, Flowering Vine, Embroidered Ridge, Material Absence, Acceptance and Rejection, Bright Mouth, Radiant Well, Sun Renewing, Heavy Wheel, Wealth and Splendor.
This section means that the Linzhong chapter contains twenty-seven rites, each with significant names, such as "Humble Service," "Cultivating Virtue," and "Walking the Path," which sound like noble virtues or codes of conduct. Other terms, like "Splendor Vanishing," "Clouds Spreading," and "Radiant Well," are abstract concepts that require contextual understanding.
"The Right Lin Zhong chapter has thirty-four laws. The Yi Ze chapter has twenty-seven laws. The names of these laws in Yi Ze, such as *Sheng Shang*, *Qing Shuang*, and *Qi Jing*, appear to depict specific states or scenarios, and some bear resemblance to terms associated with rituals or ceremonies, such as *Yu Xu*, *Xian Xing*, and *Hui Dao Gui Ren.* These terms are quite archaic and require in-depth study to understand their exact meanings.
The Right Yi Ze chapter also contains twenty-seven laws. The Nan Lü chapter has twenty-seven laws: *Bai Lü,* *Juan Xiu,* *Dun Shi,* *Su Feng,* *Jing Wu,* *Qiu Ren,* *Jie Gong,* *Fei Dun,* *Lei Zhong,* *Sheng Yin,* *Kang Jie Wei Yuan,* *You Jie,* *Gui Qi,* *Zhong De,* *Wang You,* *Yun Sai,* *Ru Shou,* *Zun Pei,* *Yao Luo,* *Wei Yin,* *Zhi Sui,* *Fen Man Dao Xin,* *Zhen Jian,* *Xu Zhi,* *Gui Cang,* *Yi Han,* *Jun Yi,* *Yue Shi,* *Wang Lao,* *Jiu You,* *Guang Ben.* The names in Nan Lü also consist of twenty-seven laws, and their names like *Bai Lü,* *Dun Shi,* and *Su Feng* feel somewhat similar in style to Lin Zhong and Yi Ze, with both descriptions of natural phenomena and social conditions, as well as some more abstract concepts.
The Right Nan Lü chapter contains thirty-four laws. The Wu She chapter has twenty-seven laws: *Si Chong,* *Huai Qian,* *Gong Jian,* *Xiu Lao,* *Xu Nong,* *Xiao Xiang,* *Bi Yan,* *Jiang Lou,* *Cang Sui,* *Ri Zai,* *Xuan Chun Yan Cang,* *Ming Kui,* *Lin Qi,* *Gui Zhong,* *Da Xu,* *Se Lian,* *Xia Ji,* *Xi Jian,* *Wu Bian,* *Qi Bao,* *Yan Nian,* *Qiu Shen Ye Se,* *Xuan Yue,* *Cheng Tian.* The names in the Wu She chapter seem to represent moral principles or life attitudes, including terms related to agriculture and nature, such as *Xu Nong,* *Xuan Chun,* and *Qiu Shen Ye Se.*"
"The chapter 'Without Shooting' contains twenty-seven rules. The chapter 'Ying Zhong' contains twenty-eight rules. The names of the rules in 'Ying Zhong,' such as 'divide,' 'ancestral micro,' 'success achieved,' seem to describe a process or outcome, while some are more abstract concepts, such as 'peaceful,' 'non-action,' and 'Gu She.' In summary, these rule names are quite poetic and require careful consideration to understand their deeper meanings.
It is said in 'Records of the Grand Historian': 'Yu the Great of the Xia dynasty used his own body to establish measurements and sound to define musical notes.' The 'Book of Rites' also states: 'A man's open palm width is one chi.' 'Zhou Guan' writes: 'Measurements are determined by the edge of a jade bi.' Zheng Sinong explains: "'Xian' means length. The diameter of this bi is one chi and serves as a standard." The 'Yi Wei Tong Gua Yan' also states: 'The width of ten horse tails is considered one fen.' 'Huainanzi' mentions: 'During the autumn equinox, the awns of rice spikes are formed; when the awns are formed, the rice is ripe. The twelve musical units correspond to a grain of rice, and twelve grains of rice equal one inch.' Here, 'rui' refers to the awn on the rice spike. 'Shuo Yuan' says: 'The standard of measurements begins with grains, where one grain of rice equals one fen.' 'Sunzi Arithmetic' states: 'The silk spit by silkworms is called a hulu, ten hulu is one miao, ten miao is one hao, ten hao is one li, and ten li is one fen.' These are ancient accounts regarding the origins of measurement, which contain some contradictions."
However, the description in "Han Shu · Zhi" is more reliable: "Measurement refers to tools used for measuring length. It was originally determined based on the length of the Huangzhong musical note. The width of a medium-sized grain of millet corresponds to the length of the Huangzhong, with the width of ninety grains of millet being the length of the Huangzhong. The width of one grain of millet is one fen, ten fen is one cun, ten cun is one chi, ten chi is one zhang, and ten zhang is one yin, thus establishing five standard lengths." Later scholars formulated measurements based on this description, and various methods were derived from the length of millet, so these methods can corroborate each other. The problem is that the size of millet varies, and the size of grains can vary between abundant and poor harvest years. The standards of ancient measurements were often adjusted, and errors inevitably occurred in the transmission among the people, leading to deviations over time. Now let me briefly introduce fifteen historical scales and their similarities and differences:
The Liu Xin bronze hu chi from the Wang Mang period as recorded in "Han Shu · Zhi";
The bronze chi during the Jianwu period of the Later Han Dynasty;
The lu chi established by Xun Xu in the tenth year of the Taishi period of the Jin Dynasty, which was the standard before the Jin Dynasty;
The bronze chi passed down from Zu Chongzhi.
During the ninth year of the Taishi period of Emperor Wu of Jin, Xun Xu, the supervisor of the imperial library, inspected the court's musical instruments and found that the eight tones were out of harmony. Upon investigation, it was discovered that the rulers from the Eastern Han to the Wei Dynasties were more than four fen longer than the ancient chi. Xun Xu had the copyist Liu Gong remake the chi according to the standards of the "Zhou Li," which represented the true ancient chi.
Then, they used this ancient chi to recast the bronze lu lu, used to adjust the musical instruments' tones. When using this chi to measure ancient musical instruments, the dimensions perfectly matched the inscriptions on the instruments! What's even more remarkable is that someone in Jijun stole a tomb, dug up the tomb of King Wei Xiang, and unexpectedly discovered jade musical scales and chimes from the Zhou Dynasty that surprisingly sounded similar to the newly cast lu lu! At that time, old bells from the Han Dynasty were gradually unearthed in various locations. When tested with the newly cast lu lu, the sounds were in perfect harmony! This indicates that Xun Xu's restoration of the ancient chi was remarkably accurate.
Emperor Wu of the Liang Dynasty recorded in "Zhonglüwei" that Zu Chongzhi had a bronze ruler with an inscription containing eighty-two characters. The inscription reads: 'In the tenth year of the Taishi period of the Jin Dynasty, the Grand Minister of the Palace Examination compared ancient measuring instruments and determined that it was four and a half fen longer than the current ruler. There are seven ancient standards used for comparison: Guxi Yulü, Xiaolu Yulü, Xijing Copper Niao, Jincuo Niao, Copper Hu, Ancient Coins, and Jianwu Copper Ruler. Guxi Yulü is slightly longer, while Xijing Copper Niao is slightly shorter, and the rest are the same as this ruler.' This bronze ruler was newly crafted by Xun Xu. The ruler we currently use was made by Du Kui. Lei Cizong and He Yinzhi also drew the "Zhonglütu", which documents Xun Xu's corrections to the ancient ruler, exactly matching the inscription on this bronze ruler. However, Xiao Ji's music score states that these seven standards were determined by the Liang Dynasty, which is inaccurate. Now we use this bronze ruler as the standard to correct rulers from previous generations. The rulers from the Liang Dynasty are one chi and seven fen longer than those from before the Jin Dynasty.
Once upon a time, an old farmer unearthed a jade ruler from the Zhou Dynasty, which was said to be the standard ruler of the world. Xun Xiu used it for calibration and discovered that all the gold, stone, silk, and bamboo items he had crafted were one meter shorter than the standard. The "Bell and Music Weights" by Emperor Wu of Liang also recorded that the palace had been passing down a bronze ruler and eight ancient jade musical measuring instruments from the Zhou Dynasty. It was discovered that the Zhou ruler had been used as a token by the Eastern Huan Marquis but later went missing. Regarding the jade musical notes, one was lost, while the remaining seven were embedded with bell music and still bore the old inscriptions. Thus, Emperor Wu of Liang commissioned a new ruler based on these jade musical notes for calibration purposes. They employed fine hairs and millet for precise measurement, recalibrating repeatedly; this was the most accurate measurement method of the time, differing from Zu Chongzhi's ruler by only half a millimeter. Subsequently, they used this new ruler to create four musical instruments known as "Tong." They also crafted a flute based on the new ruler and used it to calibrate the ancient bells. By playing the notes inscribed on the bells in accordance with the "Yize" pitch using the flute, they found that the "Yize" pitch matched perfectly with the bell sound. It appeared that the lengths of these two rulers were very similar. Xiao Ji noted in his book, "This method originates from the 'Sima Method.'" He was referring to the bronze gnomon devised by Feng Zhaoqing. Later, after the fall of the Chen Dynasty, this gnomon was brought into the court. During the Daye era of the Sui Dynasty, a proposal was made to restore the ancient standards, thereby using this gnomon to calibrate musical scales and produce eight types of musical instruments, including bells and chimes. In the Jin Dynasty, an ancient bronze ruler was unearthed in Shiping County.
In Xiao Ji's "Scores," it is stated: "During the reign of Emperor Zhang of Han, the literary historian Xi Jing from Lingling excavated a set of jade rulers under the Shun Temple in Lingdao County, which he later used to craft this ruler." Fu Chang's "Praise of the Officials of the Jin Dynasty" also notes: "The bell measurements made by Ge Xu were considered very precise by contemporaries, but only Ruan Xian from Chenliu said its tone was too high. Later, an ancient bronze ruler was excavated in Shiping County, which was ancient and nearly decayed; when compared with Xun Xu's ruler, it was a quarter of a foot shorter. People at that time believed Ruan Xian was correct." The lengths of these two rulers are also similar.
In the fourth year of Wei Jingyuan, during the reign of King Chenliu, Liu Hui mentioned in his commentary on "Nine Chapters on the Mathematical Art" that the hu and ruler used during Wang Mang's era were four and five-tenths shorter compared to the rulers of that time; compared to the rulers of the Wei Dynasty, the depth of the hu was nine inches and five and a half tenths. This is the same as what Xun Xu of the Jin Dynasty said, that "Du Kui's ruler is four and a half tenths longer than the current ruler."
Xiao Ji noted that this ruler was utilized in the Jiangdong region during the Jin Dynasty. The market ruler of the Later Zhou was one foot, nine-tenths, and three-hundredths shorter than the jade ruler. The official iron ruler was one foot and two inches long.
From the early Wei Dynasty until the division of Eastern and Western Wei, and until the Later Zhou stopped using jade rulers, people commonly mixed these rulers.
It is said that in "Zhen Luan's Arithmetic," it is recorded: "The market ruler of the Zhou Dynasty was equivalent to nine-tenths and two-hundredths of the jade ruler." It is said that this ruler was made by a Daoist named Zhi Gong during the Liang Dynasty, and it was sent to the Zhou Dynasty and was associated with an old man with a long beard. Both Zhou Taizu and Sui Gaozu thought this ruler was crafted for their benefit, and as a result, the common folk of the Zhou Dynasty continued to utilize it. By the time of the Kaihuang period in the Sui Dynasty, the government officially designated it as the official ruler, used by all departments, and it remained in use until the Ren Shou period. During the Daxie period of Emperor Yang of Sui, some common folk also used it for personal purposes.
This jade ruler actually belonged to Yuan Yanming, a lieutenant of the Wei Dynasty, made from the width of half a zhou of millet, which was followed by the Qi Dynasty. Wei Shou wrote in "Wei Shu · Lu Li Zhi": "Gongsun Chong remade a new ruler during the Yongping period, using the length of one millet length as the unit of measurement. Later, the Taichang Qing Liu Fang was ordered to repair the music, using the width of medium-sized millet as one unit. And Zhongwei Yuan Kuang used the width of one millet plus two millet gaps to determine one unit. The three parties had differing opinions and argued for a long time with no conclusion. It wasn't until the nineteenth year of Dahe that Gaozu ordered one millet width as one unit, and ninety millets in length as the length of the Huangzhong, to define the bronze ruler. The relevant departments reported that Liu Fang's ruler was the same as the one established by Gaozu, so it was used to revise the inscriptions on gold and stone. Until the Wuding period of the Northern Wei Dynasty, no one discussed the law anymore."
Eleventh, Cai Yong's bronze yue ruler. The jade ruler from the Later Zhou Dynasty was longer by one foot, one inch, five fen, and eight li than those used before the Jin Dynasty. From ancient times to the present, there has always been a bronze yue, decorated with silver inlays on top, with the inscription: "Yue, the palace of the Huangzhong, nine inches long, nine fen in circumference, capable of holding one thousand two hundred millets, weighing twelve zhu, two of which make one he. Three fen loss or gain, transforming into twelve laws." Zuxiaosun said: "It is said that this is Cai Yong's bronze yue."
During the Baoding period of the Later Zhou Emperor, the emperor ordered Dazongbo Lu Jingxuan, Shangdang Gong Changsun Shaoyuan, and Qiguo Gong Husizheng to make rulers using millet, but the dimensions were not consistent. Later, when repairing the granary and digging the ground, an ancient jade container was unearthed, which was used as a standard reference for creating laws, measurements, quantities, and weights. Thus, with this ruler, a general amnesty was proclaimed, the Tianhe era was changed, and all departments began to use it, until the end of the Zhou Xuandi Da Xiang era. The Huangzhong in this set of laws corresponds to Cai Yong's ancient yue.
Twelfth, the Song Dynasty's measuring stick (chi). It is actually longer by one foot, six inches, and four tenths of an inch compared to the measuring stick used before the Jin Dynasty. There are also the measuring stick from Qian Lezhi's armillary sphere, the iron measuring stick from the Later Zhou, the measuring stick used for adjusting clocks and music during the early Kaihuang period, and the water measuring stick used when adjusting clocks and music after pacifying the Chen Dynasty.
This measuring stick is the type commonly used by ordinary people during the Song Dynasty, which later spread to the Qi, Liang, and Chen Dynasties, where they also used this measuring stick to make musical instruments and determine musical pitch. The length is roughly the same as that of the measuring stick used after the Jin Dynasty, the measuring stick used in the folk customs of the Liang Dynasty, and the measuring stick on Liu Yao's armillary sphere. This is likely because it was used by ordinary people, and over time, it inevitably wore down and varied.
In the sixth year of the Zhou Dynasty's Jiande era, after defeating the Qi Dynasty, this measuring stick was adopted as the standard measurement and promoted nationwide. Later, in the reign of Emperor Xuan, Daxi Zhen and Niu Hong discussed:
I carefully studied the history of weights and measures. In order to govern the country well, it is necessary to carefully examine the ancient standards of weights and measures and find the most suitable solution. I consulted the iron ruler currently in use, which was crafted by Shangshu Su Chao, and it was used as the standard ruler of the former Zhou Dynasty. I found its length to be exactly the same as that of the Song Dynasty ruler, which was used at that time to calibrate the bell pitch and measure land area. Now I use the millet produced in Yangtou Mountain in the Shangdang area to measure according to the method recorded in "Hanshu·Lüli Zhi." If a large millet is piled up to exactly one foot in height, it must be shaken hard to fit into the yellow bell pitch pipe. If a medium-sized millet is piled up to one foot, although it looks a little sparse, when placed into the yellow bell pitch pipe, it can be filled without shaking. The difference between these two cases is mainly due to the different sizes of millet and the slightly inaccurate measurement method, so there is a discrepancy with the standard of the iron ruler. Moreover, the millet in the Shangdang area is different from other places, especially black in color, round and heavy in shape. It is definitely not accidental to use it for measurement. Because the rainfall and droughts in different places vary, and the fertility of the land differs, the size of millet is also uneven, making it challenging to determine an average size. Xu Shen explained that the millet grains are quite large and different from ordinary millet. I believe that these large millet grains are now the standard millet grains of ancient times, one hundred of them piled up exactly one foot, which aligns with ancient standards. In addition to filling the pitch pipe, there are still more than ten millet grains left, which may be due to a slight error in the diameter of the pitch pipe or the inaccurate method of making pitches in ancient times. Even if it requires a vigorous shake to fill it, theoretically, it also makes sense.
Now we have verified the ancient coins of the Zhou and Han dynasties, and their sizes are in accordance with the standards; the measurements of the Song dynasty's hunyin are also correct. Additionally, according to the records in "Huainanzi," twelve grains of millet stacked together measure one inch. This indicates that the ancient emperors established laws and standards with great detail and seriousness, using a ruler for measurement, which had consistent significance. The "Book of Han: Treatise on Food and Money" records: "A piece of gold measuring one inch square weighs one catty." We are currently minting gold to verify this, and an iron ruler is the closest to the standard. Based on historical documents and practical calculations, there are many instances that conform to the standards. Moreover, the use of iron rulers began during the Pingqi period, and we now use this as our standard, which aligns more closely with actual conditions. Regarding the measurement of millet with a jade ruler, where width is used as length, there would still be excess after filling it, and it would not fill the ruler completely; examining both ancient and modern practices, this method seems impractical. The harmony of the eight musical tones symbolizes a wise and prosperous reign, and a unified system of weights and measures is the governing strategy of a sagacious ruler. We have carefully studied previous documents and combined them with current situations, concluding that the use of an iron ruler is the most appropriate.
Before we had time to finalize this, Emperor Gaozu passed away. Niu Hong, Xin Yanzhi, Zheng Yi, He Tuo, and others discussed it for a long time but reached no conclusion. After pacifying the Chen dynasty, the emperor felt that the music from Jiangdong was quite good and said, "This is the ancient music of Huaxia; although it has changed somewhat with the times, it largely follows the ancient methods." Ancestor Xiaosun said, "After pacifying the Chen dynasty, we abolished the jade rulers and legal standards of the Zhou dynasty and replaced them with this iron ruler and laws, stipulating that one foot and two inches is equal to the market foot."
In the tenth year of the Kaihuang era, the musical ruler made by Wanbaochang is actually longer by one foot, one inch, and eight point six inches than the rulers from before the Jin dynasty. "The current Taiyue Bureau and the copper rulers produced are made by Wanbaochang, named the water ruler. It is said that its Huangzhong pitch corresponds to the double sound of the southern lu when measured with the iron ruler. The southern lu is the Huangzhong yu, hence it is called the water ruler."
The set of bronze instruments now retrieved from the library and the palace is made by Wan Baochang, known as the Water Ruler. The book says that its Huangzhong law is equivalent to the double pitch of the Nanlu scale of the iron ruler. Nanlu is the "feather" tone in the Huangzhong scale, so it is called the Water Ruler.
Fourteen, various types of rulers, such as the Tu Gui ruler from the Zhao Liu Yao celestial sphere, are four inches and three tenths longer than the official ruler of the Liang Dynasty, and actually one foot and five inches longer than the ruler used prior to the Jin Dynasty. "Miscellaneous rules Zhao Liu Yao celestial sphere Tu Gui ruler, longer than the Liang Fa ruler by four inches and three tenths, actually one foot and five inches longer than the ruler before the Jin Dynasty."
Fifteen, the ruler commonly used among the people during the Liang Dynasty is six inches and three tenths longer than the official ruler of the Liang Dynasty, two inches longer than the ruler on the Liu Yao celestial sphere, and actually one foot seven inches and one tenth shorter than the ruler before the Jin Dynasty. Emperor Liang Wu said in "Zhong Lu Wei": "After Emperor Wu of the Song Dynasty pacified the Central Plains, he sent the celestial sphere and the Tu Gui, saying it was made by Zhang Heng. However, after checking the inscriptions on the celestial sphere, it was cast in the fourth year of Guangchu, and the Tu Gui was made in the eighth year of Guangchu. Both were made by Liu Yao, not by Zhang Heng. He used it to make rulers, the length of which is four inches and three tenths longer than the current standard ruler, and two inches shorter than the ruler commonly used among the people." Here, the term "new ruler" refers to the official ruler of the Liang Dynasty. "Liang Dynasty common ruler longer than the Liang Fa ruler by six inches and three tenths, two inches longer than the Liu Yao celestial sphere ruler, actually one foot seven inches and one tenth shorter than the ruler before the Jin Dynasty. Liang Wu's 'Zhong Lu Wei' says: 'Song Wu Pingzhongyuan, sent the celestial sphere Tu Gui, saying it was made by Zhang Heng. Check the celestial sphere inscription, cast in the fourth year of Guangchu, and Tu Gui was made in the eighth year of Guangchu. It is also made by Liu Yao, not Zhang Heng. The ruler is made with a length of four inches and three tenths longer than the new ruler, and two inches shorter than the common ruler.' The new ruler refers to the Liang Fa ruler."
Wow, this article discusses ancient measurements, from the Zhou Dynasty through to the Han Dynasty; it's incredibly informative! Let's break it down sentence by sentence.
First of all, in the "Book of Rites," it is mentioned that the Bi family is responsible for making measuring vessels, which are one chi deep, square on the inside and round on the outside, with a capacity of one mu; the bottom is one cun, with a capacity of one dou; the ears are three cun, with a capacity of one sheng; total weight is one jun; the sound matches that of the Huang bell; no tax is required. The inscription on it reads: "When the time comes for contemplation, it reaches its extreme. The excellent measure is completed, to observe the four states. Forever opened thereafter, this vessel is maintained." The inscription does not need to be translated. It is quite majestic.
Then, in the "Zuo Zhuan," it mentions the four measuring vessels used in the Qi state: dou, qu, mu, and zhong. Four sheng make one dou, and so forth, culminating in mu. One mu is six dou and four sheng; ten mu make one zhong, which is sixty-four dou. Zheng Xuan posited that the volume of a square chi should be a thousand cun, slightly different from the "Nine Chapters on Mathematical Procedures," where Zu Chongzhi calculated it to be 1562.5 cun. This section discusses the variations in ancient measurement units and calculation methods.
In the "Nine Chapters on the Mathematical Art," it states that the volume of one hu of millet is 2700 cun, one hu of rice is 1620 cun, and one hu of beans, sorghum, hemp, and wheat is 2430 cun. Depending on the type of grain, the volume varies, but the price stays consistent, reflecting the refinement and fairness of the ancient measurement system. It also mentions that these volume calculations are based on the hu of rice, which aligns closely with the records found in the "Book of Han."
In the "Sunzi's Art of Calculation," it says that six grains of millet constitute one gui, ten gui make one miao, ten miao make one cuo, ten cuo make one shao, and ten shao make one he. Ying Shao and Meng Kang have slightly different interpretations of "gui," with Ying Shao describing it as a natural shape, representing the beginning of yin and yang; Meng Kang states that sixty-four grains of millet make one gui.
In the "Book of Han," it mentions measuring vessels such as yue, he, sheng, dou, and hu, used for measuring quantities. These measuring vessels were originally designed according to the musical scale of the Huang bell, made of bronze, square with a round exterior, and with handles on the side. It resembles a jue, featuring the hu on top, the dou below, the sheng on the left, and the he and yue on the right. This design is not only practical but also reflects philosophical concepts like yin and yang and the five elements. It weighs two jun, with a capacity of 11520. Its sound matches that of the Huang bell.
Finally, the inscription on the Han dynasty measure is: "The regulated measure, square inside but round outside, has a side length of nine li (厘) and five hao (毫), a depth of one foot, a volume of one thousand six hundred twenty inches, and a capacity of ten dou (斗)." The calculations of Zu Chongzhi and Liu Xin for this measure show slight differences, which may be due to Liu Xin's less precise calculation method. In summary, this article provides a detailed introduction to the understanding and calculation methods of measurements and weights from different periods and schools in ancient times, truly enlightening!
In the year 226, when Liu Hui commented on "The Nine Chapters on the Mathematical Art: Commercial Work," he said: The current measure used by the Minister of Agriculture has a diameter of one foot, three inches, five tenths, and five li (厘), and a depth of one foot, with a volume of one thousand four hundred forty-one inches divided by thirty. The bronze measure used during the Wang Mang period, calculated by today's standards, has a depth of nine inches, five tenths, and five hundredths, and a diameter of one foot, three inches, six tenths, eight hundredths, and seven milligrams. Using my calculation method, the measure used during the Wang Mang period, when converted to modern standards, is only a little over nine dou (斗), seven sheng (升), and four he (合). Therefore, the measure during the Wei dynasty was larger in comparison, and the ruler was longer; the measure during the Wang Mang period was relatively smaller, and the ruler was shorter.
The Liang and Chen dynasties continued to use the ancient system of measurement. The Qi dynasty stipulated that five ancient sheng (升) equal one dou (斗).
In May of the first year of the reign of Emperor Wudi of the Later Zhou Dynasty (561 AD), while the Jin State was constructing a granary, they discovered an ancient jade measuring vessel. In October of the fifth year of Bao Ding (565 AD), Emperor Wudi issued a decree to re-establish bronze laws and weights and measures, ultimately achieving unification and balance. When millet was piled for measurement, the measurements exactly matched those of the jade vessel, and there was no difference compared to other weighing and measuring tools. Consequently, a bronze measuring vessel was created based on the standard of the jade vessel and distributed nationwide. The inscription on the bronze vessel stated: "Inner diameter seven inches one fen, depth two inches eight fen, weight seven jin eight liang. In the second year of Tianhe (567 AD), on the first day of the first month, the fifteenth day, it was officially established by the local government." The inscription on the jade vessel read: "In the first year of Bao Ding of the Great Zhou, under the reign of Chongguang, in the month of Luibin, the officials of the Jin State repaired the granary, obtained the ancient jade vessel, and the design and specifications were accurate, akin to the esteemed measurements of the ancients. The Grand Preceptor of the Jin State reported this to the Heavenly Treasury, and in the fifth year, under the reign of Xieqia, the Emperor decreed to measure with precision, examine the gray laws, without deviating from the standard and consistent with millet measurements. It was then cast in gold, distributed throughout the country, to achieve harmony and balance in weights and measures." If calculated using numbers, the volume of the jade vessel is approximately 111 inches and 8 fen, and the volume of the measuring vessel is approximately 1,185 inches, 7 fen, 3 li, and 9 seconds. Zhen Luan stated in "Arithmetic": "One jade vessel is equivalent to one official vessel plus three he plus four shao." In other words, the jade vessel is larger than the official vessel. According to calculations, the official vessel referenced by Zhen Luan from the Later Zhou Dynasty has an approximate volume of 97 inches, and the measuring vessel has a volume of approximately 977 inches. The Later Zhou's jade vessel, along with the matching gold and bronze vessels, as well as the gold and bronze vessel inscribed in the sixth year of Jiande (577 AD), were all calibrated using millet. When measured with jade weights, the weight of one vessel is six jin and thirteen liang. During the Kaihuang era, it was established that three ancient vessels equate to one modern vessel. By the time of the Daye era, the capacity of the ancient vessel was reinstated.
A weighing device is a tool used to weigh objects and achieve balance; a weight refers to the weights used to measure weight. The weighing device is designed to hold the weights, allowing the weights of items to reach equilibrium. The principle of making a weighing device is similar to that of a compass, ensuring it is vertically and horizontally aligned. Counterclockwise rotation aligns with the compass, while clockwise rotation aligns with the square. The use of weighing devices in astronomy is to assist the armillary sphere (an ancient astronomical instrument) in accurately determining the seasonal markers, used to coordinate the seven luminaries (the sun, moon, and five planets), hence it is called "Jade Balance." The units for weights include such as zhu, liang, jin, jun, and shi, among others. By using a scale to weigh objects, one can know their weight. In ancient times, there were also units of weight such as shu, zhi, chui, ci, huan, gou, luo, and yi, but these standards have varied over time, and I am not entirely certain about the details. The "Book of Han" states that the origin of these weights was determined based on the weight of the yellow bell. One yue can hold 1,200 grains of millet and weighs 12 zhu. Twenty-four zhu equal one liang, sixteen liang equal one jin, thirty jin equal one jun, and four jun equal one shi. The production of these five types of weights is highly precise, determined based on principles and calibrated against physical objects. Other size differences are adjusted based on weight. The weights are made in a ring shape, with their thickness greater than their width, creating a continuous loop. When the weight and the object reach equilibrium, a weighing device is produced; when the weighing device operates, it produces a compass; the roundness of the compass leads to the creation of a square; the square produces a rope; and the straightness of the rope produces a standard. When the standard is correct, the weighing device can balance, and the measurements can be accurate. These are the five fundamental principles, all embodied in the weighing device, serving as the fundamental principles for its production.
According to the records of "Zhao Shu", in the 18th year of the reign of Shi Le, in July, when the Jiande Hall was built, a round stone resembling a water pestle was discovered. The stone was inscribed with the words: "Standard stone, weighing four qing, used for standardizing weights and measures. Made by the Xin family." It was believed to be from the time of Wang Mang. During the Jingming period of Northern Wei, Wang Xianda from Bingzhou presented an ancient bronze weight with an inscription of eighty-one characters. The inscription read: "Standard stone, weighing four qing." It also stated: "Yellow Emperor, the first ancestor, virtue passed down through Yu. Yu, the first ancestor, virtue passed down through Xin. In the year of the Great Liang, the dragon gathered in Wuchen. Wuchen directly determined the will of heaven. Based on the virtue of the earth, the rightful name is true. Changing to establish Chou, longevity and prosperity. Standardizing weights and measures, examining the present people. The dragon in Jisi, the year is truly sinking, initially ruling the world, all nations will forever follow. Descendants for generations, enjoying the tradition for billions of years." This weight was also made during the time of Wang Mang. At that time, the Minister of Music, Gong Sun Chong, based on the "Book of Han", first revised the standard of measurement, and when he saw this weight, he measured it with the new ruler and found it to weigh 120 qing. The new ruler and weight matched perfectly. Therefore, the responsibility of adjusting the music was given to Gong Sun Chong. During the reign of Emperor Xiaowen, measures and scales were made according to the standards of the "Book of Han". During the Liang and Chen dynasties, they used old weights. At that time, in the Qi dynasty, one catty was eight taels, and the saying "one catty and eight taels" has persisted to this day. In the Zhou dynasty, a jade weight of four taels was equivalent to four and a half ancient weights. During the Kaihuang period of the Sui dynasty, it was officially stipulated that three ancient weights were equivalent to one new weight. During the Daye period of the Sui dynasty, the ancient weights were restored. In this way, the ancient weight system varied across dynasties, sometimes heavier and sometimes lighter, truly making it a bit confusing. The saying "one catty and eight taels" has persisted to this day, used to describe things that are fairly equal.
