Since the dawn of time, people have established rulers to manage and established various rules and regulations for governing the state, drawing inspiration from the laws of the universe and following the principle of yin-yang harmony, exploring the mysteries of the cosmos, ultimately forming a precise system of music theory. Therefore, they could manage the country well and produce a variety of goods.
In ancient times, people had already begun to engage with music; during the era of Nuwa, instruments like the sheng and huang had already appeared, which is regarded as the origin of music. Later sages inherited and expanded upon this musical heritage, conducting more in-depth studies on it. Ling Lun invented the twelve pitches, and exquisite jade instruments emerged during the time of Yu Shun. Thus, the "Book of Documents" states: "The four seasons, the waxing and waning of the moon, and the length of days must all align with the standards of musical scales, weights, and measures." It also says: "I want to hear the music of the six pitches, five tones, eight sounds, and seven beginnings to convey my ideas." All of this indicates that the ancients, through the study of music theory, established various standards to regulate society, coordinate the relationship between humans and nature, and achieve the goals of upholding morality and worshiping the heavens. Therefore, music can move the heavens and earth, harmonize human hearts, change customs, examine gains and losses, and judge success and failure.
During the Xia and Shang dynasties, there were no significant changes to the musical traditions. By the Zhou dynasty, the "Rites of Zhou" recorded that officials in charge of music needed to master the harmony of the six pitches to distinguish the yin and yang sounds from the four directions and to create musical instruments. When King Jing of Zhou cast bells, he specifically consulted Lingzhou Jiu regarding musical scales, to which Lingzhou Jiu replied: "Musical scales are used to determine standards." Here, "jun" refers to the concept of standards; with these five standards, the measures, rules, and norms are complete. Thus, the "Book of Songs" affirms: "The Grand Master of the Yin family holds the standards of the state, and the Son of Heaven follows his advice, ensuring that the people do not lose their way." This exemplifies the principle.
Sima Qian said in the "Records of the Grand Historian": "The emperors establish various systems and set various standards, all in accordance with the principles of the six tones; this is the foundation of all matters. Especially in military affairs, great importance is placed on musical pitch. Thus it is said: 'By observing the enemy's situation, one can judge fortune and misfortune; by listening to sounds, one can judge victory and defeat.' This is an unchanging principle for emperors throughout the ages." After the fall of the Qin Dynasty, the study of musical pitch gradually declined. When the Han Dynasty was first established, Chancellor Zhang Cang was the first to propose the need to reform musical pitch, but it was not perfected. It wasn't until the reign of Emperor Wu of Han that a dedicated official position was created to oversee musical pitch. Sima Qian meticulously documented the interdependent relationships between the tones. By the time of Wang Mang, scholars began to study musical pitch once more; Liu Xin authored a memorial, which was later recorded by Ban Gu. Cai Yong also documented discussions regarding musical pitch following the Jianwu period, and Sima Shao compiled and continued to supplement these materials. Later, as dynasties changed and chaos engulfed the land, musicians fled and dispersed, resulting in the loss of standards for musical instruments and pitch.
After Cao Cao, the Wei Wu Emperor, acquired Du Kui, he had him establish musical scales. Du Kui, based on the standards of the time, was able to compile some regulations. By the time of the Jin Wu Emperor, the musical scales established by Du Kui remained in use unchanged. It wasn't until the tenth year of Tai Shi that Grand Minister of Ceremonies Xun Xiu petitioned for the creation of new musical scales and the reforging of the tuning pipes. During the Yuan Kang era, Xun Xiu's son Xun Fan inherited his father's work. Before the project could be completed, the Yongjia Rebellion broke out, and all the court's regulations were destroyed in the chaos caused by Shi Le. Later, when the emperor moved south, everything had to start over, and the ceremonial system and musical instruments were all lost. Although some materials were gradually collected later, most were lost or scattered beyond recovery. By the Song and An Dynasties, the musical scale system still had not been fully restored. During the Song Dynasty, Qian Yue compiled the sixty scales of Jing Fang and added to them, turning them into three hundred sixty scales; during the Liang Dynasty, scholar Shen Zhong documented the names of these scales. Scholars from the Later Wei, Zhou, Qi, and other dynasties also discussed musical scales. In this article, I will compile and organize the measurements of musical scales from various dynasties based on Ban Gu's "Book of Han."
The "Book of Han" states that musical scales must achieve "first, to be complete in numbers; second, to have harmonious sounds; third, to be precise in measures; fourth, to have commendable quantities; fifth, to have balanced weights." Since the Wei and Jin periods, the musical scale system has continuously evolved. Below, I will outline the key aspects of these additions and changes.
The five fundamental numbers are one, ten, hundred, thousand, and ten thousand. Ancient texts say, "Shapes come into being only after there are things, and quantities arise only after growth and reproduction." Thus, the system begins with the establishment of the first note, with the starting value of the Huang Zhong scale set at one. Then, for each time unit (or hour), it is multiplied by three. After nine time units, at the You time, the result is nineteen thousand six hundred eighty-three, completing all five numbers, which establishes the foundation of the system. Continuing to multiply by three until reaching the end of the Hai time, a total of twelve time units yields one hundred seventy-seven thousand one hundred forty-seven, which represents the total of the time units, serving as the accumulation of the system. Dividing the total accumulation by the foundational numbers results in nine inches, which is the length of the Huang Zhong Gong scale.
Therefore, it is said that numbers emerge from the law, and the law is realized through numbers; thus, they can be used to calculate all things and summarize various phenomena. When calculating, bamboo rods are used, each measuring two fen in width and three inches in length, with each rod divided into three layers, totaling two hundred and sixteen rods, forming a hexagon, which is the counting method for the Qian hexagram. Another method uses four layers of bamboo rods, totaling one hundred and forty-four rods, forming a square, which is the counting method for the Kun hexagram. Both the hexagon and the square have twelve sides, which is the greatest number in the cosmos. Therefore, when exploring profound and mysterious matters, tracing their origins, every instance utilizes these methods. One, ten, hundred, thousand, and ten thousand are their common foundation; law, degree, measure, balance, calendar, and rate are their different uses.
Thus, when measuring lengths, using measures and balances will ensure accuracy; when things have quantities, containing them in vessels will ensure precision; when things have weights, weighing them on scales will ensure accuracy; when sounds are crisp or muffled, using musical scales to coordinate them will ensure pitch accuracy; the movements of the sun, moon, and stars are recorded using calendars, ensuring precise timing; various matters intertwined are managed using ratios, ensuring they do not violate their nature. Therefore, profound and subtle conditions and changes can all be summarized and categorized using these methods.
The so-called ratios consist of nine types: 1. Square fields, for calculating the area and boundaries of land; 2. Grains, for calculating the trade and exchange of food; 3. Diminishing fractions, for calculating taxes based on wealth and status; 4. Volume, for calculating the volume and area of piled objects; 5. Work progress, for calculating the progress and actual quantity of projects; 6. Equal distribution, for calculating labor and costs in different regions; 7. Surplus and deficiency, for calculating hidden, complex, and interrelated situations; 8. Equations, for calculating intricate positive and negative numbers; 9. Measurements, for calculating distances of height, depth, width, and range. These methods employ multiplication to distribute and division to consolidate, using a unified method to connect them. Now there are methods to link them together. Thus, this encompasses the entirety of the calculation methods.
In ancient calculations, pi was roughly approximated as 3, which made the circumference 3. This method was quite unreliable. Later, notable figures like Liu Xin, Zhang Heng, Liu Hui, Wang Fan, and Pi Yanzong came up with new algorithms, but none could find a universally accepted standard answer.
It wasn't until the late Song Dynasty that a genius named Zu Chongzhi, a government official from Southern Xuzhou, invented a highly effective calculation method. He set the diameter of a circle to be one hundred million and calculated the precise value of the circumference to be approximately between 3 zhang, 1 chi, 4 cun, 1 fen, 5 li, 9 hao, 2 miao, and 7 huo and 3 zhang, 1 chi, 4 cun, 1 fen, 5 li, 9 hao, 2 miao, and 6 huo. He provided two representations of pi: an exact value with a diameter of 113 and a circumference of 355, and an approximate value with a diameter of 7 and a circumference of 22. He also studied many complex mathematical problems, such as square roots and cubes, and integrated them with circle calculations. His results were remarkably accurate and regarded as the most impressive of his time.
Zu Chongzhi documented his research findings in a book called "Zhuishu." Unfortunately, the book was too complex for the officials of that time to grasp, so it was shelved and ignored.
According to legend, Emperor Huangdi instructed Ling Lun to cut bamboo, which was three inches and nine-fen long, and then blow on it to produce the sound of the Huangzhong mode, known as "Hanshao." He then made twelve bamboo pipes to capture the Phoenix's song, used to distinguish between the yin and yang musical tones and to divide the musical scale. The tones influenced one another, starting with the Huangzhong. The "Book of Yu" states: "When leaves fall, when the moon is full, when the sun is at its peak, all should conform to the same laws, measurements, and weights." After Yu the Great received his mandate, he also employed sounds to set laws and used the body to define measurements. The "Rites of Zhou" stipulates that musical instruments should follow the twelve pitches as the standard. Sima Qian noted in the "Book of Music": "The Huangzhong is eight inches and one seventh long, the Taicu is seven inches and two sevenths long, the Linzhong is five inches and three sevenths long, and the Yingzhong is four inches and two thirds long." These three musical tones form the basis of the twelve pitches.
Ban Gu and Sima Biao said in "Lv Zhi": "Huang Zhong is nine inches long, with the deepest sound; Tai Zhu is eight inches long; Lin Zhong is six inches long; Ying Zhong is four inches and seven minutes and four fen, with the clearest sound." Zheng Xuan in "Li Ji · Yue Ling Zhu," Cai Yong in "Yue Ling Zhang Ju," as well as Du Kui, Xun Xu, and others, although there are slight differences in measurements, the measurements of the twelve tones are basically the same. "Han Shu" records that Jing Fang also proposed the theory of "growing every eight," starting from Huang Zhong and ending at Zhong Lu, completing the cycle of the twelve tones. Zhong Lu gives birth to Huang Zhong, which is less than nine inches long, a term known as "Zhi Shi," and leads to extinction downwards. The tones are mutually generated, and eventually reach Nan Lu, adding an additional forty-eight tones, totaling sixty tones. Its operating rules correspond to the twenty-eight constellations, encompassing everything upwards, arranged in nine octaves after the winter solstice. The clarity of Ying Zhong was diminished due to the division of tones and delays.
During the Song and Yuan Dynasties, Taishi Qian Lezhi further developed on the basis of Jing Fang's Zhong Lu, adding three hundred more tones, eventually reaching An Yun, which is four inches and four minutes long. There were a total of three hundred and sixty tones. Each day corresponds to a bamboo tube, with the tones of Gong, Zhi, Xuan, Yun, and others arranged in order. He Chengtian said in "Lifa Zhiyi": "The tones are mutually generated, with a one-third gain or loss; this is the ancient simple method. Just like the ancient calendar with three hundred and sixty-five degrees and a quarter, later modifications of the system are different from it. Jing Fang misunderstood this point and mistakenly believed that there were only sixty tones." He Chengtian also established a new set of musical regulations; from Zhong Lu, one can return to Huang Zhong, with twelve Gong tones cycling, harmonious and complete tones. Huang Zhong is nine inches long, Tai Zhu is eight inches and two fen long, Lin Zhong is six inches and one fen long, Ying Zhong is four inches and seven minutes and nine fen long. The added fractions from Zhong Lu upwards yield one hundred and seventy-seven thousand one hundred and forty-seven, which aligns with the twelve Earthly Branches.
In the early period of the Liang Dynasty, there were no significant changes because the system of the Jin, Song, and Qi dynasties was followed. Later, Emperor Wu wrote a piece called "Zhong Lu Wei," discussing the gains and losses of the previous dynasties; the essence of his discussion is as follows:
According to the calculations of Lü Lü, the syllables Jing, Ma, Zheng, and Cai, extending all the way to Rui Bin, are derived from Da Lü. However, Ban Gu stated in the "Record of Music and Calendars" that after Rui Bin, the order of syllable generation goes downwards. According to Ban Gu, the length of Jia Zhong would only measure three inches and a little over seven points. If the musical scale is too short, the sound of Jia Zhong would turn into a single tone, and Zhong Lü would deviate from this tone by half, which would not constitute a proper musical scale. In early spring and midsummer, when all things are growing, the climate is mild and gentle, which does not allow for short sounds. Therefore, based on sound and practical considerations, Ban Gu's statement is incorrect. Zheng Xuan used the theory of Yin and Yang to explain the order of musical scale generation, believing that Yin and Yang are generated successively. However, according to Zheng Xuan's theory, if Yin and Yang are generated successively, how can we account for the rise and fall of Yang? Looking at the divination numbers, Qian Gua uses Jia and Ren, calculated from left to right, and Kun Gua uses Yi and Gui, calculated from right to left, so Yin and Yang have the principle of rise and fall. The succession of Yin and Yang reflects the inherent laws of nature, and the rise and fall of the six positions is deduced according to mathematics. Zheng Xuan's reliance on mathematics to explain natural laws renders his argument unconvincing. He claimed that nine and six generate each other, yet he failed to clarify how the twelve musical scales are interconnected. Zheng Xuan did not consider this thoroughly, which is very obvious.
The sixty musical scales of Jing Fang are accurately calculated according to the rules. However, the results calculated by Lü Lü have five or six sounds, which do not follow the rules. Moreover, the division and generation, as well as the repeated occurrence of late inner generation and Sheng Bian, and the separation after Sheng Bian, do not follow the rules. Jing Fang applied the principles of Yin and Yang with great ingenuity, indicating there must be valid reasons for his approach. If it is not due to some profound reasoning that is hard to grasp, it is because later generations have not studied seriously.
After careful study and comparison, I still can’t tell who’s right and who’s wrong. In my spare time, I tried to figure out the principles involved, referencing old musical instruments and the ancient Jiazhong Yulü, and recreated a new ruler that’s precise down to the tiniest detail. I made four instruments, called "Tong." These four instruments have strings that are nine feet long, with an additional inch and two tenths. The Huangzhong has strings that are also nine feet long, made up of two hundred seventy strands of silk, while the other pitches are calculated using the principle of three parts gain and loss to determine the number and length of the strings. Based on the months corresponding to each pitch pipe, the interaction of the five elements, as well as the order and beginning of the musical scales, I called these instruments "Tong." The "Tong" instruments have three strings, which can determine the musical scales for each month, and the results are spot on. Testing with the Jiazhong Yulü also showed complete agreement.
I had someone make twelve flutes for playing various tones. These twelve flutes were made according to the "Jiazhong" tuning method, completely consistent with the standard pitch, without any difference. The "Record of Shanqian" states: "The three bells in front of the palace were all cast by King Jing of Zhou as 'Wushizhong'." I had the musicians try blowing the flutes tuned in the current "Wushi" tuning, but the pitches were off. When they tried the flutes tuned in the "Yize" tuning, the sound was harmonious. We also checked the inscription on the bell at the main gate and confirmed that it was tuned in "Yize" as well. The bell from the west wing was relocated to the east during the Tianjian era. When we tried blowing it with the current flutes, it was tuned in "Nanlv." Upon careful examination of the inscription, it turned out to be tuned in "Taicu," two pitches lower than those of the current flutes. I instructed the Grand Master of Music, Si Xuan Da, to take another close look and found chisel marks on the bell, inside and out. Later, I found out from some old friends that during the Taishi period of the Song Dynasty, Zhang Yong had damaged it by removing a lot of copper, causing the pitch to drop. Through this incident, we can gain insight into the methods used for ancient bell tuning. After Emperor Wu of the Song Dynasty pacified the Central Plains, General Chen Qing offered three bells, one large and two smaller ones. The two bells in front of the Taiji Hall and the one outside the main gate are from that time. The inscription on the bell in the west reads "Qingmiao Zhongzhong;" the Qin Dynasty didn’t have a Qingmiao, indicating it belonged to the Zhou Dynasty. The inscription on the other bell reads "Taicu Zhongzheng," which is the Linzhong Palace's tuning. The Jingfang deduced the tuning based on this. The inscriptions on these bells don’t mention the Qin and Han dynasties, only "Yize" and "Taicu," which is enough to prove they are not from the Qin and Han periods. It was common for the ancients to inscribe the names of servants on objects, and these bell inscriptions also prove they are not recent. Furthermore, adjusting regulations based on sound, the five tones and six pitches need to be spot on. Craftsmen are responsible for the tuning, scholars for recording, but over the years, the two have drifted apart and struggle to communicate. Even if the musical instruments are preserved, they cannot be used. The Zhou Dynasty's hymns and the Han Dynasty's songs all record their respective merits; how can they be used for later emperors, misusing their legacy? I've detailed my thoughts and asked all officials to collaborate and find the right answer.
The restructuring had not been completed when the Hou Jing rebellion broke out. The rulers of the Chen Dynasty failed to implement reforms.
In the first year of the Western Wei Emperor's abdication, Emperor Wen of Zhou took over the regency. He then ordered the Minister of Personnel, Su Chao, to thoroughly examine the musical scales. Su Chao found a ruler’s measuring tool from the Song Dynasty, using it to determine the lengths of various pipes. This work was just beginning when Emperor Min abdicated, political power shifted to the Prime Minister, and the invasion of the Qi Kingdom occurred, which ultimately left this matter incomplete. Later, during the excavation of Tai Cang, ancient jade measuring tools were discovered and used to manufacture musical instruments and scales, but this matter was largely forgotten.
At the beginning of the Kaihuang period, the Emperor commissioned Niu Hong, a scholar from the Taichang, to study and establish musical scales. Niu Hong widely gathered scholars to discuss methods for establishing musical scales, but they still couldn't reach a conclusion. Later, in the vicinity of Pingjiang (present-day Suzhou, Jiangsu), twelve musical pipes from the Chen family were discovered and handed over to Niu Hong. The Emperor dispatched music experts, such as Mao Shuang, the Prefect of Chenshanyang, Cai Ziyuan, the Prefect of Taiyue, and Yu Puming, to create the "Lu Pu" based on the solar terms. At that time, Mao Shuang was old and went to see the Emperor in casual clothes. The Emperor appointed him Governor of Huaizhou, but he declined. Therefore, the Emperor sent the Associate Lu Lang Zu Xiaosun to learn the law from Mao Shuang. Niu Hong then played these musical pipes to determine their pitch.
After the unification of the country, musical instruments from various dynasties were gathered at the Yuelu, and music scholars researched these instruments to establish the standard for the musical scale. Then, the instruments were remade to perform the "Fourteen Songs of Huangxia." Emperor Gaozu and the courtiers listened together, and the Emperor remarked, "This music is loud, harmonious, and elegant, making people feel relaxed and at ease."
However, without the five elements (metal, wood, water, fire, earth), nothing in the world—be it things or human affairs—can arise, form, or perish. Therefore, if one uses the fire ruler to gauge the five tones, the outcome will favor fire; if one uses the metal ruler, it will lead to war; if one uses the wood ruler, there will be funerals; if one uses the earth ruler, the world will fall into chaos; only by using the water ruler can harmony in music be achieved and peace in the world maintained. The Wei, Zhou, and Qi dynasties all sought the length of fabric, thus using the earth ruler. The music currently follows the water measure. The ruler from Jiangdong is shorter than the earth ruler but longer than the water ruler. Common folks don’t grasp these distinctions; they simply call a ruler made of jade a jade ruler and one made of iron an iron ruler. The emperor ordered the use of the "water ruler" to establish musical standards, melting down all prior metal and stone instruments to silence the debates.
In the fourth year of the Ren Shou era, Liu Zhao wrote a memorial to the crown prince discussing Zhang Zhou Xuan's calendar and also touched upon musical standards. His main point was that the essence of music is in its pitch, and the determination of pitch relies on standard tuning tubes. If the pitch is not accurate, harmony cannot be achieved. Thus, the length of the tuning tubes and the size of the bells must be precise. However, previous calculation methods were not accurate enough. They could never return to the starting point. During the Han dynasty, Jing Fang made a mess by creating sixty pitches, while Qian Lezhi during the Song dynasty produced three hundred sixty pitches. A close look at ancient texts shows there were no such claims; making hasty changes could cause issues. This would not only lead to discrepancies in pitch lengths but also errors in the sizes and numbers of the tuning tubes. Liu Zhao corrected all of these, in hopes of clarifying the situation. He calculated that the actual length of the Huangzhong pipe is sixty-three fen, then reduced each tuning tube by three fen, with seven fen equating to one cun. By this calculation, the Huangzhong is nine cun long, the Taizhu is eight cun one fen four li, the Linzhong is six cun, and the Yingzhong is four cun two fen eight li seven fen of four. That year, Emperor Gaozu passed away, and Emperor Yang had just ascended to the throne, not having had time to address this issue, which was then put on hold, leading to the loss of relevant texts.
In the second year of Daye, Emperor Yang of Sui ordered the use of the standard tuning of the Liang Dynasty to remake instruments such as bells and chimes, which was more in line with ancient standards than those of previous dynasties. However, the relevant systems, documents, and laws established by Mao Shuang were destroyed in a fire in Jiangdu. The "Book of Han" states: "The circumference of the Yellow Bell is nine fen, the Linzhong is six fen, and the Taicu is eight fen." The "Continuation of the Book of Han" and Zheng Xuan both state: "The diameter of the twelve pitch pipes is three fen, and the circumference is nine fen." Later, Prince Anfeng of Northern Wei, following the records of Ban Gu's "Book of Han," stated that the circumference of the Linzhong pitch pipe was six fen, and the Taicu pitch pipe was eight fen. As a result, using these pitch pipes did not produce tones that matched those of the Yellow Bell, Shang, and Zhi. Only when using pitch pipes with a circumference of nine fen could they match the standard bell instruments. After the pacification of the Chen Dynasty in the ninth year of Kaihuang, people such as Niu Hong, Xin Yanzhi, Zheng Yi, and He Tuo, referred to ancient pitch standards and, based on the actual situation at the time, each made a Yellow Bell pipe with a diameter of three fen and a length of nine inches. Variations in measurement caused the sound to differ in pitch; due to variations in diameter and measurement, the capacity of grains would also differ. I will now list these numbers:
- Before the Jin Dynasty, the Yellow Bell pipe could hold eight hundred and eight millet grains.
- Under the Sui Dynasty's legal measurement, the Yellow Bell pipe could hold eight hundred and twenty-eight millet grains.
- During the Sui Dynasty, the standard measurement of the Yellow Bell pipe had three types: one could hold nine hundred and twenty-five millet grains, one could hold nine hundred and ten millet grains, and one could hold one thousand one hundred and twenty millet grains.
- Under the measurement used by Han Dynasty officials, the Yellow Bell pipe could hold nine hundred and thirty-nine millet grains.
- On an antique silver vessel inscribed with "Huangzhong Yue can hold one thousand two hundred," indicating that the vessel could hold one thousand two hundred millet grains.
- During the Song Dynasty, they used an iron measuring stick to measure, and the Yellow Bell had two standards of capacity: one could hold one thousand two hundred millet grains, and the other could hold one thousand and forty-seven millet grains.
Looking back to the early Northern Wei Dynasty, at that time the Yellow Bell could hold one thousand one hundred and fifteen millet grains.
During the Northern Zhou Dynasty, when measured with a jade ruler, the volume of the Huangzhong was measured at one thousand two hundred and sixty-seven grains. During the Northern Wei period, the measurement results varied across different periods. In the middle period, the Huangzhong could contain one thousand five hundred and fifty-five grains, while in the later period, it could hold one thousand eight hundred and nineteen grains. By the time of the Eastern Wei Dynasty, the volume of the Huangzhong had further increased, able to hold two thousand eight hundred and sixty-nine grains of millet. There was also a type called the "Wanbao Changshuichi," which could measure the Huangzhong Lu Mu at one thousand three hundred and twenty grains of millet. The standard rulers and iron rulers of the Liang Dynasty had the same length and diameter, but could hold different amounts of millet grains. This was because the craftsmen intentionally altered the vessels to manipulate their volume. During the Northern Qi period, there was a man named Xindufang, who served as a military officer in the Tiancao of Emperor Shengwu's court. He was particularly clever and could predict the weather just by observing the clouds. One time, he pointed to the sky and said to someone, "The season of early spring has arrived!" When others checked his weather-predicting pipe, they found that the ash had already reacted. His monthly weather predictions were always very accurate. He also created a twenty-four-leaf wheel fan, buried underground, to track the twenty-four solar terms. At each solar term, one leaf of the fan would turn automatically while the others stayed still, perfectly aligned with the reaction of the ash in the pipe, as if it had been predetermined.
In the ninth year of the Kaihuang reign, after defeating the Chen dynasty, Emperor Yang Jian sent Mao Shuang, Cai Ziyuan, and Yu Puming to observe the solar terms. Following the previous method, they set up twelve wooden tables in a triple-locked chamber, with each table representing one of the Earthly Branches. Then, they placed the tuning pipes in order of the twelve Earthly Branches on the tables, buried them in soil with a level surface, filled the tables with reed ash, and covered the mouths of the tuning pipes with thin silk. Each month, when the solar term arrived and matched the sound of the tuning pipes, the reed ash would fly up, breaking through the silk and scattering outside. However, the responses of the solar terms varied, with different amounts of ash flying out at different times. Some responded right at the start of the month, while others only reacted in the latter part of the month. Some ash flew out in three to five days, while others took the entire month to fly out a little bit.
Yang Jian found this very strange and asked Niu Hong. Niu Hong replied, "When half of the ash flies out, it indicates a mild solar term; when all the ash flies out, it indicates a strong solar term; when no ash flies out, it indicates a weak solar term. A mild solar term corresponds to a peaceful political climate, a strong solar term corresponds to unruly officials, and a weak solar term corresponds to a tyrannical ruler." Yang Jian argued, "If officials are unruly and the ruler is tyrannical, the political climate will certainly not be peaceful, regardless of the month. The responses of the twelve tuning pipes in a year vary as well, so how can we say that tyrants and unruly officials will always be like this?" Niu Hong was at a loss for words. So Yang Jian asked Mao Shuang and his team to develop a systematic approach. Mao Shuang reviewed historical records and wrote a treatise titled "Lüpu." The treatise outlined the following points:
According to Minister Mao Shuang's research, the Yellow Emperor sent the Linglun family to gather bamboo in the Xian Valley and created the twelve tones under the nest of the phoenix bird. This enabled the qi of heaven and earth to be perceived, marking the origin of numerical systems. The yang tubes represent the pitches, while the yin tubes represent the lü; their qi can be utilized to forecast the four seasons, and their numbers can be used to record all things. Yun Li was the pioneer in developing numerical systems, which likely serves as the foundation of legal principles. The numbers one, ten, hundred, thousand, ten thousand, hundred million, and trillion are all derived from this, and measurements and standards also originate from it. Thus, the Yu clan used pitches to coordinate sounds, and Zou Yan modified it, establishing the five beginnings. The distinctions of the calendar and clothing colors also originated from this. The Xia Dynasty was based on humanity, the Yin Dynasty was based on the land, and the Zhou Dynasty was based on heaven. Confucius said, "I have mastered the calendar of the Xia Dynasty," meaning he grasped the essence of qi and numbers.
When the Han Dynasty was first established, Zhang Cang formulated laws and promoted the theory of "Five Victories," believing that the virtue of water should prevail. In fact, this was all due to the negligence of officials during the Warring States period, and the Qin Dynasty's destruction of scholarship led to a decline in this field of knowledge. Zhang Cang merely assembled fragmented information without fully understanding it. It was not until the reign of Emperor Wu of Han that the role of music coordinator was officially created, appointing Li Yannián as the chief coordinator. Although Li Yannián understood some new musical pieces and methods of modulation, he had not completely grasped the fundamental principles of music theory, so the dimensions and colors of musical instruments had yet to be established at that time.
By the time of Emperor Yuan of Han, he himself understood music theory, and his official Jing Fang was also well-versed in it. Emperor Yuan had Wei Xuan and others conduct experiments and research together with Jing Fang. Jing Fang explained, "I study Jiao Yanshou's theories using the method of six tones generating one another. The tones above generate those below; all follow the pattern of three generating two; the tones below generate those above, all follow the pattern of three generating four. Yang tones generate Yin tones downward, and Yin tones generate Yang tones upward; this is the proper method of the palace scale." Later, Liu Xin organized this material, writing a memorial that recorded the origins and developments of music theory, and his understanding of it deepened. Ban Gu's "Book of Han: Treatise on Music and Calendars" was entirely based on the materials compiled by Liu Xin, while Sima Biao's "Continuation of the Book of Han: Treatise on Music and Calendars" mainly referenced Jing Fang's theories.
By the Eastern Han, the dimensions of musical scales increased slightly. During the Wei Dynasty, Du Kui also established musical scales to observe the climate, reportedly so precise that ashes no longer flew. Xun Xu, a Grand Minister during the Jin Dynasty, obtained some ancient bronze pipes and compared them with the scales established by Du Kui, discovering that the ancient bronze pipes were a quarter longer than those established by Du Kui, which revealed that Du Kui's scales were inaccurate because of a measurement error. Therefore, Xun Xu recreated ancient measuring rods according to the records in the "Rites of Zhou" to determine the dimensions of musical instruments, ensuring that the tuning of the instruments was accurate.
After the Jin Dynasty, there were gradually errors in musical tuning. During the time of Emperor Wu of Liang, there was the jade tuning of Ji Zhong, and during the time of Cangwu County in the Song Dynasty, someone adapted the jade tuning for use with transverse flutes, but the dimensions of these instruments were largely retained. I dedicated over thirty years to studying and gained a solid understanding of musical tuning. Later, when I became the Minister of Imperial Sacrifices, responsible for managing music, I brought the jade ruler and the ruler from the Song Dynasty’s Grand Historian to the court for presentation. The court ordered craftsmen to make instruments according to the style of these objects. From that point on, musical tuning was precise and consistent. During Hou Jing's rebellion, my brother Xi acquired these instruments from the Bureau of Music. Later, when Emperor Chen Xuan was held hostage in Jingzhou and witnessed the defeat of Emperor Yuan of Liang, my brother died in Northern Zhou. He was preparing to report to the court, but shortly after, Emperor Chen Wu ascended the throne. Emperor Chen Wu derived sixty tunings from the twelve original tunings, privately studied climate change, and made some notable discoveries. During the reign of Emperor Chen Taijian, my brother became the Minister of Personnel and wanted to report these items to the court. Eventually, Emperor Chen Xuan passed away, and Empress Chen took the throne, demoting my brother to serve as the interior minister in Yongjia. He left these items at home to be passed down to his descendants. When the Chen Dynasty fell, these items were ultimately lost.
The twelve tuning pipes used in the palace are determined based on the fluctuations of yin and yang. Starting from the Yellow Bell, the yang energy gradually weakens, while the yin energy gradually increases, culminating in Zhonglv, where all the seasonal changes throughout the year are included. After Zhonglv comes Zhishi, followed by Qu Mie, and finally Nanshi, encompassing a total of sixty musical tones. During the eleventh month of late autumn, the Yellow Bell should be sounded. The Yellow Bell marks the beginning of winter solstice, symbolizing the beginning of yang energy. Its length is nine inches, corresponding to the days of the year. In the eleventh month, the Yellow Bell should be sounded to express the six qi and balance the nine virtues. After that, the standards set by Jing Fang for the tuning pipes are followed, with the lengths of Gong and Zhi tones being used in order each day. There are a total of twelve tuning pipes, each governing specific tonal ranges, extending to form sixty musical tones. This can also be inferred from the Eight Trigrams, transforming into sixty-four hexagrams. The tuning pipes that generate one another will transform; for example, the Yellow Bell pipe will generate the Lin Bell pipe, because yang energy generates yin energy, so there will be changes. The tuning pipes that interact with each other will resonate, for example, the Zhonglv pipe governs the harmony of all things, akin to a mother guiding her children. Therefore, the tuning pipes that change will resonate at different times, while the tuning pipes that resonate will harmonize in sequence within the same month. If the harmonization times are not consistent, it is not the primary tuning pipe's solar terms, but rather the subordinate tuning pipes resonating with each other via the primary tuning pipe. These tuning pipes were also lost during the end of the Daye period when Emperor Yang of Sui was killed in Jiangdu (Yangzhou).
In the Song Dynasty, Qian Lezhi, while researching Jing Fang's theories on music and tuning, also created the 360 musical notes himself. In his "Treatise on Tuning," the scholar Shen Zhongzhong from the Liang Dynasty remarked: "The 'Book of Changes' uses 360 hexagrams to correspond to the number of days in a year; this is the numerical system of music and the calendar. The 'Huainanzi' says: 'One pitch produces five sounds, twelve pitches produce sixty sounds, and multiplying by six gives three hundred sixty sounds, corresponding to the number of days in a year. The numerical system of music and the calendar reflects the laws of the operation of heaven and earth.' Therefore, the method for calculating these 360 musical notes has remained unchanged since ancient times." Shen Zhongzhong, based on the numbers in the "Huainanzi," employed Jing Fang's method to calculate the 360 musical notes. He took the fundamental note for each month as a baseline, used the unit's number as the numerator, and the number of days included in each solar term as the denominator, determining the fractional value corresponding to each note based on the number of days. Then, he allocated the seven sounds accordingly; for example, the note for the winter solstice is Huangzhong, with Huangzhong as the Gong, Taicu as the Shang, Linzhong as the Zhi, Nanlü as the Yu, Guxi as the Jiao, Yingzhong as the Bian Gong, and Ruibin as the Bian Zhi. In this way, the five sounds and seven tones are now complete. The notes for other days follow this sequence. The note for the day corresponds to the Gong sound, with the Shang, Zhi, and other sounds arranged in order. By studying sound and air, one can discern the sequence of time; all living things grow according to the laws of solar terms.
From Huangzhong to Zhuangjin, there are a total of 150 musical notes generated using the method of three parts reduced by one; from Yixing to Yizhao, there are a total of 209 musical notes generated using the method of increasing three parts by one; only the Anyun note is final, with no new notes generated. The lengths of these musical notes are calculated based on the actual length of Huangzhong, which is 177,147, using a ratio of three to nine, dividing by this base to obtain the lengths in cun, fen, and small fen, discarding the remainder, resulting in the length for each note. The revision of the pitch system is arranged in the order of upper and lower generation, Gong and Zhi. Now, I will briefly mention the names of the musical notes in the Huangzhong section: Huangzhong: Baoyu, Hanwei, Dide, Guangyun, Xiaji, Kezhong, Zhishi, Wojian, Chishu, Huangzhong, Tongsheng, and Qiansheng.
Yinpu, Jingsheng, Zimeng, Guangbei, Xianheng, Naiwen, Naisheng, Weiyang, Fendong, Shengqi, Yunfanyuyan
Shengyin, Tunjie, Kaiyuan, Zhiwei, Xiemei, Bujian, Xuanzhong, Yuzhu, Diaofeng
One section of the Right Yellow Bell, consisting of thirty-four pitches. (Each melody consists of thirty-four sections, with each section corresponding to one of the thirty-one days in a month.)
Great Lu:
It is said that the movement of heaven and earth has its own rules, just like music has its highs and lows, its fast and slow. The ancients summarized these rules into five types of music, namely Great Lu, Tai Cu, Jia Zhong, Gu Xi, Zhong Lu, and Rui Bin. This isn't merely music; it holds significance for a nation's rise and fall, as well as the flourishing of all living things.
First is Great Lu, a total of twenty-seven melodies, each corresponding to a day, with an extra third of a day. Great Lu, much like a strong leader, aids the king in ruling the nation, distinguishing right from wrong and fostering prosperity. Even in tough times, it must uphold justice, remain strong and unyielding, and shine like the sun, discerning right from wrong, uniting as one, adapting to the times, being flexible, talking less and acting more, working together with the king in unity and inheriting and promoting the glorious tradition. "Gao Dong, Shi Zan, Da You, Kun Yuan, Fu Shi, Kuang Bi, Fen Fou, You Fan, Wei Wei, Qi Wang, Shu Ji Zhi Yi, Bing Qiang, Ling Yin, Lu Yang, Shi Shen, Ji Xi, Zhi Dao, Shi Shi, Quan Bian, Shao Chu, A Heng, Tong Yun Cheng Ming, Shan Shu, Xiu Guang." These twenty-seven melodies embody the cyclical nature of the seasons, perpetually renewing themselves throughout the year.
Then comes Tai Cu, a total of thirty-four tones, seven more than Da Lü! This Tai Cu is like the beginning of all things' growth, from unnoticed sprouting to standing tall, and then to flourishing with abundant branches and leaves, experiencing the changes of spring, summer, autumn, and winter. "Unknown its own meaning builds pavilion wind fits beginning time breath reaches life gourd plays initial horn gentle young yang soft paddle merchant sound bends aligns supports weakness bears aligns moves plants all uplifted combines mountain stops speed follows period dragon leaps hooks brightness regulates green essentials binds calyx extends spreads punishes advances distinguishes order east makes praises shows stagnation falls." You see, from weakness to strength, from growth to maturity, these thirty-four tones depict the vibrant scene of nature.
Next is Jia Zhong, with twenty-seven tones, which resembles the harmonious scene between heaven and earth, where yin and yang combine, and all things grow. "Bright multitude harmonizes partners yin praises the wind governs myriad changes opens time shakes virtue rides strings fragrance disperses gentle air wind rushes carefree joy dry party four gaps breeds nature indulges in leisure benevolent might competes south dawn morning grows fulfilled groups divide pure and new." Look at how all things thrive and flourish, what a beautiful scene!
Now let's look at Gu Xia, with thirty-four tones, which is like the return of spring to the earth, a scene of all things reviving. "South bestows embraces comes to test the spirit reveals carries horn washes away the old and brings joy uplifts outstanding energy begins to rise gentle clouds charms mountains clears paths road time daily journey truly sinks scorching wind first section soft strings aligns initial beginning aligns things flourish reforms tender shoots abundant and robust ascends brightness strong progress (lower gives peace and fortune) depends on action (upper brings nurturing) young choices follow the path of vermilion and black, embodying integrity." These tones depict the scene from spring to summer, where everything thrives and flourishes.
Zhong Lu, with its twenty-seven tunes, evokes the image of a summer in full bloom. "Zhu Ming Qi Yun Jing Feng Chu Huan Yu Wu Si Fen Nan Zhong Li Chun Shuai Nong You Cheng Nan E Jing Zhi Zhi Xiang Qu Nei Zhen Zhu Cao Han Hui Qu Yi Yao Chou Si Qi Qing He Wu Ying Jie Mou Huang Luo Zhen Zhen Tian Ting Zuo Zhou." This tune vividly depicts the ripening and harvesting of all things.
Finally, we have Rui Bin, which symbolizes a grand spectacle, with a total of thirty-four tunes. "Nan Shi (Jing Fang Zhong Lu) Mi Jing Ze Xuan Bu E Man Lei Qian Dong Sheng Bian Bin An Huai Yuan Sheng Ji Gui Tong Hai Shui Xi Li Gong An Zhuang Chong Ming Yuan Tiao Sheng Zhong Feng Zhu Zhao Yang Zhi Shi Rui Tong Chun Huo Yi Ci Gao Yan Qi Huang." It is like a grand festival, an auspicious atmosphere permeates, creating a scene of thriving prosperity.
In conclusion, these five sets of music tunes are not only music but also the ancient people's deep understanding of the laws governing the cosmos, as well as their philosophical thinking on the harmonious unity of the universe.
The following is a collection of musical tunes, divided into five parts: Rui Bin, Lin Zhong, Yi Ze, Nan Lu, and Ying Zhong. The Rui Bin section has twenty-seven tunes.
The Lin Zhong section consists of thirty-four tunes, which include themes of humility and service, advocating virtue, and following reason, which can lead to strength; as Yin energy rises, evil is absent; eliminating destruction, radiance fades, friends celebrate, clouds fill the sky; average responsibility, admire achievements; tolerance and peace, pass calmly; virtues are average, no obstacles; sufficient etiquette, profound wisdom; serious responsibility, pure loyalty; return to beauty, wonderful sound, warm wind, follow the solar terms; Mimosa flowers bloom, embroidered mountain peaks; there is no distinction between good and bad among all things; Jing Kou, Yao Jing; the sun shines brightly, the heavy wheel turns; wealth and prosperity.
The Yize section has twenty-seven rules. The content is: ascending merchant tune, clear and refreshing, with refined charm, female virtue, white concealment, following laws, reducing penalties, maintaining purity and self-restraint, Golden Heaven, Liu Shan, returning to the path of benevolence; accompanied by yin energy, leaving the south, sunlight fading, gentle and diligent; prolonging the second element, harmonizing the seventh element, without weeds, distributing savings, Kong Xiu; the nine virtues are all pure and everyone believes that it will bring peace to the world.
The Nanlu section has thirty-four rules. The content is: white Lu, abandoning luxury, being sincere and simple, elegant atmosphere, strong objects; the leader thrives, united and self-disciplined; fertile and retiring, among the weak; grand yin energy, opposing moral rigidity, intimidating distant lands; moderation, returning home; the Doctrine of the Mean, the strategy of the king; truly blocking the way, reaping the rewards; reining in excess, letting go; without any marks, quality is balanced; distributing fullness, the Way of the Heart; purity and strength, savings ceased; returning to storage, Yihan; justice and equality, making people happy, without toil; the nine continents, shining brightly.
The Wusha section has twenty-seven rules. The content is: thoughtful impulse, embracing humility, respectful frugality, nurturing life, caring for the people; eliminating disasters, blocking concealment; the descent of Lou, deeply hidden; the sun shines perpetually, spring cycles, castration is hidden; bright Kui stars, neighbors are equal, following the masses; large-scale savings, frugal restraint; lifting up the people, easing burdens; endless, aspiring for preservation; longevity; deep autumn scenery, mysterious moon, clear sky.
Finally, the Yingzhong section has a total of twenty-eight rules. The content is: clear allocation, tracing the minutiae, grasping the beginning, achieving success and fame, ensuring stability, calm and peaceful; slowly restrained, ruling through non-action; Gushan, heavy and obscure, turbulent and quiet; recruitment, not yet pregnant; countless opportunities, boundless longevity, lasting forever; restoration, slow timing, establishing systems, never stopping; the nine fields, the eight wildernesses, countless trillions, peaceful functioning.
The "Records of the Grand Historian" mentions: "Emperor Yu of the Xia used his own body as a standard for measurements and used sound to determine musical pitch." The "Book of Rites" mentions: "The length of a man's outstretched palm is one foot." The "Rites of Zhou" records: "The edge of a bi disk is used to determine measurements." Zheng Sinong explained: "'Xian' refers to length. This means that the diameter of the bi disk is one foot, which is used to establish the measurement standard." The "I Ching" says: "The length of ten horse tails is one 'fen'." The "Huainanzi" states: "At the autumn equinox, the awns on the grain heads take shape, indicating that the grain is ripe. The number of musical pitches corresponds to the length of twelve awns being equal to one grain of millet, and twelve grains of millet equal one inch." Here, "緌" refers to the awns on the grain heads. The "Shuoyuan" states: "The standard for measurements originates from millet, with one grain of millet equaling one 'fen'." The "Suanzi Arithmetic" states: "The silk produced by silkworms is called 'hu', ten 'hu' equal one 'miao', ten 'miao' equal one 'hao', ten 'hao' equal one 'li', and ten 'li' equal one 'fen'." These are all records of the origins of measurements, but the accounts contradict each other. Only the "Book of Han·Treatise on Music" records a more reliable account: "Measurements are tools used to measure length and are fundamentally derived from the length of the yellow bell. Using medium-sized millet—jiu-shu, the width of one grain of jiu-shu is one 'fen', and the length of ninety grains of jiu-shu is the length of the yellow bell. One jiu-shu is one 'fen', ten 'fen' equal one inch, ten inches equal one foot, ten feet equal one zhang, and ten zhang equal one yin, thus establishing five measurement units." Later authors based their measurement standards on this account, and because the size of jiu-shu varies, various different methods emerged, but the basic principle is the same. The issue is that the size of jiu-shu varies, and there are differences between years of good and poor harvests. The calibration of ancient measurements frequently experienced discrepancies, and errors that arose during the oral transmission gradually resulted in deviations. Next, I will briefly list fifteen different scales from various dynasties, along with their similarities and differences. The "Book of Han·Treatise on Music" records the copper ruler of Liu Xin during the Wang Mang period; the copper ruler from the Jianwu period of the Eastern Han; the law ruler defined by Xun Xu in the tenth year of the Taishi era of the Jin Dynasty, which is the ruler before the Jin Dynasty; and the copper ruler handed down by Zu Chongzhi.
In the ninth year of the Tai Shi era of the Jin Wu Emperor, the Director of the Central Secretariat, Xun Xu, inspected the eight tones of the court orchestra and found that the pitch was incorrect. After careful research, he discovered that the rulers from the Han Dynasty to the Wei Dynasty were over four fen longer than the ancient rulers. Xun Xu then had the official Liu Gong create a ruler that conformed to the ancient standard based on the records in the "Rites of Zhou," which is known as the "ancient ruler."
They then used this ancient ruler to recast the bronze instruments for tuning the musical instruments. When measuring ancient musical instruments with this ancient ruler, they found that the dimensions matched perfectly with those recorded in the inscriptions. Interestingly, someone in Ji County excavated the tomb of King Xiang of Wei and surprisingly found jade lü and bells from the Zhou Dynasty that produced sounds remarkably similar to the newly cast instruments! At that time, some old Han Dynasty bells were also discovered in various locations, and when tested with the new instruments, the pitches matched perfectly.
The "Bell and Pitch Treatise" records that Zu Chongzhi also obtained a bronze ruler inscribed with inscriptions, which documented that in the tenth year of Tai Shi, the Central Secretariat examined ancient artifacts and compared them to the rulers of that time, discovering that they were four and a half fen longer. The inscriptions mentioned seven ancient artifacts used for calibration: first, the Gu Xie jade lü; second, the Xiao Lü jade lü; third, the copper Wang Nian from Xi Jing; fourth, the gold-corrected Wang Nian; fifth, the copper hu; sixth, the ancient coins; and seventh, the Jianwu copper ruler. The inscription stated that the pitch of the Gu Xie jade lü was a bit higher, the pitch of the copper Wang Nian from Xi Jing was slightly lower, and the others were all consistent with the standard of this bronze ruler. There are a total of eighty-two characters in this inscription, and this bronze ruler is the one newly made by Xun Xu. The ruler currently in common use is the one made by Du Kui.
The "Bell and Pitch Diagram" by Lei Cizong and He Yinzhi records Xun Xu's correction of the ancient ruler, which matches the content of this inscription. However, Xiao Ji's music scores incorrectly state that the seven artifacts were verified during the Liang Dynasty. We should now use this bronze ruler as the standard for calibrating the rulers of different dynasties.
During the Liang Dynasty, the Liang Dynasty's ruler was actually one foot and seven tenths longer than the previous rulers before the Jin Dynasty. It is said that a farmer once dug up a jade ruler from the Zhou Dynasty in the field, which was considered the standard ruler of the world. Xun Xu used it to calibrate musical instruments and found that the metal strings and bamboo instruments he crafted were all one meter shorter than the jade ruler. The "Zhong Lu Wei Cheng" written by Emperor Wu of Liang also recorded that the Zhou Dynasty's bronze ruler and eight ancient jade metrics were preserved in the imperial palace. Later, it was found that the Zhou Dynasty's ruler was taken by the Dong Hunhou as a token of goodwill and given away, resulting in the ruler being lost. One of the jade metrics was broken, and the remaining seven were clamped with the laws of the bells, which still bore earlier inscriptions. So Emperor Wu of Liang commissioned the creation of a new ruler based on these jade metrics for cross-verification. They repeatedly measured with fine hairs and millets, accurately determining the length of the ruler, making it the most accurate ruler of the time, half a tenth longer than that of Zu Chongzhi. Then they crafted four musical instruments called "Tong" with this new ruler. They also crafted a flute based on this new ruler and used it to calibrate the ancient bells, determining the pitch of the Yi Ze sound in accordance with the scale, and using the flute to determine the pitches of the Yin and Yun, which were found to match. You see, the lengths of these two rulers are almost the same. Xiao Ji noted in his writings: "This method originates from the 'Sima Fa'. The Liang Dynasty engraved the scale of the ruler on the shadow table to measure the sun's shadow." It refers to the bronze guibiao crafted by Zu Tan at the emperor's behest. Later, when the Chen Dynasty was destroyed, this guibiao was brought into the court. During the Sui Dynasty, someone proposed the restoration of ancient standards, thus, it was utilized to calibrate musical scales and create eight instruments, including bells and chimes. While digging in Shiping (now Xianyang City, Shaanxi Province), someone discovered an ancient bronze ruler.
Xiao Ji's "Musical Scores" records: "During the reign of Emperor Zhang of Han, Xi Jing, a literary historian from Lingling, unearthed a set of jade measuring rods under the Shun Temple in Lingdao County and made this ruler according to it." Fu Chang's "Praise of the Jin Public" says: "The bell measures made by Ge Xu were considered very precise at the time, but only Ruan Xian from Chenliu criticized its pitch as too high. Later, Shimiping dug up that ancient bronze ruler, which was almost rotting, compared it with Xun Xu's ruler, and discovered it was a quarter of a foot shorter. People at that time all agreed with Ruan Xian." The lengths of these two measuring rods were also comparable.
In the fourth year of Jin's Prince Jingyuan of Chenliu, Liu Hui commented in "Nine Chapters on Arithmetic" that the measures used by Liu Xin during the Wang Mang period were four and a half inches shorter than the rulers at the time. Compared to the rulers of the Wei Dynasty, the depth of the measure was nine inches and five-tenths. This is the same as what Xun Xu of the Jin Dynasty said, "Du Kui's ruler is four and a half inches longer than the current ruler."
Xiao Ji noted that these were the measuring rods used in the Jiangdong region during the Jin Dynasty. The market measuring rod of the Later Zhou Dynasty was one foot nine inches shorter than the jade measuring rod. The official measuring rod during the Kaihuang period, known as the iron ruler, is one foot two inches long. Before this, from the early Wei Dynasty to the division of the Eastern and Western Wei, until the Later Zhou Dynasty stopped using the jade ruler, people commonly used these measuring rods interchangeably.
It is said that the "Zhenluan Arithmetic" states that the market measuring rod of the Zhou Dynasty was a jade measuring rod of nine inches and two-tenths. Some say that this was crafted by a Daoist named Zhigong during the Liang Dynasty, sent to the Zhou Dynasty, and given to an old man with a long beard. Both Emperor Taizu of the Zhou Dynasty and Emperor Gaozu of the Sui Dynasty believed that this measuring rod was created for them! In any case, people of the Zhou Dynasty were using this ruler. By the time of the Kaihuang period of the Sui Dynasty, the official standard was formally established as the official measuring rod, which all departments had to use, and it continued to be used until the Renshou period. By the time of the Daye period of the Sui Dynasty, some common folk also began using it privately.
This jade ruler was actually made by Yuan Yanming, a lieutenant of the Wei Dynasty, using the width of half a grain of millet as his standard. The Qi Dynasty subsequently adopted this ruler. Wei Shou wrote in the "Book of Wei: Treatise on Law and Calendar": "Gong Sun Chong created a new ruler during the Yongping years, using the length of one grain of millet to define a 'cun' (inch). Later, the Minister of Rites Liu Fang was ordered to revise the music standards and used the width of a medium-sized grain of millet to define a 'fen' (one-tenth of a 'cun'). As for Lieutenant Yuan Kuang, he defined a 'fen' using the width of one grain of millet plus the width of two seams. The three interpretations differed, and they debated for a long time without reaching a conclusion. In the 19th year of the Dahe era, Emperor Gaozu ordered that the width of one grain of millet be used as a 'fen', with ninety grains of millet equating to the length of the Huangzhong, a type of ancient musical instrument, thus determining the length of the bronze ruler. Relevant departments reported that Liu Fang's ruler was the same as the one established by Emperor Gaozu, so Liu Fang's ruler was used to revise the standards for bronze and stone measurements. Until the Wuding era, legal matters were no longer discussed."
Then, there was also a bronze ruler made by Cai Yong. The jade ruler from the Later Zhou was longer by one foot, one inch, and five-tenths of a centimeter compared to rulers before the Jin Dynasty.
From ancient times to the present, a bronze ruler has been handed down, with silver-inlaid characters. The inscription reads: "Ruler, the realm of Huangzhong, nine inches long, nine-tenths in circumference, capable of holding twelve hundred grains of millet, weighing twelve zhu (a unit of weight), two of which make one he. Three-tenths of gain or loss, converting into twelve pitches." Ancestor Xiaosun claimed that this item is the bronze ruler made by Cai Yong.
During the Baoding years of the Later Zhou Emperor, the emperor commanded Grand Minister Lu Jingxuan, Duke of Shangdang Changsun Shaoyuan, and Duke of Qiguo Husi Zheng to make rulers using millet, but they were unable to reach a consensus on its length. Later, while repairing the granary, they unearthed an ancient jade dou (a measuring vessel) and used it as a standard reference, establishing laws and measurements based on the jade dou. As a result, this ruler was adopted, and a general amnesty was subsequently declared, changing the era to Tianhe. All departments used this ruler until the end of the Daxiang era. The Huangzhong measurement of this ruler matched that of Cai Yong's ancient ruler.
Finally, there is the Song Dynasty ruler, which is one foot, six inches, and four tenths longer than the rulers used before the Jin Dynasty. There are also Qian Lezhi's celestial sphere ruler, the iron rulers from the Later Zhou, the rulers used for adjusting musical measurements during the early Kaihuang period, and the water rulers used to adjust musical measurements after pacifying the Chen Dynasty.
This ruler was commonly used by the people during the Song Dynasty and later spread to the Qi, Liang, and Chen dynasties for making musical instruments and determining musical scales. The length is similar to the rulers used after the Jin Dynasty, the rulers used by the people in the Liang Dynasty, and the ruler on Liu Yao's celestial sphere. This is likely due to long-term usage by the people, which inevitably led to some additions and changes over time.
In the sixth year of Jiande during the Zhou Dynasty, after pacifying the State of Qi, this ruler was adopted as the standard measurement and promoted for nationwide use. Later, during Emperor Xuan's reign, Daxi Zhen, Niu Hong, and others discussed:
I carefully studied the weighing and measuring of these things; governing a country requires good rules, and it is essential to thoroughly examine previous materials and verify them to find the most suitable ones. I checked the iron ruler currently in use, which was made based on a ruler created by Minister Su Chuo under Emperor Taizu. It was carefully inspected at the time and was used during the previous Zhou dynasty. I compared it and found that the length of this iron ruler matches that of the Song dynasty ruler, which was used to calibrate musical pitches and measure land area. Right now, I'm using millet from Yangtoudshan in the Shangdang region for my measurements, following the method outlined in the "Book of Han: Treatise on Law and Calendars." If I stack large millet to fill one foot according to quantity, I have to shake it vigorously to make it full when measuring the Huangzhong tuning. If I use medium-sized millet to fill one foot, even though it looks a bit sparse, I can fill it without shaking when measuring the Huangzhong tuning. These two situations are different mainly because of the uneven size of the millet, so using the iron ruler for measurement will ultimately have a slight error. Moreover, the millet from the Shangdang region is different from that of other places; it is particularly dark and plump, and using it for measurement is certainly not a random choice. This mainly stems from the varying rainfall and drought conditions across regions, along with differences in soil fertility, which makes it hard to find an average size. Xu Shen explained in "Shuowen Jiezi" that large millet is very big and different from ordinary millet. I believe that the large millet we have now is the same as that large millet from back then; stacking one hundred of them to fill one foot is in accordance with ancient methods. Besides the millet that fills one foot, there are still about a dozen left, which could be due to errors in measuring the millet's diameter, or maybe the laws themselves aren't precise enough. Of course, the idea that you need to shake it vigorously to fill it also makes sense.
We have now examined the ancient coins of the Zhou and Han dynasties, and they are similar in size; the armillary sphere of the Song dynasty also has no issues with its scale. In addition, according to the records of the "Huainanzi," twelve grains of millet stacked together equal one inch. This indicates that the laws set by ancient emperors, meticulously calculated through legal means, are consistent. The "Book of Han, Treatise on Food and Money" states: "A square inch of gold weighs one catty." When we now melt gold for testing, we find that the iron ruler is the closest match. Based on historical records and actual calculations, there are many points of agreement. Furthermore, it has been in use, and confirming it now aligns better with reality. As for measuring millet with a jade ruler, taking width as length, when piled up, there is still some left over, but in reality, it cannot be filled completely. We have consulted ancient and modern materials, and it seems that this method may not be feasible. The rulers of the Jin and Liang dynasties were too short and small; when filled with millet, they simply cannot contain it, and using this for calibration would certainly lead to a pitch that is too high and sharp. Harmonizing the eight musical tones is the ideal of a wise and virtuous ruler, and unifying laws and measurements is the guiding principle for future generations. We carefully examined previous literature and weighed the pros and cons based on current circumstances, and we believe that using an iron ruler is the most appropriate choice.
Before we could finalize our decision, Gaozu passed away. Niu Hong, Xin Yanzhi, Zheng Yi, and He Tuota discussed for a long time but couldn't reach a conclusion. After pacifying the Chen dynasty, the emperor thought that the music of Jiangdong was good, saying: "This is the ancient music of Huaxia; although it has changed with the times, it still largely follows the ancient ways." Zuxiaosun said: "After pacifying the Chen dynasty, we abolished the jade ruler and laws of the Zhou dynasty and adopted this iron ruler and its laws, stipulating that one foot two inches is defined as the market foot."
In the tenth year of the Kaihuang reign, the water ruler made by Wanbaochang was actually one foot one inch eight fen and six li longer than the ruler before the Jin dynasty. "The current Taiyue treasury and the internal bronze law, a section of which was made by Wanbaochang, is referred to as the water ruler law. It is said that its Huangzhong pitch should be twice as loud as that of the iron ruler Nanlv. Nanlv, also known as Huangzhongyu, is why it is called the water ruler law."
Now, both the Tai Le Ku and the imperial court have taken out a set of bronze tuning pipes, crafted by Wanbao Chang, called the Water Ruler tuning. It is said that its Huangzhong pitch is equivalent to twice the Nanlü pitch of the Iron Ruler. Nanlü is the Huangzhong Yu sound, so it is called the Water Ruler tuning.
Fourteen, various measuring rods, such as the Tu Gui measuring rod on the Zhao Liu Yao Hun Tianyi, are four fen and three li longer than the official measuring rod of the Liang Dynasty, and five fen longer than the ruler before the Jin Dynasty. "The Tu Gui measuring rod on the Zhao Liu Yao Hun Tianyi is longer than the Liang official measuring rod by four fen and three li, actually longer than the ruler before the Jin by one foot and five fen."
Fifteen, the measuring rod commonly used by the people of the Liang Dynasty is six fen and three li longer than the official measuring rod of the Liang Dynasty, two fen longer than the ruler on Liu Yao's celestial sphere, and seven fen and one li longer than the ruler before the Jin Dynasty. Emperor Wu of Liang said in the "Zhong Lu Wei": "After Emperor Wu of Song pacified the Central Plains, he sent the Hun Tianyi and Tu Gui, claiming they were crafted by Zhang Heng. However, according to the inscription on the Hun Tianyi, 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 Zhang Heng. The measuring rod he produced is four fen and three li longer than the current standard measuring rod, and two fen shorter than the measuring rod commonly used by the people." Here, the "new measuring rod" refers to the official measuring rod of the Liang Dynasty. "The customary measuring rod of the Liang Dynasty is six fen and three li longer than the Liang official measuring rod, two fen longer than Liu Yao's celestial measuring rod, actually longer than the ruler before the Jin Dynasty by one foot and seven fen and one li. Liang Wu's 'Zhong Lu Wei' says: 'Song Wu pacified the Central Plains, sent Hun Tianyi and Tu Gui, claiming it was made by Zhang Heng. The inscription on the Hun Tianyi 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 Zhang Heng. The measuring rod is four fen and three li longer than the current new measuring rod, and two fen shorter than the measuring rod commonly used by the people.' The new measuring rod refers to the Liang official measuring rod."
Wow, this article discusses ancient measurement systems, packed with information! Let's go through it step by step.
First, the "Zhou Li" describes a measuring instrument called "Lu," which is one foot deep, square on the inside and round on the outside, with a volume equivalent to one Lu; its base (bottom) is one inch, with a volume of one Dou; its ears (handles) measure three inches and have a volume of one Sheng; it weighs one Jun (a unit of weight); its sound is in harmony with the Huang Zhong; and it is exempt from taxation. Inscribed on it are the words: "After careful consideration, it has achieved perfection. The fine measure is established, to observe the four nations. Forever to be initiated, this instrument serves as a standard." This inscription is quite grand, meaning that after repeated study, this perfect measuring instrument has finally been created for the world to see, allowing future generations to use it as a standard.
The "Zuo Zhuan" mentions that in ancient Qi, there were four types of measuring instruments: Dou, Qu, Lu, and Zhong. Four Sheng equal one Dou, and using this analogy, one can derive the volumes of Lu and Zhong. One Lu is equal to six Dou and four Sheng, and ten Lu equal one Zhong, equivalent to sixty-four Dou. Zheng Xuan's calculation of Lu's volume differs slightly from that in the "Nine Chapters on the Art of Calculation," and Zu Chongzhi, using a more accurate method, determined Lu's volume to be 1562.5 cubic inches. Based on its square inside and round outside shape, the diameter is approximately one foot, four inches, one-tenth of an inch, and two-tenths of a hair's breadth, with a depth of one foot, which closely resembles the size of the ancient Hu.
The "Nine Chapters on Commercial Calculation" records the volumes of Hu for various grains, such as millet, beans, hemp, and wheat, which vary in volume based on their differing densities, yet their prices remain consistent. Using the rice Hu (a standard measure) as a reference aligns with the records in the "Book of Han." The "Sunzi Suanshu" discusses even smaller units: six Su equal one Gui, ten Gui equal one Miao, ten Miao equal one Cuo, ten Cuo equal one Shao, and ten Shao equal one He. Ying Shao and Meng Kang provide slightly different definitions of Gui, yet both pertain to millet.
"The Book of Han" states that in ancient times, there were measuring tools such as the yue (龠), he (合), sheng (升), dou (斗), and hu (斛), which were used to measure quantities. The standards for these measuring tools were determined based on the musical pitch of the Huang Zhong, made of bronze, with a square interior and round exterior, shaped like a jue (an ancient wine vessel). On top was the hu, below was the dou, and on the left and right were the sheng, he, and yue. These measuring tools were intricately designed, embodying concepts like yin, yang, and the five elements. They weighed two jin, had specific volume standards, and their sound harmonized with the Huang Zhong.
Finally, "The Book of Han" also records the inscription on the hu: "The musical scale of the hu is square on the inside and round on the outside, with a side measurement of nine li and five hao, a circumference of one hundred and sixty-two cun, and a depth of one chi, with a total volume of one thousand six hundred and twenty cun, equivalent to ten dou." Zu Chongzhi's calculation based on the value of pi slightly differs from the inscription, while Liu Xin's calculation differs even more, possibly due to his less precise method.
In conclusion, ancient people conducted detailed studies on weights and measures, from simple measuring tools to complex calculation methods, showcasing the ingenuity of ancient laborers. These records provide valuable information for understanding ancient societies.
In the year 226 AD, Liu Hui noted that the hu used by the Grand Minister of Agriculture today has a diameter of one chi three cun five fen five li, a depth of one chi, and a volume of one thousand four hundred and forty-one cun and three fen. The bronze hu dating back to the Wang Mang era, calculated using today's measurements, had a depth of nine fen five li, a diameter of one chi three cun six fen eight li seven hao. By my calculation, the volume of the bronze hu from the time of Wang Mang was only slightly more than nine dou, seven sheng, and four he of today's standards. Therefore, the hu of the Wei Dynasty was larger, with longer measurements, while the hu from Wang Mang's time was smaller, with shorter measurements.
The Liang and Chen Dynasties continued to use the ancient system of weights and measures that had been in place. The Qi Dynasty established that five ancient sheng equaled one dou.
In the first year of the Bao Ding era of the Later Zhou (561 AD), in May, when the Jin State was building granaries, an ancient jade measure was discovered. By the fifth year of Bao Ding (565 AD), in October, Emperor Wu ordered the re-establishment of copper laws for weights and measures, ultimately achieving a unified and harmonious system of measurement. When millet was piled up, the volume measured was completely consistent with this jade measure, showing no discrepancies compared to other weighing and measuring tools. Thus, a copper sheng was cast using this jade measure as the standard and promulgated for use across the country. The inscription on the copper sheng reads: "In the first year of the Bao Ding era of the Later Zhou, in the year of Chongguang, in the month of Ruibin, when the Jin State government repaired the granaries, this ancient jade measure was discovered. Its shape and specifications are very standardized, akin to the ancient standard measuring instruments. The Grand Master of the Jin State reported this matter, and the Emperor ordered that the jade measure be kept in the Tianfu. By the fifth year of Bao Ding, in the year of Xieqia, the Emperor ordered careful measurements using tools such as ropes, rules, and standards, checking repeatedly to ensure there were no errors whatsoever. The jade measure was subsequently melted down and cast into a copper sheng, promulgated throughout the land to achieve a standard of measurement for a peaceful and prosperous era." This is how the inscription on the copper sheng is written. The inscription on the jade sheng reads: "In the first year of the Bao Ding era of the Great Zhou, in the year of Chongguang, in the month of Ruibin, when the Jin State government repaired the granaries, an ancient jade sheng was obtained, its form and structure are correct, like the ancient excellent measures. The Grand Master of the Jin State reported this, and it was ordered to be kept in the Tianfu. In the fifth year, in the year of Xieqia, the Emperor then decreed to measure using ropes and rules, examining the standards carefully, ensuring there were no discrepancies in weight or volume. Thus, it was melted down and cast, and promulgated throughout the world to align with the standards of peaceful measurement." Now, if calculated numerically, the volume of this jade sheng is approximately 110.8 inches, equivalent to a volume of 1,185.739 cubic inches for a hu. Zhen Luan recorded in "Arithmetic": "One jade sheng equals one official dou, three he, and four ladles." Therefore, the jade sheng is comparatively larger, while the official dou is relatively smaller. Based on numerical calculation, the official dou of the Later Zhou referenced by Zhen Luan has a volume of approximately 97 inches, equivalent to a volume of about 977 inches for a hu. The jade dou from the Later Zhou, along with the corresponding gold-inlaid copper dou and the gold-inlaid copper dou from the sixth year of Jiande (577 AD), were all determined using millet for volume. When using the jade dou for weighing, one sheng of grain consistently weighs six jin and thirteen liang.
During the Kaihuang period, it was stipulated that the ancient "three sheng" was equivalent to the modern "one sheng." By the Daye period, the ancient unit "dou" was reinstated. "Heng" means balance; "quan" means weight. The function of "heng" is to support "quan," used to measure the weight of objects. Its principle is akin to using a base to ensure the straightness of a plumb line; turning left reveals a compass, while turning right reveals a square. The role of "heng" in the cosmos is to assist the celestial sphere, to deliberate on the construction of pointers, facilitating the harmonious movement of the seven celestial bodies (the sun, moon, and five planets). Hence, it is called "yu heng." "Quan," on the other hand, refers to weight units such as zhu, liang, jin, jun, and shi, which are used to measure the weight of items with scales to determine their lightness or heaviness. In ancient times, there were also weight units like shǔ, gēng, chuí, zī, huán, gōu, luò, and yǐ, each dynasty having different standards, and I am not fully informed on the specifics. The "Book of Han" states that the origin of "quan" is based on the weight of the yellow bell. One yue can hold 1,200 grains of millet, which weighs 12 zhu; 24 zhu equals one liang; 16 liang equals one jin; 30 jin equals one jun; and 4 jun equals one shi. The manufacturing of these five types of "quan" is highly precise, and their manufacturing standards are determined based on principles, calibrated with objects. Other size differences are adjusted according to weight. They are made into a circular shape, ensuring that the thicker parts function optimally, continuing indefinitely. When "quan" matches the object, "heng" is generated; the functioning of "heng" produces a compass, the compass's circular form gives rise to a square, the square yields a rope, and the straight line of the rope produces a plumb line. When the plumb line is straight, "heng" is balanced, and "quan" is accurate. These five principles are all embodied in the weighing instrument, serving as the main standard.
According to the records in the "Zhao Shu," in July of the eighteenth year of Shi Le's reign, while constructing the Jiande Hall, a round stone shaped like a water trough was found. It was inscribed with the following words: "Law Stone, weighing four jun, used for law and measurement. Made by the Xin family." The subsequent discussion pertains to the Wang Mang era. During the Jingming period of the Northern Wei, a man named Wang Xianda from Bingzhou presented an ancient copper weight, which was inscribed with 81 characters. The inscription stated: "Law Stone, weighing four jun." It also said: "The Yellow Emperor, the first ancestor, had virtues comparable to those of Yu. The ancestor of Yu had virtues equal to Xin. In the year of Daliang, the dragon gathered in Wuchen. Wuchen was determined, heaven commanded people. According to the virtue of the land, the correct title is authentic. Correct the flaws, leading to long life and prosperity. With the same law of measurement, we check against the current people. The dragon is in Jisi, and the year is Shichen, initially establishing order in the world, all nations will forever comply. Generations of descendants will enjoy and pass down their legacy for countless years." This was also made during the Wang Mang period. At that time, the Grand Music Officer Gongsun Chong first adjusted the weights and measures based on the "Book of Han," and upon seeing this weight, used the new measure to weigh it, resulting in a weight of one hundred and twenty jin. The new measure and weight matched perfectly. Thus, it was handed over to Gongsun Chong to adjust the musical scale. During the reign of Emperor Xiaowen, the 斗 (dou) and 尺 (chi) were crafted strictly according to the standards set in the "Book of Han."
It is said that during the Liang and Chen dynasties, the old weights were still in use. At that time, one jin of the Qi dynasty was equivalent to eight liang today. The jade weights of the Zhou dynasty had one jin equivalent to four liang, which was about four and a half liang of the old weights. By the Kaihuang period of the Sui dynasty, it was mandated that one jin equaled three jin of the old weights. During the Daye period of the Sui dynasty, the old weights were used again.
Thus, the ancient weights were all over the place, sometimes this way and sometimes that way, truly confusing. The Liang and Chen dynasties used the old weights; one jin of the Qi dynasty was eight liang, and the jade weights of the Zhou dynasty had one jin equal to only four liang, which was similar to four and a half liang of the old weights. During the Kaihuang period of the Sui dynasty, a new standard came out, stating that one jin was equal to three jin of the old weights, but by the Daye period, it changed back to the old weights. These ancient measurement units are indeed quite baffling!
