By Norah Han

The study of harmony, in both ancient East and West, was occasionally related to the belief of the harmonic consonance between heaven and earth. The Pythagoreans discovered that the ratio 2:3 between two music tones gives a pleasant sound and thus resembles the harmonic echo between heaven and earth.¹ Similarly, in ancient Chinese literature the connection between music and the cosmos, 2:3, was regarded as the interval of harmony and symbolizes the reverberation between earth and heaven, as 2 was regarded as the number of the earth, and 3 represented the three points that form the sky.² Starting from the period of the Han Dynasty (206 B.C – 220 A.D.), each lunar month was associated with one musical pitch (律, lü) , as people believed that music was an intangible link between humans and heaven. During each month, all ritual performances were carried out in the specific lü assigned to the month.³

The calculation of musical pitches in ancient China can be traced back to its earliest written form in the books 吕氏春秋 (Master Lü’s Spring and Autumn Annals) written by scholars funded by 吕不韦 (Lü Buwei) in 241 B.C. and 管子 (Guanzi) written by 管子 (Guan Zi) in the fourth century B.C.. In his book, Guan Zi recorded the method of adding and subtracting a third ( 三分损益法) and applied it to the number 81, an auspicious number in the Chinese tradition. Through his pitch-generating formula, Guan Zi was able to generate other four numbers: 54, 72, 48, and 64, respectively. Together with the number 81, the five numbers correspond to the musical notes do-so-re-la-mi in today’s common music notation system. These five notes form the fundamental pentatonic scale of the Chinese music tradition, which are gōng 宫 (do), shāng 商 (re), jué 角 (mi), zhǐ 徵 (so) and yǔ 羽 (la).

Image 1: The pentatonic scale (CDEGA) commonly used in traditional Chinese music 

The concept of lü, which consists of 12 tones that make up the chromatic series had long been used in ceremonial rituals ever since the Shang-Zhou era (circa 1766-256 B.C.) in ancient China, with the fundamental note 黄钟 (huangzhong, just like the C in today’s standard chromatic scale).⁴  However, a tuning problem occurred. In the traditional Chinese music theory literature, such problem is referred to as “the huangzhong cannot be restored –  if one tunes the 12 notes of the chromatic scale in perfect fifths, the 13th note relative to the fundamental note, which is supposed to be an exact octave higher than the fundamental note, is actually a bit higher than expected. Similarly, there exists an ancient  Greek counterpart to the tuning problem, commonly referred to as the “Pythogorean comma” of the Pythogorean tuning system.

 The “huangzhong cannot be restored” problem, or the “Pythogorean comma,” was not resolved until the establishment of a new tuning system, the 12-tone equal temperament. The equal temperament tuning system is constructed via dividing an octave into 12 equal parts, each corresponding to a note in the chromatic scale using the 2^1/12 specific ratio defined using the notion of irrationality. In other words, the system can not do without the discovery of irrational numbers. Such an advancement in mathematics was made almost simultaneously in the East and the West, as with the emergence of the equal temperament system. 

In the history of Chinese music theory literature, Zhu Zaiyu was the first person to come up with an exact mathematical formula that gives the system of 12 equally tempered tones, approximately around 1581.⁵ Zhu Zaiyu (1536-1611) was a Ming Dynasty (1368-1644) Chinese scholar, musician, and poet. His areas of research included mathematics, astronomy, metrology, music theory, choreography, etc. He also created musical instruments, composed songs, and wrote poems. Born into a noble family, Zhu Zaiyu did not live a privileged life, as his father criticized the emperor, which enraged the emperor who imprisoned Zhu’s father. After his father’s case was redressed and Zhu inherited the title the Prince of Zheng, Zhu continued to live in the mountain ranges and conducted his research in mathematics, physics, music theory, and dance theory. Though Zhu was a prolific scholar and writer, his books were ignored by the emperor as well as the scholarly community at that time. 

The fundamental pitch that generates all the other 11 pitches is referred to as huangzhong in Zhu’s works. Though being the first person in ancient Chinese history to propose a theory that is analogous to today’s equal temperament tuning system, Zhu did not invent the concept of huangzhong. Huangzhong first appeared in its written form in 汉书 (Han Shu) (105), a historical chronology compiled by 班固 (Ban Gu) of the Han Dynasty, which is an expansion on the works by 刘歆 (Liu Xin). On the other hand, according to the legend, the origin of huangzhong can be dated back to 黄帝 (Huang Di), who is deemed as the founding figure of the Chinese civilization. Huang Di sent 伶伦 (Ling Lun) to the mountain ranges within the 昆仑 (Kun Lun). Ling Lun returned with a bamboo that produced the huangzhong pitch, and according to the legend, Ling Lun was inspired by the voice of  凤凰 (Fenghuang), an immortal bird in Chinese mythology. As mentioned earlier, music’s main role in ancient China was to accompany rituals, which were deemed as ways to communicate with the gods. Moreover, the consonant, correctly tuned pitches used in rituals were perceived as the harmonious relationship with the cosmos. Thus, when a new dynasty replaced the old one, the former huangzhong pitch was always replaced by a new pitch, as it was believed that it was the improper music ritual system that partially led to the demise of the former dynasty.⁶

Zhu’s definition of the huangzhong, on the other hand, is slightly different from his predecessors’. Instead of defining the length of huangzhong as 9 寸 cun, Zhu’s huangzhong is 1 尺 chi (10 cun ~ 255 mm) long. His fixed huangzhong would correspond to today’s concert pitch E5 (650 Hz).⁷ In his work《律呂精义》, Zhu gives the following formula that defines his 新法密律 (the new law for density rate), or equivalent to its western counterpart, the equal temperament system.

l_n+1= 2^-1/12l_n,

where l_n denotes the length of each of the 12 fundamental pitches. The length of the fundamental pitch, huangzhong, is defined as 1 chi. In his book 《乐律全书》, Zhu gives an illustration of the relationship between the 12 pitches via a picture with explanations on the side: “长律下生短律，下生者皆左旋。短律上生长律，上生者皆右旋。（In the descending direction, the higher pitch gives rise to the lower pitch, and the counterclockwise direction of the graph labels the pitches in the descending direction. In the ascending direction, the lower pitch gives rise to the higher pitch, and the clockwise direction of the graph labels the pitches in ascending direction.)”

Image 2: 左旋右旋相生之图 the picture of the generation of pitches through left and right rotation, from 《乐律全书》. 

By using this formula, Zhu constructed his instrument  律准, equivalent to a monochord that has 12 strings, each corresponding to a pitch in his 12-pitch system.

Zhu’s 律准, “monochord” from 《乐律全书》

Moreover, Zhu constructed pitch pipes in order to apply his formula more accurately. Through constructing pipes that have different diameters,  Zhu defined in total 12 pitches, 36 notes, spanning 3 octaves in his book 《律呂精义》. Moreover, he classified them as the clear tones, the central tones, and the murky tones. A paragraph from 《律呂精义》 in which Zhu elaborates on his understanding of the concept of the clear tones, the central tones, and the murky tones, gives the readers of today a peek into the philosophy behind Zhu’s music system:

十二律皆中声也 […] 夫何为中声耶歌出自然虽髙而不至于揭不起虽低而不至于咽

不出此所謂中声也中声之上則有半律是为清声中声之下则有倍律是为浊声

Translation: The twelve pitches are all central tones… They are regarded as the central tones, as the pitches are neither too high nor too low for the singers to perform. Higher than the central tones are the half pitches called the clear tones. Lower than the central tones are the double pitches called the murky tones.

Zhu was able to define the accurate calculation of the equal temperament system that is the most dominant tuning system used eversince the western classical era compositions and performances till today’s worldwide repertoire. His calculation of the notes were astonishingly accurate—to 24 digits after the decimal point. Moreover, his formula and the definition of 密律 (density ratio), 2^1/12, is the same ratio used in the equal temperament tuning system developed by Flemish mathematician Simon Stevin around 1600 in his unpublished treatise Van De Spiegheling der singconst (On the Theory of Music) under the section “Arithmetical Division of the Monochord”. 

Zhu Zaiyu’s works were heavily influenced by his uncle 何瑭 He Tang’s work 《乐律管见》. Zhu also worked with his father, whose interest in music influenced him ever since childhood. In his short article introducing his work 《律呂精义》to the emperor, Zhu wrote:

“进献书籍，述家学，成父志，……臣父恭王厚烷存日，颇好律历。其悟性所得，虽与先儒不无异同，而臣愚以为，其义或可补先儒所未发。”

Translation: I hereby offer my book to describe my family’s study and to complete my father’s goal, … my father Zhu Houwan, the King of Gong, is quite interested in the study of musical pitches and calendar. His study of these topics, though is quite similar to his predecessors, I humbly think that his study can supplement the aspects overlooked by his predecessors.

This article marks several important milestones of the development of music systems in ancient China, but the story is far from complete. First of all, the scholars’ attempt to find an underlying law that governs the voices that are considered as music is undeniably important, as constructing a system that builds consensus among people makes the production and communication of music easier. However, a brief review of the history of music laws developed by the ancient Chinese does not present a complete picture of the practice of music in ancient China. Moreover, the readers should note that music in ancient China did not solely exist in the court ceremonies, but also in the daily life rituals as folk music, festivity accompaniment, music for life events like funerals and weddings, etc.

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¹ Cho, Gene Jinsiong. 2003.”Music and Man,” The discovery of musical equal temperament in China and Europe in the sixteenth century,1-15. Lewiston, NY: Mellen.

² Ibid.

³ Woo, Shingkwan. “The Ceremonial Music of Zhu Zaiyu.” Doctoral dissertation, The State University of New Jersey, 2017. 73-99.

⁴ Ibid.

⁵ Woo, Shingkwan. “The Ceremonial Music of Zhu Zaiyu.” Doctoral dissertation, The State University of New Jersey, 2017. 201-228.

⁶ Woo, Shingkwan. “The Ceremonial Music of Zhu Zaiyu.” Doctoral dissertation, The State University of New Jersey, 2017.73-99.

⁷ Woo, Shingkwan. “The Ceremonial Music of Zhu Zaiyu.” Doctoral dissertation, The State University of New Jersey, 2017. 201-228.

Sources

Stanevičiūtė, Rūta, Nick Zangwill, and Rima Povilionienė. 2019. Of essence and context: between music and philosophy. Springer EBooks. https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=2149659.

吴鸿雅 Wu, Hongya. 朱载堉新法密率的人文理解研究 [A Study on Zhu Zaiyu’s Density Rate Law ]. Ziran bianzhengfa tongxun 自然辨证法通讯, no.2( 2006).

Zhu, Zaiyu 朱载堉. 《律呂精义》. 人民音乐出版社, 2006.

Zhu, Zaiyu 朱载堉. 《乐律全书》. 电子科技大学出版社,2017.