Chromatic Scale Construction in Ancient China

Guangming Li

Music theory is an essential part of Chinese music and culture. Its centrality to understandings of cosmology and social order may seem familiar to music theorists trained in the European tradition. Yet because the distinct characteristics of Chinese music theory have only rarely been incorporated into discussions within the history of Western music theory, exploring corresponding approaches to questions such as tuning and scale generation remains a meaningful point of departure. This was demonstrated in the Global Perspectives in Histories of Music Theory conference at Columbia University in February, which showcased the impressive scale and potential of engaging across cultural contexts.

In addition to learning from a range of excellent scholars, I had the opportunity to present my own research on the construction of the chromatic gamut as first recorded in an exchange between the King Jing (d. 520 BCE) of the Eastern Zhou period (770 BCE – 256 BCE) and the court music official Ling Zhoujiu. In this exchange, Zhoujiu mentions the creator of the chromatic scale, describes the purpose for creating pitch reference, the name of the tuning apparatus, and the twelve disyllabic names, among other things. Yet, Zhoujiu offers no direct explanation for how the pitch reference was created and how the twelve disyllabic names were selected.

At present, Chinese music theorists rely on a method called the “a-third-removing-extending” (or ATRE) technique to explain the process of generating a chromatic gamut. Akin to the cycle of fifths, the ATRE method was first recorded in Guanzi, an encyclopedic treatise that dates from the seventh century BCE.  While the ATRE can generate a complete chromatic scale, it cannot account for the scale introduced in Zhoujiu’s exchange with the King.  For instance, it fails to address the two major-third-chains in the pitch name pattern of the chromatic gamut in the process of generating fourths and fifths.

For the purpose of this post, I will explain the basics of the ATRE method and briefly discuss the motivation for my own research on a new understanding of the ancient method for generating the twelve standard pitches.

“A-Third-Removing-Extending” Method

The details of this classical method for generating the pentatonic scale are found in Guanzi, which is named after the philosopher Guan Zhong (719-645 BCE).  Famed for his achievements as the Prime Minister of the Qi State, Guang Zhong came to be called the “Pioneer of the Legalists,” “The Teacher of the Saints,” “First Prime Minister of the Land,” and worshipped as a divine figure by Daoists.  The mathematical process of constructing a pentatonic scale, known as “三分损益,” which I translate as the “a-third-removing-extending” (ATRE) method, is found in a chapter entitled “Diyuan” of Guanzi. The five monosyllabic names for each of the scale degrees in the pentatonic scale are gong, zhi, shang, yu, and jiao.

This process can be illustrated as the following: begin by stopping on a horizontal string at any point no less than 1/4 of the original string length from the left, and pluck the string on the right. Take the tone from the vibrating portion of the string as the initial length, gong (arbitrarily assigned to C4). To obtain the next pitch, zhi (G3), extend ⅓ of the string length of gong (C4) leftward; to obtain the next pitch shang (D4), stop at ⅓ of the string length of the zhi (G4) from the left; to obtain the next pitch yu (A3), extend ⅓ of the string length of the shang (D4) leftward; to obtain the last pitch jiao (E4), stop at the point ⅓ of the length of the yu (A3) from the left.

Arranging these sequences of ratios in an ascending order based on the ratio results in an ascending pentatonic scale: zhi(G3)-yu(A3)-gong(C4)-shang(D4)-jiao(E4).

Figure 1. “A-Third-Removing-Extending” Method (designed by G. Li)

Limitations of “A-Third-Removing-Extending” Method

Meanwhile, in his reply to King Jing, Zhoujiu introduces the names of twelve standard disyllabic pitch names in a complete chromatic scale.  They are: huangzhong, dalü, taicu, jiazhong, guxian, zhonglü, ruibin, linzhong, yize, nanlü, quyi, yingzhong. The relationship among these pitch names in the Western chromatic scale are C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab, A, A#/Bb, and B. The process of applying the ATRE method to construct a complete chromatic scale commonly found in Chinese classics (e.g. in Lüshi Chunqiu, 241BCE) is as follows:

huangzhong (C ) – linzhong (G) – taicu (D) – nanlü (A) – guxian (E) – yingzhong (B) – ruibin (F#/Gb) – dalü (C#/Db) – yize (G#/Ab) – jiazhong (D#/Eb) – wuyi (A#/Bb) – zhonglü (F).

Figure 2. Chromatic scale based on the ATRE method and the issue of major third chains (designed by G. Li)

However, this method cannot account for the appearance of major third chains in the pitch-naming pattern of the gamut. Two major third chains are found among pitches at even number positions in the gamut, and assigned a common suffix-like morpheme in their names. The pitches jiazhong (4), lingzhong (8), and yingzhong (12) share a common morpheme zhong and they form a chain of major thirds Eb4-G4-B4; the pitches da (2), zhong (6), and nan (10) share a common morpheme and they form a chain of major third Db4-F4-A4.

Given that the series of pitch names appeared in an official record that captured dialogue between the King and his music director, the particular sequence of major thirds is not likely to be the music director’s spontaneous creation, but rather a reflection of a well-established convention.  Further, the fact that the music director did not refer to the ATRE method suggests the possibility that a different method of directly building on major thirds and the chromatic gamut could have been applied.

By drawing together a linguistic analysis of ancient texts with acoustic principles and string performance techniques, I offer a different method for constructing the chromatic gamut called the “harmonic-and-stopped-tone-mutual-converting” (or HSTMC) method. Based on ancient Chinese zither design and existing performance techniques, the HSTMC method could have easily been obtained on a monochord. This is based on the fact that a single-stringed zither or yixianqin (akin to the monochord) was commonly known in antiquity.  And given that the historical text describes the creator of the chromatic pitch system as a “divinely blind” music master, the HSTMC method requires only understanding the physical behavior of the string through haptic touch, without the aid of mathematical calculation or the ATRE method.  When applied, HSTMC better corresponds with Zhoujiu’s numerological description of pitch system construction and better aligns with the tenets of Chinese natural philosophy.

Details of the HSTMC method for generating the chromatic gamut remain forthcoming in a paper titled, “New Findings on Non-mathematical Methods of Constructing the 12-tone Chromatic Scale with the Monochord in Ancient China.” This study not only enriches understandings of music theory and music culture in ancient China, but also affords a new perspective on understanding Western music theory beyond exercises in abstract mathematical contemplation.

 Special thanks to Leon Chisholm, Carmel Raz, Nori Jacoby, and Lan Li for their assistance and feedback in writing this post. Special thanks also to the Heyman Center for the Humanities at Columbia University and the Global Perspectives in Histories of Music Theory Conference.

Li picture blog

Guangming Li, PhD (UCLA), is a scholar, performing artist, and educator.  His areas of research include issues in music archaeology and history, music theory, music cognition, and music aesthetics.


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