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*Lü shu* 律書 is a chapter (ch. 25) of Sima Qian's 司馬遷 (145-c. 86 BCE) history book *Shiji* 史記, where it belongs to the eight treatises (*shu* 書). The book describes the measures of pitch pipes on a cosmological background. This aspect brings it into relation with the following chapter, *Li shu* 曆書, on calendric calculation (see also *Lülizhi* 律曆志).

The author of the treatise is not the first music theorist whose writings have survived. Reports and explanations of pitch pipes can be found in earlier works, from the *Guanzi* 管子 to the *Lüshi chunqiu* 呂氏春秋 and the Daoist treatise *Huainanzi* 淮南子, but also military tracts such as the *Liutao* 六韜.

Yet the treatise on music theory by Sima Qian in the *Shiji* is, however, the first and only one that describes the natural symbolism of the pipes in such a precise and detailed manner. Its etymology seems far-fetched in many respects, but it gives a deep insight into the attempts of ancient thinkers to understand the cosmos in its contexts.

The treatise on musical theory and tuning consists of five parts, of which the first one is flowing seamlessly into the second one. The first part describes the meaning of pitch pipes in warfare as a means to exploit, by blowing the pipes, the current situation of the own army and that of the enemy. The different theories about this which can be found in some older books differ from one another and cannot be interpreted quite clearly because there are descriptions of what the different tones of the pipes can reveil, but not in what concrete way these interpretations are made. The passage from a military tract quoted in Zheng Xuan's 鄭玄 (127-200) commentary on the ritual classic *Zhouli* 周禮 is completely opaque, because how could an army music master tell his army by music - or even by single notes - that things are going badly? In this sense, army music would not be signal music, but rather a band for mystical moods.

The military treatise *Liutao* includes similar statements: The sound of drums (*baogu* 枹鼓) in the enemy's camp corresponded to the tone *jue* 角, the light of inimical fire the tone *zhi* 徵, the sound of bronze and iron the tone *shang* 商, the sighing of people the tone *yu* 羽, and silence to the tone *gong* 宮.

The second part deals with warfare from the semi-mythological Xia dynasty 夏 (21th-17th cent. BCE) to Emperor Wen 漢文帝 (r. 180-157 BCE) of the Han dynasty 漢 (206 BCE-220 CE) to whom Sima Qian commemorates his peaceful government and praises him as a role model. The praise is introduced with the summarizing words "the great historiographer (or great astronomer) says" and thus marks a clear turning point towards the third part, which abruptly goes into the subject of pitch pipes.

The third part is the actual main part of the treatise and describes the pitch pipes in their dependence on the cardinal directions, winds, cyclical characters, and seasons. After going through the entire annual cycle of seasons, months, and winds, the text abruptly turns to the the physical features of the pitch pipes and their calculation. The transmitted version of the *Shiji* includes serious errors in the numeric dimensions.

不周風居西北，主殺生。東壁居不周風東，主辟生氣。而東之至於營室。營室者，主營胎陽氣而產之。東至于危。危，垝也。言陽氣之危(=垝)，故曰危。 | The Buzhou wind "Not a Whole Round" originates in the northwest and is the master of the destruction of life. The constellation Dongbi "Eastern Wall" is east of the Buzhou wind, it is master of the development of life. Further to the east is the Yingshi constellation "Army Camp", the lord of the formation and production of the Yang energy. Further east is the constellation Wei "Danger", which means "broken wall", namely the broken wall of the Yang energy. |

十月也，律中應鍾。應鍾者，陽氣之應，不用事也。其於十二子為亥。亥者，該也。言陽氣藏於下，故該也。 | In the tenth month, the pipe Yingzhong "Corresponding Bell" is appropriate, which corresponds to the Yang energy, but [the Yang energy at that time] has no effect. Among the Twelve Terrestrial Branches, it corresponds to the hai (/ɣɒi˥/) Branch, which means "missing" (/kɒi˥˩/), that is, the Yang energy is still hidden under the earth, so it is "missing". |

The section begins with the determination of the largest basic pipe with 81 (9 × 9) units without these numbers being derived or explained beforehand. These units represent an apparent volume calculated purely on the basis of numeric games (circumference in *fen* times length in *cun*, see weights and measures). A paragraph later the text only deals with the length ratios of the pitch pipes, with the longest pipe, the Yellow Bell pipe (*huangzhong* 黃鐘) measuring nine *cun* "inches". But even in between there are many inexplicable number associations in these passages which were copied or even inserted by copyists and which constituted serious problems for commentators of later ages.

生鍾分： | Fraction to produces the "bell" (i.e. pitch pipes) [series]: |

子，一分。 | [The celestial stem] zi [correlating to the pipe Huangzhong 黃鍾] corresponds to an integral number. |

丑，三分二。 | Chou [pipe Dalü 大呂] corresponds to ^{2}/_{3} [= 0.666]. |

寅，九分八。 | Yin [pipe Taicu 泰蔟] corresponds to ^{8}/_{9} [= 0.888]. |

卯，二十七分十六。 | Mao [pipe Jiazhong 夾鍾] corresponds to ^{16}/_{27} [= 0.593]. |

辰，八十一分六十四。 | Chen [pipe Guxian 姑洗] corresponds to ^{64}/_{81} [= 0.70]. |

巳，二百四十三分一百二十八。 | Si [pipe Zhonglü 中呂] corresponds to ^{128}/_{243} [= 0.527]. |

午，七百二十九分五百一十二。 | Wu [pipe Ruibin 蕤賓] corresponds to ^{512}/_{729} [= 0.702]. |

未，二千一百八十七分一千二十四。 | Wei [pipe Linzhong 林鍾] corresponds to ^{1,024}/_{2,187} [= 0.468]. |

申，六千五百六十一分四千九十六。 | Shen [pipe Yize 夷則] corresponds to ^{4,096}/_{6,561} [= 0.624]. |

酉，一萬九千六百八十三分八千一百九十二。 | You [pipe Nanlü 南呂] corresponds to ^{8,192}/_{19,683} [= 0.416]. |

戌，五萬九千四十九分三萬二千七百六十八。 | Xu [pipe Wuyi 無射] corresponds to ^{32,768}/_{59,049} [= 0,555]. |

亥，十七萬七千一百四十七分六萬五千五百三十六。 | Hai [pipe Yingzhong 應鍾] corresponds to ^{65,536}/_{177,147} [= 0.369]. |

生黃鍾術曰： | The method of determining the [the length of the other pipes starting with the] Yellow Bell pipe is as follows: |

以下生者，倍其實，三其法。以上生者，四其實，三其法。 | For the next lower pipe, the numerator is doubled and the denominator tipled. For the next higher pipe, the numerator is quadrupled and the denominator tripled. |

上九，商八，羽七，角六，宮五，徵九。 | The [method of the] Upper Nine means: the tone shang 8, the tone yu 7, the tone jue 6, the tone gong 5, and the tone zhi 9. |

置一而九三之以為法。實如法，得長一寸。凡得九寸，命曰「黃鍾之宮」。故曰音始於宮，窮於角；數始於一，終於十，成於三；氣始於冬至，周而復生。 | Set 1 and then [multiply] 9 times by 3 to get the denominator [19386]. If the numerator and denominator are the same, the result is 1 ("one inch"). This corresponds to a total length of 9 inches, which is called "the tone of the Huangzhong pipe" according to the [heavenly] order. Therefore it is said: The notes begin with the tone gong and end with jue. The numbers begin at 1, end at 10, and find their completion at 3. The energy [of this cycle] begins at the winter solstice and begins anew when an entire cycle has been completed. |

The concluding part is a metaphysical discussion about the divine origin of sounds from nature and how the saint man preserves this knowledge. Pitch pipes and the calendar, Sima Qian emphasizes again in his closing words, are the basis for all earthly measures and guidelines.

Sima Qian's essay is hardly significant in terms of music theory, since it contains a huge amount of errors and inexplicable passages. However, it is the first literary source to provide more details on the calculation of the individual pitch pipes.

Starting from the longest pipe, one follows the "added and subtracted third" method. For the second pipe, subtract one third of the length of the first, for the next pipe add one third of the length of the second pipe, and so on. Graphically, this method results in a downward zigzag line. Sima Qian does not say which tones these pipes produce, especially the "intermediate pipes" with non-integer lengths.

The second meaning of the *Lü shu* treatise lies in the field of music philosophy: music is part of nature, is integrated into it and therefore also expresses natural processes. The calculation of the individual pipe lengths follows, without having been explicitly said, numbers that are understood as symbols of nature, namely the recurring yang numbers three and nine in the denominator of the mathematical fractions, and the yin numbers two and four in the counters. What is missing, however, is an explanation of whether these numerical ratios are based on experiments or are pure number play.

A reference to the latter explanation of the numerical relationships is the fact that the observation of intervals is completely absent, an interesting phenomenon in China's early music history. Sima Qian describes with great meticulousness (whereby the comments from the Tang period 唐, 618-907, try to supplement the elements he omitted) the connection between cardinal directions, the prevailing winds and constellations there, the applicable cyclical characters from the terrestrial branches and the celestial stems and the corresponding pitch pipes.

The first and only western translation of the treatise on music theory is that of Édouard Chavannes in his *Mémoires historiques de Se-ma Ts’ien* from 1895.

Sources:

Dong Shuyan 董樹岩, Dai Nianzu 戴念祖, Luo Lin 羅琳 (1994). "Shiji Lü shu lüshu kuangzheng: Jian lun Xian-Qin guanlü 《史記•律書》律數匡正──兼論先秦管律", *Ziran kexue shi yanjiu* 自然科學史研究, 1994 (1).

Huang Datong 黃大同 (2013). "Shiji Lü shu shenglü xiangpei guanxi de wusheng zhi mi shijie 《史記•律書》聲律相配關係中的五聲之謎釋解", *Yinyue yanjiu* 音樂研究, 2013 (7).

Liu Caonan 劉操南 (1996). "Shiji Lü shu Li shu kaoshi 《史記•律書•曆書》考釋", *Guji zhengli yanjiu xuekan* 古籍整理研究學刊, 1996 (1).

Liu Wenrong 劉文榮 (2009). "Shiji Lü shu zhong de shenglüfa yanjiu 《史記•律書》中的生律法研究", *Huanghe zhi sheng* 黃河之聲, 2009 (4)

Tan Yingxue 譚映雪 (2010). "Shiji Lü shu zhong lüshu yu sheng Huangzhong shu 《史記•律書》中“律數”與“生黃鐘術”", *Tianjin Yinyue Xueyuan xuebao* 天津音樂學院學報, 2010 (2).

Yuan Jiefang 袁芥芳 (1990). "Shiji Lü shu suanjie 《史記•律書》算解", *Xinghai Yinyue Xueyuan xuebao* 星海音樂學院學報, 1990 (7).