Lü shu 律書

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.

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.

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.