Lǜlǚ xīn shū 律呂新書
New Treatise on the Pitch-Pipes by 蔡元定 (Cài Yuándìng)
About the work
A two-juan treatise on the mathematical theory of the twelve pitch-pipes and their derivation, by Cài Yuándìng of the Jiànyáng Cài family — Zhū Xī’s chief collaborator on xiàngshù matters. Juàn shàng (the Lǜlǚ běnyuán 律呂本原, “Foundations of the Pitch-Pipes”) gives the entire system in 13 short, programmatic chapters: the huángzhōng 黃鐘 and its volumetric area, the generation of the eleven other pitches from it, the biànlǜ 變律 (derived pitches), the diagrams of the five tones plus the biànlǜ, the 84-tone diagram, the 60-mode diagram, the wind-watching ceremony, and the four standardizations of length, capacity, weight, and balance. Juàn xià (the Lǜlǚ zhèngbiàn 律呂証辨, “Critical Discrimination of the Pitch-Pipes”) is the explanatory and polemical companion in 10 essays. The work is the foundational treatise of Sòng Dàoxué music theory and was completed in close conversation with Zhū Xī, who wrote its preface and refers to its method in his Yǔlèi and correspondence.
Tiyao
[Your servants] respectfully report: Lǜlǚ xīn shū in 2 juàn, by Cài Yuándìng of the Sòng. Yuándìng’s zì was Jìtōng; he was a man of Jiànyáng. During the Qìngyuán era he was caught in the dǎngjìn 黨禁 (“party proscription”) and exiled to Dàozhōu, where he died; his life is given in the Sòngshǐ Dàoxué zhuàn. Master Zhū praises this work as a lǜ shū of “deep precision,” saying that none of the recent Confucians can equal it; he also says, “Jìtōng’s understanding of musical pitch-pipes shows great mental power: he has read so many books.” When Zhū wrote the preface, he said: “The volumetric numbers of the huángzhōng’s circumference and diameter can be derived from the volumetric fēn of the Hàn hú measure; that the cùn takes nine fēn as its method, the doctrines of Huáinánzǐ, the Grand Astronomer, and Xiǎo Sīmǎ provide. The numbers of the five tones plus the two derived tones, and the use of derived pitch-pipes for half-tones, are fully laid out in Dù [Yòu]‘s Tōngdiǎn. That the biàngōng and biànzhǐ cannot serve as principal modes is amply demonstrated in Kǒng [Yǐngdá]‘s Lǐshū. As for first seeking the origin of sound-and-breath and from the pitch-pipes generating the foot-rule, this is the truly outstanding part — and is also amply attested in the regulations of the two Hàn dynasties, in the doctrines of Cài Yōng, in our dynasty’s Huìyào, and in the words of Master Chéng and Master Zhāng. The book is in fact one which Zhū and Cài, master and disciple, brought to completion together; this is why it received such uniquely high praise. The book is in two juàn. The first is the Lǜlǚ běnyuán in 13 essays: 1. Huángzhōng; 2. The volumetric basis of the huángzhōng; 3. The generation of the eleven pitches from the huángzhōng; 4. The volumetric basis of the twelve pitches; 5. The biànlǜ (derived pitches); 6. Diagram of the generation of the five tones from the pitch-pipes; 7. Generation of the biàn tones; 8. Diagram of the eighty-four sounds; 9. Diagram of the sixty modes; 10. Wind-watching; 11. Standardization of length; 12. Standardization of capacity; 13. Standardization of weights and balances. The second is the Lǜlǚ zhèngbiàn in 10 essays: 1. Casting of the pitch-pipes; 2. The numbers of the pitch-pipes’ length and circumference; 3. The volumetric basis of the huángzhōng; 4. Sānfēn sǔnyì, “thirds, increase and decrease, mutual generation”; 5. Concord of sound; 6. The succession of the five tones great and small; 7. The biàngōng and biànzhǐ; 8. The sixty modes; 9. Wind-watching; 10. Length, capacity, weights and balance. — Now, examining Yuándìng’s procedure: he commonly cuts a piece of bamboo to fit the huángzhōng pipe, takes its length to be 9 cùn, and measures circumference and diameter as in the huángzhōng method, alternating in turn so as to obtain the central tone and the central seasonal qì. The cutting-of-the-pipe must thus rest on wind-watching, but the doctrine of wind-watching is the most extravagantly speculative of all. The records of the Hànshū, HòuHànshū, Jìnshū, and Suíshū on this point disagree among themselves: one account says you take a wooden tray and lay the pitch-pipes on top of it; another says you bury them flush with the ground; another says you lay them on the tray and bury both in earth flush with the ground — three different specifications for the depth and height of the placement. One account says you stuff reed-membrane ash into the inside ends of the pipes, and when the qì arrives the ash is dispelled; another says you pack the pipe with bamboo-membrane ash and cover the mouth with a silk-gauze, and when the qì arrives it blows the ash and shifts the gauze — and again, the gauze may shift greatly, slightly, or not at all; another says the ash flies and the white silk scatters outward, and that the qì-response is variously early or late, the ash-flight variously much or little. These accounts also do not agree. So if the wind-watching cannot be relied on, and human voices afford no independent verification, then the so-called “origin of sound and breath” of Master Cài is also nothing more than fine listening, not something demonstrable in actual fact. Liú Xīn’s bronze hú-measure is fully described in the Hànshū yuèlǜzhì, and the Suíshū yuèlǜzhì repeats its inscription: “The standard jiāliáng (good measure) is the hú: square in cùn, round in its outer rim — protruding off-center 9 lí 5 háo, area 162 square cùn, depth 1 cùn, volume 1620 cùn, capacity 10 dǒu.” Zǔ Chōngzhī criticized Liú Xīn for this, judging the inscription on the Hàn hú an absurdity, “the egregious blunder of an arithmetical age.” Yuándìng nevertheless follows the Hànshū in setting the volumetric basis of the huángzhōng at 810 fēn — why? — Xún Xù’s foot-rule, the so-called “early-Jìn foot” of the Suíshū, was already in Jìn times derided by Ruǎn Xián as too high. Yuándìng holds that this foot derives from the lǜ of the Jízhǒng [tomb] cache and from Liú Xīn’s hú, and is closest to antiquity, with a high tense music-pitch — but who knows what the diameter and circumference were in those days? When the ancients said “diameter 3 fēn, circumference 9 fēn” they meant a circular diameter of 3 with a perimeter of 9, where “kòng wéi” (cavity-circumference) is just yuánzhōu (the perimeter of a circle). Hú Yuán suspected the pipe so derived was too narrow to hold 1200 millet grains, and so enlarged the diameter to 4 lí 6 háo with a circumference of 10 fēn 3 lí 8 háo — which is also a 3:9 ratio. He took the cavity-perimeter to be the area inside the pipe, namely 9 square fēn. Yuándìng followed Hú in this, and using the round-field (circle area) algorithm produced his huángzhōng volumetric basis on too large a scale: this is an error of imprecise computation. — Again, holding that the six derived pitches of the huángzhōng do not respond to the principal-mode tones and ignoring the role of the clear-tones; or holding that the èr biàn (the two derived tones) cannot form a principal mode, while ignoring that the èr biàn mode is itself five-tone complete (and that if one took the two derived tones as the seven tones of every principal mode they would take you out of the principal mode) — all such matters Yuándìng has not worked through, and so we specially note them here so as to correct his errors. Respectfully edited and presented in the eleventh month of Qiánlóng 46 (1781). Editor-Generals: Jì Yún, Lù Xīxióng, Sūn Shìyì. Editor-in-chief: Lù Fèichí.
Abstract
The Lǜlǚ xīn shū is the foundational Dàoxué music treatise — Cài Yuándìng’s mature systematization of the mathematical theory of the twelve pitch-pipes, written in close collaboration with Zhū Xī. Zhū contributed the preface, refers repeatedly to the work in the Yǔlèi, and characterized it as the deepest treatment of music in the recent tradition. The internal architecture is exemplary: a thirteen-essay positive exposition of the system (Lǜlǚ běnyuán) followed by a ten-essay critical-defensive companion (Lǜlǚ zhèngbiàn). The work introduces in Sòng-systematic form the biànlǜ (derived-pitches) doctrine, the 60-mode chart, and the strict sānfēn sǔnyì “third-up, third-down” generative method for deriving the eleven pitches from huángzhōng. Although composed in the Southern Sòng (the work was complete by the time Zhū Xī wrote his preface in 1187), the volumetric mathematics relies on Liú Xīn’s Hàn hú-measure as transmitted by the Hànshū yuèlǜzhì — a choice the Sìkù compilers (writing six centuries later, with access to the corrected mathematical traditions of Zǔ Chōngzhī and to Qīng-period precision metrology) judge erroneous, since Hú Yuán and Cài Yuándìng have effectively confused the cavity-perimeter with cavity-area, producing a huángzhōng volumetric basis too large by approximately the ratio 9:π. Despite these technical reservations, the work was the standard SòngYuánMíng treatise on pitch-pipe theory and is referenced by virtually every later Chinese musical treatise. Cài was caught in the Qìngyuán dǎngjìn of 1196 and died in exile in Dàozhōu in 1198; the Lǜlǚ xīn shū survived as part of the Zhū-school canon. Date 1187 reflects the year of Zhū Xī’s preface.
Translations and research
- Kenneth J. DeWoskin. 1982. A Song for One or Two: Music and the Concept of Art in Early China. Ann Arbor: Center for Chinese Studies. — Substantial discussion of the Sòng lǜ-lǚ tradition with reference to Cài.
- Joseph Needham, Science and Civilisation in China, vol. IV.1 (Physics). Cambridge: CUP, 1962. Chapter 26h on acoustics treats the Lǜ-lǚ xīn shū as the canonical Sòng treatment.
- 戴念祖. 1994. 中國聲學史. 河北教育出版社. — Standard Chinese-language history of acoustics; takes the Lǜ-lǚ xīn shū as the Sòng-period anchor.
- 楊蔭瀏. 1981. 中國古代音樂史稿. — Detailed analysis of the biàn-lǜ doctrine.
Other points of interest
The mathematical relationship between the work and Zhū Xī is unusually complex: the project was already underway when Zhū wrote, but Zhū’s preface and the CàiZhū correspondence document the back-and-forth. The text is the principal vehicle by which the sānfēn sǔnyì doctrine became the standard SòngYuánMíng treatment of pitch-pipe generation; subsequent reformers (including Zhū Zàiyù in KR1i0009) measured their innovations against Cài’s framework.