Lǜlǚ chǎnwēi 律呂闡微
Bringing Forth the Subtleties of the Pitch-Pipes by 江永 (Jiāng Yǒng)
About the work
A ten-juan music treatise by Jiāng Yǒng dedicated principally to expounding 朱載堉’s mathematical method of equal temperament (the xīnfǎ mìlǜ). The book opens with a one-juan compilation called Huángyán dìngshēng 皇言定聲, citing five doctrinal pronouncements of the Kāngxī emperor on music; this serves as the imperial framing of the rest of the book. (Jiāng notes that he had not seen the imperial Lǜlǚ zhèngyì in five juan, hence does not engage with the Western five-line / six-name / eight-symbol / three-tempo notation it introduces.) The remaining nine juan systematically expound Zhū Zàiyù’s mathematical derivation of equal temperament — using gōugǔ (right-triangle) extraction, square-rooting, cube-rooting, and continuous proportional ratios to derive the twelve pitches as the twelfth root of two. Jiāng’s distinctive contribution is to use Zǔ Chōngzhī’s high-precision mìlǜ (close-ratio, π = 22/7 to better) for the circle-square computations, which yields slightly different numerical values from Zhū Zàiyù’s, and to add the previously-unsolved derivation of jiāzhōng from the doubled ruíbīn.
Tiyao
[Your servants] respectfully report: Lǜlǚ chǎnwēi in 10 juàn, by Jiāng Yǒng of our dynasty. The book opens with a one-juan compilation citing five doctrinal pronouncements of the Sage-Ancestor August-Emperor Kāngxī on music — Huángyán dìngshēng — to crown the whole work. Jiāng has not seen the imperially-composed Lǜlǚ zhèngyì in 5 juan, and so on the Western five-line / six-name / eight-symbol / three-tempo apparatus he is largely unable to gloss. — His thesis: he takes the Míng Zhèngshìzǐ Zhū Zàiyù as his lodestar, with only minor differences in his use of close-ratio (mìlǜ) for circumferences and diameters and in his initial computation. — Zhū Zàiyù’s book has been mostly misread and rashly criticized. Now upon examination: Zàiyù’s setting huángzhōng equal to 1 chǐ is borrowing the 1-chǐ unit to start the gōugǔ and square-extraction ratios; he is not adding to the 9-cùn pipe. He says: “the huángzhōng pipe is 9 cùn long — by vertical-millet fēn-units; cùn of 9 fēn each, total 81 fēn: this is the foundation of the pitch-pipe and the basic count of huángzhōng. The huángzhōng is 10 cùn long — by horizontal-millet fēn-units; cùn of 10 fēn each, total 100 fēn: this is the foundation of the foot-rule. The pitch-pipe by vertical millet and the foot-rule by horizontal millet have different names and dimensions, but the proportion is in fact the same.” Most lucid — but the muddled critic still pulls “9 cùn” out to argue against him. Is this not delusion? — Examining the Kǎogōng jì’s Lìshì wéi liáng: “interior-square 1 chǐ and rounding it outside” — this means circle-diameter equals square-diagonal. The square-diagonal-extraction technique is identical with the equal-sided right-triangle hypotenuse-extraction technique. Now setting interior-square 1 chǐ as the huángzhōng length: both gōu and gǔ are 1 chǐ, each squared and summed; square-rooted gives the hypotenuse — the diagonal of the interior square, which is also the diameter of the exterior circle, which is also the ratio for the doubled ruíbīn pitch. The principle of square-and-round mutual containment: a square’s interior circle is the half of its exterior; its exterior circle is the double of its interior; a circle’s interior square is half its exterior, the exterior square is double its interior. Now the circle’s interior square has edge 1 chǐ and area 100; the exterior square has edge 2 chǐ and area 400. To extract the diagonal of the interior 1-chǐ square: square-it (100), double (200), then extract square root — i.e. the square-of-the-diagonal is 200, equal to double of the interior and half of the exterior. The doubled-ruíbīn pipe-length squared is the double of huángzhōng normal and the half of the doubled huángzhōng. Hence using the circle’s interior square as the huángzhōng normal and the exterior square as the doubled huángzhōng, the diagonal is the doubled ruíbīn. From there, multiplying by gōu and extracting the square root gives doubled nánlǚ; multiplying by gōu and gǔ and extracting the cube root gives doubled yīngzhōng. Once we have yīngzhōng, each pitch can be derived by multiplying by huángzhōng normal value 10 cùn and dividing by the doubled yīngzhōng value, and so on. — His pitches derived by gōugǔ multiplication-division-square-root differ from the old pitches by only fractions of a millimetre, slightly larger; left-to-left mutual generation can solve the puzzle of “going without returning”; the twelve pitches’ diameters and circumferences differ but the half-huángzhōng still responds to the principal huángzhōng — which solves the puzzle of why a same-diameter huángzhōng does not respond to its half but does to the half of tàicù. Jiāng Yǒng’s exposition of Zàiyù’s book is thoroughly principled and structured, and his single innovation — the doubled-ruíbīn-generates-jiāzhōng method — supplements what the original work lacked. — Only on the cube-root extraction of doubled yīngzhōng: while he understands the four-rate proportional comparison for nánlǚ, he does not penetrate the foundational principle of cube-root extraction and so does not state it freely. The principle: in continuous-proportion four-rate sequences, the first-rate squared-and-multiplied-by-the-fourth-rate equals the second-rate squared-and-multiplied. Setting huángzhōng normal as first rate, yīngzhōng doubled as second rate, wúyì doubled as third rate, nánlǚ doubled as fourth rate, then squaring huángzhōng normal and multiplying by doubled nánlǚ and cube-rooting yields the second rate, doubled yīngzhōng. Zhū Zàiyù’s purpose was to make zhònglǚ generate back to huángzhōng, hence took huángzhōng normal as first rate, huángzhōng doubled as last rate, listed the thirteen rates by pitch-length succession, with yīngzhōng as 2nd rate, nánlǚ as 4th, ruíbīn as 7th. The multiplication / division / square-rooting / cube-rooting are all expressions of the continuous-proportion principle, but he hides this underlying principle behind the rhetoric of “circle-square gōugǔ”, and so Jiāng Yǒng has not seen it. Respectfully edited and presented in the tenth month of Qiánlóng 46 (1781). Editor-Generals: Jì Yún, Lù Xīxióng, Sūn Shìyì. Editor-in-chief: Lù Fèichí.
Abstract
The Lǜlǚ chǎnwēi is the principal mid-Qing Chinese exposition of Zhū Zàiyù’s mathematical equal-temperament. Jiāng Yǒng, the leading Wǎnpài kǎozhèng mathematician of his generation, undertook the work explicitly because Zhū Zàiyù’s Yuèlǜ quán shū (KR1i0009) had been so widely misread (by 毛奇齡 and others as well as by orthodox Sòng-Confucians) that the actual mathematical content was being lost. Jiāng’s exposition decodes the gōugǔ (right-triangle) language of Zhū’s mathematics and reveals the underlying procedure: derivation of the twelfth root of two by iterated square- and cube-root extraction. The Sìkù compilers credit Jiāng with successfully expounding the algebra and supplying the missing jiāzhōng derivation, but identify a remaining gap: Jiāng has correctly read the four-rate proportional principle for the nánlǚ derivation but not for the yīngzhōng cube-root step. The Sìkù tíyào itself supplies the missing principle (continuous-proportion four-rate identity), revealing that the underlying mathematical structure of Zhū’s work is neither gōugǔ nor iterative root-extraction but the algebraic identity in continuous proportion. This is one of the most mathematically substantive single tíyào in the entire Sìkù. Jiāng’s Lǜlǚ chǎnwēi is the necessary companion to his own Lǜlǚ xīn lùn (KR1i0020): the Xīn lùn is Jiāng’s positive doctrine, which rejects equal temperament in favour of uniform-segmentation along the calendar; the Chǎnwēi is his exposition of Zhū Zàiyù’s opposing doctrine. The two works together represent the most sophisticated mid-Qing engagement with the central problem in Chinese music theory. Composition: not precisely datable; the bracket 1740–1755 spans Jiāng’s mature period and is consistent with internal references to the Lǜlǚ zhèngyì hòubiān of Yǒngróng et al.
Translations and research
- Joseph Needham. 1962. Science and Civilisation in China, vol. IV.1. — Treats Jiāng Yǒng’s Chǎn-wēi as the principal mid-Qing exposition of Zhū Zàiyù.
- Gene Cho. 2003. The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. Lewiston: Edwin Mellen Press. — Detailed treatment of Jiāng Yǒng’s reception of Zhū Zàiyù.
- Robinson, Kenneth. 1980. A Critical Study of Chu Tsai-yü’s Contribution to the Theory of Equal Temperament in Chinese Music. Wiesbaden: Steiner.
- 戴念祖. 1986. 朱載堉 ── 明代的科學和藝術巨星. 人民出版社. — Treats Jiāng Yǒng as the most important early reader of Zhū Zàiyù.
- 楊蔭瀏. 1981. 中國古代音樂史稿. — Notes the work’s exposition of Zhū’s equal temperament.
Other points of interest
The Lǜlǚ chǎnwēi is the source for the standard story that Zhū Zàiyù’s mathematical method of equal temperament went largely uncomprehended in his own century and recovered as a working theory only in the eighteenth century by Jiāng Yǒng. The Sìkù tíyào’s commentary, supplying the underlying continuous-proportion principle, is one of the rare cases where the Sìkù compilers contribute substantive mathematical content rather than merely cataloguing.