Yuèlǜ quán shū 樂律全書
Complete Treatise on Music and Pitch by 朱載堉 (Zhū Zàiyù)
About the work
The collected works of the Míng prince and polymath Zhū Zàiyù 朱載堉 (1536–1611) on music, mathematics, calendrics, and ritual dance — assembled and presented to the Wànlì throne in 1606 (with internal pieces dating from 1581 onwards). The catalog gives 42 juàn; the Míng shǐ yìwénzhì gives 40. The SKQS recension contains twelve internal works, of which only four — Lǜlǚ jīngyì 律呂精義 (inner and outer chapters, 10 juàn each), Lǜxué xīnshuō 律學新說 (4 juàn), and Xiāngyǐn shī yuèpǔ 鄉飲詩樂譜 (6 juàn) — are formally divided into juàn; the others (Yuèxué xīnshuō, Suànxué xīnshuō, Cāomàn gǔyuè pǔ, Liùdài xiǎowǔ pǔ, Bāyì zhuìzhào tú, Língxīng xiǎowǔ pǔ, Xuángōng héyuè pǔ) are unstructured. The work is the most comprehensive single Chinese treatise on music and is internationally famous as the place where Zhū derived the world’s first mathematically explicit equal temperament (shíèr píngjūn lǜ 十二平均律, his “xīnfǎ mìlǜ” 新法密率), computing the twelfth root of two to twenty-five-decimal-place accuracy a decade before Simon Stevin’s parallel European derivation.
Tiyao
[Your servants] respectfully report: Yuèlǜ quán shū in 42 juàn, by Zhū Zàiyù of the Míng. Zàiyù was the shìzǐ (Heir-Apparent) of Prince Gōng of Zhèng, Hòuwǎn. The book was presented to the court in the Wànlì era. The Míng shǐ yìwénzhì gives 40 juàn; the present recension contains twelve works in all, of which only the Lǜlǚ jīngyì (inner and outer chapters, 10 juàn each), Lǜxué xīnshuō (4 juàn), and Xiāngyǐn shī yuèpǔ (6 juàn) are divided into juàn; the other seven are not — and the discrepancy with the Yìwénzhì is presumably the Míng shǐ’s error. — Zàiyù devoted his life to lǜshù (pitch-mathematics) and laboured for decades to complete this work. The volume is vast, but the substance is fully expounded in the Lǜlǚ jīngyì. — His doctrine: length is grounded in the huángzhōng pipe; this huángzhōng divided into ten equal cùn of ten fēn each (i.e. 100 fēn total) corresponds to 100 horizontal millet-grains, and is the foot-rule of length (dù chǐ); divided into 8.1 cùn of nine fēn each (i.e. 81 fēn total) corresponds to 81 vertical millet-grains, and is the foot-rule of pitch (lǜ chǐ); 100 horizontal grains = 81 vertical grains = 90 diagonal grains. Hence the huángzhōng is 1 chǐ by horizontal-millet measure (100 fēn), 0.81 chǐ by vertical-millet measure (81 fēn), and 0.9 chǐ by diagonal measure (90 fēn). For the lengths of the twelve pitches he relied on Liú Xīn’s specification “nèi fāng (square inside) of 1 chǐ and yuán (round) outside,” reading “circumference is the square’s diagonal” — that is, taking the principal huángzhōng as 1 chǐ and using the right-triangle (Pythagorean) hypotenuse-finding method to obtain the hypotenuse, which is the doubled ruíbīn (ruíbīn bèilǜ). For huáng normal as the cathetus and ruíbīn doubled as hypotenuse; for ruíbīn normal as cathetus and huáng normal as hypotenuse — the two pitches are reciprocally cathetus-and-hypotenuse. The other pitches (nánlǚ, yīngzhōng, etc.) cannot be derived by mere gōugǔ; he derives them via the method of zhūchéngfāng bǐlì xiāngqiú — proportional extraction of successive nth roots. Zàiyù admits as much: “the gōugǔ method is a flowering of words.” The pipe-length depends on the foot-rule chosen; calling the huángzhōng “9 cùn” is just a calculational convention — like Zhèng Kāngchéng’s gloss on the twelve pitches, breaking 1 cùn into 10 fēn is the correct method of length-determination, and Tàishǐgōng’s rounding 10 to 9 was for convenience of sǔnyì and was a provisional convention. Some critics object that taking 1 chǐ as the huángzhōng contradicts the “9 cùn” pronouncement; one may say they have not penetrated his meaning. — That zhònglǚ generates back to huángzhōng — Hé Chéngtiān, Liú Zhuó, and Hú Yuán all so held; Cài’s [Cài Yuándìng’s] critique was that only the huángzhōng of one pitch becomes a real pitch and the other eleven do not — failing to grasp that pitches are generated by sound, not by number. When you blow the pipe and the sound responds, that is a pitch. If you bend the sound to fit the number, then even the five tones will be discordant; how could it then be called a pitch? — Some critics object that taking square roots and powers leaves residual decimals as a flaw. But where reason demands square-root extraction or power-raising, residual decimals — even in the smallest grain — do no harm. Suppose by the gōugǔ method one finds the hypotenuse and the squared cathetus and squared base summed extracts as a non-perfect square: are we to say there is no hypotenuse? This is the cavil of one who does not know the mathematics. — His proposition that the foot of 100 horizontal millet-grains corresponds to 81 vertical, his proposition that half the huángzhōng does not respond to the huángzhōng but half the tàicù does — these are subtle. The Sòng zǔ rénhuángdì [Kāngxī’s] Lǜlǚ zhèngyì fully adopts these propositions; one cannot dismiss them merely because they differ from Master Cài. — His twelve-pitch reciprocal generation method: take the principal huángzhōng as 1 chǐ (first ratio, 1st of 13 ratios); take the doubled huángzhōng (2 chǐ, the 13th ratio); then the doubled ruíbīn is the 7th ratio; thus zhònglǚ can generate back to huángzhōng, and the rotation may proceed in either direction by direct extraction of successive ratios — that is, by power-roots and proportional successive ratios. Setting out the 13 ratios numerically: 1 (1st), 2 (square root, 2nd), 4 (3rd), 8 (4th), 16 (5th), 32 (6th), 64 (7th), 128 (8th), 256 (9th), 512 (10th), 1024 (11th), 2048 (12th), 4096 (13th). Multiplying the first ratio (1) by the last (4096), extracting the square root, gives 64 — the 7th ratio: this is the method of huángzhōng finding ruíbīn. Multiplying the 7th ratio (64) by the first (1), extracting the square root, gives 8 — the 4th ratio: this is ruíbīn finding nánlǚ. Multiplying the first (1) by itself and then by the 4th (8), extracting the cube root, gives 2 — the 2nd ratio (square edge): this is nánlǚ finding yīngzhōng. Multiplying the 4th (8) by itself and then by the first (1), extracting the cube root, gives 4 — the 3rd ratio: this is nánlǚ finding wúyì. The proportional relations: the 1st is to the 2nd as the 2nd is to the 3rd, as the 3rd is to the 4th, and so on through the 13th — all the same proportion. Or alternately: the 1st separated by one, the 2nd by two or three, equals the proportions in the reverse direction — the same. This is the method by which each pitch finds the next. — The work does not openly state the foundational principle; furthermore, the multiplication of the principal huángzhōng by its doubled form and the resulting square-extraction belongs to one type with the gōugǔ method (the cathetus-finds-hypotenuse, the side-finds-diagonal); from the ruí-bīn-finds-nán-lǚ step downwards, the gōugǔ method does not apply, yet he still presents it as gōugǔ — overly-secretive, intended to dazzle the reader. Jiāng Yǒng’s Lǜlǚ chǎnwēi (KR1i0021) is dedicated to explaining Zàiyù’s method; though Jiāng Yǒng was the most penetrating mathematician of his time, he could not fully recover Zàiyù’s intent. The rest may be inferred. (Tiyao recovered from Kyoto Zinbun digital Sìkù tíyào №0079901; the source _000.txt carries instead the three Qiánlóng imperial colophons of 1786, 1787, and 1789 attacking Zhū Zàiyù’s qín notation and the work’s musicology.)
Abstract
The Yuèlǜ quán shū is the most ambitious single work in the East Asian science-of-music tradition. Its principal scientific achievement — Zhū Zàiyù’s mathematical derivation of equal temperament in the Lǜxué xīnshuō (1584) and the Lǜlǚ jīngyì (1596) — predates Simon Stevin’s European derivation by roughly a decade and constitutes the first mathematically explicit demonstration of the twelfth root of two as the irrational division of the octave in any tradition. The accompanying Suànxué xīnshuō contributes a precise method for computing nth roots and proportional ratios using the abacus. The Liùdài xiǎowǔ pǔ and Bāyì zhuìzhào tú are the first systematic Chinese notations of court ritual dance with foot-by-foot dancer-position diagrams, anticipating later European chorégraphie. The Xiāngyǐn shī yuèpǔ gives full notation for the eighteen Shī odes used in the xiāngyǐn jiǔ ritual. Reception: despite the technical mastery of the work, Zhū Zàiyù’s positions on (a) using a 100-fēn huángzhōng (rather than the orthodox 9-cùn / 90-fēn), (b) abandoning sānfēn sǔnyì in favour of equal-temperament ratios, (c) abandoning the orthodox xuángōng doctrine of clear-tones, and (d) the qín notation using vernacular gōngchě characters and a sixteen-stroke ornamentation system, all gave deep offense to the Qiánlóng emperor. Three full imperial colophons (Qiánlóng 51 [1786], Qiánlóng 52 [1787], Qiánlóng yǐmǎo [1795]) appended to the front matter denounce the work as “sīxīn fēigǔ” (“teaching itself, not the ancients”) and direct that all errors be itemized in commentaries appended to the tíyào — a treatment given to no other work in the entire SKQS yuè section. Zhū Zàiyù’s huángzhōng mathematics nevertheless became the substantive basis of the KāngxīYōngzhèng Lǜlǚ zhèngyì (KR1i0010) — a fact the Qiánlóng colophons studiously avoid. Composition spans 1584 (the Lǜxué xīnshuō preface) to 1606; the work was presented to the throne in Wànlì 24 (1596). The catalog meta date convention “1536–1611” gives the author’s lifedates rather than the work’s composition window.
Translations and research
- Joseph Needham, Wang Ling, and Kenneth Robinson. 1962. Science and Civilisation in China, vol. IV.1 (Physics). Cambridge: CUP. — The classic English-language treatment; declares Zhū “the prince of musicology.” Sections 26h and the appendix to chapter 26 give detailed mathematical analysis.
- Gene Cho. 2003. The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. Lewiston: Edwin Mellen Press. — Comprehensive comparative monograph.
- Robinson, Kenneth. 1980. A Critical Study of Chu Tsai-yü’s Contribution to the Theory of Equal Temperament in Chinese Music. Wiesbaden: Steiner. — The standard monograph on the mathematics.
- 戴念祖. 1986. 朱載堉 ── 明代的科學和藝術巨星. 人民出版社. — Full Chinese-language biography and intellectual study.
- 楊蔭瀏. 1981. 中國古代音樂史稿. — Treats the Yuè-lǜ quán shū as the climax of the Chinese music-theoretical tradition.
- Standaert, Nicolas. 2006. “Ritual dances and their visual representations in the Ming and Qing.” East Asian Library Journal 12.1. — Treats the dance-notation portions of the Yuè-lǜ quán shū.
- Lenoir, Yves and Nicolas Standaert. 2005. Les danses rituelles chinoises d’après Joseph-Marie Amiot. Brussels: Lessius.
Other points of interest
The Qiánlóng colophons are an unusually direct and personal critique. The first (Qiánlóng 51, 1786) directs the Sìkù editors — Yǒngróng, Débǎo, Zōu Yìxiào — to itemize every divergence between the Yuèlǜ quán shū and the Kāngxī Lǜlǚ zhèngyì and append the list to the tíyào. The list, dozens of items long, is preserved in the SKQS as _000.txt of this entry and is itself a major Qing-period musicological document. The third colophon, dated Yǐmǎo (1795), is occasioned by Qiánlóng’s review of two later qín anthologies (Yán Chéng’s Sōngxián guǎn qínpǔ and Chéng Xióng’s Sōngfēng gé qínpǔ) and constitutes the most extensive imperial intervention in Chinese music theory of the late Qīng. Qiánlóng’s argument that ancient music was one-character-one-tone (yī zì yī yīn) — and his rejection of all elaborate fingerings — became the official doctrine of late-Qīng court music.