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Numbers with Histories: Li Chunfeng on Harmonics and Astronomy

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Monographs in Tang Official Historiography

Part of the book series: Why the Sciences of the Ancient World Matter ((WSAWM,volume 3))

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Abstract

Focusing on Li Chunfeng’s 李淳風 (602–670) Sui shu 隋書 ‘Lü-li zhi’ 律曆志 (Treatise on Harmono-metrology and Mathematical Astronomy) and its predecessor in the Han shu 漢書, this chapter questions the universality of the marriage between these fields in Chinese thought, arguing instead that it is the product of specific compilers’ grappling with generic conventions and the messy course of history. The joint treatise appears first in the Han shu, a natural consequence of its reliance on Liu Xin’s 劉歆 (ca. 50 BCE–23 CE) synthetic writings, and culminates in Li Chunfeng’s treatises, after which subsequent histories abandon the model. Whatever the initial strength of this marriage, we argue, the common impulse to accuracy and empirical testing began to drive the two fields apart from both one another and from the promise of an elegant universal order as early as the Eastern Han. Framing his presentation on Liu Xin’s, and drawing heavily from his and others’ work, we attempt to show how Li Chunfeng’s editorial hand acts to address the unravelling of this order by imposing a telos upon the history of harmono-metrology and shifting the basis of the two fields from Book of Changes number symbolism to Nine Chapters number ratios, freeing number from the realm of timeless, petrified truth to allow it its own history in the face of progress.

Résumé

Consacré au « Lü-li zhi » 律曆志 (traité sur l’harmono-métrologie et sur l’astronomie mathématique) que Li Chunfeng 李淳風 (602–670) écrivit pour le Sui shu 隋書 ainsi qu’à son prédécesseur du Han shu 漢書, ce chapitre met en question l’universalité du mariage entre ces deux champs du savoir en Chine, et soutient au contraire que ce mariage est à la fois l’œuvre de compilateurs bien précis et de celui du cours impétueux de l’histoire. Ce traité double apparaît pour la première fois dans le Han shu,—conséquence naturelle du fait qu’il est fondé sur les écrits synthétiques de Liu Xin 劉歆 (ca. 50 av. n. è.-23 ap. n. è.)—et pour la dernière dans les traités de Li Chunfeng, après quoi le modèle est abandonné. Quel que fût la force initiale de ce mariage, croyons-nous, un certain mouvement général vers la précision et l’expérimentation empirique commença dès les Han orientaux à séparer les deux domaines l’un de l’autre et à les éloigner des promesses d’un bel ordre universel. Nous tentons de montrer comment Li Chunfeng, en élaborant sa présentation à partir de celle de Liu Xin et en se fondant considérablement sur son œuvre et sur celles d’autres auteurs, l’adapta pour tenir compte de l’effondrement de l’ordre qu’elle épouse. Pour ce faire, nous montrons qu’il imposa un telos à l’histoire de l’harmono-métrologie et échangea le fondement de ces deux champs en passant d’un symbolisme numérique provenant du Classique des mutations à une idée de proportion et de proportionnalité tirée des Neuf chapitres sur les procédures mathématiques, ce qui eut pour effet de libérer les nombres d’une modalité de vérité atemporelle et pétrifiée et de leur attribuer leur propre histoire face au fait historique du progrès.

The third and last distinction between heavenly movements, I mean the lateral, should be related to modulations between tonoi. … Among them we should compare the Dorian tonos, which is right in the middle of the others, with the middle positions of their lateral movements… the Mixolydian and the Hypodorian, as being the extremes, with the most northerly and southerly positions, conceived in the guise of tropics; and the remaining four tonoi, which are between the ones mentioned, with those falling on the parallels between the tropics and the celestial equator, these being themselves four in number, because of the division of the slantwise circle [i.e. the ecliptic] into twelve, corresponding to the twelve parts of the zodiac.

—Ptolemy, Harmonics iii/12

Tr. Barker (1984–1989: vol. 2, 386).

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Notes

  1. 1.

    Kassler (1982).

  2. 2.

    An ‘origin’ (yuan 元) refers to the coincidence of new moon, winter solstice, midnight and day jiazi.01 of the sexagenary day count as well as the period of its reoccurrence. Liu Xin uses the value of 4617 years, dividing the ‘origin’ into three 1539-year ‘concordances’ (tong 統)—‘heaven’, ‘earth’ and ‘man’—marking the coincidence of new moon, winter solstice and midnight on different sexagenary days. For an explanation of these periods, see Sivin (1969: 12–13).

  3. 3.

    In other words, (1 × 311) ÷ (1 × 39) = 32. Liu Xin’s operation gets us to the quotient 9, which is the canonical cun-length of the huangzhong pitch pipe and a yang number in Book of Changes numerology, whence ‘nine in the first’, referring to a solid yang-line in the first line of a hexagram which changes to a broken yin-line.

  4. 4.

    In other words, (9 × 6) ÷ 9 = 6, six being the canonical cun-length of the linzhong pitch pipe and the value of a changing yin line in Changes numerology.

  5. 5.

    In other words, the pitch pipe length of the next tone ‘up’ (in the circle of fifths) is ‘generated’ from pitch pipe length x as (x × 6 × 2) ÷ 9, while, one octave ‘down’, it is ‘generated’ by ‘decreasing’ x × 6 ÷ 9. These formulae, which we may express as x(3 + 1) ÷ 3 and x(3 − 1) ÷ 3, respectively, are known in lüHM as sanfen sunyi 三分損益 (decrease/increase [one of] three parts).

  6. 6.

    The ‘six lü and six lǚ’ refer to the six tones/pitch pipes of the chromatic scale, huangzhong (DO)—yingzhong (ALT.DO). The ‘twelve chronograms’ (chen 辰) refer to the heavenly branches (zi.B01–hai.B12) as used as counters of astronomical time and space.

  7. 7.

    Cit. Guoyu, 3.5a.

  8. 8.

    In other words, in the sexagenary cycle, the ten heavenly stems (jia.S01–gui.S10) repeat six times, each series beginning at jia.S01; and the twelve earthly branches (zi.B01–hai.B12) repeat five times, each series beginning at zi.B01; add these heavenly (5) and earthly (6) numbers together, and one gets 11.

  9. 9.

    Han shu, 21A.980–981. In the Triple Concordance li, the ‘day divisor’ is one component in Liu Xin’s value for the length of the synodic month: 2392/81 = 29 43/81 days. This value is taken from the Taichu li (see below).

  10. 10.

    On the history of harmonics and metrology in China, see Cheung (1974: vol. 1, 52–62), Qiu (1992), Keightley (1995), Bagley (2005) and Goodman (2010: 215–277).

  11. 11.

    On the categories li and tianwen, see Morgan (2017: 10–25) and the other contributions to this volume.

  12. 12.

    On Cai Yong and Liu Hong’s authorship of the Hou Han shu treatise, see Mansvelt Beck (1990: 56–63). Modern critical editions of the Song shu also possess a ‘Lü-li zhi’, but this is nothing more than a product of modern harmonisation, as pre-modern editions universally devote distinct treatises to the two subjects as per the model of the Shiji. We thank Li Liang 李亮 for opening our eyes to this point.

  13. 13.

    The Song shi’s resurrection of the ‘Lü-li zhi’ is a natural consequence, it seems, of the brief flurry of lüHM activity inspired by the renewed classicist orientation of Song 宋 (960–1279) intellectual culture. Prior to this, the treatise tells us, the field seems to have dried up: ‘From [627/649] to [954], five dynasties rose and fell and 300 years passed; gentlemen of broad learning, for their part, set about meticulously collecting [the remnants] of what had been lost, but the lüHM treatises were all lacking. At the beginning of the Song, all the realm was united, gentleman experts were gathered, and the systems of state and monarchy were all restored to the ancient dao’ 暨唐貞觀迄周顯德, 五代隆替, 踰三百年, 博達之士頗亦詳緝廢墜, 而律志皆闕。 宋初混一 内, 能士畢舉, 國經王制, 悉復古道 (Song shi, 68.1493).

  14. 14.

    The six lüHM titles listed under these headings are as follows: (1) anon., Lü-li shu fa 律曆數法 in 3 juan (Han shu, 30.1766); (2) anon., Lü-li zhu jie 律曆注解 in 1 juan (Sui shu, 34.1024); (3) anon., Huangzhong suan fa 黃鍾算法 in 38 and 40 juan (Sui shu, 34.1026; Xin Tang shu, 59.1546); (4) anon., Suan lü-lǚ fa 算律呂法 in 1 juan (Sui shu, 34.1026); (5) anon., Tui Han shu Lü-li zhi shu 推漢書律曆志術 in 1 juan (Sui shu, 34.1024; Xin Tang shu, 59.1546); and (6) Xindu Fang 信都芳 (sixth century), Qi zhun 器準 in 3 juan (Xin Tang shu, 59.1546).

  15. 15.

    Han shu, 21A.955.

  16. 16.

    Han shu, 21A.979.

  17. 17.

    On Liu Xin’s lüHM work, see Vogel (1994). On his Santong li, see Nōda and Yabuuti (1947), Teboul (1983), Kawahara (1996: 148–195) and Cullen (2017a, 32–137, b, 123–178). On his archaeoastronomical work on the Zuo Tradition via the Santong li in his Shijing, see Cullen (2001). On his life and times, see Loewe (2000: 383–386) and Xu Xingwu (2005).

  18. 18.

    See Kalinowski (2011).

  19. 19.

    Han shu, 21A.975–976.

  20. 20.

    On the political and ideological context of the Taichu li and the reform of 104 BCE, see Cullen (1993).

  21. 21.

    律則成物, 曆則編時, 律曆交道, 聖人以謀, Taixuan jing, 7.6b; cf. the first-century BCE ‘Zengzi tianyuan’ 曾子天圓, Da Dai Liji, 5.17b–18b. Note that the Song shu 宋書 treatise posits that Yang wrote the Taixuan jing on the basis of Liu Xin’s work: ‘Liu Xin’s Triple Concordance system was particularly and repeatedly loose and wide (inaccurate)… Yang Xiong’s heart-mind was confused by its rhetoric and so adopted it for the Taixuan’ 劉歆三統法尤復疏闊…揚雄心惑其說, 采為太玄 (Song shu, 12.231).

  22. 22.

    On Han office titles, see Bielenstein (1980). On the history and function of the short-lived office of Xi-He, see Yoshino (2003). On the legitimation of Wang Mang’s ascendancy to the Han throne, see Sukhu (2005–2006).

  23. 23.

    On the life, times and work of Li Chunfeng see Chap. 2, this volume.

  24. 24.

    Note that the Wei shu follows the Han shu in this sense, if only for the reason that its harmono-metric contents—a précis and two historical anecdotes—are too sparse to constitute their own juan.

  25. 25.

    See Jin shu, 17.474.

  26. 26.

    This passage does not appear in the Book of Documents as we currently have it. Yan Shigu 顏師古 (581–645) explains that ‘this is [from] a lost document; it means to say that in mustering his patrimony, the king should first establish the calculation of numbers to command the hundred affairs’ 逸書也; 言王者統業, 先立算數以命百事也 (Han shu, 21A.957 (commentary)).

  27. 27.

    I.e., zi.B01 = 1; chou.B02 = 3 × 1 = 3; yin.B03 = 32 × 1 = 9 … hai.B12 = 311 × 1 = 177 147.

  28. 28.

    Described in the ‘Xici zhuan’ 繫辭傳 commentary, the ‘great expansion’ is the combinatory method used to produce hexagram line numbers through the manipulation of milfoil stalks. For an explanation of the divinatory procedure of the Book of Changes, see Nielsen (2003: 39–43). For an explanation of the significance of the number, shape and geometric configuration of counting rods as presented in the Li Chunfeng and Liu/Ban treatises, see Li Yan (1955). On the use of positive and negative counting rods, see Zhu Yiwen (2010). We thank Zhu Yiwen 朱一文 for sharing with us his understanding of the counting-rod-related passages in Li and Liu’s treatises and directing us to the studies cited here (personal communication, October 2013).

  29. 29.

    These are very small units of measure.

  30. 30.

    Han shu, 21A.956.

  31. 31.

    Here the Zuo Tradition actually reads, ‘creatura (wu) exhibit form only upon birth, reproduction only upon form, and number only upon reproduction’ 物生而後有象, 象而後有滋, 滋而後有數 (Chunqiu Zuo zhuan zhushu, 13.17a).

  32. 32.

    In other words, 311 ÷ 39 = 32 = 9; see Note 27.

  33. 33.

    See Note 28.

  34. 34.

    The last few lines of this paragraph are modified from Hou Han shu, zhi 1, 2999, the main difference being that Li Chunfeng has inserted lü 率 into the list of ‘applications’.

  35. 35.

    Here Li Chunfeng is citing the Chap. headings of The Nine Chapters of Mathematical Procedures (Jiuzhang suanshu 九章算術) and the purport of each chapter as summarised in Liu Hui’s commentary thereto; see Chemla and Guo (2004).

  36. 36.

    Here Li Chunfeng is citing Liu Hui’s commentary to The Nine Chapters, which, after introducing the procedure for adding fractions, opines, ‘Multiply to disaggregate them, simplify to assemble them, homogenise and equalise to make them communicate, how could those not be the key-points (gangji) of computations/mathematics (suan 算)?’ 乘以散之, 約以聚之, 齊同以通之, 此其筭之綱紀乎 (Jiuzhang suanshu, 1.9 (commentary); tr. Chemla (2010: 277)).

  37. 37.

    Sui shu, 16.387–388. On the length measures zhang, chi, cun, fen, etc., see Chap. 1, Fig. 1.2, this volume.

  38. 38.

    See Note 28.

  39. 39.

    今依班志編錄五代聲律度量, Sui shu, 16.386.

  40. 40.

    Hou Han shu, zhi 1, 3016; Song shu, 11.208–209; Jin shu, 16.489–490.

  41. 41.

    Song shu, 11.212–220; Jin shu, 16.480–485; cf. Goodman (2010: 228–232).

  42. 42.

    The Jingchu li, preface and procedure text, is found in Song shu, 12.232–258, and Jin shu, 18.535–562. For an excellent comparison and synthetic treatment of the Song shu and Jin shu liMA chronicles, see Hasebe (1991). As to Li Chunfeng’s source, Morgan (2017: 197) suggests that it may also have been He Chengtian.

  43. 43.

    Sui shu, 16.386, 16.397, 17.416; Jin shu, 17.498, 17.503.

  44. 44.

    余所謂「述故事,整齊其世傳」, 非所謂「作」也, Shiji, 130.3299–3300. By the 170s–180s and on through the Southern Dynasties, the politicization of historiographical writing had become acute, and participants, though not creating new genres per se, pushed their compilational and editing agendas in the spirit of a certain type of literary creativity; on arguments over chronology and editing in the third century CE, see Goodman (2010: 334–346).

  45. 45.

    Particularly remarkable in this regard is Li’s conspicuous failure to mention Shen Yue 沈約 (441–513) and He Chengtian’s Song shu treatise(s) whatsoever, though it is the single source from which he seems to have appropriated the majority of his Jin shu treatise; see Morgan (2017: 179–202). Note that Li does however refer to He’s ideas in his Sui shu LüHM juan (Sui shu, 16.389), and that there were lacunae in the transmission of He’s private writings soon after his death in 447; see Goodman (2015).

  46. 46.

    Han shu, 21A.955.

  47. 47.

    29 43/81 = 29.53086, cf. the modern value at epoch of 29.53059 days. 29 499/940 = 29.53085 days is the value adopted by Bian Xin 編訢 and Li Fan 李梵 in the Sifen li 四分曆 of 85/86 CE, and 29 773/1457 = 29.53054 days is that of Liu Hong’s Qianxiang li 乾象曆 of circa 206 CE. Note that the mean synodic month is but one element determining the predictive accuracy of an astronomical system, and that the Sifen li’s perceived success in predicting lunar phenomena over the Santong li in the late first century CE had more to do with accumulated error from system origin than this particular value. Note also that while the Qianxiang li maintains the ‘day divisor’ as the denominator of the mean synodic month, the Sifen li names this instead the ‘obscuration months’ (bu yue 蔀月), having reappropriated the term ‘day divisor’ as the denominator of the solar year. For historical values of the mean synodic month, see Chen (1995: 212–232).

  48. 48.

    For further criticism of the equation of yin-yang, five-agents and Book of Changes analogical reasoning as the one, true ‘Chinese thought/cosmology’, see Graham (1986: 8–11), Saussy (2000), Brown (2006), Martzloff (2009: 38–41) and Nylan (2010). For an excellent example of compartmentalisation, see the example of Ge Hong’s 葛洪 (283–343) argument against the metaphysical case for ‘umbrella heaven’ 蓋天 cosmology by appeal to experiential falsification from shadows, motions, apparent sizes and optics as well as prophecy and more sophisticated metaphysics: Sui shu, 19.509–511, and Jin shu, 11.281–284, tr. Ho Peng Yoke (1966: 55–58); cf. Morgan (2017: 203–205).

  49. 49.

    See Goodman (2005, 2010) and, on the Ji Tomb, Shaughnessy (2006: 131–184).

  50. 50.

    This is according to the master list in Qu Anjing (2008: 629–633); cf. Yabuuti (1963) and Sivin (2009: 42–53, Table 2.1). Note that these tables tend not include every li ‘astronomical system’ mentioned in received sources, focusing on those leading to government reforms; by our count, the middle period between the Han and Tang produced something more like forty-three new li.

  51. 51.

    On public debate, see also Cullen (2007).

  52. 52.

    For He Chengtian, see Song shu, 12.261; for Zhang Zixin, see Sui shu, 20.561.

  53. 53.

    The former—the first extant example of the sort of datasets applied to the ranking of competing liMA systems—occurs in the context of a failed reform at the Cao-Wei court circa 226; see Jin shu, 17.500–502; cf. Morgan (2017: 140–176). The latter occurs in the context of debate surrounding the late-Sui Daye li 大業曆; see Sui shu, 17.429–434.

  54. 54.

    On the Book of Changes hexagram line numbers ‘nine, six, seven and eight’, see Nielsen (2003: 294).

  55. 55.

    Hou Han shu, zhi 2, 3030.

  56. 56.

    Chen Meidong (2007: 551–565).

  57. 57.

    Cf. Sivin (1986).

  58. 58.

    The following analysis is taken from Morgan (2017: 45–48, 140–212). On the conceptual language of accuracy, see Hashimoto (1979).

  59. 59.

    Hou Han shu, zhi 2, 3041; tr. modified from Sivin (1986: 158).

  60. 60.

    古人所未達, Yixing 一行 (683–727) , cited in Xin Tang shu, 27A.593. For similar historical criticism and apology in the astral sciences, see Morgan (2016).

  61. 61.

    Jin shu, 17.498; cf. Song shu, 12.231–232. On Liu Hong’s Qianxiang li, see Cullen (2017a: 235–355; 2017b: 325–392). Note that Liu Hong’s use of Book of Changes number symbolism marks the last attempt of its kind until Yixing in the eighth century.

  62. 62.

    荀勖造新鍾律, 與古器諧韵, 時人稱其精密, Jin shu, 16.491; cf. Goodman (2010: 204–205) (worded differently).

  63. 63.

    Axiom 6 we deduce from the strategy of ear-testing mentioned in the following quotation from Liu Xin; for the others, see above.

  64. 64.

    Here, Liu Xin is citing the ‘Xici zhuan’ commentary to the Book of Changes, which reads ‘The three and five operations are undertaken in order to obtain a change. Divisions and combinations of the numbers are made’ 參伍以變錯綜其數 (Zhouyi zhushu, 11.35a; tr. Wilhelm (1967: 314)).

  65. 65.

    Han shu, 21A.956.

  66. 66.

    Sui shu, 16.394.

  67. 67.

    Sui shu, 16.402.

  68. 68.

    On the practice of hou qi, see Bodde (1959) and Huang and Chang (1996). Suffice it to say that pre-Song sources explicitly describe the practice as a method for generating omens, e.g. ‘if it matches: harmony; otherwise: omen’ 效則和,否則占 (Hou Han shu, zhi 1, 3024), which comports with the long-standing connection between harmonics and divination. On harmonics divination, see Kalinowski (2011).

  69. 69.

    He does something similar in the ‘Tianwen zhi’, on which see Chap. 6, this volume.

  70. 70.

    今略諸代尺度一十五度, 并異同之說如左, Sui shu, 16.402.

  71. 71.

    Sui shu, 16.404.

  72. 72.

    For studies of Li Chunfeng’s instrument catalogue, see Ma Heng’s 馬衡, ‘Suishu Lüli zhi shiwu deng chi’ 隋書律曆志十五等尺, translated in Ferguson (1941); see also Qiu Guangming et al. (2001: 305–317) and Goodman (2010: 197–206), which examine the back-stories of certain of Li’s instruments. To compare with ancient Greek instrument catalogues of this nature, see Creese (2010: 50 ff., esp. 118–122).

  73. 73.

    參考古律度, Sui shu, 16.392–394.

  74. 74.

    詔施用水尺律樂, 其前代金石, 並鑄毀之, 以息物議, Sui shu, 16.392.

  75. 75.

    Sui shu, 16.402.

  76. 76.

    Goodman (2010: 189–190, 198–200) analyses comments in several sources that cast doubt on the discovery of bells or chimes in the Ji Tomb, or, at least, that Xun Xu or his son ever used them in any tests. In fact Xun Xu had already begun, even before the Ji Tomb discovery, to ‘call in’ (or hunt down) ancient foot-rules and bells in others’ collections around the country.

  77. 77.

    Goodman (2010: passim). The political crises in the background of Xun’s and his peers’ work concerning precision in ritual arts never directly used the vocabulary or findings associated with a pure Zhou standard (‘prisca Zhou’), even though the latter impacted the court’s music and lüHM, nor were such empirical findings ever directly expressed in an ideology of delegitimation; Goodman (2010: 374–382) explains that Xun’s peers generally rejected this ritual precision based mostly on ad hominem grounds.

  78. 78.

    For an analysis of how Xun Xu was judged in the Jin shu biography written for him in the Tang, see Goodman (2010: 114–118). For Taizong’s interest in the Jin as a political model, although a delicate one, see McMullen (2013).

  79. 79.

    Ptolemy, Harmonics i.13–14; tr. Barker (1984–1989: vol. 2, 304–305).

  80. 80.

    Jin shu, 16.491.

  81. 81.

    Jiuzhang suanshu, 1.18; tr. Chemla and Guo (2004: 167). For more on lü in the context of mathematics, see ibid., 119–219, 956–959.

  82. 82.

    順性命之理, Han shu, 21A.956.

  83. 83.

    自伏戲畫八卦, 由數起, Han shu, 21A.955.

  84. 84.

    See Chemla (2010).

  85. 85.

    Jiuzhang suanshu, preface; tr. modified from Chemla and Guo (2004: 127).

  86. 86.

    Not only is suan 算 ‘mathematics’ either omitted from or subordinate within such headings, so too are mathematical texts like The Nine Chapters listed after titles concerning the various facets of liMA; see Han shu, 30.1765–1767; Sui shu, 34.1022–1026; Jiu Tang shu, 47.2037–2039. On bibliographical classification, see Chap. 11, this volume.

  87. 87.

    Song shi, 71.1608.

  88. 88.

    See Lam (2006).

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  1. Laboratoire SPHERE, UMR 7219, CNRS, Université Paris Diderot, Case 7093, 5 Rue Thomas Mann, 75205, Paris Cedex 13, France

    Daniel P. Morgan

  2. Seattle, WA, USA

    Howard L. Goodman

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    Daniel Patrick Morgan

  2. Centre de recherche sur les civilisations de l’Asie orientale, Paris, France

    Damien Chaussende

Appendix

Appendix

See Tables 3.1 and 3.2.

Table 3.1 Han shu ‘Lü-li zhi’ table of contents
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Table 3.2 Sui shu ‘Lü-li zhi’ table of contents
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Morgan, D.P., Goodman, H.L. (2019). Numbers with Histories: Li Chunfeng on Harmonics and Astronomy. In: Morgan, D., Chaussende, D. (eds) Monographs in Tang Official Historiography. Why the Sciences of the Ancient World Matter, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-18038-6_3

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