1
後漢書志第一律曆上律準候氣
Treatise I of the Later Han: musical pitch (upper scroll)—regulators for tuning pipes and the qi-detection method.
2
古之人論數也，曰：「物生而後有象，像而後有滋，滋而後有數」。 然則天地初形，人物既著，則筭數之事生矣。 記稱大橈作甲子[1]，隸首作數[2]。 二者既立，以比日表[3]，以管萬事。 夫一、十、百、千、萬，所同用也； 律、度、量、衡、歷，其別用也。 故體有長短，檢以度[4]； 物有多少，受以量[5]； 量有輕重，平以權衡[6]； 聲有清濁，協以律呂； 三光運行，紀以歷數：然後幽隱之情，精微之變，可得而綜也[7]。
Ancient thinkers defined reckoning in stages: "beings yield forms, forms yield proliferation, and only then do numbers apply." Once heaven, earth, and humanity existed, mathematics followed inevitably. Tradition credits Da Nao with the stem-branch cycle note one and Li Shou with systematic numbering note two. Those tools calibrated gnomon shadows note three and ordered every administrative reckoning. Digits from one through myriad underpin every tally; but pitch, linear measure, volume, balance, and celestial reckoning each branch separately. Bodies are gauged with rulers note four; bulk is poured into standard measures note five; weight rests on beam balances note six; sound divides into clear and muddy tones harmonized by the pitch-cycle; sun, moon, and stars ride calendrical mathematics note seven until even arcane shifts yield to systematic grasp.
3
漢興，北平侯張蒼首治律歷。 孝武正樂，置協律之官。 至元始中，博徵通知鐘律者，考其意義，羲和劉歆典領條奏，前史班固取以為志。 而元帝時，郎中京房，房字君明，知五聲之音，六律之數。 上使太子太傅韋玄成、字少翁諫議大夫章，雜試問房於樂府。 房對：「受學故小黃令焦延壽。 六十律相生之法：以上生下，皆三生二，以下生上，皆三生四，陽下生陰，陰上生陽，終於中呂，而十二律畢矣。 中呂上生執始，執始下生去滅，上下相生，終於南事，六十律畢矣。 夫十二律之變至於六十，猶八卦之變至於六十四也。 宓羲作易，紀陽氣之初，以為律法。 建日冬至之聲，以黃鐘為宮，太蔟為商，姑洗為角，林鐘為徵，南呂為羽，應鐘為變宮，蕤賓為變徵[8]。 此聲氣之元，五音之正也。 故各終一日。 其餘以次運行，當日者各自為宮，而商徵以類從焉[9]。
At the dynastic founding Zhang Cang of Beiping Marquis reorganized harmonics and astronomy. Emperor Wu standardized court music and appointed specialists to tune the bells. Yuanshi-era commissions collected tuning theory; Liu Xin’s Directorate of Astronomy codified it for the throne, and Ban Gu folded that dossier into his monographs. Palace attendant Jing Fang (Junming), contemporary with Yuan Emperor, knew the five tones and six-pitch arithmetic inside out. The throne tasked Wei Xuancheng and Counselor Zhang with cross-examining him at the imperial conservatory. Fang answered: “I studied under former Xiaohuang magistrate Jiao Yanshou. Sixty-pitch derivation uses three-halves downward steps and four-thirds upward steps until Zhonglü closes the duodecimal set. Zhonglü ascends to Zhishi, Zhishi descends to Qumie, and the ladder finishes at Nanshi—sixty names in all. Twelve pipes blooming into sixty notes mirrors eight trigrams blooming into sixty-four hexagrams. Fuxi’s Zhou yi anchors yang qi’s primal emergence—the same logic pitch computation follows. On the jian day of midwinter you tune Huangzhong to keynote gong, slot Taicu as shang, Guxian as jue, Linzhong as zhi, Nanlü as yu, Yingzhong as biangong, Ruibin as bianzhi note eight. That cluster forms the acoustic foundation and canonical pentatonic template. Each configuration therefore governs an entire day. Other days rotate the pattern; the daily tonic becomes gong while shang and zhi shift by analogy note nine.
4
禮運篇曰『五聲、六律、十二管還相為宮』，此之謂也[10]。 以六十律分朞之日，黃鐘自冬至始，及冬至而復，陰陽寒燠風雨之占生焉。 於以檢攝羣音，考其高下，苟非草木之聲，則無不有所合。 虞書曰『律和聲』，此之謂也。」
The Liyun line about twelve tubes cyclically serving as keynote captures exactly this note ten. Splitting sixty tones across the calendar begins and ends at Huangzhong on midwinter day, feeding weather divination. Against that grid every musical tone lines up unless it is mere rustling vegetation. “Thus,” he concluded, “the Documents command pitch to harmonize voice.”
5
房又曰：「竹聲不可以度調，故作準以定數。 準之狀如瑟，長丈而十三弦，隱閒九尺，以應黃鐘之律九寸； 中央一弦，下有畫分寸，以為六十律清濁之節。」 房言律詳於歆所奏，其術施行於史官，候部用之。 文多不悉載。 故總其本要，以續前志。
Fang continued: “Because bamboo defies abstract calibration, I built a tuning regulator to nail the ratios. It resembles a se: thirteen strings over one zhang, nine chi of vibrating length answering Huangzhong’s nine-cun tube; its middle string carries engraved fen and cun used to mark every bright or dull step among sixty tones." His theory ran deeper than Liu Xin’s submission and became standard for historians and seasonal qi observers. Most detail is omitted here for length. This section condenses the core to continue the earlier monograph.
6
律術曰：陽以圓為形，其性動。 陰以方為節，其性靜。 動者數三，靜者數二。
Harmonic theory: yang manifests as round motion and tends toward movement. Yin squares itself into boundaries and stays quiescent. Active numbers ride three; static numbers ride two.
7
以陽生陰，倍之； 以陰生陽，四之：皆三而一。 陽生陰曰下生，陰生陽曰上生。
Yang generating yin doubles the prior value; yin generating yang multiplies by four before the universal division by three. Down-generation lowers the pipe; up-generation raises it.
8
上生不得過黃鐘之清濁，下生不得及黃鐘之數實。 皆參天兩地，圓蓋方覆，六耦承奇之道也。 黃鐘，律呂之首，而生十二律者也[11]。 其相生也，皆三分而損益之。 是故十二律之，得十七萬七千一百四十七，是為黃鐘之實[12]。 又以二乘而三約之，是為下生林鐘之實。 又以四乘而三約之，是為上生太蔟之實。 推此上下，以定六十律之實。 以九三之，數萬九千六百八十三為法。 律為寸，於準為尺。 不盈者十之，所得為分。 又不盈十之，所得為小分。 以其餘正其強弱。
Ascending steps cannot surpass Huangzhong’s upper bound; descending steps cannot undershoot its canonical mass-number. The arithmetic embodies heaven-three earth-two, dome over square plane, even couples lifting odd centers. Huangzhong seeds every subsequent pipe note eleven. Each link applies the classical subtract-one-third or add-one-third rule. Twelve iterations therefore pile to 177,147—the canonical Huangzhong “solid” note twelve. Two-thirds of that mass defines down-generated Linzhong. Four-thirds steps back upward to Taicu’s mass. Repeat the ladder to enumerate sixty tonal solids. Use 19,683—nine cubed times three—as the modular divisor. Pipe lengths translate to chi on the regulator at ten-to-one scale. Fractional remainders become fen by decimal promotion. Smaller scraps refine again into micro-fen. Residual micro-units mark acoustic strong or weak bias.
9
黃鐘，十七萬七千一百四十七。 下生林鐘。 黃鐘為宮，太蔟商，林鐘徵。 一日律九寸，準九尺。
Huangzhong solid: 177,147. Down-generates Linzhong. Mode: Huangzhong tonic, Taicu shang, Linzhong zhi. Nine cun on the pipe, nine chi on the regulator.
10
色育，十七萬六千七百七十六。 下生謙待。 色育為宮，未知商，謙待徵。 六日。
Seyu solid: 176,776. Down-generates Qiandai. With Seyu as tonic the matching shang pipe goes unrecorded; Qiandai supplies zhi. Occupies six days in the cycle.
11
律，八寸九分小分八微強。 準，八尺九寸萬五千九百七十三。 執始，十七萬四千七百六十二。 下生去滅。 執始為宮，時息商，去滅徵。 六日。 律，八寸八分小分七大強。 準，八尺八寸萬五千五百一十六。 丙盛，十七萬二千四百一十。
Pipe measure 8.98+ strong micro-fen. Regulator line 8 chi 9 cun and 15,973 residual ticks. Zhishi solid: 174,762. Down-generates Qumie. Mode: Zhishi gong, Shixi shang, Qumie zhi. Six-day span. Pipe 8.87+ marked strong. Regulator 8 chi 8 cun, remainder 15,516. Bingsheng solid: 172,410.
12
下生安度。 丙盛為宮，屈齊商，安度徵。 六日。 律，八寸七分小分六微弱。 準，八尺七寸萬一千六百七十九。 分動，十七萬八十九。 下生歸嘉。 分動為宮，隨期商，歸嘉徵。 六日。 律，八寸六分小分四強。 準，八尺六寸八千一百五十二。
Down-generates Andu. Mode: Bingsheng gong, Juqi shang, Andu zhi. Six days. Pipe 8.76 slightly weak. Regulator 8 chi 7 cun, remainder 11,679. Fendong solid: 170,089. Down-generates Guijia. Mode: Fendong gong, Suiqi shang, Guijia zhi. Six-day allocation. Pipe 8.64 strong. Regulator 8 chi 6 cun, remainder 8,152.
13
質末，十六萬七千八百。 下生否與。 質末為宮，形晉商，否與徵。 六日。 律，八寸五分小分二強。 準，八尺五寸四千九百四十五。
Zimo solid: 167,800. Down-generates Fouyu. Mode: Zimo gong, Xingjin shang, Fouyu zhi. Six days. Pipe 8.52 strong. Regulator 8 chi 5 cun, remainder 4,945.
14
大呂，十六萬五千八百八十八。 下生夷則。 大呂為宮，夾鐘商，夷則徵。 八日。
Dalü solid: 165,888. Down-generates Yize. Mode: Dalü gong, Jiazhong shang, Yize zhi. Eight-day span.
15
律，八寸四分小分三弱。 準，八尺四寸五千五百八。 分否，十六萬三千六百五十四。 下生解形。 分否為宮，開時商，解形徵。 八日。 律，八寸三分小分一強。
Pipe 8.43 weak. Regulator 8 chi 4 cun, remainder 5,508. Fenfou solid: 163,654. Down-generates Jiexing. Mode: Fenfou gong, Kaishi shang, Jiexing zhi. Eight days. Pipe 8.31 strong.
16
準，八尺三寸二千八百五十一。 凌陰，十六萬一千四百五十二。 下生去南。 凌陰為宮，族嘉商，去南徵。 八日。 律，八寸二分小分一弱。 準，八尺二寸五百一十四。 少出，十五萬九千二百八十。 下生分積。 少出為宮，爭南商，分積徵。
Regulator: 8 chi 3 cun plus 2,851 micro-units. Lingyin solidity: 161,452. Down-generates Qunan. Mode: Lingyin tonic, Zujia shang, Qunan zhi. 8-day span in the cycle. Pipe length recorded as 8 cun 2 fen minor-fen 1 weak. Regulator: 8 chi 2 cun plus 514 micro-units. Shaochu solidity: 159,280. Down-generates Fenji. Mode: Shaochu tonic, Zhengnan shang, Fenji zhi.
17
六日。 律，八寸小分九強。 準，八尺萬八千一百六十。 太蔟，十五萬七千四百六十四。 下生南呂。 太蔟為宮，姑洗商，南呂徵。 一日。 律，八寸。 準，八尺。
6-day span in the cycle. Pipe length recorded as 8 cun minor-fen 9 strong. Regulator: 8 chi plus 18,160 micro-units. Taicu solidity: 157,464. Down-generates Nanlü. Mode: Taicu tonic, Guxian shang, Nanlü zhi. 1-day span in the cycle. Pipe length recorded as 8 cun. Regulator: 8 chi.
18
未知，十五萬七千一百三十四。 下生白呂。 未知為宮，南授商，白呂徵。 六日。
Weizhi solidity: 157,134. Down-generates Bailü. Mode: Weizhi tonic, Nanshou shang, Bailü zhi. 6-day span in the cycle.
19
律，七寸九分小分八強。 準，七尺九寸萬六千三百八十三。 時息，十五萬五千三百四十四。 下生結躬。 時息為宮，變虞商，結躬徵。 六日。 律，七寸八分小分九少強。 準，七尺八寸萬八千一百六十六。 屈齊，十五萬三千二百五十三。
Pipe length recorded as 7 cun 9 fen minor-fen 8 strong. Regulator: 7 chi 9 cun plus 16,383 micro-units. Shixi solidity: 155,344. Down-generates Jiegong. Mode: Shixi tonic, Bianyu shang, Jiegong zhi. 6-day span in the cycle. Pipe length recorded as 7 cun 8 fen minor-fen 9 modestly strong. Regulator: 7 chi 8 cun plus 18,166 micro-units. Quqi solidity: 153,253.
20
下生歸期。 屈齊為宮，路時商，歸期徵。 六日。 律，七寸七分小分九弱。 準，七尺七寸萬六千九百三十九。 隨期，十五萬一千一百九十。 下生未卯。 隨期為宮，形始商，未卯徵。 六日。 律，七寸六分小分八強。 準，七尺六寸萬五千九百九十二。 形晉，十四萬九千一百五十五。 下生夷汗。 形晉為宮，依行商，夷汗徵。 六日。 律，七寸五分小分八弱。 準，七尺五寸萬五千三百二十五。
Down-generates Guiqi. Mode: Quqi tonic, Lushi shang, Guiqi zhi. 6-day span in the cycle. Pipe length recorded as 7 cun 7 fen minor-fen 9 weak. Regulator: 7 chi 7 cun plus 16,939 micro-units. Suiqi solidity: 151,190. Down-generates Weimao. Mode: Suiqi tonic, Xingshi shang, Weimao zhi. 6-day span in the cycle. Pipe length recorded as 7 cun 6 fen minor-fen 8 strong. Regulator: 7 chi 6 cun plus 15,992 micro-units. Xingjin solidity: 149,155. Down-generates Yihan. Mode: Xingjin tonic, Yihang shang, Yihan zhi. 6-day span in the cycle. Pipe length recorded as 7 cun 5 fen minor-fen 8 weak. Regulator: 7 chi 5 cun plus 15,325 micro-units.
21
夾鐘，十四萬七千四百五十六。 下生無射。 夾鐘為宮，中呂商，無射徵。 六日。
Jiazhong solidity: 147,456. Down-generates Wuyi. Mode: Jiazhong tonic, Zhonglü shang, Wuyi zhi. 6-day span in the cycle.
22
律，七寸四分小分九強。 準，七尺四寸萬八千一十八。 開時，十四萬五千四百七十。 下生閉掩。 開時為宮，南中商，閉掩徵。 八日。 律，七寸三分小分九微強。 準，七尺三寸萬七千八百四十一。 族嘉，十四萬三千五百一十三。 下生鄰齊。 族嘉為宮，內負商，鄰齊徵。 八日。 律，七寸二分小分九微強。
Pipe length recorded as 7 cun 4 fen minor-fen 9 strong. Regulator: 7 chi 4 cun plus 18,018 micro-units. Kaishi solidity: 145,470. Down-generates Biyan. Mode: Kaishi tonic, Nanzhong shang, Biyan zhi. 8-day span in the cycle. Pipe length recorded as 7 cun 3 fen minor-fen 9 slightly strong. Regulator: 7 chi 3 cun plus 17,841 micro-units. Zujia solidity: 143,513. Down-generates Linqi. Mode: Zujia tonic, Neifu shang, Linqi zhi. 8-day span in the cycle. Pipe length recorded as 7 cun 2 fen minor-fen 9 slightly strong.
23
準，七尺二寸萬七千九百五十四。 爭南，十四萬一千五百八十二。 下生期保。
Regulator: 7 chi 2 cun plus 17,954 micro-units. Zhengnan solidity: 141,582. Down-generates Qibao.
24
爭南為宮，物應商，期保徵。 八日。 律，七寸一分小分九強。 準，七尺一寸萬八千三百二十七。 姑洗，十三萬九千九百六十八。 下生應鐘。 姑洗為宮，蕤賓商，應鐘徵。 一日。 律，七寸一分小分一微強。 準，七尺一寸二千一百八十七。
Mode: Zhengnan tonic, Wuying shang, Qibao zhi. 8-day span in the cycle. Pipe length recorded as 7 cun 1 fen minor-fen 9 strong. Regulator: 7 chi 1 cun plus 18,327 micro-units. Guxian solidity: 139,968. Down-generates Yingzhong. Mode: Guxian tonic, Ruibin shang, Yingzhong zhi. 1-day span in the cycle. Pipe length recorded as 7 cun 1 fen minor-fen 1 slightly strong. Regulator: 7 chi 1 cun plus 2,187 micro-units.
25
南授，十三萬九千六百七十。 下生分烏。 南授為宮，南事商，分烏徵。
Nanshou solidity: 139,670. Down-generates Fenwu. Mode: Nanshou tonic, Nanshi shang, Fenwu zhi.
26
六日。 律，七寸小分九大強。 準，七尺萬八千九百三十。 變虞，十三萬八千八十四。 下生遲內。 變虞為宮，盛變商遲內徵。 六日。 律，七寸小分一半強。
6-day span in the cycle. Pipe length recorded as 7 cun minor-fen 9 strongly. Regulator: 7 chi plus 18,930 micro-units. Bianyu solidity: 138,084. Down-generates Chinei. Mode: Bianyu tonic, Shengbian shang, Chinei zhi. 6-day span in the cycle. Pipe length recorded as 7 cun minor-fen 1 half strong.
27
準，七尺三千三十。 路時，十三萬六千二百二十五。 下生未育。 路時為宮，離宮商，未育徵。 六日。 律，六寸九分小分二微強。 準，六尺九寸四千一百二十三。 形始，十三萬四千三百九十二。 下生遲時。 形始為宮，制時商，遲時徵。
Regulator: 7 chi plus 3,030 micro-units. Lushi solidity: 136,225. Down-generates Weiyu. Mode: Lushi tonic, Ligong shang, Weiyu zhi. 6-day span in the cycle. Pipe length recorded as 6 cun 9 fen minor-fen 2 slightly strong. Regulator: 6 chi 9 cun plus 4,123 micro-units. Xingshi solidity: 134,392. Down-generates Chishi. Mode: Xingshi tonic, Zhishi shang, Chishi zhi.
28
五日。 律，六寸八分小分三弱。 準，六尺八寸五千四百七十六。 依行，十三萬二千五百八十二。 上生色育。 依行為宮，謙待商，色育徵。 七日。 律，六寸七分小分三大強。 準，六尺七寸七千五十九。
5-day span in the cycle. Pipe length recorded as 6 cun 8 fen minor-fen 3 weak. Regulator: 6 chi 8 cun plus 5,476 micro-units. Yihang: 132,582. Generates Seyu. Mode: Yihang tonic, Qiandai shang, Seyu zhi. 7-day span in the cycle. Pipe length recorded as 6 cun 7 fen minor-fen 3 strongly. Regulator: 6 chi 7 cun plus 7,059 micro-units.
29
中呂，十三萬一千七十二。 上生執始。 中呂為宮，去滅商，執始徵。 八日。 律，六寸六分小分六弱。 準，六尺六寸萬一千六百四十二。 南中，十二萬九千三百八。 上生丙盛。 南中為宮，安度商，丙盛徵。 七日。 律，六寸五分小分七微弱。
Zhonglü: 131,072. Generates Zhi-shi. Mode: Zhonglü tonic, Qumie shang, Zhi-shi zhi. 8-day span in the cycle. Pipe length recorded as 6 cun 6 fen minor-fen 6 weak. Regulator: 6 chi 6 cun plus 11,642 micro-units. Nanzhong: 129,308. Generates Bing-sheng. Mode: Nanzhong tonic, An-du shang, Bing-sheng zhi. 7-day span in the cycle. Pipe length recorded as 6 cun 5 fen minor-fen 7 micro-weak.
30
準，六尺五寸萬三千六百八十五。 內負，十二萬七千五百六十七。 上生分動。
Regulator: 6 chi 5 cun plus 13,685 micro-units. Neifu: 127,567. Generates Fen-dong.
31
內負為宮，歸嘉商，分動徵。 八日。 律，六寸四分小分八強。 準，六尺四寸萬五千九百五十八。 物應，十二萬五千八百五十。 上生質末。 物應為宮，否與商，質末徵。 七日。 律，六寸三分小分九強。 準，六尺三寸萬八千四百七十一。 蕤賓，十二萬四千四百一十六。 上生大呂。 蕤賓為宮，夷則商，大呂徵。
Mode: Neifu tonic, Gui-jia shang, Fen-dong zhi. 8-day span in the cycle. Pipe length recorded as 6 cun 4 fen minor-fen 8 strong. Regulator: 6 chi 4 cun plus 15,958 micro-units. Wuying: 125,850. Generates Zimo. Mode: Wuying tonic, Fouyu shang, Zimo zhi. 7-day span in the cycle. Pipe length recorded as 6 cun 3 fen minor-fen 9 strong. Regulator: 6 chi 3 cun plus 18,471 micro-units. Ruibin: 124,416. Generates Dalü. Mode: Ruibin tonic, Yi-ze shang, Dalü zhi.
32
一日。 律，六寸三分小分二微強。 準，六尺三寸四千一百三十一。
1-day span in the cycle. Pipe length recorded as 6 cun 3 fen minor-fen 2 slightly strong. Regulator: 6 chi 3 cun plus 4,131 micro-units.
33
南事，十二萬四千一百五十四。* (下) **[不]*生。 南事窮，無商、徵，不為宮。
Nanshi: 124,154. (lower) does not generate. When southern affairs are exhausted, there is no shang or zhi note, and it does not serve as gong.
34
七日。 律，六寸三分小分一弱。 準，六尺三寸一千五百* (三) **[一]*十一。 盛變，十二萬二千七百四十一。 上生分否。 盛變為宮，解形商，分否徵。 七日。
7-day span in the cycle. Pipe length recorded as 6 cun 3 fen minor-fen 1 weak. Regulator: 6 chi 3 cun plus 1,500 micro-units. Editorial parenthesis marking alternate reading note 3. eleven. Shengbian: 122,741. Generates Fenfou. Mode: Shengbian tonic, Jiexing shang, Fenfou zhi. 7-day span in the cycle.
35
律，六寸二分小分三* (大) **[半]*強。 準，六尺二寸七千六十四。 離宮，十二萬一千八* (百一) *十九。 上生凌陰。 離宮為宮，去南商，凌陰徵。 七日。 律，六寸一分小分五微強。 準，六尺一寸萬二百二十七。 制時，十一萬九千四百六十。 上生少出。 制時為宮，分積商，少出徵。 八日。 律，六寸小分七弱。 準，六尺萬三千六百二十。 林鐘，十一萬八千九十八。 上生太蔟。 林鐘為宮，南呂商，太蔟徵。 一日。 律，六寸。 準，六尺。
Pipe length recorded as 6 cun 2 fen minor-fen 3. (greater) a little more than half. Regulator: 6 chi 2 cun plus 7,064 micro-units. Ligong: 121,800... Editorial parenthesis marking alternate reading note 101. The manuscript omits characters before the figure 19. Generates Lingyin. Mode: Ligong tonic, Qunan shang, Lingyin zhi. 7-day span in the cycle. Pipe length recorded as 6 cun 1 fen minor-fen 5 slightly strong. Regulator: 6 chi 1 cun plus 10,227 micro-units. Zhishi: 119,460. Generates Shaochu. Mode: Zhishi tonic, Fenji shang, Shaochu zhi. 8-day span in the cycle. Pipe length recorded as 6 cun minor-fen 7 weak. Regulator: 6 chi plus 13,620 micro-units. Linzhong: 118,098. Generates Taicu. Mode: Linzhong tonic, Nanlü shang, Taicu zhi. 1-day span in the cycle. Pipe length recorded as 6 cun. Regulator: 6 chi.
36
謙待，十一萬七千八百五十一。 上生未知。 謙待為宮，白呂商，未知徵。 五日。
Qiandai: 117,851. Generates Weizhi. Mode: Qiandai tonic, Bailü shang, Weizhi zhi. 5-day span in the cycle.
37
律，五寸九分小分九弱。 準，五尺九寸萬七千二百一十三。 去滅，十一萬六千五百八。 上生時息。 去滅為宮，結躬商，時息徵。 七日。 律，五寸九分小分二弱。 準，五尺九寸三千七百八十三。 安度，十一萬四千九百四十。 上生屈齊。
Pipe length recorded as 5 cun 9 fen minor-fen 9 weak. Regulator: 5 chi 9 cun plus 17,213 micro-units. Qumie: 116,508. Generates Shixi. Mode: Qumie tonic, Jiegong shang, Shixi zhi. 7-day span in the cycle. Pipe length recorded as 5 cun 9 fen minor-fen 2 weak. Regulator: 5 chi 9 cun plus 3,783 micro-units. An-du: 114,940. Generates Quqi.
38
安度為宮，歸期商，屈齊徵。 六日。 律，五寸八分小分四*[微]*弱。 準，五尺八寸七千七百八十六。 歸嘉，十一萬三千三百九十三。 上生隨期。 歸嘉為宮，未卯商，隨期徵。 六日。 律，五寸七分小分六微強。 準，五尺七寸萬一千九百九十九。 否與，十一萬一千八百六十七。 上生形晉。 否與為宮，夷汗商，形晉徵。 五日。 律，五寸六分小分八強。 準，五尺六寸萬六千四百二十二。
Mode: An-du tonic, Guiqi shang, Quqi zhi. 6-day span in the cycle. Pipe length recorded as 5 cun 8 fen minor-fen 4 weak. Regulator: 5 chi 8 cun plus 7,786 micro-units. Gui-jia: 113,393. Generates Suiqi. Mode: Gui-jia tonic, Weimao shang, Suiqi zhi. 6-day span in the cycle. Pipe length recorded as 5 cun 7 fen minor-fen 6 slightly strong. Regulator: 5 chi 7 cun plus 11,999 micro-units. Fouyu: 111,867. Generates Xingjin. Mode: Fouyu tonic, Yihan shang, Xingjin zhi. 5-day span in the cycle. Pipe length recorded as 5 cun 6 fen minor-fen 8 strong. Regulator: 5 chi 6 cun plus 16,422 micro-units.
39
夷則，十一萬五百九十二。 上生夾鐘。 夷則為宮，無射商，夾鐘徵。 八日。 律，五寸六分小分二弱。 準，五尺六寸三千六百七十二。 解形，十* (一) *萬九千一百三。 上生開時。 解形為宮，閉掩商，開時徵。 八日。 律，五寸五分小分四強。
Yi-ze: 110,592. Generates Jiazhong. Mode: Yi-ze tonic, Wuyi shang, Jiazhong zhi. 8-day span in the cycle. Pipe length recorded as 5 cun 6 fen minor-fen 2 weak. Regulator: 5 chi 6 cun plus 3,672 micro-units. Jiexing: ten... Editorial parenthesis marking alternate reading note 1. The manuscript omits characters before the figure 19,103. Generates Kaishi. Mode: Jiexing tonic, Biyan shang, Kaishi zhi. 8-day span in the cycle. Pipe length recorded as 5 cun 5 fen minor-fen 4 strong.
40
準，五尺五寸八千四百六十五。 去南，十萬七千六百三十五。 上生族嘉。 去南為宮，鄰齊商，族嘉徵。 八日。 律，五寸四分小分六大強。 準，五尺四寸萬三千四百六十八。 分積，十萬六千一百八十* (八) **[七]*。 上生爭南。 分積為宮，期保商，爭南徵。 七日。 律，五寸三分小分九半強。 準，五尺三寸萬八千六百* (八) **[七]*十一。 南呂，十萬四千九百七十六。 上生姑洗。 南呂為宮，應鐘商，姑洗徵。 一日。 律，五寸三分小分三強。 準，五尺三寸六千五百六十一。
Regulator: 5 chi 5 cun plus 8,465 micro-units. Qunan: 107,635. Generates Zujia. Mode: Qunan tonic, Linqi shang, Zujia zhi. 8-day span in the cycle. Pipe length recorded as 5 cun 4 fen minor-fen 6 strongly. Regulator: 5 chi 4 cun plus 13,468 micro-units. Accumulated fraction: 106,180... Editorial parenthesis marking alternate reading note 8. seven. Generates Zhengnan. Mode: Fenji tonic, Qibao shang, Zhengnan zhi. 7-day span in the cycle. Pipe length recorded as 5 cun 3 fen minor-fen 9 half strong. Regulator: 5 chi 3 cun plus 18,600 micro-units. Editorial parenthesis marking alternate reading note 8. seventy-one. Nanlü: 104,976. Generates Guxian. Mode: Nanlü tonic, Yingzhong shang, Guxian zhi. 1-day span in the cycle. Pipe length recorded as 5 cun 3 fen minor-fen 3 strong. Regulator: 5 chi 3 cun plus 6,561 micro-units.
41
白呂，十萬四千七百五十六。 上生南授。 白呂為宮，分烏商，南授徵。 五日。
Bailü: 104,756. Generates Nanshou. Mode: Bailü tonic, Fenwu shang, Nanshou zhi. 5-day span in the cycle.
42
律，五寸三分小分二強。 準，五尺三寸四千三百* (七) **[六]*十一。 結躬，十萬三千五百六十三。 上生變虞。 結躬為宮，□內商，變虞徵。 六日。 律，五寸二分小分六* (少) *強。 準，五尺二寸萬二千一百一十四。 歸期，十萬二千一百六十九。 上生路時。 歸期為宮，未育商，路時徵。 六日。 律，五寸一分小分九微強。 準，五尺一寸萬七千八百五十七。 未卯，十萬七百九十四。 上生形始。
Pipe length recorded as 5 cun 3 fen minor-fen 2 strong. Regulator: 5 chi 3 cun plus 4,300 micro-units. Editorial parenthesis marking alternate reading note 7. sixty-one. Jiegong: 103,563. Generates Bianyu. Mode: Jiegong tonic, □Nei shang, Bianyu zhi. 6-day span in the cycle. Pipe length recorded as 5 cun 2 fen minor-fen 6. (lesser) a little more. Regulator: 5 chi 2 cun plus 12,114 micro-units. Guiqi: 102,169. Generates Lushi. Mode: Guiqi tonic, Weiyu shang, Lushi zhi. 6-day span in the cycle. Pipe length recorded as 5 cun 1 fen minor-fen 9 slightly strong. Regulator: 5 chi 1 cun plus 17,857 micro-units. Weimao: 100,794. Generates Xingshi.
43
未卯為宮，□時商，形始徵。 六日。 律，五寸一分小分二微強。 準，五尺一寸四千* (八十) **[一百]*七。 夷汗，九萬九千四百三十七。 上生依行。 夷汗為宮，色育商，依行徵。 七日。 律，五寸小分五強。 準，五尺萬二百二十。
Mode: Weimao tonic, □Shi shang, Xingshi zhi. 6-day span in the cycle. Pipe length recorded as 5 cun 1 fen minor-fen 2 slightly strong. Regulator: 5 chi 1 cun plus 4,000 micro-units. Editorial parenthesis marking alternate reading note 80. one hundred and seven. Yihan: 99,437. Generates Yihang. Mode: Yihan tonic, Seyu shang, Yihang zhi. 7-day span in the cycle. Pipe length recorded as 5 cun minor-fen 5 strong. Regulator: 5 chi plus 10,220 micro-units.
44
無射，九萬八千三百四。 上生中呂。 無射為宮，執始商，中呂徵。 八日。 律，四寸九分小分九強。 準，四尺九寸萬八千五百七十三。 閉掩，九萬六千九百八十。 上生南中。 閉掩為宮，丙盛商，南中徵。 八日。 律，四寸九分小分三弱。
Wuyi: 98,304. Generates Zhonglü. Mode: Wuyi tonic, Zhi-shi shang, Zhonglü zhi. 8-day span in the cycle. Pipe length recorded as 4 cun 9 fen minor-fen 9 strong. Regulator: 4 chi 9 cun plus 18,573 micro-units. Biyan: 96,980. Generates Nanzhong. Mode: Biyan tonic, Bing-sheng shang, Nanzhong zhi. 8-day span in the cycle. Pipe length recorded as 4 cun 9 fen minor-fen 3 weak.
45
準，四尺九寸五千三百三十三。 鄰齊，九萬五千六百七十五。 上生內負。 鄰齊為宮，分動商，內負徵。 七日。 律，四寸八分小分六微強。 準，四尺八寸萬一千九百六十六。 期保，九萬四千三百八十八。 上生物應。 期保為宮，質末商，物應徵。 八日。 律，四寸七分小分九* (微) **[半]*強。 準，四尺七寸萬八千七百七十九。 應鐘，九萬三千三百一十二。 上生蕤賓。 應鐘為宮，大呂商，蕤賓徵。 一日。 律，四寸七分小分四微強。 準，四尺七寸八千十九。
Regulator: 4 chi 9 cun plus 5,333 micro-units. Linqi: 95,675. Generates Neifu. Mode: Linqi tonic, Fen-dong shang, Neifu zhi. 7-day span in the cycle. Pipe length recorded as 4 cun 8 fen minor-fen 6 slightly strong. Regulator: 4 chi 8 cun plus 11,966 micro-units. Qibao: 94,388. Generates Wuying. Mode: Qibao tonic, Zimo shang, Wuying zhi. 8-day span in the cycle. Pipe length recorded as 4 cun 7 fen minor-fen 9. (wei) a little more than half. Regulator: 4 chi 7 cun plus 18,779 micro-units. Yingzhong: 93,312. Generates Ruibin. Mode: Yingzhong tonic, Dalü shang, Ruibin zhi. 1-day span in the cycle. Pipe length recorded as 4 cun 7 fen minor-fen 4 slightly strong. Regulator: 4 chi 7 cun plus 8,019 micro-units.
46
分烏，九萬三千一百一十* (七) **[六]*。 上生南事。 分烏窮次，無徵，不為宮。
Fenwu: 93,110... Editorial parenthesis marking alternate reading note 7. six. Generates Nanshi. When Fenwu is exhausted in sequence, there is no zhi note, and it does not serve as gong.
47
七日。 律，四寸七分小分三微強。 準，四尺七寸六千五十九。 □內，九萬二千五十六。 上生盛變。 □內為宮，分否商，盛變徵。 八日。 律，四寸六分小分八弱。 準，四尺六寸萬五千一百四十二。 未育，九萬八百一十七。 上生離宮。 未育為宮，凌陰商，離宮徵。 八日。 律，四寸六分小分一少強。 準，四尺六寸二千七百五十二。 □時，八萬九千五百九十五。 上生制時。 □時為宮，少出商，制時徵。 六日。 律，四寸五分小分五強。 準，四尺五寸萬二百一十五。
7-day span in the cycle. Pipe length recorded as 4 cun 7 fen minor-fen 3 slightly strong. Regulator: 4 chi 7 cun plus 6,059 micro-units. □Nei: 92,056. Generates Shengbian. Mode: □Nei tonic, Fenfou shang, Shengbian zhi. 8-day span in the cycle. Pipe length recorded as 4 cun 6 fen minor-fen 8 weak. Regulator: 4 chi 6 cun plus 15,142 micro-units. Weiyu: 90,817. Generates Ligong. Mode: Weiyu tonic, Lingyin shang, Ligong zhi. 8-day span in the cycle. Pipe length recorded as 4 cun 6 fen minor-fen 1 modestly strong. Regulator: 4 chi 6 cun plus 2,752 micro-units. □Shi: 89,595. Generates Zhishi. Mode: □Shi tonic, Shaochu shang, Zhishi zhi. 6-day span in the cycle. Pipe length recorded as 4 cun 5 fen minor-fen 5 strong. Regulator: 4 chi 5 cun plus 10,215 micro-units.
48
截管為律，吹以考聲，列以物氣，道之本也[13]。 術家以其聲微而體難知，其分數不明，故作準以代之。 準之聲，明暢易達，分寸又粗。 然弦以緩急清濁，非管無以正也。 均其中弦，令與黃鐘相得，案畫以求諸律，無不如數而應者矣。
Cutting pipes to make pitch-pipes, blowing them to test the sounds, and arranging them according to the qi of things is the root of the method [13]. Because specialists found their sounds subtle and their substance hard to know, and because the fractional measurements were unclear, they made the zhun to replace them. The sound of the zhun is clear, resonant, and easy to grasp, and its measurements in fen and cun are also coarser. Yet strings depend on slackness and tightness for clear and turbid tones; without pipes there is no way to correct them. When its central string is tuned to correspond with Huangzhong, the marked divisions can be used to seek the several pitch-pipes, and none fails to respond according to the numbers.
49
音聲精微，綜之者解。 元和元年，待詔候鐘律殷肜上言：「官無曉六十律以準調音者。 故待詔嚴崇具以準法教子男宣，宣通習。 願召宣補學官，主調樂器。」
Sounds are subtle and minute; those who gather them together understand them. In the first year of Yuanhe, Yin Rong, Expectant Appointee for Observing the Bells and Pitch-Pipes, submitted: "Among the officials, no one understands the sixty pitch-pipes well enough to tune sounds with the zhun. The former Expectant Appointee Yan Chong fully taught the method of the zhun to his son Xuan, and Xuan mastered it. I request that Xuan be summoned and appointed to an academic office, to take charge of tuning musical instruments."
50
詔曰：「崇子學審曉律，別其族，協其聲者，審試。 不得依托父學，以聾為聰。
The edict said: "If Chong's son has studied and truly understands the pitch-pipes, distinguishes their families, and harmonizes their sounds, he should be carefully tested. He must not rely on his father's learning and treat deafness as hearing.
51
聲微妙，獨非莫知，獨是莫曉。 以律錯吹，能知命十二律不失一，方為能傳崇學耳。」 太史丞弘試十二律，其二中，其四不中，其六不知何律，宣遂罷。 自此律家莫能為準施弦，候部莫知復見[14]。 熹平六年，東觀召典律者太子舍人張光等問準意。 光等不知，歸閱舊藏，乃得其器，形制如房書，猶不能定其弦緩急，音不可書以時人，知之者欲教而無從，心達者體知而無師，故史官能辨清濁者遂絕。 其可以相傳者，唯大搉常數及候氣而已。
Sound is subtle and wondrous: if one alone says it is wrong, no one knows; if one alone says it is right, no one understands. Only if the pitch-pipes are blown in mixed order and he can identify the twelve pitch-pipes without missing one may he be considered able to transmit Chong's learning." Assistant Grand Scribe Hong tested the twelve pitch-pipes: two were correct, four were incorrect, and for six he did not know which pitch-pipes they were, so Xuan was dismissed. From then on, no pitch specialist could string the zhun, and the Office of Observations no longer knew how to see it restored [14]. In Xiping 6, the Eastern Lodge summoned Zhang Guang, a Crown Prince's retainer who handled the pitch-pipes, and others to ask about the meaning of the zhun. Guang and the others did not know the method. Searching the old storehouses, they found the instrument and saw that its form matched Fang's description, but they still could not set the strings properly. Its tones could not be captured in writing: those who knew the practice had no way to teach it, and those who sensed it by ear had no master. Thus the officials who could distinguish clear from turbid pitches disappeared. What could be transmitted were only the rough constant numbers and the methods for observing qi.
52
夫五音生於陰陽，分為十二律，轉生六十，皆所以紀鬥氣，效物類也。 天效以景，地效以響，即律也。 陰陽和則景至，律氣應則灰除。 是故天子常以日冬夏至御前殿，合八能之士，陳八音，聽樂均，度晷景，候鐘律，權土灰，放陰陽。 冬至陽氣應，則樂均清，景長極，黃鐘通，土灰輕而衡仰。 夏至陰氣應，則樂均濁，景短極，蕤賓通，土灰重而衡低[15]。 進退於先後五日之中，八能各以候狀聞，太史封上。 效則和，否則占[16]。
The five notes arise from yin and yang, divide into the twelve pitch-pipes, and turn to generate sixty; all serve to record the Dipper's qi and model the categories of things. Heaven gives its model through shadows, and earth gives its model through resonance; these are the pitch-pipes. When yin and yang are harmonious, the shadow arrives; when the qi of the pitch-pipes responds, the ashes are expelled. On the winter and summer solstices, the Son of Heaven therefore went to the front hall, assembled specialists in the eight arts, set out the eight classes of instruments, checked the tuning, measured the gnomon shadow, tested bells and pitch-pipes, weighed earth and ash, and modeled the balance of yin and yang. At the winter solstice, when yang qi responds, the musical tuning is clear, the shadow reaches its maximum length, Huangzhong is open, and the earth and ashes are light, so the balance rises. At the summer solstice, when yin qi responds, the musical tuning is turbid, the shadow reaches its maximum shortness, Ruibin is open, and the earth and ashes are heavy, so the balance sinks [15]. Within five days before or after the appointed time, each of the men of the eight abilities reported the observed conditions, and the Grand Scribe sealed and submitted them. If the signs matched, there was harmony; if not, divination was made [16].
53
候氣之法，為室三重，戶閉，塗釁必周，密佈緹縵。 室中以木為案，每律各一，內庳外高，從其方位，加律其上，以葭莩灰抑其內端[17]，案歷而候之。
For the method of observing qi, a chamber of three layers was made, the doors were closed, the plastering and sealing were made complete, and red silk curtains were densely arranged. In the chamber, each pitch-pipe rested on its own wooden stand, lower toward the center and higher outward according to its direction. Reed-membrane ash was packed against the inner end, [17] and the pipes were watched according to the calendar.
54
氣至者灰* (去) **[動]*。 其為氣所動者其灰散，人及風所動者其灰聚。 殿中候，用玉律十二。 惟二至乃候靈台，用竹律六十。 候日如其歷[18]。
When the qi arrived, the ashes... (were expelled) moved. When the ashes were moved by qi, they scattered; when they were moved by people or by wind, they gathered together. Observations in the palace used twelve jade pitch-pipes. Only at the two solstices were observations made at the Numinous Terrace, using sixty bamboo pitch-pipes. The days of observation followed the calendar [18].
55
校勘記
Collation Notes
56
二九九九頁一０行作子丑以名* (日) **[月]*集解引盧文弨說，謂「日」當為「月」。 案子丑等亦謂十二辰，則當繫於月明矣。 後人因下有枝幹相配，以成六旬，遂改為「日」，泥甚。 今據改。
Page 2999, line 10: "made zi and chou to name..." (day) The Collected Explanations cites Lu Wenchao, who says that "day" should be "month." Since zi and chou are also the twelve earthly branches, this should clearly be attached to the months. Later people, because the text below says that branches and stems are paired to complete the cycle of sixty, changed it to "day," which is very forced. It is now changed accordingly.
57
二九九九頁一三行以粟生之* (十) **[一]*粟為一分集解引盧文弨說，謂「以粟」說苑作「以黍」，無「十粟」二字。 按：校補謂「十」當作「一」，粟猶黍也。 雖說苑亦無「一黍」二字，然不別出數，即是就一黍言。 前書律歷志云「一黍之廣，度之九十分，黃鐘之長。 一為一分」。 夫黃鐘長九寸，一黍之廣當之長九十分之一，亦即是一黍為一分，故知此一粟為一分矣。 今據校補說改。
Page 2999, line 13: "generated it from millet..." Editorial parenthesis marking alternate reading note 10. The Collected Explanations cites Lu Wenchao, who says that for "one grain of millet is one fen," the Garden of Persuasions has "one grain of panicled millet" and lacks the words "ten grains of millet." Note: the Supplement to the Collation says that "ten" should be "one," and that su is the same as shu, panicled millet. Although the Garden of Persuasions also lacks the words "one grain of panicled millet," since it gives no separate number, it is speaking in terms of a single grain. The Treatise on Pitch-Pipes and the Calendar in the Former Han says, "The width of one grain of panicled millet is measured as one ninetieth of the length of Huangzhong. One is one fen." Now Huangzhong is nine cun long, and the width of one grain of panicled millet corresponds to one ninetieth of that length. This is precisely one grain of panicled millet as one fen, so we know that here one grain of millet is one fen. It is now changed according to the Supplement to the Collation.
58
三０００頁二行十粟重一圭十圭重一銖按：集解引盧文弨說，謂說苑「十粟」作「十六黍」，「十圭」作「六圭」。
Page 3000, line 2, "ten grains of millet weigh one gui; ten gui weigh one zhu": the Collected Explanations cites Lu Wenchao, who says that in the Garden of Persuasions, "ten grains of millet" is "sixteen grains of panicled millet," and "ten gui" is "six gui."
59
三０００頁四行廣於萬按：集解引盧文弨說，謂前志「廣」作「衍」。
Page 3000, line 4, "broad at ten thousand": the Collected Explanations cites Lu Wenchao, who says that in the Former Han Treatise, "broad" is written "overflowing."
60
三０００頁六行郎中京房* (房字君明) *知五聲之音六律之數「房字君明」四字據集解引盧文弨說刪，盧說見下。 又集解引盧文弨說，謂「五聲之音，六律之數」通典作「五音六十律之數」。 王先謙謂晉、宋志並作「五音六十律」，此文鬥也。 今按：「六律」一詞於此泛用作律呂解亦可通，今不改。
Page 3000, line 6, "Gentleman of the Palace Jing Fang..." (Fang, courtesy name Junming) The four characters "Fang, courtesy name Junming" are deleted according to Lu Wenchao, as cited in the Collected Explanations; Lu's statement appears below. The Collected Explanations also cites Lu Wenchao, who says that where this text has "the sounds of the five notes and the numbers of the six pitch-pipes," the Comprehensive Institutions has "the numbers of the five notes and sixty pitch-pipes." Wang Xianqian says that the Jin and Song Treatises both have "five notes and sixty pitch-pipes," and that this text is corrupt. My note: the term "six pitch-pipes" can also be understood here in the broad sense of pitch-pipes and pitch-standards, so it is not changed.
61
三０００頁七行上使太子太傅* (韋) *玄成* (字少翁) *按：集解引盧文弨說，謂甄鸞五經算術無「韋」字，與下王章亦不書姓合。 下「字少翁」三字亦無。 蓋閱者偶作旁記，而誤寫入正文，與上「房字君明」並當刪去，不可以史記有「解揚字子虎」相比例。 今據刪。
Page 3000, line 7, "The emperor had the Crown Prince's Grand Tutor..." (Wei) Xuancheng... (courtesy name Shaoweng) Note: the Collected Explanations cites Lu Wenchao, who says that Zhen Luan's Arithmetic of the Five Classics lacks the graph "Wei," which agrees with the fact that Wang Zhang below is also written without his surname. The three characters "courtesy name Shaoweng" below are also absent. Probably a reader casually wrote them as marginal notes, and they were mistakenly copied into the main text. Together with "Fang, courtesy name Junming" above, they should be deleted. They cannot be compared by analogy with the Shiji phrase "Xie Yang, courtesy name Zihu." They are now deleted accordingly.
62
三０００頁一二行故各* (終) **[統]*一日按：集解引惠棟說，謂「終」禮記正義引作「統」，北史牛宏傳同。 又引盧文弨說，謂算術亦作「統」。 今據改。
Page 3000, line 12: "therefore each..." (end) For "governs one day," the Collected Explanations cites Hui Dong, who says that the Correct Meaning of the Book of Rites quotes "end" as "govern," and the Biography of Niu Hong in the Northern Histories is the same. It also cites Lu Wenchao, who says that the Arithmetic likewise has "govern." It is now changed accordingly.
63
三０００頁一五行苟非* (草) **[革]*木之聲集解引盧文弨說，謂「草」當依算術作「革」。 今據改。 按：王先謙謂晉志作「草」。
Page 3000, line 15: "if it is not..." (grass) For "the sounds of hide and wood," the Collected Explanations cites Lu Wenchao, who says that "grass" should follow the Arithmetic and read "hide." It is now changed accordingly. Note: Wang Xianqian says that the Jin Treatise has "grass."
64
三００一頁六行以* (制) **[律]*長短為制據汲本改。
Page 3001, line 6: "by..." (regulate) For "using the lengths and shorts of the pitch-pipes as the standard," this is changed according to the Ji edition.
65
三００一頁一四行上生不得過黃鐘之* (清) *濁下生不得及黃鐘之* (數實) **[清]*集解引盧文弨說，謂「清」字衍。 「之數實」當作「之清」，依算術改正。 今據改。 盧又謂「及」上脫「不」字。 今按：上生不得過黃鐘之濁者，意即所生之音不得低於黃鐘本律，下生不得及黃鐘之清者，意即所生之音不得高於或等於黃鐘半律，「過」與「及」字異而義同，非有脫字也。 今不改。
Page 3001, line 14: "upper generation may not exceed the... of Huangzhong" (clear) "turbid; lower generation may not reach the... of Huangzhong" (number and substance) The Collected Explanations cites Lu Wenchao, who says that the graph "clear" is intrusive. "Its number and substance" should be "its clear tone," and is corrected according to the Arithmetic. It is now changed accordingly. Lu also says that the graph "not" is missing above "reach." My note: "upper generation may not exceed Huangzhong's turbid tone" means the generated note must not fall below the original Huangzhong pitch. "Lower generation may not reach Huangzhong's clear tone" means it must not rise to, or above, the Huangzhong half-pitch. The verbs differ, but their force is parallel; no graph is missing. It is not changed.
66
三００一頁一六行而生十一律者也「十一」汲本、殿本作「十二」。 集解引盧文弨說，謂通鑒注引「十二」作「十一」。 今按：作「十二」者鬥，參閱下條校記自明。
Page 3001, line 16, "and thereby generates the eleven pitch-pipes": the Ji and Palace editions read "twelve" for "eleven." The Collected Explanations cites Lu Wenchao, who says that a Tongjian note quotes "twelve" as "eleven." My note: the reading "twelve" is corrupt; this is clear by consulting the next collation note.
67
三００一頁一六行是故十二律之得十七萬七千一百四十七按：「十二律之」語意不明，疑有脫鬥。 依文當作「十一三之」，蓋以三自乘十一次，所得之數為十七萬七千一百四十七也。
Page 3001, line 16, "therefore the twelve pitch-pipes obtain 177,147": the phrase "the twelve pitch-pipes" is unclear and probably has omissions or corruption. According to the context it should read "eleven threes," meaning that three is multiplied by itself eleven times, producing the number 177,147.
68
三００二頁二行* (數) **[得]*萬九千六百八十三為法據集解引盧文弨說改。
Page 3002, line 2: (number) For "obtains 19,683 as the divisor," this is changed according to Lu Wenchao as cited in the Collected Explanations.
69
三００二頁二行*[於]*律為寸集解引盧文弨說，謂「律」上脫「於」字，算術有。 今據補。
Page 3002, line 2: for "is cun in the pitch-pipe," the Collected Explanations cites Lu Wenchao, who says that the graph "in" is missing above "pitch-pipe"; the Arithmetic has it. It is now supplied accordingly.
70
三００二頁四行以為黃鐘之管按：集解引盧文弨說，謂前志「管」作「宮」。
Page 3002, line 4, "taking it as the tube of Huangzhong": the Collected Explanations cites Lu Wenchao, who says that the Former Han Treatise reads "gong" for "tube."
71
三００二頁五行比黃鐘之音按：集解引盧文弨說，謂前志「音」作「宮」。
Page 3002, line 5, "compared with the sound of Huangzhong": the Collected Explanations cites Lu Wenchao, who says that the Former Han Treatise reads "gong" for "sound."
72
三００二頁一一行故滋萌於子按：王先謙謂前志「滋」作「孳」。
Page 3002, line 11, "therefore things sprout at zi": Wang Xianqian says that the Former Han Treatise writes the first graph of "sprout" differently.
73
三００二頁一一行振羨於辰前志「羨」作「美」。 按：王念孫謂「美」當為「羨」，字之鬥也。
Page 3002, line 11, "shake forth at chen": in the Former Han Treatise, "forth" is written "beautiful." Note: Wang Niansun says that "beautiful" should be "forth," and that the graph is corrupt.
74
三００二頁一一行昧曖於未按：王先謙謂前志「曖」作「薆」。
Page 3002, line 11, "grow dim at wei": Wang Xianqian says that in the Former Han Treatise, the final graph is written differently.
75
三００二頁一二行大成於丁按：集解引盧文弨說，謂前志「成」作「盛」。
Page 3002, line 12, "reach completion at ding": the Collected Explanations cites Lu Wenchao, who says that the Former Han Treatise reads "flourish" for "complete."
76
三００二頁一二行豐茂於戊按：王先謙謂前志「茂」作「楙」。
Page 3002, line 12, "grow luxuriantly at wu": Wang Xianqian says that the Former Han Treatise writes the final graph with a variant form.
77
三００二頁一三行而變化之情則可見矣按：王先謙謂前志無「則」字。
Page 3002, line 13, "and the nature of change can be seen": Wang Xianqian says that the Former Han Treatise lacks the graph "then."
78
三００三頁一行色育集解引盧文弨說，謂「色」隋志及律呂新書俱作「包」，當是也。 算術、禮記正義並作「色」。
Page 3003, line 1, "se yu": the Collected Explanations cites Lu Wenchao, who says that in the Sui Treatise and New Book of Pitch-Pipes, "se" is both written "bao," which should be correct. The Arithmetic and the Correct Meaning of the Book of Rites both write "se."
79
三００三頁二行下生謙待按：隋志「謙待」作「謙侍」。 下同。
Page 3003, line 2, "lower generation qian dai": in the Sui Treatise, "qian dai" is written "qian shi." The same applies below.
80
三００三頁六行律八寸八分小分七大強集解引惠棟說，謂「七大強」一作「八弱」。 今按：禮記正義作「小分八弱」。 又按：集解引盧文弨說，謂「大」當作「太」。
Page 3003, line 6, "the pitch-pipe is eight cun, eight fen, and seven small parts, greatly strong": the Collected Explanations cites Hui Dong, who says that "seven, greatly strong" is also written "eight, weak." My note: the Correct Meaning of the Book of Rites has "eight small parts, weak." Another note: the Collected Explanations cites Lu Wenchao, who says that "great" should be written "grand."
81
三００三頁一０行分動集解引惠棟說，謂「動」一作「勳」。 今按：隋志作「動」。 下同。
Page 3003, line 10, "the fraction moves": the Collected Explanations cites Hui Dong, who says that "moves" is also written "merit." My note: the Sui Treatise has "moves." The same applies below.
82
三００三頁一三行質末集解引盧文弨說，謂隋志﹑禮運正義「末」作「未」。 按：殿本作「未」。 下同。
Page 3003, line 13, "zhi mo": the Collected Explanations cites Lu Wenchao, who says that in the Sui Treatise and the Correct Meaning of the Liyun, "mo" is written "wei." Note: the Palace edition writes "wei." The same applies below.
83
三００三頁一四行形晉按：隋志「形」作「刑」。 下同。
Page 3003, line 14: for "xing Jin," the Sui Treatise reads the graph xing as "punishment." The same applies below.
84
三００三頁一五行小分二*[半]*強集解引盧文弨說，謂算術「強」上有「半」字，是。 今據補。
Page 3003, line 15, "two small parts, a little more than half": the Collected Explanations cites Lu Wenchao, who says that in the Arithmetic, the graph "half" appears above "strong," which is correct. It is now supplied accordingly.
85
三００四頁五行下生解形按：隋志「解形」作「解刑」。 下同。
Page 3004, line 5, "lower generation Jiexing": in the Sui Treatise, Jiexing is written with the graph for "punishment." The same applies below.
86
三００四頁七行凌陰按：集解引盧文弨說，謂隋志﹑正義「凌」俱作「陵」。
Page 3004, line 7, "Lingyin": the Collected Explanations cites Lu Wenchao, who says that in the Sui Treatise and the Correct Meaning, Ling is written with the graph for "mound."
87
三００四頁八行族嘉按：隋志作「佚喜」。 下同。
Page 3004, line 8, "Zujia": the Sui Treatise has "Yixi." The same applies below.
88
三００五頁一一行下生未卯按：隋志「未卯」作「未卬」。
Page 3005, line 11, "lower generation Weimao": in the Sui Treatise, Mao is written as Ang.
89
三００五頁一三行十四萬九千一百五十* (五) **[六]*集解引盧文弨說，謂「五十五」算術作「五十六」，是。 今據改。
Page 3005, line 13, "149,150..." Editorial parenthesis marking alternate reading note 5. The Collected Explanations cites Lu Wenchao, who says that where this text has "fifty-five," the Arithmetic has "fifty-six," which is correct. It is now changed accordingly.
90
三００五頁一四行下生夷汗按：隋志「夷汗」作「夷污」。 下同。 又按：「夷」原斗「無」，逕改正。
Page 3005, line 14, "lower generation Yihan": in the Sui Treatise, Han is written Wu. The same applies below. Another note: "Yi" was originally corruptly written "Wu," and is corrected directly.
91
三００五頁一五行七尺五寸萬五千三百* (二) **[三]*十五按：各本並作「二十五」，今據算理改。
Page 3005, line 15, "seven chi, five cun, 15,300..." Editorial parenthesis marking alternate reading note 2. thirty-five. Note: all editions have "twenty-five"; it is now changed according to the mathematical principle.
92
三００六頁五行下生閉掩按：隋志「閉掩」作「閉奄」。 下同。
Page 3006, line 5, "lower generation Biyan": in the Sui Treatise, Yan is written An. The same applies below.
93
三００六頁六行小分九微* (弱) **[強]*按：集解引盧文弨說，謂算術作「微強」，是。 今據改。
Page 3006, line 6, "nine small parts, slightly..." (weak) strong. Note: the Collected Explanations cites Lu Wenchao, who says that the Arithmetic has "slightly strong," which is correct. It is now changed accordingly.
94
三００六頁八行內負按：隋志「負」作「貞」。 下同。
Page 3006, line 8, "Neifu": in the Sui Treatise, Fu is written Zhen. The same applies below.
95
三００七頁一行十三萬九千六百七十*[四]*集解引錢大昕說，謂當作「七十四」，脫「四」字。 又引盧文弨說，謂算術有「四」字。 今據補。
Page 3007, line 1, "139,674": the Collected Explanations cites Qian Daxin, who says that it should read "seventy-four" and that the graph "four" is missing. It also cites Lu Wenchao, who says that the Arithmetic has the graph "four." It is now supplied accordingly.
96
三００七頁二行下生分烏按：隋志「分烏」作「分焉」。 下同。
Page 3007, line 2, "lower generation Fenwu": in the Sui Treatise, Wu is written Yan. The same applies below.
97
三００七頁一０行形始按：隋志「形」作「刑」。
Page 3007, line 10: for "xing began," the Sui Treatise reads the graph xing as "punishment."
98
三００七頁一五行小分三* (大) **[半]*強集解引盧文弨說，謂算術作「半強」，是。 今據改。
Page 3007, line 15, "three small parts..." (greater) a little more than half. The Collected Explanations cites Lu Wenchao, who says that the Arithmetic has "half strong," which is correct. It is now changed accordingly.
99
三００八頁九行小分八*[微]*強集解引盧文弨說，謂算術作「微強」，是。 今據補。
Page 3008, line 9, "eight small parts, slightly strong": the Collected Explanations cites Lu Wenchao, who says that the Arithmetic has "slightly strong," which is correct. It is now supplied accordingly.
100
三００九頁二行* (下) **[不]*生集解引錢大昕說，謂十二律之變窮於南事，安得雲下生乎？ 疑「下」為「不」字之鬥。 又引盧文弨說，謂「下生」當作「不生」。 今據改。
Page 3009, line 2: (lower) does not generate. The Collected Explanations cites Qian Daxin, who says that the transformations of the twelve pitch-pipes are exhausted at Nanshi, so how could it say "lower generation"? He suspects that "lower" is a corruption of "not." It also cites Lu Wenchao, who says that "lower generation" should read "does not generate." It is now changed accordingly.
101
三００九頁三行六尺三寸一千五百* (三) **[一]*十一按：各本作「三十一」，今據算理改。
Page 3009, line 3, "six chi, three cun, 1,500..." Editorial parenthesis marking alternate reading note 3. eleven. Note: all editions have "thirty-one"; it is now changed according to the mathematical principle.
102
三００九頁六行小分三* (大) **[半]*強集解引盧文弨說，謂算術作「半強」，是。 今據改。
Page 3009, line 6, "three small parts..." (greater) a little more than half. The Collected Explanations cites Lu Wenchao, who says that the Arithmetic has "half strong," which is correct. It is now changed accordingly.
103
三００九頁七行十二萬一千八* (百一) *十九集解引錢大昕說，謂當云「一千八十九」。 又引盧文弨說，謂「百一」二字誤衍，算術無。 今據刪。
Page 3009, line 7, "121,800..." Editorial parenthesis marking alternate reading note 101. nineteen. The Collected Explanations cites Qian Daxin, who says that it should read "1,089." It also cites Lu Wenchao, who says that the two characters "hundred and one" are mistakenly intrusive and are absent from the Arithmetic. They are now deleted accordingly.
104
三０一０頁九行小分四*[微]*弱集解引盧文弨說，謂算術作「微弱」，是。 今據補。
Page 3010, line 9, "four small parts, slightly weak": the Collected Explanations cites Lu Wenchao, who says that the Arithmetic has "slightly weak," which is correct. It is now supplied accordingly.
105
三０一一頁四行十* (一) *萬九千一百三集解引錢大昕說，謂當云「十萬」。 又引盧文弨說，謂「十」下「一」字衍，算術無。 今據刪。
Page 3011, line 4, "ten..." Editorial parenthesis marking alternate reading note 1. "...ten thousand, nine thousand, one hundred and three": the Collected Explanations cites Qian Daxin, who says that it should read "one hundred thousand." It also cites Lu Wenchao, who says that the graph "one" after "ten" is intrusive and absent from the Arithmetic. It is now deleted accordingly.
106
三０一一頁一０行十萬六千一百八十* (八) **[七]*集解引錢大昕說，謂當云「八十七」。 又引盧文弨說，謂「八」鬥，算術「七」。 今據改。
Page 3011, line 10, "106,180..." Editorial parenthesis marking alternate reading note 8. seven. The Collected Explanations cites Qian Daxin, who says that it should read "eighty-seven." It also cites Lu Wenchao, who says that "eight" is corrupt, and that the Arithmetic has "seven." It is now changed accordingly.
107
三０一一頁一二行小分九半強集解引盧文弨說，謂算術無「半」字，當作「少強」。 今按：依算理當作「半弱」。
Page 3011, line 12, "nine small parts, half strong": the Collected Explanations cites Lu Wenchao, who says that the Arithmetic lacks the graph "half" and that it should read "slightly strong." My note: according to the mathematical principle, it should read "half weak."
108
三０一一頁一二行五尺三寸萬八千六百* (八) **[七]*十一按：各本作「八十一」，今據算理改。
Page 3011, line 12, "five chi, three cun, 18,600..." Editorial parenthesis marking alternate reading note 8. seventy-one. Note: all editions have "eighty-one"; it is now changed according to the mathematical principle.
109
三０一二頁三行五尺三寸四千三百* (七) **[六]*十一按：各本作「七十一」，今據算理改。
Page 3012, line 3, "five chi, three cun, 4,300..." Editorial parenthesis marking alternate reading note 7. sixty-one. Note: all editions have "seventy-one"; it is now changed according to the mathematical principle.
110
三０一二頁六行小分六* (少) *強集解引盧文弨說，謂算術作「微強」，案止當作「強」。 今據刪。
Page 3012, line 6, "six small parts..." (lesser) strong. The Collected Explanations cites Lu Wenchao, who says that the Arithmetic has "slightly strong"; by the case, it should simply read "strong." It is now deleted accordingly.
111
三０一二頁一二行五尺一寸四千* (八十) **[一百]*七按：各本作「四千八十七」，今據算理改。
Page 3012, line 12, "five chi, one cun, 4,000..." Editorial parenthesis marking alternate reading note 80. one hundred and seven. Note: all editions have "4,087"; it is now changed according to the mathematical principle.
112
三０一三頁一二行小分九* (微) **[半]*強集解引盧文弨說，謂算術作「半強」，是。 今據改。
Page 3013, line 12, "nine small parts..." (slightly) a little more than half. The Collected Explanations cites Lu Wenchao, who says that the Arithmetic has "half strong," which is correct. It is now changed accordingly.
113
三０一四頁一行九萬三千一百一十* (七) **[六]*集解引錢大昕說，謂當作「一十六」。 又引盧文弨說，謂算術作「六」。 今據改。
Page 3014, line 1, "93,110..." Editorial parenthesis marking alternate reading note 7. six. The Collected Explanations cites Qian Daxin, who says that it should read "sixteen." It also cites Lu Wenchao, who says that the Arithmetic has "six." It is now changed accordingly.
114
三０一四頁一三行列以物氣集解引惠棟說，謂晉志「物」作「效」。 今按：作「效」似合。
Page 3014, line 13, "arranging them according to the qi of things": the Collected Explanations cites Hui Dong, who says that in the Jin Treatise, "things" is written "model." My note: the reading "model" seems suitable.
115
三０一四頁一四行然弦以緩急清濁集解引張文虎說，謂「弦以」之「以」疑當作「之」，或「緩怎」下脫「為」字。
Page 3014, line 14, "yet strings depend on slackness and tightness for clear and turbid tones": the Collected Explanations cites Zhang Wenhu, who says that the word "by" in "strings by" may be "of," or that the word "make" is missing after "slackness and tightness."
116
三０一五頁一行泠道縣按：「泠」原斗「冷」，逕改正。
Page 3015, line 1, "Lingdao County": Ling was originally corruptly written Leng, and is corrected directly.
117
三０一五頁三行故待詔嚴崇按：晉﹑宋志「崇」並作「嵩」，魏志亦作「嵩」。 集解引錢大昕說，謂古文崇嵩通，漢武帝改嵩高山為「崇高」。
Page 3015, line 3, "the former Expectant Appointee Yan Chong": in the Jin and Song Treatises, Chong is both written Song, and the Wei Treatise also writes Song. The Collected Explanations cites Qian Daxin, who says that in ancient writing Chong and Song were interchangeable, and that Emperor Wu of Han changed Mount Songgao to "Mount Chonggao."
118
三０一五頁五行方為能傳崇學耳「方」原斗「力」，逕改正。 按：晉﹑宋志並作「乃」。
Page 3015, line 5, "only then may he be considered able to transmit Chong's learning": fang was originally corruptly written li, and is corrected directly. Note: the Jin and Song Treatises both read "then."
119
三０一五頁八行音不可書以* (時) **[曉]*人王先謙謂晉志作「音不可書以曉」，宋志作「音不可以書曉」，蓋「書以」誤倒，明「時」字誤。 按：王氏以「曉」字為句，「人」字連下讀。 今依晉志改「時」為「曉」，而以「人」字屬上讀。
Page 3015, line 8, "sound cannot be written to..." (time) "make people understand": Wang Xianqian says that the Jin Treatise reads "sound cannot be written to make clear," and the Song Treatise reads "sound cannot by writing be made clear"; probably "written to" is reversed and the graph "time" is mistaken. Note: Wang takes the graph "understand" as the end of the clause and reads "people" with what follows. It is now changed from "time" to "understand" according to the Jin Treatise, while reading "people" with the preceding phrase.
120
三０一五頁一二行皆不應* (日) **[月]*律據汲本﹑殿本改。
Page 3015, line 12, "all do not correspond to..." (day) "monthly pitch-pipes": changed according to the Ji and Palace editions.
121
三０一五頁一二行乃能順天地按：汲本﹑殿本「順」作「感」。
Page 3015, line 12, "only then can they accord with Heaven and Earth": in the Ji and Palace editions, "accord" is written "resonate with."
122
三０一五頁一五行可順上天之明* (待) **[時]*隋書音樂志下引「待」作「時」。 今據改。
Page 3015, line 15, "may accord with Heaven's bright..." (waiting) time. In the Treatise on Music in the Book of Sui, "waiting" is quoted as "time." It is now changed accordingly.
123
三０一六頁二行皆所以紀鬥氣按：「斗」字疑有誤，或當作「卦」。
Page 3016, line 2, "all serve to record the Dipper's qi": the graph "Dipper" is suspect and may be "hexagram."
124
三０一六頁四行權土* (灰) **[炭]*集解引惠棟說，謂晉灼引蔡邕律歷記作「土炭」，漢書律歷志亦云「懸土炭」。 今據改，下同。
Page 3016, line 4, "weighing earth..." (ashes) charcoal. The Collected Explanations cites Hui Dong, who says that Jin Zhuo quotes Cai Yong's Record of Pitch-Pipes and the Calendar as "earth and charcoal," and the Treatise on Pitch-Pipes and the Calendar in the Han says "hanging earth and charcoal." It is now changed accordingly, and likewise below.
125
三０一六頁四行* (放) **[效]*陰陽集解引惠棟說，謂「放」一作「效」，晉志作「效」。 今據改。
Page 3016, line 4: (imitate) "model yin and yang": the Collected Explanations cites Hui Dong, who says that "imitate" is also written "model," and the Jin Treatise writes "model." It is now changed accordingly.
126
三０一六頁八行氣至者灰* (去) **[動]*集解引錢大昭說，謂閩本作「動」。 王先謙謂殿本作「動」，晉志作「去」。 今按：下云「其為氣所動者其灰散」，則作「去」者非，今據改。
Page 3016, line 8, "when the qi arrives, the ashes..." (depart) move. The Collected Explanations cites Qian Dazhao, who says that the Min edition has "move." Wang Xianqian says that the Palace edition has "move," while the Jin Treatise has "depart." My note: the text below says, "when moved by qi, the ashes scatter," so the reading "depart" is wrong. It is now changed accordingly.
127
三０一七頁一行*[孔]*徑三分據御覽十六補。
Page 3017, line 1, "bore diameter three fen": supplied according to Imperial Readings, chapter 16.
128
三０一七頁一行其餘皆* (補) **[漸]*短集解引惠棟說，謂李氏本「補」作「漸」。
Page 3017, line 1, "all the rest are..." (supplemented) gradually shortened. The Collected Explanations cites Hui Dong, who says that the Li edition writes "supplemented" as "gradual."
129
呂氏春秋曰：「黃帝師大橈。」 博物記曰：「容成氏造歷，黃帝臣也。」 月令章句：「大橈探五行之情，占斗綱所建，於是始作甲乙以名日，謂之幹，作子丑以名月，謂之枝，枝幹相配，以成六旬。」
The Springs and Autumns of Master Lu says, "The Yellow Emperor took Darao as his teacher." The Record of Things Broadly Considered says, "Rongcheng made the calendar; he was a minister of the Yellow Emperor." The Monthly Ordinances, Chapters and Sentences says: "Darao investigated the nature of the five phases and observed what the Dipper handle established. Thereupon he first made jia and yi to name the days, calling them stems, and made zi and chou to name the months, calling them branches. Branches and stems are paired to complete the cycle of sixty."
130
博物記曰：「隸首，黃帝之臣。」 一說，隸首，善筭者也。
The Record of Things Broadly Considered says, "Lishou was a minister of the Yellow Emperor." One account says that Lishou was skilled in calculation.
131
表即晷景。
The biao is the gnomon shadow.
132
說苑曰：「以粟生之，十粟為一分，十分為一寸，十寸為一尺，十尺為一丈。」
The Garden of Persuasions says, "Measures are generated from millet: ten grains of millet make one fen, ten fen make one cun, ten cun make one chi, and ten chi make one zhang."
133
說苑曰：「千二百粟為一籥，十籥為一合，十合為一升，十升為一斗，十斗為一斛。」
The Garden of Persuasions says, "One thousand two hundred grains of millet make one yue, ten yue make one ge, ten ge make one sheng, ten sheng make one dou, and ten dou make one hu."
134
說苑曰：「十粟重一圭，十圭重一銖，二十四銖重一兩，十六兩重一斤，三十斤重一鈞，四鈞重一石。」
The Garden of Persuasions says, "Ten grains of millet weigh one gui, ten gui weigh one zhu, twenty-four zhu weigh one liang, sixteen liang weigh one jin, thirty jin weigh one jun, and four jun weigh one shi."
135
前志曰：「夫推歷生律，制器規圓矩方，權重衡平，準繩嘉量，探賾索隱，鉤深致遠，莫不用焉。 度長短者不失毫釐，量多少者不失圭撮，權輕重者不失黍累。 紀於一，協於十，長於百，大於千，廣於萬。」
The Former Han Treatise says, "In calculating calendars and generating pitch-pipes, making instruments by compass and square, weighing heaviness and balancing level, standardizing cords and approved measures, probing the subtle and seeking the hidden, reaching the deep and extending to the distant, there is nothing for which these are not used. Those who measure length do not miss by a hao or li; those who measure quantity do not miss by a gui or cuo; those who weigh lightness and heaviness do not miss by a grain or thread. They are recorded in one, harmonized in ten, lengthened in a hundred, enlarged in a thousand, and broadened in ten thousand."
136
月令章句曰：「以姑洗為角，南呂為羽，則微濁也。」
The Monthly Ordinances, Chapters and Sentences says, "Using Guxian as jue and Nanlu as yu makes the sound slightly turbid."
137
月令章句曰：「律，率也，聲之管也。 上古聖人本陰陽，別風聲，審清濁，而不可以文載口傳也。 於是始鑄金作鐘，以主十二月之聲，然後以效升降之氣。 鐘難分別，乃截竹為管，謂之律。 律者，清濁之率法也。 聲之清濁，以長短為制。」
The Monthly Ordinances, Chapters and Sentences says, "A pitch-pipe is a standard, the tube of sound. The sages of high antiquity took yin and yang as the root, distinguished the sounds of the wind, and examined clarity and turbidity, but these could not be carried in writing or transmitted orally. Thereupon they first cast metal to make bells, to govern the sounds of the twelve months, and only afterward used them to model the qi of rising and falling. Since bells were difficult to distinguish, they cut bamboo to make tubes and called them pitch-pipes. Pitch-pipes are the standard method for clarity and turbidity. The clarity and turbidity of sound are regulated by length and shortness."
138
鄭玄曰：「宮數八十一，黃鐘長九寸，九九八十一也。 三分宮去一生徵，徵數五十四，林鐘長六寸，六九五十四也。 三分徵益一生商，商數七十二，太蔟長八寸，八九七十二也。 三分商去一生羽，羽數四十八，南呂長五寸三分寸之一，五九四十五又三分寸之一，為四十八也。 三分羽益一生角，角數六十四，姑洗長七寸九分寸之一，七九六十三又九分寸之一，為六十四也。 三分角去一生變宮，三分變宮益一生變徵。 自此已後，則隨月而變，所謂『還相為宮。』」
Zheng Xuan said: "The number of gong is eighty-one; Huangzhong is nine cun long, and nine times nine is eighty-one. Taking one third from gong generates zhi. The number of zhi is fifty-four; Linzhong is six cun long, and six times nine is fifty-four. Adding one third to zhi generates shang. The number of shang is seventy-two; Taicu is eight cun long, and eight times nine is seventy-two. Taking one third from shang generates yu. The number of yu is forty-eight; Nanlu is five cun and one third of a cun long, and five times nine is forty-five plus one third of a cun, making forty-eight. Adding one third to yu generates jue. The number of jue is sixty-four; Guxian is seven cun and one ninth of a cun long, and seven times nine is sixty-three plus one ninth of a cun, making sixty-four. Taking one third from jue generates altered gong; adding one third to altered gong generates altered zhi. After this, they change with the months. This is what is meant by "returning to take one another as gong.""
139
前書曰：「黃帝使伶倫，自大夏之西，崑崙之陰，取竹之嶰谷生，其竅厚均者，斷兩節閒而吹之，以為黃鐘之管。 制十二筩以聽鳳之鳴，其雄鳴為六，雌鳴亦六，比黃鐘之音，而皆可以生之，是為律本。 至治之世，天地之氣合以生風。 天地之風氣正，十二律乃定。」
The Former Han says, "The Yellow Emperor sent Ling Lun west from Daxia to the northern slope of Kunlun. He took bamboo growing in Xie Valley, selected pieces whose cavities and thickness were even, cut between two joints, and blew them, making them the tube of Huangzhong. He made twelve tubes to listen to the phoenix's cry: the male cry made six and the female cry also made six. They were compared with the sound of Huangzhong, and all could be generated from it; this was the root of the pitch-pipes. In an age of perfect order, the qi of Heaven and Earth joins to produce wind. When the wind and qi of Heaven and Earth are correct, the twelve pitch-pipes are then fixed."
140
前書曰：「太極元氣，函含三為一。 極，中也。 元，始也。 行於十二辰，始動於子。 參之於丑，得三。 又參之於寅，得九。 又參之於卯，得二十七。 又參之於辰，得八十一。 又參之於巳，得二百四十三。 又參之於午，得七百二十九。 又參之於未，得二千一百八十七。 又參之於申，得六千五百六十一。 又參之於酉，得萬九千六百八十三。 又參之於戌，得五萬九千四十九。 又參之於亥，得十七萬七千一百四十七。 此陰陽合德，氣鐘於子，化生萬物者也。 故滋萌於子，紐牙於丑，引達於寅，冒茆於卯，振羨於辰，巳盛於巳，咢佈於午，昧曖於未，申堅於申，留孰於酉，畢入於戌，該閡於亥，出甲於甲，奮軋於乙，明炳於丙，大成於丁，豐茂於戊，理紀於己，斂更於庚，悉新於辛，懷任於壬，陳揆於癸。 故陰陽之施化，萬物之終始，既類旅於律呂，又經歷於日辰，而變化之情則可見矣。」
The Former Han says: "The original qi of the Supreme Ultimate enfolds three as one. Ji means center. Yuan means beginning. It moves through the twelve earthly branches, first stirring at zi. It is tripled at chou, producing three. It is again tripled at yin, producing nine. It is again tripled at mao, producing twenty-seven. It is again tripled at chen, producing eighty-one. It is again tripled at si, producing two hundred forty-three. It is again tripled at wu, producing seven hundred twenty-nine. It is again tripled at wei, producing two thousand one hundred eighty-seven. It is again tripled at shen, producing six thousand five hundred sixty-one. It is again tripled at you, producing nineteen thousand six hundred eighty-three. It is again tripled at xu, producing fifty-nine thousand forty-nine. It is again tripled at hai, producing one hundred seventy-seven thousand one hundred forty-seven. This is the joined virtue of yin and yang, with qi gathering at zi and transforming to generate the myriad things. The cycle names each stage of growth: sprouting at zi, shooting at chou, extending at yin, emerging at mao, shaking forth at chen, flourishing at si, spreading at wu, dimming at wei, hardening at shen, ripening at you, withdrawing at xu, and closing at hai. The stems likewise mark emergence at jia, bursting at yi, brightness at bing, completion at ding, luxuriance at wu, ordering at ji, contraction and change at geng, renewal at xin, pregnancy at ren, and measured display at gui. Thus the transforming action of yin and yang and the end and beginning of the myriad things are both arrayed among the pitch-pipes and standards and pass through the days and branches, so the nature of change can be seen."
141
前書注曰：「章帝時，零陵文學奚景於泠道縣舜祠下得白玉琯管。 古以玉為琯。」
The commentary to the Former Han says: "In the time of Emperor Zhang, Xi Jing, a scholar of Lingling, obtained a white jade pitch-pipe beneath the shrine to Shun in Lingdao County. In antiquity, jade was used to make pitch-pipes."
142
薛瑩書曰，上以太常樂丞鮑鄴等上樂事，下車騎將軍馬防。 防奏言：「建初二年七月，鄴上言：『王者飲食，必道須四時五味，故有食舉之樂，所以順天地，養神明，求福應也。 移風易俗，莫善於樂。 樂者天地之和，不可久廢。 今官樂但有太蔟，皆不應月律。 可作十二月均，各應其月氣，乃能順天地，和氣宜應。 明帝始令靈台六律候，而未設其門。 樂經曰十二月行之，所以宣氣豐物也。 月開斗建之門，而奏歌其律。 誠宜施行。 願與待詔嚴崇及能作樂器者共作治，考工給所當。』 詔下太常。 太常上言：『作樂器直錢百四十六萬，請太僕作成上。』 奏寢。 今明詔下臣防，臣輒問鄴及待詔知音律者，皆言聖人作樂，所以宣氣致和，順陰陽也。 臣愚以為可順上天之明待，因歲首令正，發太蔟之律，奏雅頌之音，以立太平，以迎和氣。 其條貫甚備。」 詔書以防言下三公。
Xue Ying's letter says that, because Music Assistant Bao Ye and others of the Grand Music Office had submitted matters concerning music, the emperor sent the matter down to General of Chariots and Cavalry Ma Fang. Fang memorialized: "In the seventh month of Jianchu 2, Ye wrote: 'A ruler's food and drink should accord with the four seasons and the five flavors. Hence there is music for presenting food: it follows Heaven and Earth, nourishes the spirits, and seeks auspicious response. For transforming customs and changing practices, nothing is better than music. Music is the harmony of Heaven and Earth and cannot long be abandoned. Now official music has only Taicu, and none of it corresponds to the monthly pitch-pipes. The tunings for the twelve months should be made, each corresponding to the qi of its month; only then can they accord with Heaven and Earth and make harmony and qi respond properly. Emperor Ming first ordered observations of the six pitch-pipes at the Numinous Terrace, but had not yet established their gates. The Classic of Music says that it is performed through the twelve months in order to spread qi and enrich things. Each month opens the gate established by the Dipper and performs songs in its pitch-pipe. It truly ought to be put into practice. I request that, together with Expectant Appointee Yan Chong and those able to make musical instruments, we jointly make and repair them, with the Office of Works supplying what is needed.' The edict sent this down to the Grand Music Office. The Grand Music Office submitted: 'Making the musical instruments will cost 1,460,000 cash. We request that the Grand Coachman have them completed and submitted.' The memorial was set aside. Now that Your enlightened edict has been sent down to me, Fang, I have accordingly asked Ye and the Expectant Appointees who know music and pitch-pipes. They all say that sages made music in order to spread qi, bring harmony, and accord with yin and yang. In my foolish opinion, we should follow Heaven's clear timing: at the year's beginning issue the proper commands, sound the Taicu pitch-pipe, and perform the Odes and Hymns, thereby establishing great peace and welcoming harmonious qi. Its regulations are very complete." The edict took Fang's words and sent them down to the Three Excellencies.
143
淮南子曰：「水勝故夏至濕，火勝故冬至燥。 燥故灰輕，濕故灰重。」
The Huainanzi says, "Because water prevails, the summer solstice is damp; because fire prevails, the winter solstice is dry. Because it is dry, the ashes are light; because it is damp, the ashes are heavy."
144
易緯曰：「冬至人主不出宮，寢兵，從樂五日，擊黃鐘之磬。 公卿大夫列士之意得，則陰陽之晷如度數。 夏至之日，如冬至之禮。 冬至之日，樹八尺之表，日中視其晷。 晷如度者其歲美，人民和順。 晷不如度者則歲惡，人民多斗言，政令為之不平。 晷進則水，晷退則旱。 進一尺則日食，退一尺則月食。 月食則正臣下之行，日食則正人主之道。」
The Apocrypha of the Changes says, "At the winter solstice the ruler does not leave the palace, rests the troops, follows the music for five days, and strikes the chime-stone of Huangzhong. When the intentions of dukes, ministers, grandees, and ranked officers are fulfilled, the yin-yang shadow accords with the degrees and numbers. On the day of the summer solstice, the same rites were performed as at the winter solstice. On the day of the winter solstice, an eight-chi gnomon was set up, and at midday its shadow was observed. When the shadow accords with the degrees, the year is good and the people are harmonious and compliant. When the shadow does not accord with the degrees, the year is bad, the people have many disputes, and government orders become uneven. If the shadow advances, there is flood; if the shadow retreats, there is drought. If it advances by one chi, there is a solar eclipse; if it retreats by one chi, there is a lunar eclipse. When there is a lunar eclipse, the conduct of ministers and subordinates is corrected; when there is a solar eclipse, the way of the ruler is corrected."
145
葭莩出河內。
Reed membrane comes from Henei.
146
月令章句曰：「古之為鐘律者，以耳齊其聲。 後不能，則假數以正其度，度數正則音亦正矣。 鐘以斤兩尺寸中所容受升斗之數為法，律亦以寸分長短為度。 故曰黃鐘之管長九寸，徑三分，圍九分，其餘皆補短，雖大小圍數無增減。 以度量者可以文載口傳，與觿共知，然不如耳決之明也。」
The Monthly Ordinances, Chapters and Sentences says, "In antiquity, those who made bells and pitch-pipes used the ear to align their sounds. Later, when this could no longer be done, they borrowed numbers to correct their measures; when the measures and numbers were correct, the sounds were also correct. For bells, the method was based on their jin and liang, their chi and cun, and the number of sheng and dou they could contain; pitch-pipes likewise used cun and fen, length and shortness, as their measure. Therefore it is said that the tube of Huangzhong is nine cun long, three fen in diameter, and nine fen in circumference; all the rest are made up from the shorter ones, and although they differ in size, their circumferential numbers do not increase or decrease. What is measured can be carried in writing, transmitted orally, and known by the multitude, but it is not as clear as deciding by the ear."