1
道體為一，天地之元，萬物之祖也。 散而為氣，則有陰有陽； 動而為數，則有奇有偶； 凝而為形，則有剛有柔； 發而為聲，則有清有濁，其著見而為器，則有律、有呂。 凡禮樂、刑法、權衡、度量皆出於是。 自周衰樂壞，而律呂候氣之法不傳。 西漢劉歆、揚雄之徒，僅存其說。 京房作準以代律，分六十聲，始于南事，終於去滅。 然聲細而難分，世不能用。 曆晉及隋、唐，律法微隱。 《宋史》止載律呂大數，不獲其詳。 今掇仁宗論律及諸儒言鐘律者記於篇，以補續舊學之闕。
The Way is unified in essence; it is the root of Heaven and Earth and the source of all things. Diffused as breath, it yields yin and yang; set in motion as number, it yields odd and even; condensed into form, it yields hard and soft; Expressed as sound, it yields clear and muddy tones; made manifest in instruments, it becomes the pitch pipes and the secondary pipes. Ritual, music, penal codes, weights, and measures all stem from this principle. After Zhou's decline and the collapse of proper music, the method of using pitch pipes to observe the seasonal qi was lost. In the Western Han, Liu Xin, Yang Xiong, and their circle kept only the theory alive. Jing Fang devised a tuning standard to substitute for the pitch pipes, dividing sound into sixty degrees from Nanshi to Qumie. But the gradations were too fine to distinguish, and the world could not use them. From Jin through Sui and Tang, the science of pitch grew increasingly obscure. The History of Song records only the broad figures for pitch pipes and secondary pipes, not the full particulars. Here we gather Emperor Renzong's treatise on pitch together with scholars' writings on bell pitch, to fill the gaps in the ancient tradition.
2
仁宗著《景祐樂髓新經》，凡六篇，述七宗二變及管分陰陽、剖析清濁，歸之於本律。 次及間聲，合古今之樂，參之以六壬遁甲。
Emperor Renzong wrote the New Classic of the Marrow of Music of the Jingyou reign in six sections, explaining the seven root tones and two variant tones, assigning yin and yang to the pipes, distinguishing clear from turbid sounds, and tracing them all back to the fundamental pitch standards. It goes on to treat intermediate tones, reconciling ancient and modern music and correlating them with the six-ren and dunjia divination systems.
3
其一、釋十二均，曰：「黃鐘之宮為子、為神後、為土、為雞緩、為正宮調，太簇商為寅、為功曹、為金、為般頡、為大石調，姑洗角為辰、為天剛、為木、為嗢沒斯、為小石角，林鐘徵為未、為小吉、為火、為雲漢、為黃鐘徵，南呂羽為酉，為從魁、為水、為滴、為般涉調，應鐘變宮為亥、為登明、為日、為密、為中管黃鐘宮，蕤賓變徵為午、為勝先、為月、為莫、為應鐘徵。 大呂之宮為大吉、為高宮，夾鐘商為大沖、為高大石，仲呂角為太一、為中管小石調，夷則徵為傳送、為大呂徵，無射羽為河魁、為高般涉，黃鐘變宮為正宮調，林鐘變徵為黃鐘徵。 太簇之宮為中管高宮，姑洗商為高大石，蕤賓角為歇指角，南呂徵為太簇徵，應鐘羽為中管高般涉，大呂變宮為高宮，夷則變徵為大呂徵。 夾鐘之宮為中呂宮，仲呂商為雙調，林鐘角在今樂亦為林鐘角，無射徵為夾鐘徵，黃鐘羽為中呂調，太簇變宮為中管高宮，南呂變徵為太簇徵。 姑洗之宮為中管中呂宮，蕤賓商為中管商調，夷則角為中管林鐘角，應鐘徵為姑洗徵，大呂羽為中管中呂調，夾鐘變宮為中呂宮，無射變徵為夾鐘徵。 仲呂之宮為道調宮，林鐘商為小石調，南呂角為越調，黃鐘徵為中呂徵，太簇羽為平調，姑洗變宮為中管中呂宮，應鐘變徵為姑洗徵。 蕤賓之宮為中管道調宮，夷則商為中管小石調，無射角為中管越調，大呂徵為蕤賓徵，夾鐘羽為中管平調，中呂變宮為道調宮，黃鐘變徵為仲呂徵，林鐘之宮為南呂宮，南呂商為歇指調，應鐘角為大石調，太簇微為林鐘徵，姑洗羽為高平調，蕤賓變宮為中管道調宮，大呂變徵為蕤賓徵。 夷則之宮為仙呂，無射商為林鐘商，黃鐘角為高大石調，夾鐘徵為夷則徵，仲呂羽為仙呂調，林鐘變宮為南呂宮，太簇變徵為林鐘徵。 南呂之宮為中管仙呂宮，應鐘商為中管林鐘商，大呂角為中管高大石角，姑洗徵為南呂徵，蕤賓羽為中管仙呂調，夷則變宮為仙呂宮，夾鐘變徵為夷則徵。 無射之宮為黃鐘宮，黃鐘商為越調，太簇角為變角，仲呂徵為無射徵，林鐘羽為黃鐘羽，南呂變宮為中管仙呂宮，姑洗變徵為南呂徵。 應鐘之宮為中管黃鐘宮，大呂商為中管越調，夾鐘角為中管雙角，蕤賓徵為應鐘徵，夷則羽為中管黃鐘羽，無射變宮為黃鐘宮，仲呂變徵為無射徵。」
The first section explains the twelve scales: "The Yellow Bell gong is zi, Shendu, earth, Jihuan, and the Zhenggong mode; the Great Cluster shang is yin, Gongcao, metal, Banjie, and the Dashi mode; the Maiden Washing jue is chen, Tiangang, wood, Womosi, and the Xiaoshi jue; the Forest Bell zhi is wei, Xiaoji, fire, Yunhan, and the Yellow Bell zhi; the Southern Pipe yu is you, Congkui, water, Di, and the Banshe mode; the Responding Bell bian gong is hai, Dengming, the sun, Mi, and the mid-pipe Yellow Bell gong; the Embellished Guest bian zhi is wu, Shengxian, the moon, Mo, and the Responding Bell zhi. The Great Pitch gong is Daji and the Gaogong mode; the Pinched Bell shang is Dachong and the Gaodashi mode; the Second Pitch jue is Taiyi and the mid-pipe Xiaoshi mode; the Ethereal Pitch zhi is Chuansong and the Great Pitch zhi; the Wushe yu is Hekui and the Gaobanshe mode; the Yellow Bell bian gong is the Zhenggong mode; the Forest Bell bian zhi is the Yellow Bell zhi. The Great Cluster gong is the mid-pipe Gaogong mode; the Maiden Washing shang is the Gaodashi mode; the Embellished Guest jue is the Xiezhi jue; the Southern Pipe zhi is the Great Cluster zhi; the Responding Bell yu is the mid-pipe Gaobanshe mode; the Great Pitch bian gong is the Gaogong mode; the Ethereal Pitch bian zhi is the Great Pitch zhi. The Pinched Bell gong is the Zhonglü gong mode; the Second Pitch shang is the Shuangdiao mode; the Forest Bell jue in current music is likewise the Forest Bell jue; the Wushe zhi is the Pinched Bell zhi; the Yellow Bell yu is the Zhonglü mode; the Great Cluster bian gong is the mid-pipe Gaogong mode; the Southern Pipe bian zhi is the Great Cluster zhi. The Maiden Washing gong is the mid-pipe Zhonglü gong mode; the Embellished Guest shang is the mid-pipe shang mode; the Ethereal Pitch jue is the mid-pipe Forest Bell jue; the Responding Bell zhi is the Maiden Washing zhi; the Great Pitch yu is the mid-pipe Zhonglü mode; the Pinched Bell bian gong is the Zhonglü gong mode; the Wushe bian zhi is the Pinched Bell zhi. The Second Pitch gong is the Daodiao gong mode; the Forest Bell shang is the Xiaoshi mode; the Southern Pipe jue is the Yuediao mode; the Yellow Bell zhi is the Zhonglü zhi; the Great Cluster yu is the Pingdiao mode; the Maiden Washing bian gong is the mid-pipe Zhonglü gong mode; the Responding Bell bian zhi is the Maiden Washing zhi. The Embellished Guest gong is the mid-pipe Daodiao gong mode; the Ethereal Pitch shang is the mid-pipe Xiaoshi mode; the Wushe jue is the mid-pipe Yuediao mode; the Great Pitch zhi is the Embellished Guest zhi; the Pinched Bell yu is the mid-pipe Pingdiao mode; the Second Pitch bian gong is the Daodiao gong mode; the Yellow Bell bian zhi is the Second Pitch zhi. The Forest Bell gong is the Southern Pipe gong mode; the Southern Pipe shang is the Xiezhi mode; the Responding Bell jue is the Dashi mode; the Great Cluster zhi is the Forest Bell zhi; the Maiden Washing yu is the Gaoping mode; the Embellished Guest bian gong is the mid-pipe Daodiao gong mode; the Great Pitch bian zhi is the Embellished Guest zhi. The Ethereal Pitch gong is Xianlü; the Wushe shang is the Forest Bell shang; the Yellow Bell jue is the Gaodashi mode; the Pinched Bell zhi is the Ethereal Pitch zhi; the Second Pitch yu is the Xianlü mode; the Forest Bell bian gong is the Southern Pipe gong mode; the Great Cluster bian zhi is the Forest Bell zhi. The Southern Pipe gong is the mid-pipe Xianlü gong mode; the Responding Bell shang is the mid-pipe Forest Bell shang; the Great Pitch jue is the mid-pipe Gaodashi jue; the Maiden Washing zhi is the Southern Pipe zhi; the Embellished Guest yu is the mid-pipe Xianlü mode; the Ethereal Pitch bian gong is the Xianlü gong mode; the Pinched Bell bian zhi is the Ethereal Pitch zhi. The Wushe gong is the Yellow Bell gong mode; the Yellow Bell shang is the Yuediao mode; the Great Cluster jue is the bian jue; the Second Pitch zhi is the Wushe zhi; the Forest Bell yu is the Yellow Bell yu; the Southern Pipe bian gong is the mid-pipe Xianlü gong mode; the Maiden Washing bian zhi is the Southern Pipe zhi. The Responding Bell gong is the mid-pipe Yellow Bell gong mode; the Great Pitch shang is the mid-pipe Yuediao mode; the Pinched Bell jue is the mid-pipe shuang jue; the Embellished Guest zhi is the Responding Bell zhi; the Ethereal Pitch yu is the mid-pipe Yellow Bell yu; the Wushe bian gong is the Yellow Bell gong mode; the Second Pitch bian zhi is the Wushe zhi."
4
二、明所主事，調五聲為五行、五事、四時、五帝、五神、五嶽、五味、五色，為生數一二三四五、成數六七八九十，為五藏、五官及五星。
Second, it explains what each tone governs, correlating the five tones with the five phases, five affairs, four seasons, five emperors, five spirits, five sacred peaks, five flavors, and five colors, with the generating numbers one through five and completing numbers six through ten, and with the five viscera, five sense organs, and five planets.
5
三、辯音聲，曰：「宮聲沈厚粗大而下，為君，聲調則國安，亂則荒而危。 合口通音謂之宮，其聲雄洪，屬平聲，西域言'婆陀力'。 一曰婆陀力。
Third, it distinguishes the tones: "The gong tone is deep, full, broad, and heavy, descending in pitch; it represents the ruler. When it is in tune, the state is secure; when it falls into disorder, the realm grows desolate and endangered. A closed-mouth, sustained tone is called gong; its sound is bold and resonant, of the level tone; in the Western Regions it is called 'Potuoli.' That is, Potuoli.
6
商聲勁凝明達，上而下歸於中，為臣，聲調則刑法不作，威令行，亂則其宮壞。 開口吐聲謂之商，音將將、倉倉然，西域言'稽識'。 '稽識'，猶長聲也。 角聲長而通徹，中平而正，為民，聲調則四民安，亂則人怨。 聲出齒間謂之角，喔喔、確確然，西域言'沙識'，猶質直聲也。 徵聲抑揚流利，從下而上歸於中，為事，聲調則百事理，亂則事隳。 齒合而唇啟謂之徵，倚倚、栎戲栎戲然，西域言'沙臘'。 '沙臘'，和也。 羽聲喓喓而遠徹，細小而高，為物，聲調則倉稟實、庶物備，亂則匱竭。 齒開唇聚謂之羽，詡、雨、酗、芋然。 西域言'般瞻'。 變宮，西域言'侯利箑'，猶言'斛律'聲也。 變徵聲，西域言'沙侯加濫'，猶應聲也。」
The shang tone is forceful, firm, bright, and clear, falling from above toward the center; it represents the minister. When it is in tune, penal law is not abused and commands are obeyed; when it falls into disorder, the gong tone is ruined. An open-mouth, exhaled tone is called shang; its sound is ringing and resonant; in the Western Regions it is called 'Jishi.' 'Jishi' means a drawn-out tone. The jue tone is long and penetrating, centered and even; it represents the people. When it is in tune, the four classes of society are at peace; when it falls into disorder, the people grow resentful. A tone produced between the teeth is called jue; its sound is woowo and queque; in the Western Regions it is called 'Shashi,' a plain, straightforward tone. The zhi tone rises and falls in flowing cadence, ascending from below toward the center; it represents affairs. When it is in tune, the hundred affairs are well ordered; when it falls into disorder, affairs collapse. With teeth closed and lips parted is called zhi; its sound is yiyi and lixi-lixi; in the Western Regions it is called 'Shala.' 'Shala' means harmonious. The yu tone is yaoya and far-carrying, fine and high in pitch; it represents things. When it is in tune, the granaries are full and all things are supplied; when it falls into disorder, stores run empty. With teeth open and lips gathered is called yu; its sound resembles xu, yu, xu, and yu. Within the Western Regions it is called 'Banzhan.' The bian gong—in the Western Regions it is called 'Houliqie,' akin to the sound 'Hulü.' The bian zhi tone—in the Western Regions it is called 'Shahoujialan,' like a responsive tone."
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其四、明律呂相生，祭天地宗廟，配律陽之數，曰：「太空，育五太：太易、太初、太始、太素、太極也。 分為七政，陽數七，所以齊律呂、均節度，不可加減也。 以育六甲，六甲，天之使，行風雹，筴鬼神。 為歲日時有善惡，故為九宮。 九者，陽數變化之道也。 為四正卦、五行、十幹，陰陽錯綜，律呂相葉，命宮而商者應，修下而高者降，下生隔八，上生隔六，皆圖于左。」
Fourth, it explains how pitch pipes and secondary pipes generate one another, the rites of sacrifice to Heaven, Earth, and the ancestral temples, and the yang numbers assigned to the pitch pipes: "The Great Void gives rise to the five Greats—Great Change, Great Beginning, Great Start, Great Simplicity, and the Supreme Ultimate. It divides into the seven regulators; seven is the yang number by which pitch pipes and secondary pipes are harmonized and measures standardized—nothing may be added or subtracted. From this the six jia are produced; the six jia are Heaven's emissaries, driving wind and hail and divining the spirits. Because years, days, and hours hold good and ill fortune, there are the nine palaces. Nine is the path of transformation within yang numbers. It yields the four cardinal hexagrams, the five phases, and the ten heavenly stems; yin and yang intertwine, pitch pipes and secondary pipes answer one another, the gong is set and the shang responds, the long below and high above descend, downward generation skips eight steps and upward generation skips six—all are charted on the left."
8
其五、著十二管短長。
Fifth, it records the lengths of the twelve pipes.
9
其六、出度量衡，辯古今尺龠。 律呂真聲，本陰陽之氣，可以感格天地，在於符合尺寸短長，宜因聲以定之。 因聲定律，則庶幾為得； 以尺定聲，則乖隔甚矣。
Sixth, it presents measures, weights, and balances, and distinguishes ancient and modern foot-rules and pitch tubes. The true tones of pitch pipes and secondary pipes arise from yin and yang qi and can move Heaven and Earth; the essential point is that their dimensions match their lengths, and they should be determined from sound. When pitch is set from sound, one may nearly get it right; but when sound is fixed by the foot-rule, the error is very great.
10
初，馮元等上《新修景祐廣樂記》時，鄭保信、阮逸、胡瑗等奏造鐘律，詔翰林學士丁度、知制誥胥偃、右司諫高若訥、韓琦，取保信、逸、瑗等鐘律詳考得失。 度等上議曰：「保信所制尺，用上黨秬黍圓者一黍之長，累而成尺。 律管一，據尺裁九十黍之長，空徑三分，空圍九分，容秬黍千二百。 遂用黍長為分，再累成尺，校保信尺、律不同。 其龠、合、升、鬥深闊，推以演算法，類皆差舛，不合周、漢量法。 逸、瑗所制，亦上黨秬黍中者累廣求尺，制黃鐘之律。 今用再累成尺，比逸、瑗所制，又復不同。 至於律管、龠、合升、鬥、斛、豆、區、鬴亦率類是。 蓋黍有圓長、大小而保信所用者圓黍，又首尾相銜，逸等止用大者，故再考之即不同。 尺既有差，故難以定鐘、磬。 謹詳古今之制，自晉至隋，累黍之法，但求尺裁管，不以權量參校，故歷代黃鐘之管容黍之數不同。 惟後周掘地得古玉鬥，據鬥造律，兼制權量，亦不同周、漢制度。 故《漢志》有備數、和聲、審度、嘉量、權衡之說，悉起于黃鐘。 今欲數器之制參互無失，則《班志》積分之法為近。 逸等以大黍累尺、小黍實龠，自戾本法。 保信黍尺以長為分，雖合後魏公孫崇所說，然當時已不施用，況保信今尺以圓黍累之，及首尾相銜，有與實龠之黍再累成尺不同。 其量器，分寸既不合古，即權衡之法不可獨用。」 詔悉罷之。
When Feng Yuan and others first submitted the Newly Revised Comprehensive Record of Music of the Jingyou reign, Zheng Baoxin, Ruan Yi, Hu Yuan, and others memorialized to cast bells and pitch pipes. The throne ordered Hanlin Academician Ding Du, Drafting Edict Officer Xu Yan, Right Remonstrance Officer Gao Ruone, and Han Qi to examine the bells and pitch pipes made by Baoxin, Yi, Yuan, and others in detail for their merits and defects. Du and the others submitted a memorial stating: "The foot-rule made by Baoxin used the length of one round black millet grain from Shangdang, stacked to form a foot. One pitch pipe, cut to the length of ninety millet grains by the foot-rule, with an inner diameter of three fen and inner circumference of nine fen, holding twelve hundred black millet grains. Then, using the length of millet as fen and stacking again to form a foot, this differed from Baoxin's foot-rule and pitch pipe. The depth and width of the yue, he, sheng, and dou vessels, calculated by computational methods, were all in error and did not accord with Zhou and Han measures. What Yi and Yuan made also used medium black millet grains from Shangdang, stacked in width to obtain a foot, and fashioned the Yellow Bell pitch pipe. Now, using stacked grains again to form a foot and comparing with what Yi and Yuan made, they again differ. The pitch pipes, yue, he, sheng, dou, hu, dou, qu, and fu vessels were generally the same in this respect. Millet comes in round and long forms and in large and small sizes; Baoxin used round millet with head and tail linked end to end, while Yi and the others used only the large grains—so when reexamined they differed. Because the foot-rule already differed, bells and chime-stones could not be reliably fixed. We have carefully examined ancient and modern systems: from Jin through Sui, the method of accumulating millet sought only to cut pipes by the foot-rule and did not cross-check with weights and measures; thus the number of millet grains held by the Yellow Bell pipe differed from dynasty to dynasty. Only in Later Zhou, when an ancient jade dou was excavated, pitch pipes were made according to the dou and weights and measures were also fashioned—but these too differed from the Zhou and Han systems. Thus the Han Treatise speaks of complete numbers, harmonious sound, careful measurement, excellent capacity, and weights and balances—all originating from Yellow Bell. If one now wishes the systems of numbers and vessels to cross-check without error, the method of accumulated parts in the Ban Treatise is the closest approach. Yi and others used large millet to stack the foot and small millet to fill the yue, departing from the original method on their own. Baoxin's millet foot-rule used length as fen; although this matched what Gongsun Chong of Later Wei said, it was already obsolete even then—how much more so now, when Baoxin's foot-rule stacks round millet with head and tail linked end to end, which differs from using the millet that fills the yue to stack again into a foot. As for the capacity vessels, since the fen and cun already did not accord with antiquity, the method of weights and balances cannot be used by itself." An edict ordered all of them abolished.
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又詔度等詳定太府寺並保信、逸、瑗所制尺，度等言：
Another edict ordered Du and others to examine in detail the foot-rules of the Court of the Imperial Treasury and those made by Baoxin, Yi, and Yuan. Du and the others stated:
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尺度之興尚矣，《周官》璧羨以起度，廣徑八寸，袤一尺。
The origin of foot-rules and measures goes far back; the Rites of Zhou uses the bi-yan to establish measure, eight cun in width and diameter and one chi in length.
13
《禮記》布手為尺，《淮南子》十二粟為一寸，《孫子》十厘為分，十分為寸，雖存異說，其可適從。 《漢志》，元始中，召天下通知鐘律者百餘人，使劉歆典領之。 是時，周滅二百餘年，古之律度當有考者。 以歆之博貫藝文，曉達曆算，有所製作，宜不凡近。 其審度之法云：「一黍之廣為分，十分為寸，十寸為尺。」 先儒訓解經籍多引以為義，曆世祖襲，著之定法。 然而歲有豐儉，地有磽肥，就令一歲之中，一境之內，取以校驗，亦復不齊。 是蓋天物之生，理難均一，古之立法，存其大概爾。 故前代制尺，非特累黍，必求古雅之器以雜校焉。 晉泰始十年，荀勖等校定尺度，以調鐘律，是為晉之前尺。 勖等以古物七品勘之，一曰姑洗玉律，二曰小呂玉律，三曰西京銅望臬，四曰金錯望臬，五曰銅斛，六曰古錢，七曰建武銅尺。 當時以勖尺揆校古器，與本銘尺寸無差，前史稱其用意精密。 《隋志》所載諸代尺度，十有五等，然以晉之前尺為本，以其與姬周之尺、劉歆銅斛尺、建武銅尺相合。
The Book of Rites takes the span of the hand as a foot; the Huainanzi says twelve grains make one cun; the Sunzi says ten li make one fen and ten fen make one cun—although differing accounts survive, which should be followed? This Han Treatise records that in the Yuanshi era, more than a hundred persons throughout the realm versed in bells and pitch were summoned, and Liu Xin was put in charge. At that time, more than two hundred years had passed since the fall of Zhou; the pitch pipes and measures of antiquity ought still to have been investigable. Given Xin's broad mastery of arts and letters and his clear understanding of calendrics and computation, whatever he produced ought not to have been ordinary. His method of careful measurement states: "The width of one millet grain is one fen; ten fen make one cun; ten cun make one chi." Earlier scholars, in annotating the classics, often cited this as authoritative, and successive emperors inherited it and established it as fixed law. Yet years vary between abundance and scarcity, and lands vary between barren and fertile; even within a single year and a single region, grains taken for verification still do not match. This is because things born of Heaven are by nature hard to make uniform; the ancients, in establishing law, preserved only the general outline. Thus in former ages, when making foot-rules, they did not rely on accumulating millet alone but necessarily sought ancient elegant vessels for cross-verification. In the tenth year of Taishi of Jin, Xun Xu and others corrected and fixed the foot-rule and measure to tune bells and pitch pipes—this is the earlier Jin foot-rule. Xu and others checked against seven kinds of ancient objects: first, the Maiden Washing jade pitch pipe; second, the Small Pitch jade pitch pipe; third, the Western Capital bronze wangnao; fourth, the gold-inlaid wangnao; fifth, the bronze hu; sixth, ancient coins; seventh, the Jianwu bronze foot-rule. At the time, using Xu's foot-rule to measure and verify against ancient vessels, there was no discrepancy with the dimensions on their inscriptions; earlier histories praise the precision of his approach. The Sui Treatise records foot-rules and measures of various dynasties in fifteen grades, yet takes the earlier Jin foot-rule as the standard, because it accords with the Zhou of Ji foot-rule, Liu Xin's bronze hu foot-rule, and the Jianwu bronze foot-rule.
14
竊惟周、漢二代，享年永久，聖賢製作，可取則焉。 而隋氏銷毀金石，典正之物，罕復存者。 夫古物之有分寸，明著史籍，可以酬驗者，惟有法錢而已。 周之圜法，曆載曠遠，莫得而詳。 秦之半兩，實重八銖； 漢初四銖，其文亦曰半兩。 孝武之世始行五銖，下暨隋朝，多以五銖為號，既歷代尺度屢改，故大小輕重鮮有同者，惟劉歆置銅斛。 世之所鑄錯刀並大泉五十，王莽天鳳元年改鑄貨布、貨泉之類，不聞後世復有兩者。 臣等檢詳《漢志》、《通典》、《唐六典》云：「大泉五十，重十二銖，徑一寸二分。 錯刀環如大泉，身形如刀，長二寸。 貨布重二十五銖，長二寸五分，廣一寸，首長八分有奇，廣八分，足股長八分，間廣二分，圍好徑二分半。 貨泉重五銖，徑一寸。」 今以大泉、錯刀、貨布、貨泉四物相參校，分寸正同。 或有大小輕重與本志微差者，蓋當時盜鑄既多，不必皆中法度，但當較其首足、肉好長廣、分寸，皆合正史者用之，則銅斛之尺從可知矣。 況經籍制度皆起周世，以劉歆術業之博，祖沖之算數之妙，荀勖揆較之詳密，校之既合周尺，則最為可法。 兼詳隋牛弘等議，稱後周太祖敕蘇綽造鐵尺，與宋尺同，以調中律，以均田度地。 唐祖孝孫雲，隋平陳之後，廢周玉尺，用此鐵尺律，然比晉前尺長六分四氂。 今司天監影表尺，和峴所謂西京銅望臬者，蓋以其洛都舊物也。 晉荀勖所用西京銅望臬者，蓋西漢之物，和峴謂洛陽為西京，乃唐東都爾。
We consider that the Zhou and Han dynasties endured for long ages, and the works of sages and worthies may be taken as models. Yet the Sui house melted down metal and stone, and canonical standard objects rarely survive. Among ancient objects whose dimensions are clearly recorded in historical works and can be used for verification, there are only standard coins. Zhou's round coin standard extends across vast ages and cannot be known in detail. The Qin dynasty 's half-liang actually weighed eight zhu; the early Han four-zhu coin also bore the inscription half-liang. In the era of Emperor Wu of Han the five-zhu coin was first issued; down to the Sui dynasty, most coins bore the five-zhu designation. Since foot-rules and measures changed repeatedly through the ages, size and weight rarely matched—only Liu Xin's bronze hu. Of coins cast in the world are the cuodao and the daquan wushi; in the first year of Tianfeng of Wang Mang, huobu and huoquan and the like were recast—neither is heard of again in later ages. We have examined in detail the Han Treatise, the Comprehensive Institutions, and the Six Codes of Tang, which state: "The daquan wushi weighed twelve zhu and had a diameter of one cun two fen. The cuodao had a ring like the daquan and a body shaped like a knife, two cun long. This huobu weighed twenty-five zhu, was two cun five fen long and one cun wide; the head was eight fen and a fraction long and eight fen wide; the foot and shank were eight fen long; the gap was two fen wide; the rim and hole had a diameter of two and a half fen. This huoquan weighed five zhu and had a diameter of one cun." Now, using the daquan, cuodao, huobu, and huoquan—the four objects—for mutual cross-checking, the dimensions match exactly. Some may differ slightly in size and weight from the original record, for illicit casting was widespread at the time and not every coin necessarily conformed to standard measure—but when the head, foot, rim, hole, length, and width all match the orthodox histories, they may be used, and the foot-rule of the bronze hu may then be inferred. Moreover, the institutions recorded in the classics all originated in the Zhou era; given the breadth of Liu Xin's learning, the subtlety of Zu Chongzhi's computation, and the meticulousness of Xun Xu's measurements—when verified they accord with the Zhou foot-rule and are most fit to be taken as model. In addition, examining in detail the discussion of Niu Hong and others of Sui, who said that Emperor Taizu of Later Zhou ordered Su Chuo to make an iron foot-rule identical to the Song foot-rule, to tune the central pitch pipe and to measure fields and land. Zu Xiaosun of Tang said that after Sui pacified Chen, the Zhou jade foot-rule was abolished and this iron foot-rule pitch pipe was used, yet it was six fen four li longer than the earlier Jin foot-rule. This shadow-table foot-rule of the Directorate of Astronomy today, what He Ning called the Western Capital bronze wangnao, is probably an old object from the Luoyang capital. This Western Capital bronze wangnao used by Xun Xu of Jin was probably an object of Western Han; He Ning called Luoyang the Western Capital, but that was Tang's eastern capital.
15
今以貨布、錯刀、貨泉、大泉等校之，則景表尺長六分有奇，略合宋、周、隋之尺。 由此論之，銅斛、貨布等尺寸昭然可驗。 有唐享國三百年，其間製作法度，雖未逮周、漢，然亦可謂治安之世矣。
Now, using the huobu, cuodao, huoquan, daquan, and the like to verify against it, the shadow-table foot-rule is six fen and a fraction longer, roughly matching the foot-rules of Song, Zhou, and Sui. Starting from this it follows that the dimensions of the bronze hu, huobu, and the like are clearly verifiable. The Tang dynasty held the realm for three hundred years; although the measures and institutions made in that period did not reach those of Zhou and Han, it may still be called an age of peace and good order.
16
今朝廷必求尺之中，當依漢錢分寸。 若以為太祖膺圖受禪，創制垂法，嘗詔和峴等用影表尺與典修金石，七十年間，薦之郊廟，稽合唐制，以示詒謀，則可且依影表舊尺，俟有妙達鐘律之學者，俾考正之，以從周、漢之制。 王樸律准尺比漢錢尺寸長二分有奇，比影表尺短四分，既前代未嘗施用，復經太祖朝更易。 其逸、瑗、保信及照所用太府寺等尺，其制彌長，出古遠甚，又逸進《周禮度量法議》，欲且鑄嘉量，然後取尺度權衡，其說疏舛，不可依用。 謹考舊文，再造影表尺一、校漢錢尺二並大泉、錯刀、貨布、貨泉總十七枚上進。
Should the court now must seek the mean in foot-rules, it should follow the fen and cun of Han coins. If one considers that the Founding Emperor received the Mandate and established institutions for posterity, and once ordered He Ning and others to use the shadow-table foot-rule in managing metal and stone ritual objects—for seventy years it was offered at suburban and temple rites, checked against Tang institutions, and shown as a legacy—then for the time being the old shadow-table foot-rule can be followed, awaiting a master versed in bells and pitch pipes to examine and correct it and return to the Zhou and Han systems. The scholar Wang Pu's pitch-standard foot-rule was two fen and a fraction longer than the Han coin dimensions and four fen shorter than the shadow-table foot-rule; it had never been employed in former ages and had again been changed in the Founding Emperor's reign. The foot-rules of Yi, Yuan, Baoxin, and Zhao, and those of the Court of the Imperial Treasury and the like, were made progressively longer and departed far from antiquity; Yi also submitted the Discourse on the Measures of the Rites of Zhou, wishing first to cast the excellent capacity vessel and then take foot-rule, measure, and balance—his theory was loose and erroneous and could not be relied upon. We carefully examined the old texts and remade one shadow-table foot-rule, two Han coin foot-rules for verification, and seventeen coins in all—daquan, cuodao, huobu, and huoquan—and submitted them.
17
詔度等以錢尺、影表尺各造律管，比驗逸、瑗並太常新舊鐘磬，考定音之高下以聞。
The edict ordered Du and others to make pitch pipes from the coin foot-rule and shadow-table foot-rule each, compare and verify against Yi's and Yuan's bells and chime-stones and the old and new ones of the Court of Imperial Sacrifices, examine and fix the pitch heights, and report.
18
度等言：「前承詔考太常等四尺，定可用者，止按典故及以《漢志》古錢分寸參校影表尺，略合宋、周、隋之尺，謂宜准影表尺施用。 今被旨造律管驗音高下，非素所習，乞別詔曉音者總領校定。」 詔乃罷之。 而若訥卒用漢貨泉度尺寸，依《隋書》定尺十五種上之，藏于太常寺：一、周尺，與《漢志》劉歆銅斛尺、後漢建武中銅尺、晉前尺同； 二、晉田父玉尺，與梁法尺同，比晉前尺為一尺七氂； 三、梁表尺，比晉前尺為一尺二分二氂一毫有奇； 四、漢官尺，比晉前尺為一尺三分七毫； 五、魏尺，杜夔之所用也，比晉前尺為一尺四分七氂； 六、晉後尺，晉江東用之，比晉前尺為一尺六分三厘； 七、魏前尺，比晉前尺為一尺一寸七厘； 八、中尺，比晉前尺為一尺二寸一分一厘； 九、後尺，同隋開皇尺、周氏尺，比晉前尺為一尺二寸八分一厘； 十、東魏後尺，比晉前尺為一尺三寸八毫； 十一、蔡邕銅龠尺，同後周玉尺，比晉前尺為一尺一寸五分八厘； 十二、宋氏尺，與錢樂之渾天儀尺、後周鐵尺同。 比晉前尺為一尺六分四厘； 十三、太府寺鐵尺，制大樂所裁造尺也； 十四、雜尺，劉曜渾儀土圭尺也，比晉前尺為一尺五分； 十五、梁朝俗尺，比晉前尺為一尺七分一厘。 太常所掌，又有後周王樸律准尺，比晉前尺長二分一厘，比梁表尺短一厘； 有司天監影表尺，比晉前尺長六分三厘，同晉後尺； 有中黍尺，亦制樂所新造也。
The official Du and others said: "Previously, receiving an edict to examine the four foot-rules of the Court of Imperial Sacrifices and the like and determine which could be used, we found only by historical precedent and by cross-checking the shadow-table foot-rule against the ancient coin fen and cun of the Han Treatise that it roughly matched the foot-rules of Song, Zhou, and Sui, and said the shadow-table foot-rule should be adopted. Now, charged with making pitch pipes to verify pitch height, which was not our usual study, we beg that another edict appoint one versed in sound to take overall charge of verification." An edict then abolished the project. But Ruone, before his death, used Han huoquan to measure dimensions and, according to the History of Sui, fixed fifteen kinds of foot-rules and submitted them, stored in the Court of Imperial Sacrifices: first, the Zhou foot-rule, the same as Liu Xin's bronze hu foot-rule in the Han Treatise, the bronze foot-rule of Jianwu in Later Han, and the earlier Jin foot-rule; second, the field-father jade foot-rule of Jin, the same as the Liang standard foot-rule, one chi seven li compared with the earlier Jin foot-rule; third, the Liang table foot-rule, one chi two fen two li one hao and a fraction compared with the earlier Jin foot-rule; fourth, the Han official foot-rule, one chi three fen seven hao compared with the earlier Jin foot-rule; fifth, the Wei foot-rule, that used by Du Kui, one chi four fen seven li compared with the earlier Jin foot-rule; sixth, the later Jin foot-rule, used in the Jiangdong region of Jin, one chi six fen three li compared with the earlier Jin foot-rule; seventh, the earlier Wei foot-rule, one chi one cun seven li compared with the earlier Jin foot-rule; eighth, the middle foot-rule, one chi two cun one fen one li compared with the earlier Jin foot-rule; ninth, the later foot-rule, the same as the Kaihuang foot-rule of Sui and the Zhou foot-rule, one chi two cun eight fen one li compared with the earlier Jin foot-rule; tenth, the later Eastern Wei foot-rule, one chi three cun eight hao compared with the earlier Jin foot-rule; eleventh, Cai Yong's bronze yue foot-rule, the same as the jade foot-rule of Later Zhou, one chi one cun five fen eight li compared with the earlier Jin foot-rule; Twelfth, the Song foot-rule, the same as Qian Yuezhi's armillary foot-rule and the iron foot-rule of Later Zhou. One chi six fen four li compared with the earlier Jin foot-rule; Thirteenth, the iron foot-rule of the Court of the Imperial Treasury, the foot-rule cut and made by the Office of Great Music; fourteenth, the miscellaneous foot-rule, the armillary earth-square foot-rule of Liu Yao, one chi five fen compared with the earlier Jin foot-rule; fifteenth, the popular foot-rule of the Liang dynasty, one chi seven fen one li compared with the earlier Jin foot-rule. Under the Court of Imperial Sacrifices there was also Wang Pu's pitch-standard foot-rule of Later Zhou, two fen one li longer than the earlier Jin foot-rule and one li shorter than the Liang table foot-rule; there was the shadow-table foot-rule of the Directorate of Astronomy, six fen three li longer than the earlier Jin foot-rule, the same as the later Jin foot-rule; there was the middle-millet foot-rule, also newly made by the Office of Music.
19
其後宋祁、田況薦益州進士房庶曉音，祁上其《樂書補亡》三卷，召詣闕。 庶自言賞得古本《漢志》，云：'度起于黃鐘之長，以子穀秬黍中者一黍之起，積一千二百黍之廣，度之九十分，黃鐘之長，一為一分。 '今文脫'之起積一千二百黍'八字，故自前世以來，累黍為尺以制律，是律生於尺，尺非起于黃鐘也。 且《漢志》'一為一分'者，蓋九十分之一，後儒誤以一黍為分，其法非是。 當以秬黍中者一千二百實管中，黍盡，得九十分，為黃鐘之長，九寸加一以為尺，則律定矣。」 直秘閣範鎮是之，乃為言曰：「照以縱黍累尺，管空徑三分，容黍千七百三十； 瑗以橫黍累尺，管容黍一千二百，而空徑三分四厘六毫：是皆以尺生律，不合古法。 今庶所言，實千二百黍于管。 以為黃鐘之長，就取三分以為空徑，則無容受不合之差，校前二說為是。 蓋累黍為尺，始失之于《隋書》，當時議者以其容受不合，棄而不用。 及隋平陳，得古樂器，高祖聞而歎曰：'華夏舊聲也！ '遂傳用之。 至唐祖孝孫、張文收，號稱知音，亦不能更造尺律，止沿隋之古樂，制定聲器。 朝廷久以鐘律未正，屢下詔書，博訪群議，冀有所獲。 今庶所言，以律生尺，誠眾論所不及，請如其法，試造尺律，更以古器參考，當得其真。」 乃詔王洙與鎮同于修制所如庶說造律、尺、龠：律徑三分，圍九分，長九十分； 龠徑九分，深一寸； 尺起黃鐘之長加十分，而律容千二百黍。 初，庶言太常樂高古樂五律，比律成，才下三律，以為今所用黍，非古所謂一稃二米黍也。 尺比橫黍所累者，長一寸四分。
Afterward Song Qi and Tian Kuang recommended Fang Shu, a jinshi of Yizhou versed in sound; Qi submitted his Supplement to the Lost Music Book in three chapters, and he was summoned to court. Shu himself said he had been rewarded with an ancient copy of the Han Treatise, which states: 'Measure begins from the length of Yellow Bell; taking the medium black millet grain of zigu, one grain as the start, accumulating the width of twelve hundred millet grains, nine-tenths of the length of Yellow Bell—one is one fen.' 'The present text lacks the eight characters 'as the start, accumulating twelve hundred millet grains,' so from former ages onward, accumulating millet to make a foot-rule to fix pitch pipes meant pitch pipes were born from the foot-rule—the foot-rule did not begin from Yellow Bell. Moreover, in the Han Treatise 'one is one fen' means one-ninth of one-tenth; later scholars mistakenly took one millet grain as one fen—the method is incorrect. A single should fill a pipe with twelve hundred medium black millet grains; when the millet is exhausted, nine-tenths are obtained as the length of Yellow Bell; adding one to nine cun makes the foot-rule, and then pitch pipes are fixed." Direct Secretariat Associate Fan Zhen agreed and spoke on his behalf, saying: "Zhao accumulated millet vertically to make a foot-rule; the pipe had an inner diameter of three fen and held seventeen hundred thirty millet grains; Yuan accumulated millet horizontally to make a foot-rule; the pipe held twelve hundred millet grains, with an inner diameter of three fen four li six hao: both derive pitch pipes from the foot-rule and do not accord with ancient method. That which Shu now says is to fill twelve hundred millet grains in a pipe. By taking that as the length of Yellow Bell and then taking three fen as the inner diameter, there is no discrepancy of capacity mismatch; compared with the two earlier theories, this is correct. The practice of accumulating millet to make a foot-rule first went wrong in the History of Sui; at the time the debaters, because capacity did not match, abandoned it and did not use it. Whenever Sui pacified Chen and obtained ancient musical instruments, Emperor Gaozu heard them and sighed: 'The old sounds of Huaxia!' They were then transmitted and used. Extending to Zu Xiaosun and Zhang Wenshou of Tang, who were called masters of sound, they also could not remake foot-rules and pitch pipes, but only followed Sui's ancient music and fixed sound vessels. The court, for long because bells and pitch pipes were not corrected, repeatedly issued edicts and broadly sought group discussion, hoping for some gain. What Shu now says—deriving the foot-rule from pitch pipes—is truly beyond what the many discussions reached; please according to his method try making foot-rules and pitch pipes, and further cross-check with ancient vessels—then the true can be obtained." An edict then ordered Wang Zhu and Zhen together at the Office of Revision to make pitch pipes, foot-rules, and yue according to Shu's theory: pitch pipes with diameter three fen, circumference nine fen, length nine-tenths; yue with diameter nine fen, depth one cun; the foot-rule beginning from the length of Yellow Bell plus one-tenth, and the pitch pipe holding twelve hundred millet grains. Initially Shu said the music of the Court of Imperial Sacrifices was five pitch pipes higher than ancient music; when the pitch pipes were completed, they were only three pitch pipes lower, because the millet now used was not the one-grain-two-seeds millet of antiquity. This foot-rule compared with that accumulated from horizontal millet was one cun four fen longer.
20
庶又言：「古有五音，而今無正徵音。 國家以火德王，徵屬火，不宜闕。 今以五行旋相生法，得徵音。」 又言：「《尚書》'同律、度、量、衡'，所以齊一風俗。 今太常、教坊、鈞容及天下州縣，各自為律，非《書》同律之義。 且古者帝王巡狩方嶽，必考禮樂同異，以行誅賞。 謂宜頒格律，自京師及州縣，毋容輒異，有擅高下者論之。」 帝召輔臣觀庶所進律尺、龠，又令庶自陳其法，因問律呂旋相為宮事，令撰圖以進。 其說以五正、二變配五音，迭相為主，衍之成八十四調。 舊以宮、徵、商、羽、角五音，次第配七聲，然後加變宮、變徵二聲，以足其數。 推以旋相生之法謂五行相戾非是，當改變徵為變羽，易變為閏，隨音加之，則十二月各以其律為宮，而五行相生，終始無窮。 詔以其圖送詳定所。 庶又論吹律以聽軍聲者，謂以五行逆順，可以知吉凶，先儒之說略矣。
Shu further said: "In antiquity there were five tones, but today there is no proper zhi tone. This state takes Fire as its virtue; zhi belongs to fire and ought not to be lacking. At present, by the method of cyclical mutual generation of the five phases, the zhi tone is obtained." He also said: "The Documents says 'uniform pitch pipes, measures, capacity, and balances'—this is to unify customs. At present, the Court of Imperial Sacrifices, the Music Teaching Office, the Junrong Guard, and prefectures and counties throughout the realm each make pitch pipes on their own—this is not the meaning of uniform pitch pipes in the Book. Further, in antiquity when emperors toured the sacred peaks, they necessarily examined whether ritual and music were the same or different, to impose rewards and punishments. It was fitting to promulgate pitch standards; from the capital to prefectures and counties, none may arbitrarily differ—whoever presumptuously raises or lowers pitch shall be prosecuted." The emperor summoned chief ministers to view the pitch pipes, foot-rules, and yue submitted by Shu, and also ordered Shu himself to explain his method; he was asked about pitch pipes and secondary pipes cyclically generating modes, and ordered to compose a diagram and submit it. His theory paired five proper and two variant tones with the five tones, each in turn serving as root, and extended them to form eighty-four modes. In former times, gong, zhi, shang, yu, and jue—the five tones—were assigned in sequence to the seven sounds, and then bian gong and bian zhi were added to complete the number. Extending by the method of cyclical mutual generation, he said the five phases conflicting was wrong; bian zhi should be changed to bian yu, and yi changed to run, added according to the tone—then each of the twelve months takes its pitch pipe as gong, and the five phases generate one another in endless succession. The edict ordered his diagram sent to the Detailed Examination Office. Shu also discussed blowing pitch pipes to listen to army sounds, saying that by the reverse and direct order of the five phases, good and evil can be known—the earlier scholars' theory was incomplete.
21
是時瑗、逸制樂有定議，乃補庶試秘書省校書郎，遣之。 鎮為論於執政日：
By that time Yuan and Yi had fixed opinions on making music, so Shu was supplemented as proofreader of the Secretariat and sent. Zhen addressed in discussion to the chief ministers, saying:
22
今律之與尺所以不得其真，累黍為之也。 累黍為之者，史之脫文也。 古人豈以難曉不合之法，書之于史，以為後世惑乎？ 殆不然也。 易曉而必合也，房庶之法是矣。 今庶自言其法，依古以律而起尺，其長與空徑、與容受、與一千二百黍之數，無不合之差。 誠如庶言，此至真之法也。
This reason pitch pipes and foot-rules today do not obtain the true is that millet is accumulated to make them. The practice of accumulating millet to make them is a lacuna in the histories. Could the ancients really have written in the histories a method hard to understand and not matching, to confuse later ages? Surely not. Plain and to understand and necessarily matching—Fang Shu's method is this. At present, Shu himself explains his method: following antiquity, deriving the foot-rule from pitch pipes—the length, inner diameter, capacity, and the number twelve hundred millet grains all match without discrepancy. Indeed, as Shu says, this is the most true method.
23
且黃鐘之實一千二百黍，積實分八百一十，於演算法圓積之，則空徑三分，圍九分，長九十分，積實八百一十分，此古律也。 律體本圓。 圓積之是也。 今律方積之，則空徑三分四厘六毫，比古大矣。 故圍十分三厘八毫，而其長止七十六分二厘，積實亦八百一十分。 律體本不方，方積之，非也。 其空徑三分，圍九分，長九十分，積實八百一十分，非外來者也，皆起於律也。 以一黍而起於尺，與一千二百黍之起於律，皆取於黍。 今議者獨於律則謂之索虛而求分，亦非也。 其空徑三分，圍九分，長九十分之起於律，與空徑三分四厘六毫，圍十分三厘八毫，長七十六分二厘之起於尺，古今之法，疏密之課，其不同較然可見，何所疑哉？ 若以謂工作既久而復改為，則淹引歲月，計費益廣，又非朝廷製作之意也。 其淹久而計費廣者，為之不敏也。 今庶言太常樂無姑洗、夾鐘、太簇等數律，就令其律與其說相應，鐘磬每編才易數三，因舊而新，敏而為之，則旬月功可也，又向淹久而廣費哉？
Further, the substance of Yellow Bell is twelve hundred millet grains; accumulated substance is eight hundred ten parts; calculating by circular area in computational method, the inner diameter is three fen, circumference nine fen, length nine-tenths, accumulated substance eight hundred ten parts—this is the ancient pitch pipe. This body of a pitch pipe is originally round. Circular area calculation is correct. At present, pitch pipes are calculated by square area, so the inner diameter is three fen four li six hao—larger than antiquity. In this way, circumference is ten fen three li eight hao, but the length is only seventy-six fen two li, and accumulated substance is also eight hundred ten parts. The body of a pitch pipe is originally not square; square area calculation is incorrect. This inner diameter three fen, circumference nine fen, length nine-tenths, accumulated substance eight hundred ten parts are not external—they all arise from the pitch pipe. Starting from one millet grain to make a foot-rule and beginning from twelve hundred millet grains to make a pitch pipe both take from millet. At present, debaters alone, regarding pitch pipes, call it seeking emptiness to obtain parts—this too is incorrect. This inner diameter three fen, circumference nine fen, length nine-tenths arising from pitch pipes, versus inner diameter three fen four li six hao, circumference ten fen three li eight hao, length seventy-six fen two li arising from the foot-rule—ancient and modern methods, loose and dense reckonings—their difference is clearly visible; what is there to doubt? Should one says that because work has already continued long and then is changed again, months and years are drawn out and expense increases—this too is not the intent of the court's production. The drawing out of out time and increasing expense comes from lack of promptness in doing it. At present, Shu says the music of the Court of Imperial Sacrifices lacks several pitch pipes such as Maiden Washing, Pinched Bell, and Great Cluster; let his pitch pipes correspond to his theory—bells and chime-stones need only change a few numbers in each set; adapting the old to make new, doing it promptly, the work can be done in a month or so—why draw out time and increase expense?
24
執政不聽。
The chief ministers did not listen.
25
四年，鎮又上書曰：
Within the fourth year, Zhen again submitted a memorial, saying:
26
陛下制樂以事天地、宗廟，以揚祖宗之休，茲盛德之事也。 然自下詔以來，及今三年，有司之論紛然未決，蓋由不議其本而爭其末也。 竊惟樂者，和氣也。 發和氣者，聲音也。 聲音之生，生於無形，故古人以有形之物傳其法，俾後人參考之，然後無形之聲音得而和氣可道也。 有形者，秬黍也，律也，尺也，龠也，鬴也，斛也，算數也，權衡也，鐘也，磬也，是十者必相合而不相戾，然後為得，今皆相戾而不相合，則為非是矣。 有形之物非是，而欲求無形之聲音和，安可得哉？ 謹條十者非是之驗，惟裁擇焉！
The Majesty is making music to serve Heaven and Earth and the ancestral temples, to display the blessings of the ancestors—this is a matter of great virtue. Still, from the issuing of the edict until now, three years have passed, and the discussions of the responsible officials remain unsettled—this is because they do not debate the root but contend over the branches. It is our view that that music is harmonizing qi. What releases harmonizing qi is sound. The birth of sound arises from the formless; thus the ancients used formed objects to transmit their method, so that later people might cross-check—then formless sound can be obtained and harmonizing qi may be conveyed. This formed objects are black millet, pitch pipes, foot-rules, yue, fu, hu, computation, weights and balances, bells, and chime-stones—these ten must all match and not conflict; only then is correctness obtained. Now all conflict and do not match—then they are wrong. If the formed objects are wrong, how can one expect formless sound to harmonize—how may that be achieved? We respectfully set forth proof that all ten are wrong—may Your Majesty decide!
27
按《詩》「誕降嘉種，維秬維秠。」 誕降者，天降之也。 許慎云：「秬，一稃二米。」 又云：「一秬二米。」 後漢任城縣產秬黍二斛八斗，實皆二米，史官載之，以為嘉瑞。 又古人以秬黍為酒者，謂之秬鬯。 宗廟降神，惟用一尊； 諸侯有功，惟賜一卣，以明天降之物，世不常有而可貴也。 今秬黍取之民間者，動至數百斛，秬皆一米，河東之人謂之黑米。 設有真黍，以為取數至多，不敢送官，此秬黍為非是，一也。
On examination, the Odes: 'Heaven sent down excellent seed, black millet and double-grained millet.' Sent down' means sent down from Heaven. Xu Shen states: 'Ju, one husk two grains.' He also says: 'One ju, two grains.'.' In Later Han, Rencheng county produced two hu eight dou of ju millet; all were double-grained; the historiographers recorded it as an auspicious omen. Further, when the ancients made wine from ju millet, they called it ju chang. Within the ancestral temple, when descending the spirit, only one zun is used; when feudal lords had merit, only one you was bestowed—to show that what Heaven sent down is not common in the world and is precious. At present, ju millet taken from among the people often reaches several hundred hu; the ju are all single-grain; people east of the Yellow River call them black millet. Even if there were true millet, because the quantity taken would be very great, one would not dare send it to the court—this ju millet is incorrect; first.
28
又按先儒皆言律空徑三分，圍九分，長九十分，容千二百黍，積實八百一十分。 今律空徑三分四厘六毫，圍十分二厘八毫，是為九分外大其一分三厘八毫，而後容千二百黍，除其圍廣，則其長止七十六分二厘矣。 說者謂四厘六毫為方分，古者以竹為律，竹形本圓，今以方分置算，此律之為非是，二也。
Further, earlier scholars, all say pitch pipes have inner diameter three fen, circumference nine fen, length nine-tenths, capacity twelve hundred millet grains, accumulated substance eight hundred ten parts. At present, pitch pipes have inner diameter three fen four li six hao, circumference ten fen two li eight hao—this is one fen three li eight hao larger outside the nine fen, and then they hold twelve hundred millet grains; subtracting the circumference and width, the length is only seventy-six fen two li. Exponents say four li six hao are square parts; the ancients used bamboo for pitch pipes, and bamboo is originally round; now square parts are placed in calculation—this pitch pipe is incorrect; second.
29
又按《漢書》，分、寸、尺、丈、引本起黃鐘之長，又雲九十分黃鐘之長者，據千二百黍而言也。 千二百黍之施於量，則曰黃鐘之龠； 施于權衡，則曰黃鐘之重； 施于尺，則曰黃鐘之長。 今遺千二百之數，而以百黍為尺，又不起于黃鐘，此尺之為非是，三也。
Further, the History of Han, fen, cun, chi, zhang, and yin originally began from the length of Yellow Bell; it also says nine-tenths the length of Yellow Bell—this refers to twelve hundred millet grains. Applying twelve hundred millet grains to capacity, it was called the yue of Yellow Bell; applying it to weights and balances, it is called the weight of Yellow Bell; applying it to the foot-rule, it is called the length of Yellow Bell. At present, abandoning the number twelve hundred and using one hundred millet grains as a foot-rule, and not beginning from Yellow Bell—this foot-rule is incorrect; third.
30
又按《漢書》言龠，其狀似爵，爵謂爵曁戔，其體正圓。 故龠當圓徑九分，深十分，容千二百黍，積實八百一十分，與律分正同。 今龠乃方一寸，深八分一厘，容千二百黍，是亦以方分置算者，此龠之非是，四也。
Further, the History of Han on the yue: its form resembles a jue; jue means jue and jian, and its body is truly round. In this way, the yue should have round diameter nine fen, depth ten fen, capacity twelve hundred millet grains, accumulated substance eight hundred ten parts—exactly the same as pitch pipe parts. At present, the yue is square one cun, depth eight fen one li, capacity twelve hundred millet grains—this too uses square parts in calculation; this yue is incorrect; fourth.
31
又按《周禮》鬴法：方尺，圓其外； 深尺，容六斗四升。 方尺者，八寸之尺也； 深尺者，十寸之尺也。 何以知尺有八寸、十寸之別？ 按《周禮》：「璧羨度尺，好三寸以為度。」 璧羨之制，長十寸，廣八寸，同謂之度尺。 以為尺，則八寸、十寸俱為尺矣。 又《王制》云：「古者以周尺八尺為步，今以六尺四寸為步。」 八尺者，八寸之尺也； 六尺四寸者，十寸之尺也。 同謂之周尺者，是周用八寸、十寸尺明矣。 故知八寸尺為鬴之方，十寸尺為鬴之深，而容六斗四升，千二百八十龠也。 積實一百三萬六千八百分。 今鬴方尺，積千寸，此鬴之非是，五也。
Further, the fu method of the Rites of Zhou: one chi square, rounding the outside; one chi deep, capacity six dou four sheng. A single chi square is the eight-cun foot-rule; one chi deep is the ten-cun foot-rule. How does one know foot-rules have the distinction of eight cun and ten cun? On examination, the Rites of Zhou: 'The bi-yan measures the foot-rule; the hole three cun serves as measure.' The form of the bi-yan is ten cun long and eight cun wide—both are called measure foot-rules. Taken as foot-rules, both eight cun and ten cun were foot-rules. Moreover, the Royal Regulations states: 'In antiquity eight chi of Zhou foot-rule made one pace; now six chi four cun make one pace.' Eight chi is the eight-cun foot-rule;. Six chi four cun is the ten-cun foot-rule. Both called Zhou foot-rules—Zhou used eight-cun and ten-cun foot-rules was clear. In this way, one knows the eight-cun foot-rule is the square of the fu, the ten-cun foot-rule is the depth of the fu, and capacity is six dou four sheng, twelve hundred eighty yue. Accumulated substance was one million three hundred six thousand eight hundred parts. At present, the fu is one chi square, accumulated one thousand cun—this fu is incorrect; fifth.
32
又按《漢書》斛法：方尺，圓其外，容十斗，旁有庣焉。 當隋時，漢斛尚在，故《隋書》載其銘曰：「律嘉量斛，方尺圓其外，庣旁九厘五毫，冪百六十二寸，深尺，容一斛。」 今斛方尺，深一尺六寸二分，此斛之非是，六也。
Further, the hu method of the History of Han: one chi square, rounding the outside, capacity ten dou, with a tilted side. Within Sui times the Han hu still existed; thus the History of Sui records its inscription: 'Pitch-standard excellent-capacity hu, one chi square rounding the outside, tilted side nine li five hao, area one hundred sixty-two cun, one chi deep, capacity one hu.' Now the hu is one chi square, one chi six cun two fen deep—this hu is incorrect; sixth.
33
又按演算法，圓分謂之徑圍，方分謂之方斜，所謂「徑三、圍九、方五、斜七」是也。 今圓分而以方法算之，此算數非是，七也。
Further, computational method: circular parts are called diameter and circumference; square parts are called square and diagonal—the so-called 'diameter three, circumference nine, square five, diagonal seven' is this. At present, using square method to calculate circular parts—this computation is incorrect; seventh.
34
又按權衡者，起千二百黍而立法也。 周之鬴，其重一鈞，聲中黃鐘； 漢之斛，其重二鈞，聲中黃鐘。 鬴、斛之制，有容受，有尺寸，又取其輕重者，欲見薄厚之法，以考其聲也。 今黍之輕重未真，此權衡為非是，八也。
Further, weights and balances: they begin from twelve hundred millet grains to establish law. This fu of Zhou weighs one jun and its sound matches Yellow Bell; the hu of Han weighs two jun and its sound matches Yellow Bell. This forms of fu and hu have capacity and dimensions; also taking their light and heavy, one wishes to see the method of thin and thick to examine their sound. At present, the light and heavy of millet is not true—this weights and balances is incorrect; eighth.
35
又按：「鳧氏為鐘：大鐘十分，其鼓間之，以其一為之厚； 小鐘十分，其鉦間之，以其一為之厚。」 今無大小薄厚，而一以黃鐘為率，此鐘之非是，九也。
Further, examining: 'The Fu clan makes bells: for large bells, ten parts—the space between the drums, one part of it is the thickness; for small bells, ten parts—the space between the zheng, one part of it is the thickness.' Now there is no distinction of large and small, thin and thick, but all take Yellow Bell as rate—this bell is incorrect; ninth.
36
又按：「磬氏為磬，倨句一矩有半，其博為一，股為二，鼓為三。」 蓋各以其律之長短為法也。 今亦以黃鐘為率，而無長短厚薄之別，此磬之非是，十也。
Further, examining: 'The Qing clan makes chime-stones: the obtuse angle one and a half right angles; the width is one, the shank two, the drum three.' Each takes the length of its pitch pipe as standard. At present, too Yellow Bell is taken as rate, with no distinction of long and short, thin and thick—this chime-stone is incorrect; tenth.
37
前此者，皆有形之物也，可見者也。 使其一不合，則未可以為法，況十者之皆相戾乎？ 臣固知其無形之聲音不可得而和也。 請以臣章下有司，問黍之二米與一米孰是？ 律之空徑三分與三分四厘六毫孰是？ 律之起尺與尺之起律孰是？ 龠之圓制與方制孰是？ 鬴之方尺圓其外，深尺與方尺孰是？ 斛之方尺圓其外，庣旁九厘五毫與方尺深尺六寸二分孰是？ 算數之以圓分與方分孰是？ 權衡之重以二米秬黍與一米孰是？ 鐘磬依古法有大小、輕重、長短、薄厚而中律孰是？ 是不是定，然後制龠、合、升、鬥、鬴、斛以校其容受； 容受合，然後下詔以求真黍； 真黍至，然後可以為量、為鐘磬； 量與鐘磬合於律，然後可以為樂也。 今尺律本末未定，而詳定、修制二局工作之費無慮千萬計矣，此議者所以云云也。 然議者不言有司論議依違不決，而願謂作樂為過舉，又言當今宜先政令而禮樂非所急，此臣之所大惑也。 儻使有司合禮樂之論，是其所是，非其所非，陛下親臨決之，顧於政令不已大乎。
This foregoing are all formed objects that can be seen. If even one does not match, it could not serve as standard—how much more when all ten conflict? Your subject firmly knows that formless sound could not be obtained and harmonized. We ask that send my memorial below to the responsible officials and ask: two-grain millet versus single-grain—which is correct? Pitch pipe inner diameter three fen versus three fen four li six hao—which was correct? Pitch pipe deriving foot-rule versus foot-rule deriving pitch pipe—which was correct? Yue round form versus square form—which was correct? Fu one chi square rounding the outside, one chi deep versus one chi square—which was correct? The scholar Hu one chi square rounding the outside, tilted side nine li five hao versus one chi square one chi six cun two fen deep—which is correct? Computation using circular parts versus square parts—which was correct? Weight of weights and balances using two-grain ju millet versus single-grain—which was correct? Bells and chime-stones following ancient method with large and small, light and heavy, long and short, thin and thick matching pitch—which was correct? Whenever right and wrong are fixed, then make yue, he, sheng, dou, fu, and hu to verify their capacity; when capacity matches, then issue an edict to seek true millet; when true millet arrives, then capacity vessels and bells and chime-stones can be made; when capacity and bells and chime-stones match pitch pipes, then music can be made. At present, foot-rules and pitch pipes, root and branch, are not yet fixed, yet the work expenses of the Detailed Examination and Revision offices are estimated at no less than ten million—this is why debaters talk thus. Still, debaters do not say the responsible officials' discussions vacillate undecided, but wish to say making music is excessive action, and also say that today one should first attend to government orders and ritual and music are not urgent—this is what your subject greatly puzzles over. Should only the responsible officials would combine discussions of ritual and music, affirm what should be affirmed and reject what should be rejected, and Your Majesty personally decided—would that not be greater than government orders?
38
昔漢儒議鹽鐵，後世傳《鹽鐵論》。 方今定雅樂以求廢墜之法，而有司論議不著盛德之事，後世將何考焉？ 顧令有司，人人各以經史論議條上，合為一書，則孰敢不自竭盡，以副陛下之意？ 如以臣議為然，伏請權罷詳定、修制二局，俟真黍至，然後為樂，則必得至當而無事于浮費也。
In former times, Han scholars debated salt and iron; later ages transmitted the Discourse on Salt and Iron. At present, fixing elegant music to seek lost methods—yet the responsible officials' discussions do not record a matter of great virtue; what will later ages examine? We ask that order the responsible officials, each person submitting discussion based on classics and histories item by item, combined into one book—then who would dare not exhaust themselves to fulfill Your Majesty's intent? Should my memorial is approved, I respectfully beg temporarily to suspend the Detailed Examination and Revision offices; wait until true millet arrives, then make music—then the most fitting will surely be obtained without wasteful expense.
39
詔送詳定所。 鎮說自謂得古法，後司馬光數與之論難，以為弗合。 世鮮鐘律之學，卒莫辯其是非焉。
The edict ordered it sent to the Detailed Examination Office. Zhen's theory claimed to have obtained ancient method; afterward Sima Guang several times debated with him, holding that it did not match. This age rarely had scholars of bells and pitch pipes; in the end none could distinguish right and wrong.
40
宋興百餘年，司天數改曆，其說曰：「曆者歲之積。 歲者月之積，月者日之積，日者分之積，又推餘分置閏，以定四時，非博學妙思弗能考也。 夫天體之運，星辰之動，未始有窮，而度以一法，是以久則差，差則敝而不可用，曆之所以數改造也。 物銖銖而較之，至石必差，況於無形之數哉？」 乾興初，議改曆，命司天役人張奎運算，其術以八千為日法，一千九百五十八為半分，四千二百九十九為朔，距乾興元年壬戌，歲三千九百萬六千六百五十八為積年。 詔以奎補保章正。 又推擇學者楚衍與曆官宋行古集天章閣，詔內侍金克隆監造曆，至天聖元年八月成，率以一萬五百九十為樞法，得九钜萬數。 既上奏，詔翰林學士晏殊制序而施行焉，命曰《崇天曆》。 曆法曰演紀上元甲子，距天聖二年甲子，歲積九千七百五十五萬六千三百四十。 上考往古，歲減一算； 下驗將來，歲加一算。
Since Song's rise for more than a hundred years, the Directorate of Astronomy repeatedly revised the calendar; its theory states: 'The calendar is the accumulation of years. Years are the accumulation of months, months the accumulation of days, days the accumulation of parts; further pushing remainder parts to set intercalation to fix the four seasons—without broad learning and subtle thought one could not examine it. This motion of the heavenly body and the movement of stars and constellations have never begun to have an end; yet measuring them by one method, over long time there is discrepancy, discrepancy then decay and unusability—this is why calendars are repeatedly remade. Comparing things grain by grain, when it reaches a shi there must be discrepancy—how much more with formless numbers?' At the beginning of Qianxing, calendar revision was discussed; the Directorate of Astronomy laborer Zhang Kui was ordered to compute; his method used eight thousand as day divisor, one thousand nine hundred fifty-eight as half-part, four thousand two hundred ninety-nine as new moon; from Qianxing first year renshen, years three million nine hundred six thousand six hundred fifty-eight as accumulated years. The edict supplemented Kui as Director of Calendar Preservation. In addition, selecting the scholar Chu Yan and calendar officials Song Xinggu gathered at the Hall of Heavenly Patterns; an edict ordered Inner Attendant Jin Kelong to supervise calendar making; by Tiansheng first year eighth month it was completed, generally using ten thousand five hundred ninety as pivot divisor, obtaining nine huge myriad numbers. After submission, an edict ordered Hanlin Academician Yan Shu to compose the preface and put it into effect; it was named the Chongtian Calendar. The calendar method states: calculating the upper origin jiazi, from Tiansheng second year jiazi, accumulated years nine thousand seven hundred fifty-five million six thousand three hundred forty. On examination, antiquity upward, subtract one count per year; verifying the future downward, add one count per year.
41
步氣朔
Calculating qi and new moon.
42
《崇天》樞法：一萬五百九十。
Chongtian pivot divisor: ten thousand five hundred ninety.
43
歲周：三百八十六萬七千九百四十。
Year cycle: three million eight hundred sixty-six thousand seven hundred ninety.
44
歲餘：五萬五千五百四十。
Year remainder: fifty-five thousand five hundred forty.
45
氣策：一十五、餘五千三百一十四、秒六。
Qi divisor: fifteen, remainder five thousand three hundred fourteen, parts six.
46
朔實：三十一萬二千七百二十九。
New moon constant: three hundred twelve thousand seven hundred twenty-nine.
47
歲閏：一十一萬五千一百九十二。
Year intercalation: one hundred fifteen thousand one hundred ninety-two.
48
朔策：二十九、餘五千六百一十九。
New moon divisor: twenty-nine, remainder five thousand six hundred nineteen.
49
望策：一十四、餘八千一百四、秒一十八。
Full moon divisor: fourteen, remainder eight thousand one hundred four, parts eighteen.
50
弦策：七、餘四千五十二、秒九。
String constant: seven, remainder 4,052, parts 9.
51
中盈分：四千六百二十八、秒一十二。
Middle surplus constant: 4,628, parts 12.
52
朔虛分：四千九百七十一。
New moon void constant: 4,971.
53
閏限：三十萬三千一百二十九、秒二十四。
Intercalation limit: 303,129, parts 24.
54
秒法：三十六。
Parts divisor: 36.
55
旬周：六十三萬五千四百。
Ten-day cycle constant: 635,400.
56
紀法：六十。
Era divisor: 60.
57
推天正冬至：置距所求積年，以歲周乘之，為氣積分； 滿旬周去之，不盡，以樞法約之為大餘，不滿為小餘。 大餘命甲子，算外，即所求年天正冬至日辰及餘。 若以後合用約分，即以樞法退除為分秒，各以一百為母。
To calculate the heavenly standard winter solstice: take the accumulated years from the sought point, multiply by the year cycle to obtain the qi accumulated parts; remove full ten-day cycles; what remains, divide by the pivot divisor for the great remainder, and what does not fill is the small remainder. Count the great remainder from jiazi, outside the count, and that gives the day, double-hour, and remainder of the heavenly standard winter solstice for the sought year. If using later combined reduced parts, then divide back by the pivot divisor for parts and seconds, each with one hundred as denominator.
58
求次氣：置天正冬至大、小餘，以氣策秒累加之，秒盈秒法從小余，小余滿樞法從大余，滿紀法去之，不盡，命甲子，算外，即各得次氣日辰及餘秒。
To find the next qi: set the great and small remainders of the heavenly standard winter solstice, add the qi divisor and parts cumulatively; when parts fill the parts divisor they advance the small remainder, when the small remainder fills the pivot divisor it advances the great remainder, and when the era divisor is full it is removed; what remains, count from jiazi, outside the count, and each next qi day, double-hour, and remainder in parts and seconds is obtained.
59
推天正十一月經朔：置天正冬至氣積分，朔實去之，不盡為閏餘； 以減天正冬至氣積分，為天正十一月經朔加時積分； 滿旬周去之，不盡，以樞法約之為大餘，不滿為小餘。 大餘命甲子，算外，即所求年天正十一月經朔日辰及餘。
To calculate the heavenly standard eleventh month standard new moon: set the qi accumulated parts of the heavenly standard winter solstice and subtract the new moon constant; what does not fill is the intercalation remainder; subtract this from the qi accumulated parts of the heavenly standard winter solstice to obtain the accumulated parts of the heavenly standard eleventh month standard new moon at the hour; remove full ten-day cycles; what remains, divide by the pivot divisor for the great remainder, and what does not fill is the small remainder. Count the great remainder from jiazi, outside the count, and that gives the day, double-hour, and remainder of the heavenly standard eleventh month standard new moon for the sought year.
60
求弦望及次朔經日：置天正十一月經朔大、小餘，以弦策累加之，去命如前，即各弦、望及次朔經日及餘秒。
To find quarter moons, full moons, and the next standard new moon days: set the great and small remainders of the heavenly standard eleventh month standard new moon, add the string divisor cumulatively, count as before, and each quarter moon, full moon, and next standard new moon day and remainder in parts and seconds is obtained.
61
求沒日：置有沒之氣小餘，三百六十乘之，其秒進一位，從之，用減歲周，余滿歲餘為日，不滿為餘。 命其氣初日，算外，即其氣沒日日辰。 凡二十四氣小余滿八千二百六十五、秒三十以上為有沒之氣。
To find the disappearance day: set the small remainder of the qi that has disappearance, multiply by 360, advance the parts one place and add, use this to subtract from the year cycle; the remainder filling the year remainder gives days, and what does not fill is the remainder. Count from the first day of that qi, outside the count, and that gives the day and double-hour of that qi's disappearance day. Whenever among the twenty-four qi the small remainder reaches 8,265 and parts 30 or above, it is a qi that has disappearance.
62
求減日：置有減經朔小餘，三十乘之，滿朔虛分為日，不滿為餘。 命經朔初日，算外，即為其朔減日日辰。 凡經朔小餘不滿朔虛分為有減之朔。
To find the reduction day: set the small remainder of the standard new moon that has reduction, multiply by thirty; when full new moon void parts it gives days, and what does not fill is the remainder. Count from the first day of the standard new moon, outside the count, and that gives the day and double-hour of that new moon's reduction day. Whenever the standard new moon small remainder does not fill the new moon void parts, it was a new moon that has reduction.
63
步發斂
Calculating emission and retraction.
64
候策：五、餘七百七十一、秒一十四。
Phenological constant: five, remainder 771, parts 14.
65
卦策：六、餘九百二十五、秒二十四。
Hexagram constant: six, remainder 925, parts 24.
66
土王策：三、餘四百六十二、秒三十。
Earth-king constant: three, remainder 462, parts 30.
67
辰法：八百八十二半。
Double-hour divisor: 882.5.
68
刻法：一千五十九。
Quarter divisor: 1,059.
69
秒法：三十六。
Parts divisor: 36.
70
推七十二候：各因中節大、小餘命之，為其氣初候日也； 以候策加之，為次候； 又加之，為末候。
To calculate the seventy-two phenological periods: each assigns from the great and small remainders of the middle node to obtain the first phenological day of that qi; add the phenological divisor to obtain the next phenological period; add again to obtain the last phenological period.
71
求六十四卦：各因中氣大、小餘命之，為公卦用事日； 以卦策加之，得次卦用事日； 以土王策加諸侯之卦，得十有二節之初外卦用事之日。
To find the sixty-four hexagrams: each assigns from the great and small remainders of the middle qi to obtain the day when the duke hexagram takes effect; add the hexagram divisor to obtain the day when the next hexagram takes effect; add the earth-king divisor to the feudal lords' hexagram to obtain the day when the outer hexagram of the twelve nodes takes effect.
72
推五行用事日：各因四立日大、小餘命之，即春木、夏火、秋金、冬水首用事日； 以土王策減四季中氣大、小餘，命甲子，算外，即其月土始用事日。
To calculate the days when the five phases take effect: each assigns from the great and small remainders of the four establishment days to obtain the first day when spring wood, summer fire, autumn metal, and winter water take effect; subtract the earth-king divisor from the great and small remainders of the middle qi of the four seasons, count from jiazi, outside the count, and that gives the day when earth begins to take effect in that month.
73
七十二候及卦日與《應天》同。
The seventy-two phenological periods and hexagram days are the same as in the Yingtian Calendar.
74
求發斂去經朔：置天正十一月閏餘，以中盈及朔虛分累益之，即每月閏餘； 滿樞法除之為閏日，不盡為小餘，即各得其月中氣去經朔日及餘秒。 其餘閏滿閏限至閏，仍先見定朔大小。 其月內無中氣，乃為閏月。
To find emission and retraction from the standard new moon: set the intercalation remainder of the heavenly standard eleventh month, add the middle surplus and new moon void parts cumulatively, and that gives each month's intercalation remainder; divide by the pivot divisor for intercalation days, what does not fill is the small remainder, and each month's middle qi distance from the standard new moon day and remainder in parts and seconds is obtained. Whenever the remainder fills the intercalation limit up to intercalation, still first determine the fixed new moon size. When a month contains no middle qi, it is an intercalary month.
75
求卦候去經朔：各以卦、候策及餘秒累加減之，中氣前以減，中氣後以加。 即各得卦、候去經朔日及餘秒。
To find hexagram and phenological periods from the standard new moon: each uses hexagram and phenological divisors and remainder parts to add and subtract cumulatively; before the middle qi subtract, after the middle qi add. Then each hexagram and phenological period distance from the standard new moon day and remainder in parts and seconds was obtained.
76
求發斂加時：置小餘，以辰法除之為辰數，進一位，滿刻法為刻，不滿為刻分。 其辰數命子正，算外，即各加時所在辰、刻及分。
To find emission and retraction at the hour: set the small remainder, divide by the double-hour divisor for the number of double-hours, advance one place, when full quarter divisor it gives quarters, and what does not fill is quarter parts. Count the number of double-hours from zi midnight, outside the count, and that gives each double-hour, quarter, and part at which the hour occurs.