Volume 81 Treatises 34: Measures and Calendar 14

中原既失，禮樂淪亡。高宗時，胡銓著《審律論》，曰：

Once the Central Plains were lost, the rites and music of the dynasty were lost as well. During the reign of Emperor Gaozong, Hu Quan composed the Treatise on Examining Pitch Standards, in which he wrote:

臣聞司馬遷有言曰：「六律為萬事根本，其于兵械尤所重，望敵知吉凶，聞聲效勝負，百王不易之道也。」臣嘗深愛遷之言律於兵械為尤重，而深惜後之談兵者止以戰鬥、擊刺、奇謀，此律之所以汨陳而學者未嘗道也。

I have heard Sima Qian say: "The six pitch pipes are the foundation of all things; in warfare and weaponry they are especially important—by observing the enemy one can discern good and ill fortune, and by hearing sounds one can tell victory from defeat. This is the unchanging principle of the hundred kings." I have long admired Qian's insistence that pitch standards matter most of all in military affairs, yet I deeply regret that later military theorists speak only of combat, thrusting, and clever stratagems. That is why pitch standards have been obscured in the literature and scholars have scarcely discussed them.

夫律、度、量、衡，古也淵源于馬遷，濫觴于班固，劉昭挹其流，孟康、京房、錢樂之之徒汨其泥而揚其波。遷之言曰：「黃鐘之實八十一以為宮，而以九為法，實如法，得長一寸，則黃鐘為九寸矣。黃鐘之實十七萬七千一百四十七，而以一萬九千六百八十三為法，實如法，亦得長一寸，亦黃鐘為九寸也。然則十七萬七千一百四十七與夫所謂八十一者，雖多少之不同，而其實一也；萬九千六百八十三與夫所謂九者，雖多少之不同，而其法一也。又曰，丑二，寅八，卯十六，辰六十四。夫丑與卯，陰律也；寅與辰，陽律也。生陰律者皆二，所謂下生者倍其實；生陽律者皆四，所謂上生者四其實。遷之言財數百，可謂簡矣，而後之言律者祖焉，是不亦淵源于馬遷乎？

Pitch standards, measures, quantities, and balances: the ancients traced their deepest source to Sima Qian and their first stream to Ban Gu; Liu Zhao drew from their current, while Meng Kang, Jing Fang, Qian Yuezhi, and others stirred the mud and sent the waves higher. Qian writes: "The volumetric standard of Yellow Bell is eighty-one, which serves as the foundation tone; with nine as the divisor, when volume and divisor are in proportion one obtains a length of one inch—thus Yellow Bell measures nine inches. The volumetric standard of Yellow Bell is 177,147, with 19,683 as the divisor; when volume and divisor are in proportion one still obtains a length of one inch—Yellow Bell is still nine inches. Thus 177,147 and the figure called eighty-one differ in magnitude, yet their substance is one and the same; and 19,683 and the figure called nine differ in magnitude, yet their method is one and the same. He also states: for Chou, two; for Yin, eight; for Mao, sixteen; for Chen, sixty-four. Chou and Mao are yin pitch pipes; Yin and Chen are yang pitch pipes. Those that generate yin pipes all use two—what is called "lower generation" doubles the volumetric standard; Those that generate yang pipes all use four—what is called "upper generation" quadruples the volumetric standard. Qian's remarks on pitch standards run to several hundred words and are remarkably concise, yet later theorists of pitch all treat him as their founder—does this not show that the deepest source lies with Sima Qian?

固之言曰：黃鐘之實，八百一十分。蓋遷意也。然以林鐘之實五百四十，而乃以為六百四十，林鐘、太蔟之實以其長自乘，則聲雖有，小同于黃鐘之宮耳。然則魏柴玉制律，而與黃鐘商、徵不合，其失兆此矣。夫自子一分，終於亥十七萬七千一百四十七分，蓋遷術也。而固亦曰，太極元氣，函三為一，始動於子，參之於丑，歷十二辰之數，而得黃鐘之實，以為陰陽合德，化生萬物。其說蓋有本矣。然其言三分蕤賓損一，下生大呂，而不言夫所謂濁倍之變何？夫蕤賓之比于大呂，則蕤賓清而大呂濁，今又損二分之一以生大呂，則大呂之聲乃清於蕤賓，是不知夫倍大呂之濁。然則蕭衍之論，至於夾鐘而裁長三寸七分，其失兆此矣。是不亦濫觴于班固乎？

Ban Gu states: The volumetric standard of Yellow Bell is 810 parts. This follows Sima Qian's intent. Yet he gives the volumetric standard of Forest Bell as 540 but treats it as 640; when the standards of Forest Bell and Great Cluster are multiplied by their respective lengths, tones are produced, but they differ only slightly from the Yellow Bell foundation tone. Thus when Chai Yu of Wei fashioned pitch pipes, they failed to match the commercial and fifth tones derived from Yellow Bell—the root of his error lay here. From one part at Zi to 177,147 parts at Hai—this is Sima Qian's computational method. Ban Gu likewise writes that the primordial qi of the Supreme Ultimate, containing three within one, first stirs at Zi, is tripled at Chou, passes through the twelve earthly branches, and thereby yields the volumetric standard of Yellow Bell—by which yin and yang unite in virtue to generate the myriad things. His theory does have a foundation. Yet he describes diminishing Flaring Guest by one-third to lower-generate Great Clarity, without explaining what is meant by the muddy doubled variant—why? Relative to Great Clarity, Flaring Guest is the higher tone and Great Clarity the lower; yet if one diminishes by half again to generate Great Clarity, Great Clarity would sound higher than Flaring Guest—this shows ignorance of the muddy doubled form of Great Clarity. Thus in Xiao Yan's theory, when he reached Pinched Bell he cut the length to three inches and seven-tenths—the root of his error lay here. Does this not also trace its first stream to Ban Gu?

昭之言曰：推林鐘之實至十一萬八千九十八、太蔟之實至十五萬七千四百六十四，二乘而三約之者，為下生之實；四乘而三約之者，為上生之實。此遷、固之意，昭則詳矣。然以蕤賓為上生大呂，而大呂乃下生夷則，何也？蓋昭之說陽生陰為下生，陰生陽為上生。今以蕤賓為上生大呂，則是陽生陰，乃上生也；以大呂為下生夷則，是陰生陽，乃下生也。其蔽亦由不知夫大呂有濁倍之變，則其視遷、固去本遠矣。是不亦挹其流于劉昭乎？

Liu Zhao writes: Projecting the volumetric standard of Forest Bell to 118,098 and of Great Cluster to 157,464—what is multiplied by two and reduced by three yields the standard for lower generation; what is multiplied by four and reduced by three yields the standard for upper generation. This is the intent of Sima Qian and Ban Gu; Liu Zhao elaborated it in full. Yet he has Flaring Guest upper-generate Great Clarity, while Great Clarity lower-generates Leveling Tone—how can this be? Liu Zhao's theory holds that when yang generates yin it is lower generation, and when yin generates yang it is upper generation. Now if Flaring Guest upper-generates Great Clarity, yang generating yin is treated as upper generation; and if Great Clarity lower-generates Leveling Tone, yin generating yang is treated as lower generation. His error likewise stems from ignorance of Great Clarity's muddy doubled variant—thus his reading of Sima Qian and Ban Gu strays far from the root. Does this not also show him drawing from Liu Zhao's current?

若夫孟康、京房、錢樂之之徒，則又大不然矣。夫班固以八十一分為黃鐘之實，起十二律之周徑，度其長以容其實，初末嘗有徑三圍九之說也。康之徒惑於八十一分之實，以一寸為九十分，而不察方圜之異，於是有徑三圍九之論興焉。天律之形圜，如以為徑三圍九，則刓其四用之方，而不足於九分之數，以之容黍，豈能至於千二百哉！然則所謂圍九，方分也。何以知之？知龠之方，則知黃鐘之分亦方也。固雖無明說，其論洛下閎起曆之法曰：「律容一龠，積八十一寸，則一日之分也。」夫八十一寸者，是乃八百一十分，以千二百黍納之龠中，則不搖而自滿，是無異黃鐘之容也。龠之制，方寸而深八分。一龠之方，則黃鐘之分，安得而不方哉！圍九方分而圜之，則徑不止於三分矣。故夫徑三圍九之說，孟康為之也。

As for Meng Kang, Jing Fang, Qian Yuezhi, and their followers, they were even further astray. Ban Gu took eighty-one parts as the volumetric standard of Yellow Bell, derived the circumference and diameter of the twelve pitch pipes, and measured their length to contain their volume—he never advanced the theory of "diameter three, circumference nine." Meng Kang and his followers, misled by the eighty-one-part standard, treated one inch as ninety parts without noticing the difference between square and circular measures—thus the "diameter three, circumference nine" theory arose. The pitch pipes of heaven are circular; if one applies "diameter three, circumference nine," one cuts away the square usable on four sides and falls short of the ninety-part measure—how could such a vessel hold 1,200 grains of millet! Thus what is called "circumference nine" refers to square parts. How do we know this? Once one knows the yue measure is square, one knows the fractional parts of Yellow Bell must be square as well. Ban Gu offers no explicit statement, but in discussing Luoxia Hong's method for establishing the calendar he writes: "The pitch pipe holds one yue; accumulated to eighty-one inches, this yields one day's fractional measure." Eighty-one inches are in fact 810 parts; when 1,200 grains of millet are placed in the yue, it fills without shaking—no different from the capacity of Yellow Bell. The yue is one inch square and eight-tenths of an inch deep. If a single yue is square, how could the fractional parts of Yellow Bell be anything but square! If "circumference nine" square parts were made circular, the diameter would exceed three parts. Thus the "diameter three, circumference nine" theory was Meng Kang's invention.

然由律生呂，數十有二，止矣；京氏演為六十，錢樂之廣為三百六十，則與黃帝之說悖矣。蓋樂之用《淮南》之術，一律而生五音，十二律而為六十音，而六之，故三百六十音，以當一歲之日。以黃鐘、太蔟、姑洗、林鐘、南呂生三十有四，以大呂、夾鐘、中呂、蕤賓、夷則、無射生二十有七，應鐘生二十有八，始于包育，而終於安運。然由黃鐘迄於壯進百有五十，則三分損一焉以下生；由依行迄於億兆二百有九，則三分益一焉以上生；惟安運為終而不生。其言與黃帝之法大相牴牾。自遷、固而下，至是雜然莫適為主，至五季王樸而後議少定，沈括、蔣之奇論之當矣。是不亦汨其泥而揚其波乎？

Yet from the pitch standards the twelve pitch pipes are generated, and there the count stops; Jing Fang's school expanded them to sixty, and Qian Yuezhi broadened them to 360—both contradict the Yellow Emperor's doctrine. Music adopted the method of the Huainanzi: one pitch standard generates five tones; twelve standards yield sixty tones; multiplied by six, this gives 360 tones to correspond to the days of the year. Yellow Bell, Great Cluster, Maiden Wash, Forest Bell, and Southern Flaring generate thirty-four; Great Clarity, Pinched Bell, Middle Clarity, Flaring Guest, Leveling Tone, and Unpitched generate twenty-seven; Responding Bell generates twenty-eight—beginning with Bao Yu and ending with An Yun. From Yellow Bell through Zhuang Jin there are 150 tones generated by diminishing one-third for lower generation; from Yi Xing through Yi Zhao there are 209 tones generated by augmenting one-third for upper generation; only An Yun is terminal and generates no further tone. Their doctrine sharply contradicts the Yellow Emperor's method. From Sima Qian and Ban Gu onward, theories proliferated with no settled authority until Wang Pu in the Five Dynasties brought debate somewhat to rest; Shen Kuo and Jiang Zhiqi then argued the matter correctly. Did they not stir the mud and send the waves still higher?

嗚呼！律也者，固以實為本而法為末，陛下修其實于上，而有司方定其法於下，以協天地中和之聲，則夫數子者，其說有可考焉，臣敢輕議哉！

Alas! Pitch standards take volumetric substance as fundamental and computational method as secondary; Your Majesty is refining the substance above while officials below are establishing the method, to harmonize with the balanced tones of heaven and earth. The theories of those masters may then be examined—how dare I presume to judge them lightly!

淳熙間，建安布衣蔡元定著《律呂新書》，朱熹稱其超然遠覽，奮其獨見，爬梳剔抉，參互考尋，推原本根，比次條理，管括機要，闡究精微。其言雖多出於近世之所未講，而實無一字不本于古人之成法。其書有《律呂本原》、《律呂證辨》。《本原》者，《黃鐘》第一，《黃鐘之實》第二，《黃鐘生十二律》第三，《十二律之實》第四，《變律》第五，《律生五聲圖》第六，《變聲》第七，《八十四聲圖》第八，《六十調圖》第九，《候氣》第十，《審度》第十一，《嘉量》第十二，《謹權量》第十三。《證辨》者，《造律》第一，《律長短圍徑之數》第二，《黃鐘之實》第三，《三分損益上下相生》第四，《和聲》第五。權臣既誣元定以偽學，貶死舂陵，雖有其書，卒為空言，嗚呼惜哉！

During the Chunxi era, Cai Yuanding of Jian'an, a man of no official rank, wrote the New Book on Pitch Pipes. Zhu Xi praised his far-reaching vision, his bold independent insights, his meticulous sifting of evidence, his cross-comparison of sources, his tracing of doctrines to their roots, his orderly arrangement of topics, his grasp of essentials, and his exposition of subtle points. Though much of what he wrote addressed topics neglected in recent times, not a single word failed to rest on the established methods of antiquity. The work comprises Origins of Pitch Pipes and Evidential Analysis of Pitch Pipes. Origins contains thirteen chapters: Yellow Bell; the Volumetric Standard of Yellow Bell; Yellow Bell Generates the Twelve Pitch Pipes; the Standards of the Twelve Pipes; Variant Pitch Pipes; Diagram of Pitch Pipes Generating the Five Tones; Variant Tones; Diagram of Eighty-four Tones; Diagram of Sixty Modes; Awaiting Qi; Examining Measures; Fine Capacity; and Careful Weights and Measures. Evidential Analysis contains five chapters: Making Pitch Pipes; Numbers for Length, Circumference, and Diameter; the Volumetric Standard of Yellow Bell; Three-part Diminishment and Augmentation with Upper and Lower Generation; and Harmonious Sounds. Powerful ministers denounced Yuanding for heterodox learning; he was exiled and died at Chunling. Though his book survives, it remained mere words on the page—alas, what a loss!

久之，宜春歐陽之秀復著《律通》，其自序曰：

Long afterward, Ouyang Zhixiu of Yichun wrote Comprehensive Pitch Theory, in whose preface he states:

自律呂之度數不見於經，而釋經者反援《漢志》以為據，蓋濫觴于《管子》、《呂氏春秋》，流衍於《淮南子》、司馬遷之書，而波助于劉歆、京房之學。班固《漢志》，盡歆所出也；《司馬彪志》，盡房所出也。後世協律者，類皆執守以為定法。歷代合樂，不為無人，而終不足以得天地陰陽之和聲，所以不能追還于隆古之盛者，大抵由三分損益之說拘之也。夫律固不能舍損益之說以求之，由其有損有益，而後有上生下生之異。至其專用三分以為損益之法則失之，未免乎聲與數之不相合，有非天成之自然耳。

Because the numerical measures of pitch pipes do not appear in the canonical classics, commentators have instead cited the Han Treatises as authority—a doctrine that first trickled from Guanzi and Lüshi Chunqiu, spread through Huainanzi and Sima Qian, and was carried forward by Liu Xin and Jing Fang. Ban Gu's Han Treatises were entirely the work of Liu Xin; Sima Biao's Treatises were entirely the work of Jing Fang. Later harmonizers of pitch pipes all clung to these as fixed law. Each dynasty had its experts in harmonizing music, yet none succeeded in attaining the balanced tones of heaven and earth's yin and yang; the reason they could not restore the splendor of high antiquity lay largely in their bondage to the three-part diminishment-and-augmentation theory. Pitch pipes cannot be derived without diminishment and augmentation; it is through diminishment and augmentation that upper and lower generation arise. But to rely exclusively on one-third as the method of diminishment and augmentation is an error; sound and number inevitably fail to align, falling short of what heaven endowed by nature.

蓋嘗因其損益、上下生之義，而去其專用三分之蔽，乃多為分法以求之，自黃鐘以往，其下生者盈十，而上生者止一而已。此其數之或損或益，出於自然，而與舊法固不侔矣。若謂相生之法，一下必一上，既上而復下，則其法之窮也，於蕤賓、大呂間見之。夫黃鐘而降，轉以相生，至於姑洗則下生應鐘，而應鐘之上生蕤賓者，法也。今乃蕤賓之生大呂，又從而上生焉，此《班志》所載，所以變其說為下生大呂，而大呂之長遂用倍法矣。夫律之相生而用倍法，猶為有理，獨專用三分以為損益，則律之長短，不中乎天地自然之數爾。

I have followed the principles of diminishment, augmentation, and upper and lower generation while discarding the error of relying solely on one-third, and devised many fractional methods of calculation. From Yellow Bell onward, lower generations number more than ten, while upper generation occurs only once. Whether these numbers are diminished or augmented, they arise from nature and are fundamentally unlike the old method. If one insists that generation must alternate lower and upper without exception, the method reaches its limit—and that limit appears between Flaring Guest and Great Clarity. From Yellow Bell downward, pipes are generated in succession; at Maiden Wash one lower-generates Responding Bell, and Responding Bell upper-generates Flaring Guest—this is the standard method. Yet here Flaring Guest generates Great Clarity by upper generation again—this is what Ban's Treatises record, which is why they revised the theory to lower-generate Great Clarity and applied the doubling method to Great Clarity's length. Using the doubling method in pitch generation is still defensible; it is exclusive reliance on one-third for diminishment and augmentation that makes pipe lengths fail to match heaven and earth's natural proportions.

生律之分，蓋不止於三分損益之一端，以一律而分為三，此生律之極數，特一求徵聲之法耳。苟以三分損益，一下生而一上生，則聲律殆無窮矣，何至於十二而止也乎。夫十二律之生也，十律皆下生，一律獨上生。唯其下生者，損之極也，而後上生者益焉。上生則律窮矣，此窮上反下、窮下反上之理也。琴一弦之間具十二律，皆用下生之法，而末以上生法終之。若以七弦而緊慢之為旋宮之法，則應鐘一均之律，宮聲之外，多用倍法生一律矣。此天地聲音自然而然，不可拘於一而不知通變也。故正律止於十二而已。

The fractional divisions for generating pitch pipes are not limited to three-part diminishment and augmentation alone; dividing one pipe into three parts is the ultimate fractional division—merely one method for deriving the fifth tone. If one applied three-part diminishment and augmentation with strict alternation of lower and upper generation, tones would be nearly infinite—how could the series stop at twelve? Of the twelve pitch pipes, ten are generated by lower generation and only one by upper generation. Only when lower generation has diminished to the extreme does upper generation augment. Upper generation then exhausts the series—this is the principle that when the upper limit is reached one returns downward, and when the lower limit is reached one returns upward. On a single qin string all twelve pitch pipes are present, all produced by lower generation, with upper generation applied only at the end. If the seven strings are tightened and loosened as the method of mode rotation, then within the Responding Bell scale, beyond the foundation tone the doubling method is often used to generate additional pipes. This reflects how the sounds of heaven and earth arise naturally—one cannot cling to a single fixed rule without understanding adaptation. Thus the canonical pitch pipes number twelve and no more.

竊意十二律之度數，當具于《周禮》之《冬官》，如《考工記》鳧氏為鐘、磬氏為磬之類，各有一職。然《冬官》一篇既亡，則世無以考其度數之詳，而三分損益之說散見於書傳者，恐或得之目擊而不及識其全，或得之口授而未能究其誤，或求諸耳決而不能究其真，因是遂著為定論。夫人皆以為法之盡善矣，豈知三分損益所生之律，乃僅得其聲之近似而未真。蓋非師曠之聰，則耳不能齊，其聲之近似者，足以惑人之聽，是以不復求其法之未盡善者。此蔡邕所以不如耳決之明者，亦不能盡信其法也。

I suspect the numerical measures of the twelve pitch pipes were fully recorded in the Winter Offices section of the Rites of Zhou, as in the Artificers' Record where the Wild-Goose clan fashioned bells and the Chime-stone clan fashioned sonorous stones—each craft had its specialist. But with the Winter Offices chapter lost, the world could no longer verify the detailed measures. The three-part diminishment-and-augmentation theory scattered through transmitted texts may have been grasped by eyewitness without understanding the whole, received orally without tracing errors, or judged by ear without reaching the truth—thus it was enshrined as settled doctrine. Everyone assumed the method was perfect—who knew that pitch pipes generated by three-part diminishment and augmentation only approximate the true sound? Without Shi Kuang's acuity of hearing, the ear cannot discriminate precisely; sounds that merely approximate are enough to deceive the listener—so no one sought to improve a method that was not truly perfect. This is why even Cai Yong, who fell short of the clarity of ear-based judgment, could not fully trust the method.

後世之制樂者，不知律法之固有未善，而每患其聲音高下之不協，以至取古昔遺亡之器而求之，蓋亦不知本矣。聲以數而傳，數以聲而定，二者皆有自然之則。如侈者聲必咋，弇者聲必郁，高者數必短，下者數必長。侈弇者，數也，未聞其聲而已知其有咋鬱之分；高下者，聲也，未見其數已知其有長短之異。故不得其自然之聲，則數不可得而考；不得其自然之數，則聲不可得而言。今之制律者，不知出此，而顧先區區於秬黍之縱橫、古尺之修短、斛鬥之廣狹、鐘磬之高下謀之，是何足以得其聲之和哉！

Later composers of music, unaware that pitch theory was inherently flawed, worried instead that high and low tones failed to harmonize, and even sought out ancient instruments long lost—missing the root of the problem entirely. Sound is conveyed through number, and number is established through sound—each follows natural principles. An enlarged pipe produces a harsh tone; a constricted pipe a muffled one; a high tone corresponds to a short measure; a low tone to a long one. Enlargement and constriction are matters of measure—before hearing the sound one can tell whether it will be harsh or muffled; high and low are matters of sound—before examining the measure one knows whether it will be long or short. Thus without obtaining the natural sound, the numerical measure cannot be verified; without obtaining the natural measure, sound cannot be properly described. Today's pipe-makers fail to grasp this principle, yet fuss over the orientation of black millet grains, the length of ancient rulers, the capacity of hu and dou measures, and the height of bells and chime-stones—how could such concerns ever yield harmonious sound!

邵雍曰：「世人所見者，漢律曆耳。」然則三分損益之法為未善，亦隱然矣。近世蔡元定特著一書，可謂究心，然其說亦有可用與否。其可用者，多其所自得，而又有證于古，凡載於吾書者可見矣；其否者，皆由習熟於三分上下生之說，而不於聲器之近似者察之也。豈嘗察之而未有法以易之乎？此《律通》之所以作也。

Shao Yong remarked: "What the world knows is only the Han pitch standards and calendar." Thus it is implicitly clear that the three-part diminishment-and-augmentation method is imperfect. In recent times Cai Yuanding devoted himself to a special treatise, which shows genuine dedication; yet his theories too have their sound and unsound parts. What is sound in his work largely reflects his own insights, corroborated by antiquity, as my book demonstrates; what is unsound stems from his familiarity with three-part upper-and-lower generation theory, without scrutinizing how closely sounds and instruments actually match. Had he truly examined the matter and found no method to replace it? This is why I wrote Comprehensive Pitch Theory.

蓋律之所以長短，不止乎三分損益之一端，自四分以往，推而至於有二十分之法。管之所以廣狹，必限於千二百黍之定數，因其容受有方分、圜分之異，與黍體不相合，而遂分辨其空龠有實積、隙積之理。其還相為宮之法，有以推見其為一陰一陽相繼之道，而非一上一下相生之謂也。

Pipe lengths are determined by more than three-part diminishment and augmentation alone—from four parts onward, the method extends even to twenty-part divisions. Tube width must be tied to the fixed count of 1,200 millet grains; because capacity differs between square and circular measures and does not match the grain's form, one must distinguish between solid volume and interstitial volume within the empty yue. The method of mode rotation reveals a succession of yin and yang tones—not mere alternation of upper and lower generation.

嗟乎！觀吾書者，能知其數之出於自然而然，則知由先漢以前至於今日，上下幾二千年，凡史傳所述三分損益一定之說者，可以刪而去之矣。使其說之可用也，則累世律可協、樂可和，何承天、劉焯輩不改其法矣。故京房六十律不足以和樂，而況錢樂之衍為三百六十之非法，徒增多而無用乎？是其數非出於自然之無所加損，而徒欲傅會於當期之日數雲爾。

Alas! Readers who grasp that these numbers arise from natural spontaneity will see that for nearly two thousand years, from pre-Han times to the present, all historical accounts treating three-part diminishment and augmentation as fixed doctrine may be discarded. If that theory were sound, successive generations would have achieved harmonious pitch and music—why then did He Chengtian, Liu Zhuo, and others revise the method? Jing Fang's sixty pitch pipes failed to harmonize music—how much less useful was Qian Yuezhi's expansion to an irregular 360, which merely multiplied numbers to no purpose? Their numbers did not arise from nature without addition or diminution, but merely forced a fit with the day-count of their own era.

古之聖人所以定律止於十二者，自然之理數也。苟不因自然之理數，則以三分損益之法衍之，聲律殆不特三百六十而已也，而況京房之六十乎！且房之律，吾意其自為之也，而托言受之焦延壽，以欺乎人，以售其說。使律法之善，何必曰受諸人？律法不善矣，雖焦延壽何益哉！所謂善不善者，亦顧其法之可用與否耳。曩者，魏漢津嘗創用指尺以制律，乃竊京房之故智，上以取君之信，下以遏人之議，能行之於一日，豈能使一世而用之乎？

The reason the ancient sages fixed pitch pipes at twelve lies in the natural order of numbers. Without following natural numerical principles, extending the three-part method would yield far more than 360 tones—let alone Jing Fang's sixty! Moreover, I suspect Jing Fang devised his pitch pipes himself, then claimed to have received them from Jiao Yanshou in order to deceive the public and promote his theory. If pitch theory is sound, why claim it was received from another? If pitch theory is unsound, what good is Jiao Yanshou's authority? Whether a theory is sound depends solely on whether its method works. Formerly Wei Hanjin innovated by using finger-length to fashion pitch pipes—merely stealing Jing Fang's old trick to win the ruler's trust above and silence critics below. He could enforce it for a day, but could he make the world adopt it?

今《律通》之作，其數之損益可以互相生，總為百四十四以為之體，或變之，又可得二百一十有六以為之用，乾坤之策具矣。世不用則已，用則聲必和，亦因古黃鐘九寸法審之，以人物之聲而稍更定之耳。或曰：律止十二，胡為復衍百四十四律乎？」應之曰：「十二者，正聲也；百四十四者，變聲也。使不為百四十四者，何以見十二宮七聲長短之有定數，而宮、商、角、徵、羽清濁之有定分乎？其要主於和而已。故有正聲則有變聲也，通其變然後可與論律矣。」

In Comprehensive Pitch Theory, the diminished and augmented numbers mutually generate: 144 form the substance, and with transformation yield 216 for practical use—the numbers of Qian and Kun are complete. If the world ignores it, so be it; if adopted, sound will harmonize. The method follows the ancient nine-inch Yellow Bell standard, adjusted slightly according to actual human and instrumental tones. Someone asked: "Pitch pipes number only twelve—why derive 144 additional pipes?" He replied: "Twelve are the primary tones; 144 are variant tones. Without the 144, how could one discern the fixed proportions of length among the twelve modes and seven tones, or the fixed distinctions of clear and muddy among gong, shang, jue, zhi, and yu? The essential point is harmony alone. Where there are primary tones there must be variant tones; only by mastering the variants can one truly discuss pitch theory."

《律通》上、下二篇：《十二律名數》第一，《黃鐘起數》第二，《生律分正法》第三，《生律分變法》第四，《正變生律分起演算法》第五，《十二宮百四十四律數》第六，《律數傍通法》第七，《律數傍通別法》第八；《九分為寸法辨》第九、第十，《五十九律會同》第十一，《空圍龠實辨》第十二，《十二律分陰陽圖說》第十三，《陽聲陰聲配乾坤圖》第十四，《五聲配五行之序》第十五，《七聲配五行之序》第十六，《七聲分類》第十七，《十二宮七聲倡和》第十八，《六十調圖說》第十九，《辨三律聲法》第二十。真德秀、趙以夫皆盛稱之。

Comprehensive Pitch Theory comprises two parts. The first eight chapters are: Names and Numbers of the Twelve Pitch Pipes; Starting Numbers from Yellow Bell; Correct Method of Fractional Generation; Variant Method of Fractional Generation; Algorithm for Deriving Correct and Variant Fractions; Numbers of 144 Pipes in Twelve Modes; Method of Interpenetrating Pitch Numbers; and Separate Method of Interpenetrating Pitch Numbers; followed by: Discourse on Nine Parts per Inch (chapters 9–10); Concordance of Fifty-nine Pitch Pipes; Discourse on Empty Circumference and Yue Volume; Diagram of Twelve Pipes Divided into Yin and Yang; Diagram Matching Yang and Yin Tones to Qian and Kun; Sequence of Five Tones and the Five Phases; Sequence of Seven Tones and the Five Phases; Classification of Seven Tones; Mutual Responsiveness of Twelve Modes and Seven Tones; Diagram of Sixty Modes; and Discerning Three Pitch Sound Methods. Zhen Dexiu and Zhao Yifu both praised it highly.

舒州桐城縣丞李如篪作《樂書》，評司馬光、範鎮所論律，曰：

Li Ruchi, assistant magistrate of Tongcheng in Shuzhou, wrote the Book of Music, in which he evaluates Sima Guang and Fan Zhen's debates on pitch standards:

鎮得蜀人房庶言尺法，庶言：「嘗得古本《漢書》，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。 '今文脫去'之起積一千二百黍'八字，故自前世累黍為之，縱置之則太長，橫置之則太短。今新尺橫置之不能容一千二百黍，則大其空徑四厘六毫，是以樂聲太高，皆由儒者誤以一黍為一分，其法非是。不若以千二百黍實管中，隨其短長斷之，以為黃鐘九寸之管九十分，其長一為一分，取三分以度空徑，數合則律正矣。」鎮盛稱此論，以為先儒用意皆不能到。其意謂制律之法，必以一千二百黍實黃鐘九寸之管九十分，其管之長一為一分，是度由律起也。光則據《漢書》正本之「度起于黃鐘之長。以子穀秬黍中者，一黍之廣，度之九十分，黃鐘之長，一為一分。」本無「之起積一千二百黍」八字。其意謂制律之法，必以一黍之廣定為一分，九十分則得黃鐘之長，是律由度起也。

Fan Zhen obtained the foot-measure method from Fang Shu of Shu, who said: "I once obtained an ancient edition of the Book of Han stating: 'Measure arises from the length of Yellow Bell; using medium black millet from Zi Valley, the height of one grain, accumulated to the breadth of 1,200 grains, measured to ninety parts—the length of Yellow Bell, one part. The present text omits the eight characters 'height accumulated to 1,200 grains.' Hence from former times people stacked grains to measure; placed vertically the result was too long, horizontally too short. The new foot measure laid horizontally cannot hold 1,200 grains, so they enlarged the inner diameter by 0.046 units—making musical tones too high—all because scholars mistakenly treated one grain as one part. Their method is wrong. Better to fill the pipe with 1,200 grains, cut it to the appropriate length, and take the nine-inch Yellow Bell pipe as ninety parts, one part equaling one unit of length; use three parts to measure the inner diameter—when the numbers align, the pitch is correct." Fan Zhen praised this theory highly, believing earlier Confucian scholars had never grasped the point. His view is that pitch pipes must be made by filling the nine-inch Yellow Bell pipe (ninety parts, one part equaling one unit of length) with 1,200 grains—measure arises from pitch. Sima Guang relied on the standard edition of the Book of Han: "Measure arises from the length of Yellow Bell. Using medium black millet from Zi Valley, the breadth of one grain, measured to ninety parts—the length of Yellow Bell, one part." The original text contains no phrase 'height accumulated to 1,200 grains.' His view is that pitch pipes must be made by fixing the breadth of one grain as one part; ninety parts yield the length of Yellow Bell—pitch arises from measure.

《書》云：「同律、度、量、衡。」先言律而後及度、量、衡，是度起於律，信矣。然則鎮之說是，而光之說非也。然庶之論積一千二百黍之廣之說則非，必如其說，則是律非起於度而起於量也。光之說雖非先王作律之本，而後之為律者，不先定其分寸，亦無以起律。又其法本之《漢志》之文，則光之說亦不得謂其非是也。

The Book of Documents states: "Harmonize pitch, measure, quantity, and weight." Pitch is named before measure, quantity, and weight—measure arises from pitch, beyond doubt. Thus Fan Zhen is right and Sima Guang wrong. Yet Fang's theory of accumulating the breadth of 1,200 grains is wrong; if accepted, pitch would arise not from measure but from quantity. Sima Guang's method, though not how the ancient kings originally fashioned pitch pipes, is nonetheless necessary for later makers who cannot establish pitch without first fixing fractional measures. Moreover, since his method rests on the Han Treatises, Sima Guang's view cannot simply be dismissed as wrong.

故嘗論之，律者，述氣之管也。其候氣之法，十有二月，每月為管，置於地中。氣之來至，有淺有深，而管之入地者，有短有長。十二月之氣至，各驗其當月之管，氣至則灰飛也。其為管之長短，與其氣至之淺深，或不相當則不驗。上古之聖人制為十二管，以候十二辰之氣，而十二辰之音亦由之而出焉。以十二管較之，則黃鐘之管最長，應鐘之管至短；以林鐘比于黃鐘，則短其三分之一；乙太簇比之林鐘，則長其三分之一；其餘或長或短，皆上下於三分之一之數。其默符于聲氣自然之應者如此也，當時惡睹所謂三分損益哉！又惡睹夫一千二百黍實黃鐘容受之量與夫一黍之廣一為一分之說哉！古之聖人既為律矣，欲因之以起度、量、衡之法，遂取秬黍之中者以實黃鐘之管，滿龠傾而數之，得黍一千有二百，因以制量；以一黍之廣而度之，得黃鐘管九十分之一，因以起度；以一龠之黍之重而兩之，因以生衡。去古既遠，先王作律之本始，其法不傳，而猶有所謂一千二百黍為一龠容受之量與夫一黍之廣一為一分者可考也。推其容受而度其分寸，則律可得而成也。先王之本於律以起度、量、衡者，自源而生流也；後人以度、量、衡而起律者，尋流而及源也。

I have argued that pitch pipes are tubes that convey qi. The method of awaiting qi employs twelve tubes, one for each month, buried in the earth. Qi arrives at varying depths, and the tubes are buried to varying lengths. When each month's qi arrives, the corresponding tube is tested—when qi arrives, the ash flies up. If the tube's length does not match the depth at which qi arrives, the test fails. The sages of high antiquity fashioned twelve tubes to await the qi of the twelve earthly branches, and the tones of those branches issued from them. Comparing the twelve tubes, Yellow Bell's is longest and Responding Bell's shortest; Forest Bell is one-third shorter than Yellow Bell; Great Cluster is one-third longer than Forest Bell; the remaining pipes vary in length, all by increments of one-third. They silently matched the natural correspondence of sound and qi—how could they have known anything of three-part diminishment and augmentation! How could they have known the theory of 1,200 grains filling Yellow Bell, or of one grain's breadth as one part! Having fashioned pitch pipes, the ancient sages wished to derive measure, quantity, and weight from them. They took medium black millet, filled the Yellow Bell pipe, poured it into the yue and counted—obtaining 1,200 grains—and thereby established quantity; measured the breadth of one grain to obtain one-ninetieth of the Yellow Bell pipe, and thereby established measure; doubled the weight of millet in one yue, and thereby established weight. The original method by which the former kings fashioned pitch pipes is lost, yet the capacity of 1,200 grains per yue and the breadth of one grain as one part can still be traced. By inferring capacity and measuring fractional dimensions, pitch pipes can be fashioned. The former kings derived measure, quantity, and weight from pitch—flowing from source to stream; later scholars who establish pitch from measure, quantity, and weight trace the stream back to the source.

光、鎮爭論往復，前後三十年不決，大概言以律起度，以度起律之不同。鎮深辟光以度起律之說，不知後世舍去度數，安得如古聖人默符聲氣之驗，自然而成律也哉？至若庶之增益《漢志》八字以為脫誤，及其他紛紛之議，皆穿鑿以為新奇，雖鎮力主之，非至當之論有補於律法者也。

Sima Guang and Fan Zhen debated back and forth for thirty years without resolution—the dispute centered on whether measure arises from pitch or pitch from measure. Fan Zhen fiercely rejected Sima Guang's theory of pitch-from-measure, unaware that without numerical measures, later generations could never achieve the ancient sages' silent verification of sound and qi, by which pitch arose naturally. Fang Shu's addition of eight characters to the Han Treatises as supposed omissions, and other disputed points, are forced novelties. Though Fan Zhen championed them, they are not sound contributions to pitch theory.

如篪書曰《樂本》，曰《樂章》。

Ruchi's book includes Origins of Music and Chapters of Music.

沙隨程迥著《三器圖議》，曰：「體有長短，所以起度也；受有多寡，所以生量也；物有輕重，所以用權也。是器也，皆准之上黨羊頭山之秬黍焉。以之測幽隱之情，以之達精微之理。推三光之運，則不失其度；通八音之變，則可召其和。以辨上下則有品，以分隆殺則有節。凡朝廷之出治，生民之日用，未有頃刻不資焉者也。古人以度定量，以量定權，必參相得，然後黃鐘之律可求，八音五聲從之而應也。皇祐中，阮逸、胡瑗累黍定尺，既大於周尺，姑欲合其量也，然竟于權不合，乃謂黍稱二兩，已得官稱一兩，反疑史書之誤。及韓琦、丁度詳定，知逸、瑗之失，亦莫能以三器參相考也。」

Cheng Jiong of Shasui wrote Illustrated Discourse on the Three Instruments, stating: "Bodies have length and shortness, by which measure is established; capacity has more and less, by which quantity is generated; things have lightness and heaviness, by which weight is employed. These instruments all take as their standard the black millet of Yangtou Mountain in Shangdang. With them one probes hidden truths and reaches subtle principles. Projecting the motions of sun, moon, and stars, one does not miss their proper measure; mastering the transformations of the eight timbres, one can summon harmony. To distinguish high and low there are grades; to mark prosperity and decline there are rhythms. Court governance and daily life alike depend on them at every moment. The ancients fixed quantity by measure and weight by quantity; all must align. Only then can Yellow Bell pitch be determined and the eight timbres and five tones respond in harmony. During Huangyou, Ruan Yi and Hu Yuan stacked millet to fix the foot measure. Though larger than the Zhou foot, they tried to match quantity—but weight still failed to align. They claimed two ounces of millet equaled one official ounce and suspected the historical records were wrong. When Han Qi and Ding Du made a detailed review, they recognized Yi and Yuan's error, yet still could not reconcile the three instruments by cross-comparison."

先是，鎮上封事曰：「樂者，和氣也；發和氣者，音聲也。音聲生於無形，故古人以有形之物傳其法，俾後人參考之。有形者何？秬黍也、律也、尺也、龠也、鬴也、斛也、算數也、權稱也、鐘也、磬也，是十者必相合而不相戾，而後為得也。」迥謂：「以黍定三器，則十者無不該。三者，尺為之本。周尺也者，先儒考其制，吻合者不一。至宋祁取《隋書》大業中歷代尺十五等，獨以周尺為之本，以考諸尺。韓琦嘉祐累黍尺二，其一亦與周尺相近。司馬備刻之于石。光舊物也。苟以是定尺，又以是參定權量，以合諸器，如挈裘而振其領，其順者不可勝數也。」

Earlier, Fan Zhen submitted a memorial stating: "Music is harmonious qi; what releases harmonious qi is sound; Sound arises from the formless; the ancients therefore transmitted their method through tangible objects for later generations to consult. What are these tangible objects? Black millet, pitch pipes, the foot measure, the yue, the fu, the hu, counting rods, the balance, bells, and chime-stones—these ten must align without contradiction before the system is complete. Jiong replied: "Fix the three instruments by millet, and all ten are encompassed. Of the three, the foot measure is fundamental. Former scholars examined the Zhou foot measure, but their reconstructions did not agree. Song Qi drew on the Book of Sui's fifteen grades of historical foot measures from the Daye era, taking the Zhou foot alone as the standard for comparison. Han Qi's two Jiayou millet-stacked foot measures—one of which also approximates the Zhou foot. Sima Guang had them carved in stone for preservation. These were Guang's own artifacts. If the foot is fixed thereby, and weight and quantity cross-calibrated to harmonize all instruments—like lifting a fur coat by its collar—the resulting alignments would be beyond counting."

迥博學好古，朱熹深禮敬之。其後江陵府學教授廬陵彭應龍，既注《漢·律曆志》，設為問答，著《鐘律辨疑》三卷，至為精密，發古人所未言者。

Jiong was erudite and devoted to antiquity; Zhu Xi treated him with deep respect. Later Peng Yinglong of Luling, a professor at the Jiangling prefectural school, annotated the Han Treatises on Pitch Standards and Calendar, composed a question-and-answer treatise, and wrote three volumes of Discerning Doubts on Bell Pitch Standards—remarkably precise, articulating points the ancients had never addressed.

宋曆在東都凡八改，曰《應天》、《乾元》、《儀天》、《崇天》、《明天》、《奉元》、《觀天》、《紀元》。星翁離散，《紀元曆》亡，紹興二年，高宗重購得之，六月甲午，語輔臣曰：「曆官推步不精，今曆差一日，近得《紀元曆》，自明年當改正，協時月正日，蓋非細事。」是歲，始議制渾儀。十一月，工部言，《渾儀法要》當以子午為正，今欲定測樞極，合差局官二員。詔差李繼宗等充測驗定正宮，俟造畢進呈日，同參詳指說制度官丁師仁、李公謹入殿安設。三年正月壬戌，進呈渾儀木樣。壬申，太史局令丁師仁等言，省識東都渾儀四座：在測驗渾儀刻漏所曰至道儀，在翰林天文局曰皇祐儀，在太史局天文院曰熙寧儀，在合台曰元祐儀，每座約銅二萬餘斤，今若半之，當萬餘斤。且元祐製造，有兩府提舉。時都司覆實，用銅八千四百斤。詔工部置物料，臨安府傭工匠，仍令工部長、貳提舉。

At the Eastern Capital the Song revised its calendar eight times: Responding to Heaven, Qianyuan, Yitian, Chongtian, Mingtian, Fengyuan, Guantian, and Jiyuan. Star chroniclers had scattered, and the Jiyuan Calendar was lost. In Shaoxing year 2, Emperor Gaozong paid heavily to recover it. On the sixth month's jiawu day he told his ministers: "The calendar officials' computations are imprecise—the calendar is now one day off. I have recovered the Jiyuan Calendar; from next year it will be corrected to harmonize seasons, months, and the correct day. This is no small matter." That year they first discussed constructing an armillary sphere. In the eleventh month the Ministry of Works reported that the Essentials of the Armillary Sphere Method requires the meridian as reference; to fix measurement of the celestial pole, two Astrological Bureau officials should be appointed. An edict appointed Li Jizong and others to test and correct the instrument; when finished, Ding Shiren and Li Gongjin, officials responsible for explaining the design, would install it in the hall. In the third year, first month, day renxu, the wooden model of the armillary sphere was presented. On day renshen, Astrological Bureau Director Ding Shiren reported four Eastern Capital armillary spheres: the Zhidao Instrument at the Testing Office; the Huangyou Instrument at the Hanlin Bureau; the Xining Instrument at the Astronomy Court; and the Yuanyou Instrument at the Secretariat—each requiring over 20,000 jin of copper; half that would be over 10,000 jin. The Yuanyou manufacture had been supervised by both metropolitan offices. The Capital Bureau verified actual usage at 8,400 jin of copper. An edict ordered the Ministry of Works to prepare materials, Lin'an Prefecture to hire craftsmen, and the ministry director and deputy to supervise.

五年，日官言，正月朔旦日食九分半，虧在辰正。常州布衣陳得一言：當食八分半，虧在巳初。其言卒驗。侍御史張致遠言：「今歲正月朔日食，太史所定不驗，得一嘗為臣言，皆有依據。蓋患算造者不能通消息、盈虛之奧，進退、遲疾之分，致立朔有訛。凡定朔小餘七千五百以上者，進一日。紹興四年十二月小餘七千六百八十，太史不進，故十一月小盡；今年五月小餘七千一百八十，少三百二十，乃為進朔，四月大盡。建炎三年定十一月三十日甲戌為臘，陰陽書曰：臘者，接也，以故接新，在十二月近大寒前後戌日定之，若近大寒戌日在正月十一日，若即用遠大寒戌日定之，庶不出十二月。如宣和五年十二月二十七日丙午大寒，後四日庚戌，雖近，緣在六年正月一日，此時以十九日戊戌為臘。得一於歲旦日食，嘗預言之，不差厘刻。願詔得一改造新曆，委官專董其事。仍盡取其書，參校太史有無，以補遺闕。擇曆算子弟粗通了者，授演撰之要，庶幾日官無曠，曆法不絕。」二月丙子，詔秘書少監朱震即秘書省監視得一改造新曆。八月，曆成，震請賜名《統元》，從之。詔翰林學士孫近為序，以六年頒行，遷震一秩，賜得一通微處士，官其一子。道士裴伯壽等受賞有差。

In year 5 calendar officials predicted a solar eclipse of nine-tenths on the first day of the first month, with obscuration at the fifth double-hour of chen. Chen Deyi, a commoner of Changzhou, predicted an eight-tenths eclipse with obscuration at the beginning of si. His prediction proved correct. Censor Zhang Zhiyuan memorialized: "This year's new-moon eclipse was mispredicted by the Astrological Bureau, yet Deyi had told me his prediction with sound basis. The problem is that calendar makers fail to grasp the mysteries of syzygy, lunar phases, and planetary motion, leading to errors in fixing the new moon. When the fractional remainder for the new moon reaches 7,500 or above, one day must be advanced. In Shaoxing the fourth year, month 12, the small remainder was 7,680—the bureau failed to advance, so month 11 ended short; this year in month 5 the remainder was 7,180—320 less—triggering an advanced new moon; month 4 ended long. In Jianyan year 3 they fixed day jiaxu of month 11, day 30, as La. Yin-yang texts state that La means "connecting" old and new, fixed on a xu day near Great Cold in month 12; if that xu day falls in month 1, the more distant xu day is used so La never falls outside month 12. For example, in Xuanhe year 5, day bingwu of month 12, day 27, was Great Cold; four days later was gengxu—though near, it fell on day 1 of year 6, so day wuxu of day 19 was taken as La. Deyi had predicted the new-year solar eclipse in advance, accurate to the finest measure. I request an edict commissioning Deyi to create a new calendar under dedicated official supervision. His complete works should be collected and collated against the bureau's holdings to fill gaps. Select youths with basic calendar skills and teach them computation, so the bureau is never understaffed and methods are not lost." On month 2, day bingzi, an edict ordered Vice Director Zhu Zhen to supervise Deyi's calendar work at the Secretariat. In month 8 the calendar was finished; Zhen requested the name Unifying Origin, which was approved. An edict commissioned Academician Sun Jin to write the preface; promulgated in year 6; Zhen was promoted; Deyi received the title Erudite Recluse of Penetrating Subtlety and an office for his son. The Daoist Pei Boshou and others received graded rewards.

得一等上推甲子之歲，得十一月甲子朔夜半冬至日度起於虛中以為元。著《歷經》七卷，《曆議》二卷，《立成》四卷，《考古春秋日食》一卷，《七曜細行》二卷，《氣朔入行草》一卷，詔付太史氏，副藏秘府。

Deyi and colleagues traced back to the jiazi year, finding the jiazi new moon of month 11 at midnight winter solstice, with the sun's position at Emptiness as the epoch. He produced seven volumes of Calendar Classic, two of Calendar Discourse, four of Establishment Tables, one on ancient solar eclipses, two on the seven luminaries, and one draft on qi and new moons—deposited with the Astrological Bureau and the secret repository.

紹興九年，史官重修神宗正史，求《奉元曆》不獲，詔陳得一、裴伯壽赴闕補修之。

In Shaoxing year 9, revising Shenzong's standard history, historians could not find the Fengyuan Calendar; Chen Deyi and Pei Boshou were summoned to restore it.

十四年，太史局請制渾儀，工部員外郎謝伋言：「臣嘗詢渾儀之法，太史官生論議不同，鑄作之工，今尚闕焉。臣愚以為宜先詢訪制度，敷求通曉天文歷數之學者，參訂是非，斯合古制。」蘇頌之子應詔赴闕，請訪求其父遺書，考質制度。宰相秦檜曰：「在廷之臣，罕能通曉。」高宗曰：「此闕典也，朕已就宮中製造，範制雖小，可用窺測，日以晷度、夜以樞星為則，非久降出，第當廣其尺寸爾。」於是命檜提舉。時內侍邵諤善運思，專令主之，累年方成。

In year 14 the Astrological Bureau requested an armillary sphere; Xie Ji of the Ministry of Works reported: "I have inquired into armillary methods, but bureau students disagree and skilled casters are lacking. He proposed first establishing the design, consulting experts in astronomy and calendrics, and reconciling disputes to match ancient standards." Su Song's son Yingcan came to court seeking his father's surviving texts to verify the design. Chief Councilor Qin Hui said: "Few court ministers understand these matters thoroughly." Gaozong replied: "This classic is missing; I have already had a model made in the palace—small but usable for observation by day via the gnomon and by night via the pole star. It will soon be issued; only its size need be enlarged." He then ordered Hui to supervise. Eunuch Shao E, skilled in design, was put in charge; completion took many years.

《統元曆》頒行雖久，有司不善用之，暗用《紀元》法推步，而以《統元》為名。乾道二年，日官以《紀元曆》推三年丁亥歲十一月甲子朔，將頒行，裴伯壽詣禮部陳《統元曆》法當進作乙丑朔，於是依《統元曆》法正之。

Though the Unifying Origin Calendar had long been in use, officials poorly applied it, secretly computing by Jiyuan methods while retaining the Unifying Origin name. In Qiandao year 2, using Jiyuan methods, officials computed the jiazi new moon of month 11, dinghai year 3, for promulgation; Pei Boshou argued Unifying Origin required yichou new moon, and correction followed.

光州士人劉孝榮言：「《統元曆》交食先天六刻，火星差天二度。嘗自著曆，期以半年可成，願改造新曆。」禮部謂：「《統元曆》法用之十有五年，《紀元曆》法經六十年，日月交食有先天分數之差，五星細行亦有二三度分之殊。算造曆官拘于依經用法，致朔日有進退，氣節日分有誤，于時宜改造。」伯壽言：「造曆必先立表測景驗氣，庶幾精密。」判太史局吳澤私于孝榮，且言銅表難成、木表易壞以沮之。乃詔禮部尚書周執羔提領改造新曆，執羔亦謂測景驗氣，經涉歲月。孝榮乃采萬分曆，作三萬分以為日法，號《七曜細行曆》，上之。三年，執羔以歷來上，孝宗曰：「日月有盈縮，須隨時修改。」執羔對曰：「舜協時月正日，正為積久不能無差，故協正之。」孝宗問曰：「今曆與古曆何如？」對曰：「堯時冬至日在牽牛，今冬至日在鬥一度。」

Scholar Liu Xiaorong of Guang Prefecture said the Unifying Origin Calendar predicted eclipses six quarters too early and Mars two degrees off. He had composed his own calendar, expecting completion in six months, and sought to create a new one. The Ministry of Rites noted that Unifying Origin had been used fifteen years and Jiyuan sixty; eclipses showed fractional error and the five planets differed by two or three degrees. Officials bound to canonical methods caused new-moon and solar-term errors—a remake was timely. Boshou said a new calendar requires gnomon measurements to verify qi before precision is possible. Bureau judge Wu Ze favored Xiaorong privately and claimed copper gnomons were impractical and wooden ones perishable. Minister Zhou Zhigao was ordered to supervise; he too said gnomon testing would take months. Xiaorong adopted a myriad-part system, using thirty thousand as the day divisor, titled Detailed Motion of the Seven Luminaries Calendar, and submitted it. In year 3 Zhigao submitted the calendar; Xiaozong said sun and moon vary and must be adjusted in season. Zhigao replied that Shun harmonized seasons and the correct day because accumulated error requires periodic correction. Xiaozong asked how the new calendar compared with antiquity. He answered: "At Yao's time winter solstice fell in Ox; today it falls at Dipper, one degree."

孝榮《七曜細行曆》自謂精密，且預定是年四月戊辰朔日食一分，日官言食二分，伯壽並非之，既而精明不食。孝榮又定八月庚戌望月食六分半，候之，止及五分。又定戊子歲二月丁未望月食九分以上，出地，其光復滿。伯壽言：「當食既，復滿在戌正三刻。」

Xiaorong's calendar claimed precision and predicted a one-tenth eclipse on the wuchen new moon of month 4; officials said two-tenths; Boshou rejected both—the eclipse did not occur. He predicted a six-tenths lunar eclipse on the gengxu full moon of month 8; observation showed only five-tenths. He predicted a lunar eclipse over nine-tenths on the dingwei full moon of month 2, wuzi year, with the moon emerging and light restored. Boshou said it should be total, with full restoration at the third quarter of the xu period.

侍御史單時言：「比年太史局以《統元曆》稍差而用《紀元曆》，《紀元》浸差，邇者劉孝榮議改曆，四月朔日食不驗，日官兩用《統元》、《紀元》以定晦朔，二曆之差，歲益已甚，非所以明天道、正人事也。如四月朔之日不食，雖為差誤，然一分之說，猶為近焉。八月望之月食五分，新曆以為食六分，亦為近焉。聞欲以明年二月望月食為驗，是夜或有陰晦風雨，願令日官與孝榮所定七政躔度其說異同者，俟其可驗之時，以渾象測之，察其稍近而屢中者，從其說以定曆，庶幾不致甚差。」詔從之。十一月，詔國子司業權禮部侍郎程大昌、監察御史張敦實監太史局驗之。時孝宗務知曆法疏密，詔太史局以高宗所降小渾儀測驗造曆。四年二月十四日丁未望，月食生光復滿，如伯壽言。

Censor Shan Shi memorialized that the bureau, finding Unifying Origin slightly off, used Jiyuan, which also drifted; Xiaorong's new calendar failed the month-4 eclipse test; using both calendars worsened discrepancy yearly—this failed to clarify heaven's way. Though month 4's eclipse failed, the one-tenth prediction was relatively close. The month-8 full-moon eclipse was five-tenths; the new calendar predicted six—also relatively close. He proposed comparing Xiaorong's and official predictions at the next verifiable eclipse, using the armillary sphere, and adopting whichever proves more accurate. An edict approved. In month 11 Cheng Dachang and Zhang Dunshi were ordered to supervise bureau testing. Xiaozong, seeking to judge calendrical precision, ordered testing with Gaozong's small armillary sphere. On the fourth year, month 2, day 14, dingwei full moon, the eclipse restored fully—as Boshou predicted.

時等又言：「去年承詔，十二月癸卯、乙巳兩夜監測太陰、太白，新曆為近。今年二月十四日望月食，臣與大昌等以渾儀定其光滿，則舊曆差近，新曆差遠。若遽以舊曆為是，則去年所測四事皆新曆為近，今者所定月食，乃復稍差，以是知天道之難測。儒者莫肯究心，一付之星翁曆家，其說又不精密。願令繼宗、孝榮等更定三月一日內七政躔度之異同者，仍令臣等往視測驗而造曆焉。」三月，詔時與大昌同驗之。太史局止用《紀元曆》與新曆測驗，未嘗參以《統元曆》。臣等先求判太史局李繼宗、天文官劉孝榮等《統元》、《紀元》、新曆異同，於三月初九日夜、十一日早、十四日夜、二十日早詣太史局，召三曆官上臺，用銅儀窺管對測太陰、木、火、土星昏晨度經歷度數，參稽所供，監視測驗。初九日昏度：舊曆太陰在黃道張宿十二度八十七分，在赤道張宿十度；新曆在黃道張宿十四度四十分，在赤道張宿十五度太。臣等驗得在赤道張宿十五度半。今考之新曆稍密，舊曆皆疏。十一日早晨度：木星在黃道室宿十五度七分，在赤道室宿十三度少；土星在黃道虛宿七度三分，在赤道虛宿七度強。新曆木星在黃道室宿十五度四十四分，在赤道室宿十四度少弱；土星在黃道虛宿六度二十一分，在赤道虛宿六度少弱。臣等驗得五更三點，土星在赤道虛宿六度弱；五更五點，木星在赤道室宿十四度。今考之新曆稍密，舊曆皆疏。十二日，都省令定驗《統元》、《紀元》及新曆疏密。《統元曆》昏度，太陰在黃道氐宿初度九十四分，在赤道氐宿三度少；《紀元曆》在黃道氐宿初度八十三分，在赤道氐宿二度太；新曆在黃道亢宿八度七十一分，在赤道亢宿九度少弱。三曆官以渾儀由南數之，其太陰北去角宿距星二十一度少弱。新舊曆官稱昏度亢宿未見，祗以窺管測定角宿距星，復以曆書考東方七宿，角占十二度，亢占九度少；既亢宿未見，當除角宿十二度，即太陰此時在赤道亢宿九度少弱。今考之新曆全密，《紀元》、《統元曆》皆疏。二十日早晨度：《統元曆》太陰在黃道鬥宿十一度九十一份，在赤道鬥宿十二度少；火星在黃道危宿七度九十一分，在赤道危宿七度少；土星在黃道虛宿八度八十二分，在赤道虛宿八度太強。《紀元曆》太陰在黃道鬥宿十一度四十分，在赤道鬥宿十一度半；火星在黃道危宿六度，在赤道危宿六度太；土星在黃道虛宿七度三十九分，在赤道虛宿七度半弱。新曆太陰在黃道鬥宿十度六十一分，在赤道鬥宿十度少；火星在黃道危宿七度二十分，在赤道危宿六度；土星在黃道虛宿六度五十三分，在赤道虛宿六度半。三曆官驗得太陰在赤道鬥宿十度，火星在赤道危宿六度強，土星在赤道虛宿六度半。今考之太陰，《紀元曆》疏；火星，新曆、《紀元曆》全密，《統元曆》疏；土星，新曆全密，《紀元》、《統元曆》疏。」

Shi reported that on guimao and yisi nights of month 12 the previous year, the new calendar matched the moon and Venus better. For this year's month-2 day-14 eclipse, they found the old calendar slightly nearer for full restoration. He noted inconsistency: last year favored the new calendar; this eclipse favored the old—heaven's way is hard to measure. Confucians neglect the matter, leaving it to star chroniclers whose theories lack precision. He requested Jizong and Xiaorong to reconcile planetary positions by month 3 day 1 and authorize further observation. In month 3 Shi and Dachang were ordered to verify together. The bureau tested only Jiyuan and the new calendar, not Unifying Origin. They compared all three calendars on nights of month 3 days 9, 11, 14, and 20 using the bronze armillary sighting tube. Night of day 9 dusk: old calendar placed the moon at Extended 13°87' ecliptic, 10° equator; new calendar at Extended 14°40' ecliptic, 15°+ equator. Verification showed equator Extended 15.5°. The new calendar proved slightly more accurate; old calendars were loose. Morning of day 11: Jupiter at Encampments 15°7' ecliptic, 13°− equator; Saturn at Emptiness 7°3' ecliptic, 7°+ equator. New calendar: Jupiter Encampments 15°44' ecliptic, 14°− equator; Saturn Emptiness 6°21' ecliptic, 6°− equator. At fifth watch, third point: Saturn equator Emptiness 6°−; fifth watch, fifth point: Jupiter equator Encampments 14°. Again the new calendar was denser; old calendars sparse. On day 12 the Secretariat ordered final comparison of all three calendars. Unifying Origin dusk: moon Base 1°94' ecliptic, 3°− equator; Jiyuan: Base 1°83' ecliptic, 2°+ equator; new: Horn 8°71' ecliptic, 9°− equator. Officers measured the moon 21°− north of Horn's distance star. Horn was not visible; officers measured Horn's distance star and deduced from lodge widths— subtracting Horn's 12° placed the moon at Horn neighbor 9°− equator. The new calendar was fully accurate; Jiyuan and Unifying Origin were loose. Morning day 20 Unifying Origin: moon Dipper 11°91' ecliptic, 12°− equator; Mars Rooftop 7°91' ecliptic, 7°− equator; Saturn Emptiness 8°82' ecliptic, 8°++ equator. Jiyuan: moon Dipper 11°40' ecliptic, 11.5° equator; Mars Rooftop 6° ecliptic, 6°+ equator; Saturn Emptiness 7°39' ecliptic, 7.5°− equator. New calendar: moon Dipper 10°61' ecliptic, 10°− equator; Mars at Rooftop 7°20' ecliptic, 6° equator; Saturn at Emptiness 6°53' ecliptic, 6.5° equator. Officers verified the moon at Dipper 10° equator, Mars at Rooftop 6°+ equator, Saturn at Emptiness 6.5° equator. For the moon, Jiyuan Calendar was loose; Mars: new and Jiyuan calendars fully accurate, Unifying Origin loose; Saturn: new calendar fully accurate, Jiyuan and Unifying Origin loose."

又詔時與尚書禮部員外郎李燾同測驗，時等言：「先究《統元》、《紀元》、新曆異同，召三曆官上臺，用銅儀窺管對測太陰、土、火、木星晨度經歷度數，參稽所供，監視測驗。二十四日早晨度：《統元曆》太陰在黃道危宿十一度九十分，在赤道危宿九度；木星在黃道室宿十八度一十五分，在赤道壁宿初度少；火星在黃道危宿十度七十分，在赤道危宿十度；土星在黃道虛宿八度九十五分，在赤道虛宿九度。《紀元曆》太陰在赤道危宿十度五十三分，在赤道危宿八度半；木星在黃道室宿十七度六十八分，在赤道室宿十四度少；火星在黃道危宿九度八十四分，在赤道危宿九度；土星在黃道留在虛宿七度四十分，在赤道虛宿七度半。新曆太陰在黃道危宿十三度五分，在赤道危宿十二度；木星在黃道室宿十八度一十分，在赤道室宿十六度半強；火星在黃道危宿十度八分，在赤道危宿九度；土星在黃道虛宿六度六十分始留，在赤道虛宿六度半強始留。三曆官驗得太陰在赤道危宿十度，木星在赤道室宿十六度太，火星在赤道危宿九度半，土星在赤道虛宿六度半弱。今考之太陰，《統元曆》精密、《紀元曆》、新曆皆疏；木星，新曆稍密，《紀元》、《統元曆》皆疏；火星，《紀元》、新曆皆稍密，《統元曆》疏；土星，新曆稍密，《紀元》、《統元曆》皆疏。二十七日早晨度：《統元曆》木星在黃道壁宿初度四十六分，在赤道壁宿初度太強；火星在黃道危宿十二度九十二分，在赤道危宿十二度強；土星留在黃道虛宿八度九十八分，在赤道虛宿九度。《紀元曆》木星在黃道壁宿初度二十五分，在赤道壁宿初度分空；火星在黃道危宿十二度九十七分，在赤道危宿十一度；土星留在黃道虛宿七度四十八分，在赤道虛宿七度半。新曆木星在黃道壁宿初度四十四分，在赤道壁宿初少強；火星在黃道危宿十二度二十二分，在赤道危宿十一度半；土星留在黃道虛宿六度六十分，在赤道虛宿六度半強。三曆官驗得木星在赤道壁宿初度少，火星在赤道危宿十一度，土星在赤道虛宿六度半。今觀木星，新曆稍密，《紀元》、《統元曆》皆疏；火星，《紀元曆》全密，《統元》、新曆皆疏；土星，新曆稍密，《紀元》、《統元曆》皆疏。」

Another edict ordered Shi and Li Tao of the Ministry of Rites to test jointly. They reported: first compare the three calendars, summon officers to the platform, and measure dawn positions of the moon and planets with the bronze armillary. Morning of day 24 Unifying Origin: moon Rooftop 11°90' ecliptic, 9° equator; Jupiter Encampments 18°15' ecliptic, Wall 0°− equator; Mars Rooftop 10°70' ecliptic, 10° equator; Saturn Emptiness 8°95' ecliptic, 9° equator. Jiyuan: moon Rooftop 10°53' ecliptic, Rooftop 8.5° equator; Jupiter Encampments 17°68' ecliptic, 14°− equator; Mars Rooftop 9°84' ecliptic, 9° equator; Saturn stationary at Emptiness 7°40' ecliptic, 7.5° equator. New calendar: moon Rooftop 13°5' ecliptic, 12° equator; Jupiter Encampments 18°10' ecliptic, 16.5°+ equator; Mars Rooftop 10°8' ecliptic, 9° equator; Saturn beginning station at Emptiness 6°60' ecliptic, 6.5°++ equator. Officers verified moon Rooftop 10° equator, Jupiter Encampments 16°+ equator, Mars Rooftop 9.5° equator, Saturn Emptiness 6.5°− equator. For the moon, Unifying Origin was precise; Jiyuan and the new calendar were loose; Jupiter: new calendar slightly accurate; Jiyuan and Unifying Origin loose; Mars: Jiyuan and new slightly accurate; Unifying Origin loose; Saturn: new slightly accurate; Jiyuan and Unifying Origin loose. Morning day 27 Unifying Origin: Jupiter Wall 0°46' ecliptic, 0°++ equator; Mars Rooftop 12°92' ecliptic, 12°+ equator; Saturn stationary Emptiness 8°98' ecliptic, 9° equator. Jiyuan: Jupiter Wall 0°25' ecliptic, Wall 0° equator; Mars Rooftop 12°97' ecliptic, 11° equator; Saturn stationary Emptiness 7°48' ecliptic, 7.5° equator. New calendar: Jupiter Wall 0°44' ecliptic, 0°−+ equator; Mars Rooftop 12°22' ecliptic, 11.5° equator; Saturn stationary Emptiness 6°60' ecliptic, 6.5°+ equator. Officers verified Jupiter Wall 0°− equator, Mars Rooftop 11° equator, Saturn Emptiness 6.5° equator. Jupiter: new slightly accurate; Jiyuan and Unifying Origin loose; Mars: Jiyuan fully accurate; Unifying Origin and new loose; Saturn: new slightly accurate; Jiyuan and Unifying Origin loose."

由是朝廷始知三曆異同，乃詔太史局以新舊曆參照行之。禮部言：「新舊曆官互相異同，參照實難，新曆比之舊曆稍密。」詔用新曆，名以《乾道曆》，己丑歲頒行。

The court thus learned how the three calendars differed and ordered the Astrological Bureau to implement them by cross-reference. The Ministry of Rites reported that new and old officers disagreed, making cross-reference difficult, though the new calendar was slightly more accurate. An edict adopted the new calendar, named the Qiandao Calendar, promulgated in the jichou year.

孝榮有《考春秋日食》一卷，《漢魏周隋日月交食》一卷，《唐日月交食》一卷，《宋朝日月交食》一卷，《氣朔入行》一卷，《強弱日法格數》一卷。

Xiaorong also authored Investigating Spring and Autumn Solar Eclipses, Han through Sui Eclipses, Tang Eclipses, Song Eclipses, Qi and New Moon Entry, and Strong-Weak Day Methods and Grid Numbers—one volume each.

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