1
卷一十六志第六
Volume 16, Treatise Six.
2
律曆志上
Treatise on the Pitch Pipes and the Calendar, Part One
3
《易》曰：「形而上者謂之道，形而下者謂之器。」 夫神道廣大，妙本於陰陽； 形器精微，義先於律呂。 聖人觀四時之變，刻玉紀其盈虛，察五行之聲，鑄金均其清濁，所以遂八風而宣九德，和大樂而成政道。 然金質從革，侈弇無方； 竹體圓虛，修短利制。 是以神瞽作律，用寫鐘聲，乃紀之以三，平之以六，成於十二，天之道也。 又葉時日於晷度，效地氣于灰管，故陰陽和則景至，律氣應則灰飛。 灰飛律通，吹而命之，則天地之中聲也。 故可以範圍百度，化成萬品，則《虞書》所謂「葉時月正日，同律度量衡」者也。 中聲節以成文，德音章而和備，則可以動天地，感鬼神，道性情，移風俗。 葉言志於詠歌，鑒盛衰於治亂，故君子審聲以知音，審音以知樂，審樂以知政，蓋由茲道。 太史公律書云：「王者制事立物，法度軌則，一稟於六律。 六律為萬事之本，其于兵械尤所重焉。 故云望敵知吉凶，聞聲效勝負，百王不易之道也。」
The "Book of Changes" distinguishes the Way, which transcends concrete form, from "vessels"—the patterned instruments that embody it. That numinous Way is boundless, yet its subtle foundation is still the interplay of yin and yang. Concrete forms and ritual implements are refined down to the smallest detail, and their meaning begins with the regulation of pitch pipes and scale degrees. By watching the turning seasons they cut jade tallies for fullness and decline; by listening to the five phases in sound they cast bronze to balance bright and dull registers—so the eight winds could be brought into accord, the nine virtues proclaimed, court music perfected, and governance given its proper tone. Bronze, however, yields to the worker's hand—its bore widens or narrows without a single fixed rule. Bamboo tubes are naturally round and hollow, so their length can be cut to precise measure. The blind masters therefore framed the pipes to write down the Yellow Bell's note: they grouped it in threes, evened it in sixes, and finished the cycle at twelve pipes—the same numerology Heaven itself uses. They matched the calendar to gnomon readings and read the breath of the soil in thermoscopic tubes, so harmony of yin and yang showed in the solstitial shadow, while the monthly pipe's breath showed when the light ash sprang. Once the ash leaps and the pipe answers, a single sustained breath names the central tone that stands between Heaven and Earth. That is why it can standardize every measure and give form to every thing—the very sense, in the "Canon of Shun," of "harmonizing the seasons, months, and intercalary days" and "making the pipes, lengths, pecks, and balances one system." When that central tone is shaped into patterned music and virtue rings through it in full harmony, it can stir Heaven and earth, move spirits, speak to human feeling, and remake the temper of a people. Poetry lodges the mind in song; order and chaos leave their echo in what we hear. That is why a cultivated man studies sound to grasp tone, tone to grasp music, and music to grasp how the realm is governed—always along this same path. Sima Qian's treatise on the pipes insists that the king, whenever he fixes rites or fashions tools—every statute, gauge, and model—grounds them all in the six yang pipes. Those six pipes underlie every undertaking; they weigh heaviest of all on arms and equipment. Men say you can read an army's luck by scanning the foe and judge victory or defeat by the note that rides the wind—a principle no dynasty has seen fit to abandon."
4
及秦氏滅學，其道浸微。 漢室初興，丞相張蒼首言律，未能審備。 孝武帝創置協律之官，司馬遷言律呂相生之次詳矣。 及王莽際，考論音律，劉歆條奏，大率有五：一曰備數，一、十、百、千、萬也； 二曰和聲，宮、商、角、徵、羽也； 三曰審度，分、寸、尺、丈、引也； 四曰嘉量，龠、合、升、鬥、斛也； 五曰權衡，銖、兩、斤、鈞、石也。 班固因而志之。 蔡邕又記建武已後言律呂者，至司馬紹統采而續之。 漢末天下大亂，樂工散亡，器法堙滅，魏武始獲杜夔，使定樂器聲調。 夔依當時尺度，權備典章。 及武帝受命，遵而不革。 至泰始十年，光祿大夫荀勖奏造新度，更鑄律呂。 元康中，勖子籓嗣其事，未及成功，屬永嘉之亂，中朝典章，咸沒于石勒。 及元帝南遷，皇度草昧，禮容樂器，掃地皆盡，雖稍加采掇，而多所淪胥，終於恭、安，竟不能備。 今考古律相生之次，及魏武已後言音律度量者，以聲明於篇雲。
When Qin burned the books, that tradition nearly died away. At the founding of Han, Chancellor Zhang Cang reopened the discussion of pitch, but his work still left gaps. Emperor Wu of Han added a harmonics bureau, and Sima Qian traced, pipe by pipe, how each tone generates the next. Under Wang Mang the court re-examined acoustics, and Liu Xin's memorial broke the system into five rubrics. The first was "numbering in full"—the powers from one through ten thousand; the second was "matching the five notes"—gong, shang, jue, zhi, and yu; the third was "measured lengths"—from the fen up through the yin; the fourth was "honest dry and wet measures"—from the yue up to the hu; the fifth was "weights and steelyards"—from the zhu to the shi stone. Ban Gu folded that scheme into his "Treatise on Pitch and Calendar." Cai Yong continued the story for the Eastern Han after Jianwu, and Sima Biao later collected and extended every note on the subject. When Han collapsed, players fled and died, carrying their craft to the grave. Only with Cao Cao did the court recover Du Kui and task him with resetting the orchestra. Du Kui worked from contemporary foot-rules and patched the code back into service. When Emperor Wu of Jin took the throne, he kept Du Kui's tuning unchanged. In the tenth year of Taishi the household minister Xun Xi asked to cast new bronze pipes from a revised measure. In Yuankang his son Xun Fan took over, but Yongjia's sack of Luoyang swallowed the capital's libraries before the work was finished, and Shi Le seized what survived. After Emperor Yuan fled south the court barely had a wardrobe or band left. Later reigns scraped together odds and ends, yet most pieces never surfaced again, and by the reigns of Gong and An the old orchestra could not be restored. What follows collates the classical cycle of generating pipes with every major Jin-era discussion of acoustics and metrology from Cao Cao on.
5
《傳》云：「十二律，黃帝之所作也。 使伶倫自大夏之西，乃之昆侖之陰，取竹之嶰穀生，其竅厚均者，斷雨節間長三寸九分而吹之，以為黃鐘之宮，曰含少。 次制十二竹筒，寫鳳之鳴，雄鳴為六，雌鳴亦六，以比黃鐘之宮，皆可以生之以定律呂。 則律之始造，以竹為管，取其自然圓虛也。」 又云「黃帝作律，以玉為管，長尺，六孔，這二十月音。 至舜時，西王母獻昭華之琯，以玉為之。」 及漢章帝時，零陵文學奚景於泠道舜祠下得白玉琯。 又武帝太康元年，汲郡盜發六國時魏襄王塚，亦得玉律。 則古者又以玉為管矣。 以玉者，取其體含廉潤也。 而漢平帝時，王莽又以銅為之。 銅者，自名也，所以同天下，齊風俗也。 為物至精，不為燥濕寒暑改節，介然有常，似士君子之行，故用焉。
The lore runs: "The Yellow Emperor fashioned the twelve pipes. He sent Ling Lun beyond Great Xia to Kunlun's north slope, chose even-gauge bamboo from the Xie Valley gorge, cut a tube three cun nine fen between nodes, sounded it, and fixed the Yellow Bell's fundamental under the name Hánshǎo. He then bored twelve bamboo pipes, imitating the cock and hen phoenix—six tones each—and keyed them to the Yellow Bell so the whole chromatic set could be spun off from that root. That is why the earliest pipes were bamboo: the material is already a perfect hollow cylinder." Another line adds that the Yellow Emperor cut jade pipes a foot long with six holes—one for each month of the year—though the wording in our manuscript is damaged here. At Shun's court the Queen Mother of the West sent the "Radiant Splendor" jade flutes. During Han Zhangdi's reign the classicist Xi Jing of Lingling pulled a white jade pitch pipe from Shun's temple at Lengdao. In Jin Wudi's Taikang 1 plunderers who broke into King Xiang of Wei's Warring States tomb at Ji commandery likewise turned up jade tuning pipes. So the ancients sometimes tuned in jade as well as bamboo. Jade was prized because its body seems to drink in a cool, even gleam. Under Han's child emperor Ping, Wang Mang ordered a bronze set. Bronze advertised a single public standard—metal the whole country could copy so customs would line up. It is a stuff so stable that heat, damp, and frost barely pull its note—steady as a gentleman's bearing—which is why later courts preferred it.
6
《周禮》太師掌六律、六呂，以合陰陽之聲。 六律陽聲，黃鐘、太蔟、姑洗、蕤賓、夷則、無射也； 六呂陰聲，大呂、應鐘、南呂、林鐘、仲呂、夾鐘也。 又有太師則執同律以聽軍聲，而詔以吉凶。 其典同掌六律之和，以辯天地四方陰陽之聲，以為樂器，皆以十有二律而為之數度，以十有二聲而為之齊量焉。
The "Rites of Zhou" puts the Grand Master of Music in charge of six yang and six yin pipes so the bright and dark registers answer each other. The six yang pipes are Yellow Bell, Tàicù, Gǔxǐ, Ruíbīn, Yízé, and Wúyì; the six yin pipes are Dàlǚ, Yìngzhōng, Nánlǚ, Línzhōng, Zhònglǚ, and Jiázhōng. On campaign the same officer carries a uniform pipe to listen to the troops' hum and read Heaven's verdict. The Diǎntóng office blends the six pipes, sorts the tones of heaven, earth, and the four directions into yin and yang, and builds instruments whose lengths and volumes all count out in twelves.
7
及周景王將鑄無射，問律於泠州鳩，對曰：「夫六，中之色，故名之曰黃鐘，所以宣養六氣九德也。 由是第之。 二曰太蔟，所以金奏贊陽出滯也。 三曰姑洗，所以羞潔百物，考神納賓也。 四曰蕤賓，所以安靜神人，獻酬交酢也。 五曰夷則，所以詠歌九德，平人無貳也。 六曰無射，所以宣佈哲人之令德，示人軌儀也。 為之六間，以揚沈伏而黜散越也。 元間大呂，助宣物也，二間夾鐘，出四隙之細也。 三間中呂，宣中氣也。 四間林鐘，和展百事，俾莫不任肅純恪也。 五間南呂，贊陽秀也。 六間應鐘，均利器用，俾應復也。」 此皆所以律述時氣效節物也。
Planning the Wúyì bell for Zhou, King Jing questioned Lěng Zhōujiū, who answered: "Yellow stands for the central element—hence the fundamental pipe is named Yellow Bell, the voice that broadcasts and feeds the six vapors and nine virtues. On that basis he ordered the scale. Next comes Tàicù, the pipe that drives bronze fanfares, lifts the yang qi, and shakes off stagnation. Third is Gǔxǐ, used to rinse offerings clean, draw spirits down, and welcome guests. Fourth is Ruíbīn, the pitch of quiet worship and of pledging cup after cup. Fifth is Yízé, the tone on which the nine virtues are hymned and common folk kept single-hearted. Sixth is Wúyì, the pipe that proclaims a sage-king's shining edicts and shows the people their compass. Six intercalary pipes sit between them to lift what lies too deep and trim what flies too loose. The first gap is Dàlǚ, pushing the year's growth outward; the second is Jiázhōng, threading the subtlest filigree through the four seasons. Zhònglǚ is the third slot, venting the qi that pools at the year's hinge. Fourth is Línzhōng, harmonizing every task of state so each is met with reverent care. Fifth is Nánlǚ, coaxing the yang into full bloom. Sixth is Yìngzhōng, evening implements and echoes so the circuit can close and reopen." Every one of those pipes is a calendar in sound, pegging the month's qi to a note.
8
及秦始皇焚書蕩覆，典策缺亡，諸子瑣言時有遺記。 呂不韋《春秋》言：黃鐘之宮，律之本也，下生林鐘，林鐘上生太蔟，太蔟下生南呂，南呂上生姑洗，姑洗下生應鐘，應鐘上生蕤賓，蕤賓下生大呂，大呂下生夷則，夷則上生夾鐘，夾鐘下生無射，無射上生中呂。 三分所生，益其一分以上生； 三分所生，去其一分以下生。 後代之言音律者多宗此說。
Qin Shihuang's bonfire scattered the classics; only stray glosses survived in the philosophers. The "Lüshi chunqiu" still preserves the full circle: Yellow Bell down to Forest Bell, Forest Bell up to Tàicù, and so on through the twelve—each third generates the next by adding or subtracting a third of its length. When a pipe "ascends," you lengthen the tube by one third of its predecessor; when it "descends," you shorten it by one third. Later acousticians nearly all hew to that rule of thirds.
9
及漢興，承秦之弊，張蒼首治律曆，頗未能詳。 故孝武帝正樂，乃置協律之官，雖律呂清濁之體粗正，金石高下之音有准，然徒捃采遺存，以成一時之制，而數猶用五。
The Han began in Qin's rubble: Zhang Cang reopened pitch and calendrics but left the fine points fuzzy. When Emperor Wu "rectified" music he added harmonizers—enough to sort bronze and stone into something like tune—but the court was really collating debris, and the math still ran on fives.
10
時淮南王安延致儒博，亦為律呂。 雲黃鐘之律九寸而宮音調，因而九之，九九八十一，故黃鐘之數立焉，位在子。 林鐘位在未，其數五十四。 太蔟其數七十二，南呂之數四十八，姑洗之數六十四，應鐘之數四十二，蕤賓之數五十七，大呂之數七十六，夷則之數五十一，夾鐘之數六十八，無射之數四十五，中呂之數六十，極不生。 以黃鐘為宮，太蔟為商，姑洗為角，林鐘為徽，南呂為羽。 宮生徵，徵生商，商生羽，羽生角，角生應鐘，不比正音，故為和； 應鐘生蕤賓，不比正音，故為繆。 日冬至，音比林鐘浸以濁。 日夏至，音比黃鐘浸以清。 十二律應二十四時之變。 甲子，中呂之徵也。 丙子，夾鐘之羽也。 戊子，黃鐘之宮也。 庚子，無射之商也。 壬子，夷則之角也。 其為音也，一律而生五音，十二律而為六十音。 因而六之，六六三十六，故三百六十音以當一歲之日。 故律曆之數，天地之道也。
Meanwhile Liu An of Huainan patronized scholars and wrote his own pipe treatise. There the Yellow Bell tube is nine cun long and anchors gong; nine times nine gives eighty-one, fixing its numerology on the branch zi. Forest Bell sits on wèi with fifty-four as its tally. The chain runs Tàicù seventy-two, Southern Regulator forty-eight, Gǔxǐ sixty-four, Yìngzhōng forty-two, Ruíbīn fifty-seven, Dàlǚ seventy-six, Yízé fifty-one, Jiázhōng sixty-eight, Wúyì forty-five, Zhònglǚ sixty, ending where the cycle cannot generate another pipe. Those pipes map to the pentatonic set: gong on Yellow Bell, shang on Tàicù, jue on Gǔxǐ, zhi on Forest Bell, yu on Southern Regulator. From gong you derive zhi, then shang, yu, and jue; jue slips down to Yìngzhōng, outside the core pentatonic, so the Huainan writers dub that link "harmony"; Yìngzhōng in turn feeds Ruíbīn—again off the main five—so they call that second step "wavering." On the longest night the pitch slides toward Forest Bell and thickens toward mud. On the shortest night it slides toward Yellow Bell and thins toward ice. Twelve pipes answer the twenty-four qi nodes of the year. A jiǎzǐ day rides Zhònglǚ's zhi voice. A bǐngzǐ day rides Jiázhōng's yu voice. A wùzǐ day sits on Yellow Bell's gong. A gēngzǐ day carries Wúyì's shang. A rénzǐ day takes Yízé's jue. Each fundamental spawns the five degrees; twelve fundamentals spin sixty changing notes. Sixfold iteration—six times six thirty-six—maps three hundred sixty mutations onto the three hundred sixty days. That is why acoustics and the calendar share one mathematics—the weave of Heaven and earth.
11
司馬遷八書言律呂，粗舉大經，著于前史。 則以太極元氣函三為一，而始動於子，十二律之生，必所起焉。 於是參一於醜得三，因而九三之，舉本位合十辰，得一萬九千六百八十三，謂之成數，以為黃鐘之法。 又參之律於十二辰，得十七萬七千一百四十七，謂之該數，以為黃鐘之實。 實如法而一，得黃鐘之律長九寸，十一月冬至之氣應焉。 蓋陰陽合德，氣鐘於子，而化生萬物，則物之生莫不函三。 故十二律空徑三分，而上下相生，皆損益以三。 其術則因黃鐘之長九寸，以下生者倍其實，三其法：以上生者，四其實，三其法。 所以明陽下生陰，陰上生陽。
Sima Qian's eight monographs already roughed out the theory, as the earlier standard histories show. There the primordial taiji breath wraps triads in unity; motion begins at the branch zi, and the twelve pipes must germinate from that gate. Tripling one against chǒu, then compounding nines across the stem cycle, lands on 19,683—the "full divisor" canonized for Yellow Bell. Run the same ladder against all twelve branches and you reach 177,147, the "total dividend" stored inside Yellow Bell. Divide the inner total by the outer and the sounding length comes out nine cun—the pipe that answers the eleventh month's winter solstice breath. Yin and yang meet their virtue in zi, where qi knits the myriad things, each triply seeded. Hence every classical pipe keeps a three-fen bore, and each generation up or down the cycle lengthens or shortens by a third. The algorithm says: from the nine-cun Yellow Bell, descend by doubling the numerator while tripling the denominator, or ascend by quadrupling while still tripling the denominator— which is how yang notes beget yin and yin notes climb back to beget yang.
12
起子，為黃鐘九寸，一。
At zi the Yellow Bell line starts: a tube nine cun long, taken as unity.
13
醜，三分之二。
For chǒu the ratio is two-thirds.
14
寅，九分之八。
For yín the ratio is eight-ninths.
15
卯，二十七分之十六。
For mǎo it is sixteen parts out of twenty-seven.
16
辰，八十一分之六十四。
For chén, sixty-four over eighty-one.
17
巳，二百四十三分之一百二十八。
For sì, one hundred twenty-eight parts in two hundred forty-three.
18
午，七百二十九分之五百一十二。
For wǔ, five hundred twelve over seven hundred twenty-nine.
19
未，二千一百八十七分之一千二十四。
For wèi, one thousand twenty-four over two thousand one hundred eighty-seven.
20
申，六千五百六十一分之四千九十六。
For shēn, four thousand ninety-six over six thousand five hundred sixty-one.
21
酉，一萬九千六百八十二分之八千一百九十二。
For yǒu, eight thousand one hundred ninety-two parts in nineteen thousand six hundred eighty-two.
22
戌，五萬九千四十九分之三萬二千七百六十八。
For xū, thirty-two thousand seven hundred sixty-eight over fifty-nine thousand forty-nine.
23
亥，十七萬七千一百四十七分之六萬五千五百三十六。
For hài, sixty-five thousand five hundred thirty-six over one hundred seventy-seven thousand one hundred forty-seven.
24
如是周十二辰，在六律為陽，則當位自得而下生陰，在六呂為陰，則得其所衡而上生於陽，推算之術無重上生之法也。 所謂律取妻，呂生子，陰陽升降，律呂之大經也。 而遷又言十二律之長，今依淮南九九之數，則蕤賓為重上。 又言五音相生，而以宮生角，角生商，商生徵，徵生羽，羽生宮。 求其理用，罔見通途。
Run the circuit of twelve branches and you see the rule: each yang pipe sits in its own seat and drops a third to sire yin; each yin pipe reaches across to its partner and climbs a third to sire yang—and nowhere does the math allow a second consecutive ascent. That is the old formula 'yang pipes wed, yin notes bear children'—the rise and fall of yin and yang that governs every pipe relationship. Sima Qian adds that, measured by the Huainan nine-times table, Ruíbīn would be asked to 'ascend twice,' which breaks the rule. He also orders the pentatonic cycle as gong → jue → shang → zhi → yu → gong—a sequence that does not match the thirds method. Applied to practice, that account never lines up end to end.
25
及元始中，王莽輔政，博征通知鐘律者，考其音義，使羲和劉歆典領調奏。 班固《漢書》采而志之，其序論雖博，而言十二律損益次第，自黃鐘長九寸，三分損一，下生林鐘，長六寸。 三分益一，上生太蔟而左旋，八八為位。 一上一下，終於無射，下生中呂。 校其相生所得，與司馬遷正同。 班固采以為志。
Under Wang Mang's Yuanshi regency the court cast a wide net for acousticians, reviewed their theories, and put Liu Xin of the Xihe office in charge of the tuning project. Ban Gu folded Liu Xin's work into the Hanshu monograph: his introduction is learned, yet the step-by-step thirds—Yellow Bell at nine cun, subtract a third down to Forest Bell at six cun— then add a third up to Tàicù, wheel counterclockwise, and let eight-times-eight mark each station— each step alternating up and down until Wúyì, then the last descent lands on Zhònglǚ. Carry the arithmetic through and you land on the same lengths Sima Qian gave. Ban Gu took that scheme as his standard.
26
元帝時，郎中京房知五音六十律之數，上使太子傅玄成、諫議大夫章雜試問房于樂府，房對：「受學于故小黃令焦延壽。 六十律相生之法：以上生下，皆三生二； 以下生上，皆三生四。 陽下生陰，陰上生陽，終於中呂，而十二律畢矣。 中呂上生執始，執始下生去滅。 上下相生，終於南事，而六十律畢矣。 夫十二律之變至於六十，猶八卦之變至於六十四也。 宓犧作《易》，紀陽氣之初以為律法。 建日冬至之聲，以黃鐘為宮，太蔟為商，姑洗為角，林鐘為徽，南呂為羽，應鐘為變宮，蕤賓為變徵，此聲氣之元，五音之正也。 故各統一日，其餘以次運行，當日者各自為宮，而商角徽羽以類從焉。 《禮運》曰「五聲、六律、十二管還相為宮」，此之謂也。 以六十律分期之日，黃鐘自冬至始，及冬至而復，陰陽、寒燠、風雨之占生焉。 於以檢攝群音，考其高下，苟非革木之聲，則無不有所合。 《虞書》曰「律和聲，此之謂也。」
Emperor Yuan had Jing Fang—who had mastered the sixty-pipe extension of the pentatonic set—interrogated at the imperial music bureau by Xuan Cheng and Zhang Za; Fang said he had learned the art from Jiao Yanshou, the former Lesser Huang prefect. Fang's rule for stepping down: from a longer parent pipe, three always begets two; for stepping up, three always begets four. Yang drops to yin, yin climbs back to yang, and the ring closes on Zhònglǚ with the classical twelve. From Zhònglǚ the cycle continues up to Zhǐshǐ and down to Qùmiè— —until Nánshì finishes the fifth layer of splits and the full sixty names are filled out. Twelve pipes unfolding to sixty parallels the eight trigrams unfolding to sixty-four lines. Fuxi wrote the "Zhouyi" by pinning the first yang breath of the year—the same logic that underlies the pipe numbers. He keyed the winter solstice to Yellow Bell on gong, Tàicù on shang, Gǔxǐ on jue, Forest Bell on zhi, Southern Regulator on yu, with Yìngzhōng and Ruíbīn as the two altered steps—the seed tones of the whole system. Each pitch governs its own day while the others march around it; whichever pitch heads the day acts as temporary gong, and the four companions fall in by class. That is the sense of the "Record of Rites" line that the five voices and twelve tubes 'revolve as gong.' Mapping the sixty mutations across the year begins at the winter solstice node on Yellow Bell; when the node comes round again, you read yin-yang, temperature, and weather from the same cycle. With that grid you can police every instrument: aside from hide and wood drums, every timbre should click into one of the slots. Shun's canon meant exactly this when it said the pipes "harmonize the sounds."
27
京房又曰：「竹聲不可以度調，故作準以定數。 准之狀如瑟，而長丈，十三弦，隱間九尺，以應黃鐘之律九寸。 中央一弦，下有畫分寸，以為六十律清濁之節。」 房言律詳於歆所奏，其術施行于史官，候部用之，文多不悉載。 截管為律，吹以考聲，列以效氣，道之本也。 術家以其聲微而體難知，其分數不明，故作準以代之。 准之聲明暢易達，分寸又粗，然弦以緩急清濁，非管無以正也。 均其中弦，令與黃鐘相得，案畫以求諸律，則無不如數而應者矣。 《續漢志》具載其六十律准度數，其相生之次與《呂覽》、《淮南》同。
Fang added that bamboo notes cannot be cut by abstract measure alone, so he built a string "standard" (zhun) to nail the fractions down. The device looks like a small se, a full zhàng in length, strung thirteen times across a sounding length of nine chǐ—mirroring the nine-cun Yellow Bell tube. The middle string is ruled in fen and cun so every semitone in the sixtyfold cycle has a visible notch. Fang's treatise outruns Liu Xin's memorial; the observatories adopted his zhun even though the full proof is too long for this chapter. Sawing a pipe, blowing to hear it, lining it up to read the qi—that is still the bedrock method. Because a tube whispers and its bore is hard to measure, technicians reach for the string standard instead. Strings speak clearly and coarse marks are easy to read, yet only a finished pipe can correct what the ear accepts as true pitch. Match the middle string to Yellow Bell, trace the inked ticks outward, and every derived length falls on the predicted number. Fan Ye's continuation of the Han treatise preserves every zhun length; the breeding order matches both the "Lüshi chunqiu" and the "Huainanzi."
28
漢章帝元和元年，待詔候鐘律殷肜上言：「官無曉六十律以准調音者。 故待詔嚴崇具以准法教子男宣，原召宣補學官，主調樂器。」 詔曰：「崇子學審曉律，別其族，協其聲者，審試。 不得依託父學，以聾為聰。 聲微妙，獨非莫知，獨是莫曉。 以律錯吹，能知命十二律不失一，乃為能傳崇學耳。」 試宣十二律，其二中，其四不中，其六不知何律，宣遂罷。 自此律家莫能為准。
In Han Zhangdi's Yuanhe 1 the expectant official Yin Rong, charged with bells and pipes, reported that no musician left could align the sixty mutations with Fang's string standard. Yan Chong had already drilled the method into his son Xuan; Yin asked the throne to call Xuan in as a student officer to tune the orchestra. The rescript answered: if the son really knows intervals, can sort families of notes, and can lock them in ensemble, test him rigorously— —and do not let him hide behind his father's reputation. Acoustics is subtle: outsiders never hear the fault, insiders alone notice the fit. Have him blind-test the twelve pipes; only if he names every length without a slip may he inherit Yan Chong's chair. Xuan flubbed the exam—two hits, four misses, six pipes he could not name—so the court sent him home. After that debacle nobody could credibly wield the zhun.
29
靈帝熹平六年，東觀召典律者太子舍人張光等問准意，光等不知，歸閱舊藏，乃得其器。 形制如房書，猶不能定其弦緩急。 音，不可書以曉人，知之者欲教而無從，心達者體知而無師，故史官能辨清濁者遂絕。 其可以相傳者，唯候氣而已。
In Lingdi's Xiping 6 the Eastern Pavilion called Zhang Guang and other experts to explain the zhun; they confessed ignorance, ransacked the archives, and finally turned the hardware up. It looked exactly like Fang's drawing, yet they still could not agree how taut each string should be. Tone refuses to live on paper: masters who know cannot lecture it, savants who feel it cannot copy it—so the last archivists who could hear clean from muddy simply disappeared. All that survived as a teachable rite was the ash-in-jar solstice test.
30
漢末紛亂，亡失雅樂。 魏武時，河南杜夔精識音韻，為雅樂郎中，令鑄銅工柴玉鑄鐘，其聲均清濁多不如法，數毀改作，玉甚厭之，謂夔清濁任意，更相訴白于魏武王。 魏武王取玉所鑄鐘雜錯更試，然後知夔為精，於是罪玉。
When Han collapsed, courtly "ya" music went missing in the chaos. Cao Cao's court relied on Du Kui of Henan, a connoisseur of timbre, as gentleman of ya music; Du told the founder Chai Yu to cast ritual bells, but the bronze kept missing the thirds rule and had to be smashed and remelted until Chai accused Du of arbitrary tuning and both men appealed to Cao Cao. Cao Cao personally shuffled Chai's bells back into the set, listened again, and decided Du had been right all along—Chai paid for the false accusation.
31
泰始十年，中書監荀勖、中書令張華出禦府銅竹律二十五具，部太樂郎劉秀等校試，其三具與杜夔及左延年律法同，其二十二具，視其銘題尺寸，是笛律也。 問協律中郎將列和，辭：「昔魏明帝時，令和承受笛聲以作此律，欲使學者別居一坊，歌詠講習，依此律調。 至於都合樂時，但識其尺寸之名，則絲竹歌詠，皆得均合。 歌聲濁者用長笛長律，歌聲清者用短笛短律。 凡弦歌調張清濁之制，不依笛尺寸名之，則不可知也。」
In Jin Wudi's tenth year of Taishi Xun Xi and Zhang Hua hauled twenty-five bronze and bamboo measures from the palace vault for Liu Xiu to audition: three matched Du Kui and Zuo Yanian's law, while twenty-two bore flute dimensions carved on the shank. They pressed General Lie He, who said Emperor Ming of Wei once told him to derive the pitch set from flute fingerings and house students in a separate compound to practice songs to those fingerings. In full ensemble you only read the length labels and every string, pipe, and voice lines up. Dark voices take the long bore; bright voices take the short. Anyone tuning zithers or voices without those flute marks is working blind.
32
勖等奏：「昔先王之作樂也，以振風蕩俗，饗神祐賢，必協律呂之和，以節八音之中。 是故郊祀朝宴，用之有制，歌奏分獻，清濁有宜。 故曰「五聲、十二律還相為宮」，此經傳記籍可得知者也。 如和對辭，笛之長短無所象則，率意而作，不由曲度。 考以正律，皆不相應； 吹其聲均，多不諧合。 又辭'先師傳笛，別其清濁，直以長短。 工人裁制，舊不依律。 '是為作笛無法。 而和寫笛造律，又令琴瑟歌詠，從之為正，非所以稽古先哲，垂憲於後者也。 謹條牒諸律，問和意狀如左。 及依典制，用十二律造笛象十二枚，聲均調和，器用便利。 講肄彈擊，必合律呂，況乎宴饗萬國，奏之廟堂者哉？ 雖伶夔曠遠，至音難精，猶宜儀形古昔，以求厥衷，合乎經禮，於制為詳。 若可施用，請更部笛工選竹造作，下太樂樂府施行。 平議諸杜夔、左延年律可皆留，其禦府笛正聲、下徽各一具，皆銘題作者姓名，其餘無所施用，還付禦府毀。」 奏可。
Xun Xi answered with a long memorial: the sages invented music to move wind and custom, to feed spirits and honor worthies, and they always locked the eight timbres to the twelve pipes. Hence suburban rites and palace feasts follow fixed registers—anthem and libation each get their own cup, bright and dark ranges stay distinct. That is why the classics repeat that the five degrees and twelve tubes "take turns as gong." Lie He's own confession is that his flutes follow no canonical length—they are guesswork, not measure. Set them against the bronze pipes and none lines up; blow them and the averages sound smooth yet refuse to lock with the orchestra. He also admitted that teachers "passed flutes down" by cutting longer or shorter tubes— —and workshop cutters never used the pitch law at all. In other words, there was no method. Yet from those flutes he wrote the nominal pitches and forced zithers, se, and singers to obey—hardly the way to honor antiquity or leave a pattern for later kings. We therefore tabulate every pipe and put He's claims to the test point by point. Following the classics we bored twelve bamboo flutes, one for each pipe; they tune evenly and are easy to play. If they hold for lessons and chamber music, they will surely hold when the feudatories banquet or the temple orchestra plays. Ling Lun is millennia gone and perfection is elusive, yet the court should still mimic the old forms, seek the middle course, and hew to canonical ritual—that is the thorough path. If the plan stands, commission bamboo cutters, build the set, and issue it through the Grand Music bureau. The collegial verdict: keep every pitch template by Du Kui and Zuo Yanian; from the palace flutes retain only one standard-length and one lower-register instrument, each stamped with its maker; melt the useless remainder back at the vault. The emperor approved.
33
勖又問和：「作笛為可依十二律作十二笛，令一孔依一律，然後乃以為樂不？」 和辭：「太樂東廂長笛正聲已長四尺二寸，今當復取其下征之聲。 於法，聲濁者笛當長，計其尺寸乃五尺有餘，和昔日作之，不可吹也。 又，笛諸孔雖不校試，意謂不能得一孔輒應一律也。」 案太樂四尺二寸笛正聲均應蕤賓，以十二律還相為宮，推法下征之孔當應律大呂。 大呂笛長二尺六寸有奇，不得長五尺餘。 輒令太樂郎劉秀、鄧昊等依律作大呂笛以示和，又吹七律，一孔一校，聲皆相應。 然後令郝生鼓箏，宋同吹笛，以為雜引、《相和》諸曲。 和乃辭曰：「自和父祖漢世以來，笛家相傳，不知此法，而令調均與律相應，實非所及也。」 郝生、魯基、種整、硃夏皆與和同。
Xun pressed again: must twelve holes on twelve tubes—one hole per pipe—be the definition of correct music? Lie answered that the eastern-gallery concert flute is already four chǐ two cùn, yet the court now wants its subdominant register as well. A dull register demands a longer bore—over five chǐ by the math—and the prototype I once cut could not be blown at all. Besides, he said, without empirical tests you cannot expect every finger hole to land on a separate pipe tone. Auditors replied: the four-chi-two concert flute centers on Ruíbīn; if you rotate the twelve pipes as temporary gong, the lower-zhi vent should answer Dàlǚ. A Dàlǚ bore is only a little past two chǐ six cùn—nowhere near five. The throne told Liu Xiu and Deng Hao to cut a Dàlǚ flute to spec, then pipe the seven degrees hole by hole—each matched the law. They paired Hao Sheng's zheng with Song Tong's flute on preludes and the "Harmony" suites. Lie He finally conceded that Han-era flute families had never heard of such precision, yet the new instruments sang dead true—beyond anything he could match. Hao Sheng, Lu Ji, Zhong Zheng, and Zhu Xia sided with Lie He.
34
又問和：「笛有六孔，及其體中之空為七，和為能盡名其宮商角徵不？ 孔調與不調，以何檢知？」 和辭：「先師相傳，吹笛但以作曲，相語為某曲當舉某指，初不知七孔盡應何聲也。 若當作笛，其仰尚方笛工依案舊像訖，但吹取鳴者，初不復校其諸孔調與不調也。」 案《周禮》調樂金石，有一定之聲，是故造鐘磬者先依律調之，然後施於廂懸。 作樂之時，諸音皆受鐘磬之均，即為悉應律也。 至於饗宴殿堂之上，無廂懸鐘磬，以笛有一定調，故諸弦歌皆從笛為正，是為笛猶鐘磬，宜必合於律呂。 如和所對，直以意造，率短一寸，七孔聲均，不知其皆應何律，調與不調，無以檢正，唯取竹之鳴者，為無法制。 輒部郎劉秀、鄧昊、王豔、魏邵等與笛工參共作笛，工人造其形，律者定其聲，然後器象有制，音均和協。
The board asked whether he could even name gong, shang, jue, and zhǐ for the six holes plus the bore. And how would he prove each vent was in tune? Lie admitted masters only traded finger charts—"lift the third finger here"—with no theory tying each hole to a pitch name. Imperial flute makers still copy the old outline and quit as soon as the tube whistles, never calibrating the holes. The "Rites of Zhou" says bronze and stone must be tuned to fixed pitches before they hang in the bell galleries. In performance every color listens back to the bells and chimes, which means everything is tied to the twelve pipes. When the banquet hall lacks suspended bells, the flute becomes the temporary standard: strings and singers follow it exactly as they would follow the gallery chimes—so that lone bamboo must still obey the twelve pipes. Lie He's method amounts to shaving every tube roughly a cun short, drilling seven even holes, and never checking which pipe each hole should speak—if it blows, he ships it. That is not a method at all. The court then paired Liu Xiu, Deng Hao, Wang Yan, and Wei Shao with palace flute makers: carpenters cut the bamboo while acousticians set the scale, so each tube had both proper form and true pitch.
35
又問和：「若不知律呂之義作樂，音均高下清濁之調，當以何名之？」 和辭：「每合樂時，隨歌者聲之清濁，用笛有長短。 假令聲濁者用三尺二笛，因名曰此三尺二調也； 聲清者用二尺九笛，因名曰此二尺九調也。 漢魏相傳，施行皆然。」 案《周禮》奏六樂，乃奏黃鐘，歌大呂； 乃奏太蔟，歌應鐘，皆以律呂之義，紀歌奏清濁。 而和所稱以二尺，三尺為名，雖漢魏用之，俗而不典。 部郎劉秀、鄧昊等以律作笛，三尺二寸者應無射之律，若宜用長笛，執樂者曰請奏無射； 二尺八寸四分四厘應黃鐘之律，若宜用短笛，執樂者曰請奏黃鐘。 則歌奏之義，若合經禮，考之古典，於制為雅。
They pressed Lie He again: without knowing what the twelve pipes mean, what name could you give to "bright" or "dark" tuning? Lie said the band simply grabbed a longer or shorter flute to match the vocalist. A thick voice gets a three-chi-two tube, and everyone nicknames that gauge "the three-chi-two mode"; a bright voice gets a two-chi-nine tube, nicknamed "the two-chi-nine mode." That folk habit, he claimed, ran unchanged from Han through Wei. The "Rites of Zhou" orders the six suites as "sound Yellow Bell, sing Dàlǚ"; then "sound Tàicù, sing Yìngzhōng"—every pairing is named from the pipes, not from bamboo lengths. Lie He's foot-and-inch nicknames may have been common, but they are not canonical nomenclature. Liu Xiu's team shows that a three-chi-two tube matches Wúyì, so the director can honestly call, "Play Wúyì" when the chart demands the long pipe; while two chǐ eight cùn four fēn four lí matches Yellow Bell, so the call becomes "Play Yellow Bell" when the short bore is needed. Song and accompaniment named that way line up with the ritual classics—exactly the elegant usage the canon expects.
36
《書》曰：「予欲聞六律、五聲、八音，在治忽。」 《周禮》、《國語》載六律六同，《禮記》又曰「五聲、十二律還相為宮」。 劉歆、班固撰《律曆志》亦紀十二律，惟京房始創六十律。 至章帝時，其法己絕，蔡邕追紀其言，亦曰今無能為者。 依案古典及今音家所用，六十律者無施于樂。 謹依典記，以五聲、十二律還相為宮之法，制十二笛象，記注圖側，如別，省圖，不如視笛之孔，故復重作蕤賓伏孔笛。 其制云：
The "Canon of Documents" says Shun wanted "the six pipes, the five notes, and the eight timbres"—the acoustic mirror of good and bad government. The "Rites of Zhou" and "Discourses of the States" pair six lǜ with six tóng; the "Record of Rites" repeats that five voices and twelve tubes rotate as gong. Liu Xin and Ban Gu's monograph still tracks twelve pipes; Jing Fang alone extended the cycle to sixty. By Zhangdi's reign the sixtyfold art was dead; Cai Yong copied Jing Fang's boast yet admitted no living master could rebuild it. Between ancient manuals and modern orchestras, the sixty mutations never reach the stage. We therefore built twelve demonstration flutes keyed to the classical "five over twelve" cycle, sketched them with marginal notes, and—because ink diagrams mislead the eye—recut a Ruibīn flute with its muted vent to show the holes directly. The specifications run as follows:
37
黃鐘之笛，正聲應黃鐘，下徵應林鐘，長二尺八寸四分四厘有奇。 正聲調法，以黃鐘為宮，則姑洗為角，翕笛之聲應姑洗，故以四角之長為黃鐘之笛也。 其宮聲正而不倍，故曰正聲。
The Yellow Bell tube's concert register centers on Yellow Bell, its subdominant on Forest Bell, and its sounding length is two chǐ eight cùn four fēn four lí plus a remainder. In concert tuning, Yellow Bell is gong and Gǔxǐ is jue; the half-stopped bore must speak Gǔxǐ, so the tube's length equals the sum of the Yellow Bell and Gǔxǐ pipe lengths measured from the mouth end. Because the fundamental is neither doubled nor halved, musicians call this register the "true" or concert pitch.
38
正聲調法：黃鐘為宮，第一孔也。 應鐘為變宮，第二孔也。 南呂為羽，第三孔也。 林鐘為徵，第四孔也。 蕤賓為變徵，第五附孔也。 姑洗為角，笛體中聲。 太蔟為商。 笛後出孔也。 商聲濁於角，當在角下，而角聲以在體中，故上其商孔，令在宮上，清于宮也。 然則宮商正也，餘聲皆倍也； 是故從宮以下，孔轉下轉濁也。 此章記笛孔上下次第之名也。 下章說律呂相生，笛之制也。 正聲調法，黃鐘為宮。 作黃鐘之笛，將求宮孔，以始洗及黃鐘律，從笛首下度之，盡二律之長而為孔，則得宮聲也。 宮生徵，黃鐘生林鐘也。 以林鐘之律從宮孔下度之。 盡律作孔，則得徵聲也。 徵生商，林鐘生太蔟也。 乙太蔟律從徵孔上度之，盡律以為孔，則得商聲也。 商生羽，太蔟生南呂也。 以南呂律從商孔下度之，盡律為孔，則得羽聲也。 羽生角，南呂生姑洗也。 以姑洗律從羽孔上行度之，盡律而為孔，則得角聲也。 然則出於商孔之上，吹笛者左手所不及也。 從羽孔下行度之，盡律而為孔，亦得角聲，出於商附孔之下，則吹者右手所不逮也，故不作角孔。 推而下之，復倍其均，是以角聲在笛體中，古之制也。 音家舊法，雖一倍再倍，但令均同，適足為唱和之聲，無害于曲均故也。 《國語》曰，匏竹利制，議宜，謂便於事用從宜者也。 角生變宮，姑洗生應鐘也。 上句所謂當為角孔而出於商上者，墨點識之，以應鐘律。 從此點下行度之，盡律為孔，則得變宮之聲也。 變宮生變徵，應鐘生蕤賓也。 以蕤賓律從變宮下度之，盡律為孔，則得變征之聲。 十二笛之制，各以其宮為主，相生之法，或倍或半，其便事用，例皆一也。
Hole one, counting from the mouth, is gong on Yellow Bell. Hole two is the altered gong on Yìngzhōng. Hole three is yu on Nánlǚ. Hole four is zhǐ on Línzhōng. Ruibīn's altered zhǐ sits on the fifth side hole. Gǔxǐ's jue sounds from the air column inside the bore. Tàicù's shang vents through the hole behind the player's hands. Shang should sit lower than jue, yet jue lives inside the tube, so the shang vent is punched above gong to keep it brighter than the fundamental. Gong and shang stay on the true pipe; every other step borrows an octave duplicate; so each hole farther from the mouth sounds duller than the last. This passage only names the holes from crown to foot. The following section explains how the pipes beget one another—the geometry of the flute. Concert method again: Yellow Bell on gong. Mark gong by measuring from the blowing edge the combined lengths of the Gǔxǐ and Yellow Bell tubes; where that tally ends, drill gong. Gong generates zhǐ, so Yellow Bell spawns Forest Bell. Drop the Forest Bell length below the gong vent. Drill zhǐ where that measurement stops. Zhǐ generates shang, so Forest Bell spawns Tàicù. Measure up from the zhǐ hole by the Tàicù length; the next stop is shang. Shang generates yu, so Tàicù spawns Nánlǚ. Down from shang by the Nánlǚ length comes yu. Yu generates jue, so Nánlǚ spawns Gǔxǐ. Up from yu by the Gǔxǐ length lands jue. That point would sit above the shang hole—where the player's left hand cannot reach. Measuring down from yu by the same length also yields jue, but the vent would fall under the thumb hole where the right hand cannot cover it, so no separate jue hole is bored. Doubling the air column downward parks jue inside the bore—exactly the classical layout. Older masters doubled or halved lengths freely as long as ensemble balance held, because duets only need matching color, not a fresh hole for every partial. The "Discourses" line that gourd and bamboo "yield to fit and follow convenience" means just this: build what the band can actually finger. Jue spawns altered gong, so Gǔxǐ leads to Yìngzhōng. Where theory wants a jue hole above shang, draftsmen ink a dot sized to the Yìngzhōng length. Drill downward from that dot by the Yìngzhōng pipe and you capture altered gong. Altered gong leads to altered zhǐ, Yìngzhōng to Ruibīn. Drop the Ruibīn length from altered gong to locate altered zhǐ. Each of the twelve model flutes uses its namesake as temporary gong; whether you double or halve lengths, the workshop rule stays the same. Subdominant tuning makes Forest Bell gong on the fourth hole.
39
下徵調法：林鐘為宮，第四孔也。 本正聲黃鐘之徵。 徵清，當在宮上，用笛之宜，倍令濁下，故曰下徵。 下徵更為宮者，《記》所謂「五聲，十二律還相為宮」也。 然則正聲清，下徵為濁也。 南呂為商，第三孔也。 本正聲黃鐘之羽，今為下征之商也。 應鐘為角，第二孔也。 本正聲黃鐘之變宮，今為下征之角也。 黃鐘為變徵，下征之調，林鐘為宮，大呂當為變徵，而黃鐘笛本無大呂之聲，故假用黃鐘以為變徵也。 假用之法，當為變征之聲，則俱髮黃鐘及太蔟、應鐘三孔。 黃鐘應濁而太蔟清，大呂律在二律之間，俱發三孔而徵磑蒦之，則得大呂變征之聲矣。 諸笛下徵調求變征之法，皆如此也。 太蔟為徵，笛後出孔。 本正聲之商，今為下征之徵也。 姑洗為羽，笛體中翕聲。 本正聲之角，今為下征之羽。 蕤賓為變宮。 附孔是也。 本正聲之變徵也，今為下征之變宮也。 然則正聲之調，孔轉下轉濁，下征之調，孔轉上轉清也。
That hole was zhǐ in the concert Yellow Bell chart. True zhǐ should sit above gong, but the flute doubles it down an octave for ergonomics, so players call the mode "lower zhǐ." Letting that lowered zhǐ rotate into gong is precisely the "five notes, twelve tubes take turns as gong" rule from the "Record of Rites." Concert pitch stays bright; the lowered mode is the darker twin. Hole three becomes shang on Nánlǚ. It used to be yu in the concert chart; in lower zhǐ it is shang. Hole two is jue on Yìngzhōng. Formerly altered gong in the concert set; now jue in the lowered set. Lower zhǐ needs Dàlǚ as altered zhǐ while Forest Bell is gong, but the Yellow Bell bore has no Dàlǚ slot, so acousticians fake the step with Yellow Bell itself. The trick is to lift Yellow Bell, Tàicù, and Yìngzhōng together when altered zhǐ is required. Yellow Bell speaks dull, Tàicù bright; Dàlǚ lives between them. Half-covering all three vents and shading the airstream splits the difference, approximating Dàlǚ's altered zhǐ—though the manuscript is corrupt at the verb describing the embouchure. Every lower-zhǐ flute uses the same three-hole shading for its borrowed altered zhǐ. Tàicù supplies zhǐ on the rear hole. It was shang in the concert chart; now zhǐ in the lowered chart. Gǔxǐ's yu rings from the stopped column inside the bore. It was jue in the concert chart; now yu in the lowered chart. Ruibīn hosts altered gong. That is the side vent. It was altered zhǐ upstairs; downstairs it becomes altered gong. So concert tuning darkens as you march toward the foot, while lowered zhǐ brightens as you march back toward the crown. The "clear jue" recipe makes Gǔxǐ the temporary gong—that is, the stopped air column inside the tube.
40
清角之調：以姑洗為宮，即是笛體中翕聲。 于正聲為角，於下徵為羽。 清角之調乃以為宮，而哨吹令清，故曰清角。 惟得為宛詩謠俗之曲，不合雅樂也。 蕤賓為商，正也。 林鐘為角，非正也。 南呂為變徵，非正也。 應鐘為徵，正也。 黃鐘為羽，非正也。 太蔟為變宮。 非正也。 清角之調，唯宮、商及徵與律相應，餘四聲非正者皆濁，一律哨吹令清，假而用之，其例一也。
The "clear jue" key treats the stopped column inside the bore as temporary gong on Gǔxǐ. In the concert chart that column is jue; in the lowered-zhǐ chart it is yu. Players treat that column as gong and overblow it bright, which is why musicians call the mode "clear jue." It works for street songs and common airs, not for court ya music. Ruibīn as shang is the canonical assignment. Forest Bell as jue is already a compromise step. Nánlǚ as altered zhǐ is another non-canonical rung. Yìngzhōng as the zhi degree is the proper assignment here. Yellow Bell as yu is again an expedient, not the classical mapping. Tàicù supplies the altered gong step. Again, not a canonical pairing. Under the clear-jue key only gong, shang, and zhǐ lock to the bronze standard; the other four steps stay dark until a player overblows them into tune.
41
凡笛體用律，長者八之，蕤賓、林鐘也。 短者四之。 其餘十笛，皆四角也。 空中實容，長者十六。 短笛竹宜受八律之黍也。 若長短大小不合於此，或器用不便聲均法度之齊等也。 然笛竹率上大下小，不能均齊，必不得已，取其聲均合。 三宮，一曰正聲，二曰下徵，三曰清角也。 二十一變也。 宮有七聲，錯綜用之，故二十一變也。 諸笛例皆一也。 伏孔四，所以便事用也。 一曰正角，出於商上者也，二曰倍角，近笛下者也，三曰變宮，近于宮孔，倍令下者也； 四曰變徵，遠于徵孔，倍令高者也。 或倍或半，或四分一，取則於琴徽也。 四者皆不作其孔，而取其度，以應退上下之法，所以協聲均，便事用也。 其本孔隱而不見，故曰伏孔也。
Long-chamber flutes scale by eight against the pipe length—Ruibīn and Forest Bell fall in that class. Short chambers multiply by four instead. The other ten models all take the "corner" ratio as their yardstick. Interior volume for the long set follows the sixteenfold rule. Short-flute culms need a bore that holds eight pipes' worth of millet by measure. If you ignore those ratios or pick clumsy bamboo, timbre and measure fall out of step, and real culms taper from fat crown to narrow foot—when theory and wood disagree, chase an even sound rather than perfect arithmetic. The three usable modes are concert pitch, lowered zhǐ, and brightened jue. Together they yield twenty-one modulating steps. Because gong carries seven usable degrees that can be permuted, the full set of modulations comes to twenty-one. Every flute model follows the same pattern. Four "buried" vents are cut for practical fingering. First comes true jue venting above shang, then doubled jue down by the foot, then altered gong tucked beside the gong hole and dropped an octave— and finally altered zhǐ offset beyond the zhǐ hole and raised an octave. The offsets follow zither frets: double, half, or add a quarter— —without visible bores, only measured lengths that obey the up-and-down rules so the scale evens out and the hands stay nimble. Because those roots stay invisible, builders call them "hidden" vents.
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大呂之笛，正聲應大呂，下徵應夷則，長二尺六寸六分三厘有奇。
The Dàlǚ template answers Dàlǚ in concert pitch and Yízé in lowered zhǐ, with a tube of two chǐ six cùn six fēn three lí plus a remainder.
43
太蔟之笛，正聲應太蔟，下徵應南呂，長二尺五寸三分一厘有奇。
The Tàicù pipe locks to Tàicù and Nánlǚ at two chǐ five cùn three fēn one lí plus a remainder.
44
夾鐘之笛，正聲應夾鐘，下徵應無射，長二尺四寸。
The Jiázhōng model answers Jiázhōng and Wúyì at two chǐ four cùn.
45
姑洗之笛，正聲應姑洗，下徵應應鐘，長二尺二寸三分三厘有奇。
The Gǔxǐ flute answers Gǔxǐ and Yìngzhōng at two chǐ two cùn three fēn three lí plus a remainder.
46
蕤賓之笛，正聲應蕤賓，下徵應大呂，長三尺九寸九分五厘有奇。 變宮近宮孔，故倍半令下，便於用也。 林鐘亦如之一。 林鐘之笛，正聲應林鐘，下徵應太蔟，長三尺七寸九分七厘有奇。
The Ruibīn bore answers Ruibīn and Dàlǚ at three chǐ nine cùn nine fēn five lí plus a remainder. Altered gong hugs the main hole, so the pattern drops it an octave for the player's thumb. Forest Bell uses the same altered-gong trick. The Forest Bell template answers Línzhōng and Tàicù at three chǐ seven cùn nine fēn seven lí plus a remainder.
47
夷則之笛，正聲應夷則，下徵應夾鐘，長三盡六寸。 變宮之法，亦如蕤賓，體用四角，故四分益一也。
The Yízé pipe answers Yízé and Jiázhōng; the received text's "three exhaust six cun" is emended to three chǐ six cùn. Altered gong on this flute copies Ruibīn's corner-length body, so the math adds a fourth.
48
南呂之笛，正聲應南呂，下徵姑洗，長三尺三寸七分有奇。
The Nánlǚ model answers Nánlǚ and Gǔxǐ at three chǐ three cùn seven fēn plus a remainder (the line omits the usual "answers" verb before Gǔxǐ).
49
無射之笛，正聲應無射，下徵應中呂，長三尺二寸。
The Wúyì bore answers Wúyì and Zhònglǚ at three chǐ two cùn.
50
應鐘之笛，正聲應應鐘，下徵應蕤賓，長二尺九寸九分六厘有奇。
The Yìngzhōng flute answers Yìngzhōng and Ruibīn at two chǐ nine cùn nine fēn six lí plus a remainder.
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五音十二律
Five degrees and twelve classical pipes.
52
土音宮，數八十一，為聲之始。 屬土者，以其最濁，君之象也。 季夏之氣和，則宮聲調。 宮亂則荒，其君驕。 黃鐘之宮，律最長也。
The earth class is gong, tallied at eighty-one, where acoustics begins. Earth sounds stay darkest, like the sovereign's steady voice. When the last month of summer's qi is balanced, gong rings true. When gong slips, the land feels empty and the king turns proud. Yellow Bell's gong carries the longest tube in the set.
53
火音徵，三分宮去一以生，其數五十四。 屬火者，以其徵清，事之象也。 夏氣和，則徵聲調。 徵亂則哀，其事勤也。
The fire class is zhǐ, born by taking a third from gong, numbered fifty-four. Fire sounds are bright, like the busy work of government. When midsummer qi is balanced, zhǐ rings true. When zhǐ slips, music turns mournful and every task feels like drudgery.
54
金音商，三分徵益一以生，其數七十二。 屬金者，以其濁次宮，臣之象也。 秋氣和，則商聲調。 商亂則詖，其官壞也。
Metal is shang, born by adding a third to zhǐ, numbered seventy-two. Metal sits just below gong in darkness, like the minister's supporting voice. When autumn qi is balanced, shang rings true. When shang slips, factions form and the bureaucracy rots.
55
水音羽，三分商去一以生，其數四十八。 屬水者，以為最清，物之象也。 冬氣和，則羽聲調。 羽亂則危，其財匱也。
Water is yu, born by taking a third from shang, numbered forty-eight. Water sounds are the brightest in the pentad, like the myriad things in clarity. When winter qi is balanced, yu rings true. When yu slips, danger looms and the treasury runs dry.
56
木音角，三分羽益一以生，其數六十四。 屬木者，以其清濁中，人之象也。 春氣和，則角聲調。 角亂則憂，其人怨也。
Wood is jue, born by adding a third to yu, numbered sixty-four. Wood timbre mediates bright and dark, like common folk between ruler and minister. When spring qi is balanced, jue rings true. When jue slips, anxiety spreads and the people grow bitter.
57
凡聲尊卑，取象五行，數多者濁，數少者清； 大不過宮，細不過羽。
Pitch nobility mirrors the five phases: big tallies sound dull, small tallies bright— the bass anchor never passes gong, the top shimmer never passes yu.
58
十一月，律中黃鐘，律之始也，長九寸。 仲冬氣至，則其律應，所以宣養六氣九德也。 班固三分損一，下生林鐘。
The eleventh month sits on Yellow Bell, the head of the twelve pipes, nine cùn long. Midwinter's breath makes that pipe speak, broadcasting and feeding the six vapors and nine virtues. Ban Gu's cycle subtracts a third down to Forest Bell.
59
十二月，律中大呂，司馬遷未下生之律，長四寸二百四十三分寸之五十二，倍之為八寸二百四十三分寸之一百四。 季冬氣至，則其律應，所以助宣物也。 三分益一，上生夷則； 京房三分損一，下生夷則。
The twelfth month takes Dàlǚ, Sima Qian's "not yet down-generated" length of four cùn 52/243, doubled to eight cùn one hundred four parts. Late winter's qi wakes Dàlǚ so it can push the year's growth outward. Add a third upward and you reach Yízé; Jing Fang instead subtracts a third downward to Yízé.
60
正月，律中太蔟，未上生之律，長八寸。 孟春氣至，則其律應，所以贊陽出滯也。 三分損一，下生南呂。
The first month keys to Tàicù, the "up from wèi" generated class, eight cùn long (the manuscript graph is the wei stem character, here read as the wèi-branch class). Early spring's qi hits Tàicù so the yang is praised and stagnation shaken loose. Subtract a third downward to reach Nánlǚ.
61
二月，律中夾鐘，酉下生之律，長三寸二千一百八十七分寸之一千六百三十一，倍之為七寸二千一百八十七分寸之一千七十五。 仲春氣至，則其律應，所以出四隙之細也。 三分益一，上生無射； 京房三分損一，下生無射。
The second month takes Jiázhōng, the "down from yǒu" length of three cùn one thousand six hundred thirty-one two-thousand-one-hundred-eighty-sevenths, doubled to seven cùn one thousand seventy-five parts. Mid-spring's qi answers Jiázhōng, threading subtlety through the four gaps of the year. Add a third upward to reach Wúyì; Jing Fang subtracts a third downward to Wúyì.
62
三月，律中姑洗，酉上生之律，長七寸九分寸之一。 季春氣至，則其律應，所以修絜百物，考神納賓也。 三分損一，下生應鐘。
The third month keys to Gǔxǐ, the "up from yǒu" class, seven cùn and one-ninth cùn long. Late spring's qi strikes Gǔxǐ to rinse offerings, draw spirits, and welcome guests. Drop a third downward to Yìngzhōng.
63
四月，律中中呂，亥下生之律，長三寸萬九千六百八十三分寸之六千四百八十七，倍之為六寸萬九千六百八十三分寸之萬二千九百七十四。 孟夏氣至，則其律應，所以宣中氣也。
The fourth month takes Zhònglǚ, the "down from hài" length of three cùn six thousand four hundred eighty-seven nineteen-thousand-six-hundred-eighty-thirds, doubled to six cùn twelve thousand nine hundred seventy-four parts. Early summer's qi hits Zhònglǚ to vent the qi pooled at the year's hinge.
64
五月，律中蕤賓，亥上生之律，長六寸八十一分寸之二十六。 仲夏氣至，則其律應，所以安靜人神，獻酬交酢也。 三分損一，下生大呂； 京房三分益一，上生大呂。
The fifth month keys to Ruibīn, the "up from hài" pipe six cùn and twenty-six eighty-firsts long. Midsummer's qi answers Ruibīn, the pitch of quiet worship and pledging cup after cup. Subtract a third downward to Dàlǚ; Jing Fang adds a third upward to Dàlǚ.
65
六月，律中林鐘，醜下生之律，長六寸。 季夏氣至，則其律應，所以和展百物，俾莫不任肅純恪也。 三分益一，上生太蔟。
The sixth month takes Línzhōng, the "down from chǒu" pipe six cùn long. Late summer's qi strikes Línzhōng to harmonize every task in reverent care. Add a third upward to Tàicù.
66
七月，律中夷則，醜上生之律，長五寸七百二十九分寸之四百五十一。 孟秋氣至，則其律應，所以詠歌九則，平百姓而無貸也。 三分損一，下生夾鐘； 京房三分益一，上生夾鐘。
The seventh month keys to Yízé, the "up from chǒu" pipe five cùn four hundred fifty-one seven-hundred-twenty-ninths long. Early autumn's qi answers Yízé, the pipe that hymns the nine statutes, steadies the commoners, and brooks no slack (the phrase is "nine canons," not "nine virtues"). Subtract a third downward to Jiázhōng; Jing Fang adds a third upward to Jiázhōng.
67
八月，律中南呂，卯下生之律，長五寸三分寸之一。 仲秋氣至，則其律應，所以贊陽秀也。 三分益一，上生姑洗。
The eighth month takes Nánlǚ, the "down from mǎo" pipe five cùn and one-third cùn long. Mid-autumn's qi answers Nánlǚ, coaxing the yang into full bloom. Add a third upward to Gǔxǐ.
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九月，律中無射，卯上生之律，長四寸六千五百六十一分寸之六千五百二十四。 季秋氣至，則其律應，所以宣佈哲人之令德，示人軌儀也。 三分損一，下生中呂； 京房三分益一，上生中呂。
The ninth month keys to Wúyì, the "up from mǎo" pipe four cùn six thousand five hundred twenty-four six-thousand-five-hundred-sixty-firsts long. Late autumn's qi hits Wúyì to broadcast a sage's shining edicts and show the people their compass. Subtract a third downward to Zhònglǚ; Jing Fang adds a third upward to Zhònglǚ.
69
十月，律中應鐘，巳下生之律，長四寸二十七分寸之二十。 孟冬氣至，則其律應，所以均利器用，俾應復也。 三分益一，上生蕤賓。
The tenth month takes Yìngzhōng, the "down from sì" pipe four cùn and 20/27 long. Early winter's qi answers Yìngzhōng, evening tools and echoes so the circuit can close. Add a third upward to Ruibīn.
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淮南、京房、鄭玄諸儒言律曆，皆上下相生，至蕤賓又重上生大呂，長八寸二百四十三分寸之百四； 夷則上生夾鐘，長七寸千一百八十七分寸之千七十五； 無射上生中呂，長六寸萬九千六百八十三分寸之萬二千九百七十四； 此三品于司馬遷、班固所生之寸數及分皆倍焉，餘則並同。 斯則泠州鳩所謂六間之道，揚沈伏，黜散越，假之為用者也。 變通相半，隨事之宜，贊助之法也。 凡音聲之體，務在和均，益則加倍，損則減半，其於本音恆為無爽。 然則言一上一下者，相生之道； 言重上生者，吹候之用也。 於蕤賓重上生者，適會為用之數，故言律者因焉，非相生之正也。
Huainan, Jing Fang, Zhèng Xuán, and others all teach upper/lower thirds, but at Ruibīn they "ascend again" to Dàlǚ—eight cùn one hundred four two-hundred-forty-thirds— Yízé ascends to Jiázhōng at seven cùn one thousand seventy-five one-thousand-one-hundred-eighty-sevenths; Wúyì ascends to Zhònglǚ at six cùn twelve thousand nine hundred seventy-four nineteen-thousand-six-hundred-eighty-thirds; Those three steps double the inch fractions Sima Qian and Ban Gu gave; every other length matches. That is the six-interstice path Lěng Zhōujiū described: lift the sunken, trim the loose—borrowed for practical tuning. Half-step swaps follow whatever the rite requires—they are aids, not the core thirds. Sound wants balance: raise it and you double, lower it and you halve—against the fundamental it must never lie. Talking of "one up, one down" names the true third-generation road— while "ascend again" names a listening trick for ash-tube tests. Ruibīn's double ascent lands on a handy tally for workshop use, so acousticians quote it, yet it is not canonical breeding.
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楊子雲曰：「聲生於日，謂甲己為角，乙庚為商，丙辛為徵，丁壬為羽，戊癸為宮也。 律生於辰，謂子為黃鐘，醜為大呂之屬也。 聲以情質，質，正也。 各以其行本情為正也。 律以和聲，當以律管鐘均和其清濁之聲。 聲律相協而八音生。 協，和也。」 宮、商、角、徵、羽，謂之五聲。 金、石、匏、革、絲、竹、土、木，謂之八音。 聲和音諧，是謂五樂。
Yáng Xióng said sound maps to the ten stems: jiǎ/jǐ as jue, yǐ/gēng as shang, bǐng/xīn as zhǐ, dīng/rén as yu, wù/guǐ as gong. Pitch maps to the twelve branches—zǐ as Yellow Bell, chǒu as Dàlǚ, and so down the ring— Sound rides on feeling for its body, and "body" here means being true to type— each degree staying loyal to the element that fathers it. Pipes exist to tune sound: bronze tubes and matched bells must pull bright and dark voices into one line. Pitch and timbre must agree before bronze, stone, silk, or bamboo can speak as the eight voices. Here xie simply means "bring into tune." The five named scale degrees are gong, shang, jue, zhi, and yu. The eight material sources are metal, stone, gourd, leather, silk, bamboo, clay, and wood. That mutual tuning of voice and material is what the canon calls the "five musics."
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夫陰陽和則景至，律氣應則灰除。 是故天子常以冬夏至日御前殿，合八能之士，陳八音，聽樂均，度晷景，候鐘律，權土灰，效陰陽，冬至陽氣應則灰除，是故樂均清，景長極，黃鐘通，土灰輕而衡仰。 夏至陰氣應則樂均濁，景短極，蕤賓通，土灰重而衡低。 進退於先後五日之中，八能各以候狀聞，太史令封上。 效則和，否則占。
Solstitial harmony shows in the gnomon; the monthly pipe's breath shows when the ash flies. On both solstices the king convenes the eight experts in the forward hall: they sound the eight colors of instrument, check ensemble balance, read the sundial, listen to bells and pipes, weigh thermoscopic ash, and read yin against yang—at midwinter yang answers so the ash flies, the pitch turns bright, the shadow stretches to its yearly maximum, Yellow Bell opens, the ash pan lightens, and the steelyard's beam lifts. At midsummer yin answers: the band slides toward duller color, the gnomon throws its shortest mark, Ruibīn opens, the ash tray weighs heavy, and the balance sinks. Across the ten-day solstice window each master files his reading, and the Grand Astrologer forwards the sealed bundle. Agreement means peace; mismatch becomes divination.
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候氣之法，為室三重，戶閉，塗釁周密，布緹幔。 室中以木為案，每律各一，內房中外高，從其方位，加律其上，以葭莩灰抑其內端，案曆而候之：氣至者灰去； 其為氣所動者，其灰散； 人及風所動者，其灰聚。 殿中候用玉律十二，惟二至乃候。 靈台用竹律。 楊泉記云：「取弘農宜陽縣金門山竹為管，河內葭莩為灰。」 或云以律著室中，隨十二辰埋之，上與地平，以竹莩灰實律中，以羅縠覆律呂，氣至吹灰動縠。 小動為和，大動，君弱臣強； 不動，君嚴暴之應也。
Breath-testing needs a triple room: sealed doors, pasted joints, red curtains drawn. Inside stand twelve tables, each bearing a pipe oriented to its compass point; pack reed ash in the bore, watch the calendar, and when the month's breath hits, the ash puffs out. A true cosmic puff scatters the grains. Human breath or a draft merely clumps the ash. The throne hall keeps twelve jade pipes for the two solstices alone. The observatory relies on bamboo tubes instead. Yang Quan notes bamboo cut on Mount Jīnmén in Yiyang and reed ash from Henei. Another rite buries the twelve tubes flush with the floor by the twelve branches, fills them with membrane ash, veils them in gauze, and waits for the puff that lifts the cloth. A tremble means balance; a gale hints ministers overpowering the throne. Dead stillness foretells a cruel king.
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起度之正，《漢志》言之詳矣。 武帝泰始九年，中書監荀勖校太樂，八音不和，始知後漢至魏，尺長於古四分有餘。 勖乃部著作郎劉恭依《周禮》制尺，所謂古尺也。 依古尺更鑄銅律呂，以調聲韻。 以尺量古器，與本銘尺寸無差。 又，汲郡盜發六國時魏襄王塚，得古周時玉律及鐘、磬，與新律聲韻闇同。 于時郡國或得漢時故鐘，吹律命之皆應。 勖銘其尺曰：「晉泰始十年，中書考古器，揆校今尺，長四分半。 所校古法有七品：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。 姑洗微強，西京望臬微弱，其餘與此尺同。」 銘八十二字。 此尺者勖新尺也，今尺者杜夔尺也。
Ban Gu already spelled out the true foot-rule in the Han treatise. In Jin Wudi's ninth year of Taishi Xun Xi retuned the court band and found Eastern Han and Wei had stretched the foot more than four fēn past the old standard. He ordered Liu Gong to cut the "Zhou foot" that antiquarians prize. From that measure he cast fresh bronze pipes. Every heirloom bronze measured true to its own inscription. Plunderers who cracked King Xiang's tomb at Ji turned up Zhou jade pipes and bells that sang in tune with Xun Xi's new set. When stray Han bells surfaced in the provinces, each spoke correctly to the new tubes. He stamped the ruler with a Taishi-10 colophon comparing it to the bloated contemporary foot. Seven relics anchored the proof: Gǔxǐ and lesser lǚ jade pipes, two Han gnomons, the standard hu, old cash, and the Jianwu foot. Only Gǔxǐ read a hair bright and the Han gnomon a hair flat—everything else lined up. The colophon holds eighty-two graphs. That ruler is Xun Xi's new measure; the older court foot is Du Kui's.
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荀勖造新鐘律，與古器諧韻，時人稱其精密，惟散騎侍郎陳留阮咸譏其聲高，聲高則悲，非興國之音，亡國之音。 亡國之音哀以思，其人困。 今聲不合雅，懼非德正至和之音，必古今尺有長短所致也。 會咸病卒，武帝以勖律與周漢器合，故施用之。 後始平掘地得古銅尺，歲久欲腐，不知所出何代，果長勖尺四分，時人服咸之妙，而莫能厝意焉。
The court admired Xun's crisp bells, yet Ruǎn Xiǎn sneered that they rang sharp—and sharp music, he said, is the whine of dying dynasties. A dying state's airs are sad and thoughtful; its people are cornered. Ruǎn adds that the new pitch feels un-ya: unless the foot itself has drifted, you cannot get the morally "level" tone the classics demand. Ruǎn died young; Wudi still adopted Xun's law because it matched Zhou and Han bronze. Later diggers at Shǐpíng unearthed a corroded old ruler four fēn longer than Xun's—vindicating Ruǎn's ear while the debate raged on.
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史臣案：「勖於千載之外，推百代之法，度數既宜，聲韻又契，可謂切密，信而有徵也。 而時人寡識，據無聞之一尺，忽周漢之兩器，雷同臧否，何其謬哉！ 《世說》稱「有田父于野地中得周時玉尺，便是天下正尺，荀勖試以校己所治金石絲竹，皆短校一米」。 又，漢章帝時，零陵文學史奚景於泠道舜祠下得玉律，度以為尺，相傳謂之漢官尺。 以校荀勖尺，勖尺短四分； 漢官、始平兩尺，長短度同。 又，杜夔所用調律尺，比勖新尺，得一尺四分七厘。 魏景元四年，劉徽注《九章》云：王莽時劉歆斛尺弱於今尺四分五厘，比魏尺其斛深九寸五分厘； 即荀勖所謂今尺長四分半是也。 元帝后，江東所用尺，比荀勖尺一尺六分二厘。 趙劉曜光初四年鑄渾儀，八年鑄土圭，其尺比荀勖尺一尺五分。 荀勖新尺惟以調音律，至於人間未甚流布，故江左及劉曜儀錶，並與魏尺略相依准。
The annalists still call Xun Xi's reconstruction razor-sharp and well attested. Yet shallow contemporaries waved one stray ruler, ignored two Han standards, and parroted the same verdict—sheer folly. The Shishuo adds a farmer's Zhou jade ruler that made Xun's entire orchestra look a meter short. Han Zhangdi's Lingling clerk Xi Jing also cut a "Han office foot" from Shun's jade pipe. Stacked against the Han office ruler, Xun's foot comes up four fēn short. The Han office foot and the Shǐpíng find match each other. Du Kui's tuning foot against Xun's new foot works out to one chǐ four fēn seven lí. Wei Jingyuan 4: Liu Hui notes Wang Mang's hú foot falls four fēn five lí short of the Jin foot, leaving the hú nine cùn five fēn deep against Wei measure. That is exactly Xun Xi's complaint that the current foot overshoots by four and a half fēn. Post-Yuan Jiangnan adopted a foot one chǐ six fēn two lí longer than Xun's. Liu Yao's Zhao court cast instruments with a foot one chǐ five fēn longer than Xun's. Because Xun's foot stayed inside the music bureau, southern instruments and Liu Yao's observatory still hewed to the longer Wei foot.
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《周禮》：「栗氏為量，鬴深尺，內方尺而圓其外，其實一鬴。 其臀一寸，其實一豆。 其耳三寸，其實一升。 重一鈞，其聲中黃鐘。 概而不稅。 其銘曰：'時文思索，允臻其極。 嘉量既成，以觀四國。 永啟厥後，茲器維則。 '」《春秋左氏傳》曰：「齊舊四量，豆、區、釜、鐘。 四升曰豆，各自其四，以登於釜。」 四豆為區，區鬥六升也。 四區為釜，六斗四升也。 釜十則鐘，六十四斗也。 鄭玄以為釜方尺，積千寸，比《九章粟米法》少二升八十一分升之二十二。 以算術考之，古斛之積凡一千五百六十二寸半，方尺而圓其外，減傍一厘八毫，其徑一尺四寸一分四毫七秒二忽有奇，而深尺，即古斛之制也。
The "Rites of Zhou" puts the foundry officer in charge of the standard fu: a square foot inside, walls rounded outside, one foot deep, one fu in volume. The pedestal adds one cùn; that compartment holds one dǒu. The lifting ears add three cùn and hold one shēng. The whole piece weighs one jūn and rings Yellow Bell true. Scrape the strike flat without shaving extra duty into the measure. The lid proclaims that thoughtful kings hit the utmost mark. When the honest measure was cast, he displayed it to the four quarters. May your heirs take this vessel as their pattern forever. The "Zuo" adds that old Qi used four jars—dòu, ōu, fǔ, and the zhōng. Four shēng fill a dòu; each tier climbs by fours toward the fǔ. Four dòu fill an ōu, and that ōu levels out at six shēng. Four ōu make a fǔ, which is six dǒu and four shēng. Ten fǔ make a zhōng, totaling sixty-four dǒu. Zhèng Xuán's cubic-foot fǔ tallies a thousand cubic cun—two shēng and twenty-two eighty-firsts shy of the "Nine Chapters" grain reckoning. Math shows the classic hu at one thousand five hundred sixty-two and a half cubic cun: a square with a circle struck outside, nibbling one lí eight háo from the flank, yields diameter one chǐ four cùn one fēn four háo seven miǎo two hū with remainder for a depth of one chǐ—the canonical hu.
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《九章商功法》程粟一斛，積二千七百寸； 米一斛，積一千六百二十七寸； 菽荅麻麥一斛，積二千四百三十寸。 此據精粗為率，使價齊，而不等其器之積寸也。 以米斛為正，則同於《漢志》。 魏陳留王景元四年，劉徽注《九章商功》曰：「當今大司農斛，圓徑一尺三寸五分五厘，深一尺，積一千四百四十一寸十分寸之三。 王莽銅斛，於今尺為深九寸五分五厘，徑一尺三寸六分八厘七毫。 以徽術計之，於今斛為容九斗七升四合有奇。」 魏斛大而尺長，王莽斛小而尺短也。
The "Nine Chapters" labor chapter sets one hú of grain at two thousand seven hundred cubic cun; one hú of polished rice at one thousand six hundred twenty-seven; and one hú of beans, peas, hemp, or wheat at two thousand four hundred thirty. Those figures weight coarse against fine so market prices line up even when the jars differ. Using the rice hú as baseline lands on the same number Ban Gu gave. In Wei Jingyuan 4 Liu Hui glossed the "Nine Chapters": the minister's hu measures one chǐ three cùn five fēn five lí across, one chǐ deep, one thousand four hundred forty-one and three-tenths cubic cun inside. Wang Mang's bronze hú, converted to Jin feet, is nine cùn five fēn five lí deep and one chǐ three cùn six fēn eight lí seven háo across. Liu Hui's π method makes the same vessel hold nine dǒu seven shēng four gě plus a trace. So the Wei measure runs large with a long foot, while Wang Mang's runs small with a short foot.
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衡權者，衡，平也； 權，重也。 衡所以任權而均物，平輕重也。 古有黍、壘、錘、錙、鐶、鈞、鋝、鎰之目，歷代參差。 《漢志》言衡權名理甚備，自後變更，其詳未聞。 元康中，裴頠以為醫方人命之急，而稱兩不與古同，為害特重，宜因此改治權衡，不見省。 趙石勒十八年七月，造建德殿，得圓石，狀如水碓，銘曰：「律權石，重四鈞，同律度量衡。 有辛氏造。」 續咸議，是王莽時物。
Weights and balances: héng means even; quán names the counterweight. The beam carries the weight and evens the load—true light against true heavy. Older metrologies listed shǔ, lěi, chuí, zī, huán, jūn, zhuó, and yì—each dynasty tweaked the names. Ban Gu's treatise already explains weights exhaustively; later tweaks are not copied in this chapter. Under Yuankang Pei Yì pleaded to realign medical weights—people's lives hung on every liǎng—but the court tabled his memo. Shi Le's builders turned up a round "law stone" counterweight marked four jūn, claiming it harmonized pitch, length, and weight. The inscription credits the Xin house. Xù Xián read the stone as another Wang Mang relic.