郑世子曰∶律吕相生，左旋隔八，则右转隔六；右转隔八，则左旋隔六。何谓左旋隔八，
右转隔六？如黄生林，林生太，太生南，南生姑，姑生应，应生蕤，蕤生大，大生夷，夷生夹，
夹生无，无生仲，仲生黄是也。何谓右转隔八，左旋隔六？如黄生仲，仲生无，无生夹；夹生
夷，夷生大，大生蕤，蕤生应，应生姑，姑生南，南生太，太生林，林生黄是也。右转左旋，左
右逢源，周而复始，循环无端，乃律吕之妙。古人算律，往而不返，但晓右旋，不知右转，此
所以未密也。律管有大小，大生小为下生，小生大为上生，一言尽之矣。黄钟至大而应钟
至小，故为上下之终始也。
天地之间，声之大者如雷霆，小者如蠓蚁，皆不得其和。故圣人设音律以调之，而后声
之大者不过宫，声之小者不过羽，其于和阴阳、赞化育之道，莫善于此，乃为三分损益之法
以正五音。然音止于五，犹不足以尽其变，由是截竹为管，作十二律以应十二月，而亦以三
分损益之法正之。如黄钟为宫，宫者音之君也，一阳之律也，阳生于子而数始于九，因而九
之，九九八十一而黄钟之数立焉。阳下生阴，长管生短管也，三损其一则为短；阴上生阳，
短管生长管也，三益其一则为长。如黄钟九寸，分九为三而去其一，则为六寸，便为隔八下
生林钟六月之管。又三分林钟之六而益其一，以二加六得八寸，便为上生太簇正月之管。
余律亦然。又以宫数数之，九九八十一，宫音也；三分去其一分二十七，则得五十四为征
音；以五十四为三分，而又添一分十八，则得七十二为商音；以七十二为三分，而损去一分
二十四，则得四十八为羽音；以四十八为三分，而又添一分十六，则得六十四为角音。此音
律三分损益之数，皆出于自然而然。
古人算律有三种∶其一，以黄钟为九寸，每寸九分，共计八十一分；其二，以黄钟为十
寸，每寸十分，共计百分；其三，以黄钟为九寸，每寸十分，共计九十分。
其一，出淮南子及晋书、宋书。（此纵黍度也。）
黄钟之数八十一。（一曰八寸一分，下同。宫音数也。）
林钟之数五十四。（征音同。）
太簇之数七十二。（商音同。）
南吕之数四十八。（羽音同。）
姑洗之数六十四。（角音同。）
应钟之数四十三。
蕤宾之数五十七。
大吕之数七十六。
夷则之数五十一。
夹钟之数六十八。
无射之数四十五。
仲吕之数六十。
其二，出太史公律书。（此横黍度也。）
黄钟长十寸。（整一百分，如前三分损益相生，得林钟之数长六寸六分六厘六毫，太簇之数长八寸
八分八厘八毫，而十二律之数自见。）
其三出京房律准及后汉志。（此斜黍度也。）
黄钟长九寸。（每寸十分，如前三分损益相生，各得十二律之数。）
上古法，下生者三分减一，三减其一则为二，故用二因三归；上生者三分添一，以三添
一则为四，故用四因三归。别法，下生者五十乘之，七十五除之；上生者一百乘之，七十五
除之。所得与古同而算法不同。
假如黄钟长十寸。古法置黄钟为实，下生者二因三归得林钟。别法以五十乘之，七十五除之，亦得林钟。
又如林钟长六寸六分六厘六毫。古法置林钟为实，上生者四因三归得太簇。别法以
一百乘之，七十五除之，亦得太簇。余仿此。
郑世子说：律吕的相生，向左旋转时每隔八位，则向右旋转时每隔六位；向右旋转时每隔八位，则向左旋转时每隔六位。什么叫向左旋转隔八、向右旋转隔六呢？例如黄钟生林钟，林钟生太簇，太簇生南吕，南吕生姑洗，姑洗生应钟，应钟生蕤宾，蕤宾生大吕，大吕生夷则，夷则生夹钟，夹钟生无射，无射生仲吕，仲吕再生黄钟，这就是向左旋转隔八（即下生）。什么叫向右旋转隔八、向左旋转隔六呢？例如黄钟生仲吕，仲吕生无射，无射生夹钟，夹钟生夷则，夷则生大吕，大吕生蕤宾，蕤宾生应钟，应钟生姑洗，姑洗生南吕，南吕生太簇，太簇生林钟，林钟再生黄钟，这就是向右旋转隔八（即上生）。向右和向左旋转，左右逢源，周而复始，循环无端，这是律吕的精妙之处。古人计算律管长度，只往不返，只知道右旋（上生），不知道左转（下生），这就是不够精密的原因。律管有大小之分，大管生小管称为下生，小管生大管称为上生，一句话就可以概括了。黄钟最大而应钟最小，所以它们是上下生的终始。
天地之间，声音大的如雷霆，小的如蚊蝇，都不能达到和谐。所以圣人设立音律来调和它们，之后声音大的不超过宫音，声音小的不超过羽音，这对于调和阴阳、辅助化育之道，没有比这更好的了，于是采用三分损益法来校正五音。然而音只有五种，还不足以穷尽其变化，因此截取竹子做成律管，制作十二律来对应十二月，也用三分损益法来校正它们。例如以黄钟为宫音，宫是音的主君，是一阳之律，阳气生于子时且数字始于九，将九乘以九，得到八十一，黄钟的数值就确立了。阳气下生阴气，即长管生短管，减去三分之一就是短管；阴气上生阳气，即短管生长管，增加三分之一就是长管。例如黄钟长九寸，将九分为三份并去掉一份，就是六寸，这便是隔八下生得到的林钟（六月之管）。又将林钟的六寸分为三份并增加一份，用二加上六得到八寸，这便是上生得到的太簇（正月之管）。其余律管也是如此。再用宫音的数值来计算：九九八十一，是宫音的数值；将八十一分为三份并去掉一份二十七，得到五十四，这是徵音的数值；以五十四分为三份，再增加一份十八，得到七十二，这是商音的数值；以七十二分为三份，并去掉一份二十四，得到四十八，这是羽音的数值；以四十八分为三份，再增加一份十六，得到六十四，这是角音的数值。这些音律三分损益的数值，都是出于自然规律。
古人计算律管有三种方法：第一种，以黄钟为九寸，每寸九分，共计八十一分；第二种，以黄钟为十寸，每寸十分，共计百分；第三种，以黄钟为九寸，每寸十分，共计九十分。
第一种方法，出自《淮南子》及《晋书》、《宋书》。（这是纵黍度。）
黄钟的数值是八十一。（一说八寸一分，下同。这是宫音的数值。）
林钟的数值是五十四。（徵音相同。）
太簇的数值是七十二。（商音相同。）
南吕的数值是四十八。（羽音相同。）
姑洗的数值是六十四。（角音相同。）
应钟的数值是四十三。
蕤宾的数值是五十七。
大吕的数值是七十六。
夷则的数值是五十一。
夹钟的数值是六十八。
无射的数值是四十五。
仲吕的数值是六十。
第二种方法，出自司马迁《史记·律书》。（这是横黍度。）
黄钟长十寸。（整一百份，如前所述通过三分损益相生，得到林钟的长度为六寸六分六厘六毫，太簇的长度为八寸八分八厘八毫，而十二律的数值自然显现。）
第三种方法，出自京房《律准》及《后汉书·律历志》。（这是斜黍度。）
黄钟长九寸。（每寸十分，如前所述通过三分损益相生，各得十二律的数值。）
上古的方法，下生者减去三分之一，减三分之一即剩下三分之二，所以用二乘三除；上生者增加三分之一，加三分之一即变成原来的四分之三，所以用四乘三除。另一种算法，下生者乘以五十除以七十五；上生者乘以一百除以七十五。所得结果与古法相同但计算方法不同。
假如黄钟长十寸。古法将黄钟作为被除数，下生者用二乘三除得到林钟。别法用五十乘之，七十五除之，也得到林钟。
又如林钟长六寸六分六厘六毫。古法将林钟作为被除数，上生者用四乘三除得到太簇。别法用一百乘之，七十五除之，也得到太簇。其余仿照此例。
Zheng Shizi (Prince of Zheng) stated: In the generation of Lü (pitch pipes), rotating left involves skipping eight positions, whereas rotating right involves skipping six positions; rotating right involves skipping eight positions, whereas rotating left involves skipping six positions. What is meant by "rotating left skipping eight" and "rotating right skipping six"? For instance, Huangzhong generates Linzhong, Linzhong generates Taicu, Taicu generates Nanlü, Nanlü generates Guxian, Guxian generates Yingzhong, Yingzhong generates Ruibin, Ruibin generates Dailü, Dailü generates Yize, Yize generates Jiazhong, Jiazhong generates Wuyi, Wuyi generates Zhonglü, and Zhonglü generates Huangzhong again. This constitutes the method of generating downwards (Xia Sheng, producing a shorter pipe from a longer one) by skipping eight positions to the left. What is meant by "rotating right skipping eight" and "rotating left skipping six"? For instance, Huangzhong generates Zhonglü, Zhonglü generates Wuyi, Wuyi generates Jiazhong, Jiazhong generates Yize, Yize generates Dailü, Dailü generates Ruibin, Ruibin generates Yingzhong, Yingzhong generates Guxian, Guxian generates Nanlü, Nanlü generates Taicu, Taicu generates Linzhong, and Linzhong generates Huangzhong again. This constitutes the method of generating upwards (Shang Sheng, producing a longer pipe from a shorter one) by skipping eight positions to the right. Rotating right and left, finding sources on both sides, cycling endlessly without beginning or end, this is the subtlety of the Lü system. Ancient scholars calculating the Lü only went forward without returning, knowing only of right rotation (Shang Sheng) but not left rotation (Xia Sheng); this is why their calculations were not precise enough. The Lü pipes vary in size; generating a smaller pipe from a larger one is called Xia Sheng (downward generation), and generating a larger pipe from a smaller one is called Shang Sheng (upward generation). This can be summarized in one phrase. Huangzhong is the largest, and Yingzhong is the smallest; thus, they mark the beginning and end of upward and downward generations.
Between heaven and earth, sounds as loud as thunder or as quiet as mosquitoes and ants fail to achieve harmony. Therefore, the sages established musical pitch standards to regulate them. Consequently, the loudest sound does not exceed Gong (the fundamental tone), and the quietest does not exceed Yu (the fifth tone). Regarding the way of harmonizing Yin and Yang and assisting the processes of transformation and nourishment, nothing is superior to this. Thus, the method of San Fen Sun Yi (Three-Part Diminution and Augmentation) was employed to rectify the Wu Yin (Five Tones). However, since there are only five tones, this is insufficient to exhaust all variations. Hence, bamboo tubes were cut to create the Twelve Lü (Twelve Pitch Pipes) corresponding to the twelve months, also regulated by the method of San Fen Sun Yi. For example, taking Huangzhong as Gong. Gong is the ruler of tones and represents the law of Yang. Yang arises at Zi (midnight) and numbers begin with nine. Multiplying nine by nine yields eighty-one, establishing the numerical value of Huangzhong. Yang generates Yin downward (Shang Sheng produces Xia Sheng contextually here referring to length reduction logic in text, though strictly Shang/Xia refers to pitch/length direction): a long pipe generates a short pipe; reducing it by one-third makes it short. Yin generates Yang upward: a short pipe generates a long pipe; augmenting it by one-third makes it long. For instance, Huangzhong is nine cun. Dividing nine into three parts and removing one part leaves six cun; this is the Linzhong pipe for the sixth month, generated downward by skipping eight positions. Again, dividing Linzhong's six cun into three parts and adding one part results in eight cun (adding two to six); this is the Taicu pipe for the first month, generated upward. The same applies to other Lü. Calculating using the Gong number: 9x9=81 is the Gong tone; reducing 81 by one-third (27) yields 54, which is the Zhi tone; dividing 54 into three parts and adding one part (18) yields 72, which is the Shang tone; dividing 72 into three parts and removing one part (24) yields 48, which is the Yu tone; dividing 48 into three parts and adding one part (16) yields 64, which is the Jue tone. These numerical values of the musical Lü derived from San Fen Sun Yi arise naturally.
There are three methods used by ancients to calculate Lü:
First, taking Huangzhong as nine cun, with each cun divided into nine fen, totaling eighty-one fen.
Second, taking Huangzhong as ten cun, with each cun divided into ten fen, totaling one hundred fen.
Third, taking Huangzhong as nine cun, with each cun divided into ten fen, totaling ninety fen.
The first method appears in *Huainanzi*, *Book of Jin*, and *Book of Song*. (This uses longitudinal millet measurement.)
Huangzhong number: 81. (Also said to be 8 cun 1 fen, same below. Number of Gong tone.)
Linzhong number: 54. (Same as Zhi tone.)
Taicu number: 72. (Same as Shang tone.)
Nanlü number: 48. (Same as Yu tone.)
Guxian number: 64. (Same as Jue tone.)
Yingzhong number: 43.
Ruibin number: 57.
Dailü number: 76.
Yize number: 51.
Jiazhong number: 68.
Wuyi number: 45.
Zhonglü number: 60.
The second method appears in Sima Qian’s *Records of the Grand Historian: Treatise on Music* (*Shiji Lü Shu*). (This uses transverse millet measurement.)
Huangzhong length: 10 cun. (Exactly 100 units. As described above via San Fen Sun Yi generation, Linzhong length is 6 cun 6 fen 6 li 6 hao, Taicu length is 8 cun 8 fen 8 li 8 hao, and the numbers of the Twelve Lü become apparent.)
The third method appears in Jing Fang’s *Lü Zhun* and the *Book of Later Han: Treatise on Astronomy and Calendar* (*Hou Hanshu Lizhi*). (This uses oblique millet measurement.)
Huangzhong length: 9 cun. (Each cun is 10 fen. As described above via San Fen Sun Yi generation, each obtains the numbers of the Twelve Lü.)
In ancient methods, for downward generation (Xia Sheng), one reduces by one-third. Reducing by one-third leaves two-thirds, so one multiplies by two and divides by three. For upward generation (Shang Sheng), one adds one-third. Adding one-third results in four-thirds, so one multiplies by four and divides by three. An alternative method: for downward generation, multiply by fifty and divide by seventy-five; for upward generation, multiply by one hundred and divide by seventy-five. The results are the same as the ancient method, but the calculation differs.
For example, if Huangzhong is 10 cun long. The ancient method takes Huangzhong as the dividend; for downward generation, multiply by two and divide by three to obtain Linzhong. The alternative method multiplies by fifty and divides by seventy-five, also obtaining Linzhong.
Again, if Linzhong is 6 cun 6 fen 6 li 6 hao long. The ancient method takes Linzhong as the dividend; for upward generation, multiply by four and divide by three to obtain Taicu. The alternative method multiplies by one hundred and divides by seventy-five, also obtaining Taicu. The rest follows this pattern.