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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.