《易数钩隐图遗论九事(刘牧)》

Taihao (that is, Fuxi) received a heavenly revelation as a dragon-horse arrived bearing the Hetu on its back. The first chapter discusses this event.

This concerns the legend in the Yijing scholarly tradition of Fuxi and the Hetu: according to the tale, a dragon-horse emerged from the Yellow River with diagrammatic patterns on its back, and Fuxi created the Bagua (Eight Trigrams) on this basis.

In former times, when Taihao Fuxi (the legendary sage-king of high antiquity) ruled the world, he perceived an auspicious omen: a dragon-horse bearing the numbers of heaven and earth emerged from the Yellow River. This is what is called the Dragon Chart (Longtu). The numbers on the chart were arranged with nine above, one below, three on the left and seven on the right; two and four stood like shoulders on either side, six and eight like a pair of feet distributed below, and five occupied the center of the abdomen. Whether added vertically, horizontally or diagonally, the sum of the numbers is always fifteen. This is precisely what the Xici commentary of the Yizhuan means by “san wu yi bian, cuo zong qi shu,” that is, inferring numerical principles through mutual cross-referencing and transformed arrangements.

Taihao Fuxi took images from the Dragon Chart and then determined the numbers of the Five Phases (wuxing) according to the proper positions of the four directions. He regarded yang Qi (vital energy) as beginning to sprout from the zi position (jianzi, the month of the winter solstice), serving as the root of the growth of all things; yin Qi as beginning to arise at the wu position (jianwu, the month of the summer solstice), serving as the foundation for the withering of all things. The two Qi of Yin-Yang interact, and thus all things undergo change, able both to grow and to perish.

Heavenly number one is paired with the Kan trigram, earthly number two with the Li trigram; heavenly number three occupies the Zhen position, earthly number four the Dui position, and heavenly number five is governed from the center—these are the generating numbers of the Five Phases. Yin alone cannot generate and yang alone cannot issue forth, so earthly number six is paired with the zi position, heavenly number seven with the wu position, earthly number eight with the mao position, heavenly number nine with the you position, and earthly number ten with the center. These numbers together constitute the completing numbers of the Five Phases. From this the images of the Bagua (Eight Trigrams) are determined, and the Eight Trigrams are then superimposed in pairs to form the sixty-four hexagrams and three hundred eighty-four yao. This is the profound purport of the sages in creating hexagram images and observing the emblematic signs of things.

Fuxi was separated from King Wen of Zhou by tens of thousands of years. When King Wen ascended the throne it was the ninth year of the reign of King Zhou of Shang. Was not the author of the Yijing (Classic of Changes) King Wen, who found himself in a time of hardship? More than five hundred years after King Wen’s death Confucius was born. Confucius elucidated the principles of the Yi, saying: “The Yellow River produced the Dragon Chart, the Luo River produced the Luo Writing, and the sages took their models from them.” From this we know that the dragon-horse’s auspicious manifestation could be sensed and made clear only by Fuxi; the numerical principles of the Hetu (River Chart) were then deduced and expounded by Confucius.

Inquiring into these interwoven and synthesized numerical principles, they can encompass heaven, earth and Yin-Yang above and command all things in the middle; the transformations of the Dao of Heaven and of human affairs, as well as the profound mysteries of ghosts and spirits, are all contained therein. Confronted with such a chart of numbers, how can one lightly deny it? Like the words of the sages, its numerical principles can reach up to heaven, earth and Yin-Yang, command all things, and reveal the changes of heaven and humanity together with the mysteries of ghosts and spirits. The original text has missing characters at the end, so the meaning of the sentence is incomplete.

The sages observed the forms of Heaven, Earth, and the myriad things, drawing the gua according to the interweaving, transforming, and combining numerical principles of the Dragon Chart (Longtu). They looked upward to observe celestial phenomena and downward to examine the patterns of the earth; they took analogies from what was near in their own bodies and from what was distant in all things. Once the hexagram images composed of six yao (lines) were established, the principles of Heaven, Earth, and Humanity were complete. Thus the hexagrams can exhaust the changes arising from the four operations and deduce the numerical principles of all things.

Tracing the origins of the Bagua (Eight Trigrams), they arise from the Four Images. The Four Images correspond to the generation-and-completion numbers of the Five Phases (Wood, Fire, Earth, Metal, Water): the number of Water is six; three strokes are removed to form the Kan gua, and the remaining three strokes are arranged at the Hai position to constitute the Qian gua; the number of Metal is nine; three strokes are removed to form the Dui gua, and the remaining six strokes are arranged at the Wei position to constitute the Kun gua; the number of Fire is seven; three strokes are removed to form the Li gua, and the remaining four strokes are arranged at the Si position to constitute the Xun gua; the number of Wood is eight; three strokes are removed to form the Zhen gua, and the remaining five strokes are arranged at the Yin position to constitute the Gen gua. This is what is called “the Four Images give birth to the Eight Trigrams.”

Here Metal, Wood, Water, and Fire are specially listed while Earth is omitted, because each of Metal, Wood, Water, and Fire corresponds to and presides over one of the four seasons. The sages apprehended the hexagram images from within the formless, and from beyond the images they grasped their meanings; they then arranged and drew the gua, and by means of yarrow-stalk calculations they stacked the numbers of the yao to form complete hexagrams.

The stalk number of the Qian gua is two hundred sixteen, that of the Kun gua one hundred forty-four; together they total three hundred sixty, corresponding exactly to the number of days in a year and also each corresponding to the circulation of the four seasons. The total number of stalks used for the hexagram lines of the upper and lower parts is eleven thousand five hundred twenty, corresponding to the number of the myriad things. This arrangement is truly vast and exquisitely subtle. The numbers seven and nine of the yang yao and the numbers six and eight of the yin yao are all numbers naturally formed by Heaven and Earth, not something human wisdom could create. Even Fuxi (the legendary sage-king of high antiquity who created the Eight Trigrams), though endowed with spirit-like wisdom from birth, merely acted on Heaven’s behalf in completing this creation.

The Number of the Great Expansion is Fifty, Third (next section title: The number of the Great Expansion is fifty.)

“The number of the Great Expansion is fifty, of which forty-nine are actually used.” Here the “fifty” refers to the wondrous function contained within yarrow-stalk divination: it manifests the numbers of Yin-Yang, determines the yarrow-stalk counts of Qian and Kun, and thereby develops into the sixty-four hexagrams and three hundred eighty-four lines. The “forty-nine” symbolizes setting “Heaven-One” aside unused, thereby indicating that Taiji (Supreme Ultimate) is profound and unfathomable. “Fifty-five” is the total of the numbers of Heaven and Earth, by which change is accomplished and the Dao (the Way) of ghosts and spirits is set in motion.

Regarding the number of the Great Expansion, ancient scholars have offered various explanations; Ma Jichang, Zheng Kangcheng and others each had their own views. I understand it according to what is said in the Xici: there are five Heavenly numbers, namely the odd numbers one, three, five, seven, and nine; there are five Earthly numbers, namely the even numbers two, four, six, eight, and ten. Heavenly numbers and Earthly numbers pair with each other, forming five positions, and they combine with one another, thereby generating Metal, Wood, Water, Fire, and Earth of the Five Phases (Wood, Fire, Earth, Metal, Water).

The so-called “five positions” are the positions where odd and even numbers pair; the so-called “combining” is the combining of Yin-Yang. Yin-Yang combine to generate the Five Phases, therefore in each group of paired numbers one number is to be set aside unused, in order to represent the function at the generation of the Five Phases. For example, Heaven-One and Earth-Six combine to generate Water: the combining numbers are the “parents,” Water is the “child” generated thereby. Setting aside one number in this group symbolizes the generating function of Water, and is similar in principle to setting one number unused in the number of the Great Expansion.

The remaining groups of paired numbers are likewise: Earth-Two and Heaven-Seven, Heaven-Three and Earth-Eight, Earth-Four and Heaven-Nine, Heaven-Five and Earth-Ten, when combining to generate the Five Phases each must set aside one number. In this way the numbers of the Five Phases become complete. Setting aside one number from each group is not a deficiency; rather the five groups of “parental numbers” conceal the generating function of the Five Phases, thereby accomplishing change and setting ghosts and spirits in motion.

The Five Phases assist Heaven and Earth in generating the myriad things; their function is subtle and unfathomable, and the changes of Yin-Yang are even more something human effort cannot fully know, therefore it is said that the generating function of the Five Phases is concealed therein. If it were not so, from where would the numbers of the Five Phases come? The numbers of Wood and Fire govern the growth of the myriad things, the numbers of Metal and Water govern the harvest of the myriad things; Earth has no fixed direction, lodging in and presiding over the four seasons. This has the same meaning as what the sage said: “Jing (essence) and Qi (vital energy) cohere to form things, hun-soul Qi wanders and disperses to become change.”

The odd and even numbers between Heaven and Earth pair with each other and thus generate the Five Phases. If one only sees the pattern of numerical combination yet does not understand how the Five Phases are generated from this, one cannot explain how the Five Phases come into being. Therefore the five groups of numbers each setting aside one number precisely embodies the principle of the generation of the Five Phases and their latent function. From this it can be known that the number of the Great Expansion is fifty.

The images of the Yi are formed through four operations, and after eighteen changes a hexagram is completed. The Bagua (Eight Trigrams) are only the initially formed images and have not yet been expanded into a complete hexagram system; only by further derivation from these can the sixty-four hexagrams take shape. The principles of the Three Powers (Heaven, Earth, and Humanity), the origins of the myriad things, the numerical principles of Yin-Yang, and the deep and unfathomable transformations of ghosts and spirits can all be roughly perceived from the hexagram images.

If the transformations of all sixty-four hexagrams were fully enumerated, the writing would become exceedingly voluminous. Therefore we begin here solely with the Qian hexagram in order to illustrate the rest. Qian belongs to Heaven; the seven hexagrams that successively transform from it are Heaven-Wind Gou, Heaven-Mountain Dun, Heaven-Earth Pi, Wind-Earth Guan, Mountain-Earth Bo, Fire-Earth Jin, and Fire-Heaven Dayou. The remaining hexagrams may likewise be derived according to this pattern.

The Yi states that “the wandering soul constitutes change.” When a hexagram reaches the seventh stage of transformation it is called the “wandering soul”; when it reaches the eighth stage it is called the “returning soul.” “Returning soul” means that the hexagram image returns to the form in which it was originally generated. The transformations of the other hexagrams all follow this same rule.

Distinguishing Yin and Yang Hexagrams, Section Five

Qian symbolizes Heaven and is therefore called Father; the three trigrams Xun, Li, and Dui symbolize daughters and successively transform from the Qian hexagram. Kun symbolizes Earth and is therefore called Mother; the three trigrams Zhen, Kan, and Gen symbolize sons and are successively generated from the Kun hexagram.

Yang hexagrams contain a greater number of yin lines: Zhen, Kan, and Gen each consist of one yang line and two yin lines, resembling a single ruler leading two subjects. Yin hexagrams contain a greater number of yang lines: Xun, Li, and Dui each consist of two yang lines and one yin line, resembling two rulers leading a single subject.

A yang line has only one stroke and occupies the ruling position; a yin line has two strokes and occupies the subordinate position. This sequence of ruler and subject accords with constant principle and is therefore called the Way of the Superior Person. When a yin line of two strokes occupies the ruling position while a yang line of one stroke occupies the subordinate position, the sequence is regarded as contrary to constant principle and is therefore called the Way of the Inferior Person.

Recovering the Heart of Heaven and Earth, Section Six

The Upper Discourse holds that Fuxi (the legendary sage-king of high antiquity), among the numbers he observed from the Dragon Chart, saw “six” at the Hai position (Hai among the earthly branches, corresponding to the tenth lunar month). The Hai position is where the Qi (vital energy) of old yin resides in the tenth month; at this time yin Qi is dark and has developed to its utmost flourishing. “Six” is the number that represents yin.

The hexagrams Qian and Kun are the masters of Yin-Yang change. The saying “return after seven days” cannot be separated from the Yin-Yang bodies represented by Qian and Kun: Qian represents yang, numbered nine; Kun represents yin, numbered six. From the Jianzi month (the calendar month taking the Zi month as the start of the year), when yang Qi first arises, through the Si month, all belong to Qian yang; from the Jianwu month, when yin Qi first arises, through the Hai month, all belong to Kun yin. Zi and Wu are separated by the “six” of the Hai position, that is, six days. “Six” is the number of old yin; when young yang returns, the meaning of “seven days” becomes apparent.

The waxing and waning of Yin-Yang is the Dao (the Way). Although yin Qi in the tenth month has reached its extreme, yang Qi still lies hidden beneath it, so the liting grass sprouts. In the fourth month, when yang Qi is in command, yin Qi likewise remains hidden beneath it, so the micao (a class of weak grasses) withers. Kong Yingda said that the Hai position of the tenth month corresponds to thirty days. The sage does not say that yang Qi is reborn only after a full month has passed, but rather that although yin Qi has reached its utmost, yang Qi is still within it; when yang Qi is in command in the fourth month, yin Qi still lurks beneath it.

Some hold that one hexagram corresponds to six days and seven fen. The author opposes this view and illustrates with an example: the line statement of six in the second place of the Jiji hexagram says, “The wife loses her hair ornament; do not pursue it, in seven days it will be obtained.” Jiewei explains that after seven days the line changes at the fifth position and becomes the Fu hexagram, so “seven days” is used to convey the turning and returning of Yin-Yang. Lu Zi likewise said that the waxing and waning of Yin-Yang, going and returning, often takes seven days as a cycle. From this one can see the principle of “return after seven days.”

When the Jianzi month arrives, the celestial wheel moves leftward, the earthly axis turns rightward, the two Qi of Yin and Yang interact, and the myriad things thereby grow. At the winter solstice of the following year, the myriad things each return to their root. The “root” spoken of here is the heart. In terms of the two Qi of Yin and Yang, it is yang Qi gradually advancing and yin Qi gradually receding. At the summer solstice yang Qi returns to the Si position; at the winter solstice yin Qi returns to the Hai position; therefore it is called “returning to the root.”

The Lower Discourse is not appended with Jing (scripture) text.

The Book of Changes says: “Thunder is stored beneath the earth, symbolizing that the activity of all things has temporarily come to rest; the Fu hexagram manifests the original mind of Heaven and Earth, symbolizing that all things return to their root.” Heaven and Earth take purity and stillness as the foundation for nurturing the myriad things. They do not act deliberately, and the myriad things grow of themselves; they do not create deliberately, and the myriad things arise of themselves. Heaven and Earth remain silent and unmoving—this is the original mind of Qian and Kun. At the winter solstice, Yin qi (vital energy) reaches its peak, and Yang qi again sprouts forth from beneath the earth; therefore this is called “returning to the root.”

The Book of Changes is a book about change: the firm and the yielding transform into each other, the myriad things move without cease, and there is no absolute beginning. Yang qi plays the leading role: when Yin qi increases, Yang qi withdraws and conceals itself; when Yang qi is reborn, Yin qi recedes. Day arrives and night withdraws; night falls and day returns again—there is not a single moment without change. From this the sage observes the transformations, the births and extinctions of the myriad things, and the cyclical patterns of coming and going, and knows that the process of return will not go astray. Therefore some call the rebirth of Yang qi the original mind of Heaven and Earth.

Yet the original mind of Heaven and Earth can be seen only as it manifests through the myriad things; if one seeks it deliberately, one cannot obtain it. Confucius said: “What is there for those under Heaven to ponder? Although the paths of the world differ, they ultimately lead to the same destination; the destination is one, yet thoughts are of a thousand kinds.” The sage does not take his individual mind as mind, but unites with the mind of Heaven and Earth, taking the needs of the myriad things as his own intention—why then cling to his own ideas? Therefore, to pursue something deliberately may not succeed; to pursue nothing may not succeed either. The great sage clings neither to having a mind nor to having no mind, but follows the natural intention of Heaven and Earth and acts within it.

Hexagram conclusion: Wei Ji (Not Yet Complete) Hexagram (Seventh)

The Yijing is divided into an Upper Jing and a Lower Jing. According to the Qianzuodu (a work that interprets the Yijing), Confucius said: “The number of yang is three and the number of yin is four; these are the numbers that each occupy their proper positions.” Therefore the sixty-four hexagrams formed by eight eights are divided into upper and lower sections in order to symbolize Yin-Yang. Yang Qi (vital energy) is pure and belongs to the odd numbers, so the Upper Jing contains thirty hexagrams; yin Qi is not purely one and also involves transformation, so the Lower Jing contains thirty-four hexagrams.

The Upper Jing opens with Qian and Kun because Heaven and Earth are the root of creation and transformation and the source of the ten thousand things. Next come Kan and Li, which symbolize the sun and moon. The sun and moon hang in Heaven and appear on Earth, and they are able to drive the beginning-to-end transformations of all things. The Lower Jing opens with Xian and Heng because they treat the mutual influence of man and woman and the Dao of husband and wife getting along. The Dao of husband and wife can continue the sacrifices of the ancestral temples and becomes the foundation of the ethical order of human relations between Heaven and Earth. It closes with Jiji and Weiji, thereby making manifest the warning of flourishing and decline, rectifying the righteousness between ruler and minister, and also showing that only by clearly discerning principles and acting with caution can the kingly Dao be preserved.

In the Jiji hexagram the nine in the third place and the nine in the fifth place fail to maintain the proper distinction of above and below, thereby warning against employing petty people. In the Weiji hexagram the nine in the fourth place and the six in the fifth place embody the way in which ruler and minister relate to one another and reveal the luminous De of the junzi. How profound indeed are the teachings of the sages! When a matter has already been completed one must think of the hidden perils of what remains incomplete; when a matter is not yet completed one should discern clearly, act cautiously, and dwell at ease while awaiting the proper time. Therefore the opening of Weiji continues from the close of Jiji. At the end of Jiji the head has already been soaked; when Weiji opens, the tail too will inevitably be soaked. Head and tail are joined, beginning and end alternate in transformation, circulating without cease and moving together with the two Qi of Heaven and Earth Yin-Yang (the “two yi”). Alas, when things reach a state of completion and fullness like that of Jiji, is it not worth taking warning?

The statement of the nine in the fourth place says: “Stirring into motion to attack the Guifang, after three years one receives the reward of a great state.” Master Lu explained: “Three years is the number of the opening of yang Qi.” The principle of the Yi is to use the year to govern the months, the year to govern the days, the month to govern the ten-day weeks, and the day to govern the hours.

The nine in the fourth place says: “Zhen (sending forth troops on a campaign) campaigns against the Guifang, lasting three years, and receives rewards from a great state.”

Lu Zi explains that “three years” is the number by which yang Qi (vital energy) opens. The calculation method of the Yi uses the year to govern the months, the year to govern the days, the month to govern the ten-day periods, and the day to govern the double-hours. Therefore, when the classic says “day,” one stalk represents one double-hour; when it says “year,” one stalk represents one month. In this way, both “three days” and “three years” correspond to one stalk of the number nine. “Seven days” corresponds to one stalk of the numbers one, nine, two, and six; “ten-day period” and “ten years” correspond to one stalk of the numbers nine, six, seven, and eight.

Months have the deficiency of the new-moon day, while years have the surplus of intercalary months. Therefore, when referring to months, the numbers seven and eight are combined and then halved, to symbolize the number of ten-day periods in one lunation; when referring to years, the numbers of the nine and six lines are taken and used together, to symbolize the number of days in one intercalary month. Three ten-day periods constitute one lunation, and the number of ten-day periods in eight months corresponds to the number when yin Qi is at its fullest, totaling twenty-four. There is one intercalation every three years, and the days of the intercalary month correspond to the three hundred eighty-four lines in the two parts of the Yi text. Therefore, three hundred sixty stalks correspond to the regular number of days in a year, with the remainder also contained therein; the three hundred eighty-four lines correspond to the days of the intercalary month, with the added portion also gathered therein.

This is the subtle intention of the sages; it is not a deliberate detour to analyze or a circuitous elucidation. Otherwise, why would the odd and even numbers of Yin-Yang naturally and secretly correspond with heaven and earth?

The Method of Accumulating Numerical Tallies (Eighth)

The number of the Great Expansion is fifty, of which forty-nine are actually used, because one number is set aside unused. This unused number is precisely the key that enables the functioning to run through. Its intention is subtle; it is not a deliberate detour to analyze or a circuitous elucidation. Otherwise, why would the odd and even numbers of Yin-Yang naturally and secretly correspond with heaven and earth?

The number of the Great Expansion is fifty, but in actual use forty-nine are employed, one being left unused. This unused “one” is nevertheless the basis that allows the derivation to cohere throughout; although it itself is not reckoned among the numbers, it is what enables the numbers of the hexagrams to take form.

When generating a hexagram, the forty-nine yarrow stalks (the plant stems used in milfoil divination) are gathered together and held in both hands, symbolizing the chaotic, undifferentiated state before Heaven and Earth have separated. The stalks are then divided into two heaps, left and right, symbolizing the Two Modes of Heaven and Earth. Next one stalk is taken from the left-hand heap and placed between the little finger of the left hand, symbolizing the Three Powers of Heaven, Earth, and Humanity.

The remaining stalks in the left hand are counted off in groups of four, symbolizing the four seasons. After this counting, the leftover stalks are placed together with the one previously set aside; this is called “returning the odd remainder to the space between the fingers,” and it symbolizes the intercalary month in the calendar. The ancients held that two intercalary months occur in every five years, so a further portion is taken from the right-hand heap of stalks and likewise counted in groups of four; its remainder is also placed beside the hanging stalks. This step is called “a second remainder and then hanging.”

In this way the first counting is completed, yielding a result of five or nine; the second and third countings then each yield four or eight. Once the three countings are finished, one line (a yin or yang line of the hexagram) is obtained; after eighteen such changes a complete hexagram is formed.

In ancient times, the Yellow Emperor sent Ling Lun to Kunlun Mountain west of Daxia, where he selected the longest bamboo tubes and fashioned twelve pitch pipes to discern the cries of the phoenix: the male phoenix uttered six calls and the female phoenix uttered six calls. Thereby the twelve pitches were established so that sounds could generate one another, with Huangzhong as the foundation of the pitches.

Thus the initial nine of the Qian hexagram corresponds to the first of the pitch pipes and marks the starting point of the transformation of yang qi (Qi, vital energy). Calculated sixfold and taking nine as the standard, one obtains Linzhong: the length of Huangzhong multiplied by nine and then by six yields fifty-four. Dalü corresponds to the initial six of the Kun hexagram, serving as the first of the lü pipes and the starting point of the transformation of yin qi. All these calculations follow the principle of “three for heaven and two for earth” (taking three as heaven’s number and two as earth’s number).

Nine and six embody the relations of Yin-Yang, of husband and wife, and of mother and son. Generation between different categories is called mother and son: for example, Huangzhong generates Linzhong, yet still requires the aid of Dalü to bring about generation. Combination within the same category is called husband and wife: for example, Dalü pairs with Huangzhong, uniting like husband and wife, and only then generates the sixth month. This is the regular operation of heaven and earth.

Yang qi initially returns to the Wu-Ji Qinggong (a term related to the central direction and the gong tone), which is the mother of Huangzhong. When it intersects with Ruibin (one of the twelve pitches) in the fifth month, the pitch of Huangzhong has already been entrusted to the eldest son, awaiting activation at the winter solstice. Huangzhong begins to take effect from the eleventh month, when yang qi first arises, and is therefore the root of the pitches. Yet in the fifth month, as one yin first arises, after Huangzhong it returns to the Qinggong to be gathered in, and only then generates Zhonglü.

According to the Book of Jin, Jing Fang of the Han dynasty was well-versed in the numerical principles of the six lü and five tones, as well as the method of generating the sixty pitches: downward generation is always “three generates two”; upward generation is always “three generates four.” Yang pitches generate yin lü downward, and yin lü generate yang pitches upward, proceeding until Zhonglü, at which the twelve pitches are complete. Zhonglü then generates Zhishi upward, Zhishi generates Qumie downward, and continuing in this manner of upward and downward generation until Nanshi, the sixty pitches are fully derived. The extension of the twelve pitches into sixty pitches resembles the derivation of the Bagua (Eight Trigrams) into sixty-four hexagrams.

Huangzhong pairs with Dalu to generate Linzhong; Taicu pairs to generate… (the original text here reads “qu mie,” whose meaning is unclear and which may involve a textual omission or error). Proceeding in this way according to the sequential order of mutual generation above and below, the derivation continues until Nanlü, at which point the sixty lü are fully obtained. The transformation of the twelve lü into sixty lü is like the derivation of the sixty-four hexagrams from the Bagua (Eight Trigrams).

The method of yang generating yin below and yin generating yang above

Huangzhong pairs with Dalu to generate Linzhong; Taicu pairs with Zhonglü to generate Nanlü; Linzhong takes Ruibin as its pair to generate Taicu; Nanlü takes Yize as its pair to generate Guxian; Yuanshe intersects and resonates with Yingzhong to generate Jiazhong; Jiazhong takes Taicu as its pair to generate Yize; Yize pairs with Nanlü to generate Dalu; Dalu generates Ruibin. Ruibin intersects with Wu and Ji to form the “Qinggong”; the Qinggong is then transmitted to the eldest son. Here Huangzhong, Taicu, and the rest are the names of the ancient twelve lü-lü; “yang generating yin below, yin generating yang above” refers to the computational rules governing the mutual generation of the lü-lü.

Huangzhong (the fundamental pitch among the ancient twelve lülü) measures nine cun in length and is the basis for determining the pitches. Subtracting one-third of it according to the method of “losing one-third” (sanfen sunyi) yields Linzhong. The pitches generate one another in sequence. When the derivation reaches the fifth month, Ruibin intersects with Wu and Ji, then returns to Huangzhong, thereby producing Qinggong.

The longest pipe measures one chi two cun three fen. Subtracting nine cun leaves three cun three fen, from which Zhonglü is generated. Dalü measures three cun seven fen; increasing it by one-third according to the method of “gaining one-third” (sanfen yiyi) generates Ruibin upward. Taicu measures eight cun; subtracting one-third according to “losing one-third” generates Nanlü downward. Jiazhong measures four cun two fen; increasing by one-third generates Yize upward. Guxi measures seven cun one fen; subtracting one-third generates Yingzhong downward.

Zhonglü intersecting with Taicu yields “Husband Taicu,” with a pipe length of three cun three fen; increasing by one-third generates Zhishi. Ruibin measures four cun nine fen, beginning from Shaoyang and ending at Houfu; subtracting one-third and then intersecting with Wu and Ji, it returns into Qinggong. Linzhong measures six cun; increasing by one-third generates Taicu downward. Yize measures five cun six fen; subtracting one-third generates Dalü upward. Nanlü measures five cun three fen; increasing by one-third generates Guxi downward.

Yuanshe intersecting with its wife Yingzhong yields a pipe length of six cun three fen; subtracting one-third generates Jiazhong. Yingzhong measures four cun seven fen; increasing by one-third and then intersecting with its “husband” Yuanshe, it serves as the leading singer.

Zhishi is generated below the Huangzhong section and above Zhonglü. Now it further generates Mie downward, located below Linzhong, and therefore must intersect with Nanshi. At this point the cycle of derivation is completed. Nanshi She serves as the leading singer; Zhishi, generated below the Huangzhong section and above Zhonglü, now generates Mie downward located below Linzhong and must intersect with Nanshi. Thus the cycle is completed. Nanshi in turn generates the pitch located beside Dianbin.

Here “lü” refers to the ancient musical pitches; “losing one-third” and “gaining one-third” are methods of calculating the pitches according to the proportional lengths of the pipes, while “generating upward” and “generating downward” indicate the sequential derivation of the next pitch. Some of the names and phrases in the text are suspected of having been corrupted in transmission; the foregoing follows the original order in a direct translation.