The Mathematics of the Tai Xuan Jing

The Tai Xuan Jing, or Canon of Supreme Mystery, is a four-line, base 3 oracle that is clearly patterned after the I Ching.  In addition to solid and once-broken lines, a third, twice-broken line, is added, so that in combination they represent heaven, earth, and man.  It contains 34, or 81, tetragrams, or Shou, reminiscent of the number of chapters in the Tao Te Ching.  Most of the texts, depending on which edition you are using, are accompanied by nine Apprasials, or Tsan, which resemble the line texts of the I Ching.

The first nine tetragrams:


Two English translations (Walters and Nylan; see the bottom of the page) are commercially available; and even though they are ostensibly about the same subject, they are completely different books.  Nylan’s is based on the “standard edition,” and the text is more structured, possibly reflecting more editing; while Walters says his is based on a simpler, alternative version found in an old woodcut book in the British Library.  The books take a completely different approach to consulting the oracle as well, which will be analyzed in detail below.  Concerning the overall process, Nylan is very clear that the tetragram lines are generated from the top down, unlike the I Ching, while Walters says to do it from the bottom up.

remainders sum
0 (1) 0 (1) 1 3
0 (1) 1 0 (1) 3
0 (1) 2 2 5
1 0 (1) 0 (1) 3
1 1 2 4
1 2 1 4
2 0 (1) 2 5
2 1 1 4
2 2 0 (1) 5

The yarrow stalk method

In addition to horoscopic and numerological methods along the lines of the Plum Blossom method, Walters describes a method of determining the Shou using yarrow stalks.  64 stalks are divided into three bundles, and each bundle counted off by threes until 0, 1, or 2 is left.  If the remainder is 0, one stalk is added from its bundle to make 1.  The remainders are combined; the total will be 3, 4, or 5, which represent solid, once-broken, and twice-broken lines, respectively.  The process is repeated three times to generate four lines.

The remainder possiblities are displayed in the table to the right.  The relative probabilities of the three lines are 3:3:3, or in other words, equal.  A simpler method would be to just divide a bundle of stalks (big enough to divide randomly, exact number not important) and count off one half by threes until 1, 2, or 3 is left.  Let 1 represent a line of one segment, or unbroken; 2 represent a line of two segments, or once-broken; and 3 represent a line of three segments, or twice broken.

To determine the Tsan, one counts off a bundle of stalks by nines after creatively dividing it.  Nine is a relatively large number when it comes to a handful of yarrow stalks, and it may be hard to make a simple division of a bundle without coming up with remainders that tend to cluster near each other.  So one divides the bundle into three, saves the right-hand bundle, and combines the left-hand and central bundles to again divide into three.  The new right-hand bundle is added to the original right-hand bundle.  A total of four threefold divisions is made, so that the bundle to be counted consists of the original right-hand bundle with three smaller bundles added.  The latter add some additional randomness to the outcome.  (One might wonder if making two divisions of a bundle before counting off by threes would similarly enhance the process.)

Another method of determining the Tsan is described, that of performing the original yarrow stalk division above two more times to generate two more lines, and reading them as numbers.  Like the tetragram numbers themselves, they are base 3, with 1 added so that they begin with 1 rather than 0.

1 2 3 4 5 6 7 8 9

heads per throw total
1 1 2
1 1 2
1 2 3
1 1 2
1 1 2
1 2 3
2 1 3
2 1 3
2 2 4

The four-coin method

Nylan describes two coin methods for determining the tetragram lines, one simple and one rather complex.  In the simple method, one first throws two coins.  If both come up tails, throw them again until at least one comes up heads.  The resulting possiblities are:  first coin heads, second coin heads, or both heads.  Repeat this process with another two coins.  The total number of heads showing determines the tetragram line:  2 for solid, 3 for once-broken, and 4 for twice-broken.  (These lines are assigned values of 7, 8, and 9, respectively, reminiscent of the I Ching line numbers.  Add 5 to the number of heads to arrive at these values; or, what I would do, just subtract 1 to indicate lines of 1, 2, or 3 segments.)

The possiblities are displayed in the table to the right.  Curiously, the relative line probabilities with this method are 4:4:1 for solid, once-broken, and twice-broken.  A coin method which yields equal odds for each line like the yarrow stalk method above would be to throw two coins in succession.  Again, if both coins are tails, start over.  Then treat the coins as a two-digit binary number, tails = 0 and heads = 1.  Let 01 represent a line of one segment, 10 (2 in base 10) a line of two segments, and 11 (3 in base 10) a line of three segments.

first division, 32
remainders total coins left
1 1 2 29
2 3 5 26
3 2 5 26
second division, 29
remainders total coins left
1 1 2 27
2 3 5 24
3 2 5 24
second division, 26
remainders total coins left
1 1 2 24
2 3 5 21
3 2 5 21

The 36-coin method

I call this method “36-coin” even though four coins are removed at the outset and never contribute to the outcome.  To summarize, 32 coins are divided by tossing them one at a time.  The heads and tails are each counted off by threes until 1, 2, or 3 coins are left.  The sum of the two remainders will be 2 or 5; they, plus one more, are removed from the total.  The coins that are left, which number either 26 or 29, are once more divided by tossing, then counted off as they were above.  The remainders (again 2 or 5) are removed, leaving either 21, 24, or 27 coins left.  Divide the total by 3 to indicate solid (7), once-broken (8), and twice-broken (9) lines, respectively.

What are the odds?  The possible results of the divisions are displayed in the table to the right.  Since the second division of 26 has double the chance of happening as that of 29, the list of final outcomes is 27, 24, 24; 24, 21, 21; 24, 21, 21.  The odds are thus 4:4:1, the same as for the four-coin method above.

This method uses so many coins, it looks like it ought to use yarrow stalks instead.  And in fact, an equivalent yarrow method, said to be taken from the Numbers commentary, is described in An Introduction to Yang Hsiung’s “Canon of Supreme Mystery” by Michael Nylan and Nathan Sivin.  But The Elemental Changes (below) describes only the coin method above.  One problem might be that 32 stalks is too small a starting number to make two good random divisions.  Since the outcome is intended to be one of the numbers (7×3, 8×3, or 9×3) above, tossing coins one at a time may be preferred for a better random division of a relatively small number.

According to Nylan, the Appraisals are read according to the time of day when the divination is carried out:  1, 5, and 7 for morning; 3, 4, and 8 for evening; and 2, 6, and 9 for the “median” times.

The T’ai Hsuan Ching by Roger Clough

The Alternative I Ching translated by Derek Walters

The Elemental Changes:  The Ancient Chinese Companion to the I Ching translated by Michael Nylan

Having said all this, I personally think the Tai Xuan Jing is kind of weird, and I don’t really use it.  What I find interesting are the varied ways that the ancients found to manipulate numbers and associate them with meaning.

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