wen derivatives

8/9/2019 Wen Derivatives

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introduction>xplain how this paper attempts to build on the foundation of McKennas FOD integers, while it largely ignores the outgrowths of his work,amely timewave and novelty theory, and calendric correspondences.

ditors and commentators of the I Ching highlight the importance of the six line-places. The I Ching itself makes little or no direct mention of he significance of the individual line-places (or gua/kua), but individuals who have edited the I Ching or rendered translations of it tend toonvey the significance thereof. Yin lines are at home in the 2 nd , 4 th , and 6 th (even) places; while yang lines are best suited to the oddlaces. Also, there is said to be resonance between corresponding lines in the upper and lower halves of any given hexagram. The bestituations obtain when yin and yang each occupy their proper places; thus yang in the 1 st place responds to yin in the 4 th place, for examp

xcerpted passages from Terrence McKennas Derivation of the Timewave from the King Wen Sequence of Hexagrams

The earliest arrangement of the hexagrams of the I Ching is the King Wen Sequence. It was this sequence that I chose to study as a possible basis for a newmodel of the relationship of time to the ingression and conservation of novelty. In studying the kinds of order in the King Wen Sequence of the I Ching Imade a number of remarkable discoveries. It is well known that hexagrams in the King Wen sequence occur in pairs. The second member of each pair isobtained by inverting the first. In any sequence of the sixty-four hexagrams there are eight hexagrams which remain unchanged when inverted. In the KingWen Sequence these eight hexagrams are paired with hexagrams in which each line of the first hexagram has become its opposite, (yang changed to yinand vice -versa ).

The question remains as to what rule or principle governs the arrangement of the thirty-two pairs of hexagrams comprising the King Wen Sequence. Myintuition was to look at the first order of difference, that is, how many lines change as one moves through the King Wen sequence from one hexagram tothe next. The first order of difference will always be an integer between one and six [excluding five]. When the first order of difference within pairs isexamined it is always found to be an even number. Thus all instances of first order of difference that are odd occur at transitions from one pair of hexagrams to the next pair. When the complete set of first order of difference integers generated by the King Wen Sequence is examined they are foundto fall into a perfect ratio of 3 to 1, three even integers to each odd integer. The ratio of 3/1 is not a formal property of the complete sequence but was acarefully constructed artifact achieved by arranging hexagram transitions between pairs to generate fourteen instances of three and two instances of one .Fives were deliberately excluded. The fourteen threes and two ones constitute sixteen instances of an odd integer occurring out of a possible sixty-four.This is a 3/1 ratio exactly.


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The set of McKenna First-Order Difference integers:

6, 2, 4, 4, 4, 3, 2, 4, 2, 4, 6, 2, 2, 4, 2, 2, 6, 3, 4, 3, 2, 2, 2, 3, 4, 2, 6, 2, 6, 3, 2, 3,4, 4, 4, 2, 4, 6, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 4, 4, 1, 6, 2, 2, 3, 4, 3, 2, 1, 6, 3, 6, 3

By extending McKennas basic work, the first-order difference (FOD), the author believes to have uncovered subtler constructs within theWen sequence. Once derived, the FOD sequence may be expanded to detail the exact line-places (gua/kua) at which change occurs. Thisxpansion yields an n-tuple for each FOD integer. The data obtained can be presented in several modes; the author has chosen to representhese data the quantity, location, and direction of change, in a 64x6 matrix of binary elements, where each of the 64 rows corresponds toxactly one FOD integer, and the 6 columns correspond to the 6 kua (line-places). Alternately, each derivative can be consistentlyepresented as a 6-tuple, with naughts in the non-changing places.

The author has introduced the term Wen derivative to describe the additional attributes of the set of FOD integers. Unlike traditionalerivatives, which are often mathematical functions, the Wen derivatives comprise 64 n-tuples (alternately, 6-tuples) which denote therecise quantity, location, and direction of change occurring between successive pairs of I Ching hexagrams when they are ordered in theraditional (King Wen) sequence.

Derivative 5, depicted graphically just below, is a 6-tuple indicating that four instances of change occur in the transition from Hsu (5) to Su6): specifically in the 6 th , 4 th , 3 rd, and 1 st places; the first two changes are to yang, the latter pair to yin. A formal and concise notation, like [+64 -3 -1] or [+6 0 +4 -3 0 -1] suggests itself. Additional explanatory terms are introduced further below.

5 th derivative)


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The following chart displays the set of FOD integers, and the Wen derivatives in graphical fashion; it also shows their derivation and therecise quantity and locations of change. The distribution and recurrence of the first Wen derivatives is also shown. Finally, the chartisplays the direction of change or polarity for each place that changes between successive pair of hexes: e.g., derivative 63 details changen all six places, upward-pointing white arrowheads indicate yin changing to yang; downward-pointing black arrowheads denoting yanghanging to yin.

There are 214 total changes over the complete collection of derivatives: 107 of these changes show yang alternating to yin, 107 of thesehanges show yin alternating to yang

Changes in the six places between successive hexes over 64 pairs of hexes (King Wen arrangement):yang --> yin yin --> yang

: 38 19 19: 36 18 18: 34 17 17: 36 18 18: 32 16 16: 38 19 19otal: 214

The following chart displays the Wen derivatives of integers in graphical fashion, also showing their derivation and the precise locations of hange. The distribution and recurrence of the FOD integers is also shown. Finally, the chart displays the direction of change for eachhanging place between successive pair of hexes: e.g., Wen derivative 63 details change in all six places: upward-pointing white arrowheadsenote yin changing to yang; downward-pointing black arrows denote yang changing to yin. The traditional associations of high, bright, etc.or yang; and low, dark, etc. for yin supplied the inspiration for the symbols. The canonical names were taken from the Eranos edition, anxcellent research reference with complete concordance.


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hart 1

canonical name base-2 base-10 indexsort

index index occurs # place/direction

force 111111 63 01 1 1 1 1 1 101 01 1 of 9 6

field 000000 0 02 0 0 0 0 0 002 02 1 of 2 2

sprouting 010001 17 03 0 1 0 0 0 103 03 1 of 4 4

enveloping 100010 34 04 1 0 0 0 1 004 04 1 of 1 4

attending 010111 23 05 0 1 0 1 1 105 05 1 of 4 4

arguing 111010 58 06 1 1 1 0 1 006 06 1 of 2 3

legions/leading 000010 2 07 0 0 0 0 1 007 07 1 of 4 2

grouping 010000 16 08 0 1 0 0 0 008 08 1 of 3 4

small accumulating 110111 55 09 1 1 0 1 1 109 09 1 of 5 2

treading 111011 59 10 1 1 1 0 1 110 10 1 of 1 4

pervading 000111 7 11 0 0 0 1 1 111 01-2 2 of 9 6

obstruction 111000 56 12 1 1 1 0 0 012 11 1 of 2 2

concording people 111101 61 13 1 1 1 1 0 113 07-2 2 of 4 2

great possessing 101111 47 14 1 0 1 1 1 114 12 1 of 1 4

humbling 000100 4 15 0 0 0 1 0 015 09-2 2 of 5 2

providing-for 001000 8 16 0 0 1 0 0 016 02-2 2 of 2 2

following 011001 25 17 0 1 1 0 0 117 01-3 3 of 9 6

corrupting 100110 38 18 1 0 0 1 1 018 13 1 of 1 3

nearing 000011 3 19 0 0 0 0 1 119 03-2 2 of 4 4

viewing 110000 48 20 1 1 0 0 0 020 14 1 of 2 3

gnawing bite 101001 41 21 1 0 1 0 0 121 09-3 3 of 5 2

adorning 100101 37 22 1 0 0 1 0 122 11-2 2 of 2 2

stripping 100000 32 23 1 0 0 0 0 023 15 1 of 4 2

returning 000001 1 24 0 0 0 0 0 124 06-2 2 of 2 3

without embroiling 111001 57 25 1 1 1 0 0 125 16 1 of 4 4

great accumulating 100111 39 26 1 0 0 1 1 126 17 1 of 2 2

jaws/swallowing 100001 33 27 1 0 0 0 0 127 01-4 4 of 9 6

great exceeding 011110 30 28 0 1 1 1 1 028 09-4 4 of 5 2


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gorge 010010 18 29 0 1 0 0 1 029 01-5 5 of 9 6

radiance 101101 45 30 1 0 1 1 0 130 18 1 of 1 3

sensitivity 011100 28 31 0 1 1 1 0 031 07-3 3 of 4 2

persevering 001110 14 32 0 0 1 1 1 032 19 1 of 1 3

retiring 111100 60 33 1 1 1 1 0 033 03-3 3 of 4 4

great invigorating 001111 15 34 0 0 1 1 1 134 08-2 2 of 3 4

prospering 101000 40 35 1 0 1 0 0 035 05-2 2 of 4 4

brightness hiding 000101 5 36 0 0 0 1 0 136 20 1 of 1 2

dwelling people 110101 53 37 1 1 0 1 0 137 16-2 2 of 4 4

polarizing 101011 43 38 1 0 1 0 1 138 01-6 6 of 9 6

limping 010100 20 39 0 1 0 1 0 039 16-3 3 of 4 4

taking-apart 001010 10 40 0 0 1 0 1 040 21 1 of 2 3

diminishing 100011 35 41 1 0 0 0 1 141 07-4 4 of 4 2

augmenting 110001 49 42 1 1 0 0 0 142 22 1 of 1 4

parting 011111 31 43 0 1 1 1 1 143 15-2 2 of 4 2

coupling 111110 62 44 1 1 1 1 1 044 23 1 of 1 3

clustering 011000 24 45 0 1 1 0 0 045 16-4 4 of 4 4

ascending 000110 6 46 0 0 0 1 1 046 24 1 of 1 3

confining 011010 26 47 0 1 1 0 1 047 09-5 5 of 5 2

the well/welling 010110 22 48 0 1 0 1 1 048 25 1 of 1 3

skinning 011101 29 49 0 1 1 1 0 149 03-4 4 of 4 4

the vessel/holding 101110 46 50 1 0 1 1 1 050 08-3 3 of 3 4

shake 001001 9 51 0 0 1 0 0 151 05-3 3 of 4 4

bound 100100 36 52 1 0 0 1 0 052 26 1 of 1 1

infiltrating 110100 52 53 1 1 0 1 0 053 01-7 7 of 9 6

converting maidenhood 001011 11 54 0 0 1 0 1 154 17-2 2 of 2 2

abounding 001101 13 55 0 0 1 1 0 155 15-3 3 of 4 2

soujourning 101100 44 56 1 0 1 1 0 056 27 1 of 1 3

ground 110110 54 57 1 1 0 1 1 057 05-4 4 of 4 4

open 011011 27 58 0 1 1 0 1 1


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58 21-2 2 of 2 3

dispersing 110010 50 59 1 1 0 0 1 059 15-4 4 of 4 2

articulating 010011 19 60 0 1 0 0 1 160 28 1 of 1 1

centering conforming 110011 51 61 1 1 0 0 1 161 01-8 8 of 9 6

small exceeding 001100 12 62 0 0 1 1 0 062 14-2 2 of 2 3

already fording 010101 21 63 0 1 0 1 0 1

63 01-9 9 of 9 6 not-yet fording 101010 42 64 1 0 1 0 1 0

64 29 1 of 1 3

force 111111 99 1 1 1 1 1 1


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his chart attempts to represent several schemes for classifying the Wen derivatives:

niqueness: greena Wen derivative occurs singly across the entire series of 64, it is termed unique . 15 such derivatives exist.

rder: arranged by column; Wen derivative fall into one of five different orders , indicating the number of places in which change occurs for that set. Anrder will contain one or more families of Wen derivatives. Each family will contain one or more instances of Wen derivatives. No derivatives of order 0 or 5xist when the 64 hexes are placed in the traditional sequence.

amily: each order contains one or more families whose members are similar in that the set of changing places are identical, though the polarity of thesehanges may differ in one or more places

iscreteness: bold . indicates the first incidence of a Wen derivative that occurs multiple times. Of the full series of 64 Wen derivatives, 29 discrete instancesxist; each of the 64 sets will fall into one of these. The notion of discreteness should be revised since component polarities often differ between otherwisedentical derivatives.

ddness / evenness of index: whether or not some inference may be drawn by this quality of the index is not now known, though the fact that when indexedy the full series, the indices of all 15 unique Wen derivatives are evenly divisible by 2 is notable. The ratio of evenly-indexed derivatives to oddly-indexed

will also be investigated

hart 2

6 4 3 2 1

-6 5 -4- 3 -2 -1 1 -6 -5 4 +3 10 -6 -5 -4 6 +6 +5 36 +6 60

+6 +5 +4 -3 -2 -1 11 -6 +5 +3 +1 4 +6 +5 +4 24 -6 +1 23 +5 52

+6 -5 -4 +3 -2 -1 17 +6 -5 +2 -1 3 +6 +5 -2 32 +6 -1 43

-6 +5 +4 +3 +2 -1 27 +6 +5 -2 -1 19 -6 +5 -1 30 +6 -1 55

+6 -5 +4 +3 -2 +1 29 -6 -5 +2 +1 33 +6 -4 +1 40 -6 +1 59

-6 +5 -4 +3 -2 -1 38 +6 -5 +2 -1 49 +6 -4 -1 58 +5 -2 7

-6 -5 +4 -3 +2 +1 53 -6 +4 +3 +2 42 -6 -3 -2 44 -5 +2 13

-6 -5 +4 +3 -2 -1 61 +6 +4 -3 -1 5 -6 -3 +1 18 -5 +2 31

+6 -5 +4 -3 +2 -1 63 -6 -4 +3 +1 35 +5 +4 -3 46 +5 -2 41

+6 -4 +3 -1 51 +5 -4 +2 56 +5 +1 2

-6 +4 -3 +1 57 -5 +4 +1 20 +5 +1 16

-6 -4 -2 -1 14 +5 -4 +1 62 +4 -3 9

+6 +3 +2 +1 8 +5 +3 +1 64 +4 -3 15

+6 -3 -2 -1 34 +4 -2 +1 48 -4 +3 21

-6 -3 -2 +1 50 -4 -3 28

-5 -4 +3 +2 25 -4 +3 47

-5 +4 -3 +2 37 -3 -2 26

-5 +4 -3 +2 39 +3 -2 54

-5 -4 +3 +2 45 +3 +1 12

-3 -1 221 9 8 19 11 14 7 20 2 2

even:odd 1:8 7:12 14:0 8:12 2:0

ua: a line-place (whose index may range {1..6}), denoting the exact location(s) of change between successive hexagrams when arranged inhe King Wen sequence. change between successive hexagrams when arranged in the King Wen sequence is defined by McKenna as first-rder difference (FOD).

he complete set of FOD comprises sixty-four integers, each of which gives the quantity of changing lines kua for successive hexagrams (when

rranged in the King Wen sequence). the collection of kua for each member of the set of FOD (of which there are 64) yields a Wen derivativ


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Wen derivative : an n-tuple corresponding to a given FOD integer; the domain of such a set is the group of integers {+/- 1..+/- 6}; n may range1..4,6}; FOD:5, FOD:0 do not exist for the King Wen sequence.

hart 3Wen derivative by

size 6 5 4 3 2 1 0 summation

A occurrences 9 0 19 14 20 2 0 64

Bdiscrete

instances1 0 8 11 7 2 0 29

Cduplicate

occurrences8 0 11 3 13 0 0 35

E non-occurring 0 0 7 9 8 4 0 28

Dpossible

combinations1 6 15 20 15 6 1 64

hart 4

line-changes6 5 4 3 2 1 summation

HTotal #changes

per kua 38 36 34 36 32 38 214

JDups of changes

in kua21 20 21 22 20 23 127

I#changes per kua

w/o dups17 16 13 14 12 15 87

K non-occurring 10 11 14 13 15 12 75

L I + K 27 27 27 27 27 27 162

hart 5

OT: # times a set

occurs1 2 3 4 5 9 summation

PN: # sets occurring

T times15 6 1 5 1 1 29

Q T*N 15 12 3 20 5 9 64


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xamples for reading the tables:: 9 non-unique Wen derivatives of size 6: 8 discrete Wen derivatives of size 4: 13 duplicate Wen derivatives of size 2: 6C6 + 6C5 + 6C4 + 6C3 + 6C2 + 6C1 + 6C0 = 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64: 38 changes occur in the 6 th place, including duplicate sets

: 16 changes occur in the 5 th place, excluding duplicate sets: 23 changes occur in the 1 st place, strictly among duplicate sets: of all possible Wen derivatives, 28 do not appear; this is the distribution per kua/P: 15 FOD-sets are unique, 6 FOD-sets have twins, 5 Wen derivatives are 4-tuples

: 5 Wen derivatives occurring 4 times each total 20 sets

otes:he summation of row I (87) is three times the summation of row B (29)

he summation of Row P (29) is equal to the summation of row B (29)he length of a lunar month is approx 29.53 daysow I comprises the consecutive integers from 12 to 17ow K comprises the consecutive integers from 10 to 15ows I and K are ordered such that their sum is always 27

ommentary: if, as McKenna intuited, the lunar year comprises 13 months of 29.53 solar days (one lunar month: the time i t takes the moon to orbit the earth), this yields 383.89 solar days, orpproximately 384 days, one day for each line in the I Ching. The table below comprises 29 discrete elements, each occurring one or more times, totaling 64 elements. Each element is a set of placesorresponding to a single FOD. If each element stands for a single day in the lunar month, what other correspondences may exist?


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Observations and correspondences:

he complete canon of change contains 64 times 6 lines, or 384 lines.

he complete set of 64 Wen derivatives encompasses 214 line-changes between consecutive hexagrams

he subset of 35 non-discrete Wen derivatives encompasses 127 line changes

he subset of 29 discrete Wen derivatives encompasses 87 line-changes

he subset of 15 unique Wen derivatives encompasses 44 line-changes.

he subset of 14 Wen derivatives encompasses 43 line-changes

=====================================================84 lines (complete canon) less 214 changing-lines (all Wen derivatives) leave 170 unchanging lines

14 changing-lines (all Wen derivatives) less 44 changing-lines (the subset of 15 unique) leave 170 lines

84 (complete canon) less 44 changing-lines (unique Wen derivatives) leaves 340 lines: twice 170

what is the significance of 170 in this context?

=====================================================he subset of 29 discrete Wen derivatives encompasses 87 line-changes

he subset of 15 unique Wen derivatives encompasses 44 line-changes.

wice 15 is one more than 29, twice 44 is one more than 87

=====================================================hanges in the six places over the subset of 29 discrete Wen derivatives:

6: 175: 164: 143: 132: 121: 15-----

otal: 87

hange occurs 87 times over each of the six places (29*3)Additionally, there are 87 instances of non-change over the six places

(places) multiplied by 29 (hexagrams) equals 174 potential changes in total.

what is the likelihood that a derivation of the King Wen arrangement would result in an even distribution of change and non-change over the subset of 29Wen derivatives?

hange occurs 44 times over the set of 15 unique Wen derivativesAdditionally, there are 46 instances of stasis over the six places

(places) multiplied by 15 (hexagrams) equals 90 potential changes in total.

Over the Wen sequence of 64 hexes, there are 384 lines (6*64); already shown is that change between successive hexes occurs 214 times in six kua across 64exes in the King Wen arrangement. This is 0.55729 changes on the average (somewhat more than half)

irection of line-changes over the FOD integers:

here are 214 total changes over the complete collection of FOD integers07 of these changes describe yang alternating to yin07 of these changes describe yin alternating to yang

What are the odds of a perfectly-even distribution of directed line-changes?


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Qualities of the King Wen sequence:

a. the 32 ordered hexagrams pairs are composed such that the second hexagram of the pair is the mirror-image inversion of the first.

b. when the initial place-wise differentiation between successive hexagrams is taken (yielding the FOD set), the number of line-places where change occurs is always an even integer when taken within a pair odd integers only occur when change is observed between pairs

c. the ratio of even integers to odd integers within the FOD set is 3 to 1; odd integers constitute of the set

d. the integer 5 does not occur in the FOD set, indicating that five simultaneous line-changes never occur between successive hexes.

e. expanding the FOD set of integers to reveal the line-places at which change occurs shows exactly half of the 214 changes that occur in the Wenderivatives are from yang to yin; half of the changes in each of the six places move in the opposite direction.

his regular pattern of alternation between yin and yang is a partial consequence of feature (a). The ordered pairs (beginning with (1,2) ending with (63,64))ecessarily determine this alternation in the intra-pair derivatives. The sequence of the ordered pairs is responsible for the inter-pair yin-yang alternation.he ordered pairs are sequenced such that the alternation of yin to yang and back in each of the 6 places (gua/kua) is maintained over the entire sequence.

Given that there are 32! or 2.6E35 possible sequences of the pairs, and the likelihood that some of the possible sequences will not maintain the flow of yin toang to yin in each place, this too should be regarded as intelligent design just McKenna felt should be regarded the absence of any instance of 5imultaneously changing lines in the FOD set, or the 3:1 ratio of even numbers to odd numbers in the FOD set. Determination of which and how many suchequences of pairs do in fact produce the periodicity observed in the derivative set is left as a future line of inquiry.


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Hypotheses and lines of inquiry

The particular significance granted by the I Ching to the 5 th (ruler) place is hypothesized to be related to the absence of fives (5) in the FODntegers. This line of inquiry leads to the possibility that there could be five alternate orderings of the hexes according to the other fivelaces. Each of these alternate orderings would resemble the King Wen ordering in that the first differentiation would reveal the absence of ne of the digits {1, 2, 3, 4, 6}, corresponding to the absence of change in that place between consecutive hexes.f this intuition is to be proven, it remains to determine the essential ordering principle of the King Wen sequence, and use this to derivecorrect sequences for the remaining hypothetical five sequences.

n chart 3 row D above, we show a combinatorial distribution of the 57 possible Wen derivatives sized one through four, and six. (Note that wh4 Wen derivatives are theoretically possible, the nature of the Wen ordering necessarily excludes FOD(0), and we have yet to understand why FOD(5) does not exist.)

Though the complete collection of Wen derivatives comprises 64 members, there are only 29 discrete sets in the collection; the other 35 areut duplicates of these 29. Investigating the nature of the missing 28 sets may give more insight to the ordering principle of the hexagrams.

We organized the non-occurring FOD integers (identified in bold red type in chart 6) by kua. The 28 sets comprise 75 kua; we discovered that. (?)

Having observed that the regular alternation of yang and yin observed among the derivative is a construct of the arrangement of theexagrams, the questions beg, what is the significance and what is the function?

Now that direction/polarity is a component of the Wen derivative, we should re-examine the previously-defined notions of unique andiscrete as regards the Wen derivatives. Otherwise identical derivatives may now differ in terms of the changes in polarity in any of theomponents of the n-tuple. E.g., derivative 21 has two instances previously regarded as equal; the polarity of the final component in the 3-uple, however, is now seen as different between the two instances.


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his chart catalogues the Wen derivatives that never appear (bold red)

Missing FOD integers

hart 6

654321 6543 654 65 6

6542 653 64 5

6541 652 63 4

6532 651 62 3

6531 643 61 26521 642 54 1

6432 641 53

6431 632 52

6421 631 51

6321 621 43

5432 543 42

5431 542 41

5421 541 32

5321 532 31

4321 531 21

521

432

431

421

321

Copyright 2008 R. Quincy Robinson

All rights reserved

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