Quantifying Change: Uniting the Oracle with the Changes
One troublesome aspect of I Ching for the author had been that there appeared to be no connection between the yarrow-stalk oracle (ostensibly presented by Confucius) and the text. To the point, the text of I Ching does not speak in terms of the four xiang, only in terms of greater YIN [6] or greater YANG [9]. Lesser YIN [8] and lesser YANG [7] are not referenced in the text.
The divination ritual provided by Ta Chuan introduces the four digram figures denoting HEAVEN or greater YANG [⚌], FIRE or lesser YIN [⚍], WATER or lesser YANG [⚎], and EARTH or greater YIN [⚏] by the numbers [9,8,7,6] respectively. The xiang, as the digrams are called, have a long heritage within I Ching tradition. The xiang constitute all permutations of the two basic lines [⚊], [⚋] taken in pairs (2 x 2 = 4). Nine and six indicate changing conditions in a divination, but the reader of I Ching cannot possibly observe changing conditions in a hexagram figure that displays them statically.
Put differently, examining the 64 hexagram figures outside the context of divination provides little means to discern whether a given hexagram has moving lines, or the positions at which they occur – changing lines are essentially meaningless outside the divination context. Without knowledge of where and how Change occurs, one attempting to systematize or quantify the Change would be left to broadly assign 9 (or 7) in place of YANG, and 6 (or 8) in place of YIN wherever YIN and YANG lines are encountered.
An initial step taken towards quantifying and measuring Change involved adopting a shift in perspective from which it follows that Ch'ien ("all nines") and K'un ("all sixes") refer to situations with all 6 positions are changing in parallel. As situations, all hexagrams are thus dynamic contexts, not merely still images as they are appear in the text.
In Taoist cosmogony, xiang are regarded as the primordial reality that emerged after YIN [⚋] and YANG [⚊] separate and emerge from the Taiji singularity, which differentiated itself from the Way (Tao) in order that Reality might be made manifest. Thus, in the context of I Ching, Changes may also be understood as comprising three digrams which represent the Three Powers: EARTH, HUMANITY, and HEAVEN. The relevance of this will shortly be made apparent.
Initially, the expanded first-order difference integers accounted only two types of line-changes: YIN changing to YANG, and conversely, at any of the six positions. In the following depiction of Change #5, the white-spaces in the 2nd and 5th places betray flaws in the original version of the expansion procedure:
Such white-spaces invariably arose from the two non-change conditions: unchanging (or static) YANG, and unchanging (or static) YIN. It became clear that for any Wen pair and at any of the six positions, four kinds of line-changes may manifest. To omit or treat identically the two static conditions when they clearly denote different states is to commit the same errors McKenna committed with Timewave by only considering the number of line-changes while ignoring the places at which they occur and by ignoring non-changing positions.
On learning of the xiang and their suitability for representing the different kinds of line-changes, the author refined the expanded first-order difference operation (renamed as Change) by using xiang bi-grams to denote the four discrete types of line-changes. This Change procedure was iterated over the full set of 32 Wen pairs, producing thirty-two Changes. However, yao-number 144 does not occur in these thirty-two Changes produced per the rules given in Ta Chuan. Furthermore, the yao-numbers corresponding to these thirty-two Changes did not sum to 11,520.
It was determined that the only way to produce the yao-number 144 thus effecting "all sixes” changing at once is for K'un to transform into Ch'ien. This occurs when one reverses the ordering of King Wen pair #1 to indicate #2 Earth changing to #1 Heaven. This implies that the remaining Changes are produced by reverse passage through the King Wen sequence.
Initially, King Wen pairs were deemed properly-formed when an oddly-indexed hexagram is followed by an evenly-indexed hexagram; e.g., [1,2] or [29,30]. Subsequently, the rule for a properly-formed King Wen pair was expanded to include the reverses of the Wen pairs, and the procedure was amended to iterate over the additional thirty-two Wen anti-pairs (reversed Wen pairs) which completed the Canon of sixty-four Changes. The yao-numbers (11,520) were then found to sum correctly, and the yao-numbers corresponding to hexagrams #1 and #2 properly sum to 360.
One troublesome aspect of I Ching for the author had been that there appeared to be no connection between the yarrow-stalk oracle (ostensibly presented by Confucius) and the text. To the point, the text of I Ching does not speak in terms of the four xiang, only in terms of greater YIN [6] or greater YANG [9]. Lesser YIN [8] and lesser YANG [7] are not referenced in the text.
The divination ritual provided by Ta Chuan introduces the four digram figures denoting HEAVEN or greater YANG [⚌], FIRE or lesser YIN [⚍], WATER or lesser YANG [⚎], and EARTH or greater YIN [⚏] by the numbers [9,8,7,6] respectively. The xiang, as the digrams are called, have a long heritage within I Ching tradition. The xiang constitute all permutations of the two basic lines [⚊], [⚋] taken in pairs (2 x 2 = 4). Nine and six indicate changing conditions in a divination, but the reader of I Ching cannot possibly observe changing conditions in a hexagram figure that displays them statically.
Put differently, examining the 64 hexagram figures outside the context of divination provides little means to discern whether a given hexagram has moving lines, or the positions at which they occur – changing lines are essentially meaningless outside the divination context. Without knowledge of where and how Change occurs, one attempting to systematize or quantify the Change would be left to broadly assign 9 (or 7) in place of YANG, and 6 (or 8) in place of YIN wherever YIN and YANG lines are encountered.
An initial step taken towards quantifying and measuring Change involved adopting a shift in perspective from which it follows that Ch'ien ("all nines") and K'un ("all sixes") refer to situations with all 6 positions are changing in parallel. As situations, all hexagrams are thus dynamic contexts, not merely still images as they are appear in the text.
In Taoist cosmogony, xiang are regarded as the primordial reality that emerged after YIN [⚋] and YANG [⚊] separate and emerge from the Taiji singularity, which differentiated itself from the Way (Tao) in order that Reality might be made manifest. Thus, in the context of I Ching, Changes may also be understood as comprising three digrams which represent the Three Powers: EARTH, HUMANITY, and HEAVEN. The relevance of this will shortly be made apparent.
Initially, the expanded first-order difference integers accounted only two types of line-changes: YIN changing to YANG, and conversely, at any of the six positions. In the following depiction of Change #5, the white-spaces in the 2nd and 5th places betray flaws in the original version of the expansion procedure:
On learning of the xiang and their suitability for representing the different kinds of line-changes, the author refined the expanded first-order difference operation (renamed as Change) by using xiang bi-grams to denote the four discrete types of line-changes. This Change procedure was iterated over the full set of 32 Wen pairs, producing thirty-two Changes. However, yao-number 144 does not occur in these thirty-two Changes produced per the rules given in Ta Chuan. Furthermore, the yao-numbers corresponding to these thirty-two Changes did not sum to 11,520.
It was determined that the only way to produce the yao-number 144 thus effecting "all sixes” changing at once is for K'un to transform into Ch'ien. This occurs when one reverses the ordering of King Wen pair #1 to indicate #2 Earth changing to #1 Heaven. This implies that the remaining Changes are produced by reverse passage through the King Wen sequence.
Initially, King Wen pairs were deemed properly-formed when an oddly-indexed hexagram is followed by an evenly-indexed hexagram; e.g., [1,2] or [29,30]. Subsequently, the rule for a properly-formed King Wen pair was expanded to include the reverses of the Wen pairs, and the procedure was amended to iterate over the additional thirty-two Wen anti-pairs (reversed Wen pairs) which completed the Canon of sixty-four Changes. The yao-numbers (11,520) were then found to sum correctly, and the yao-numbers corresponding to hexagrams #1 and #2 properly sum to 360.
NB: While the yao-numbers of Wen pair [1,2] indeed sum to 360 (also true for anti-pair [2,1]) , we discovered that pairs do not always sum to 360. The smallest yao-number sum of a King Wen pair was observed to be 344: [9,10], [13,14], [43,44]; the largest yao-number sum observed was 376: [7,8],[15,16], [23,24].
We now have a probability distribution for the King Wen Sequence. The yao-numbers and their relative frequencies are: 144 (1), 168 (3), 172 (6), 180 (20), 184 (12), 188 (6), 192 (3), 216 (1). A graphical representation of the “King Wen Distribution” (including distribution statistics) follows.
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| King Wen's Distribution |
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