Quantifying Change: Defining the Change operation
Considering any King Wen pair of hexagrams, making comparisons across the members of the pair at each of the six positions, observing and denoting where line-changes occur, the type of line-changes, and where line-changes do not occur, describes the Change operation. The product of this operation constitutes an individual Change. Fully-iterated* over the King Wen sequence, the Change operation produces sixty-four discrete Changes. These sixty-four Changes bear a close semantic relationship to the sixty-four hexagrams.
NB: *Full-iteration means to apply the Change operation as one traverses the pairs of the King Wen sequence from beginning to end and back to the beginning.
Example: Change #28 is composed of the pair [55,56]. Change #28 describes a situation where two positions, the bottom and top, are changing in different directions:
By contrast, its anti-pair, Change #37 (composed of the hexagram pair [56,55] also has two changing lines and four stable lines, but the polarity of each changing line is the opposite of Change #28. Static lines, by nature, do not change polarity.
While a Change has the same shape as a hexagram, it must be emphasized that each Change captures two qualities at each of the six positions which describe what transpired within the pair resulting from the Change:
Polarity and state are terms used to describe qualities that describe mutually-exclusive conditions of a line. Representing them simultaneously requires (2 x 2 = 4) four values. Fortunately, we are not required decide how to represent these conditions. The appendices of I Ching include a section entitled Ta Chuan, the Great Treatise. Ta Chuan is an invaluable resource for accessing the philosophical and cosmogonical context of the Changes. Ta Chuan provides a description of the “method of threes and fives” (i.e., the yarrow stalk oracle), a divination ritual commonly used at the time of its writing. The description of the yarrow-stalk oracle includes a discussion of the ritual numbers [6,7,8,9]. These ritual numbers will serve to track polarity and state as described above.
The first thirty-two King Wen pairs (H,H') are as follows: [1,2], [3,4], ... [63,64]. The remaining thirty-two King Wen pairs are the reversals, pair-wise and sequential, of the first thirty-two pairs: [64,63] [62,61], ... [2,1]. Performing the Change operation (taking the expanded first-order difference within the sixty-four King Wen pairs), produces a sequence of sixty-four figures which record the positional state-changes that occurred within each pair as the Wen sequence is traversed to its end and back. The pair-wise Change operation is continuous over the sequence of sixty-four hexagrams.
The ritual numbers six, seven, eight, and nine are central to the divination ritual and are described as xiang, or symbols, which we employ for representing the four kinds of Change. The cosmogonical origin of the four xiang appears to stem from the initial intermingling of the primordial YIN and YANG (the Gates of Change, Ch'ien and K'un, represented by Changes #1 and #2). Indeed, this can shown geometrically:
The Cartesian plane is composed of two axes which we may use these represent Ch'ien and K'un. We may use resulting quadrants to stand for the xiang:
⚏ [6] = greater/dynamic YIN (alternates to YANG)
⚎ [7] = lesser/static YANG
⚍ [8] = lesser/static YIN
⚌ [9] = greater/dynamic YANG (alternates to YIN)
Considering any King Wen pair of hexagrams, making comparisons across the members of the pair at each of the six positions, observing and denoting where line-changes occur, the type of line-changes, and where line-changes do not occur, describes the Change operation. The product of this operation constitutes an individual Change. Fully-iterated* over the King Wen sequence, the Change operation produces sixty-four discrete Changes. These sixty-four Changes bear a close semantic relationship to the sixty-four hexagrams.
NB: *Full-iteration means to apply the Change operation as one traverses the pairs of the King Wen sequence from beginning to end and back to the beginning.
Example: Change #28 is composed of the pair [55,56]. Change #28 describes a situation where two positions, the bottom and top, are changing in different directions:
Position 6 shows YANG changing to YIN
Positions 5 and 2 show static YIN
Positions 4 and 3 show static YANG
Position 1 shows YANG alternating to YIN.
By contrast, its anti-pair, Change #37 (composed of the hexagram pair [56,55] also has two changing lines and four stable lines, but the polarity of each changing line is the opposite of Change #28. Static lines, by nature, do not change polarity.
Position 6 shows YIN changing to YANG
Positions 5 and 2 show static YIN
Positions 4 and 3 show static YANG
Position 1 shows YIN alternating to YANG.
While a Change has the same shape as a hexagram, it must be emphasized that each Change captures two qualities at each of the six positions which describe what transpired within the pair resulting from the Change:
- polarity of the position (in terms of YIN versus YANG)
- state of that position's polarity (in terms of static versus dynamic).
Polarity and state are terms used to describe qualities that describe mutually-exclusive conditions of a line. Representing them simultaneously requires (2 x 2 = 4) four values. Fortunately, we are not required decide how to represent these conditions. The appendices of I Ching include a section entitled Ta Chuan, the Great Treatise. Ta Chuan is an invaluable resource for accessing the philosophical and cosmogonical context of the Changes. Ta Chuan provides a description of the “method of threes and fives” (i.e., the yarrow stalk oracle), a divination ritual commonly used at the time of its writing. The description of the yarrow-stalk oracle includes a discussion of the ritual numbers [6,7,8,9]. These ritual numbers will serve to track polarity and state as described above.
The first thirty-two King Wen pairs (H,H') are as follows: [1,2], [3,4], ... [63,64]. The remaining thirty-two King Wen pairs are the reversals, pair-wise and sequential, of the first thirty-two pairs: [64,63] [62,61], ... [2,1]. Performing the Change operation (taking the expanded first-order difference within the sixty-four King Wen pairs), produces a sequence of sixty-four figures which record the positional state-changes that occurred within each pair as the Wen sequence is traversed to its end and back. The pair-wise Change operation is continuous over the sequence of sixty-four hexagrams.
The ritual numbers six, seven, eight, and nine are central to the divination ritual and are described as xiang, or symbols, which we employ for representing the four kinds of Change. The cosmogonical origin of the four xiang appears to stem from the initial intermingling of the primordial YIN and YANG (the Gates of Change, Ch'ien and K'un, represented by Changes #1 and #2). Indeed, this can shown geometrically:
The Cartesian plane is composed of two axes which we may use these represent Ch'ien and K'un. We may use resulting quadrants to stand for the xiang:
⚎ [7] = lesser/static YANG
⚍ [8] = lesser/static YIN
⚌ [9] = greater/dynamic YANG (alternates to YIN)
The positional changes in polarity (YIN or YANG) may be combined to produce a notion of the quantity of Change thus represented. Such a quantity may take on, for example, the values [2,4,6]. These values reflect the total number of positions undergoing change within a given pair of hexagrams when ordered in the King Wen sequence. There are other ways of interpreting the quantity of Change as well. For instance, one might count greater YIN and greater YANG individually, which would allow for values [1,2,3].
We also are able to infer some notion of direction as it regards traversal through the King Wen sequence. Forward motion through the King Wen sequence is represented by King Wen pairs #1 to #32, return motion is represented by King Wen pairs #33 to #64, which reverse the order of the pairs. Because the Change operation preserves the direction of traversal through the King Wen sequence, the Change operation appears to violate commutativity, though superficially.
The following table presents the sixty-four King Wen pairs and anti-pairs in a color-coded tabular format. Change indices, hexagram indices and yao-sums are in the various columns. The thirty-two King Wen pairs are read from upper left to lower right. The thirty-two anti-pairs are read from lower-right to upper left.
We also are able to infer some notion of direction as it regards traversal through the King Wen sequence. Forward motion through the King Wen sequence is represented by King Wen pairs #1 to #32, return motion is represented by King Wen pairs #33 to #64, which reverse the order of the pairs. Because the Change operation preserves the direction of traversal through the King Wen sequence, the Change operation appears to violate commutativity, though superficially.
The following table presents the sixty-four King Wen pairs and anti-pairs in a color-coded tabular format. Change indices, hexagram indices and yao-sums are in the various columns. The thirty-two King Wen pairs are read from upper left to lower right. The thirty-two anti-pairs are read from lower-right to upper left.
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