AS207A/KH0225 - Subsidiary Pendant Sums


Drawings:
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All Khipus' Ascher Sum FieldmarksLeft-Handed XRay
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This Khipu's Fieldmark Catalog
This Khipu's Drawing

Left Handed Sums:     # Sums = 1,  Max # Summands = 6,   (Min, Mean, Max) Sum Values = (54, 54, 54)
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Left Handed Sum Detail: - Click on column name to sort
# Color Sum Schema Sum Cord Sum Cord Value # Summands Summands
1
p195s140, 3, 1 : 54W:BB546p33: 31MB:YG + p34: 3YB:MB + p35: 1YB:MB + p37: 9YB:MB + p38: 1YB:MB + p40: 9YB:MB

Khipu Notes:
Ascher Databook Notes:
  1. Construction Note: The attachment is loose so that the pendant is movable. It may be associated with the first group of pendants.
  2. Construction Note: The pendant attachment is irregular.
  3. Construction Note: The pendant is formed into a loop at 26.0 cm.
  4. Construction Note: The top cord is tied through the pendants so that it emanates from the right side of the pendant group. All other top cords on this khipu are tied so that they emanate from the left side of the pendant group.
  5. A 3.0 cm cord fragment found with this group is color KB:W and has 1 E knot at 1.0 cm. It is probably part of pendant 165, 167, or 168.
  6. Construction Note: The pendant appears to have been added after the khipu was completed. The other pendants in the group are tight and immediately adjacent to each other. This pendant is tied over them and tightly attached. Although listed as a subsidiary, one of the cords attached to it may be a top cord since the cord is tied through the pendant attachment loop.
  7. Construction Note: M1, M2, M3 are exceptionally thick cords.
  8. The khipu can be viewed as in 3 parts. There is a single pendant then, by spacing, Part I is 9 groups of 6 to 14 pendants each, Part II is 10 groups each consisting of 6 pendants united by a top cord, and Part II is 9 groups of 4 to 17 pendants each.

    Part I

    1. Group 2 has 7 pendants united by spacing and color. The 4th and 5th pendants in the group have double units values. Keeping these unit values distinct, the first 6 pendants sum to 8 3& 2 &1. The value on the 7th pendant is 83. Thus:

      \[ P_{127} = \sum\limits_{i=1}^{6} P_{12i} \]
    2. Group 3 has 8 pendants united by spacing and color. The value of the second pendant is the sum of the next 6 pendants, that is:

      \[ P_{132} = \sum\limits_{i=3}^{8} P_{13i} \]
      Although in a different order, these 6 values are the same as those in the first group of Part II:

      P122 = P212
      P123 = P216
      P135 = P214
      P13,i+5 = P21i for i=1,2,3

      Since we hypothesize that the P21i are sums of the next 9 groups in Part II, P132 would be a sum of sums.

    3. The first 6 pendants of Group 5 have the values 1,0,9,1,0,9. The 9 pendants of group 6 are separated by color into 3 groups of 3. Each of the 9 cords have the value 1 &9. The next 2 groups have 6 pendants each. Both groups have the values 1, 9 repeated 3 consecutive times.
    4. Although the groups have 6 to 14 pendants each by spacing, within these, by color and values, there are subgroupings of 3 or 6 pendants. The 7 pendants in Group 2 are 6 values followed by their sum and the last 6 pendants in Group 3 are summed on the pendant preceeding them. By values, the 14 pendants in Group 5 seem to be subgroups of 6, 6, and 2. By color Group 6 is 3 subgroups of 3 each, Group 7 is 6 pendants, Group 8 is 6 pendants, and the 8 pendants in Group 9 are grouped by color into 2, 3, 3.

    Part II

    1. Each of these 1 0 groups is 6 pendants united by a top cord.
    2. The top cords on Groups 2 to 10 emanate from the right of the pendants they unite. The top cord in Group 1 emanates from the left.
    3. In each group, the pendants and top cord are the same color. Groups 2 and 4 share one color; Groups 8, 9, 10 share one color; and each of Groups 1, 3, 5, 6, 7 is a different color.
    4. The values on Groups 2 to 10 are quite similar. On all, position 1 has value 1; position 2 is zero-valued; position 3 has value 0, 1, or 2; position 4 has value 9; position 5 is zero-valued; and position 6 has value 0, 1, 2, or 3. The third position is the only one with subsidiaries. In each group it has 1 or 2 subsidiaries, one or none of color CB (values unknown as always broken ), and one of YB or MB (values 0, 1, or 2). Group 1 sums Groups 2-10. The summation is exact in 5 of the 6 pendant positions and off by 2 in position 4. It is also exact on one of the subsidiary positions and, due to breakage, unknown on the other. So:

      \[ P_{21j} = \sum\limits_{i=2}^{10} P_{2ij} \]
      j = 1, 2, 3, 5, 6, 3s2 j = 4 off by 2 j = 3s1 probably but unknown due to breakage.

    5. Although arranged differently, the values in Part II Group 1 are the same as those in Part I Group 3. (See observation b with Part I above).
    6. Between Groups 9 and 10, there are 2 smaller khipus tied to the main cord. We hypothesize that this was preplanned as there is space on the main cord between these 2 groups but not between other adjacent groups.

    Part III

    1. The first group of pendants have values similar to those in Part II. There, Groups 2-10 had values 1,0,0 - 2, 9,0,0 -3; here the values are 1, 0, 1, 9, 0, 0. However, this group has no subsidiaries or top cord.
    2. All of the pendants in Group 7 are the same color, each has the same color subsidiary, and each has the value 9 & 1. Since the superimposed cord shares these characterisics, we interpret it as a member of the group. We interpret the cord appended to its attachment loop as a top cord. Assuming the double units positions are distinct, the value of the top cord is the sum of the values of the pendants in the group: 6 X (9 & 1) = 543.
    3. A similar summation appears in Group 3: 3 X (9 & 1) = 273.
  9. The values 9 and 1 play an important role on this khipu and the 2 smaller associated khipus. There are repetitions of 9 & 1, 1 & 9, [1, 0, 9], [1, 9] and 9 consecutive groups with values [1, 0, 9]. The sums of these 9 groups are repeated in two different places. (Curiously, the one value that is an incorrrect sum sums 9 9's). Also, Parts I & II consist of 9 groups each and Part II has 9 groups and 1 sum group.