AS209
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OKR Name: KH0229 Original Author: Marcia & Robert Ascher Museum: Royal Ontario Museum, Toronto Museum Number: HP519 Provenance: Unknown Region: Unknown 
Total Number of Cords: 143 Number of Ascher Cord Colors: 13 Benford Match: 0.945 Similar Khipu: Previous Next 
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Khipu Notes
Ascher Databook Notes:
 Construction Note: The end of the main cord has been cut or cut and wrapped. This finishing may be intentional and the khipu complete. At 19.5 cm the main cord has been repaired or joined to another piece of the same cord.

 Two cord fragments were found associated with the first group of pendants. They are both DB. One is broken at both ends with a cluster of 2s at 1.0 cm from one end and an overall length of 3.5 cm. The other has an overall length of 38.5 cm. Beginning at its broken end, there is a cluster of 2s at 2.5 cm from the end then a space of 6.0 cm until a 1E and then 30.0 cm until a finished end. These are probably parts of 2s1 or 3s1 or 7s4.
 This pendant is broken at 12.5 cm. However, this is the only place the fragment could have come from and so is assumed to be part of this pendant.
 A cord fragment was found associated with the group of pendants 2735. The fragment is color KB. It is broken at both ends with a knot cluster of 4L at 1.5 cm from one end. Its overall length is 17.5 cm. It is probably a part of 31s1, 32s1, 33s1 or 35s1.
 Three cord fragments were found associated with the last group of pendants. All are color KB. One is just a cluster of 3s. Another is a cluster of 9L and then 2.0 cm. Both of these are broken at both ends. The last is 5.0 cm with one broken end and one finished end. These are probably parts of 54s1 or 55s1.
 By space, color and magnitude of numbers, the khipu is separated into two parts. Part I is the first 17 pendants forming two groups by spacing and color; group 1 is 8 pendants with color DB and group 2 is 9 pendants of color W. The pendant values range from 123 to 555 (and one value in the 1200's). Part II contains four groups by spacing and color. The group colors are alternately LB and W. Part II begins with a blank cord (P18) associated with group 3 but much longer than the pendants that follow it. We consider it to be serving as a marker reinforcing the separation from Part I. Nine pendants in group 4 are color W but a tenth (P36), slightly apart from the others, is of color LB. With the exception of three pendants (P44, P45, P46) whose values are 170 and below, the pendant values in Part II are 374 to 2593 (and two values above 4000). P44, P45, and P46 will be seen to be anomalous for other reasons as well. Overall, the structure is:
Part I: Group 1 An 8 pendant group of color DB Group 2 A 9 pendant group of color W Part II: Marker Group 3 An 8 pendant group of color LB Group 4 A 9 pendant group of color W, then an LB pendant. Group 5 An 8 pendant group of color LB plus three anomalous pendants between the 7th and 8th pendants in the group. Group 6 A 9 pendant group of color W.
These can be referred to as P_{ij} where i =(1,...,6).
For i=(2,4,6), j=(1,...,9); for i=(1,3,5), j=(1,...,8), (and for i=(5), also j=(1',5',6',7')).

 Each P_{ij} has one or more subsidiary for i=(1,3,5) and j=(1,...,8).
 Each P_{ij} has at least one DB subsidiary.
 The first subsidiary on each P_{3j} is color DB.
 Each P_{5j} j=(1,...,8) has at least one LB subsidiary. When there are two or three subsidiaries, the color order is consistently some or all of 0G, LB, LB or GY.
 For j=(1,...,8): value on DB subsidiary on P_{3j} = value on first LB subsidiary on P_{5j} (Note that P_{5j}, j=(5',6',7') are omitted from these patterns)
 Each P_{2j} has at least one DB subsidiary. (Note: Combining this with comment 6.a.i above, each pendant in Part I has at least one DB subsidiary.) Where there are two or three subsidiaries, the color order is consistently some or all of W, DB, DB or KB. Each P_{ij} for i=(4,6) and j=(1,2,3,4) has no subsidiaries.
 Each P_{ij} has one or more subsidiary for i=(1,3,5) and j=(1,...,8).

 P_{ij} < P_{3j} < P_{5j} for j=(1,...,8)
 P_{2j} < P_{4j} < P_{6j} for j=(1,...,9)
 Within Part II:
 The pendant values plus their subsidiary values have the same rank order in groups 3 and 5 except for a reversal between pendants 3 and 4. That is:
For i=(3,5) with subsidiaries:
P_{i8} > P_{i6} > P_{i7} > P_{i4} > P_{i5} > P_{i2} > P_{i1}
P_{i8} > P_{i6} > P_{i7} > P_{i3} > P_{i5} > P_{i2} > P_{i1}
Also, with or without subsidiaries, for i=(3,5)
P_{i8} > P_{i6} > P_{i7} > P_{i4} > P_{i5}
P_{i8} > P_{i6} > P_{i7} > P_{i3} > P_{i2} > P_{i1}
 The pendant values have the same rank order in groups 4 and 6 except for a reversal between pendants 1 and 5. That is:
For i=(4,6)
P_{i9} > P_{i3} > P_{i4} > P_{i7} > P_{i5} > P_{i6} > P_{i8} > P_{i2}
P_{i9} > P_{i3} > P_{i4} > P_{i7} > P_{i1} > P_{i6} > P_{i8} > P_{i2}
 For all j=(1,...,8) withsubsidiaries P_{5j}/P_{3j} is between 1.46 and 1.70. The ratios for j=(2,5,6,7) with subsidiaries are very close to each other and very close to 3/2: That is:
P_{5j}/P_{3j} = 1.5 ± 0.7% for j=(2,5,6,7) (with subsidiaries)
Also, the ratios for j=(2,5) differ from each other by only 0.02%.  For j=(1,...,8) P_{6j}/P_{4j} is between 1.36 and 1.73.
(For the exception j=9, P_{69}/P_{49} = 2.900. ) The ratios for j=(3,4) are very close to each other, namely both are 1.4731 ± 0.1 %.
 The pendant values plus their subsidiary values have the same rank order in groups 3 and 5 except for a reversal between pendants 3 and 4. That is:
 Within group 5, the ratios Ps6/Ps7 and Ps4/Ps1 are the same to within 0.03%. This is of interest because they are remarkably close to √2 . In fact, Ps4/Ps1 = 379/268 = √2  0.0024%. There is no reason to hypothesize that the ratios of these pendant values or √2 were of any importance to the khipumaker. For us, however, it suggests an excellent approximation for √/2.

 The number of pendant values that are multiples of P_{56'} and P_{57'} suggest their values as some kind of units:
P14 = 4 P56'
P48 = 12 P56' = 3 P14
P52 = 18 P56'
Where P56' = 45, so these are 4,12,18 times 45
P55' = 5 P57'
P35 = 11 P57'
P42 = 13 P57' = 2 P25
P46 = 4 P55' = 20 P57'
Where P57' = 34, so these are 5,11,13,20 times 34. These are also 2,10,22,26,40 times 16 and, in addition P25 = 13*17, P24=19*17
 The number 17 is prominent in that it is a factor of about 12% of the pendant values. (It and two multiples of it, 34 and 85, also appear on subsidiaries but it is not prominent among subsidiary values.)
 The number of pendant values that are multiples of P_{56'} and P_{57'} suggest their values as some kind of units:
 Sum cords:
 The last pendant in each of groups 1, 3, 5 is related to the sum of the first four pendants in its group.
G1: \[ P_{18}\; \& \;its\;subsidiaries≈\sum\limits_{j=1}^{4} P_{1j}\; \& \;their\;subsidiaries \](736 ≈ 726 + 2 broken subsidiaries) G2: \[ P_{38}\; \& \;its\;subsidiaries≈\sum\limits_{j=1}^{4} P_{3j}\;\&\;their\;subsidiaries \](2508 ≈ 2511) G3: \[ P_{58}≈ \sum\limits_{j=1}^{4} P_{5j} + \sum\limits_{j=5'}^{7'} P_{5j} \](4074 ≈ 4076)  The last pendant in group 6 is approximately equal to the sum of the last pendants in groups 3 and 5. By comment a) above, they in turn were also sums.
P69 = P38 + P58 + 2.
 We have labeled the extra LB pendant just after group 4 as P51' because by color and value it appears related to group 5. That is:
P51' = P52 + P55.
 The last pendant in each of groups 1, 3, 5 is related to the sum of the first four pendants in its group.
 Although there seems to be little consistency with position, many pendant values are the sums of two or more other pendant values. Some of the details are included here but, so far, an overall explanation or generalization is lacking.
 There are altogether thirteen values that are sums of pairs of other values. They are P13, P27, P31, P32, P35, P47, P57 (in two ways), P51', P61, P64, P65, and P67 (in two ways).
 Four values in groups 1 and 2 can be expressed as the sum of three other values. Then, with only five exceptions, each of the values in groups 36 can be expressed as the sum of three other values. Moreover, they have several such expressions. In group 2, one value can be expressed as the sum of three values in four different ways. In groups 36, two can be so expressed in two different ways, three in three different ways, one in four different ways, eight in five different ways, three in six different ways, four in seven different ways, and one each in eight, ten, and eleven different ways. Since the summands can in turn be sums, these can be combined to form longer sum chains. As one example we use P61 which has one expression as a sum of two values and five as a sum of three values. Because these sums, the sums can be extended to include six expressions as sums of four values, four expressions as sums of five values, and one expression each as sums of six and seven values.
P_{61} = P_{55}' + P_{47} = P_{55}' + (P_{15} + P_{33}) = P_{55}' + P_{15} + (P_{11} + P_{27} + P_{23}) = P_{55}' + P_{15} + P_{11} + P_{27} + (P_{57}' + P_{56}' + P_{28})
= P_{55}' + (P_{22} + P_{14} + P_{31}) = P_{55}' + P_{22} + P_{14} + (P_{21} + P_{16})
= P_{55}' + (P_{12} + P_{13} + P_{31}) = P_{55}' + P_{12} + P_{13} + (P_{21} + P_{16})
= P_{55}' + (P_{56}' + P_{34} + P_{16}) = (P_{13}) + P_{34} + P_{6}
= P_{55}' + (P_{26}' + P_{42} + P_{23}) = (P_{55}') + P_{26} + P_{42} + (P_{57}' + P_{56}' + P_{28})
= P_{48} + (P_{57}' + P_{17}) = P_{48} + (P_{25} + P_{27}) = P_{48} + P_{25} + (P_{11} + P_{12}) = P_{48} + P_{25} + (P_{56}' + P_{12} + P_{21})
= P_{18} + P_{21} + P_{35} = P_{18} + P_{21} + (P_{16} + P_{28})
 All values in groups 1 and 2 that are sums can be expressed such that the summands are only from groups 1, 2, and 5' (those three anomalous pendants in group 5).
P_{11} + P_{22} = P_{27}
P_{55}' + P_{56}' = P_{13}
P_{56}' + P_{57}' + P_{28} = P_{23}
P_{12} + P_{21} + P_{5,4+j}' = P_{j,5+j} forj=(1,2)
P_{29} = P_{17} + P_{18} + P_{26} = P_{55}' + P_{12} + P_{15} + P_{21} + P_{26} + P_{28}
P_{29} = P_{56}' + P_{18} + P_{21} + P_{23} + P_{25} = P_{56}' + P_{57}' + P_{11} + P_{13} + P_{23} + P_{25} + P_{27}
P_{29} = P_{56}' + P_{57}' + P_{12} + P_{18} + P_{21} + P_{22} + P_{26}
 The values that cannot be expressed as sums of two or three other values are
P_{ij} for i=(1,2,3); j=(2,5,8)
P_{ij} for i=(1,2); j=(1,4)
and P_{17}, P_{26}, P_{69} ·
 In several cases, alternate values in groups 1 or 2 plus a third value from some other group add to a value in group 6. These cases are:
P_{21} + P_{23} + P_{54} = P_{67}
P_{23} + P_{25} + P_{18} = P_{66}
P_{25} + P_{27} + P_{48} = P_{61}
P_{27} + P_{29} + P_{12} = P_{63}
P_{14} + P_{16} + P_{42} = P_{66}
P_{15} + P_{17} + P_{55} = P_{63}
P_{15} + P_{17} + P_{35} = P_{66}
Similar sums add to values in groups 3 5. These are:
P_{13} + P_{15} + P_{35} = P_{41}
P_{13} + P_{15} + P_{55} = P_{54}
P_{15} + P_{17} + P_{52} = P_{43}
P_{15} + P_{17} + P_{43} = P_{36}
P_{16} + P_{18} + P_{51} = P_{36}
 The majority of double and triple sums involve at least one value from groups 1,2, or 5'. This is probably because the values in these groups are generally smaller in magnitude. The distribution of pendant values is:
0500 5011000 10011500 15012500 ≥2501 G1,G2,G5' 18 1 1   G3,G4 3 9 2 3 1 G5,G6  5 5 4 3
The sums that are most unlikely to be fortuitous are those involving no group 1,2 or 5' values as summands. They are among the values larger in magnitude (all ≥ 1001) and all are in Part II. They are:
P_{33} = P_{35} + P_{53} + P_{66} = P_{31} + P_{47} + P_{61}
P_{49} = P_{41} + P_{46} + P_{52}
P_{51}' = P_{52} + P_{55}
P_{56} = P_{37} + P_{48} + P_{53} = P_{46} + P_{61} + P_{68}
P_{57} = P_{53} + P_{62} = P_{37} + P_{68}
P_{58} = P_{32} + P_{51}' + P_{64}
P_{64} = P_{48} + P_{61}
P_{65} = P_{35} + P_{51}
P_{67} = P_{51} + P_{62}
Primary Cord Notes:
Construction note: The end of the main cord has been cut or cut and wrapped. This finishing may be intentional and the quipu complete. At 19.5 cm the main cord has been repaired or joined to another piece of the same cord.