Ascher Databook Notes:
- The Museum records the provenance as Hacienda Ulluj alla y Callengo. All of the khipus AS051-AS056 have the same attribution.
- All subsidiaries are colored CB.
- All subsidiary values are less than 400; all pendant values are greater than (1?)620.
- Assuming that the missing digits on the broken cords (P2 and P3s1) are zeroes, the following relationships exist between the values:
- P_{5} = P_{1}s_{1} + P_{1}s_{2} + P_{2}
- P_{5} + P_{3} + P_{3}s_{1} + P_{6} = P_{7}
- 9*(P_{4}) = P_{1} + P_{1}s_{2} + P_{5} + P_{6}
These can be combined to relate all values on the khipu, but this involves the use of one value (P1s2) twice:
- P_{3} + P_{3}s_{1} + P_{7} = {P_{1}s_{1} + P_{1}s_{2} + P_{2} + P_{6}} = P_{5} + P_{6}
- 9*P_{4} = {P_{1}s_{1} + P_{1}s_{2} + P_{2} + P_{6}} + P_{1} + P_{1}s_{2} = P_{5} + P_{6} + P_{1} + P_{1}s_{2}
Excerpt from Ascher's Code of the Khipu (pages 149-151 Section 7.10)
Khipu example 7.7 is AS055 and
AS056. We have modified one digit of one number on AS55 by one. As noted before, we do not consider an error
of one in one knot cluster to be very significant, since errors in knotting or our errors in counting knots or in transcription are all possible.
The last khipu example is, arithmetically, the most interesting. In all, it. contains only thirteen values, but their interrelationships are intricate and can be expressed in many different ways. Physically, it is two small khipus that were found together. One khipu has seven pendants and three subsidiaries and the other has just three pendants. Ignoring, for the moment, the first pendant and the subsidiaries, the khipus form a chart of values P
_{ij} (i=1,2,3; j=1,2,3) where values for i=1,2 are on the larger khipu and those for i=3 are on the smaller one. In tabular form, the chart is:
P_{11} ((1?)620) AS056 |
P_{12} (2353) AS056 |
P_{13} (934) AS056 |
P_{21} (1118) AS056 |
P_{22} (2121) AS056 |
P_{23} (756) AS056 |
P_{31} (1421) AS053 |
P_{32} (2427) AS053 |
P_{33} (734) AS053 |
The values in the chart range from 734 to 2,427.
One attractive relationship is that the product of the values in the third row is the geometric mean of the products of the first and second rows. That is:
\[ { P_{11}P_{12}P_{13} \over P_{31}P_{32}P_{33} } = { P_{31}P_{32}P_{33} \over P_{21}P_{22}P_{23} } \]
And, the product of the values in the third row is also the same as the product of the diagonal values:
\[ P_{31}P_{32}P_{33} = P_{11}P_{22}P_{33} \]
When comparing the individual values to each other, three simple fractions keep reappearing. They are
^{11}⁄
_{14},
^{7}⁄
_{8} and
^{34}⁄
_{33}. The values in the third row, when multipled by these fractions, result in the values in the second row. And, the values in the first row, when multiplied by them in a different order (
^{7}⁄
_{8},
^{34}⁄
_{33},
^{11}⁄
_{14}), result in the third row. This cyclical relationship can be summarized by:
\[ { P_{2j} \over P_{3j} } = { P_{3,j+2} \over P_{1,j+2} }\;\;\;for\;j=(1,2,3)\;where\;addition\;is\;mod\;3 \]
The table of values can , therefore , be rewritten in terms of the fractions and only the original values in the first row:
P_{11} |
P_{12} |
P_{13} |
(^{11}⁄_{14})*(^{7}⁄_{8})*P_{12} |
(^{7}⁄_{8})*(^{34}⁄_{33})*P_{12} |
(^{34}⁄_{33})*(^{11}⁄_{14})*P_{13} |
(^{7}⁄_{8})*P_{11} |
(^{34}⁄_{33})*P_{12} |
(^{11}⁄_{14})P_{13} |
Closer scrutiny of the values in the first row shows that they, too, are related to each other by these fractions. Specifically:
\[ P_{11} = ({11 \over 14}) ({7 \over 8}) P_{12} \]
(All of the foregoing statements are to within 0.4 percent . What is more, ten out of the twelve of them deviate by at most only 0.2 percent.)
So far, we have ignored the first pendant value. Having observed, as a construction feature, that sums are frequently on individual pendants set off at the end of the main cord, we examine it to see if it is the sum of values in the chart. We find, still within 0.2 percent, that it is the sum of the values in the first row divided by one of the omnipresent fractions. Its value is: (
^{33}⁄
_{34})*(P
_{11} + P
_{12} + P
_{13}). The pendant has two subsidiaries (s1, s2) and there is also a subsidiary (s3) on the pendant with value P
_{12}. The value of s1 is related to the value of s3 through the fractions. To within 0.5 percent,
s1 = (
^{7}⁄
_{8}))
^{2}*(s3) or to within 0.9 percent,
s1 = (
^{11}⁄
_{14})*(
^{34}⁄
_{33})*s3
The fact that this last relationship can be expressed in these two different ways raises the unusual idea that the three dissimilar looking fractions could themselves be related. Actually,
^{7}⁄
_{8} is a very good approximation to
sqrt((
^{11}⁄
_{14})*(
^{34}⁄
_{33}))
The approximation is accurate to 0.2 percent. Because p11 can be expressed in terms of p12 and using the approximation (
^{7}⁄
_{8}))
^{2} = (
^{11}⁄
_{14})*(
^{34}⁄
_{33})
the nine table values can be reduced to dependence on only two of the original values and two of the fractions. To simplify their presentation, let B =
^{7}⁄
_{8}, C =
^{34}⁄
_{33}, x=P
_{12}, and y=P
_{13}· The table then has the form:
CB^{3}x |
x |
y |
C^{2}B^{6}x |
CBx |
C^{2}B^{2}x |
CB^{4}x |
Cx |
CB^{2}y |
All the relationships already described can be seen from this formulation. For example, both the product of the values on the diagonal and the product of the values in the third row are C
^{3}B
^{6}x
^{2}y. And, of course, additional ones can now be found. For example, the products of the off diagonal values p12 , p23 , p31 and of p13, p21 p32 are the same as the product of the diagonal values.
The values on this khipu pair must have resulted from intentional calculations. The interrelationship of the values depends on fractions and on logic that is more complex than is used to yield values that are consistent fractional parts of their whole. But, if the two small khipus were not together, it is unlikely that any of the relationships would be seen.