AS055/KH0066 - Catalog Information

Ascher Databook Notes:

1. The Museum records the provenance as Hacienda Ulluj alla y Callengo. All of the khipus UR1051-KH0067/html/KH0067_sketch.html">KH0067 have the same attribution.
2. All subsidiaries are colored CB.
3. All subsidiary values are less than 400; all pendant values are greater than (1?)620.
4. Assuming that the missing digits on the broken cords (P2 and P3s1) are zeroes, the following relationships exist between the values:
   • P₅ = P₁s₁ + P₁s₂ + P₂
   • P₅ + P₃ + P₃s₁ + P₆ = P₇
   • 9*(P₄) = P₁ + P₁s₂ + P₅ + P₆
   
   These can be combined to relate all values on the khipu, but this involves the use of one value (P1s2) twice:
   • P₃ + P₃s₁ + P₇ = {P₁s₁ + P₁s₂ + P₂ + P₆} = P₅ + P₆
   • 9*P₄ = {P₁s₁ + P₁s₂ + P₂ + P₆} + P₁ + P₁s₂ = P₅ + P₆ + P₁ + P₁s₂

Excerpt from Ascher's Code of the Khipu (pages 149-151 Section 7.10)

Khipu example 7.7 is AS055 and KH0067/html/KH0067_sketch.html">KH0067. 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:

P11 ((1?)620) KH0067/html/KH0067_sketch.html">KH0067	P12 (2353) KH0067/html/KH0067_sketch.html">KH0067	P13 (934) KH0067/html/KH0067_sketch.html">KH0067
P21 (1118) KH0067/html/KH0067_sketch.html">KH0067	P22 (2121) KH0067/html/KH0067_sketch.html">KH0067	P23 (756) KH0067/html/KH0067_sketch.html">KH0067
P31 (1421) AS053	P32 (2427) AS053	P33 (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:

And, the product of the values in the third row is also the same as the product of the diagonal values:

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:

The table of values can , therefore , be rewritten in terms of the fractions and only the original values in the first row:

P11	P12	P13
(11⁄14)*(7⁄8)*P12	(7⁄8)*(34⁄33)*P12	(34⁄33)*(11⁄14)*P13
(7⁄8)*P11	(34⁄33)*P12	(11⁄14)P13

Closer scrutiny of the values in the first row shows that they, too, are related to each other by these fractions. Specifically: (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)*(P₁₁ + P₁₂ + P₁₃). The pendant has two subsidiaries (s1, s2) and there is also a subsidiary (s3) on the pendant with value P₁₂. 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=P₁₂, and y=P₁₃· The table then has the form:

CB3x	x	y
C2B6x	CBx	C2B2x
CB4x	Cx	CB2y

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 C³B⁶x²y. 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.