Khipu Notes

Notes from Jon Clindaniel's 2019 PhD Thesis at Harvard:

### 4.6 Proof of Concept: Signifying "Credit," "Debit," and "Result" with colors

In 2015, Urton and Chu demonstrated the use of “fixed values” in the khipus found in Sector A at Inkawasi—that is, numerical values that were repeated over and over in different accounting records. They argue that these values may have represented something like taxes that were placed on goods coming into the storage facility, as the numbers seem to have been involved in many subtractive arithmetic operations. For instance, in the first three cords of UR267A, the numbers 106, 15, and 91 occur in sequence. This sequence of numbers can be algebraically balanced by the mathematical operation of subtraction: “106-15=91.” Similar subtractive operations occur again and again throughout the khipu and Urton and Chu suggest that these fixed values (i.e., taxes) might have indicated a quantity of goods that was to be set aside for the support of the storage facility and its personnel (2015:522). In reviewing these findings, however, I discovered that these sequences of arithmetic operations were not merely implied in the khipus by the numerical values themselves. To the contrary, the sequences of arithmetic operations were explicitly denoted through the use of cord color, by means of the principles I have already laid out in this chapter (see Table 4.2). Specifically, in cords 1-3, you can see that the color white (W) was used to designate addition (credit), the color amber brown (AB) was used to designate subtraction (debit), and the mottled color combination of amber brown and medium brown (AB:MB) was used to designate the result of the arithmetic operations. For instance, the first cord on UR267 has the value 106 recorded on it and is colored white. The second cord has the value 15 recorded on it and is colored amber brown. The third cord of the sequence is colored AB:MB (i.e. mottled) and has the resulting value 91 recorded on it. Therefore, the operations as designated by the colors would be: “+106-15=91.”

#### Table 4.2: Cord Color and Numerical Data from Inkawasi Khipu UR267A

Cord NumberCord ColorColor MeaningNumber on Cord
2ABSubtraction(-)15
3AB:MBResult(=)91
8WSubtraction(-)15
9AB:MBResult(=)140 *Broken
12WSubtraction(-)15
13AB:MBResult(=)191
15WSubtraction(-)15
16WResult(=)223

Note, however, in Table 4.2 that after establishing the sequence of arithmetic operations for the khipu with the sequence of W, AB, and AB:MB, there is an inclusion effect similar to the one described in Chapter 3. The second entry maintains the meaning of Subtraction (-) but the marked color AB (used for subtraction previously) has been replaced by the more inclusive, unmarked color W. For cords 7-13, the AB:MB “result” cord color remains unchanged, however, to establish that this number is still the result of the previous two numbers. Finally, for cords 14-16, even the AB:MB cord is included within the unmarked W color and the operations continue on in the same sequence, under the alias of the unmarked color W. It seems that the khipukamayuq used the colors initially to establish the sequence of arithmetic operations, but no longer felt the need to designate these colors after that sequence had been established. Given the tendency for light colors to be associated with unmarked categories and dark colors to be associated with marked categories in a wide variety of Andean semiotic contexts, this choice of colors makes a lot of sense: summation is, by definition, superior in an additive, constructive sense and thus unmarked in relation to the arithmetic operation of subtraction. Note that there is some precedent for the use of marked and unmarked khipu signs to designate arithmetic operations like addition and subtraction. For some present-day khipukamayuqs, S-ply is conceived as “giving” (i.e. corresponding to Subtraction (-) the marked operation), and Z-ply is seen as “receiving” (i.e. corresponding to Addition (+) the unmarked operation; see Arnold 2014:40-41). As I mentioned earlier in the chapter, color combination signs seem to have been used to signify intermediary categories between marked (signified by a dark color) and unmarked (signified by a light color) categories. Here, we have a mottled cord, which I argued is the highest ranked of the various cord color combination options available to a khipukamayuq (and, thus, probably the first to be chosen to represent an intermediary category). In this case, the mottled color combination sign seems to fittingly refer to the synthesis of addition and Subtraction (-) or the arithmetic result of the two operations.

The use of cord colors gets even more complex when we look to khipu UR255, however, which Urton and Chu argue is a “matching” khipu to UR267A (2015:522). As Urton and Chu demonstrate, the numbers in UR255 seem to be organized so as to perform the opposite sequence of arithmetic operations from those performed on UR267A. Urton and Chu propose that this would have been a way for khipukamayuqs to cross-check their calculations in UR267A. So, for example, in cords 67-69 of UR255, we see the sequence: 70, 55, 15. Here, the khipukamayuqs seem to have switched the order of arithmetic operations so that the fixed value/tax is the result: “70-55=15.” Looking at Table 4.3, the first thing you should notice is that the vast majority of the cords on khipu UR255 are colored AB—the marked color on UR267A that signified subtraction. Unlike in khipu UR267A, the color AB does not seem to have been used to signify subtraction in the UR255 khipu, however. Instead, it seems to have been juxtaposed with the color Medium Brown (MB). Since AB is the lighter color of the pairing (i.e. the unmarked color), it played the role of addition on this khipu and MB played the role of subtraction. Finally, as in khipu UR267A, AB:MB designated the result. As in khipu UR267A, these colors only seem to have been used, however, to establish the sequence of arithmetic operations where it was otherwise unclear. Note that the strings of repeating arithmetic operations listed in Table 4.3 are all signified using the color of the highest ranked color on the khipu: AB. However, when the khipukamayuqs needed to reestablish the order of arithmetic operation or clarify the meaning of a particular sequence of operations, they used the MB and AB:MB color sign vehicles. For instance, note that the order of arithmetic operations is different for cords 47-50 than for 51-69 and there is a blank, unused cord that could potentially confuse the arithmetic: 187- 15=172. To make these operations clear, the khipukamayuq designated the result “172” using the AB:MB color that we now know signifies resulting values on these khipus. Similarly, in cords 71-73, the operations were a bit out of sequence. Here, the khipukamayuq again used an AB:MB cord to signify the resulting value of additions and subtractions and added an additional MB cord onto the “result” cord to signify an additional subtraction (the marked color in comparison to unmarked AB), leading to the correct resulting value.

Such patterns begin to make some sense of the wrapped stick in Figure 4.2 (see the close-up of the thread wrappings after cleaning at the bottom of the figure). While there are many color combinations on the stick, notice that there are sequences of the colors W, AB, and MB—the color oppositions khipukamayuqs referenced in order to produce the color signs for the arithmetic operations in khipus UR267A and UR255. It seems plausible that each stick acted as a sort of unified “topic” with lists of marked/unmarked color pairings that could have been used within a particular genre to produce meaning. For instance, in UR267A, W signified the unmarked action of “addition” and the paired color AB signified the marked action of “subtraction.” In UR255, AB signified the un- marked action of “addition” and the paired MB signified the marked action of “subtraction.” For both the khipus, the color combination AB:MB signified the “result” of addition and subtraction.

#### Table 4.3: Cord Color and Numerical Data from Inkawasi Khipu UR255

Cord NumberCord ColorColor MeaningNumber on Cord
48AB0
49ABSubtraction (-)15
50AB:MBResult (=)172
52ABSubtraction (-)126
53ABResult (=)15
55ABSubtraction (-)112
56ABResult (=)15
58ABSubtraction (-)95
59ABResult (=)15
61ABSubtraction (-)133
62ABResult (=)15
63AB0
65ABSubtraction (-)186
66ABResult (=)15
68ABSubtraction (-)55
69ABResult (=)15