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
1WAddition(+)106
2ABSubtraction(-)15
3AB:MBResult(=)91
7WAddition(+)161
8WSubtraction(-)15
9AB:MBResult(=)140 *Broken
11WAddition(+)206
12WSubtraction(-)15
13AB:MBResult(=)191
14WAddition(+)238
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
47ABAddition (+)187
48AB0
49ABSubtraction (-)15
50AB:MBResult (=)172
51ABAddition (+)141
52ABSubtraction (-)126
53ABResult (=)15
54ABAddition (+)127
55ABSubtraction (-)112
56ABResult (=)15
57ABAddition (+)110
58ABSubtraction (-)95
59ABResult (=)15
60ABAddition (+)148
61ABSubtraction (-)133
62ABResult (=)15
63AB0
64ABAddition (+)201
65ABSubtraction (-)186
66ABResult (=)15
67ABAddition (+)70
68ABSubtraction (-)55
69ABResult (=)15
70ABAddition (+)92
71AB0
72ABSubtraction (-)15
73MB::ABResult (=)61
73 Subsidiary 1MBSubtraction (-)16


The use of two different color pairs to signify the same arithmetic operations (Addition (+) Subtraction (-) and their result) suggests, though, that Inka khipukamayuqs did not only utilize marked/unmarked color pairs to designate cord-level arithmetic operations. Rather, they also likely used marked/unmarked color pairs to make khipu-level distinctions, like signifying the type of calculation each khipu in a matching pair recorded: net credit or net debit calculations. Khipu UR267A, for instance, seems to have recorded operations that result in after-tax, “net credit” values. Khipu UR255, on the other hand, recorded the arithmetic operations in the opposite direction, resulting in the taxed values, or “net debit” values. As I stated above, Khipu UR267A employed W as its unmarked color to signify addition and AB to signify subtraction. UR255, in contrast, used AB as its unmarked color to signify addition and MB to signify subtraction. Note that these color pairings (W and AB as well as AB and MB) stand in a marked/unmarked relationship to one another. W is unmarked in relation to AB (as seen in UR267A) and AB is also unmarked in relation to MB (as seen in UR255). Thus, the choice of an unmarked color in a color pairing would have made a difference in how that pairing related to color pairings on other khipus through markedness relations. For instance, the use of white as the unmarked color sign in UR267A reflects the khipu’s unmarked, additive characteristics in contrast to use of amber brown as the unmarked color sign in UR255, which reflects the khipu’s marked, subtractive characteristics overall.

In this way, the overall color scheme of a khipu could act as a marked or unmarked sign, predicating the khipu as a whole with information about the type of calculations that were done on the khipu. Therefore, in brief, at a khipu-level, the use of white as the unmarked color sign in UR267A signified that the khipu recorded “net credit” calculations, whereas the use of amber brown as the unmarked color sign in UR255 signified that the khipu recorded “net debit” calculations. The combination of multiple conventionalized color binaries on a wrapped stick would have made it possible for a khipukamayuq to signify nested levels of meaning (i.e. both at the khipu-level and the individual cord-level) by hierarchically relating color pairings to one another in the way I have explicated above. Theoretically, then, these nested, color-based meanings could have been correctly interpreted by anyone familiar with the code on the relevant wrapped stick.

In summary, I have demonstrated how marked/unmarked color pairings were used at Inkawasi to signify non-numerical values (i.e. arithmetic and accounting operations). The light color of each color pairing was associated with an unmarked operation (addition) and dark color with a marked operation (subtraction), as I expected based on my aggregate-level findings earlier in the chapter. Also, as theorized, color combination cords were conceived as the synthesis, or offspring, of the two solid colors in a color pairing—in this case, the synthesis of addition and subtraction. Note, however, that to date, I have yet to identify other storehouse accounting khipus that employ this same system of color corresponding to these same arithmetic actions. It is possible that the colors on a khipu are particular to the archive at hand (and/or the wrapped stick that the colors were coded by), rather than to the genre, or some other global indicator. Perhaps the only common convention between different khipu production contexts was the use of the same marked/unmarked color pairings to signify information. The scale of conventionalized sign production very well may have been limited to the use of a particular wrapped stick, while the logic of the color sign markedness was more universal. Or perhaps, khipus UR267A and UR255 at Inkawasi belong to a unique type of accounting khipus that has not been previously found in the KDB. Whatever the case might be, it seems likely that the logic of these color signs would have carried over to other binary color pairings. Moreover, other color pairings would have likely been capable of signifying additional conceptual relationships, beyond arithmetic operations alone, using the same semiotic principles identified here. Urton notes that Spanish transcriptions of Inka tribute lists recorded on khipus recounted combinations of activities on cords: for instance, using the verbs sacar and llevar, "The gold that they took they delivered to Cuzco" (1998). He suggests on this basis that khipu cords could have recorded two different forms of action/verbs on the same cord (1998:425). As we have seen with the actions "to add" and "to subtract,” color can be used to accomplish this purpose, with the light color in a family of related colors used to signify addition and the dark color used to signify subtraction. When the two colors were mottled together on a cord, this signified the combination of "to add and to subtract,” i.e. the result of the arithmetic operations. Other color combinations could have formed additional verb combinations, such as the examples Urton provides. In summary, we can see that the colors worked as dicent symbols at Inkawasi, acting as predicates for the cords, and designating the kind of arithmetic action to be done for each number on the khipu in participle form (W: “___ was added,” AB: “___ was subtracted,” AB:MB: “___ was the result,” or “___ was added and subtracted”), rather than leaving it to memory or implication. Furthermore, in a fashion consistent with the dicent symbol sign type, the subject (the number recorded on the cord) was physically modified by the cord color, thereby linking it to the predicate.