Khipu Fieldmarks - Conclusions

From a used book store, I ended up owning a version of “A Commentary on the Dresden Codex”, signed by the great man himself SIR Eric Thompson, KBE (Knight of the British Empire). It’s a gift to his son:

    To Don (his son), whom I carefully steered away from the Maya area so that he wouldn’t join the chorus of
    â€śOld Thompson was all wet when…,”
                His father
                Eric Thompson.

Thompson did a great deal to advance Mayan studies. He also did a great deal to retard it. Convinced, almost till his dying day, that Mayan writing was logographic (it’s syllabic), that palace images were gods (they were kings), etc. he was, as one Mayan scholar not so gently said, “Right about sooo much, … except for the important things.” Thompson’s signature indicates he knew all too well of his future place in history.

In the spirit of science, when researching khipu, I have tried to only use data to come to conclusions. I have let the data guide theory, and then used theory to guide data experiments to prove (or deny) those theories. Some of my conclusions, such as the evidence that khipus are unlikely to be a form of written language are disappointing to me, and controversial to others. I sincerely hope that history will NOT face me with an (admittedly small) chorus of a similar sound that eventually faced Thompson.


It’s been an interesting journey to analyze these three types of fieldmarks: statistical, relational, and analytical.

First, let’s review a synopsis of the three types of fieldmarks: statistical, relational, and analytical. Having done that, I’ll attempt to synthesize this into a narrative description of what the data says about khipu grammar/architecture.

1 Relational Conclusions

Ascher Summation Relationships

  • An astonishing number - (482/650)~74% of khipus in the Khipu Fieldguide have 1 or more Ascher Summation Relationships. This is not a good indicator for the idea that khipus are “linguistic” in nature.
  • Summations come in two-types, right-handed and left-handed, with right-handed sums having their sum cords on the left of the khipu with their summand cords to the right of the sum cord, generally within 20 to 50 cords of the sum cord. Similarly left-handed sums, generally have their sum cords on the right side of the khipu.
  • Right-handed sums and left-handed sums generally split along 60/40 lines, for all types of pendant sum relations.
  • More than one type of sum relationship can exist in a khipu. In fact, khipus often consist of multiple layers of summation relationships.
  • “X-ray” pictures of the network of summation relationships quickly reveal the overall structure of a khipu, what cords are important, where “the action” is, etc. As an example, look at UR006, the two-year calendrical khipu first analyzed by Gary Urton. The X-ray drawings clarify and confirm the two year organization of the khipu.
  • Other than actually KNOWING, using arithmetic, that a particular cord is a sum cord, how does one identify it? As shown in the pendant pendant sum study, there are a significant number of “clues”.
  • Seriated vs Banded Sums. Summation patterns on banded khipus are different from seriated khipus. Banded khipus frequently have a set of zero valued cord clusters with no knots, interspersed between sum clusters.
  • Sum cords tend to be grouped together on one “sum” cluster. This makes intuitive sense. In these “sum” clusters, the first cord has a roughly 50% chance of having a White cord color.
  • Evidence of “Sums of Sums” is ample. The maximum compound summing depth is 9, but the usual sum depth is in the 2-5 range.
  • Sum Top Cords strongly associate (>95%) with banded clusters, and 70% of the time with 6 cord clusters. They are also quite rare, occuring in less 4.5% of the khipus.

2 Statistical Conclusions

I previously noted how fieldmarks can be used as identification traits. With that in mind, let’s look at these noteworthy statistical fieldmarks.

  • Large Cord Value Khipus
    Large (by cord value) khipus are clearly in a field of their own, as is evident in viewing them in the Khipu Fieldmark Browser, when viewing them as a connected graph, or as when examining their low ascher summation counts. Their low summation counts have been calculated and documented. Simply put, large-valued khipus, don’t have sums. When you are measuring something in the waranqa scale of 1000’s, those are unusual numbers. Large cord-value khipus appear to be more of an “executive summary” and less of a working man’s khipu.

  • Top Cords and Banded Khipus
    There is ample evidence that top cords, are infrequent, infrequent in summation relationships, and are associated with banded khipus only.

  • Recto vs Verso Cord Attachments
    A look at the graph of Recto vs Verso cord attachments says it all. The relationship between these two is lovely to look at.

  • S vs Z Knots
    A match made in heaven, a look at the graph of S vz Z Knot percentage also says it all. The relationship between these two is also lovely to look at. S and Z knots are arranged in a spatial fashion in “astronomical” khipus, and we can use this fieldmark to identify likely “astronomical” khipus.

  • Colors
    Of the 651 colors, the top 200 colors cumulate to 98% of the cords. As expected, the majority of the cords are solid colors - with close to 30% being white!. Mottled colors follow, with barberpoles eventually showing up. Barberpoles may be visually arresting, but the frequency count tells us they seldom (< 0.32%) show up in the khipu record.

    • Cord color frequency distribution rapidly declines, with a very very long tail. In fact this distribution appears to be a power law, something that often appears in linguistic contexts.

      At first glance, this appeared quite exciting. A classic example of a linguistic power law distribution is Zipf’s Law whose first use by linguist Herman Zipf was to note that a word’s length in characters was inversely proportional to it’s use in language. In light of Sabine Hyland’s Collata khipu discovery that 17th-18th Century Collata khipus/khipu boards’ colors implied a phonetic letter scheme, this power law distribution makes sense and adds to her argument.

      Unfortunately, however, when we look at the actual Zipfian distribution, we see that while it is a power curve, it is not a “natural language” Zipfian distribution. Furthermore, cord colors lack the conventional amounts of hapax legomena the only-once-occuring “words” that comprise 40-60% of the words in natural language.

    • White is the most prevalent color, followed by browns of various shades, and mottled colors made from browns and white. While white cords comprise a third of all pendant cords, and of cord clusters of white, we don’t yet know what their color significance is. One theory, that they are grammatical markers for addition, appears disproved. However, they do show interesting levels of positional significance, appearing at the end or beginning of cord clusters that contain many sums (sum clusters), and in certain spikes in larger clusters (i.e. in odd positions more than even ones).

3 Analytical Conclusions

Benford’s Law and Zipf’s Law
There are many statistical “leading indicators” that imply that khipus are most likely not linguistic. Two “power laws” are Benford’s Law, which relates to the rank and frequency of values in human counting, and Zipf’s Law, which relates to the rank and frequency of words in human language. Both can be used to analyze khipu’s relationships to human expression.

  • Application of Benford’s Law, indicates that the majority of found khipus have numbers that closely fit with distributions of numbers used in human “accounts” such as taxes, inventories, census data, etc.
  • The significant presence of Ascher Sum relationships such as pendant sums of other pendants, or cluster sums of other clusters, in low Benford match khipus, implies, fairly strongly, that khipus are not a medium of literature or writing (in the western sense of the word), but rather an accounting structure of some sort.
  • Other possible Benford Law approaches can be tried. Suppose you make the argument that khipus are literary. Then you would expect a lot of khipus with low Benford matches, and with a low fanout ratio (a large number of subsidiaries compared to the number of pendant cords). A difficulty arises, however. The graph of this relationship does not confirm this hypothesis. The only significant khipu to appear in this search is UR093. However, an examination of the Ascher sum relationships for this khipu shows significant arithmetic relationships! Getting through the language gate with khipus is a difficult exercise!
  • Application of Zipf’s Law, adds considerable statistical weight to the argument that khipus do not have a “natural language vocabulary” whether color, cord value, or overall structure. These three possible linguistic mediums all fail to follow a classic Zipfian natural language distribution.
  • While Cord Values fail to have the right coefficients for a natural language distribution, they do obey the Zipfian natural language distribution of hapax legomena - the 40-60% of vocabulary that is only used once in a “text”. The Benford Law analysis also hints at this phenomenon. However, cord colors fail miserably on this measure. Color, whether, alone or in cluster sequence, lacks the correct proportion of uniquity to ubiquity for a natural language distribution of hapax legomena.

4. A Narrative Description of Khipu

Let’s summarize all that’s been discovered so far.

4.1 Are Khipus Linguistic?

At the mile-high level, we know that the majority(> 75%) of khipus exhibit some sort of numerical summation pattern. What of the other 25%? Some are summary khipus, without any need for summation patterns, some are simply counts, and some are so broken or incomplete that they can’t form meaningful patterns. This diminishes the possibility of language even more!

If khipus are indeed linguistic, we would expect their cords to take on one of two possible linguistic structures:

  • Linear/Serial Structure: One possibility is that language is encoded serially with each cord expressing a morpheme or word. In this respect, the Zipf’s law study indicates that of the set of possibilities, cord values (and knots?) are most likely to represent words of some sort. Color is not likely to be linguistic, other than as a category label of some sort.
  • Tree Structure: The other possibility is that language is expressed in a tree fashion, using pendants and subsidiaries, like the sentence diagrams you used to do in grade school. However, the Benford’s law Fan-out study suggests that encoding in a tree fashion is highly unlikely.

This leaves us with a very slim possibility that serial cord values express language. “Still…we persist”.

4.2 Khipu Typologies

So let’s assume that khipus are not linguistic in nature, and instead are of an accounting nature. Then we know that khipus break down roughly into three classes, based on mean-cord value.

  • Banded Khipus - The most common kind of banded khipu are census khipus, with the band color representing an ayllu.
  • Seriated Khipus - Based on Jon Clindaniel’s thesis work at Harvard, we know that Seriated khipus have a higher mean cord value than banded khipus, suggesting they are a “state-wide” accounting, rather than a village-based accounting.
  • Summary Khipus - At the highest level of mean cord value are the “executive” summary khipus having few relationships of any sort.

4.3 Fieldmarks/Patterns

Exploration of the data shows that most of the fieldmarks and relationships studied apply to both banded and seriated khipus, including:

  • Most of the ascher sum relationships, with the exception of cluster cluster sum bands (Clusters whose sum of all its cords, including subsidiaries, match the sum of another cluster in the khipu.)
  • The use of mixing Recto/Verso cords, or mixing S/Z Knots to create a binary distinction, such as Hanan/Hurin, as indicated in the literature.

Additionally, for ascher sum relationships where a cord’s value is the sum of a set of cords, we know that there is a “handedness” to these sums, where most of the sums lie on the left or the right of the khipu, and where the ratio of left to right sums is approximately 60/40. This may indicate a dual accounting system of the sort mentioned in the literature such as a debit/credit system of accounting.

Banded Khipus show some special characteristics.

  • Top cords are chiefly only on Banded Khipus - remember that a top cord measures the sum of an adjacent cluster
  • Cluster cluster sum bands (Clusters whose sum of all its cords, including subsidiaries, match the sum of another cluster in the khipu.)
  • Additionally, banded khipus show a special spatial arrangement of sum clusters

5. Sum Total: Takeaways and Next Steps

By building the Khipu Field Guide, taking special care to make a well-formed database that was as clean as possible, a decent base has been provided for the statistical analysis of khipus. The analyses, from many various points of view, have helped to “focus the mind” so to speak, since they disprove some theories, while revealing many new sources for further understanding:

  • A variety of summation patterns, spatially as well as by color, occur. When displayed graphically, using the X-Ray diagrams that have been built, much of a khipu’s strucure is visually revealed.
  • From a variety of viewpoints, it has been demonstrated that from 66% to 80% of all khipus are sums of some sort, and that most khipus are not likely to be a form of natural language.
  • Color has shown to be interesting as a spatial marker as well as a possible signifier for summation clusters. It’s use as a grammatical marker is questionable. It’s application as a medium for natural language expression appears highly unlikely.

Additionally the base has been established for:

  • A creation of metrics for khipu well-formedness.
  • A typology of khipus established by their underlying fieldmarks. The graph at the top of the welcome page points the way.
  • The exploration and identification of khipu “eigenvectors”, the underlying patterns that create khipus. An example of this was provided in the Ascher color investigation.