Ernest P. Moyer
Professor David Stuart
Department of Art & Art History
The University of Texas at Austin
1 University Station D1400
Austin, TX 78712–0340
The Mayan Calendar Design
In an effort to summarize the unique nature of the Maya calendar I address the following letter to you. Because of your reputation among Maya scholars, and in the interests of scholarly integrity, I felt you would be an appropriate person.
On a Maya Decipherment web blog you originated this past October 11, 2009 at
Stanley Guenter remarked that many people interested in the 2012 end-of-time theory proposed that
. . . the Maya calendar was designed specifically to end on December 21st, 2012. There are many problems with such an idea. For one thing, it would make the Maya calendar unique in the history of humanity. Many different calendars exist or have existed in different cultures . . . These calendars all were designed to count time from a specific starting point, with no designated end date. There is no reason to believe the Maya calendar was any different.
Because Guenter appears confused about calendars, as do multitudes of other persons inside and outside the Maya community, I brought together here-to-fore unrecognized features about the Maya calendar to help clarify this situation.
I shall address the following questions:
Was the Maya calendar different from all other calendars?
Was the Maya calendar unique in the history of humanity?
Was the Maya calendar designed specifically to end on December 21, 2012?
I shall show that the answers are in the affirmative for all three questions.
Comparison with Evolutionary Calendars
In order to place Maya calendrics into a proper context I performed a survey of calendars from China, India, ancient Egypt, the Hebrews, and Rome, (now Europe, Australia, New Zealand, South Africa, and the Americas). They show a commonality in form that was not replicated by the Maya. They all were originally lunar calendars, later adapted to the solar year. (The Hebrew calendar still closely follows lunar cycles, while incorporating adaptation to the solar environment. Muslims cling to more ancient lunar cycles.)
We know that man has been interested in lunar cycles from long in the past. Alexander Marshack, and now many others, have reported scratches and markings on the walls of caves and on bone artifacts that strongly suggest people were observing cycles of the moon going back more than 25,000 years. Although many of the ancient findings have dubious interpretations the evidence is haunting. We all well know the influence of the moon upon us, whether it is shown in the behavior of a pair of lovers, or in a farmer scheduling the planting of his crops. The professional staffs of mental institutions are wary of the phases of the moon.
As man became socially organized he developed an interest in time-keeping and calendars. As the millennia passed and people became more sophisticated they adopted their lunar calendars to solar representation. We see the path of this adaptation today in our custom of designating months, a word derived from the same linguistic root as moon. At one time a month represented a moon cycle. 12 lunar synodic cycles of 354 days in a year is off by only a few days from the solar year. The evidence is similar in all the old civilizations. For example, four weeks of seven days each in a month is very close to a lunar period. Or, for the Chinese, it may be three weeks of ten days. The calendar systems of the world all had evolutionary origins that began with lunar cycles.
The Maya Avoid a Moon Calendar
I shall now provide descriptions that will be very familiar to you, but I do so for the sake of a more complete presentation.
The Maya calendar is different. They developed a regular calendar that avoided the effects of the moon on their thinking and on their psyche. Although they recorded information about the moon on their monuments, and in their codices, and were as concerned as other people in the cycles of the moon, they did not use that information to design their main calendar. Many Maya Long Count inscriptions are supplemented by a Lunar Series, which provide information on the lunar phase and position of the Moon, but this practice was an adjunct to, and not an integral part of, their regular calendar. This lack of lunar calendrics was contrary to all other civilizations. The design was absolutely unique in the world.
The Tzolkin is one part of their regular calendar. It is divided into 13 tones, with 20 glyphs. It has a most peculiar arrangement that takes it through its cycle of 260 days. It does not take 13 tones through each of 20 glyphs in turn but rather takes 13 tones part way through one rotation of glyphs until it reaches the end of its 13 tones and then returns to tone #1 and continues through with glyph #14, and so on. This complicated arrangement has no parallel in any other calendar around the world. This scheme shows undeniable evidence of the hand of an intelligent designer. Otherwise it would be natural to rotate 13 tones through one glyph, then turn to the next glyph for 13 tones, and so on, but the Tzolkin is purposely designed to avoid such a simple scheme.
Furthermore, why would the Maya chose a period of time that is significantly short of a solar year? It is customary to call this part of their calendar sacred, following Maya regard for such a strange calendar. We might borrow this sacred designation to slough off a more rigorous examination, but that act does not relieve us of an objective scientific study of its function.
The Haab is the second part of their regular calendar. It is straight forward in the division of its parts, 18 of 20 days each, with five days tacked on at the end, 360 + 5. (The five extra days is known in other civilizations, i.e. the Egyptian.) Again, the number 18 cannot be related to lunar cycles in any reasonable manner. That unusual division of the year is not found in any other calendar of the civilized world. Since both eastern and western civilizations use 12 months of (approximately) 30 days, and with regard for the compelling force of the moon upon our psyche, why did the Maya also not express a monthly lunar system? Clearly, this calendrics method again exhibits an artificial design that did not have origin in a lunar system.
Again the number 20 in the Haab suggests a vigesimal mathematical origin.
The analytical structure of both the Tzolkin and the Haab suggests a design that appears to purposefully separate an emotive attachment to an instinctive lunar environment. They both provide a more objective monitor of time. Of course the Haab is intimately related to the solar cycle, and preserves a natural relationship to what may be regarded as the "god of the heavens." But one cannot feel or appeal to the movement of the moon when one uses these calendars.
We can understand why the Maya calendar could not have been evolutionary. We, in the modern civilized world, know only calendars that were handed down to us from our evolutionary forefathers. Scientifically, we lived with them, and all the impediments they placed upon us. But somewhere along the line some great mind discarded that way of thinking and used a method that would not place similar impediments upon the Maya mind. In fact, the design of the Mayan calendar stimulated the Maya mind to broader ways of thinking, an element we do not find in our evolutionary calendars. With complexity not found in other world calendar systems the Maya calendar truly is unique in the history of humanity.
The Mathematical Design of the Maya Calendar
But even more intricate relationships are found in the Maya calendar system.
The Tzolkin and the Haab rotate together, in what is called a Calendar Round. They are locked onto one another by an initial starting point in the Long Count: 4 Ahau 8 Cumku. 4 Ahau is the description of the Tzolkin place in its rotation; 8 Cumku is the description or the Haab position in its rotation when they start off together. This locking together of three distinct calendar entities is another unique feature of the Maya calendar, not duplicated by any other calendar in the world. Furthermore, because of the different length in their cycles, 260 vs 365 days, the Tzolkin and Haab do not come back to a common starting point until (73 sacred) 52 secular years are complete. (We are minded of the Jewish 50-year jubilee cycle.) Meanwhile the Long Count continues to increment.
I shall now give formal statement to the relationship between the Haab and the Tzolkin.
When they start off together, and when they reach the common end of their cycles, they are equal to one another. This may be stated as:
Length of Haab Cycle = Length of Tzolkin Cycle
Expanding this relationship:
(Number of Haab years) X (Length of Haab year in days)
(Number of Tzolkin years) X (Length of Tzolkin year in days)
We may state this numerically as:
(52) X (365) = (73) X (260)
Decomposing these numbers into their prime elements, we obtain:
(13 X 4) X (5 X 73) = (73) X (13 X 20)
Immediately I am struck by the fact that the same prime elements appear on both sides of the equation. If I rearrange the elements I get:
13 X (4 X 5) X 73 = 13 X 20 X 73
If we express the sum of the arithmetic factors of the Haab and the Tzolkin number of days in 52 (73 sacred) years, we obtain:
Haab: [13 X 4] (52) X [5 X 73] (365) = 18980.
Tzolkin: [13 X 20] (260) X 73 = 18980.
The Haab and the Tzolkin number of days in 52 (73 sacred) years have the same identical arithmetic factors: 13 X 20 X 73. These factors are merely different in the way they are assembled to produce the different lengths of the two calendars.
Two of those are prime numbers, 13 and 73. The 20 is strongly suggestive of the Maya vigesimal arithmetic system.
Importantly, for this study, the Haab year has 5 X 73 (365) days.
More importantly, when the Tzolkin and Haab complete their Calendar Round rotations to start a new cycle they are short of a full solar (tropical) year by 13 days. (The length of the current solar year is 365.24219878 days. 0.24219878 X 52 = 12.59433656 days. More discussion below.)
At this point I not only felt that the Maya calendars were artificially and intelligently designed, I also felt that I knew how they were designed.
The number of cycles for a complete Tzolkin to match a complete Haab was taken from the prime number part of the Haab: 73.
This was a mathematical necessity. Since 73 was a prime number it could not be decomposed into smaller numerical elements. If the Haab and the Tzolkin were to be equated mathematically they both had to contain the number 73.
The Haab obtained its numeric value of 73 from the number of days; the Tzolkin obtained its numeric value of 73 from the number of cycles. But if I gave mathematical expression to the equation it did not matter from which element I obtained those numbers.
This meant that I could not design a Tzolkin that would equate to a Haab to find a mutual meeting point unless I used the number 73.
And this determined the number of Tzolkin cycles (years).
This equality between the Haab and the Tzolkin should not be confused with another property in their expression. The vigesimal base of Maya arithmetic was used in both the Haab and the Tzolkin to determine the number of time periods (not months) in each: 20 x 18 and 20 X 13. (This is where it was necessary to tack on five days to make 365. 20 X 18 + 5 = 365. But 365 = 5 X 73. This is where the transformation took place to equate the Tzolkin and the Haab.)
We can easily recognize how an intelligent designer would use his knowledge of mathematics to achieve this result.
In his design we can see that the number of rotations of the Tzolkin was fixed to meet his mathematical equality: 73. He also could not disturb the number of days in a Haab year (365). (But he could decompose the number 365 from the Haab to obtain the factors of 5 X 73.) Since the number of Haab days was fixed, and the number of cycles in the Tzolkin was now fixed, 73, the only remaining part in this relationship open to mathematical choice was the length of the Tzolkin cycle.
This may be expressed mathematically by (Haab = Tzolkin):
P X 4 X 5 X 73 = Q X 20 X 73.
where P is an element of the number of cycles in the Haab, and Q is an element in the number of days in the Tzolkin. In other words P is equal to Q. Thus if we modify Q in the Tzolkin we will always force P in the Haab to the same value. (Note that the P value is always multiplied by 4 to meet the requirements of equality, while maintaining the Haab fixed 365-day year.)
Now consider two possibilities for analytical purposes other than 13 tones: 11 and 17 for the Tzolkin cycle. I suggest these two numbers because both are prime. How would they modify the number of Haab years to obtain completion of the Calendar Round?
For the 11 possibility Haab would become [11 X 4] (44) X [5 X 73] (365) = 16060.
Tzolkin would become [11 X 20] (220) X 73 = 16060.
The Haab would complete the Calendar Round in 44 years instead of 52. The Tzolkin would have only 11 tones in each of the 20 glyphs instead of 13. But the Tzolkin would still have 73 years, now reduced in length from 13 to 11 tones. The number of 20 glyphs in the Tzolkin would remain the same, by mathematical necessity.
For the 17 possibility Haab would become [17 X 4] (68) X [5 X 73] (365) = 24820.
Tzolkin would become [17 X 20] (340) X 73 = 24820.
The Haab would complete the Calendar Round in 68 years instead of 52. The Tzolkin would have 17 tones in each of the 20 glyphs instead of 13.
Evidently, the designer could have chosen some number other than 13 tones in a Tzolkin cycle. What was he trying to accomplish?
I list here the data for the length of the solar years vs. the length of the Haab years. This shows how much the Haab calendar drifts from the solar year over a period of a Calendar Round for the three Tzolkin different tones I suggested above. We immediately recognize that the Tzolkin calendar can be used to monitor the true position of the sun for each Calendar Round, regardless of the number of tones that we use in the Tzolkin. One additional cycle of the Tzolkin tones, after completion of a Calendar Round, will mark the solar position of the sun, with some error.
|Number of Tzolkin tones||Number of Haab years||Length of Haab years in days||Length of Solar years in days||Difference between Haab years and Solar years in days||Number of Calendar Rounds in 1000 years||Error in Solar position in days in 1000 years|
For 11 tones the difference would be 0.65674632 days in 52 years. For 13 it would be 0.59433656 days; for 17 it would be 0.46951704 days. As the number of tones increases the error decreases. A choice of 13 tones seems completely arbitrary. The designer might have chosen 17. However he could not exceed the vigesimal number of 20 as a constraint he placed in the design. (11 seems a bit short.)
The choice of 13 has other ramifications I discuss in Part II.
Given that 13 was his choice, note that the sun continues to drift 13 days more for each of the rotations of the 52 Calendar Round: 13 days at the end of the first rotation, 13 more days at the end of the second rotation, and so on. The true solar position can easily be monitored by adding another rotation of the Tzolkin tones for each additional 52 years. The error through the centuries is shown in the Table above.
In 52 years the drift of the Haab from the true solar position is 0.24219878 X 52 = 12.59433656 (13) days.
The Maya calendar shows the solar drift by the Tzolkin design.
Note that the error is similar to our uncompensated error in the Gregorian calendar over that many centuries. (With our leap year calendar we compensate by adding a day every four years. However this compensation is slightly too much. We decompensate on centurial years by making them ordinary years. Unless they are divisible by 400, in which case they are leap years. This causes an additional correction on years 1700, 1800, 1900, 2100, 2200, and 2300.)
I now had a handle on how the Tzolkin was designed. It was not an arbitrary statement of numbers, obtained by some delirious rambling of the Maya mind, but a statement obtained by mathematical necessity. One purpose was to help the Maya determine the true solar position as the Calendar Rounds rolled by.
The conditioning of the Maya mind to scientific thinking is illustrated by the records they left us. This acute awareness is shown between their calendar and the cycles of the heavenly bodies, as I have illustrated above, and has been noted by researchers. For example, the following remark comes from James Q. Jacobs at
The use of the haab of 365 days results in solar drift, the dates of the year moving in relation to the seasonal stations. In 1906 Charles P. Bowditch first noticed evidence of the Mesoamericans equating 1508 haabs (29 calendar rounds) with 1507 tropical years. The evidence was in the form of calendar dates with this separation. The difference between the solar year and the haab seems to be recorded in a Temple of the Cross inscriptions at Palenque, as first decoded by Teeple and discussed by Thompson (1971). Two dates express a 1508 haab interval, equal to 1507 solar years (365 x 1508 = 1507 x 365.242203) a recognition of intercalation more accurate than the present day Gregorian calendar. . .
The calculated value of 1508 Haab years is 550420 days. The current value of the tropical year of 365.24218967 X 1507 = 550419.97983269 days. The difference between these two values is 550420 - 550419.97983269 = 0.02016731 days or less than 0.5 hours in 1500 years. This difference is truly remarkable, and far better than the Gregorian calendar. (The value for the tropical year in 500 AD may have been slightly different. When dealing with such long time periods and such refinement in measure to the ninth decimal place we must consider how the earth changes in its orbital parameters during that period. You can see what this must mean for the Maya scientific abilities.)
We can see how the Tzolkin error in monitoring the true solar position over 1500 years is about 17 days, but this is a "running" error used as a method of keeping track of the sun while the difference between 1508 Haab years and 1507 solar years is a "static" calculation.
Christopher Powell wrote a now famous dissertation on the relationship between the Maya calendar and the cycle patterns of the planets in the solar system, again exhibiting the prowess of the Maya scientists.
No matter how the Maya may have arrived at their awareness of the heavenly cycles we see that their knowledge was highly sophisticated, showing that they had liberated themselves from an emotive over control by the moon.
The Maya not only were aware of the drift of the Haab with respect to the solar year, but had performed calculations to give it a specific quantity that equals or was better than our modern calendar accuracy. That the Maya could refer to time spans of 1500 years shows that they had considerable intellectual history behind them, a history that most modern scholars fail to credit. How could they know about scientific correlations over 1500 years unless they had an opportunity to monitor and keep record of them?
To be continued
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