WHITECROW BORDERLAND

Mayan Astronomy

Note 3: Multiples of 13. 2/26/99

As difficult as it might be for a people grounded in logocentric discourse, oriented toward the primacy of the Word in communicating ideas, to shift their ideological focus away from verbal constructs and reorient themselves toward numerical concepts instead, that act must be accomplished if there is to be any hope of comprehending Maya astronomy. Number as such is far more significant in native American philosophy than mere words ever have been or ever will be. The idea that number in itself can be used to communicate, like words do, any concept at all, apart from "how many," is a notion that has never taken hold in European culture. No one in Europe can be convinced either that a savage, even prehuman, society was sophisticated enough to develop a philosophical orientation, much less that such people could have conceived of anything like a concept of number, no matter how remedial or rudimentary it might have been.

I take exception to that characterization because the most simplistic act of counting becomes necessary and indispensable as soon as any group of human animals exceeds a total of one. If you are a hunter supplying food for an indeterminable number of individuals, indeterminable simply because you cannot count, whose very existence depends on your success, how do you know when you have enough birds, rabbits, fish to feed the ones who are waiting for your return? The idea that savage humans were somehow insensitive to their own survival is belied by the fact that any of them did.

How long ago did numerical systems become complex enough to record the number of days in the moon's synodic period? Some people believe that an ability of that kind might have appeared in human culture as long ago as 20,000 years in the past. For the sake of this discussion, fortunately, the exactness in a temporal frame of the emergence of numerical utility is wholly irrelevant. As recently as 5,000 years ago, ample, if not convincing, evidence exists from many different cultures that people were counting and recording intervals of lunar motion. Since the moon is the most obvious object in the night sky, it attracted human numerical scrutiny earlier than some of the less noticeable objects did. It is also true that Jewish culture had already gone on record, as it were, if the current dating of Biblical texts is indeed accurate, in declaring its commitment to the primacy of the Word over that of the number. The God of Genesis created the world by speaking the divine Word and not by counting the number of days in the moon's synodic period, so to speak. This observation has significance because the people of Mesoamerica, at about this same time (3000 B. C.), left no written records of their philosophical orientation, and may not have possessed language yet in a written form, but did manage in several ways to indicate that they had begun the process of reducing various aspects of celestial motion to numerical expression and form.

The best evidence we have of the initiation of writing in Mesoamerica indicates that numerical expression came well before verbal constructs emerged.

Even if it cannot be factually demonstrated that a culture like the one created by the Olmecs employed counted intervals of celestial motion in their perceptions of reality as early as 3000 B. C., the sophistication of their knowledge of such things several thousand years later (1000 B. C.) does tend to suggest that considerable time and effort was spent prior to the latter date in the development of various kinds of calendrical devises derived from a numerical perception of celestial motion. A mosaic pendant, for instance, was discovered in the state of Puebla, Mexico, which shows a high degree of astronomical sophistication relative to lunar motion. The devise can be used to count intervals of 12 and 13 lunar synodic months, 354 and 384 days respectively. There are several other numerical components in the structure of the Las Bocas mosaic which have astronomical significance, the inclusion of a 260-day interval for instance, which suggest that the Olmecs, as early as 1500 B. C., were already using that interval as a calendrical sequence. Again, even if that calendrical sequence did not appear in actual use until several hundred, or even a thousand, years after the mosaic was constructed, it is obvious that the Olmecs were beginning to evaluate its potential utility at least as early as 3,500 years ago. The Mayas, of course, were certainly using the 260-day almanac by the beginning of the first millennium.

In Maya numerical philosophy, and I certainly mean to distinguish here between Eurocentric, Logos-oriented philosophy and the kind the Mayas practiced, the central, controlling feature of the 260-day calendar is the number 13. The question, then, is: what does "13" mean? In Eurocentric discourse there is no way to ask that question because Europeans do not recognize the possibility that a single number has any meaning whatsoever. Numbers don't mean anything; all they do is tell you how many. At the very best, any European would tell you that "13" is an unlucky number and the person who tells you that will also, and in the same breath, try to convince you that the Mayas, because they were and remain animistic, were primarily driven by superstition in the creation of whatever philosophy they might have embraced. What then does the number "13" signify in Maya philosophical thinking and why did they develop a calendar that pulls day together in 20 groups or clusters of 13-days each? I made a table of Maya astronomical values based on multiples of 13 which is meant to illustrate precisely what such a calendrical system can be used to count relative to observable periods of celestial motion:

Multiples of 13 in Maya Astronomy

My best guess here is that the significance of these values probably does not leap off the page into immediate apprehension for most people looking at them. Making it clear why any of this is significant is not a simple task. Take one example to start: the lunar value of 325 days, which is very close to 11 synodic months, with the actual decimal value set at 324.83649 days, a number by form which the Mayas did not know or employ, can be used in the almanac's structure to track lunar synodic positions (new moon, full moon, etc.) over indefinite periods of time with a high degree of accuracy. This can be done by day-name positions in the almanac's 260-day sequence. If a new moon were to occur on 1 Imix, for instance, a second new moon would fall on a day with the same numerical coefficient (1) 325 days later. Since the difference between 260 and 325 is 65 days, or one-quarter the length of the sequence itself (4 X 65 = 260), the day-name of the second new moon would be 1 Cimi. After 11 more synodic periods of the moon, the day-name for the new moon would be 1 Chuen, then it would be 1 Cib. The lunar phase would then return to its point of departure at 1 Imix. Any Mayan astronomer during the Classic Period would be able to calculate the sequence of new moon day-names in his/her head and probably without even thinking about it first. Every synodic position of the moon, throughout its entire sequential cycle of 29 days, can be regulated calendrically in this way.

The objection will be raised here by any competent European mathematician or astronomer that the discrepancy between the Maya's whole number value as a multiple of 13 will rapidly accumulate error because it is 0.2 days longer than the actual interval for 11 synodic months. That fact will quickly render this technique of prediction meaningless. Sure, that is absolutely true. Anyone looking at the sky will see the accumulation of discrepancy between the calendrical interval and the actual celestial motion of the object itself. The question is: how difficult would it be to compensate for the accumulation of error and what steps would be necessary to fix the problem? When the new moon finishes its circuit through the requisite day-names of the calendar, making its quarter advance beyond 260 with each accumulation of 11 synodic periods of the moon, a total of exactly 1,624 days will have elapsed by the time it is scheduled once again to fall on 1 Imix. In calendrical time, however, the interval is 1,625 days. How do you fix that horrible problem? You regress the day-name by 1 and count the new moon, not on 1 Imix, but on 13 Ahau, which is the day before the original point of departure. After that in the normal extension of the calendar, you hit 13 Chicchan, then 13 Oc, then 13 Men. The next regression then shifts the day-name from 13 Ahau to 12 Etznab, and so on. Any reasonably intelligent child, even one of European descent, can master this technique by the age of nine or ten. Put another way: predicting lunar synodic positions at intervals of 11 synodic months in the Maya 260-day almanac is child's play.

An interesting symbolic aspect of the lunar value here is that 325 is the sum of the two values the Mayas used in their eclipse prediction table in the Dresden Codex; that is, 177 + 148 = 325.

The values in this table that directly reflect Venus's synodic motion are very likely the underlying cause for the development of a calendrical system based on revolving multiples of 13. Venus's synodic period shows a series of 5 different variable values for its motion over time. The period from one synodic position to the next can be as brief as 580 days and as long as 588 days. This sequence of variables is repetitious and occurs over 5 consecutive synodic periods of the planet. The whole number average interval is close to 584 days. The Mayas, at least in so far as the 260-day almanac was concerned, assigned a period of 585 days to the interval of Venus's motion because they wanted to be able to regulate significant positions of the planet on specific day-names in the calendrical structure. The accumulation of error in their almanac formula could be managed by periodically changing the sequence of day-names through a process of regression like they did with the day-names of the lunar phases.

The way it worked was simple. The first day of the Venus period was, or could have been, the planet's last day of visibility before inferior conjunction. The Mayas simply chose a day that fell 6.5 days before IC and extended the count by 13 days to reach a position that approximated the planet's first day of visibility after inferior conjunction. In other words, the last and first days of visibility at inferior conjunction were designated by day-names sharing the same coefficient and were in effect consecutive in the calendar by groups of 13 days. 260 days later, the same day-name that marked the first day of visibility is recovered and the position specified in Venus's motion by that day-name is equivalent to its last day of visibility before superior conjunction. 52 days after that, the planet's first day of visibility after superior conjunction is reached and that day-name has the same coefficient as its predecessor and the name has advanced four places of 13 days each across the almanac's day-name list. That same day-name then marks the planet's last day of visibility before inferior conjunction 260 days later. In all exactly two and one-quarter almanacs have been counted in this sequence (13 + 260 + 52 + 260 = 585). A one day adjustment can then be used to compensate for the accumulation of error and the process of matching Venus positions to predictable day-names continues. One might add here the fact that at superior conjunction the last day and first day of visibility share the same coefficient with the day of conjunction itself because 26 is also a multiple of 13. The likely probability is that the Mayas began a count like this on the last day of visibility before superior conjunction creating an addition sequence for the revolution of 26 + 26 + 260 + 13 + 260 = 585. In this system, two day-names are identical and they all share the same coefficient. This addition sequence, of course, also recovers 325, and the lunar phase sequence in the almanac, because 52 + 260 + 13 = 325.

The other planetary periods in this list, with their equivalent multiples of 13, show clearly that Mercury (116/117), Mars (780/780), Jupiter (399/403), and Saturn (378/377) could have been tracked with calendrical day-names through the Maya system in much the same way that Venus and the moon were counted. Jupiter, which may appear to be the most troublesome of the group because of the 4-day discrepancy between the whole number average and the calendrical value used to express its period, however, was less a problem than either Mercury or Mars because its sequence of variables was less radical than theirs. Jupiter's period varies from 396 to 403 over time, with the average at 399, and that fact makes its variable sequence more, not less, predictable. Mercury's period varies from 106 to 129 days over time and, while its sequence of variable periods is repetitious and predictable, it did provide more challenges to calendrical regulation than the more stable planets did. Mars has a variable period that extends to as many as 46 days of variation and may not have been particularly susceptible to regulation at all. Saturn was stable and easily managed through the day-name system outlined above.

While there is much more that can be said here about the Maya perception of planetary motion, especially with regard to calendrical methodologies, the point I wanted to make has been adequately demonstrated. The Mayas perceived the visible planets as spirits. In that same context, they believed that time itself was a bridge between the real world and the spirit world. The meaning behind that concept ought to be obvious from this discussion. Time, as it is expressed by and through the calendrical system predicated on multiples of 13 in the 260-day almanac, is inseparable from the regular motion of the planetary spirits it was designed to regulate and count. Time is the same as the motion of celestial objects in Maya culture. Time is the same as the spirits that were used to create its calendrical expression. Hence, when the Mayas wrote the number 13 in their script at any point in their long complex history, they were making a profound philosophical statement about the nature of the connection that exists between the real world and the spirit world that animates it.

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To reach [Note 1]; [Note 2]; [Note 4]; [Note 5]; [Note 6]; [Note 7]; [Note 8]; [Note 9]; [Note 9a] [Note 10]; [Note 11]; [Note 12]; [Note 13]; [Note 14]; [Note 15]; [Note 16]; [Note 17]; [Note 18] in this series of thoughts.