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. THE CORRELATION QUESTION . .. |
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The long counts recorded on monuments from the Classical
period of
Maya civilization mark off the time since creation of the world.
Most Classical long counts fall in baktun 9, as reckoned by the
scribes. The baktun is a period of 144,000 days,
about
400 years. Thus Classical civilization was at its height some 3600
years
after the putative date of creation.
The Classical period is now believed to date roughly from 200 to 900 AD. This time frame is based in part on radio-carbon and other standard archaeological dating methods. These techniques are not accurate enough to precisely date the events recorded in the inscriptions, but since the monuments bear long counts, the details of Maya history can be fixed in time by correlating the long count with the European calendar. |
The
most widely accepted correlation is a variation on the oldest effort to
match the long count to the European calendar. In 1897, Joseph Goodman
(an American journalist who was Mark Twain's first editor), proposed
that
the Maya creation date, the zero long count, was in 3114 BC. Goodman's
correlation was supported by the work of a Yucatecan scholar, Juan
Martinez,
but other correlations were more popular until J. Eric Thompson
revived
interest in Goodman's correlation in 1927. His work was supported by
the
astronomical discoveries of J.E. Teeple in 1930. Thompson reviewed the
evidence again in an influential study of the question in 1937. He was
able to narrow down the range of possible base dates to three days. The
correlation he proposed is now usually referred to as the
Goodman-Martinez-Thompson
(GMT) correlation. The base dates he identified are correlation
constants
used to convert long counts to European calendar dates:
| 11 August 3114 BC (Gregorian) 6 September 3114 BC
(Julian) 12 August 3114 BC (Gregorian) 7 September 3114 BC (Julian) 13 August 3114 BC (Gregorian) 8 September 3114 BC (Julian) |
The choice between these three dates is still hotly debated, but almost all Mayanists accept one of the versions of the GMT correlation.
Nothing, as they say, is certain except death and taxes. But the GMT correlation seems nearly as certain as any deduction from the available evidence can be. Its wide acceptance survived even the drastic revision of Maya scholarship when Thompson's intellectual hold on the field was broken by a new generation of scholars. The alternatives have few supporters among Mayanists. Yet when I searched the web for information on the correlation question, I failed to turn up any account of the arguments supporting the GMT correlation. I did, however, find defenses of at least eight alternative correlations. In the result, I fear it is all too easy for new students of the Maya to get the impression that the GMT correlation is dubious, or worse, an example of academic myopia. Some internet savants even hint darkly at conspiracy.
Of course, no one should accept anything merely because it is
favoured by tenured professors. It is for just this reason that I have
put this web document together. It sets out the reasons why the GMT
correlation
has been widely accepted as definitive. The arguments are
unavoidably
technical and complicated in places, but I have tried to make my
account as readable as possible.
The correlation problem
In principle, all that is required to correlate the long count with the European calendar is a single long count date with a known equivalent in the European calendar. This would, of course, allow us to calibrate the relationship between the calendars. Unfortunately, the Maya of the post-Classical period (900 -1519 AD) gave up the practice of recording long counts of contemporary events. We do not have even one complete long count for an event that occurred after the arrival of Europeans in the Yucatan.
The correlation must therefore be deduced from incomplete and inconclusive information. Goodman relied primarily on sources from shortly after the Spanish conquest. At the time of the conquest, the Maya still kept both the calendar round, a cycle of about 52 years, and the ukahlay katunob, the "count of the katuns", a cycle of about 256 years. Colonial sources give us the European equivalents of a few calendar round dates, and the year in which at least one katun ended. These are not enough in themselves to correlate the calendars, but long count dates on Classical monuments include the calendar round date and a count of katuns. The GMT correlation was deduced from "incomplete" conquest era dates, using arguments based on the structure of the Maya calendar.
Astronomy did not play a role in arriving at the GMT correlation, but Thompson and Teeple turned to astronomy to confirm it. Classical monuments sometimes give the long count dates of astronomical events. The Dresden Codex, a post-Classical glyph book, assigns long counts to a table of eclipses and a table of the apparitions of Venus. Floyd Lounsbury, one of the leaders of the generation of scholars who "broke the code" of Maya hieroglyphs, reviewed the astronomical evidence in detail in an important contribution to Maya calendrics in 1978. He agreed with Thompson that the astronomy in the inscriptions and the Codex is compatible with the GMT correlation.
However, astronomy does not provide the kind of unequivocal evidence many had hoped for, and most competitors to the GMT correlation have been based on interpretations of Maya astronomy. Some of these are on the remotest fringes of scholarship, or worse. Others, such as the Bohm and Wells-Fuls correlations, are based on reasoned arguments. But there is an inherent problem in any correlation theory that depends primarily on astronomy. Maya astronomy was concerned with augury, ritual, and mythology. There are still gaps in our understanding of the way the scribes used astronomy. Astronomical evidence is important, but must be used cautiously and carefully when discussing the correlation problem.
Perhaps the greatest strength of the GMT correlation is the fact
that
it was deduced from the structure of the Maya calendar, and
independently
confirmed by astronomy. Although some of the individual pieces of
evidence have weaknesses, converging lines of argument support it.
Unlike
some of its competitors, it is not a "house of cards" that will
collapse
if any one of the pieces of evidence marshaled in its favour proves to
be mistaken.
|
The long count marks off the time from creation in multiples of five periods
(9 x 144,000) + (16 x 7200) + (0 x 360) + (2 x 20) + (0 x 1) = 1411240 days The long count is followed by the calendar round date. It is actually two dates--- the day in the tzolk'in cycle of 260 days (designated by one of 13 day numbers and one of 20 day names), and the date in a 365 day year, the haab, which is divided into 18 months of 20 days, plus 5 days at year end. See Note on the Maya Calendar at this web site for further information. |
Bishop Landa and the u kahlay katunob: A starting point
The katun is 7200 days, about 20 years, in length. Rituals
were
performed on katun end dates, which were often recorded in
Classical
inscriptions. A little calendrical math will show that the tzolk'in
day name on any katun end is always Ahaw. For
example
the tzolk'in date of the katun end 9.3.0.0.0
was 2
Ahaw.
Calculation
will also show that the day number of katun ends
advances
by 11 days through the cycle of 13 day numbers from katun
to
katun.
Thus the next katun end was 9.4.0.0.0 13 Ahaw. For Maya
scribes,
relationships such as these were keys unlocking the meaning of
the
cycles of time tracked by their calendar.
| The full calendar round date of
long count
0.0.0.0.0 is 4 Ahaw 8 K'umku. (This can be
verified
by counting back from any Classical long count date. When 0.0.0.0.0 is
reached, the calendar round will stand at 4 Ahaw 8 K'umku).
The first katun after creation ended on 0.1.0.0.0 2 Ahaw.
This is 7200 = 360 x 20 days since creation. Since there are 20 day
names,
a katun includes 360 complete cycles through the 20 day names,
returning
again to a day Ahaw at katun end.
There are 13 day numbers. 553 full cycles of day numbers occur in the katun, with 11 days left over to make up the 7200 days in a katun (553 x 13 = 7189). Thus 11 days before the first katun end, the day number reached 4 again, and at katun end, the day number was 11 + 4 = 15, which must be reduced into the range of 1 to 13, giving 15-13 = 2. Haab dates need not concern us for the moment. |
The
mathematics of period end dates is particularly important because it is
the link between the Classical long count and the post-Classical
u kahlay katunob. At the time of the conquest, katuns were
named
by their tzolk'in end date rather than their long count
position.
The katun names cycle through the 13 day numbers in this
order:
2
Ahaw, 13 Ahaw, 11 Ahaw, 9 Ahaw, 7 Ahaw,
5 Ahaw, 3 Ahaw, 1 Ahaw, 12 Ahaw, 10 Ahaw,
8 Ahaw, 6 Ahaw, 4 Ahaw. The U kahlay
katunob is a full cycle of 13 katuns.
It is 13 x 20 = 260 tuns,
about 256 years in the European calendar.
The u kahlay katunob was a long enough time frame for practical purposes. The long count was used primarily to allow kings of the Classical period to link the events of their reigns to the mythological past. In the post-Classical period, political power was welded by aristocratic lineages that had little need for the expansive time frame provided by the long count.
Diego de Landa, a missionary brother who later became Bishop of Merida, described the u kahlay katunob in his Relacion de las Cosas de Yucatan, written in about 1566, and illustrated it with a katun wheel that was likely adapted from a Maya glyph book. Similar wheels can be found in post-conquest native manuscripts.
Landa reported that
| It was easy for the old man [who told him of the fall of Mayapan and other events before the Conquest] to recall events which he said had taken place 300 years years before. Had I not known of this calculation, I should not have believed it possible to recall after such a period. |
More important for present purposes, Landa also reported that
| The indians say that the Spaniards finally reached the city of Merida in the year of Our Lord's birth 1541, which is exactly the first year of the era Buluc (11) Ahau, which is in that block [of the katun wheel] in which the cross stands. . . . |
The katun which preceded katun 11 Ahaw was katun13 Ahaw. Thus if Landa's information is correct, the katun end X.X.0.0.0 13 Ahaw occurred sometime in 1541, or if he meant that 1541 was the first year wholly in the new katun, in 1540. For the moment, it is enough to fix the katun end to a date near 1540. The last known long count date recorded on a monument is a katun end, 10.4.0.0.0 12 Ahaw, from Tonina. In the northern Yucatan, the territory familiar to Landa's informants, Chichen Itza recorded its last long count in 10.3.8.14.4. These dates mark the end of the Classical period. Obviously, the conquest of the Yucatan by the Spanish, the long count position of 1540 AD, must be later. The end dates of 13 Ahaw katuns after 10.4.0.0.0 are possible candidates. These dates, each 256 years apart, and the number of years each falls after the end of the Classical era in 10.4.0.0.0, are listed below.
10.10.0.0.0
13 Ahaw 13 Mol
---118 years
11.3.0.0.0 13 Ahaw
13 Pax --- 374
years
11.16.0.0.0 13 Ahaw
8 Xul --- 630 years
12.9.0.0.0 13 Ahaw
8 K'ank'in --- 886 years
The Books of Chilam Balam: A rough correlation
Chichen Itza continued to flourish into the post-Classical period, long after it recorded its last long count, and was eventually superseded by Mayapan as the leading centre in the northern Yucatan. According to Landa's informants, Mayapan was abandoned about 120 years before he wrote. If the conquest correlates to 10.10.0.0.0, there would be no room for the post-Classical history of Chichen Itza and Mayapan. The archaeological evidence suggests that the post-Classical period lasted several centuries, but 12.9.0.0.0 is almost certainly too late to correspond to the time of the Conquest.
The choice between the intermediate dates is more difficult, but here we can get some assistance from native traditions. After the conquest, augury and history were recorded in Maya towns by the Chilam Balam, "Spokesman of the Jaguar". Books of Chilam Balam from eleven towns, written in Yucatec using the Latin script, survive. Several of the Books give an account of the history of the Yucatan from the arrival of the Itzas, the post-Classical rulers of Chichen Itza, to the time of the Conquest, and record the katun dates of the events they recount. The Books report that the Itzas settled in Chichen Itza in katun 4 Ahaw. The "League of Mayapan" was founded in the next katun, 2 Ahaw, only about 20 years later.But Chichen Itza remained the dominant power in the Yucatan until it was conquered by Hunac Ceel, the ruler of Mayapan, in katun 8 Ahaw. This was 11 katuns, about 220 years after the Itzas had arrived. Mayapan was destroyed, according to the Books, a full cycle of the u kahlay katunob, about 256 years, later, in the next katun 8 Ahaw. If, as Landa was told, and the Books confirm, the Spanish Conquest was about a century later, the entire post-Classical era lasted about 600 years. This makes 11.16.0.0.0 13 Ahaw the most probable long count date correlating to 1540 AD.
If we tentatively place 11.16.0.0.0 13 Ahaw in 1540,
the
base date of the long count is 11 baktuns
and 16 katuns (about 4652 years) earlier. The
base
date of the long count would then correlate to about 3113/3114
BC.
This is a plausible deduction, but more evidence is obviously required
before it can be accepted.
Confirmation from Juan Xiu and the Chronicle of Oxkutzcab
In 1685, Juan Xiu, a Maya of noble lineage, copied out a page "from
an ancient book", and added it to the Xiu family papers, which have
come
to be known as the Chronicle of Oxkutzcab. The
document
records events by the year in the European calendar, and also gives
both
the calendar round date of first day of the haab and of
the
tun
end
that occurred in each year.
| The entries of particular interest read: 1540 11 Ix on 1 Pop [new year]. 1542 13 Kan on 1 Pop |
 Xiu Family Tree (Chronicle of Oxkutzcab) |
The Chronicle of Oxkutzcab also provides evidence confirming
that Landa's katun 13 Ahaw corresponds to 11.16.0.0.0
rather
than an earlier or later long count position. The Chronicle
gives
the haab date 7 Xul for the tun
end in 1540.
As will be explained below, it appears that a one-day shift in the
calendar
round occurred shortly before the Spanish conquest. Thus 13 Ahaw
7 Xul in the post-conquest calendar is equivalent to 13 Ahaw
8
Xul in the Classical calendar. This is the full calendar round
date
of the Classical katun end 11.16.0.0.0. Thus the Chronicle
confirms that the long count of the conquest era katun 13 Ahaw
was 11.16.0.0.0.
Landa and the Wayeb festival: A more precise correlation
Landa supplies one more bit of useful information, which turns out to be the key to a more precise correlation. He gives a lengthy description of rituals the Spanish frairs witnessed during the Wayeb, the last days of the haab, the Maya year. He followed this with a sort of Maya ecclesiastical calendar for a complete year, beginning on the first day of the haab, 1 Pop. He also recorded the tzolk'in date of this day, 12 K'an, which he tells us was Sunday, July 16. During the time Landa was in the Yucatan, July 16 fell on Sunday only in 1553. Since we have established that 11.16.0.0.0 13 Ahaw 7 Xul (according to the post-conquest Calendar round) was sometime around 1540, if Landa's information is correct, it is only necessary to count back from 12 K'an 1 Pop on July 16, 1553 to 13 Ahaw 7 Xul to fix the exact date of the long count position 11.16.0.0.0. The katun end occurred 5004 days before 12 K'an 1 Pop. This gives us a date of Nov 3, 1539 for 11.16.0.0.0 13 Ahaw 7 Xul. The calculated base date of the long count will then be September 7, 3114 BC in the Julian Calendar in use when Landa wrote, or August 12, 3114 BC in the reformed Gregorian Calendar.
But
is Landa's correlation between the Maya new year and the European
calendar
correct? There are in fact inconsistencies between Landa's date and
dates
from other colonial sources. For example, if we count forward from 13 K'an
1
Pop,
listed as the new year calendar round date in 1542 in the Chronicle
of Oxkutzcab, we 
reach
Landa's 12 K'an 1 Pop
in 1554, not 1553. However, these inconsistencies
are more apparent than real. The key to understanding them was
discovered
by careful study of the Books of Chilam Balam.
According to the Books, the
beginning of the European year was fixed to July 16 by the Maya. There
appear to have been two reasons for this practice. First, the Maya
understood
that the European year of 365.25 days is a true solar year. Their own
measurement
of the solar year began in mid-July, when the sun reaches the zenith in
the Yucatan. Second, it chanced that the beginning of the haab
fell
on July 16 shortly after the conquest. Of course, since the haab
makes no allowance for leap year, the haab new year fell on
July
16 for only four years, but one of these appears to be the year in
which
Landa witnessed the festival welcoming the new haab. If
we
assume that the Chronicle of Oxkutzcab fixes the beginning of
the
European year to July 16, it confirms Landa's correlation of July
16, 1553 to 12 K'an 1 Pop.
| The Books of Chilam Balam are
difficult
to interpret, in part because they were recopied and reworked from the
16th to at least the late 18th Century. As the traditional
calendar
fell into disuse in the Yucatan, even the copyists failed to understand
the texts. Some of the calendar round dates in the Books
are
not possible combinations of tzolk'in and haab dates
in either
the Classical or post-Classical count.
Unfortunately, Landa's 13 K'an 1 Pop is the only complete calendar round date equated to a European calendar date we have that was definitely unaltered since the 16th Century. Fortunately, we can be quite sure that it is a real date, not merely contrived by Landa as an example. If it were contrived, he might well have chosen a day Kan since this is the first year bearer (see below), but he would likely have not assigned the specific day number 11, which varies from year to year, to it. Nor would he have also assigned the dominical letter A (used by the Church in calendrical calculations) to the date, which indicates that July 16 fell on Sunday. |
The Year-bearers: Three possible correlations
.
| The crucial 12 K'an 1 Pop date in Landa's
manuscript
is a post-conquest calendar round date. It is not compatible with
the dates recorded on Classical monuments. According to Classical
inscriptions,
the long count began on the calendar round date 4 Ahaw 8 Kumk'u.
Counting
forward from 0.0.0.0.0 4 Ahaw 8 Kumk'u will never
reach a
calendar round date that combines 12 K'an and 1 Pop.
All post- conquest calendar round dates are similarly incompatible with
Classical dates. It would appear that a slip in the alignment of the tzolk'in
and haab occurred sometime before the conquest.
The shift in the calendar has to do with the "year bearers", the tzolk'in day names on the beginning of the haab. Because the length of the haab is five days longer than a whole multiple of the twenty tzolk'in day names, the day name of new year advances by five places each year. If this year began on a day K'an, next year will begin on a day Muluk. This will be followed by new year on Ix, and the fourth new year in succession will begin on Kawak. On the fifth year, the day name will return to K'an. These four day names are the year bearers. According to Landa, the rituals performed at new year were dictated by the year bearer, which also determined whether the year would bring good or ill. Landa reported the year bearers as K'an, Muluk, Ix, and Kawak. These are also the day names of new years in the Chronicle of Oxkutzcab. |
|
The beginning of the long count on 4 Ahaw 8 Kumk'u was in the last full 20-day month of the haab. The new year 1 Pop arrived only 17 days later, on 9 Etz'nab 1 Pop. Etz'nab is not a post-Classical year bearer. The next day, however, was a day Kawak, which is a post-Classical year bearer. Thus the change between the pre-conquest and post-conquest calendars appears to have been a change in the year-bearers, which produced a one day shift in the alignment of tzolk'in and haab.
Because the calendar round was kept by peoples throughout
Mesoamerica,
all of them were aquainted with the year bearer system, and
several
sets of year bearers are known. The K'an, Muluk, Ix, and
Kawak
set
is referred to as the "Mayapan year bearers". Assuming that the
Classical
new year was celebrated on 1 Pop, the Classical year bearers
would
have been Akbal, Lamat, Ben and Etz'nab. These are
called
the "Campeche year bearers". However, in the inscriptions, the
first
day of each month begins with the "seating" (chum) of the month,
and is followed by days numbered 1-19. This is equivalent to numbering
the days of the month from 0 to 19. Chum Pop can be interpreted
as either the last day of the old year or the first of the new. If new
year was celebrated on chum Pop, the Classical year bearers
would
have been Ik, Manik, Eb, and Kaban, the "Tikal year
bearers".
The
Tikal set is used in the Paris Codex. The Dresden Codex
lists
both the Tikal and Campeche sets; which marks the beginning of the year
is uncertain.
| The relationship between the tzolk'in and haab places other constraints on possible dates. Note that in a K'an year, K'an will fall on the first day of each of the 20-day months in the haab. But because the 5 day Wayeb intervenes at year end, in the next year, it retreats 5 days, to the 16th of each month. It retreats by 5 days in each year of the cycle until on the 5th year it again falls on the 1st of the month. Thus K'an can be associated only will four days of the month. Each of the day names is similarly fixed to four possible positions in each month. Thus, for example, in the Mayapan calendar, a day Ahaw can fall only on the 2nd, 7th, 12th, or 17th of the month. The date 13 Ahaw 7 Xulin the Chronicle of Oxkutzcab is compatible with this calendar. In the Tikal calendar, a day Ahaw can fall only on the 3rd, 8th, 13th, or 18th. The Classical period end 11.16.0.0.0 13 Ahaw 8 Xul is compatible with the Tikal calendar. |
We do not know precisely when or why the year bearer system was
changed
in the Yucatan. The Dresden and Paris Codices,
which
were almost certainly produced in the post-Classical period, retain the
Classical year bearers. On the other hand, cities in the Puuc hills of
the western Yucatan abandoned the Classical year bearers during the
last
phase of the Classical period. This change may have resulted from
Mexican
influence in the western Yucatan. Munro Edmonston believes that an
obscure
passage in the Book of Chilam Balam of Tzimin records a
calendrical
compromise between the eastern and western Yucatan just before the
conquest
that led to adoption of the Mayapan calendar.
| "As the Mexican people had signs and
prophecies
of the coming of the Spaniards . . . so also did those of Yucatan. Some
years before they were conquered by Admiral Montejo, in the district of
Mani, in the province of Tutul Xiu, an Indian named Ah cambal,
filling
the office of Chilan . . . told publicly that they would
soon shift
to fresh calendar bearers, [and] be ruled by a foreign race who
would
preach a [new] God . . .".
Landa, Relacion de las Cosas de Yucatan |
For our purposes, the important question is not when or why, but how. The realignment could have occurred in several ways. The possibilities are perhaps easiest to understood by considering what might have been done on the katun end 13 Ahaw to convert the date from 13 Ahaw 8 Xul to 13 Ahaw 7 Xul. Assuming no more than a one day slip, there are three possibilities:
1. If November 3, 1539 was the Classical katun end 13 Ahaw 8 Xul, the tzolk'in date could have been left unchanged, and a day subtracted from the haab, so that this date became 13 Ahaw 7 Xul. This preserves the base date deduced above, August 12, 3114 BC.
2. If November 3 was 12 Kawak 7 Xul, the the day before the Classical katun end, the tzolk'in date could have been reset on this day to 13 Ahaw without changing the haab date. Since in this case, the Classical katun end fell on November 4, 1539, the long count base date must have been a day later than calculated above, August 13, 3114 BC.
3. If, November 3 was the day after the Classical katun end,
both tzolk'in and haab would have needed to be reset.
The
classical date on November 3 would have been 1 Imix 9 Xul.
The tzolk'in date could have been reset by one day to 13 Ahaw,
and the the haab date adjusted by two days to 7 Xul.
Since
in this case the classical katun end would have fallen on November 2,
the
long count base would have been on August 11, 3114 BC.
These three possibilities were all discussed by Thompson. For a time, August 12 was most popular with Mayanists. It seems to require the least destructive change in the calendar, leaving both the long count and the tzolk'in unchanged. The long count had fallen into disuse before the calendrical slip occurred, so its preservation may not have been an important consideration. However, the tzolk'in is another matter. It is a sacred almanac, used to time rituals and make auguries. The change in the haab could have been a consequence of the shift to numbering the days of the month from 1 to 20 instead of 0 to 19. But the truth is that we can only speculate about the motives for the change in the calendar.
August 13 is now more popular with Mayanists. It was revived by Floyd Lounsbury, and adopted by Linda Schele, David Freidel, and Vincent Malmström among others. Lounsbury based his conclusion primarily on astronomical considerations (which will be discussed below). Astronomy lends considerable support to the GMT correlation, but whether it favours the August 13, 3114 BC version is more debatable.
Logically, August 11 seems least likely because it involves changes
in both haab and tzolk'in. However, the Quiche of
highland
Guatemala still keep the tzolk'in (ch'olk'ij in
Quiche).
Barbara Tedlock, who studied the calendar of Quiche "day keepers",
reports
that the Quiche count of days coincides with Classical tzolk'in
dates only if the long count base date was August 11, 3114 BC.
Likewise,
Alfonso Caso's widely accepted correlation between the Central Mexican
calendar round and the European calendar coincides with the Classical
Maya
calendar round only if the August 11 correlation is adopted. Munro
Edmonston,
after an extensive review of Mesoamerican calendars, concluded that "no
other solution [than the August 11 correlation] is
ethnohistorically
possible without postulating a break in the continuity and
uniformity
of the universal Middle American day count". This correlation has many
supporters among anthropologists, including Victoria Bricker and Dennis
Tedlock. It is also favoured by writers, like John Major Jenkins,
who believe that Maya calendrical prophecy has modern relevance. But it
is of course possible that a break in calendrical traditions did occur,
or that the Classical Maya calendar was simply the the "odd man out"
among
Mesoamerican calendars.
| I should admit that my personal preference is the '83 correlation, even though I've used the slightly more popular '85 correlation in these pages on Maya astronomy. It seems to me that the "ethnohistorical" argument --- the match between the '83 correlation and the Quiche and Aztec counts--- makes a very strong prima facie case for '83. The '85 correlation is a bit more easily reconciled with Maya astronomical records, but it is easier for an advocate of '83 to explain this away than for an advocate of the '85 to explain away the Quiche day count. Below, I try to show that astronomy alone cannot solve the correlation problem --- Thus I am suspicious of Lounsbury's revival of the '85 correlation, replacing the choice that otherwise makes the most sense, purely on astronomical grounds. |
The Age of the Moon and the correlation problem
.
| In 1930, John Teeple demonstrated that long counts in the inscriptions are often followed by the age of the moon, the number of days elapsed since new moon. Thompson used Teeple's discovery to confirm the GMT correlation. Many of the lunar ages on the monuments are those predicted by the GMT correlation. For example, a monument from Qurigua dated 9.18.5.0.0 4 Ahaw 13 Keh correlates to September 15, 795 AD (Gregorian) if the '85 version of the GMT correlation is used. The inscription gives the age of the moon as 23 days, the value calculated using modern methods. |
 Moon age 27 days |
There are also frequent discrepancies between lunar ages and their
predicted
correlations, but most result from errors in the inscriptions
themselves.
By studying sequences of lunar dates on monuments from Copan and
Palenque,
Teeple showed that the lunar ages recorded by the scribes were often
based
on calculation rather than observation. For example, it appears that
the
scribes sometimes used the formula 6 lunar months = 177 days to date
new
moon from earlier observations. Since 6 lunar months is actually
177.18 days, this calculation accumulates an error of about 1 day
over
3 years. Although the scribes also knew more accurate
approximation
formulas, we cannot expect all the lunar ages they recorded to agree
exactly
with the GMT or any other correlation. What is possible is a
statistical
check on the GMT correlation. The average error if the GMT
correlation
is correct to about 3 days, about as close as could be expected.
| The statistical fit between the lunar
ages recorded
on the inscriptions and the ages predicted using the GMT correlation is
actually better than the bald statement that the mean error is about
than
3 days. Consider, for example, the 18 lunar series inscriptions with
readable
Classical period long counts from Palenque and Naranjo. The mean error
if the '85 version of the GMT is adopted is 3.566 days. However, the
average
is distorted by two rather large errors in the nine Palenque dates of
8.4
and 8.1 days. Thus the lunar ages cluster closer to the correct values
than the mean error may suggest. In fact, 13 of the 18 lunar ages are
correct
to within one standard deviation (1.9 days). At Naranjo, the
maximum
error is only 2.5 days, and the mean error is only 1.577 days.
It should also be noted that there is doubt about how the scribes defined "new moon". The astronomical new moon occurs when the sun and moon are in conjunction, but this moment cannot be observed directly. We do not know if Maya astronomers recorded the new moon at the moment the crescent moon first became visible after conjunction, or tried to interpolate the moment of new moon between the time it disappeared and reappeared. Depending on the acuity of the observer and atmospheric conditions, the recorded date of new moon may differ from astronomical new moon by 2 -3 days. The problem of the definition of "new moon" makes it virtually impossible to use lunar series dates to choose between the three versions of the GMT correlation. The calculation above used the '85 version and astronomical new moon. But if the '83 constant is adopted and new moon is assumed to occur when the crescent moon appeared 2 days after astronomical new moon, the results of the calculation would be similar. |
The Dresden Codex elipse table
The Dresden Codex eclipse table has provided both advocates
and
critics of the GMT correlation with ammunition. The table counts the
days
between "eclipse warning stations". The intervals between the warning
stations
are eclipse cycles, intervals at which eclipses reoccur. Similar cycles
were used by Greek and Babylonian astronomers. They predict only the
possibility
of eclipses, however. The reoccurrence of an eclipse after lapse of an
eclipse cycle may not be visible to the observer who recorded the first
eclipse. In addition, cumulative error in approximations of eclipse
cycles
is an inherent problem in any table of eclipses based on the cycles.
Nevertheless,
the eclipse table is an impressive achievement.
|
There are three long count entry points into the table.
The principle
entry date appears to be 9.16.4.10.8 12 Lamat, November
12,
755 AD if the the '85 correlation is used. The second long count is 15
days later. If 9.16.4.10.8 12 Lamat was used to predict
solar
eclipses, the second date would fall at full moon, and could be
used
as an entry point for predicting lunar eclipses. The third date is 15
days
later still, and might be a secondary entry point for predicting solar
eclipses. These entry points work reasonably well if the GMT
correlation
is correct. Harvey and Victoria Bricker have shown that all 77 solar
eclipses
(including many not visible in the Yucatan) in the 33 year run of the
table
from 9.16.4.10.8 12 Lamat occurred close to the warning
stations.(If
the '85 version of the GMT correlation is used, the mean error is .08 days). Aveni has shown that the lunar eclipse entry date works almost as well. 51 of the 69 lunar eclipses in the 33 year life of the table are close to warning stations. However, as might be expected, there are problems in interpreting the table. In particular, the status of the 9.16.4.10.8 12 Lamat entry date is uncertain. The Codex was written in the post-Classical period, likely no earlier than 1200 AD, long after 9.16.4.10.8 12 Lamat if the GMT correlation is correct. Moreover, 9.16.4.10.8 12 Lamat is not the best choice for an entry date if it correlates to November 12, 755 AD. A new moon occurred on that date, but the moon was about 15 days from its orbital node. An eclipse cannot occur if the moon is not close to the node. No eclipse occurred on Nov 12, 755 AD. A lunar eclipse did occur 15 days later, and a partial solar eclipse about 30 days later, but neither was visible in the Yucatan. Most of the solar eclipses in the 33 years after 755 AD occurred on days reached from the secondary solar eclipse predicting entry date rather than from the primary 12 Lamat entry date. |
The less than ideal performance of Nov 12, 755 AD as an entry point has been used by critics of the GMT to propose alternatives. Smiley, for example deduced a correlation by moving 9.16.4.10.8 back in time in order to match it with a date on which an eclipse occurred near the node. Vollemaere moves the long count forward in time for much the same reason. Advocates of the GMT correlation regard exercises such as these as arbitrary. There are many 12 Lamat tzolk'in dates in the span of Maya history on which the new moon was close to an orbital node. If the game is merely to match 9.16.4.10.8 12 Lamat with such a date, there are many choices.
We simply do not know enough about how the eclipse table was used in practice to assume that 9.16.4.10.8 12 Lamat need be the best possible entry point. It may be, as Harvey and Victoria Bricker argue, that a Classical eclipse table using 9.16.4.10.8 12 Lamat was actually used for solar eclipse prediction, and copied into later codices. Anthony Aveni notes that the moon was almost at the node at the lunar eclipse entry date, 15 days after 9.16.4.10.8 12 Lamat. He speculates that the table was designed primarily to predict lunar eclipses. Floyd Lounsbury suspected that 9.16.4.10.8 12 Lamat was never used as a practical base date for either solar or lunar eclipse prediction. He argued that it is throw back, calculated by subtracting multiples of the table length from a later, more accurate, base date. Cumulative error in projecting the eclipse cycles back several centuries would explain why 9.16.4.10.8 12 Lamat is only a marginally acceptable entry date.
But whether 9.16.4.10.8 12 Lamat was a practical entry
point
or a throw back from a more useful entry date, the fact remains that
the
eclipse table does do a reasonably good job of warning of
eclipses
if the GMT correlation is correct. The very nature of the table
prevents
it from doing much more, what ever entry date is chosen. Unlike
correlations
based on astronomy, the GMT correlation is not contrived to force a fit
with the eclipse table or other astronomical data. Thus the eclipse
table
offers an independent check on the GMT correlation. The eclipse table
does
not unequivocally prove the GMT correlation is correct, but if it was
based
on mistaken premises, we could hardly expect Nov 12, 755 AD to be as
good
as entry point into the table as it is in fact.
The Venus table problem
The Dresden Codex Venus table has generated even more controversy than the eclipse table. The Venus table tracks dates of the apparitions of Venus, beginning with heliacal rise (first appearance of Venus in the morning sky before sunrise). The synodic period of Venus (the time from heliacal rise to heliacal rise) is recorded at 584 day intervals in the table. This is very close to the modern figure of 583.92 days.
Like the eclipse table, the Venus table provides long counts that
appear
to be entry dates. The latest of these long counts is 9.9.9.16.0
1 Ahaw 18 K'ayab, February 9, 623 AD if the '85
version of
the GMT correlation is used. This date has been seized upon by critics
of the GMT correlation because Venus was 16 days from heliacal rise on
the date. The Bohm correlation, the most ambitious recent effort to
depose
the GMT correlation, was motivated in part by the assumption that
9.9.9.16.0
1 Ahaw
18
K'ayab
should be close to heliacal rise. Once again,
however, much depends on the status of this date.
There is clearly important ritual significance in the putative heliacal rise date 1 Ahaw. Hun (1) Ahaw is the name of one of the "hero twins" who defeated the Lords of the Underworldand made creation of the present world possible. Dennis Tedlock, translator of the Quiche Popol Vuh, which includes the story of the twins, inteprets the myth cycle as an account of the apparitions of Venus. Since heliacal rise occurs only once every 584 days, it rarely matches a day 1 Ahaw in the tzolk'in cycle, but the scribes may have had strong incentive to do so. Matching a full calendar round date is even more difficult. Lounsbury found that there is only one 1 Ahaw18 K'ayab in the Classical or post-Classical periods which was a heliacal rise date of Venus if the GMT correlation is correct. On 10.5.6.4.0 1 Ahaw 18 Kayeb = 25 November 934 AD ('85 correlation constant), Venus was within 0.1 day of heliacal rise. Equally important, this date is exactly 196 Venus cycles of 584 days after the 9.9.9.16.0 entry date recorded in the codex. Lounsbury suggests that this was a "unique event in historical time". He believes it was the date on which the Venus table was inaugurated. If he is correct, 9.9.9.16.0 1 Ahaw 18 K'ayab was a throw back, calculated by subtracting multiples of the 584 day Venus period from the heliacal rise date 10.5.6.4.0 1 Ahaw 18 K'ayeb to reach an earlier calendar round date of 1 Ahaw 18 K'ayeb. Since the Venus period is actually 593.92 days, a cumulative error resulted from projection of the table this far back in time. Thus the scribes' long-range calculation failed to reach a true heliacal rise date.
If the GMT correlation is not correct, it must be mere coincidence
that
the long count base date of the table is an exact number of Venus
cycles earlier than the heliacal rise date discovered by
Lounsbury.
Lounsbury's explanation of 9.9.9.16.0 1 Ahaw
18 K'ayab
may be too circumstantial to prove that the GMT correlation is
definately
correct, but it is certainly strong enough to make it very
difficult
to argue that the Venus table makes the GMT correlation untenable.
Unfortunately,
the nature of Maya astronomy makes it unlikely that any correlation can
be based on astronomy alone. At best astronomy can confirm or
deny
the plausibility of a correlation, but must be used cautiously even for
that purpose.
More evidence from the inscriptions
Great strides in deciphering the Maya script have been made in the last 25 years . Some critics of the GMT correlation argue that if the GMT correlation is sound, decipherment should have produced evidence that unequivocally verified it. But while absolute proof is still elusive, decipherment, particularly of astronomical references on Classical monuments, has in fact produced considerable new support for the GMT correlation.
Perhaps the most remarkable example is found in the glyphic record of the dedication ceremonies of the temples of the Cross Group at Palenque. These rituals fell within a four day period commencing 9.12.18.5.16 2 Kib 14 Mol , July 23 690 AD if the GMT correlation is correct. On this date, Mars, Jupiter, Saturn and the Moon were separated by intervals of less than 5 degrees. According to Dutting and Aveni, "[On 23 July 690 AD] all four planets were close together (a quadruple conjunction) in the same constellation, Scorpio, and they must have made quite a spectacle with bright red Antares shining but a few degrees south of the group as they straddled the high ridge that forms the southern horizon at Palenque". This astronomical event appears to be recorded in the inscription, which recounts that the wayob (spirit companions) of the gods known as the Palanque triad were "in conjunction" on 9.12.18.5.16 2 Kib 14 Mol.
| Right: "His way in conjunction, GIII". GIII is one of the gods of the Palenque triad. Conjunction is marked by the glyph inset with "crossed bands". The number three before this glyph may represent the conjunction of Mars, Jupiter and Saturn. This text begins with an account of Creation, the birth of First Mother and First Father, and of their sons, the gods of the Palenque triad. Schele and Freidel suggested that the alignment of these planets with the Moon on 23 July 690 was interpreted by Palenque scribes as the symbolic re-uniting of First Mother (the moon) and her three sons. |  |
'83, '84 or '85: A smoking gun?
The limited power of astronomy to unequivocally settle the correlation question has not prevented it from being used to erect alternative correlation theories. Nor has it prevented advocates of the GMT correlation from attempting to use astronomical data to choose between the three GMT correlation constants. What is needed is a "smoking gun", a unique astronomical event with a clear long count date. But the smoking gun is difficult to find.
Lounsbury argued that 10.5.6.4.0 1 Ahaw 18 Kayeb
was such an event. It corresponds exactly to the heliacal rise of Venus
on November 25, 934 AD only if the '85 correlation is adopted. However,
as Aveni has noted, naked-eye observation of heliacal rise is fraught
with
difficulty. Venus is lost in the glare of sunrise at the instant of
heliacal
rise. Much depends on atmospheric conditions and the acuity of the
observer,
but Venus is usually not visible until several days after true heliacal
rise, when it is about 10 degrees from the sun. Lounsbury assumed that
Venus becomes visible 4 days after true heliacal rise, but if it was
glimpsed
earlier in 934 AD, the heliacal rise date might match the '83 or '84
correlation
rather than the '85 correlation.
An eclipse might seem to be a more likely smoking gun.
Unfortunately,
however, the eclipse table fails to resolve the issue. Eclipse
prediction
is difficult. Although the eclipse cycles incorporated into the table
are
remarkably accurate, they will not always predict an eclipse with an
accuracy
of less than three days. Thus any conclusion about which the three GMT
constants is correct that is based on the eclipse table must be
suspect.
| In fact, the eclipse table appears to have a 3 day latitude build into it: Each of the tzolk'in dates at warning stations counted from the 12 Lamat entry point is flanked by the tzolk'in dates of the day before and the day after. This is likely the Classical scribe's equivalent of the modern scientist's "+ or - 1 day". |
However, what may be the single strongest piece of evidence in favour of the '85 constant is an eclipse date. Michael Closs has pointed out that Quirigua Stela E appears to record an eclipse on the katun end date 9.17.0.0.0 13 Ahaw 18 Kumk'u. If the '85 constant is correct, this correlates to January 24, 771 AD, when a partial eclipse was in fact visible in the Yucatan. Closs believes this is the only instance of a eclipse occurring on a katun end in the Classical period, which likely gave it particular significance. Certainly, this eclipse is strong evidence in favour of the GMT correlation. But does it necessarily confirm the '85 constant? Perhaps the occurrence of an eclipse within 2 days of period end was enough for the scribes at Quirigua to associate the eclipse with the end of the katun.
Other pieces of evidence can be marshaled in support of each of the
three GMT correlation constants, but the debate is unresolved. While
the
GMT correlation is almost certainly valid, it seems unlikely that the
choice
between the '83, '84, and '85 versions can be definitively resolved on
the basis of the available evidence.
|
|
| Smiley | JDN 482699 | 26 Jun 3392 BC (Gregorian) |
| Makemson | JDN 489138 | 11 Feb 3374 BC (Gregorian) |
| Spinden | JDN 489384 | 15 Oct 3374 BC (Gregorian) |
| GMT (1) | JDN 584283 | 11 Aug 3114 BC (Gregorian) |
| GMT (2) | JDN 584284 | 12 Aug 3114 BC (Gregorian) |
| GMT (3) | JDN 584285 | 13 Aug 3114 BC (Gregorian) |
| Bohm | JDN 622261 | 4 Aug 3010 BC (Gregorian) |
| Kreichgauer | JDN 626927 | 14 Mar 2997 BC (Gregorian) |
| Wells-Fuls | JDN 660208 | 27 Jun 2906 BC (Gregorian) |
| Hochleitner | JDN 674265 | 22 Dec 2867 BC (Gregorian) |
| Esalona Ramos | JDN 679108 | 27 Mar 2854 BC (Gregorian) |
| Weitzel/Vollemaere | JDN 774078 | 5 Apr 2594 BC (Gregorian) |
Ironically, the first radiocarbon dates from the Classic period, obtained from wooden beams in Yaxchilan temples, seemed to favour the Spinden correlation when the dates were reported in the 1956. These dates are still sometimes quoted to discredit the GMT correlation. However, techniques improved rapidly after 1956. In 1959, the University of Pennsylvania ran 33 samples from ten beams in a Tikal temple. The series averaged out to A.D. 746 A.D. +/- 34 years. These beams were carved with a long count equivalent to of 741 A.D. in the GMT correlation. This is of course a particularly close match. Most radiocarbon dates cluster about those expected from the GMT correlation, but typically have a wider margin of error.
The most critical date for purposes of assessing proposed
correlations
is the end of the Classical period (10.4.0.0.0 = 909 AD GMT). Samples
from
sealed Late Classical tombs at Muklebal Tzul in Belize have recently
been
dated to between 600 and 810 AD. This compares with the GMT dates for
the
Late Classical of 600-900 AD.
Bibliography
Aldana, Gerardo. "K'in in the Hieroglyphic Record: Implications of a Pattern of Dates at Copán". On-line at Mesoweb. An interesting recent article suggesting that dates at Copan that seem to refer to equinoxes, solstices, and zenith passages of the sun may be difficult to square with the GMT correlation.
Aveni, Anthony F. Skywatchers (U. of Texas Press, 2001) The best general introduction to Maya calendrics and astronomy. This is a revised edition of Skywatcers of Ancient Mexico.
------ (ed.). The Sky in Mayan Literature (Oxford University Press, Oxford, 1992)
Baaijens, Thijis. "The typical 'Landa Year' as the first step in the correlation of the maya and the christian calendar", Mexicon Vol. XVII, 1995.
Bohm B., Bohm, V. "Calculation of the Correlation of the Mayan and Christian system of Dating", Actes du XIIe Congres International des Sciences Prehistoriques et Protohistoriques. Bratislava, 1-7 Septembre 1991. (on-line version) The Bohm correlation.
Bricker, V. R. and H. M. Bricker. "Classic Maya prediction of solar eclipses", Current Anthropology, xxiv, 1-23
Paul D. Campbell, Astronomy and the Maya Calendar Correlation, Mayan Studies Series , No 5 (Aegean Park Press). Another recent alternative to the GMT correlation
Caso, Alfonso. ``Calendrical Systems of Central Mexico,'' in Wauchope (ed.) Handbook of Middle American Indians (University of Texas Press, Austin, 1965).
------. Los Calendarios Prehispánicos (México,UNAM, 1967).
Closs, Michael P. "Cognative Aspects of Ancient Maya Eclipse
Theory",
in Anthony F. Aveni, World Archaeoastronomy.
Cambridge, 1989.
.............................. "Some Parallels in the Astronomical
Events Recorded in the Maya Codices and Inscriptions", in Aveni,
ed. The Sky in Mayan Literature (Oxford, 1992).
Dutting, Deiter, and Anthony F. Aveni. "The 2 Cib 14 Mol Event in the Palenque Inscriptions". Zeitschrift fur Ethnologie 107, 1982.
Edmonson, M. S. (transl.). The Ancient Future of the Itza: The Book of Chilam Balam of Tizimin (U. of Texas Press, 1982)
--------. The Book of the Year: Middle American Calendrical Systems (University of Utah Press, 1988). An excellent survey, particularly good on controversial topics such as year bearer systems.
Gates, William. [Transciption and translation with notes of page 66 of the Cronica de Oxkutzcab] in S.G. Morley, Inscriptions at Copan (Carnegie Inst., 1921).
Goodman, J. T. The Archaic Maya Inscriptions (Taylor and Francis, London, 1897). Now only of historical interest, includes the original Goodman correlation.
Justeson, Jonn S. "Ancient Maya ethnoastronomy: an overview of hieroglyphic sources" in Aveni (ed.), World archaeoastronomy: Selected papers from the 2nd Oxford International Conference on Archaeoastronomy (Cambridge, Cambridge University Press, 1989).
Jenkins, John M. Tzolkin: Visionary Perspectives and Calendar Studies (Borderland Sciences, 1994). See Jenkin's strong defence of the '83 GMT correlation online.
Landa, Diego de. Relacion de las cosas de Yucatan (1556), English translation by W. Gates, Yucatan Before and After the Conquest (Dover Press, 1978). See an on-line translation.
Lounsbury, Floyd: "The Base of the Venus Table of the Dresden Codex and its Significance for the Calendar-Correlation Problem", in Aveni & Brotherston (eds.), Calendars in Mesoamerica and Peru: Native American Computations of Time (BAR International Series, no. 174, Oxford, 1983). Lounsbury's influential defence of the '85 GMT correlation
----. "Maya Numeration, Computation, and Calendrical Astronomy," in Dictionary of Scientific Biography, ed., Charles Coulston, 1976. Lounsbury's influential first paper on the GMT correlation and astronomy
----. "A Derivation of the Mayan-to-Julian Calendar Correlation from the Dresden Codex Venus Chronology," in Aveni (ed.), The Sky in Mayan Literature (Oxford University Press, Oxford, 1992).
Makemson, Maud W. The Maya Correlation Problem (Publications of the Vassar College Observatory, No 5, New York 1946). Makemson variously supported the '84 GMT correlation and her own correlation derived from the Dresden eclipse table.
Malmström, Vincent H. Cycles of the Sun, Mysteries of the Moon: The Calendar in Mesoamerican Civilization (University of Texas Press, 1997).
-------. "Astronomical Footnotes", Arqueología 21, Segunda Época, Enero-Junio 1999 (English version on line). Makes the interesting case that the '85 GMT correlation best fits the astronomical data if it is assumed that the Maya began the day at sunset.
Morley, S. G. "Correlation of Maya and Christian Chronology", Amer.
J. of Archaeology, 2nd ser., XIV
(1910). An early discussion of the correlation
question.
Owen, Nancy K.: "The Use of Eclipse Data to Determine the Maya Correlation Number", in Aveni (ed.), Archaeoastronomy in Pre-Columbian America (University of Texas Press, 1975). Owen rejects the GMT correlation.
Proskouriakoff, Tatiana A and J. E. S. Thompson. Maya Calendar Round Dates such as 9 Ahau 17 Mol (Notes on Middle American Archaeology and Ethnology, no. 79, Washington, 1947). Discusses the "slip" in the Maya calendar
Prufer, Keith M. "Analysis and Conservation of a Wooden Figurine Recovered from Xmuqlebal Xheton Cave in Southern Belize, C. A.", Report Submitted to FAMSI, 2001. Discusses some Late Classical radiocarbon dates.
Roys, Ralph L. (transl.). The Book of Chilam Balam of Chumayel (U. of Oklahoma Press, 1967). On-line version at the Sacred Texts web site.
Satterthwaite, Linton. Concepts and Structures of Maya Calendrical Arithmetics (Philadelphia, 1947)
--------. "Long Count Positions of Maya Dates in the Dresden
Codex
with Notes on Lunar Positions and the
Correlation Problem". Proc. 35th Int. Cong. Amer. Mexico, 1962.
Smiley, Charles H. "The Solar Eclipse Warning Table in the Dresden Codex", in Aveni, (ed.), Archaeoastronomy in Pre-Columbian America (University of Texas Press, 1975). Smiley is a leading opponent of the GMT correlation.
Smither, R. K.: "The 88 Lunar Month Pattern of Solar and Lunar Eclipses and its Relationship to the Maya Calendars", Archaeoastronomy, Vol. IX (1986).
Spinden, Herbert J.: The Reduction of Mayan Dates (Papers of the Peabody Museum of American Archaeology and Ethnology, Harvard University, vol. 6, no. 4, 1924). The most influentialearly alternative to the GMT correlation.
Stock, Anton. Astronomie der Maya-Kultur - Die Datierung der Finsternistafel aus dem Dresdner-Codex und das Korrelationsproblem. Katun Verlag, 1998. Another alternative to the GMT. See information on-line
Tedlock, Dennis. "Myth, Math, and the Problem of Correlation in Mayan Books", in Anthony F. Aveni, ed. The Sky in Mayan Literature. Oxford, 1992. Support for the '83 GMT through an analysis of astronomical references in Maya mythology
Tedlock, Barbara. Time and the Highland Maya (U. of New Mexico Press, 1982). Account of the calendar kept by modern Quiche "day keepers".
Teeple, J. E.: "Maya Astronomy", Contributions to American Archaeology (Carnegie Institution of Washington, Vol. 1, 1931).
Thompson, J. Eric S. A Correlation of the Maya and European Calendars, Field Museum of Natural History , Anthropological Series, Vol. 17, no.1, 1927.
.......... "Maya Chronology: The Correlation Question,", Contributions to American Archaeology, Volume III, Nos. 13 to 19, Carnegie Institution of Washington, No. 14, 1937. Thompson's influential review of Goodman's correlation, proposing the three versions of the GMT correlation.
------- "The Introduction of Puuc Style of Dating at Yaxchilan", Notes on Middle American Archaeology and Ethnology No. 110, May 15, 1952.
Wells, Bryan and Fuls, Andreas, Western and Ancient Maya Calendars, ESRS (West) Monograph no. 6 , Berlin 2000. Another recent alternative to the GMT correlation (see information about this correlation on-line)
Volemaere, Antoon. " La correlación maya-tolteca-azteca", Congreso internacional de americanistas, Amsterdam, 1988. Another alternative to the GMT correlation. (see information about this correlation on-line)
