The Calendrical Base of the Mesoamerican Calendar

From: Peden (r&m_at_spiderorchid.com)
Date: 06/19/04 

Date: Sat, 19 Jun 2004 14:30:51 +1000

Due to the interest shown, my paper, "The Mayan Calendar- Why 260 Days?",
has been re-formatted.
It can be seen at: www.spiderorchid.com/mesoamerica/mesoamerica.htm

It is proposed that the factual astronomical derivations are ipso facto
sufficient to demonstrate the astronomical base for the Mesoamerican
calendrical system

A summary follows.

The Mayan calendrical system uses a whole number ritual period, such that
particular multiplies of the ritual period interlock with particular
multiples of the astronomical periods, or whole number approximations to
them, making them individually commensurable. Maya interlocking cycles had
the form:

a x astronomical period = b x ritual period,

where a, b and the ritual period are whole numbers.

In other words, they replaced a multiple of a non-commensurable astronomical
period, which their arithmetic could not handle, with a corresponding
commensurable whole number cycle. For example:

52 x 365 = 73 x 260
is known as the Calendrical Round

405 LM = 46 x 260
is from the Dresden Codex. LM is the lunar month.

The most important Mesoamerican ritual period was the Tzolkin, a calendar of
great age in Mesoamerica, with a period of 260 days; made up of a repeating
sequence of the numbers one to 13, and 20 day names. Thus each day was
individually identified with a unique combination of name and number. The
count was basic to Maya ritual and sacred life. Up to now no satisfactory
explanation of the choice of 260 days has been given

The question is raised could a calendar in constant use for 2500 years
survive on ritual significance alone or must there be some underlying
astronomical reason that would account for such a long retention?

Interlocking cycles of the form:

a x astronomical period = b x interlock period,

were derived by computer analysis for the tropical year, and the synodic
periods of the moon, Venus, Mars and Jupiter, where a, b and the interlock
period are whole numbers and the interlock period is in days. In particular
it is concluded that the 260 day ritual period of Mesoamerica is the optimum
interlock period for creating such interlocking cycles. If cycles with a
common interlock period and seeking accuracies better than one day in 100
years are required, 260 days is the only choice.

The full set of derived interlocking cycles, having a 260 day interlock
period are as follows,

42 TY = 59 x 260 [0.4]

405 LM = 46 x 260 [0.3]

61 VY = 137 x 260 [0.8]

1 MY = 3 x 260 [2.8]

88 JY = 135 x 260 [0.3]

Figures in square brackets give the cycle accuracies in days in 100 years.
TY is the tropical year, LM is the lunar month, VY the Venus year, MY the
Mars year and JY the Jupiter year.

Note that in the first equation 59 is the the sum of 29 and 30, which is the
Mesoamerican whole number way of expressing an approximate lunar month of
29.5 days. Hence this interlocking cycle links the tropical year, and the
Maya approximation to the lunar month, with a 260 day interlock period. This
cycle could track the tropical year for 200 to 300 years before a one day
error accumulated and a correction would be necessary. It follows that if
this cycle is in fact a previously unknown Mesoamerican cycle, then the 260
day Tzolkin could be considered as an accurate solar calendar

It was further discovered that:-

The Long Count is a lunar calendar with an accuracy of 0.8 days in 100
years.
256 LM = 21 x 360 = 7560 days

The Accounting year of 364 days might be considered a Venus Calendar with an
accuracy of 0.3 days in 100 years.

104 years is the natural and optimum intercalation time to correct both the
solar and Venus calendars - and could be considered the basis for the
Mesoamerican Calendar Round. The solar intercalation is virtually an exact
intercalation and the Venus intercalation has an accuracy of 0.07 days in
100 years.
The solar intercalation is:

365.2423 = 365 + 63/260

= 365 + 1/4 - 1/130

= 365 + 1/4 - 1/104 + 1/520

With the tropical year given to four decimal places this is an exact
intercalation. The best correction time for the natural solar intercalation,
coincides with multiples of the 52 year Calendar Round and the ubiquitous
number 260.

It is proposed that these factual astronomical derivations are ipso facto
sufficient to demonstrate the astronomical base for the Mesoamerican
calendrical system.

See: www.spiderorchid.com/mesoamerica/mesoamerica.htm

Robert Peden