**Maya Calendrical Harmonics. (03/07/2001)**

**As I have noted elsewhere ( Ruth Nanda Anshen: The Fallacy of Opposition in Western Ideology),
the Maya created a calendrical system that depended on the extreme/
mean ratio for its articulation of time. Since the ratio itself
is harmonic, any numerical system, like the one found in the Maya
calendar, that adheres to its basic principles will necessarily
recapitulate both the ratio itself and the harmony it expresses
at the same time. As I noted in the previous discussion, when
the day-name sequence of the Maya tzolkin (260) is counted
in conjunction with the interval of the tun (360), as the
Maya always did during the Classic period, the two intervals naturally
and precisely express a primary relationship that splits between
values that preserve the terms of the extreme/ mean ratio. This
is true because the day-names of the first Maya tzolkin
are matched with the first 260 days of the tun, which fact
leaves a remainder of 100 days from the second tzolkin
that must be counted before the first 360-day interval is completed.
Hence, the second tzolkin retains 160 days in its count,
which are then matched with the first 160 day-names of the second
tun. The split here in the tzolkin between 100 and
160 is the one which preserves the extreme/ mean ratio, since
100 represents the lesser segment of the ratio, while 160 represents
the mean segment. The sum of lesser and mean segments, of course,
fixes the length of the extreme segment of the ratio at 260 days.**

**As I also noted in the previous discussion, the exact value
of the ratio, expressed as a decimal, is equal to .61803403. In
the Maya calendar, the lesser segment (100 days) divided by the
mean segment (160 days) produces a value of .625, which is .007
greater than the ideal value. At the same time, the mean segment
(160 days) divided by the extreme segment (260 days) produces
a value of .6153846, which is .003 less than the ideal value.
These facts are cause enough, and in themselves, to argue that
the Maya calendar preserves the extreme/ mean ratio when the day-name
sequence is counted in real time. What these facts do not tell
us definitively is whether or not Maya astronomers were aware
of the harmonic structure of their calendar. This conclusion is
in doubt because the Maya probably reached the values expressed
in their calendar by finding numbers that best articulated several
complex relationships between planetary motion (Mars and Venus
especially) and periodicities associated with lunar and solar
eclipse syzygies as well as with the sun's seasonal connection
to the celestial equator at equinox and solstice positions. The
best evidence in support of the possibility that the astronomers
were aware of the nature of the 260/ 360 split in their calendrical
system can be seen when the numerical values are converted directly
to an appropriate geometrical configuration.**

**In the figure given below, the extreme/ mean ratio has been
used to inscribe two concentric circles. One word of caution:
the figure was not drawn with a compass and straight-edge. As
a result, the relationships apparent in it are not exactly precise.
At the same time, however, everything seen in the figure can be
drawn with Euclidean tools to preserve the exact relationships
as I have described them below. The radius of the inner circle
is equal to the lesser segment of a given finite straight line
that has been cut in extreme/ mean ratio as Euclid prescribes
in Book VI, Proposition 30, of The Elements. The radius
of the outer circle is equal to the extreme segment of the same
finite straight line. The mean segment, of course, is that part
of the radius of the two circles that stands as the difference
between the lesser and extreme segment. In the figure, the area
shaded in blue is the mean segment of the figure. The area shaded
in red is the lesser segment. Red plus blue is the extreme segment.
As noted, the red segment is marked as being equal to 100 days;
while, the blue segment is marked as being equal to 160 days,
in order to preserve the sense of the split between the 260-day
tzolkin and the 360-day tun in the Maya calendar.**

**Also apparent in the figure is the fact that both circles have
been divided into 20 equal parts of 18 degrees each. While this
might appear arbitrary, might seem to be a fact accomplished outside
the strict application of Euclidean tools, that impression is
false, since it is true that one-half the length of the mean segment
in every instance of the use of an extreme/ mean ratio to draw
two concentric circles, regardless of the length of the given
finite straight line, always measures the length of a chord on
the outer circle that subtends exactly 18 degrees of circular
arc on its circumference. There is no conceivable exception to
this rule. Using only a compass and a straight-edge, then, it
is possible to draw the figure exactly as it appears below. What
I have added peripherally to the figure is a methodology for using
its configuration to count the day-name sequence of the Maya 260-day
tzolkin. If numerical markers are placed on the figure
at the intersection points between 18-degree angle lines and the
circumference of the inner circle, with consecutive numbers at
each place beginning with the line marked 1 Imix and extending
forward to 13 Ben, which exhausts the numerical coefficients in
the tzolkin, the next day in sequence can be named by moving
the marker for "1" across the open "horseshoe"
shaped space to the next empty position on the inner circle after
13 Ben. That creates the 14^{th} day of the sequence at
1 Ix. 2 Men is counted next as the 15^{th} day, and so
on, until all 260 days are counted in their proper and precise
order.**

**What gives this configuration its most compelling stamp of
authenticity as a genuine Maya artifact, even if no example or
representation of it has ever been found, is in the way it reverses
the order of objects classified in groups as it moves back and
forth between days and degrees in the 360-day tun and the
360-degree circle. In the tun there are 18 months (uinal)
of 20 days each. In the circle there are 20 divisions of 18 degrees
each. The difference between one thing and the other creates a
natural space where the 260-day sequence of the tzolkin
can actually be counted. If that is only the result of coincidence,
of blind chance, this case must stand as one of the most incredible
accidents that has ever occurred in human history. More probably,
this configuration rests at the deepest ground of Maya Classic
period conceptualizations of the relationship between real space
and real time. That the conception is harmonic is further emphasized
by the fact that the relationship between days in the tun
and degrees in the circle is not expressed as an opposition; rather,
these two categories are precisely the same even as they stand
as mirror-images of each other. Harmony collapses opposition by
turning the same thing around into its reverse order. What this
says is that time and space are the same thing but that they exist
as mirror-images of each other, at least in the way the Maya conceived
of them during the Classic period.**