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 14th day of the sequence at 1 Ix. 2 Men is counted next as the 15th 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.