WHITECROW BORDERLAND
Equinox and Solstice in the Maya Calendar Round.
For as long as anyone remembers Europeans have expressed the opinion that the Maya calendrical system is not equipped to count or calculate fixed positions of the sun along the eastern and western horizon (at sunrise and sunset) because it does not incorporate intercalary days for "leap" years. This "problem" with the Maya system arises because the true length of the solar year is equal to 365.2422 days and the Maya used an interval of 365 days to count that period of time. As a result, of course, one day of "error" occurs in the Maya calendar after every four years of time are counted, with one-quarter of one day being added to the deficit every year. The perception that the Maya method of counting the solar year, in the interval known as the haab, is somehow primitively based on the absence of an essential knowledge, however, is more a prejudice than it is a matter of fact. Europeans, after 1582, began adding the intercalary days to their calendar in order to "freeze" the calendrical days on which equinoxes and solstices occurred because the calendrical system they employed did not contain a calculating mechanism inherent in its structure that enabled them to predict in advance, or perceive in retrospect, the days on which these events would occur in the future, or on which they had occurred in the past. In comparison to the Maya system, then, where "freezing" the days of equinoxes and solstices became the only and inescapable sign of an advanced calendrical structure, the Maya system did not equal that standard because they omitted the mechanism of intercalary days which Europeans used to accomplish that function. Hence, in comparison to the European calendar, the Maya system is perceived as inferior to the one used in Europe and America.
The Maya, then, did use a 365-day solar year in their calendar. At the same time, however, they also employed a second period of time, known as the tzolkin, which computed time in intervals of 260 days. The 260-day almanac has always been characterized by Europeans as being a ceremonial count of the days that has no essential astronomical function whatsoever. When the day-name interval of the haab (365) is combined with the day-name interval of the tzolkin (260), as the Maya did during the Classic period, an interval of time known as the Calendar Round is produced. These two distinct sequences of day-names, in combination, count an interval equal to 18,980 days because that value represents the lowest common denominator of 260 and 365. In other words, a combination of one day-name from the tzolkin and one day-name from the haab cannot occur, or repeat, until 18,980 days have been tabulated. The Maya always counted that sequence of combined day-names in exactly the same order of occurrence in an endless "round" of days. The interval of the Calendar Round contains 52 Maya solar years and 73 Maya almanacs.
Addressing the issue of whether or not the 260-day almanac has any astronomical intent is a first order of business in this study. It is also quite easy to resolve. In the first place, 73 almanacs exactly counts 32.5 average synodic periods of the planet Venus, since 32.5 x 584 = 18,980 days. This fact has long been known to European scholars but they have always insisted that it is simply a meaningless coincidence that the CR counts a significant interval of Venus's synodic motion. The Maya used this mathematical fact to create an extremely sophisticated Venus table that was recorded in the Dresden Codex on pages 46-50. In spite of that factual evidence of the likelihood that 260 serves an obvious astronomical function, Europeans still deny the possibility of astronomical intent for the "ceremonial" interval of the tzolkin. A second obvious astronomical function of 260 surfaces in the Dresden Codex eclipse table (pp. 51a-58b), which expresses an interval equal to exactly 46 turns of the 260-day almanac (at 11,960 days) from one base-day to the next in its articulation of eclipse occurrences. In fact, a survey of base-day positions for that eclipse interval in the Maya Calendar Round demonstrates a series of 14 consecutive lunar eclipses (from 338 A. D. to 763 A. D.), most of which were visible in the Maya area during the Classic period, that absolutely confirms the utility of that interval (11,960 days) as an eclipse prediction mechanism. Those eclipses are given in the following table, where the twelfth eclipse in the sequence, at 9.16.4.10.8 12 Lamat 1 Muan, is the actual base-day as recorded in the Dresden Codex eclipse table.
| 8.17.19.2.8 12 Lamat 1 Yaxkin | Magnitude 0.645 at 4:06 PM | ||
| 8.19.12.6.8 12 Lamat 16 Uo | Magnitude 0.866 at 12:29 PM | ||
| 9.1.5.10.8 12 Lamat 16 Pax | Magnitude 0.523 at 3:11 AM | ||
| 9.2.18.14.8 12 Lamat 11 Ceh | Magnitude 0.797 at 1:31 AM | ||
| 9.4.12.0.8 12 Lamat 6 Mol | Magnitude 1.610 at 6:11 AM | ||
| 9.6.5.4.8 12 Lamat 1 Zotz | Magnitude 1.803 at 4:20 AM | ||
| 9.7.18.8.8 12 Lamat 1 Cumku | Magnitude 1.498 at 9:32 PM | ||
| 9.9.11.12.8 12 Lamat 16 Mac | Magnitude 1.949 at 11:27 PM | ||
| 9.11.4.16.8 12 Lamat 11 Ch'en | Magnitude 2.830 at 5:55 AM | ||
| 9.12.18.2.8 12 Lamat 6 Zec | Magnitude 2.913 at 3:49 AM | ||
| 9.14.11.6.8 12 Lamat 1 Pop | Magnitude 2.680 at 9:25 PM | ||
| 9.16.4.10.8 12 Lamat 1 Muan | Magnitude 2.618 at 11:25 PM | ||
| 9.17.17.14.8 12 Lamat 16 Yax | Magnitude 1.844 at 3:21 AM | ||
| 9.19.11.0.8 12 Lamat 11 Xul | Magnitude 1.871 at 9:55 PM | ||
| 10.1.4.4.7 11 Manik 5 Uo | Magnitude 2.102 at 1:40 PM |
A final mechanism built into the Maya calendrical system, and the subject of this study, is a third interval derived from a multiple of 260 that served a vital function in the calendrical astronomy of the Classic period. The interval is equivalent to 59 turns of the tzolkin and counts exactly 15,340 days. This value is simultaneously equal to exactly 42 solar years at 365.2422 days each (15,340.172 days) and as such clearly demonstrates that Maya astronomers, even as early as the time when they adopted the 260-day almanac as a primary calendrical counting device, were aware of the true length of the solar year. One could even argue that 260 was chosen as a companion to the 365-day haab in the Calendar Round precisely because 59 turns of its day-name structure creates an exact count of the solar year and therefore "corrects" the short-fall inherent in the choice they made to omit intercalary days in every four-year sequence of the haab. The interval itself "works" because any fixed position of the sun along the eastern or western horizon, which includes vernal and autumnal equinox and summer and winter solstice, must of necessity fall on the same almanac day-name after an extension of 59 almanacs because that same interval counts the exact value of 42 solar years, where the interval between sequential equinoxes and solstices is always determined by exactly 365.2422 days. This interval, furthermore, can be used for a total of 252 years (6 x 42) before a single day of regression accumulates in the formula (6 x 0.172 = 1.032 days) for predicting the day-name of the solar event in question. With respect to the haab day-name itself over this same interval of time, it must advance a total of ten days from position to position because 15,340 is equal to 42 x 365 + 10. The point here is that predicting any of these solar positions over the length of the interval requires virtually no calculation at all, since the tzolkin position remains the same, while the haab day-name simply advances by 10 days.
With respect to the CR day-names from one year to the next, the Maya calendar generates a sequence of days in its natural selection that is also completely susceptible to simplistic calculation for these same solar positions. As the following table demonstrates, using vernal equinox as an example over the course of a 42-year sequence, where every other position follows exactly the same pattern with appropriately different day-names, it is remarkably easy to predict day-names in the Maya calendrical system. For instance, in clusters of four-year intervals, before the absence of the intercalary day produces the advance to the next sequential haab designation for the event, the tzolkin day-names follow a predictable pattern. The coefficient from year to year advances by one sequential numerical value, while the day-name itself moves forward by five places. The haab designation remains constant. After four years, the first regression occurs because of the absence of the intercalary day and the fifth position moves forward by one haab day-name (from 6 Yax to 7 Yax in the example here). The tzolkin position advances by a total of two numerical coefficients (from 8 to 10 in this example), and by a total of six names (from Lamat to Ix). This pattern is always repetitiously consistent from one four-year cluster to the next. Also important to note is the fact that predicting the tzolkin designation from the beginning of one four-year cluster to the beginning of the next requires an advance of only a single named position in the sequence, since the first one here is Ben and the second one following it is Ix, while the coefficient advances by a total of five places (from 5 Ben to 10 Ix). This is also absolutely consistent over time. The point to be taken here, of course, is that anyone using this calendrical system on a daily basis as part of his/her job description would be able to make these kinds of predictions almost without thinking about it.
| Venus Superior Conjunctions | |||
| 5 Ben 6 Yax | + 16 days to Venus SC | ||
| 6 Etz'nab 6 Yax | |||
| 7 Akbal 6 Yax | |||
| 8 Lamat 6 Yax | |||
| 10 Ix 7 Yax | |||
| 11 Cauac 7 Yax | |||
| 12 Kan 7 Yax | |||
| 13 Muluc 7 Yax | |||
| 2 Men 8 Yax | +14 days to Venus SC | ||
| 3 Ahau 8 Yax | |||
| 4 Chicchan 8 Yax | |||
| 5 Oc 8 Yax | |||
| 7 Cib 9 Yax | |||
| 8 Imix 9 Yax | |||
| 9 Cimi 9 Yax | |||
| 10 Chuen 9 Yax | |||
| 12 Caban 10 Yax | + 12 days to Venus SC | ||
| 13 Ik 10 Yax | |||
| 1 Manik 10 Yax | |||
| 2 Eb 10 Yax | |||
| 4 Etz'nab 11 Yax | |||
| 5 Akbal 11 Yax | |||
| 6 Lamat 11 Yax | |||
| 7 Ben 11 Yax | |||
| 9 Cauac 12 Yax | + 10 days to Venus SC | ||
| 10 Kan 12 Yax | |||
| 11 Muluc 12 Yax | |||
| 12 Ix 12 Yax | |||
| 1 Ahau 13 Yax | |||
| 2 Chicchan 13 Yax | |||
| 3 Oc 13 Yax | |||
| 4 Men 13 Yax | |||
| 6 Imix 14 Yax | + 7 days to Venus SC | ||
| 7 Cimi 14 Yax | |||
| 8 Chuen 14 Yax | |||
| 9 Cib 14 Yax | |||
| 11 Ik 15 Yax | |||
| 12 Manik 15 Yax | |||
| 13 Eb 15 Yax | |||
| 1 Caban 15 Yax | |||
| 3 Akbal 16 Yax | + 4 days to Venus SC | ||
| 4 Lamat 16 Yax | |||
| 5 Ben 16 Yax | |||
| 6 Etz'nab 16 Yax |
The final two positions in this table, at 5 Ben 16 Yax and 6 Etz'nab 16 Yax, demonstrate the repetition of tzolkin day-names for vernal equinox that occurs after the addition of 59 turns of the day-name sequence. The interval of separation here, of course, is equal to exactly 15,340 days as the formula prescribes.
The positions of Venus included here suggest a secondary relationship between solar events and the planet's superior conjunction with the sun, specifically at vernal equinox at this point during the Classic period, that demonstrates an essential mathematical advantage for counting solar years in intervals of 365 days without resorting to intercalary days for "leap" years. As these positions demonstrate, there is a two-day regression in the Venus position relative to vernal equinox when time is counted with intercalary days included, since the interval of separation between Venus's proximity to vernal equinox is eight years in length, where two "leap" days must be added to each interval. What this means is that a calendar without intercalary days tends to "freeze" positions of Venus on the same day-names in every eight-year interval. The formula Maya astronomers employed in the Dresden Codex Venus table clearly reflects this knowledge, and this choice to omit intercalary days in their calendrical system, by virtue of the fact that 8 x 365 = 5 x 584 = 2,920 days, where eight exact solar years are equal to 2,922 days exactly. The structure of the Venus table in the Dresden Codex exactly replicates this formula, since each of the five pages expresses one synodic period of the planet divided into intervals of 236, 90, 250, and 8 days each, where every horizontal row across all five pages accounts for five synodic periods of the planet and eight Maya solar years exactly. In all, the table accounts for 104 Maya solar years (haab) and 146 turns of the almanac (tzolkin). Hence, the Venus table begins and ends on the same CR day-name-1 Ahau 18 Kayab.
The only thing not said yet concerns the fact that the Maya established orientations in their ceremonial architecture, at Copan, Honduras especially, that enabled them to sight the exact setting position of the sun, through a western-facing window in Temple 22 at that location, that always occurred 20 days after vernal equinox and 20 days prior to solar zenith passage. This fact makes it clear that the Maya of the Classic period were able to name the calendrical day on which any, or all, of the above listed vernal equinoxes occurred. Since the sun returns to the same orientation through the window in Temple 22 exactly 20 days before autumnal equinox and 20 days after solar zenith passage in the fall, those solar positions and the day-names on which they occurred were also clearly known to the Maya. The point here, of course, is that this system of naming the days of equinoxes is not just a theoretical assertion but clearly rises to the level of a practical application of the Maya calendrical system. Hence, the assertion that the 260-day almanac had no astronomical significance is clearly erroneous. Claiming, as well, that use of a 365-day interval in the haab somehow demonstrates that Maya astronomers did not know the true length of the solar year is also certainly false.
All rights reserved. Copyright Frederick Martin, December 2000.
E-Mail comments here.