This below was taken from Z.A. Simon (1984: 9-31) by permission, in a condensed form. Some of it is disputed by mainstream scholars of ancient Mesoamerica.
The Aztec calendar
Most of our largest libraries have a few dozen books on the history of Mexico, but these books usually interpret the Aztec calendar very briefly. Besides this, they often emphasize only those scientific achievements that are important for their authors. The wide range of different opinions can be illustrated by one example. A well-known expert on Mexican culture remembers Antonio de León y Gama (1736-1802), who has been called the first Mexican archaeologist. He states that the most important of León y Gama's books on archaeology is the "Descripción histórica y cronológica de las dos piedras." The sculptures of the title are two famous monoliths. Bernal adds that one of them is the Stone of the Sun, and falls into the error of attributing to the stone a calendrical significance, though he never claims that it was used as a calendar. In this situation, despite the correct but brief information of the encyclopeadias, the reader might get easily confused.
This below was taken from Z.A. Simon (1984: 9-31) by permission, in a condensed form. Some of it is disputed by mainstream scholars of ancient Mesoamerica.
In addition to this, some sources introduced certain new terms without proper consistency. Other denominations were superseded, including a few calendrical miscalculations of amateur writers. Due to the apparent lack of coherence, the new results of some interesting papers have been neglected or ignored by the editors of many popular books. Therefore, a detailed description of the Mexican (Aztec) calendar is necessary for a complete understanding of the question. Therefore we will give that description before examining what experts have said about the problems. It will be concluded with the author's solution and its implications.
It is well-known that the Mayas and Aztecs had a highly developed calendar system. The two basic time cycles that governed Mesoamerican life were the solar calendar and the ritual calendar. In other words, the Mexican calendar is twofold, and comprises a ritual calendar, with a round of 260 days, which was employed in divination and in fixing "movable feasts"; and a solar year, with a round of 365 days, according to which the seasonal feasts were held (Muser, 1978:17 and Joyce, 1970:59).
The solar calendar of 365 days, called the Vague Year (or Civil Year), was composed of 18 months of 20 days each, with a period of 5 days added at the end. The 360-day period was called "xíhuitl" by Aztecs, and "haab" or "tun" by the Maya. The final unlucky days were called "Nemontemi" in Nahuath, and "Uayeb" in Mayan.
Each month had its own special name, and the days were numbered from zero to nineteen. The days of the last month, Uayeb, were numbered from zero to four. In this calendar the Maya counted the days starting from zero rather than from one (Ivanoff, 1971: 87).
This solar calendar was inseparable from the Sacred Round, or Sacred Almanac. The priests used this ritual calendar of 260 days, called [Tonalpohualli]]by the Aztecs and tzolkin by the Maya, primarily for divinatory purposes. The concurrent permutation of the solar and ritual calendars produced the Calendar Round. An exclusively lowland Classic Maya calendar achievement was the Long Count, which permitted an infinite computation of time, backward or forward, from an established starting point (Muser, 1978: 17). By the passing centuries, this simplified system may have became dominant, but we want to know its original form.
As we have already mentioned, in both solar and ritual calendars, time elapsed in parallel fashion, simultaneously and continuously. Peter Tompkins (1976: 290) states that each day of the tzolkin was governed by a deity who was thought to influence that day for good or evil, each separate day being regarded by the Maya as an individual god, whose glyph was a stylized portrait of his attributes. The numbers 1 to 13 were also personized as the heads of the gods they represented.
The use of this 260-day calendar was in no way arbitrary. The Mesoamericans possessed the correct knowledge that 260 x 18 was the same as 360 x 13, that 260 x 7 was the same as 364 x 5, that 260 x 73 was the same as 365 x 52, and that 260 x 1461 (like the Egyptian Sothic cycle!) was the same as 365.25 x 1040. Tompkins adds that to these calendars, which all fell into the 260-day pattern, were added more refinements, in order to calculate the synodic returns of the moon and the planets.
In Mesoamerica the planet Venus looms in the dawn sky with extraordinary brilliance. Both the Maya and the Nahua astronomers devoted special attention to the planet, and particularly to its heliacal rising. Venus revolves around the sun every 224.7 days, but since the earth is moving along its own orbit, the planet appears at the same place in the sky in 584 days, called synodic period. As 5 x 584 is equal to 8 x 365, the Maya considered five Venus years equal to eight solar years. And as 365 x 104 is equal to both 146 x 260 and 65 x 584, the sacred, the solar, and the venus calendars become coincident every 37,960 days, or 104 years. That is, two Mesoamerican "centuries" of 52 years. (Actually, the Maya knew the Venus cycle to be 583.92 days, instead of a round 584, so they dropped four days every sixty-one Venus years, in order to compensate for the discrepancy and make a round number divisible by 260.)
As astronomers are quick to point out, such an accurate knowledge of the cycle of Venus, whose revolutions are by no means regular, points to a long and careful observation.
Furthermore, the Mesoamericans devised a lunar calendar that would fit with the others. Calculating that 405 lunations or 11,960 days was exactly divisible by 260 (or, 260 x 46), they obtained a lunar period of 29.53 days which is practically the same what we know today. This would give them a lunar calendar accurate within a day over a period of 300 years.
Returning to the ritual and the solar calendars, the method of naming the individual days was the same for both, and consisted in the combination of twenty pictorial signs, with the numbers one to thirteen. The signs, according to the four cardinal points, were as follows:
Joyce (1970: 61-62) states that these signs ran consecutively in the order given above, one being assigned to each day, and the series was repeated ad infinitum. Conjointly with them were repeated the numerals one to thirteen, e.g., 1 Cipactli, 2 Eecatl, 3 Calli, and so on to 13 Acatle, which was followed by 1 Ocelotl, 2 Quauhtli, etc. There being no common factor to the numbers 13 and 20, a period of 13 x 20 days, or 260, would elapse before the sign 1 Cipactli would recur. This period of 260 days constituted the divinatory or ritual calendar, known as tonalamatl. The tonalamatl was subdivided in various ways; in some manuscripts each of the twenty 13-day periods, or weeks, is shown separately, together with the figure of a god who was especially associated with the first day, but whose influence was supposed to extend over the whole "week". In some manuscripts the tonalamatl is arranged on a different system: in five long horizontal rows of 52 days each. Each row, and each vertical column of five days, is provided with a presiding deity symbol, the influence of which must be assessed.
The Mexicans reckoned 365 days to the solar year, which they divided into 18 months of twenty days each, and a nineteenth period of five days, considered extremely unlucky, at the end of the year. As the days were known by their tonalamatl names, it is obvious that the first 105 days of the year recurred at the end, after the 260-day period. However, it was possible to distinguish between two days of the same name which fell in the same year, owing to the fact that each day was associated with one of a series of nine deities, called lords of the night, a series also repeated ad infinitum, except no "lord" was assigned to any of the five unlucky days at the end of the year, which were called nemontemi or "useless days." Thus, since the number 260 is not divisible by 9, it was possible to differenciate between two days of the same name falling in one year. And since 9 goes into 360 without a remainder, the commencement of the year coincided with the beginning of the series of "lords of the night."
Nor is this all; a corresponding series of thirteen "lords of day," which, however, is not similarly composed in all manuscripts, accompanied the days (except the nemontemi), and the influences of the day- and night-lord assigned to each day respectively constituted two additional features for the consideration of the would-be interpreter of the tonalamatl.
Since each "month" consisted of twenty days, and there were twenty day-signs, it is obvious that each month in a given year started with the same sign; but that since the last month was followed by the five unlucky days, each year began with a day-sign five days later than the last. Also since 365 is divisible by 13 with 1 as remainder, it follows equally that each year began wit a day-number one in advance of the last.
Further, since there were 20 day-signs, and 5 (the highest common factor of 365 and 20) goes into twenty exactly four times, the year began with one of the four signs only. The four signs mentioned by T.A. Joyce (1970: 63-65), which gave the names to the years, are the signs Tecpatl, Calli, Tochtli and Acatl, recurring in that order. (It is more than probable that the day-number entering into the name of the year was that of the first day, as held by most authorities.) The years were named successively, 1 Tecpatl, 2 Calli, 3 Tochtli, etc., until, after a period of 52 years (that is, 13 x 4), the same sign recurred with the same number as the name-date of the year. This period of 52 years named xiuhmolpilli (meaning bundle of years) formed the shorter cycle of the Mexicans.
To understand the calculation of a 52-year cycle, imagine two wheels of time rotating simultaneously. On time when recorded days, for which there were 20 names with 13 numbers. The complete name of a day must be accompanied by the appropriate number, such as 4 Reed or 9 Eagle. The other wheel presents a somewhat parallel situation. Eighteen months of 20 days each, to which the dreaded 5-day period of bad luck was added, resulted in a 365-day cycle (18 x 20 plus 5 equals 365), corresponding to our solar year. In order to return to the very same day and month, 52 years would have to elapse. This was the basic computation of time used in Mesoamerica. See the diagrammatic illustration in Michael D. Coe, Mexico and in the books of J. Eric S. Thompson. This Calendar Round enabled unique designation within a 52-year period but difficulties arise when dealing with several cycles: one needs to know to which Calendar Round the date refers. (The Maya Long Count also used the 52-year cycle but avoided confusion by accurately recording lapsed time from an initial starting point in 3113 BCE, according to the Goodman-Martínez-Thompson correlation. Many scholars agree upon that this starting point represents a mythological event, probably the latest creation of man.)
These statements mentioned above require some adjustment. The four signs cited were actually in use during the Spanish Conquest. We should not conclude from this fact alone that this was the original form of all Mesoamerican calendars. Many of those tribes may have used the 52-year Mexican century during their whole history, using it automatically, being incapable of understanding the deeper meaning of the calendar for absolute dates. Regardless of our knowledge of the existence of the supposed infinite version of the calendar using exclusively four year-signs (like part of the Boturini manuscript with its 188 year-signs), we cannot admit that this would be the final solution of the Mesoamerican calendars. This system must have had some kind of starting point and the correct 3113 BCE of the Maya does not fit into the Aztec system. (This is the point where we have to break with the widely accepted theories, and start to build up a more understandable explanation.)
Ignacio Bernal states in The Olmec World that it must have dawned on the Olmecoids that a system in which dates were repeated every 52 years was confusing on the long run. It could be compared to our cryptic way of writing, for instance, '56. Does this refer to 1956, or 1856, or even 156? Of course at the moment of writing we know what it meant, but after many centuries it is not clear. The Long Count system (of the Maya) is far more exact and equates basically to the one that has been followed by many civilized peoples.
Seen this way, the logic behind the system is no longer apparent. Would the Mexicans have been that illogical? Did they just ignore the difficulties and the inaccuracy in such a refined institution? Our answer is definitely no. From all the facts we must not arrive at the conclusion that only 4 signs recur infinitely during the whole history of the Aztecs. Also, although the sequence of these four year signs is correct, we must emphasize that there would not be any particular reason to start with the Tecpatl sign.
If we have a good look at the picture of a conjectural Mexican calendar published by Gemelli Carreri (1645-1700), we arrive at some interesting conclusions. See A history of Mexican archaeology (1980: 55). This circular calendar is surrounded by a serpent, a symbol of both earth and time, according to Irene Nicholson (1975: 27). The orientation of the drawing itself suggests that we have to start to read it at the top. The head of the snake symbolizes the beginning. Fray Diego Durán, Book of the Gods and Rites and The Ancient Calendar refers to ann oddity that "Cipactli, which was the first figure and which means Head of Serpent," observing that "when people called it a head, I think that they understood it to be the beginning of the month, or its first day" (Durán, 1971 reprint: 394). (The English, Spanish, Hungarian and other languages have corresponding cognates for head, principal and chapter.)
We can be certain that the Head of Serpent really means the beginning in this representation. Therefore, we can assume that our count is supposed to progress from the head towards the tail of the snake. Emphasizing the correctness if this explanation, the drawing shows four loops on the serpent's body. These loops indicate the year-bearers, namely Rabbit, Reed, Flint-knife, and House. These four fields are significantly larger than the others on the drawing (altogether 52 year-signs). We can see clearly that the commencing year of a 52-year cycle was the Rabbit sign, at least at that time when the original of the drawing was made.
The best confirmation for this theory can be found in Miguel León-Portilla, Aztec Thought and Culture (1963: 54-55): The division of time gave rise to the years of the East, of the North, of the West, and of the South. In abstract terms, motion appeared as a consequence of the spatialization of time and of the orientation of the years and the days toward the four directions. Such a conclusion can be drawn from the accounts of Sahagún's Indian informants explaining the tabulation of the year-count, in which the years are spatially oriented:
William Prescott, The World of the Aztecs states that in the measurement of time, the Aztecs adjusted their civil year by the solar... A month was divided into four weeks of five days each, on the last of which was the public fair or market day. As the year is composed of nearly six hours more than 365 days, there still remained an excess, which, like other nations who have framed a calendar, they provided for by intercalation; not indeed, every fourth year as the Europeans, but at longer intervals, like some of the Asiatics. They waited till the expiration of the fifty-two vague years, when they interposed thirteen days, or rather twelve and a half, this being the number which had fallen in arrear.
Nicholson (1975: 47) repeats that some authors believe that the end of the 52-year cycles an extra or intercalary period, sometimes of twelve and sometimes of thirteen days, brought the calendar into line with the position of the sun among the stars. This is not a widely accepted theory, since its theoretical usefulness is no proof in itself. Joyce is of the following opinion: The employment by the Mexicans of a solar year of 365 days brings us to the question whether they at any time intercalated any day or days to make their year square with real solar time. It is quite obvious that a people, most of whose feasts were connected with agriculture, were bound to notice that their festivals gradually failed to correspond with the seasons, and many conjectures have been made regarding the methods which they might have used to rectify their calendar. It must be confessed that ''there is no direct evidence that days were ever intercalated in the latter, and Seler has shown that at any rate between the year of the conquest (1519) and the date of Sahagún's writing, some forty years, no intercalation had been made.
Peter Tompkins (1976: 289) asserts that Mesoamericans divided their year into 360 days plus 5 extra days on regular years and 6 on leap years or 13 every 52 years. Michael D. Coe of Yale University is categoric in his assertion that "there is no evidence that the Mesoamericans ever intercalated days or leap years." According to him, because the tropical year is 365.2422 days long, the 365-day vague year simply gained on the seasons by a factor of 13 days every 52 vague years. But the fact remains that whatever system the Mesoamericans used, the result was a calendar more accurate than ours, adds Tompkins.
Prof. Coe's theory must be right from another point of view also. Since a pictorial sign was attributed to every single day of the calendar, even to the "useless" nemontemi days, it is incontrovertible that each of these supposed intercalary days would have had its own year-sign too. All the civilized nations of Mexico and Central America seem to have been the most calendar-oriented people, as held by many authorities. Given their way of thinking, no day could have passed without being marked for posterity. If the supposed intercalary days had had their own signs, new signs besides the existing four would have had to be introduced because of the hypothetical shift of 12 or 13 days between the 52-year cycles.
The nonexistence of extra year-signs due to the intercalary days positively implies the nonexistence of an intercalary shift. On the other hand, the lack of intercalary days does not mean new signs could not have been introduced. If the Mexicans possessed 20 different symbols, why they could not have used all of them for their great time-cycles? Why did they keep repeating only four of them? We investigated this mysterious question below.
It is known that the Aztecs and the Maya used very similar systems in their calendars. Keith G. Irwin, in The 365 Days (1963: 164-167) gives a valuable observation on the Mayan calendar that might be useful for an understanding of that of the Aztecs. He noticed that there was one point about their double measuring plan of marking off days into "months" and "weeks" that we would notice in much the same way if our weeks were numbered. The month plan covered a year of 365 days while the week plan stopped at a year of 364 days. Both plans ran on continuously, so the shorter year kept drawing away from the beginning of the other year a day at a time. In 365 years the shorter year would have advanced a whole year; so at intervals 365 years apart the two "year arrangements" would have their beginnings at the same time. Irwin's ingenious theory refers to the Maya and apparently has not yet been emphasized well enough in modern publications.
In order to get further proofs for his "364-day year plan," we tried to derive a unique counting system based on the 364 days. We supposed that the Mesoamericans utilized the fact that 364 is divisible by 4, and the quotient is 91. But what kind of number is the 91? It does not seem to be a spacial figure. But if we examine it with Mesoamerican logic, it would gain enormous importance from the fact that it is divisible by 13. Thirteen days gave a Mexican "week," and otherwise 13 was held to be a fortunate number. This is just the beginning. If we add all the figures from 1 to 13 inclusive, the amount will be 91 again. This seems to be a marvellous coincidence. The ancient priests could have kept track of all the 91 days of one season easily, without confusion, in a triangular form. You can imagine it as a set-square with two angles of 45 degrees: one at the top, and the other at the bottom right. The top is one dot, the next is a horizontal line of two dots, the next is of three dots, etc. Finally, the 13th line (at the bottom, horizontally) is formed by 13 dots. This way they were able to mark one dot each day, and arrive at 91 without a confusing mess or mistake.
Representing the existence of the 364-day year, the El Castillo pyramid of Chichén Itzá, the great Mayan center in Yucatan, shows stairways of 91 steps on each side. Four times 91 equals 364! Marcel F. Homet states that in the same region the two sets of steps of the pyramid of the Sun at Teotihuacan total 364. Three hundred and sixty-four heads of serpents decorate the pyramid of Quetzalcóatl as well which indicates his connection with the Mexican calendar.
There are even more academic opinions about the existence of this 91-day period. Adrian Digby introduced such a wide-ranging technique as archaeoastronomy in Crossed trapezes: a pre-Columbian astronomical instrument in Mesoamerican Archaeology (1974:272-282). Digby wonders, "Did the possession of the instrument lead to the adoption of the enigmatic number 13, and the computing year of 364 days?"
Since the existence of both a 365 and a 364-day year seems to be proved, we can start determine the length of the period at the end of which they would coincide. So we should multiply 364 by 365, in order to get the least common multiple of 364 and 365. It will be 132,860 days. Dividing this figure by 365.242199, the number of the days in one tropical year, we get 363.7586 years.
It have been established that One rabbit was the year when the Mexicans began their latest great cycle at the time of the Conquest. However, we must figure out which were the guiding years of their preceding ages. The Aztecs attributed the invention of the calendar to their main god, Quetzalcóatl. It is known that his birthday fell on Ce äcatl or One Reed (Man across the sea, 1971: 256 and Durán, 1971 reprint: 475). Not knowing if this date referred to the birthday or the exact date of the calendar's introduction causes a slight difference. However, a life span was probably shorter than a great cycle. So we can assume that the first Great Cycle of 363.7586 years was marked as 1 Reed''. Quetzalcóatl may have wished to start at a vague date, he could have started his count from a date well-known for himself, i.e., his own birthdate. Since he was a god, he could have known the exact day, and even the hour.
Otherwise the Reed sign represents East, the cardinal point where Quetzalcóatl appeared from, and disappeared towards. East was the beginning of everything, in this divine sense, so it may have been the starting direction of their calendar, too. Let as accept that the opening year of the first Great Cycle started on One Reed, and that of the cycle in 1519 was One Rabbit.
Durán (1971: 391), a contemporary of Cortés, shows a few examples from the written Mexican records. They told him, "In the year One Reed, at the beginning of the 16th cycle, the Spaniards arrived in this land." He adds a sentence from the same paragraph, "In the year of the jubilee, which began on One Reed, the first year of the 16th cycle, there came to this land Don Hernán Cortés. We can clearly see that the 16th cycle, or certain years of it, represented the jubilee of some event, evidently of that of 1 Reed, that is, the existence of their calendar. Due to Durán's vague determination of the date, it originally might have meant, "In the year of the jubilee, which began on a day One Reed (just like the first year of the 16 cycles), at the beginning (or, in the first part) of the 16th cycle, the Spaniards arrived to Mexico." Durán may have misinterpreted this information, since he did not understand at all what the term "cycle" meant.
Many paintings from Mayan codices, the great Stone of the Sun and Gemelli's drawing show the symbols in the same sequence. If you count them one by one, beginning with the Reed sign, the 16th sign will be the Rabbit sign. However, we still do not know how long a Great Cycle was. If we suppose that it can be equated with a 52-year period, the result of our calculation from 1519 would be A.D. 739, so the Aztec calendar must have began before that date. This gives us another key: it confirms the theory that the god Quetzalcóatl and King Topiltzin were not identical beings. Perhaps Topiltzin, son of Mixcoatl, was born on a One Reed day. Bruce Hunter, A guide to Ancient Mexican Ruins (1977: 75) refers to Topiltzin, who adopted the name Quetzalcóatl, a deity well-known in Classic time. Nicholson (1975: 17, 79) and the Man across the sea (1971: 258) confirm these. Topiltzin may have reigned from 977 to 999.
The second possibility for the length of a Great Cycle would be 104 years or ce ueuetiliztli (One Old Age). This was in used in Mesoamerica as well. Using these, we would get 41 BCE for the starting point of the calendar. But Alfonso Caso, Muriel Porter Weaver (1972: 59), Curt Muser (1978: 131), and Ignacio Bernal (1973: 127) agree that Quetzalcóatl was a very ancient god, and the 52-year calendar was already in use at Monte Albán in Middle Preclassic times, which lasted from 900 to 400 BCE. Humboldt tells a legend of Puebla, in which everybody died in a great flood, exept seven giants. One of them, Xelhua, went to Cholula and erected a pyramid-shaped mountain. The gods were angered at this structure, because its summit had reached the clouds. and threw fiery stones at it. Many laborers died, and the building had to end. The work was dedicated to Quetzalcóatl (William R. Corliss, 1978: 763, and Pierre Honoré, 1964: 16). Ixtlil-Xochitl, the Mexican historiographer added that the gods created a confusion of the languages of the builders after the great flood. The date of the appearance of the first cultivated, larger size corn (and cotton) in Central America supports this: the myth tells that Quetzalcóatl was watching the ants carrying grains of maize, so he discovered it and gave it to mankind to survive.
As the third possibility, we accept the well-defined theory of the cycle of 364 years of 365 days. First we had to find the accurate year for the year 1 Rabbit that introduced the 16th cycle. Using Durán's statement, counting back from 1519, it yields 1506, 1454, or 1402 for the year One rabbit. William H. Prescott (1970: 73) informs us that the epoch, from which the Mexicans reckoned, corresponded with the year A.D. 1091. So we accept that his source remembered that 1091 was the beginning of a vague 364-year cycle.