A count built from two small cycles
The tonalpohualli, from Nahuatl words usually rendered as “count of days”, was the ritual calendar of the Aztecs (Mexica) and related peoples of central Mexico. It ran for 260 days and then began again. Unlike the months of the calendar most readers know, it was not tied to the moon or the sun. It was made by running two short cycles side by side: a cycle of thirteen numbers and a cycle of twenty named days.
The arithmetic is simple. Thirteen times twenty is 260. Because thirteen and twenty share no common factor other than one, a particular pairing, such as “7 Deer”, cannot recur until all 260 combinations have been used once. If the cycles were, say, ten and twenty, they would repeat after only twenty days. The fact that thirteen and twenty are unrelated in this way is what makes the full 260 possible.
The twenty day signs
Each day had a name, drawn from a standard list. In Nahuatl and English the usual sequence is:
- Cipactli (Crocodile)
- Ehecatl (Wind)
- Calli (House)
- Cuetzpalin (Lizard)
- Coatl (Serpent)
- Miquiztli (Death)
- Mazatl (Deer)
- Tochtli (Rabbit)
- Atl (Water)
- Itzcuintli (Dog)
- Ozomatli (Monkey)
- Malinalli (Grass)
- Acatl (Reed)
- Ocelotl (Jaguar)
- Cuauhtli (Eagle)
- Cozcacuauhtli (Vulture)
- Ollin (Movement)
- Tecpatl (Flint)
- Quiahuitl (Rain)
- Xochitl (Flower)
English renderings vary, since the Nahuatl names carry more than one sense, but the order is well attested in colonial-era manuscripts.
How the two cycles run together
Each day carries a number from one to thirteen and a sign from the list. The count starts with 1 Crocodile, then 2 Wind, 3 House, and so on up to 13 Reed. The numbers then start again while the signs continue, so the next day is 1 Jaguar, followed by 2 Eagle. The twentieth day of the cycle is 7 Flower, since 20 is one full round of thirteen plus seven more. The twenty-first day is 8 Crocodile, because the sign list has now restarted but the number has not yet returned to one.
A block of thirteen days, with the numbers one to thirteen, is called a trecena in modern writing, a Spanish term that scholars use for convenience. The 260 days contain exactly 20 trecenas. Each trecena is named for the sign on its first day. The first begins with 1 Crocodile, the second with 1 Jaguar and the third with 1 Deer, which can be checked by counting 13 days forward each time.
The Maya had a closely related count, usually called the tzolk’in, built from the same thirteen numbers and twenty days but with Maya day names. That shared structure shows how deeply the pattern was embedded in Mesoamerican numbers more widely.
How the count was used
Specialist priests, sometimes called tonalpouhque, interpreted the count. Sources written soon after the Spanish conquest, notably the work of the Franciscan friar Bernardino de Sahagun and his Nahua collaborators, describe days as being regarded as favourable, unfavourable or mixed, and report that the sign of a person’s birth day influenced how they were named and how a family viewed their prospects. These were beliefs held by the people who used the count, recorded by others. They are not facts about the days, and this article reports them as history only.
The count was written down in painted books called tonalamatl, “books of days”. Pre-conquest examples were largely destroyed, but later painted manuscripts, such as the Codex Borbonicus, show how pages of day signs, numbers and deities were laid out.
Meshing with the solar year
The Aztecs also kept a 365-day year, divided into eighteen periods of twenty days plus five extra days. Each day belonged to both counts at once. The same pair of day sign and number would return only after the two cycles lined up again, and that takes 52 years: 52 times 365 is 18,980 days, which is also 73 times 260.
In the usual reconstruction, a consequence is that only four of the twenty day signs could begin a solar year. Since 365 leaves a remainder of 5 when divided by 20, the first day of each new year moves forward five signs, so the year-starting signs cycle through Reed, Flint, House and Rabbit. These four year-bearers fit the broader fourfold thinking noted in other parts of this region.
Why 260?
Nobody knows for certain why 260 was chosen. Explanations in the literature include a link with the length of human pregnancy, a link with the span of the growing season in certain latitudes, and a link with astronomical cycles. None is secure, and it is possible that the number arose from the simple combination of thirteen and twenty rather than the other way around. The question remains open.
Comparisons
Pairing two cycles to make a longer one is a general idea. The Chinese sexagenary cycle does something similar, combining ten stems and twelve branches to reach sixty. The Aztec design differs in that its two parts are coprime and the result is not a year-length count. Likewise, the seven-day week has no counterpart in this system, because the Aztec calendar had no unit of that length. The Maya, meanwhile, added a separate linear count in the Long Count. Taken as a whole, these calendar systems show how different societies used cycles of numbers to organise time.