Changing Numerical Bases in Mesoamerican Calendrical Systems
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.
In any standard positional numeral system, a number is conventionally written as (x)y with x as the string of digits and y as its base although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)dec = 100 (in the decimal system) represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.
The vigesimal or base 20 numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten).
Among the various calendar systems in use, two were particularly central and widespread across Mesoamerica. Common to all recorded that cultures, and the most important, was the 260-day calendar, a ritual calendar with no confirmed correlation to astronomical or agricultural cycles. Apparently the earliest Mesoamerican calendar to be developed, it was known by a variety of local terms, and its named components and the glyphs used to depict them were similarly culture-specific. However, it is clear that this calendar functioned in essentially the same way across cultures, and down through the chronological periods it was maintained.
Some Mayan monuments include glyphs that record an 819-day count in their Initial Series. These can also be found in the Dresden codex.This is described in Thompson. More examples of this can be found in Kelley. Each group of 819 days was associated with one of four colors and the cardinal direction with which it was associated — black corresponded to west, red to east, white to north and yellow to south.
The 819-day count can be described several ways: Most of these are referred to using a "Y" glyph and a number. Many also have a glyph for K'awill — the god with a smoking mirror in his head. K'awill has been suggested as having a link to Jupiter. In the Dresden codex almanac 59 there are Chaacs of the four colors. The accompanying texts begin with a directional glyph and a verb for 819-day-count phrases. Anderson provides a detailed description of the 819-day count.
The second of the major calendars was one representing a 365-day period approximating the tropical year, known sometimes as the vague year. Because it was an approximation, over time the seasons and the true tropical year gradually "wandered" with respect to this calendar, owing to the accumulation of the differences in length. There is little hard evidence to suggest that the ancient Mesoamericans used any intercalary days to bring their calendar back into alignment. However, there is evidence to show Mesoamericans were aware of this gradual shifting, which they accounted for in other ways without amending the calendar itself.
These two 260- and 365-day calendars could also be synchronised to generate the Calendar Round, a period of 18980 days or approximately 52 years. The completion and observance of this Calendar Round sequence was of ritual significance to a number of Mesoamerican cultures.
A third major calendar form known as the Long Count is found in the inscriptions of several Mesoamerican cultures, most famously those of the Maya civilization who developed it to its fullest extent during the Classic period (ca. 200–900 CE). The Long Count provided the ability to uniquely identify days over a much longer period of time, by combining a sequence of day-counts or cycles of increasing length, calculated or set from a particular date in the mythical past. Most commonly, five such higher-order cycles in a modified vigesimal (b-20) count were used.
The use of Mesoamerican calendrics is one of the cultural traits that Paul Kirchoff used in his original formulation to define Mesoamerica as a culture area. Therefore the use of Mesoamerican calendars is specific to Mesoamerica and is not found outside its boundaries.
In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the usual decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A20 (the 20 means base 20), to write nineteen as J20, and the numbers between with the corresponding letters of the alphabet. This is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters "A–F". Another method skips over the letter "I", in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, and nineteen is written as K20. The number twenty is written as 1020.