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a The 260-day cycle (Tzolkin) consists of 20 day names, 13 numerals (read across, then down), each sequence continuously repeating. These sequences run simultaneously with the numbering of the days in each month, which are indicated below the Tzolkin. The month name stays the same, however, until the sequence of 20 numerals of month days is completed. Thus, the date 7 Ahau 18 Zip is followed by 8 Imix 19 Zip and then by 9 Ik "end" Zip (or 9 Ik "seating" Zotz). A spreadsheet sequence of a few years of combined Tzolkin and Haab cycles is given in the Appendix.

a The 260-day cycle (Tzolkin) consists of 20 day names, 13 numerals (read across, then down), each sequence continuously repeating. These sequences run simultaneously with the numbering of the days in each month, which are indicated below the Tzolkin. The month name stays the same, however, until the sequence of 20 numerals of month days is completed. Thus, the date 7 Ahau 18 Zip is followed by 8 Imix 19 Zip and then by 9 Ik "end" Zip (or 9 Ik "seating" Zotz). A spreadsheet sequence of a few years of combined Tzolkin and Haab cycles is given in the Appendix.

possible that each tribe named the year from the day of the festival of their god. With available data, this idea cannot be checked for most groups. The meshing of the 260-day cycle with that of the 365-day cycle has been likened to a meshing of gears, but such an analogue, although useful to us in a modern, industrialized society, should not be interpreted as

Table 12.5. The long count component of the Mayan calendar. BB.KK.TT.UU.DD, where

DD = KINS (days) UU = UINAL = 20 KINS TT = TUN = 18 UINALS = 360 KINS KK = KATUN = 20 TUNS = 360 UINALS = 7200 KINS BB = "BAKTUN" = 20 KATUNS = 400 TUNS = 7200 UINALS = 144,000 KINS

the way the Mayans thought of the interrelationship. The 52-year cycle was a bundle of years for the Mayas, and for the Aztecs, a binding of the years, a final gathering together of units of time like the tying of sheaves.

The third component of the Mesoamerican calendar, the Long Count, noted in Table 12.5, is known from a few very early monuments in the Olmec and Izapan styles and from the Mayas, but it is not part of the calendar in most other areas. It is a count from an era base, given in repeated intervals of time that have elapsed from a certain date in the remote past. This date was far remote even at the time of the earliest dated monuments. The periods of time normally used in this (or other) era counts were

(1) 144,000 days, usually called a "baktun" or Cycle and composed of 20 smaller units of 7200 days each;

(2) 7200 days, the katun, composed of 20 units of 360 days each;

(3) 360 days, the tun, composed of 18 units of 20 days each;

(5) the day, or kin—the basic unit of the long count.

The term baktun does not come directly from the Maya, but it is a logical extension of the system names by scholars; we emphasize its artificial origin here and in Table 12.5 to make clear the distinction between it and the other names. Elsewhere in the text, the quotation marks will be omitted.

Longer units were occasionally used as well. The modern analog of this calendar component is the Julian Day Number. Now, we will describe the Mayan number system and its representation.

All numbers may be written with only three symbols: a dot for 1, a bar for 5, and a shell for 0. The shell marked the completion of a series, not of 10, as in our system, but of 20 (except in the second position, where 18 units completed the series). Hence, the statement 18 + 3 = 21 would be written as

In such a case, the 18 and 21 would normally be written in red, and the interval (3) in black. The place positions in a date of the Late Classic Period of the Mayas might be written as

If one day is added to this date, it becomes

This has been called a modified vigesimal system, in which 18 units of the 2nd position (from the right) become one unit of the 3rd position, but all other changes are of 20 units becoming 1 unit of the next position. It has also been argued that the last two units are not "really" part of the system but should be considered a separate count of days (Closs 1977). Although some mathematicians may be troubled by this irregularity, there is little practical difficulty in reading numbers, for the days are included in the count without differentiation and 359 days (written: 17.19) is always followed by 360 days (written: 1.0.0). Finally, the only calculations we have deal with calendrical matters; it has been suggested that the Mayas avoided this irregularity in other forms of counting, but there are no data to support this. The Mayan codices normally contain dates and intervals written in a pure place value system, but the monuments usually include glyphs for the names of the periods.

Historically, the most important era base was a date 4 Ahau 8 Cumku, which was both the zero date of a new era and the completion of 13 baktuns from a previous era base. This base was normally written

For a long time, this was thought to mean that counts began over when they reached 13 cycles. This is untrue. It is more comparable to our practice when we say that the year 753 a.u.c. (era of the founding of Rome) was followed by the year 1 a.d. (of the Christian era), shifting from one era base to the other (cf. 4.1.5). No one would suggest that every time 753 is reached in an era count we start over:

In Figure 12.4, we show a date as it may appear in the inscriptions, written as 9 cycles or baktuns, 16 katuns, 4 tuns, 10 uinals, and 8 kins or days. Scholars usually transcribe this

Figure 12.4. A date, 9.16.4.10.8 12 Lamat 1 Muan, as it may be written in the inscriptions (left) or in the codices (right): The first part of the date is read, 9 cycles or baktuns, 16 katuns, 4 tuns, 10 uinals, and 8 kins or days. Drawing by D. Zborover.

as 9.16.4.10.8. We also show this date as it may appear in the codices, written with a place value notation as

Like all other dates, era bases such as 9 Kan 12 Kayab and 4 Ahau 8 Cumku recur every 52 years. Thus, for example, 9.9.16.0.0 4 Ahau 8 Cumku was a tun ending during the Maya Classic Period. Fifty-two Mesoamerican years later, it was 9.12.8.13.0 4 Ahau 8 Cumku. Such counts from the era base are normally referred to as Long Count dates. A parallel system has been called the Short Count. In its minimal form, this simply referred to an event as happening in a certain tun of a certain katun named for its ending day, for example, "in the 10th tun of (katun) 4 Ahau." Because Ahau can be preceded by any of 13 numbers, such a date will recur every 260 tuns (of 360 days each) in the katun order: 11 Ahau, 9 Ahau, 7 Ahau, 5 Ahau, 3 Ahau, 1 Ahau, 12 Ahau, 10 Ahau, 8 Ahau, 6 Ahau, 4 Ahau, 2 Ahau, 13 Ahau, 11 Ahau, 9 Ahau, and so on. Now, a full calendar round date recurs only at intervals of 52 Mesoamerican years (of 365 days each); so the 5th repetition of the calendar round will recur every 260 years (5 x 52 x 365) but will be off from the tun repetition (13 x 20 x 360) by 1300 days. Hence, a Short Count date, including the CR such as 6 Cauac 17 Zip in the 20th tun of katun 7 Ahau, implies the Long Count date 9.19.19.17.19 6 Cauac 17 Zip and will recur only after 18,980 tuns. This method of dating was used in inscriptions at Chichen Itza.

For a fuller discussion of how to calculate in Maya, see Sanchez (1961). This book shows how easy it is to do multiplication, division, addition, and subtraction using the Maya system, although some scholars have denied that the Maya used these simple procedures. There is, however, no evidence of the use of fractions. The Maya equivalent was to give calculations spanning a very long period of time. Whereas we may say that an average synodic period of Venus was 583.920 days, an equivalent Maya statement would be that 1000 Venus periods contained 583920 days. Unfortunately, working out Maya statements of this sort is not as simple as this example may suggest, for we usually have only the results of the calculations and these may include other factors such as reaching the nearest Venus date at a spring equinox or something of this sort.

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