Stephen P. Morse , San Francisco

The Mayan calendar is not a single calendar, but rather a series of calendars. The two described here are the Mayan Long Count and the Mayan Calendar Round. The Mayan calendar round is itself a combination of two calendars, namely the Tzolkin calendar and the Haab calendar

MAYAN LONG-COUNT CALENDAR

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A Mayan Long-Count date is specified by a sequence of five fields. The names of the fields are kin, winal, tun, katun, and baktun. They are related as follows:

1 kin is 1 day

1 winal is 20 kin, which is 20 days (a long-count month)

1 tun is 18 winal, which is 360 days (a long-count year)

1 katun is 20 tun, which is 20 long-count years

1 baktun = 20 katun, which is 400 long-count years

1 winal is 20 kin, which is 20 days (a long-count month)

1 tun is 18 winal, which is 360 days (a long-count year)

1 katun is 20 tun, which is 20 long-count years

1 baktun = 20 katun, which is 400 long-count years

The first day in the Long-Count calendar is 0 baktun, 0 katun, 0 tun, 0 winal, 0 kin. It is written as 0.0.0.0.0 and it corresponds to 11 August -3113 in the Gregorian calendar.

Note that minus sign in the year -3113. See the discussion at the end of this page about negative year numbers.

END OF THE MAYAN LONG-COUNT CALENDAR

The maximum date that can be designated in the Long-Count notation is 19.19.19.17.19. It corresponds to the Gregorian date of 12 October 4772. That is the last date of the Mayan calendar.

The popularized date of December 20, 2012 has a Long-Count value of 12.19.19.17.19. It is not the end of the calendar but is the last day having a baktun value of 12. The next day, December 21, 2012 is 13.0.0.0.0. Such end of baktuns occur approximately every 400 years, and no cataclysmic event has occurred at the end of the 12 previous baktuns.

MAYAN HAAB CALENDAR

A date in the Haab calendar is specified by a day and a month. There are 18 months of 20 days and 1 month of 5 days, making a 365 day year. The months are

Pop, Wo', Sip, Sotz', Sek,

Xul, Yaxk'in, Mol, Ch'en, Yax,

Sak', Keh, Mak, k'ank'in, Muwan,

Pax, K'ayab, Kumk'u, and Weyeb'.

Xul, Yaxk'in, Mol, Ch'en, Yax,

Sak', Keh, Mak, k'ank'in, Muwan,

Pax, K'ayab, Kumk'u, and Weyeb'.

Wayeb' is the 5-day month.

The days of each month are numbered starting from 0 instead of 1. So the first day of the year is 0 Pop and the last day is 4 Wayeb'. There is no year indication in the Haab calendar so there is no way to distinguish between an event on a date in one year from an event on that same date in another year.

MAYAN TZOLKIN CALENDAR

A date in the Tzolkin calendar is specified by a day number and a day name. The day number starts at 1 and goes up to 13. There are 20 day names, namely

Imix, Ik, Akbai, Kan, Chiccan,

Cimi, Manik, Lamut, Mulic, Oc,

Chuen, Eb, Ben, Ix, Men,

Cib, Caban, Etznab, Cauac, and Ahau

Cimi, Manik, Lamut, Mulic, Oc,

Chuen, Eb, Ben, Ix, Men,

Cib, Caban, Etznab, Cauac, and Ahau

This is different from the usual notation of a month and day in that both the day name and day number are incremented on each new day. For example, the day after 4 Ix is 5 Men.

Since there are 20 day names and 13 day numbers, the year contains 260 days. Like the Haab calender, there is no year indication in the Tzolkin calendar.

MAYAN CALENDAR ROUND

Since there is no year indication in either the Haab calendar or the Tzolkin calendar, they cannot be used to distinguish between events that occur on the same date but in different years. But by combining the dates in the two calendars, such a distinction can be made. That is, each day can be specified by the combination of its Haab date and Tzokin date. That combination starts to repeat after 52 Haab years (of 365 days) because that is the same number of days as 73 Tzolkin years (of 260 days). Both have 18,980 days. This period exceeded the expected lifetime of a person in olden days, so it was fine for recording dates unambiguously in a person's lifetime.

The calendar can be extended to additional years by including a cycle number -- that is, keeping track of the number of times that the 18,980-day cycle occurs. As noted above, the origin of the Longcount calendar is 11 August -3113. That corresponds to 4 Ahau (Tzolkin), 8 Kumk'u (Haab), 1st cycle in the Calendar Round calendar.

Note that the total combinations of Calendar Round dates is 260 (Tzolkin days) times 365 (Haab days), which comes to 94,900. But the two calendars repeat after every 18,980 days, which is exactly 1/5 of the total combinations. So only 1 in 5 combinations are actually possible, the others are invalid combinations.

ACCURACY OF THE MAYAN CALENDAR

The Mayan calendar is said to be more accurate than the Gregorian calendar, but the argument for this is a bit flakey.Here is how the calculation is done

A solar year has been accurately measured to be 365.24218 days. Now we need to calculate the average number of days in the Gregorian calendar and in the Mayan calendar, and compare them to this value.

A Gregorian year bounces around as leap years come and go, but after cycling 400 years we can calculate the average number of days per year as follows

Total number of days =

Average number of days per year =

= 14,6097 / 400 = 365.24250

365*400

+ 100 leap days for years divisible by 4

- 3 leap day for years not divisible by 400

= 14,6000 + 100 - 3 = 14,6097

+ 100 leap days for years divisible by 4

- 3 leap day for years not divisible by 400

= 14,6000 + 100 - 3 = 14,6097

Average number of days per year =

= 14,6097 / 400 = 365.24250

Now to compute the average number of days per year in the Mayan Calendar Round. The round count repeats every 52 Haab years of 365 days or 73 Tolzim years of 260 days, which corresonds to 18,980 days. Just how many Mayan years that corresponds to is hard to say since there is really no concept of a Mayan year.

But if we wait for 29 of these round-count cycles, we get 18,980 * 29 = 550,420 days. This is 29 * 52 = 1,508 Haab years, although we have cycled through the seasons 1,507 times (550,420 / 365.24218 = 1507.00009). So if we call this period 1,507 years, we get the average number of days per year to be 550,420 / 1,507 = 365.24220. And this is closer to the true value of 365.24218 than is the Gregorian value of 365.24250.

Another (and perhaps simpler) way of looking at it is as follows. After each Haab year (365 days) the earth drifts 0.24218 days relative to the seasons. That means it will drift an entire year relative to the seasons in 365.24218 / 0.24218 = 1508.14 Haab years. During this time we will have made 1507 cycles through the seasons. Dividing the total number of days in 1508 Haab years by 1507 season changes gives 1508 * 365 / 1507 = 365.24220 days per season change.

Of course we could have made the same argument for the Gregorian calendar, and the correction is even more dramatic. After each Gregorian year (365.24250 days) the earth drifts 365.24250 - 365.24218 = .00032 days relative to the seasons. That means it will drift an entire year relative to the seasons in 365.24218 / 0.00032 = 1,141,381.8 Gregorian years. During this time we will have made 1,141,383 cycles through the seasons. Dividing the total number of days in 1,141,382 Gregorian years by 1,141,383 season changes gives 1,141,382 * 365.24250 / 1,141,383 = 365.24218 days per season change.

So the bottom line is that the corrected Mayan year of 365.24220 days is closer to the true value of 365.24218 days than is the Gregorian year of 365.24250 days. But if the same correction is made to the Gregorian calendar, we get closer yet to the true value. That is why I started by saying that the argument about the Mayan calendar being more accurate than the Gregorian calendar is flakey.

NEGATIVE YEAR NUMBERS

Before I explain negative year numbers, I need to introduce some terminology. You are probably familiar with the terms BC (before Christ) and AD (Anno Domino). These terms are very religion specific, so I will use religion-neutral terms instead. The religion neutral equivalents are BCE (before the common era) and CE (common era). Year numbers in both systems are the same -- it is just the terminology that changed.

Recall that the origin of the Long-Count calendar corresponds to 11 August -3113. You might think that that corresponds to 3113 BCE, but that is not quite correct. The problem is that the BCE/CE numbering system doesn't have a year zero. The year before 1 CE was not zero but instead was 1 BCE. That is, the years go from 2 BCE to 1 BCE to 1 CE to 2 CE etc. The correspondence between the signed years and the BCE/CE years is as follows:

year 4 = 4 CE

year 3 = 3 CE

year 2 = 2 CE

year 1 = 1 CE

year 0 = 1 BCE

year -1 = 2 BCE

year -2 = 3 BCE

year -3 = 4 BCE

etc.

year 3 = 3 CE

year 2 = 2 CE

year 1 = 1 CE

year 0 = 1 BCE

year -1 = 2 BCE

year -2 = 3 BCE

year -3 = 4 BCE

etc.

So now it's clear that year -3113 is in reality 3114 BCE.