Saturday, January 14, 2012

About the Mayan Calendar Part 4: The Haab' and the Calendar Round

Last time I talked about Tzolk'in, a 260 day calendar built on cycles of 13 and 20. I'm told that all Mesoamerican cultures had a calendar like the Tzolk'in and that it was the most important of the Mesomerican calendars. That said, I'm told that by Wikipedia and yesterday I had to correct a formatting error in Wikipedia that was screwing up a page on different calendars in such a way as to make it appear that “Poseidon” was a Mayan month. I recommend taking Wikipedia with some salt.

What I can say is that whether or not the Tzolk'in was the most important Mayan calendar, it definitely wasn't the only one they had.

The Haab' is a lot like our calendar. It approximates a solar year, it has months with names, and it doesn't require one to understand modular arithmetic to make sense of it.

The Haab' has 18 twenty-day months. There's a possibility that you might be thinking, “18 twenty-day periods? That's a tun.” You'd be right if not for the fact that it also has a five day period as well. If not for the five day period the Haab' would be just like a Long Count tun and it's months just like Long Count winals.

Anyway, it's basically just a Calendar with months like our own.  There are more of them than our months, and the first day is the Seating of [month] with the day called 1 being the second day, but other than that it's pretty much the same thing. You've got Seating of Pop (which I'm probably going to call 0 Pop from here on out), 1 Pop, 2 Pop, all the way up to 19 Pop, then you're into the next month with 0 Wo', 1 Wo', 2 Wo' and so on.

The names of the months are Pop, Wo', Sip, Sotz', Tzek, Xul, Yaxk'in', Mol, Ch'en, Yax, Sak', Keh, Mak, K'ank'in, Muwan', Pax, K'ayab, and Kumk'u. The five day not-in-a-month period was called Wayeb'. (It went between Kumk'u and Pop.) If I need to refer to them I'll probably resort to numbers, so 1,1 instead of 1 Pop, but we'll see when we get there.

18 months of 20 days makes a 360 day period, and the five extra days makes it 365 which is as close to a solar year as you can get while still using whole days. The Maya knew that it wasn't exact and were content to figure how much it drifted and work with that instead of trying to fix the problem with leap days the way we would.

This brings us to a claim I've often heard which is that the Mayan calendar more accurately reflected the length of a year than ours does. Until I was looking up unrelated things for this series of posts, I never really knew where that claim was coming from. Now I do. It has to do with how we account for drift in our calendars.

It would be completely inaccurate to say that out calendar treats the length of a solar year as 365 days. It doesn't. We know that results in things getting out of whack and if we left it that way things would get way out of whack so every four years we have a leap year to knock things back into whack. If that was all you knew you'd think that our calendar thought that the length of a year was (365*4+1)/4 = 365.25 days. That's also inaccurate. (That's the Julian Calendar, we use the Gregorian one.)

One leap day every four years knocks things too far in the opposite direction, so every hundred years we skip a leap day. Which is why 1900 wasn't a leap year. So in that case you might think our calendar said the length of a year was (365*100 + 25 – 1)/100 = 365.24 days. But it turns out that that pushes things too far back in the first direction. So every 400 years we skip skipping a leap day, which is why 2000 was a leap year after all. And at that point we say, “Close enough” and stop with all this messing with leap days.

So in the end our calendar makes a claim that 400 of our years is roughly equivalent to 400 solar years. Which is to say that it claims a solar year is (365*400 + 100 – 4 + 1)/400 = 365.2425 days long.

The Maya had a completely different method of dealing with this problem. I don't pretend to know their exact reasoning, but their result looks like they said, “Ok, next year's 1 Pop is going to be a little off from this year's, and the year after's 1 Pop will be even more off. If we don't do anything it'll drift further and further in that direction. How long until it comes back around?”

What they figured out was that every 1508 Haab's there were about 1507 solar years. So where our calendar says that one solar year is 400 calendar years over 400, their calendar says that one solar year is 1508 calendar years over 1507. 1508*365/1507 = 365.2422 (I'm rounding down very slightly.)

Now there are two things worth noting here. One is that the claim is accurate, 365.2422 is closer to the length of a solar year than 365.2425. (How much closer depends on the year.) The second is that it's not by much. The difference between the two is less than .0003 days.

I don't know about the rest of you, but for me the claim that the Mayan Calendar is more accurate about the length of a year than our own calendar is a lot less impressive when you look at it in any kind of depth.

One possible reason that they didn't use a leap day is that they wanted to be able to use the Haab' in combination with other calendars. A leap day would have messed that up (unless a leap day was inserted into every single cycle with which the Haab' might interact.) I don't know if that was their reasoning. I do know that they used it in combination.

The Calendar Round was a calendar created by combining the Haab' and the Tzolk'in. 260 days in the Tzolk'in by 365 days in the Haab' would make 94900 potential days but not all of those days can actually happen.  (Remember what I said about combining cycles in post 2.)  Only one fifth of them can. This is because 260 and 365 have a factor of five in common. The Calendar Round is 52 Haab's long, or, if you prefer to think of it in a different way, it is 73 Tzolk'ins long which is 18,980 days.

The current creation is said to have started on 4 Ahau 8 Kumk'u (that's when the Long Count started), and when that date comes back around it's a Calendar Round completion.

I have heard the Calendar Round referred to as being equivalent to our century. Not in length, obviously, but as a concept. When we talk about the '20s, the '60s, '85 and such we don't have to specify which century we're talking about because it's understood which one we mean, I think that's what people are talking about when they compare the Calendar Round to a century.

Haab' years are associated with the Tzolk'in day on which they start, so now seems like a good time to say how the two fit together.

If I just pick numbers at random I risk ending up in one of the four alternate realities with days that are impossible in this one, so I'm not going to do that. I'm told that 1 Ik' (which I've been calling 01;02) falls on 0 Pop in the real world. The Haab' year that starts on 1 Ik' will be associated with 1 Ik' (which is important for Mayan prognostication which I unfortunately know nothing about) and 1 Ik' is known as the year bearer.

There are a couple of things to note. The first is that for the rest of the year the days of the Haab' month and the days of the Tzolk'in month will stay the same. Its not just 0 Pop that will be Ik', every 0 [Haab' month] will fall on an Ik'. Until the five extra days at the end of the year knock the two out of alignment Ik' will be day 0 of every month, Ak'b'al will be day 1, K'an will be day 2, and so on.

The second thing is how the year bearers cycle. 365 = 1 (mod 13).  It is also 5 mod 20. So to find out what the year bearer after 1 Ik' will be we add 01;05 to 01;02 and get 02;07 or 2 Manik'. Do it again and we get 03;12 or 3 Eb', again and it's 04;17 or 4 Kab'an. One more time and it's 05;02 which is 5 Ik'.

In other words, the year bearers work so that the number cylces from 1 to 13 increasing by one each time and the days cycle through the four days of Ik', Manik', Eb', and Kab'an.

Calendar round days are written Tzolk'in date Haab' date. So 1 Ik' 0 Pop if anyone other than I am writing it, and possibly some monstrosity like “01;02_00,01” if I happen to be writing it. I haven't decided yet.

For the Tzolk'in I wrote up a thing on how to figure out how far apart two days are, I suppose I should do something similar with this, but that's going to wait for another time. I don't even think I'll get to it this week.

A full Calendar Round is in Long Count notation, if you were wondering.

Next time the Short Count and the very little I know about Mayan prophecy and prognostication.



  1. Sir, I believe you are missing a word (different?) in this sentence: "The Mayans had a completely way of dealing..."

  2. Right you are. That has been corrected, thank you for pointing it out.