How is the accuracy of the Maya Calendar measured?

Question

It is taken as a fact in many books and specialized publications that the Maya calendrical system - the Haab' and the Tzolk'in - was more accurate and closer to the real solar year than the Gregorian calendar (even though reputable sources claim it is 365 days compared to 365.2425 days). But how is this really so? In all the studies I've read there is just a description of the calendars but not how that accuracy is obtained. How's that?

Update (05/04/2018): I've found multiple resources that support the premise on which this question is asked. However, I do recognize the issue of finding a relevant primary source (such as a Maya stela or a given codex). There was a discussion on Wikipedia regarding Maya expert Coe disproving this premise, however, since this question was asked, the author of this other question has edited it saying that Mr. Coe includes no references that support his statement in a section the user found. The question remains open since I'd like to compile more sources from other users.

Answer 1

The Gregorian, and indeed the Roman, Egyptian, and Chinese calendars all include occasional intercalary periods to make up the 0.24 of a solar day that is left over when using a 365-day calendar. These systems count solar years more accurately than the Mayan calendar.

The Tzolk'in at 260 days seems unrelated to the solar year; the Haab''s length of 365 days does seem related. However, in neither case were intercalary periods added. Doing so could complicate calculations with the Calendar Round. Maintaining continuity was a reason to stick with the inaccurate Haab'. According to David Bolles, in "The Mayan Calendar, The Solar - Agricultural Year, and Correlation Questions":

It is generally accepted by Mayanists today that the Mayan calendar was a “floating” calendar, in which no attention was given to keeping the calendar in sync with the solar - agricultural year. As Michael Coe in his book The Maya puts it, the Maya had “a ‘Vague Year’ of 365 days, so called because the actual length of the solar year is about a quarter-day more, a circumstance that leads us to intercalate one day every four years to keep our calendar in march with the sun, but which was ignored by the Maya.” Earlier Thompson wrote that “The Maya made no attempt to intercalate days in the count of the years to bring the year of 365 days into conformity with the solar year. Such a correction would have played havoc with the whole orderly plan of the calendar and would have disorganized the elaborate system of lowest multiples of different time cycles, which were of the highest importance for divinatory and ritualistic purposes.”

That said, the Maya calendar does seem more precise than the Gregorian one. Finding the number of days that elapsed between two dates is easier without leap days.

Answer 2

The Mayans used a 365 day year so is much less accurate than even the Julian calendar. The idea that it is highly accurate is based on a claim that they added 12.5 "leap days" all in one go every 52 years - see answer to this question.

For the Mayans anyway this would need good solid evidence to confirm it. It seems inconsistent with their system of the "long round".

52 long count years (360 days) are one Tzolkin (260 days) short of 52 Haab years (365 days, same as 72 Tzolkins)

So every 52 years the long count will move back one Tzolkin. After 72 of those then the long count will move back one calendar round and so will coincide again (after 73 * 52 long count years).

After 52 * 72 years or 3744 of the Haab years, or 73 * 52 = 3796 long count years, the whole cycle would repeat, you get the same long count, Haab and Tzolkin date as you got 3744 Haab years previously

They were aware of this cycle at least according to a preprint Chanier, T., 2013. The Mayan Long Count Calendar. arXiv preprint arXiv:1312.1456..

"In particular, on page 24 of the Dresden codex is written the so-called Long Round number noted 9.9.16.0.0 or 1366560 days, a whole multiple of the Tzolk’in, the Haab’, the Tun, Venus and Mars synodic periods, the Calendar Round and the Xultun number X0:LR= 1366560 = 5256×260 = 3744×365= 3796×360 = 2340×584 = 1752×780 = 72×18980= 4×341640.."

But how could they have the Long Round 9.9.16.0.0 for all three cycles to repeat, if they were inserting 12.5 "leap days" all in one go every 52 Haab years?

[This is a rhetorical question, it doesn't make sense that they would have the long round system if they were inserting 12.5 "leap days"]

Let's look at the effect of inserting those 12.5 days.

Then after 72 of the calendar rounds you would need to insert 72 *12.5 = 900 days.

Then you have 1366560 + 900 = 1367460 days in 72 of the Haab and Tzoltin cycles and 1366560 days in the 73 * 52 long count years. and they wouldn't match.

With those extra 12.5 days then two calendar rounds gives you 104*365+25 = 37985 days, then for the long count years to coincide you need a least common multiple of that with 360, which turns out to be 2,734,920 (using online lcm calculator).

It is 144 calendar rounds this time and 7597 long count years which would be notated as 18.17.19.0.0 instead of 9.9.16.0.0

Answer 3

You first have to drop the idea that a single 365-day unit is a calendar, because it isn't. The Gregorian calendar is a cycle that calculates an additional 97 days per 400 cycles of 365 days. (97/400+365 = 365.2425) But it's not a cycle of 365.2425 days, it's a cycle of 146,097 (400x365+97) days. The Haab and Tzolkin create a cycle that is 18,980 days long, but that's just the beginning.

So I offer an experiment:

If we go to a Mayan date calculator and convert the Gregorian date of "30 March 514" we get a Mayan date of "0 Pop 10 Ik'". Then, if we change the date to '31 March 2021' and convert again we get the same date. This is a 1507-Gregorian year 1508-Haab year cycle and they line up again because the extra bit of time not accounted for has added up to another 365-day cycle. All we need do is calculate the movement of the sun through those 1508 Haabs and we'll know when the growing season is for that particular cycle. You might think I've gone mad because it's actually a single day off, but if you add another 1507 years (3528) it'll work out to the same day as this year, 31 March.

So which calendar is more accurate? All they needed to do was predict the growing season and they seem to have done a good job of it.

EDIT: The answer to the question may have been lost so I'll attempt to give it again. After 1507 365-day cycles the bit of time not accounted for now equals 365 days, hence another cycle brings it up to par. This keeps the calendar accurate. The growing season will float through this 550,420 day cycle so they probably put it forward a few days for every 13 Haab cycles to keep it in the right place.

I don't know how to better answer the question or what it is that isn't understood.