Wednesday, January 11, 2012

About the Mayan Calendar Part 2: Hypothetical Calendars

Any day in the year can be uniquely identified with two numbers. For whatever reason I want the number with the smaller range to go first, so let's have an order of Month-Day. The Month number ranging from 1 to 12 and the day number ranging from 1 to 31. In an ideal world that would be all of the information you needed, but there are not 372 days in the year so some of the identifications people might make with those rules, say, “2-31,” are not real days.

But say I didn't care about that, say I didn't care about the number of days in the year at all and I just wanted to make up a calendar. Say I wanted it to be 210 days. 210 is seven times thirty, so I should be able to make a calendar that looks like X-YY where X goes from 1 to 7, and YY goes from 01 to 30. I could make one, so could you.

How would you do it?

I'm guessing that most people reading would create seven months of 30 days each. So you'd have 1-01, 1-02, 1-03, all the way up to 1-30, which would be followed by 2-01, 2-02, and so on. Some people might decide to have 30 weeks instead. So it would be 1-01, 2-01, … , 6-01, 7-01, 1-02, 2-02 and so on.

I'm guessing that because our calendar is built by dividing things up at different scales. We divide the year into 12 months, then divide each of the months into days, and each of the days into hours, and so on. Switching from one cycle to another is like zooming in or zooming out. We almost never have two different things going on at the same scale.

The Long Count I discussed in the last post is also built on that kind of thinking with each unit divided up by a unit below it, used to divide up a unit above it, or both. Never were there two things happening side by side on the same scale. As I said, I don't really find that interesting.

The Mayan calendars that I do find interesting work entirely differently.

If you wanted to make the calendar I ordered above in the style of one of those Mayan calendars you wouldn't divide and then divide the divisions. You'd simply start counting seven day weeks and thirty day months. You wouldn't have one number stay fixed while the other cycled through, you'd have both numbers moving at the same time. 1-01 would be the first day, 2-02 the second. After 7-07 would come 1-08. The thirtieth day would be 2-30, and the thirty-first 3-01.

Each day still has a unique identifier, you've still got 210 days mapped out so that you can find any day in it by being told it's X-YY, but the way we arrived there is completely different. Instead of naming months or numbering weeks or something like that, you've just got months and weeks and are letting them work alongside each other to create a larger system than either makes on its own.

In this system staying something happened on Friday the 13th narrows it down to one day every seven months, because Friday the 13th comes on a regular schedule. Which means that if 210 days is the period you're concerned with, you don't have to name the months. And if you did want to name them, you could just name them after the day they start on since each month in that period starts on a different day.

We could also imagine a system where the months were exactly the same as we're used to, but the there was no leap year (Or if leap day didn't count as a day of the week that would work too.) In that case, weeks would fit together with years such that, say, Sunday the first of January only came once every seven years. It would actually fit together quite nicely because on the next year the first of January would be Monday, then Tuesday the year after and so on. The result would be a seven year cycle. Years within the cycle wouldn't need to be numbered because the year that starts on Wednesday clearly comes two years after the one that starts on Monday.

These probably aren't the best examples, in part because I'm trying to use intervals we're familiar with, a seven day week, a thirty day month, a 365 day year.

That said, I think they're at least somewhat instructive. They show how you could set things up, they also, I think, how confusing this whole thing seems. I don't know about anyone else, but I definitely can't quickly figure out if two dates are close together or far apart. It would be pretty straightforward to know what day tomorrow or yesterday is. But if you pick two random dates in the month and week calendar I described, say Monday the 1st and Friday the 13th, it isn't immediately clear to me if they're very close together or very far apart. (I had to derive a formula for converting between a week-month calendar and one where the days were numbered 1 thorough 210 to figure it out.)

On the other hand if I say the 1st of month 5 and the 13th of month 1, you've got a much better idea of how far apart they are (though do remember that there are seven months here, not twelve.) So there's definitely something to be said for the way we do things. And when I get to the Maya's actual calendars you'll see that they seem to have realized that. Unfortunately it doesn't look like I'll be getting to it today.

So I want to close this with some other hypothetical calendars just to illustrate some points.

Seven and thirty make a calendar with 7*30 days because they don't have any factors in common. If I'd chosen six and thirty it wouldn't have worked at all. The thirtieth day would have been 6-30, and the thirty-first day would have been 1-01, right back at the beginning. The six day cycle would function as a subdivision of the 30 day month and nothing more. This is, obviously, because six divides 30. There's no remainder so the six cycle is back to the beginning after 30 days.

If I'd decided that I wanted to get closer to the number of months in a year and chosen 12 instead of seven, that wouldn't have worked quite right either. It would have been a perfectly legitimate calendar, but it would have been a much shorter one. It would have been 60 days instead of the 360 you might expect if you thought there would be 12*30 days. The thirtieth day would have been 06-30, which means that in another thirty days you'd find yourself at 12-30, and the day after would be 01-01.

I haven't mentioned this, but it's probably worth noting that you can feel completely free to add dates. If X days into a calendar is A-C, and Y days in is B-D then X+Y days in is (A+B)-(C+D). And now I'm regretting using a dash, which looks an awful lot like a minus symbol, to separate the numbers in the dates. Anyway, that works. The only thing is that the number you get might be too high. As in the last example if you add 06-30 to itself you get 12-60, but you can't have a number higher than 30 in the second slot. So you just subtract a 30 and you get 12-30, which is the actual date you're looking for. (If the number in the first slot had been to high you'd subtract a 12.)

Anyway, digression over, there is in fact a rule for how many days are in cycle created by using smaller cycles. There are as many as the least common multiple of the smaller cycles. The least common multiple of two or more numbers is just the smallest positive number (zero is not a positive number) that is a multiple of all of the numbers in question. 210 is the smallest positive number that is a multiple of both seven and thirty. 60 is the smallest positive number that is a multiple of both twelve and thirty. 30 is the smallest positive number that is a multiple of both six and thirty.

So if we decided to expand our 210 day calendar by adding a 12 cycle to go alongside the 7 and 30 cycles, the result would be a 420 day calendar, as that is the least common multiple of 7, 12, and 30.

So, right now I'm going to sleep but next time I plan to get to actual Mayan Calendars that use this sort of system.


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