pauraque_bk | Math problem.

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I need a formula that will tell me, for a given number of days in a year and a given number of days in a month, how many years it will take for the cycles to match up -- in other words, if we start with the new moon on the first day of the year, how many years until the new moon falls on that day again? (Assumption being that a month is shorter than a year.)

This is driving me crazy. Left-brainers, please help!

Current Mood: oh dear, maths.

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From:

sedesdraconis.livejournal.com

Oh, is variation about when the year starts, up to half the lunar cycle fine? 'Cause the only way I know to go about this is to vary the number of months in the year back and forth.

In this case, most years having 16 months, which moves the beginning of the year forward ~2 days each year, up until it's 10 (or 12) days sooner than it should be, then you have a year with 17 months, which sets it back to 12 (or 10) days later than it should, until it gradually catches back up again, and then starts over.

From:

sedesdraconis.livejournal.com

Some more thoughts.
1. Do the users of the calendar measure the year in any fixed way? Relatedly, how formalized is this calendar? Because you could just define the beginning of the year as the first new moon after the rainy season starts, or something similar. They wouldn't ahve to count the number of days in a year at all. You wouldn't have an 11 year cycle, though, because you wouldn't have any kind of measured cycle.

2. Is the 11 year cycle the inportant artistic point, or is it the length of the year and month. If I made a more complicated, reasonably stable 11 year cycle by changing the month length around by a few days, would you be interested?

If I did this, do you want formalized counting of the number days in a month, or is it observational; .e. the month starts with the first sighting of the new moon, whatever, regardless of how many days its been.
Does the year always begin at the beginnning of a month, or are the parallel runnning cycles that come back into alignment after 11 years, more like the relationship of our weeks to our months or years, than the relationship of our months to our years.

(I like calendar work :D)

From:

pauraque

Precessing a day every thousand years or so would be fine.

It doesn't have to be 11 years, though I wouldn't want it to be _too_ many years fewer or more than that. Same goes for the number of days in a month. (However, I'm pretty attached to 354 days in a year, though obviously I haven't decided on an exact fraction to work with.)

And no, the year definitely doesn't always start at the beginning of a month. It's as you said, more like the relationship of our weeks to our months.

Do the users of the calendar measure the year in any fixed way?

You mean fixed as in fixed to the actual solar year? No, not the way I've got it worked out now. The solstices and equinoxes would fall on different days in different years, as they do for us, but they'd notice if it changed significantly over time.

I could do a less formalized calendar, but I've done that before. The whole point of this is to do something more complicated. :)

From:

sedesdraconis.livejournal.com

And no, the year definitely doesn't always start at the beginning of a month. It's as you said, more like the relationship of our weeks to our months.

Oh, that's much, much easier. I was assuming the other way. Do you want to make something up for you, give a spreadsheet with some parameters set, that'll calculate it, or anything?

From:

pauraque

Ah, I guess I wasn't clear enough... The first day of a month is always the new moon, so if the new moon only hits new year's day every 11 years... yeah. :)

If you could make an excel spreadsheet that shows how the calculations work, I would be very grateful.

From:

sedesdraconis.livejournal.com

Sure. One thing I'm still unclear on, though. Is the new month based on the observed, or the calculated new moon?

If it's the observed new moon, that's just, someone looks up into the sky and says, "it's a new moon, I guess it's the next month now."

If it's a calculated new moon, then you've got a calendar that tells you when the new moon will be, and you'll have a leap-year like cycle of months, e.g. "most months have 24 days, but 3 months out of 10 have 25" or whatever.

Speaking of which, you probably want some kind of independent cycle of months. That is, 6 or 10 or 13 or whatever month names that you cycle through, like weekday names, yeah? The number of names / number of months in the month-cycle should probably be based on how the months get systematically corrected, i.e. in the example above, a cycle of 10 named months, and the months named "such", "so", and "t'other" have 25 days.

Ahhh. There's so many choices, and I don't know how to convey them all! But, anyway, this spreadsheet:
http://www.sedesdraconis.com/stor/calencalc.xls
you can use it to pick a year length and month length, and it won't shut down (m?)any options for the other pieces.

The documentation is in the spreadsheet, I tried to make everything crystal clear, I apologize if it's too patronizing, or if it's not clear enough.

From:

pauraque

When you "lose" a day in 1000 years, what do you do, add a day to the calendar or subtract one? And what are the "extra months"?

From:

sedesdraconis.livejournal.com

If you "lose" a day, you have to add it back in to the calendar. That is, "losing" a day means you're calednar year is slightly shorter than your solar year, and the difference has managed to accumulate up to a day, so that now the solstice (for example) is happening one calendar day too soon. So you add a calendar day, and that fixes it.

"Extra months": so you've set up the # of years in a cycle, and you've set up the approximate number of months in a year. Multiplied together, that'll give you an approximate number of months in a cycle. But if you just use that number, you wouldn't need a cycle, because there'd be an exact number of months (well, offset slighlty, cause of the fudge in year legnth, but not offset enough) in the year, whatever you put in. So you fudge it with the "extra months" cell.

So for example you've got an 11-year cycle, and about 14 months in a year. That makes 154 months in a cycle if there're (nearly) exactly 14 months in a year, but we don't want that. So you put in, say, 4 in the extra months field, so that there are 154+4=158 months in a cycle, or 14 and 4/11 months per year.