Math problem.

[pauraque_bk]

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!

Comments

[keladryb.livejournal.com]

Is a month the same as a lunar month? That is, if the moon has a 28 day cycle, is a month 28 days?

[pauraque]

Right, that's what I mean. The number of days it takes for the moon to complete its phases.

[nopejr.livejournal.com]

Nineteen.

[pauraque]

No, no, I need a formula that will tell me that for _any_ value of "days in a year" and "days in a month".

[nopejr.livejournal.com]

...as in, what's the smallest multiple of y/m that's an integer? Or, rather, what's the lowest common multiple of y and m?

[pauraque]

No, that's what I thought at first too, but it's wrong. vvvexation, answering below, got it right.

[atrata.livejournal.com]

It'd help if I had details, but I'm not sure there's a way to write the formula you want without *any* pre-existing parameters. Are you making up your own cosmology?

[pauraque]

I wasn't sure the question was answerable either, but vvvexation got it, bless her heart. :)

Are you making up your own cosmology?

Yep.

[vvvexation.livejournal.com]

If Y is the number of days in a year, and M is the number of days in a lunar month, then the cycles will match up every D days, where D is the least common multiple of Y and M--so D/Y will give you the number of years that is. To find the LCM of Y and M, you multiply them together and divide by their greatest common factor.

[vvvexation.livejournal.com]

Oh, and you don't have to assume that a month is shorter than a year for this to work.

[pauraque]

Just as you were answering, I saw that my brother was online and decided to ask him. He got the same thing, so I guess you're both right! Thank you. :)

[sedesdraconis.livejournal.com]

It depends on how you define "match up". Since in a naturalistic setting, neither the days in a month, nor the days in a year will be integers. Example:
in 19 Mean Tropical Years (average time from one equinox/solsitce to the next of the same), there are 6939.6016 (mean solar) days; in 235 Mean Synodic Months (average time from one example of a particular moon phase to the next), there are 6939.6884 days. That's well and good, and pretty close a cycle, which is why many Earth lunar calendars are based on this cycle, the Metonic Cycle.

But it's not precise. For one thing, you probably want there to be exactly 6940 days in 19 years, not 6939.6016, in which case you've got 235.0105 mean synodic months, or off by almost a third of day from being an even number synodic months. If you use this cycle of 19 years/6040 days consistently, after 20 cycles, you'll expect 380 years, 138800 days, 4700 months.

What you'll actually get in 380 years is 138792.032 days, meaning the date you think is the vernal equinox now comes 8 days after the true vernal equinox. What you'll actually get in 138800 days is 4700.211 months, meaning the full moon will come 6 days before you expect, based on the calendar.

Also, I wrote this Luna-Solar Calculator. I f you ask it for a long count, it will give the length of the cycle that matches all cycles to an error no greater than 1%.

This was written before I spent most of this last summer on astrophysics and calendars, learning about all kinds of extra complications, but they're mostly pretty fiddly.

Does this mean we'll be getting some more ConWorld work from you soon?

[sedesdraconis.livejournal.com]

Also, I wrote this Luna-Solar Calculator. If you ask it for a long count, it will give the length of the cycle that matches all cycles to an error no greater than 1%.

I changed it so you could specify your acceptable margin of error. 1% is really rather high. But it gets really slow for values like 0.1%, depending on the exact numbers.

*math-related despair*

[pauraque]

Yeah, I know I need to get some fractions in there, I'm just trying to figure out one thing at a time.

At present, I'm working with a 354-day year and a 22-day month, which gives an 11-year lunar-solar cycle. If I make them non-integers, is it going to be possible to insert "leap days" in such a way that the 11-year matchup doesn't get thrown off? I know it would be possible to just ignore the discrepancy and use the precession as another, longer cycle, but I really don't want that.

I need to see a real-world example of this, but I can't find a calendar that works this way.

And yes, this is for conworlding. Right now I'm sort of puzzling over how to give you what I have on the writing system. It seems excessive to hand you the whole dictionary (which isn't "complete", but it is fairly large)... maybe just the base characters, with examples of how the combinations work?

Re: *math-related despair*

[sedesdraconis.livejournal.com]

If I make them non-integers, is it going to be possible to insert "leap days" in such a way that the 11-year matchup doesn't get thrown off?

Um, doubtful, but it depends.

You might want to look at what I've got for my Akalet Calendar. Is that a flavor of solution that is acceptable?

maybe just the base characters, with examples of how the combinations work?

Sounds right. I'm not really sure either. Maybe something like http://www.omniglot.com/writing/chinese.htm , more or less.

Re: *math-related despair*

[pauraque]

Something in my brain says it might be possible, because though the month is based strictly on the actual phases of the moon, the year doesn't have to start on the same solar day every year. I mean, I don't want it to steadily get further and further off, but variation within a few days is no problem. (The year doesn't start at a solstice or equinox, but at the approximate beginning of the rainy season.)

Likewise, though it would be ideal for the new moon to fall on new year's day every 11 years on the nose, slight variation is fine, as long as it doesn't precess ever further off.

Am I asking for the impossible?

Re: *math-related despair*

[sedesdraconis.livejournal.com]

Pretty much any intercalary cycle will precess ever further off, for a naturalistic set up. It's just a question of how fast. Is it okay if it adds up to a day every hundred years? Thousand years? Fifty-thousand years?

If it's a hundred years, it shouldn't be too hard, but that seems too inaccurate to be content with, to me. If it's a thousand years, or a couple thousand years, you might have to do some judicious tweaking of the exact fractions to get something that'll work, but not too bad. Fifty-thousand years, you'll either have to do a ridiculous level of precision tweaking, or else have larger intercalary cycles.

But it does sound more like some I used for Akalet would be acceptable to you than I thought at first, which raises my estimation of possibility. It would make it whole lot more possible if you allowed for a second-level intercalary cycle, of say, 2-5 (or 11, or wahtever) 11 year cycles. Having it work out the same way as it would if they intergers is ... um.

Re: *math-related despair*

[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.

More thoughts

[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)

Re: More thoughts

[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. :)

Re: More thoughts

[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?

Re: More thoughts

[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.

Re: More thoughts

[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.

Re: More thoughts

[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"?

Re: More thoughts

[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.