I'm working on my setting's calendar, and I ran into a problem.
I'll give the information first, then state my issue. Hopefully someone here might be able to tell me either the formula to use, or the actual answer.
The Year is 12 (solar) months long, with an average month of 30 days, for 360 days in the year.
A week is 7 days long.
A lunar month is 32 days.
A "Cycle" begins when the lunar month begins on the first day of the week on the first day of the year. So, for argument's sake, let's say a new moon on Sunday, New Year's Day.
Can anyone tell me how long a cycle would be? Barring that can anyone tell me the formula to use?
Thanks in advance.
It's 4 years long.
I didn't do anything elegant, I brute-forced it in Excel.
Are you sure? Does that take the 7 days of the week into account? It seems like the 7 day week throws things off quite a bit, because 360/7=51.428571428571428571428571428571.
The beginning of a cycle has to be the first day of week, the first day of the year, and the first day of the lunar month.
Edit: I do appreciate your help though.
Quote from: Monster Manuel;820980Are you sure? Does that take the 7 days of the week into account? It seems like the 7 day week throws things off quite a bit, because 360/7=51.428571428571428571428571428571.
The beginning of a cycle has to be the first day of week, the first day of the year, and the first day of the lunar month.
Edit: I do appreciate your help though.
Actually, good point; I didn't take weeks into account, only 360-day years (ie, I'd forgotten that 360 isn't a multiple of 7).
If the first day of year 1 is a Sunday (First day of the week) and also lunar day 1, then the first day of year 29 is also a Sunday and lunar day 1, so the answer is 28 years.
I hate to do this to you while you're crunching numbers, but would it be easier if I just made the week 8 days long? It might make more sense to base it on the lunar month, anyway.
Quote from: Ladybird;820981Actually, good point; I didn't take weeks into account, only 360-day years (ie, I'd forgotten that 360 isn't a multiple of 7).
If the first day of year 1 is a Sunday (First day of the week) and also lunar day 1, then the first day of year 29 is also a Sunday and lunar day 1, so the answer is 28 years.
Ok. 28 years. I was hoping for a longer time frame. But thanks.
So, if I went to 8 day weeks, with 45 weeks per year, do I go back to 4 years per cycle?
I actually think that might be cool, because the moon's distance, and therefore the length of a lunar month changes at a rate of (handwavium) over the course of an Age- a truly long period of time. The moon starts out at about the distance that Mars is from Earth, and gets so close that they touch atmospheres (the End of an Age).
If the current length of a cycle is approximately 4 years, I can push the feeling that time's running out before the Age ends, and civilization is lost to the magical event called "The Fugue" which happens at the end of an Age. During this time people lose their memories and live as animals until the moon is as far away as Mars is from Earth again.
Quote from: Monster Manuel;820985Ok. 28 years. I was hoping for a longer time frame. But thanks.
Eight day weeks take it down to 4 years. 11 day weeks take it to 45 years, 13 day weeks are 52, 17 day weeks are 69.
LOL. Thanks a lot. I have a plan now and can move onto the interesting stuff.
Quote from: Monster Manuel;820984I hate to do this to you while you're crunching numbers, but would it be easier if I just made the week 8 days long? It might make more sense to base it on the lunar month, anyway.
that would make a lot more sense
Quote from: Monster Manuel;820973The Year is 12 (solar) months long, with an average month of 30 days, for 360 days in the year.
A week is 7 days long.
A lunar month is 32 days.
A "Cycle" begins when the lunar month begins on the first day of the week on the first day of the year. So, for argument's sake, let's say a new moon on Sunday, New Year's Day.
Can anyone tell me how long a cycle would be? Barring that can anyone tell me the formula to use?
For the general case, the duration of a cycle is the least common multiple of the numbers of days in a week, lunar month, and year. So, for the numbers above, it would be the LCM of 7, 32, and 360.
Reducing those numbers to their prime factors, they're [7], [2^5], and [2^3 * 3^2 * 5]. 2^3 is a factor of 2^5, so we can throw it out and the LCM is 2^5 * 3^2 * 5 * 7 days, or 10,800 days, or 28 years (as Ladybird already said).
Changing the week to 8 days shortens the cycle because 8 is already a factor of 32 (and 360, for that matter), so it gets thrown out, leaving an LCM of 2^5 * 3^2 * 5, or 1440 days/4 years.
A week of 11, 13, or 17 days lengthens the cycle because those are all prime numbers, and (by definition) have no factors in common with 32 or 360.
If you want to further extend the length of a cycle, you need to insert additional relatively prime numbers (i.e., numbers with no common factors). For example, lengthening the lunar month to 34 days ([2 * 17]) while keeping the 7-day week and 360-day year gives you 2^3 * 3^2 * 5 * 7 * 17 = 42,840 days/119 years per cycle.
Also note that 365 = 73 * 5 and 31 is prime, so using the real-world-ish numbers of a 7-day week, 31-day lunar month, and 365-day year gives you [7] * [31] * [5 * 73] = 79,205 days/217 years per cycle.
Wow. Thanks for that. A perfect explanation. I might be able to calculate the lengths of other cycles, when the orbit of the moon was longer using this information.
For what I'm doing now, I think I'm, good, but if I get ambitious later, I'll be able to work it out in much more detail.
Though the concept of a "Cycle" is not universal, I hope this thread helps others who run into similar problems.