Calendars are based on three frequencies: the rotation of the Earth on its axis, the rotation of the moon around the Earth, and the rotation of the Earth around the sun. Calendars are complicated because none of these periods is a simple multiple of any other. The ratios are certainly not integers, but they’re not even fractions with a small denominator.
As I wrote about the other day in the context of rational approximations for π, the most economical rational approximations of an irrational number come from convergents of continued fractions. The book Calendrical Calculations applies continued fractions to various kinds of calendars.
Ratio of years to days
The continued fraction for the number of days in a year is as follows.
This means that the best approximations for the length of a year are 365 days plus a fraction of 1/4, or 7/29, or 8/33, or 23/95, etc.
You could have one leap day every four years. More accurate would be 7 leap days every 29 years, etc. The Gregorian calendar has 97 leap days every 400 years.
Ratio of years to months
If we express the ratio of the length of the year to the length of the month, we get
By taking various convergents we get 37/3, 99/8, 136/11, etc. So if you want to design successively more accurate lunisolar calendars, you’d have 37 lunar months every 3 years, 99 lunar months every 8 years, etc.
This was a problem in Euler’s calculus text. Have you been reading it?
No, I haven’t seen that. I’ve been reading Calendrical Calculations. I’m supposed to write a review of it and just started reading it.
I always knew that the Jewish lunisolar calendar has a 19 year cycle in which there are 7 leap years of 13 months, leading to 235 months over 19 years. It’s fascinating that that is the best possible estimate for which the cycle takes less than centuries. Thank you for the explanation.