Calendars are Cool Part 3: Approximately Correct

In the previous post, I tried to show you the super tool behind any good mathematical calendar: continued fractions. Because of how crucial this concept is to the rest of calendar-making, I highly recommend that you have a feel for calculating and understanding continued fractions in general. I admit that my own introduction is probably not the best written, so you might want to find some outside sources to help you if I've left you perplexed. Wikipedia has a pretty good page on continued fractions.

Now, let's revisit our set-up and look at the continued fractions of each important value:

1. The solar year is the number of days it takes our planet to circle its sun.
• It is 379.11236843710856 days long.
• The first few terms in its continued fraction expansion are 379, 8, 1, 8, 1, 13.
2. The average synodic month is the number of days it take our planet's moon to go through its phases.
• It is 34.30985496045678 days long.
• The first few terms in its continued fraction expansion are 34, 3, 4, 2, 1, 1, 46.
3. The number of months per year is the quotient of these first two values.
• There are 11.049663977712754 months per year.
• The first few terms in its continued fraction expansion are 11, 20, 7.

I chose to stop the continued fractions at the number that would make the denominator too big to work with. This is a subjective decision, and you may set the bar lower or higher than mine.

In part 1, I made a few claims about these three values. Let's revisit them now that we have the relevant math.

1. The solar year length lends itself to some helpful approximation.
• Remember how higher numbers appearing in a continued fraction mean more accuracy, but also more complex denominators? The solar year has an excellent blend of high and low values that permits decent accuracy and a decent tradeoff for denominator sizes.
2. The lunar month length does not lend itself to helpful approximation.
• The sequence of values after the 34 are all low (until the 46), which means each successive approximation is only slightly better; the tradeoff for higher denominators isn't really worth the small improvements you get.
3. The months-per-year value is very close to 11.
• This is important! Sometimes the best approximation is an integer one. Also, it may encourage increased formation of lunisolar calendars. (More on that later.)
4. All of these values need "additive" corrections.
• I distinguish between "additive" corrections and "subtractive" corrections. Additive corrections involve adding days in calendars, and subtractive corrections involve removing days from calendars. The decimal parts of all three of these numbers are less than 0.5, so they're closer to their lower integer neighbor than their upper neighbor, making additive corrections necessary. (More on this later as well.)

This general overview can help you determine what calendar systems might arise naturally in your world, and which would be more common. Generally, things that are easier to approximate appear in higher frequency in calendars. In this set-up, solar calendars and lunisolar calendars seem to be favored, since both values are more easily approximated than the lunar month. However, the lunar calendar is scarily close in length to a solar calendar, which could mean that lunar calendars are common after all. It's really up to us, at this point!

In the next part, we'll finally look at a calendar I made using this system.