Calendars are Cool Part 1: My Fictional World

This series will mostly be me showcasing various calendars I made, instead of talking about calendars that exist in the real world. Why? Well, there are various advantages about talking about things that exist merely at your whim:

1. I won't accidentally offend any cultures by explaining their real calendar system.
2. I can create examples specific to what I'm trying to showcase.
3. The math is so much easier when my "measurements" can be as precise as I want.
4. It's more fun!

So, without further ado, let's look at the set-up!

...Actually, yes further ado. I was watching something on TV recently with subtitles, and they were consistently spelling "without further ado" as "without further adieu". Like, it was captioned like that every time they said it (which was actually rather a lot now that I think about it). Is that... correct? Like, it sort of makes sense, and so I wouldn't be surprised if that were the original way it was written.... I'm just shaken up now. Nothing is real in this evanescent existence.

Anyway, the world set-up is pretty simple and almost disappointingly Earth-like. One year is 379.11236843710856 solar days long, and one average synodic month (the time it takes for the moon to complete its phases) is 34.30985496045678 solar days long. For our purposes, it doesn't actually matter how long a solar day is, so you can just imagine it to be around the length of an Earth day.

Notice the hilariously precise measurements. You would never find such precision in the real world because (1) it's not useful for any calculations and (2) the lengths of years and months and days actually change at a rate that would make the last few digits useless after a few hours. In fact, I'll probably update the model at some point to include this variation in my own universe, but for now I'll stick to these figures. In practice, only the first three or four digits after the decimal point matter, and that's if you get really mathy.

Also, "average synodic month" refers to the time it takes for the phases of the moon to complete one cycle. Don't worry about the specific term because it's (usually) the only type of month you see in calendars.

This set-up is going to be constant for the foreseeable future, so I'll share some observations I made after randomly generating these values.

1. The solar year length lends itself to some helpful approximation.
2. The lunar month length does not lend itself to helpful approximation.
3. The months-per-year value is very close to 11.
4. All of these values need "additive" corrections.

All of these points are discussed in more detail in Part 3 of this series, but Part 2 describes the main strategy calendars employ to come up with approximations.