F-sharp vs G-flat in just intonation

They were not always the same note

As I gear up for teaching music theory in the fall, I'm still refining my explanation of Western music's arcane naming system for enharmonics. Why is the note between F and G sometimes called F-sharp and sometimes called G-flat? Why do we sometimes call the interval between that note and C an augmented fourth, and sometimes call it a diminished fifth? What difference does it make if they sound the same?

I had a major "aha" moment when I learned about the history of Western tuning systems, and found out that F-sharp and G-flat were originally two different and non-interchangeable notes. I have enjoyed seeing that same "aha" look on my students' faces when I explain it to them. But tuning systems are hard to understand, and my explanation still requires a lot of refining. This post is one in a series of iterations.