Do you only get FCs if you improve your score on a song on top of having no misses?

[dancinjanssen]

Hi everyone. I'm pretty new to this game, just joined about a month ago, so I apologize for the newbie question if this has been answered before. I've already FCed a handful of songs, but I've played many others with no misses but didn't beat my best score, and I didn't receive an FC for them. Then of course if I play it again with no misses and beat my score, then the FC shows up. Is that just another requirement for FCs that I didn't know about?