The 1st solo song, "Leave", is about the end of a relationship where we never even meet one half of that couple. "Falling Slowly", the breakout hit number, is a love song sung by 2 people who have only just barely spoken to each other.
I think you need to read the lyrics of "Falling Slowly." It is a song that depicts exactly the scene you described - the first few lines pretty much sum it up. "Leave" also sets up the entire conflict - heartbroken bachelor coming off a devastating breakup - vulnerable, in need of attachment. It conveys all we need to know about his character to start the show. That also seems like a convention becoming of a musical to me.
Sure, Once is an unconventional musical but I think its creators sort of get the license to decide what they intended it to be. I don't think songs being diegetic have anything to do with whether or not something is a musical, as has been pointed out with virtually all the jukeboxes. The songs (by and large) are very much a part of the plot, setting, and the conveyance of the characters' emotion. Pretty sure all of that adds up to it being a musical. The score to the show is absolutely beautiful - as was the staging. The scene on the hill has to rank as one of the best and most devastating love scenes I have seen.
Scratch and claw for every day you're worth!
Make them drag you screaming from life, keep dreaming
You'll live forever here on earth.
Updated On: 10/3/16 at 10:28 PM