This achievement takes place during the mountain area on the west side (was the 3rd boss for me)
You will come across Barb near the start of this area, stood next to a small stone wall that needs the bomb attack to open it up. Talk to Barb and they'll mention how they're going to follow you through this area in order to write a song. You will come across them at various points of this area, I talked to them at each opportunity but not sure if that's required, may as well to be safe.
After defeating the Boss at the very top of the mountain, talk to them again whilst there and you'll get told something along the lines that they need time to work on this final piece of music, and to seek them at the Stranded Sailor later on.
I believe you can simply travel there and talk right away and choose the option to listen to Barb's magnum opus. They're stood outside of the Stranded Sailor restaurant where Jefferson who gives hints for finding Shiny Items or Shrines works at.
If people still need this achievement when in the post-game, make sure you go visit Barb during the day, not during the night, otherwise they will be asleep and unable to be talked to.