To be clear, the music I'm referring to is the login screen music, and this http://na.leagueoflegends.com/en/champion-preview/wonder-above
I don't know what to search for, or where to look.
Nice! I love the Bard login screen music as well. When Sebastien (the composer and co-creator of Bard's voice) first played it for me I told him it reminded me of the 4-disc album by Nine Inch Nails called Ghosts. While every song doesn't quite match - there are underlying moods that feel similar to Bard's login.
https://www.youtube.com/watch?v=EyKNUj-AjgA&t=859
As for the stuff from the teaser site - those are the sounds that Bard makes in game while collecting his chimes. The only thing I can really think of off the top of my head is the ambient works of Brian Eno. I'm a huge Brian Eno fan myself and have no doubts that he's influenced me heavily over the years and decades.
