Nick Cave and Bad Seed Warren Ellis. The opening sequence has 'Song for Jesse' playing. It's a simple melody. A voice over tells the story of our charismatic anti-hero James, Brad Pitt, before his gunning down at the hands of a young acolyte, Robert Ford. It's like a dream.
The movie doesn't look anything like a conventional Western tale of righted wrong - more like an emotional drama. The music at least is really intense and emotionally fraught. Try listen to it. Not like Morricone at all.
This music is far from generic but this score can be used in lots of movie scenes for mood and ambience. It's haunting and really beautiful.