Cut, but not until the root. That would be compassion. There has to be a few pieces left, so it doen't stop hurting, so you don't stop remembering. He gets out of your life and conversations, but not from your memory.
That music will always make you remember him, so will that place, that nickname, that smile.
You really don't know what happened but, in the end, there was nothing, only this bitter memory trapped inside you.