The story, as it's usually told
In the version Lafcadio Hearn recorded, the snow woman kills an old woodcutter in a blizzard but spares his young apprentice, Minokichi, on condition he never speak of her. Years later he marries a pale, beautiful woman named Oyuki and raises children with her. One night he tells her the story, and she reveals she was the snow woman, spares him for the children's sake, and disappears.
What the song hears
Not a demon. A woman who chose the snow after ten years of choosing him.
