the album's second single, "butterfly", was one of the ballads carey wrote with afanasieff. carey described the song as the "favorite ballad she had ever written", one that was more personal than her previous work as the emotions conveyed through the song allude at just how meaningful the lyrics are to her. carey solely wrote the lyrics while afanasieff, who composed the music with dan shea, handled the song's instrumentals, and added a few personal r&b touches. another ballad carey wrote with afanasieff was "my all", written as a contrast to the album's general hip hop flavor. carey described the song as having "a lush sound and intense styling". it featured guitar arpeggios, which were synthetically created using sampling and keyboard notes. "the roof", "butterfly"s european single, incorporated fragments from mobb deep's "shook ones (part ii)", and was produced by carey, poke & tone and cory rooney.