Claw Finger is a song from Silent Hill. The song is track 10 in the Silent Hill soundtrack and was composed by Akira Yamaoka. It plays during Harry Mason and Dahlia Gillespie's first encounter in the Balkan Church.
The song's namesake may be that of the Mumblers, although apart from the name, the two aren't closely associated, as Mumblers aren't found anywhere near the church.