Hobbits: South of England folk music, including Morris dances.
Men of Rohan - The Border Ballads from the Scottish English border, bagpipes.
Men of Gondor - Roman Catholic unaccompanied choral music - e.g. Palestrina, Victoria etc.
Elves of Rivendell - Welsh folk music, harp etc.
Elves of Lothlorien - Byzantine chant (Greek Orthodox tradition)
Ents - Russian Orthodox choral music (featuring Basso Profundo)
Dwarves - Hmm. Something that suitable to be sung by deep, gravelly tuneless voices; perhaps simple chanting alternating between two notes of a minor third (rather like a police siren) perhaps?...
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Some Hobbits doing a Morris dance with clubs:
