See thread links above:

This FSU 'war chant' is, I believe, from the early 1900s Indian Intermezzo pop genre. It became a "riff" or entrance music for American Indians in Yank pop media.

Natoma is from classical's parallel "Indianist" school. Herbert is pretty clear he was "inspired by," but did not incorporate, American Indian melodies in his work.

Anthony Philip Heinrich (Hiawatha) and Arthur Nevin (Poia) were two fellows generally acknowledged to have spent serious time in the field but only the latter claimed to have used what he heard there, afaik.