That page I linked to has 2 books that discuss the music & the way it's done. One is Benjamin Lloyd's Hymn Book: A Primitive Baptist Song Tradition At $18 for the softcover, it's worth the purchase just for the CD.

The second is The Sound of the Dove: Singing in Appalachian Primitive Baptist Churches, by Bev It's more expensive, but being from a university press you'll have an easier time getting it through your local library.

Part of the structure is that even if the tune comes from a composed source, the note values are stretched way out & ornamentation is added. Most of these churches were small & centered on just a couple of families. The singers really learned to sing with each other to such a degree that they might not be able to sing it very well with others, even though to our ears it is quite similar. Patterson's book in particular covers it.

Note wise it's probably what often gets labelled 'mountain modal.' The intervals are not quite what you get in a 7 note diatonic scale. More likely to be pentatonic, but then 3rds might be somewhere between minor & major. Lots of sliding through notes without ever quite landing on one as well. It's hard to notate.