Cold Mountain is quite accurately showing a specific time in American music.

There was a great surge in popularity for church choirs and for writing choral music in the later 18th and early 19th c. in the US, especially in New England. Some of it seems to have come over from England, which had a similar surge in the 18th c. of what's now called West gallery Music. Some composers, like Jeremiah Ingalls and William Billings were self-taught, often carpenters, farmers, tradesmen.. In the earlier 19th c. there was a matching surge in popularity for church singing schools, where a travelling teacher would come and set up a class in a kind of solfege for a brief time. Hundred of different song books were published. The heads of notes were given ( in the most common system) four shapes- triangle, round, square, diamond, and four names- fa, sol, la, mi, and a major scale would be sung fa, sol, la, fa , sol, la ,mi, fa. It was easy to learn, and the music was pretty straightforward to sing, with simple harmonies: chromatics were uncommon, and the bass would usually be fourths, fifths and octaves. It was called variously shape-note singing, fasola singing, or Sacred Harp, after the most common song collection.

The theology expressed in the lyrics was rather Old School and often grim ( a good example is the song that opens Cold Mountain, Idumea, being sung behind the battle scene) and perhaps that and a taste for more modern harmonies meant that the tradition mostly died out in the Northern churches in the later 19th c., but stayed on in the South, where it was preserved into the 20th c. and then revived. That revival has been pretty big. If you go to fasola.org, you will likely find a group near you that meets regularly. Singers are no longer always affiliated to any church ( though some are) and it's great good fun to sing.

Because I kinda think shape note singing is fascinating, I just want to add some to u/Bodark43’s answer about where it came from and how it declined in New England.

It had been common in New England churches for a Deacon to “line out” part of a psalm to one of a standard repertoire of tunes, with the congregation then repeating it back, but deacons tended to be chosen based on social standing rather than musical ability, and the music suffered accordingly. From 1720 there started to be calls for singing from notes instead of rote to improve the state of church singing. Singing schools were organized to teach students this “rational” Regular Singing system, using tune books to teach new songs on a four note solfege before adding words. From Boston area Congregationalist churches the idea spread. Aside from any other objections, the fact that singing schools were coed remained an appeal / source of controversy. They were either tempting young people to dangerous levels of mirth or keeping them away from the temptations of secular music.

There was an explosion of new American compositions for these singing schools. Since there weren’t really any qualifications to teach a singing school except perhaps having attended one, the styles of the compositions could be unusual, including odd dissonances, open fifths and octaves. Around the turn of the century, shape note or “patent note” systems for tune books were developed to further aid in reading, even as singing schools were on the decline in New England. There were a number of factors behind this decline. In the late 18th century there had been a rising reaction to the styles of self-taught American composers like William Billings. As the economy recovered from the Revolution, there were also increasing numbers of immigrants and more resources for indulging in European cultural trends, both of which influenced urban music tastes.

To take an example, Lowell Mason (1792-1872) attended singing school in Boston in addition to lessons from a violinist and organist. While working as an organist at a church in Savannah, Georgia, he studied harmony and counterpoint with recent German immigrant Frederick L. Abel. He returned to Boston, becoming a prominent advocate for incorporating music education into public schooling and promoting more “scientific” harmony based on European composing. He visited Switzerland to learn about educator Johann Pestalozzi’s philosophy, which included the idea that children should be introduced to concepts in a progression from simple to complex. From these sorts of reformers, public school music education came to center on a gradual introduction of musical concepts and notation in seven note systems. With music being taught in public school, singing schools became redundant in urban centers. At the same time an increasing number of churches with organs allowed congregations to be dragged through unfamiliar tunes regardless of reading ability. Singing schools, however, still remained an important social function in rural areas, so singing school teachers shifted towards the more rural West and South. Rural areas also likely had less exposure to imported European compositions.

Shape singing songbooks remained popular sellers, with seven note shape systems eventually introduced into the mix. After 1855 no new four note tune books came out, but old ones such as Sacred Harp kept right on. In fact, it kept expanding with subsequent revisions:

Year	Pages
1844	263
1850	366
1859	429
1869	477

Buell Cobb’s book even has an excerpt from a Confederate soldier’s letter reassuring his family that they sing from Sacred Harp around the camp fire. I’m sure that was totally on the level.

Sources:

• John Bealle (1997) Public Worship, Private Faith: Sacred Harp and American Folksong

• Buell E. Cobb, Jr. (1978) The Sacred Harp: A Tradition and Its Music

• Michael L. Mark and Charles L. Gary (2007) A History of American Music Education