Music has always been one of the most pivotal elements of the Middle-earth universe, both in the books and the movies. Ever since the first chapter of The Hobbit, songs have been frequently used as storytelling devices to set tone, to give exposition, and even to develop many of the characters and cultures in Tolkien's tales. What's interesting about their use in the books, as Tolkien points out in his lead-in to the very first song, is that they have no actual music to accompany them in that medium. This of course leaves the door wide open for how readers can choose to hear them and has led to several different interpretations in various adaptations.
One of the best renditions of any Tolkien song is probably the song "Far Over The Misty Mountains Cold" as it's performed in An Unexpected Journey, the first of the three Peter Jackson Hobbit movies.
The song is an account of how the singers, a company of thirteen dwarves, lost their mountain home of Erebor and all of the treasure inside of it to the dragon Smaug and how they intend to reclaim it from him. Only two of Tolkien's verses are sung in the film, but the melody composed for that version is present all throughout An Unexpected Journey as something of a theme for both the company and their quest. Even the movie's end-credit song "The Song Of The Lonely Mountain" is built around the tune of it. It's about as perfect as movie themes come, being flexible enough for just about any emotional context while always creating atmosphere and hearkening back to the source material, and it's one of the most memorable pieces from the first film's soundtrack.
So why don't we hear it anywhere in the second or third film?
It's not as if the story changes in those installments. The main plot still revolves around the dwarves and their quest, so "Far Over The Misty Mountains Cold" would still be a fitting theme to use. No official explanation has ever been given for its disappearance from the trilogy, at least none that I've ever found, and this has created almost an air of mystery around the piece for fans. Seeing how the Extended Edition of An Unexpected Journey recently made its debut in theaters, this is perhaps a good time to explore this issue again and try to draw a conclusion.
Looking at the context of the song's initial performance, we see the dwarves singing it together in the home of Bilbo Baggins, the story's main protagonist, while Bilbo eavesdrops on them from another room. This is the last that Bilbo hears of the dwarves before waking the next morning to find them gone and deciding to join their quest, which at the time was being described to him as "an adventure." An Unexpected Journey is by far the most lighthearted of the three Hobbit films, so it's possible that "Far Over The Misty Mountains Cold" is being used to convey the excitement and fun of the quest from Bilbo's point of view before things grow more serious in the next installment.
If that is the reason though, then why isn't the "Far Over The Misty Mountains Cold" theme replayed at a later point in the trilogy to remind viewers of those better times? The theme for the Shire where Bilbo lives, for instance, is played in all three films for that exact reason. In addition to providing emotional throwbacks like that, the act of reprising old themes in later soundtracks is a good way to unify the music of an entire film series. Dropping a major theme one third of the way through a trilogy can feel unbalanced to a lot of listeners, as this trilogy has proven.
However, giving it the title "Misty Mountains" is an inaccuracy. The song was only called "Far Over The Misty Mountains Cold" originally because that's the first line in its lyrics. The song isn't really about the Misty Mountains; it's about the Lonely Mountain of Erebor. The fact that the end-credit song built around its melody is called "The Song Of The Lonely Mountain" only drives that point in deeper.
Furthermore, the tune of "Into The West" isn't introduced until the third and final Lord of the Rings film, so its use in only one film works fine in that trilogy. The filmmakers didn't introduce it and then exclude it from any sequels. Even if they did do that, "Into The West" is used sparingly throughout The Return of the King's soundtrack and isn't associated with any particular characters the way that "Far Over The Misty Mountains Cold" is, so its disappearance might not have been noticed had it dropped from The Lord of the Rings before the third installment.
A third possibility for the disappearance of "Far Over The Misty Mountains Cold," and probably a more realistic one, is that time may have been simply working against Howard Shore while he was composing the soundtracks for the Hobbit film trilogy.
It may be news to some people that the song's melody in the Hobbit movies wasn't actually composed by Shore. It was composed by Plan 9, a company in New Zealand that creates music for film and TV. Having helped with the soundtracks for many of Peter Jackson's other films, including The Lord of the Rings, Plan 9 was approached to write a tune for the song that the dwarves sing in Bilbo's house. Long story short, Howard Shore loved what they came up with so much that he worked its melody into the score that he had written.
What isn't news to people is that the Hobbit trilogy had a very problematic production from the start. The temporary change in director, constant pressure from the new studio, and the evolution from two movies to three partway through shooting made things so difficult for the filmmakers that it really is a miracle that the films turned out as good as they did. Is it possible that in the midst of all of these setbacks, demands, and deadlines, Shore simply didn't have time to work the "Far Over The Misty Mountains Cold" theme into the later soundtracks?
Looking at all of the above mentioned reasons, it's likely that every one of them played some part in the disappearance of "Far Over The Misty Mountains Cold." Maybe when his workload suddenly increased from two to three soundtracks, Shore decided to limit the theme to the first movie for the other discussed reasons and it just never occurred to him that people might miss it later.
Things may have turned out very differently had the production of the trilogy gone smoother, but that's not at all to say that the music replacing that theme in the later installments is bad. Howard Shore, Peter Jackson, and everyone involved in the Hobbit movies did the best job that they could with a much less favorable situation than they had with The Lord of the Rings, and they should be admired for that. The fact that Shore put "Far Over The Misty Mountains Cold" in his score at all when he didn't have to shows that his heart was in the right place.
Perhaps in hindsight, we can look on that theme's use just in An Unexpected Journey not as a loss, but as a musical equivalent of the Arkenstone in Erebor's treasure hoard; as a rare gem that should be remembered fondly but not obsessed over.