George Harrison wrote the closing theme song, but the actual job of scoring the film went to Mike Moran, not a name most movie buffs have heard of—he's only got a handful of movies to his credit.
So, how'd he get the gig? George Harrison, who was delivering the money to make this thing, had one or two teeny connections in the record industry. Moran, educated at the Royal College of Music in London, worked as a composer and a session musician. He got tapped to do the job because he worked quickly and efficiently, and he didn't need a lot of bells and whistles to get the job done. For a movie operating lean and mean like this one, he was perfect.
It didn't always seem that way, though. One of the producers, Denis O'Brien, wanted Harrison to write songs for the whole thing. Gilliam didn't agree, and though there was a fight over it, Gilliam won. The score would be orchestral music instead of songs, and Moran was the guy to compose it.
Moran's score is remarkably effective: it's memorable, but it's also good at helping this production on a budget send us to different times and places. It starts with an energetic, upbeat piano and synthesizer riff, which puts us in a nice, peppy mood and gets us ready for some swashbuckling adventures.
The music shifts gears to match each of the times and places Kevin travels to while still keeping the music linked.