George Duning composed the score for From Here to Eternity, which earned him an Oscar nomination. He was a storied composer who ended up being nominated for Oscars five different times, including for Picnic (1955). Duning also did music for tons of movies and TV shows (and music which was subsequently reused for lots of TV shows), with his work being frequently uncredited.
But the most memorable musical moment in the film is probably a song that some of the characters actually play: "Re-Enlistment Blues," which was written by James Jones (the author of the novel, From Here to Eternity), Fred Karger, and Robert Wells. The character who sings the song in the movie, "Sal Anderson," is actually played by Merle Travis, a famous Country guitarist and singer from the era.
The song itself is a bluesy lament about a soldier who's left the army but doesn't like life outside and ends up reenlisting. "Got paid out on Monday / Not a dog soldier no more / They gimme all that money / So much my pockets is sore / More dough than I can use… Re-Enlistment Blues!" The tune is also referenced in Dunning's score.
Also, the soldiers entertain themselves with music that features in the movie—the group who plays "Re-Enlistment Blues" lets Prewitt jam with them on the bugle. Plus, we see Sgt. Judson playing piano (badly) in The New Congress Club, which ends up starting a fight with Maggio.