The song Claim Jumper has a fitting country Western sound. It's fairly repetitive, but it raises in pitch a few times through each loop and then drops back to the original pitch. Since it's the only song, the title comes from the game.
The song starts when you boot the game and continues as you start playing, but it stops abruptly just after 3 minutes in. It is rendered on the Atari's POKEY chip.
The game doesn't have audio credits, but the VGMPF was contacted by Gray Chang and he had this to say about the song: