(With music setting on) When riding a minecart (minecart must be in full motion), press ESC on your keyboard, turn off music. The new minecart sound will still be playing, even when you get off the mine cart.

STEPS:
1) Ride a minecart with your sound and music both on.
2) Press ESC on your keyboard.
3) Slide the music bar all the way to off
The mine cart sound should still be playing, even if you get out of the cart.