Hi cytronicsza
I was having the same problem as you, going crazy with the fact that I can't cut all the game music off, because even if you turn the Music slider all the way down - as you said - you still get that annyoing country jingle here and there. When you do have the Music slider up, you can sometimes even hear 2 different songs playing at the same time. I was thinking how could EA have done such a massive mistake like that, but I have just finally found out what it was.
It seems that whenever you drive past a playable car - the cars parked around the map for you to jump in - this annoying song comes up, kind of in the background. Even you got your music OFF. It's the song you were referring to, with no lyrics and such.
It seems to me that the developers' intention was to give you a sign, an alert, every time you're passing near a free playable car, and the annoying music is probably supposed to be the music playing inside that playable car, from it's stereo.
When you are driving a car with no music, approach a playable car and change to it, you will suddenly hear that same annoying song going but this time from inside the car you've just left, whereas the new car you're now sitting in is silent.
Hope it helps mate. Overall, really amazing game.
