I can sympathise with having a persistent song in your head. Every single waking second, I HAVE to have some tune or rhythm in my head. The only time there isn't is if I'm already listening to music, or sometimes watching tv if it really has my attention. For years and years I've had a small number of tunes/songs that have been my constant companions, mostly annoying little tunes that make people laugh when they hear me singing it under my breathe (if I'm not singing, then I'm humming it, tapping my feet or hands to it, or even grinding my teeth to the rhythm). Ode to Joy, You Are My Sunshine, the Smurfs theme song, the chicken dance tune, a particular tune from the old Mario Bros game. Some tunes come and go for months at a time, but those ones listed are pretty damn persistent.
I don't think I know what it's like to have a quiet, non-musical mind.