How about "Johnny is a Shoemaker"? If you sing this type of song unaccompanied,it usually tends to get higher and higher as the verses progress, (or lower and lower.) A good many Hymns are like this too, and you'll find that the organ player has to work the melody back to the starting note somehow.
               Another song which does this when sung without music is an Irish ballad called "As I roved out on a bright May morning, To view the meadows and blossoms gay....." That was on a record I had by Planxty.

                                                 AVATARA