First a disclaimer: I'm very weak on music theory but, if you're referring to what I think you're referring, it strikes me that Cerdd Dant in Wales is all about counterpoint. Here are examples of Arfon Gwilym - one of our great 'tradition bearers (tho I think he'd cringe at the title) - on Sain's site . Listen to tracks 3 or 5 for cerdd dant.

I think the comment, "Clearly there isn't much counterpoint in traditional music" (crazy little woman, above) is the sort of sweeping statement that leads to trouble.

leeneia, funny you should mention parakeets. When I was searching youtube for examples the best one I could find was a satire on 'formal' cerdd dant where the singer is singing about his budgie!

sian