what a great answer...
You gave some concrete strategies, but also acknowledge that a lot of counter melody (and harmony) for that matter is intuitive, not intellectual, and probably most of the folks who do it well just HEAR it inside, and their playing is an expression of that.
So I'd just add to your response that I believe that it is possible to develop that sense and ability... through a LOT of listening to the players whose sounds inspire you, and to as broad a range of others as you can...and also to learn to listen to the sounds WITHIN...to be willing to experiment and play with it until you like what you hear. Parallel thirds is the place a lot of people start when they try to harmonize... mostly because it is so prevalent in western harmonies...we hear it so much that it comes to mind easily. But it is a very limited strategy. The more you listen to other kinds of sounds, the more you can "think outside the box". IMHO
There IS no "exactly what to play and when to play it"... if there was, this would be a science and not an art.
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Brian Dean OM #32
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Phoenix Neoclassical #256
If you're gonna walk on thin ice, you might as well dance!