I would begin with natural or unnatural, and eliminate the unnatural desires.
I'm not sure if the labels of "natural, necessary, unnecessary, unnatural" are workable for me. I like to use more words to explore things, and here is an example:
Comparing what Godfrey wrote with what Kalosyni wrote, I expect most of us are going to agree on the benefits of both methods of analysis.
The differences appear to me to come out with Godfrey's "I would begin...." because the point of division seems to be mostly the question of where to start.
Do you start with hedonic calculus analysis, or do you start with natural and necessary analysis?
No doubt some will want to say that it doesn't matter but for someone who is new to the philosophy or who just wants to be intellectually rigorous, it will matter.
In fact, a major difference came out last night, in that one view is that if we can determine that a desire is neither natural nor necessary, then there is no reason to consider evaluating that action according to the hedonic calculus. That's because it is the point of view of some that we would NEVER under any circumstances pursue a desire that is neither natural nor necessary. In our meeting last night we had specific disagreement on that.