At one point Kiranavali hummed a raga (hummed so as not to introduce confusing elements like swarams or tadarinana syllables) highlighting the special gamakas from the plain notes, which prompted Jill to remark "NOW I get it".
That would have been interesting to hear, too bad they cut that out ( on a lighter note, they played just the brief alapana portion of Shashank's Orajupuchu and then announced "That was orajupuchu in Kannada Gowla"

. May be that is a result of a final cut as well or they are just reading the liner notes ( alright, I am now picking nits ) )
I see what Varsha is getting at. The internal experience is something very hard to describe in words to others and that experience itself is a changing one over time.
Thinking this through based on the posts above, there seems to be a few different dimensions to the description. My hope and thesis is that it should be possible to describe it to someone musically knowledgeable like Pasternak. Making one experience it a much tougher assignment. ( may be that is a distinction with out a difference, but we can try ).
The working hypothesis is this: How can one resist exploring something beautiful that has a shape and emotional content. In other words, a personality. How can such a message not resonate with people who love music and art? Granted, the shape and emotional content are fungible and the boundaries of each are a bit fuzzy. But these are definitely within the grasp of any listener, Indian or Non-Indian.
With that in mind, I think there are these dimensions.
What is it?: a) It is in the realm of melody. OK that is easy. Done b) Then the next baby step: It has an identity of its own. Wow..hold on now...Well, that sounds easy but we have to mull over it. Hopefully that is a communicable and transmittable idea. People nod quite readily at that but it is hard to tell if they are relating to this amorphous identity properly. So if someone nods to that concept too readily, watch out! i would rather someone challenge me on that to explain more and engage in a discussion as to what it is.
How does that identify come about (technically)? : Here Uday's talking points on Constraints and Reinforcements is useful. I like that definition very much for its preciseness and directness. May be it is a bit technical and rigorous for most people but it should strike a chord among those who think along those lines. Constraints indeed define the form ( in many human and nature's endeavors, not just music, so it is a positive word and an organizing principle we see repeatedly ). Reinforcement of the form is obviously necessary to establish that identity. Methods like Kiranavali's humming of plain and gamaka notes definitely helps.
The 'what' and 'how' are closely related and are probably not separate but some people like to approach them separately since otherwise it may be too daunting.
Why raga? : This is I think this is the dimension Varsha and Rsachi are focusing on, rightfully so. This is monumentally important since it is about musical aestheticss. This is too fuzzy for new comers, especially for those who do not have the cultural context. I think we need to deal with it in small dosage initially. There are two potential dangers which need to be considered 1) if one makes it too fuzzy as CM folks, and artistic folks in general, tend to do, it will turn them off. It is already abstract enough. 2) And on the other hand, for those who are amenable to working with such fuzzy things it may not convey anything useful since any art form of course has aesthetics and emotional component. Personally I see this one too many times: A point that needs to be made in a clear and specific way gets smeared into a kind of generality that befuddles the newcomers. But that may be the non-artist in me speaking. In any case, I think it has to be specific enough, not too mumbo-jumbo, not too technical and not too exotic. I do not know the right balance is but this can be adjusted depending on context. But I think this part is important in the overall mix, in the right proportion.
That is what led me to rhetorically ask at the beginning how any one can resist getting to know an amorphous aesthetic thing that has a definite shape and emotional personality. A contradiction in terms alright - amorphous thing with a shape - but that is the crux of the matter. That leads me to the last dimension, the mechanics of how one goes about establishing a relationship with such a 'musical personality'.
How does one grasp/get that identity so you can recognize it when it is standing in front of you ( so to speak )? Now this is the mechanics that the listener goes through. Listen to music in a guided manner, attend concerts a few times, may be someone sitting next to you can guide you through a little bit, and what Srikant suggests in terms of playing songs in the same raga and see if they find the identity through similarity etc. As a case in point,
Sri. Nageswaran asked in another thread about identifying the raga based on a little bit of tuning by the violinist Sri. Narasimhan. Mahavishnu replied correctly in that thread. But that is qutie tough since there is just that mere suggestion of the raga there and I am not 100% sure if I would have gotten it. I tried that as an experiment with a few of my friends who are not that much into CM, but like music and know old film songs well. I asked them if they can relate that little tuning clip to a famous film song from yesteryear. I for sure knew that they know the song I have in mind. That did not work, they did not get it. I then picked out 20 seconds of a Shashank alapana in that raga ( in fact from the Sruthi Philadelphia soundcloud account that is linked from that NPR site,here is the link, quite beautiful:
http://soundcloud.com/sruti/spirit-ahir ... v-shashank ) and sent it to them. They got it this time. 'Yeah, that is like Ullathil Nalla Ullam from Karnan. Quite impressive.
Now I am curious if someone who is completely out of our cultural context can hear both the Shashank Alapana for say 20 seconds and then the starting instrumental portions of Ullathil Nalla Ullam and find them similar. I will be surprised if they don't but it will be worth an experiment to mix them with a few other unrelated snippets and see if they pick out the related pair, just to be sure.
But on this fourth count, I am not too worried or bothered if these other genre folks do not follow through to that level of experience. That is a tall order for anyone to expect that out of someone else. Hopefully the first three aspects give them enough of a motivation to get to that fourth stage, if they so wish.
BTW, I am not concluding that the above classifications will somehow directly help in this communication process. I am just trying to give some structure to the problem. Each of those still require further definition and talking points in a relate-able way, like the attempt Uday made above. We all probably had the occasion to talk about raga to others and may be that experience, successful and unsuccesful, will help in formulating them.