Profundity, or rather the discovery of profundity in something, must necessarily be experiential. We cannot find profundity in Beethoven if we do not listen to Beethoven, but if we do listen to Beethoven we only permit the possibility of finding profundity in Beethoven. For some people, it will simply not be there, no matter how much others would like to push onto them their own findings of profundity in the music of Beethoven or in any other composer. I don't doubt that most people agree that Beethoven has at least produced musical moments which we find profound in some way, but this does not necessarily mean that the music of Beethoven is profound, only that most people agree that they find it to be so. The distinction between the purely objective and the mass subjective is always contentious here, so I'm not sure if we want to get into it, but I think that is ultimately where all or most discussions of this kind lead.
So, what is musical profundity? Given that I reject the notion that there is anything like an objective measure of qualities which are found only through the process of reflection upon the listening experience after the fact, I would propose the following definition: a listening experience in which the music appears to bypass the brain and is instead felt directly by the body. This is what we might even term a spiritual experience, something that transcends the rational everyday processes of thought and enters into us directly at a deeper level than that on which we normally process musical input. Given that we are attempting to qualify and quantify, I'm not sure "spiritual" really cuts it, but we could probably make a link between what we think of as spiritual and elements in the subconscious which are sympathetic to particular musical inputs, and which reflexively expand the experience of listening to such inputs.
Also, I think it is possible to draw a distinction between two kinds of profundity in music, these being a) profundity of construction, and b) profundity of experience. In the case of a, I am talking specifically about the way in which the musical materials are treated, their interrelations, the "thesis" which they serve, and the level on which they come together. This "profundity of construction" is something that is inherent in the music, the fruits of the composer's labour, which can be gleaned through analysis of the actual score. On the other hand, b is specific to the listener, and may, as I suggested in the second paragraph, involve no ratiocination whatsoever. Both a and b are obviously somewhat dependent on the natural sympathies of the receiver; despite a being a matter of inherent musical qualities, it is entirely possible to give two people of equal experience and level of education the task of analysing the same score independently of each other, and have their findings differ, possibly wildly, because of those natural sympathies. And a similar test in the mode of b would perhaps give us even wilder differences, not only for all that is true in the case of a, but also as, in the process the mind goes through in order to recall and collect the particulars of the listening experience into an ordered account, it may invent certain things to aid in its explanation, such as narratives and images, or may conflate those things with the music through the sudden triggering of memory during the listening experience, and these retroactively become part of the experience. So, in another way, it is entirely arguable that profundity of experience is far more than just a case of "feeling the music," and may have more to do with what we add to it than what we take away from it.
It's also important, I feel, to make a further distinction between a and b, which is the fact of authorship of input. In a, the input can reasonably be assumed to be the work of the composer, yet in b we are presented with an input which comes from this source but is filtered through something else. Even if we could remove the fact of interpretation from the situation, the fact is that we are still hearing a sonic projection of a work written in a logographic system. But since in real terms it is not possible to remove interpretation from a performance, the fact of interpretation complicates things further, and it is for this reason that I say a can only be experienced through the analysis of a score rather than focused "analytical listening" to a performance or recording. Of course this may be pedantic, after all it is reasonable to assume in most cases that a given performance is generally an accurate projection of the score, and that it can therefore be analysed with a view to determining the profundity of construction, yet I feel that projection and interpretation distance the listener enough from the source that this is still impossible to properly achieve.
Take note, ladies, gentlemen, and children of all ages, the post you have just read is the result of someone thinking far too much about things that don't really matter. Enjoy your music and don't sweat the small stuff!