About the best one could do with music in such a context (IMO) is to ask to what degree it augments or subverts (comments upon...) the action...but, as noted, this function is not unique to noir. So attempting to quantify it just doesn't address its "noirness."
I think a discussion of music in noir shows the possible range of application and documenting/analyzing that would be a fascinating endeavor (and I"m not sure that some enterprising PhD student hasn't already done so).
Mark, I think a film can be rich without being great just as Manny Farber decided that "termite art" was more interesting than "white elephant art." But what Farber meant was that such art created its own peculiar logic often at odds with received notions of form and structure, and I think that Blake Edwards was too overtly derivative in his influences to completely succeed in such an endeavor. (Ralph Nelson runs into similar problems with ONCE A THIEF a few years later.)
Where it sits in the movement of 60s film and the dissolution of genre is definitely interesting, however, and I hope I managed to convey that in what was written. There is a longing for a kind of simplicity that the film sets in motion, teases us with and ultimately denies to us; that idea (that things can somehow be "simple" and that societal values can be, if not transparent, at least somewhat straightforward...) is reinvented in a more simplistic shorthand for "alienation" as it is manifested in neo-noir.