Anechoically flat means input equals output (again, along some solid angle).
Agreed. Apologies if I came across as if I thought something different.
The room is the confounding factor, of course...
Loudspeaker/room interaction can be overall beneficial or overall detrimental. The more correct the reflections, the more beneficial the loudspeaker/room interaction is likely to be, in my opinion.
One cannot get all rooms (reflected sound) right, so I'm not sure what magic sauce you think you've found. I am curious though.
Floyd Toole has said that "frequency response is the most important parameter in any audio component. If it is wrong, nothing else matters." So for now, let's just look at frequency response, because THIS is what matters the most. In particular, let's pay attention to the influence the reflected sound has on a speaker's frequency response (we can come back to the effects of reflections on other things later).
Now in the context of loudspeakers I THINK Toole means the "perceived frequency response", rather than one PARTICULAR frequency response measurement (anechoic, in-room, whatever). The direct sound is not the perceived frequency response, nor is the reflected sound, nor is the measured steady-state in-room response (which includes both). That which is
perceived to be "flat" may well include a flat direct sound plus non-flat reflected sound. The tonal balance we perceive is in effect a
weighted average of the direct and reflected sound, with that weighting probably being tipped towards the first-arrival sound by an amount which varies with the characteristics of the reflections at the listening position.
We have a body of research which shows that a combination of flat direct sound and non-flat reflected sound which sums to a steady-state in-room frequency response curve along the lines of those shown in
post #13 results in a subjectively
flat, or at least subjectively
preferred, perceived frequency response. (The "trained listeners" curve is arguably the most reliable one.)
So here is the big question: Is a combination of flat direct sound plus non-flat reflected sound the ONLY valid way to achieve our preferred perceived frequency response?
Maybe not. Let me describe one theoretical alternative approach:
Suppose we prioritize minimizing the spectral discrepancy between the direct and reflected sound, such that the direct sound's frequency response becomes MORE like the (non-flat) preferred steady-state in-room frequency response, and the reflected sound becomes MORE like the direct sound (closer to flat)? In other words, have the two approximately "meet in the middle". Might there be a subjective benefit to having the spectral balance of the reflections be virtually identical to that of the first-arrival sound? Well, we don't know - it hasn't been researched (though I can think of arguments why it
might be). Is there an obvious reason why this approach would not be viable? If so let me know, because such is not apparent to me.
Let me know your thoughts thus far, if you don't mind. (And no, I'm not necessarily talking about an omnidirectional system, though that is one possible approach.)