WmAx said:
The more linear the power response, the closer the room reverberant field will resemble the on axis response. That is a given, since the definition of power response is the acoustical power radiated off axis.
I would perhaps restate that. The more linear the power response, the more closely the sound radiated to various points in the room will resemble the on-axis response.
The actual reverberant field presented to the ears of the listener may or may not resemble what the speaker radiates. (I have asked a number of people about this and have not received an answer so far.)
As for importance, Toole has [1]shown conclusively, in statistically significant, rigorous scientific study, that all things being equal, the off axis response linearity determines perceived sound quality.
It's been a long time since I read these. I'll have to dig Toole's articles up and give them another look.
One thing that always bothered me was that speakers such as Bose designs would evidently do quite well in Toole's tests. However, they do not sound at all accurate to me. The Ohm Walsh radiators would also appear to do well, and, in my opinion, do sound better than Bose, but that's not saying much.
You can [alternatively] use high levels of acoustical treatments to remove the reverberant field from contributing much, therefor the off axis response is irrelevant, for the most part. The problem with this approach for stereo reproduction is that you end up removing the reverberant field that lends to creating an enveloping ambiance for the listener. Stereo reproduction is not a true reproduction format, and there are no standards in producing music for this format, either. The stereo format can not reproduce spatial realism on it's own. But with the assistance of contributing room reflections, it can be greatly enhanced, due to psycho-acoustic effects of the reflections(if the reflections comply with several specific conditions).
I don't think I can agree with this at all. For many years I was lucky enough to live in a house with a very large, open yard, and no close neighbors. I regularly put my stereo outside during the summer and listened to various things. I listened to a number of different speakers and electronics, and of course a wide variety of music.
I heard a fair bit of spacial "realism." In fact, I would say that a normal listening room muddles things up due to reflections.

Now, I'm sure that the "realism" I heard was not what would have been heard at the original performance, but the same could be said about speakers in a reverberant room.
The best case scenario for stereo reproduction would be as if there were a wall between you and the performance, with two holes in that wall. (Two spaced microphones with an acoustic performance, and then two loudspeakers placed at two vertices of an equilateral or isoceles triangle, with the listener at the third vertex.)
Multi-mono studio recordings can only really give us the amplitude cues for locating sounds, so they are probably flawed right there. (We do then get head transform and time cues, but they're governed by the speakers and not the original performance.
It is probably necessary to eliminate the crosstalk between the speakers, either mechanically or with transaural processing. Otherwise you hear two events at each ear instead of one.
Designs with poor off axis behaviour require substantially more room treatment to perform optimally, as compared to speakers with an even power response. As for
There is a definite difference between sound events emanating from the speaker and then that same event as reflected back to the listener by the room. There will be time differences, and, unless the room reflects all frequencies evenly, frequency response differences. (I think there will be frequency-variant attenuation due to passage through the air and distance too.)
I feel that more room treatment is a good idea in all cases.
Pat McGinty made a statement something like," When you get the time and phase response right, the frequency and power response fall right into line." The measurements of his speakers don't appear to be horrible.
But if someone ignores the current state of perceptual science as it relates to their work, they are not demonstrating anything that anyone can conclude as being especially wise.
I'm sorry. I'm just having a hard time (and have had a hard time) swallowing this because the speakers I have heard that claimed to follow Toole's precepts did not sound very good to me. Perhaps I have not heard good examples of these, or perhaps there were other factors that swayed my opinion. (And opinion it is.)
[1]Loudspeaker Measurements and Their Relationship to Listener Preferences: Part 2
Floyd E. Toole
JAES, May, 1986, Vol. 34, pages 227-235
I'll have to look and see if I have this or not.