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My best understanding of the research findings is that the direct sound has primary importance for perception of tonal balance. Get that wrong and one is behind the eight ball, almost futile to try and compensate with reflections.My understanding is that the perceived tonal balance is normally a weighted average of the direct sound and the reflections, with the exact weighting probably varying somewhat with the specific conditions (presumably including the direct-to-reflected sound ratio; the frequency response variance between the direct sound and the summed reflections; arrival times of the reflections; and decay times of the reflections across the spectrum).
Is this, or something similar, your understanding?
Once the direct sound is right, then getting the reflected sound right will be preferred to getting it wrong. And 'right' means either constant directivity or smoothly varying directivity with frequency.
One could assess the reflected sound from examining the off-axis measurements alone, or in combination with (known) direct sound.
And of course, getting the bass right (which is all reflections), and adjusted in level 'to taste', will be important to perceived tonal balance.
I'll try, but bear in mind, what I am trying to say is that falling summed FR is not automatically preferred (I think Keith stated to be axiomatic, "falling FR is preferred"), and that looking at summed FR is a clunky proxy way to assess the preferred-ness of a frequency response, and can get you into trouble when looking at non-standard gear or rooms. Did someone say horns?I'm not familiar with situations where falling summed frequency response is not preferred... the implication presumably being that flat or rising summed frequency response is preferred when the two secondary conditions (falling off-axis treble and in-room falling off-axis treble) are not met.
Could you elaborate on what sort of conditions would result in a falling summed frequency response not being preferred, and/or correct me if I've misunderstood you?
One example: live rooms vs dead rooms. A really seriously dead room (which I have seen quite a few audiophiles aim for) will be best served by an effectively flat 'summed' frequency response, because it's all direct sound energy and direct sound sounds best when flat.
Another example: in-wall speakers in very live rooms. It might take quite a few drivers, but you could put together an in-wall speaker that is close to 180° beamwidth at all frequencies. Put that speaker (or an omni speaker with flat FR) in a very live room with hard furnishings (hello modern decor), and neither the speakers nor the room will be causing much off-axis energy decay that varies with frequency. So, the assumption behind the convention for falling summed FR, is not applicable here either.
None of which is to deny the clear experimental finding that, with standard speakers in standard rooms, a listener preference test will reveal the desirability of a falling summed FR.
cheers