Any benefits to mounting a tweeter above the cabinet?

I didn't mention it because B&W didn't do it:

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https://www.stereophile.com/content/bowers-wilkins-802-d3-diamond-loudspeaker-measurements
Sure, test report: "However, the output of the tweeter is very slightly too forward in time, which suggests that the B&W's optimal listening axis will be a little below the tweeter axis."

So, tweeter effectively offset; also Focal from #7, separate offset tweeter via different tilt angles etc.
 
I'll try to watch the video @AllenB next time I (succeed) climbing the firewall. But below that post:

(Stereophile) "In my admittedly anecdotal experience, a speaker that is time-coherent (on the listening axis) does have a small edge when it comes to presenting a stereo soundstage, in terms of image focus and image depth....

"In 1990, Rodney Greenfield and Malcolm Omar Hawksford [34] used DSP-based digital filters to try to separate the akudible effects of a loudspeaker's phase error from its amplitude response error. The point was made that a semi-reverberant environment will tend to mask phase effects. In addition, when typical recordings are played, which may have undergone many phase-altering stages during production, the audibility of phase differences becomes moot.... Nevertheless, the authors "very tentatively" concluded that equalizing a loudspeaker's excess phase error modified listeners' perception of the apparent soundstage." (end excerpt)

Given that research was done 35 years ago, hasn't any diyaudio members tried to repeat/improve upon it? If measured IR pre/post phase corruption of a 1-way fullrange driver were posted I (we) can (single-blind) interpret the differences if any.

With many speakers, there tends to be some tradeoff between image specificity and soundstage depth. You run into this with loudspeaker dispersion (narrow vs wide), room placement, and room treatment.

How much of each thing you like comes down to personal preference. As the Stereophile link mentions, most speakers are not time and phase coherent, and many listeners can't tell the difference. So it's not like it's the only answer to achieving good sound. It's typically viewed as a "nice to have," though obviously some manufacturers think it's important for their designs.

https://www.stereophile.com/content/measuring-loudspeakers-part-two-page-4
"One school of thought holds that it is very important to perceived quality; another, which includes almost all loudspeaker engineers, finds it unimportant. Floyd Toole, now with Harman International but then with Canada's National Research Council, in his summary of research at the NRC into loudspeaker performance that is described in two classic 1986 papers [32, 33], concluded thusly: 'The advocates of accurate waveform reproduction, implying both accurate amplitude and phase responses, are in a particularly awkward situation. In spite of the considerable engineering appeal of this concept, practical tests have yielded little evidence of listener sensitivity to this factor...the limited results lend support for the popular view that the effects of phase are clearly subordinate to amplitude response.'

This is also my view. Of the 350 or so loudspeakers I have measured, there is no correlation between whether or not they are time-coherent and whether or not they are recommended by a Stereophile reviewer. However, I feel that if other factors have been optimized—on-axis response, off-axis dispersion, absence of resonance-related problems, and good linearity—like a little bit of chicken soup, time coherence (hence minimal acoustic phase error) cannot hurt. In my admittedly anecdotal experience, a speaker that is time-coherent (on the listening axis) does have a small edge when it comes to presenting a stereo soundstage, in terms of image focus and image depth. But time coherence does not compensate for coloration, poor presentation of instrumental timbres, a perverse frequency balance, or high levels of nonlinear distortion.

In 1990, Rodney Greenfield and Malcolm Omar Hawksford [34] used DSP-based digital filters to try to separate the akudible effects of a loudspeaker's phase error from its amplitude response error. The point was made that a semi-reverberant environment will tend to mask phase effects. In addition, when typical recordings are played, which may have undergone many phase-altering stages during production, the audibility of phase differences becomes moot: 'one is simply detecting a change in phase distortion and not a correction of it and as such preferences would most likely be personal.' Nevertheless, the authors "very tentatively" concluded that equalizing a loudspeaker's excess phase error modified listeners' perception of the apparent soundstage."
 
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