I'm not sure that's the problem.
This is a review and detailed measurements of the B&W 805S Bookshelf speaker. It is on kind loan from a member. The 805S is discontinued but seems to draw on the same principals of their speakers so makes for a good test. Looks like the original cost was US $2,500. Despite being bookshelf...
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With the smaller 805S, there are cancellation issues in the crossover area, but no 5 kHz dip even though it's using the same tweeter and cabinet design as the 804S.
It seems like the dip only occurs with speakers that have the FST midrange.
702 S2 (FST midrange and tweeter on top):
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Sidebar 3: Measurements
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705 S3 (Tweeter on top, but no FST midrange):
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Sidebar 3: Measurements I used DRA Labs' MLSSA system and a calibrated DPA 4006 microphone to measure the Bowers & Wilkins 705 S3's behavior in the farfield, and an Earthworks QTC-40 mike for the nearfield responses. I left off the grille that covers the woofer for the measurements. Fig.1 Bowers...
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I did a few more measurements today, at 30cm, in front of the phase plug (on the FST midrange).
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The result was as expected. A huge dip.
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I then blocked the tweeter using a piece of cloth and a rubber band (with the tweeter grille on

).
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Finally, I tried blocking the midrange instead (with a piece of acoustic ceiling tile).
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The tweeter response is uneven (measured at an angle), but the 5 kHz dip is gone.
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All three measurements. At this distance and angle, the tweeter and mid cancel each other out at 4.2 kHz:
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Another measurement, at 50cm:
Blue = Tweeter axis, nothing blocked
Red = Acoustic ceiling tile in front of the midrange
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Just for fun, I compared the 804S to Revel F208.
I know measuring at midrange axis, and at 30cm is unusual, but I wanted to see how the F208 performs in comparison.
F208: +/- 2.5 dB (+/- 2dB up to 10 kHz)
804S: +/- 11.8 dB (

)
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