1. The first point is that the cone material itself does not affect the sound (there is no "ultra fast" material). So if you equalize different tweeters in the free field to an identical frequency and phase response, they will sound identical.
There is a master's thesis from Germany that investigates exactly this. I have
discussed this work in detail here.
2. However, different materials have different radiation characteristics with identical tweeter designs, as the break-up behavior is different. A silk dome tweeter breaks up very early but in a controlled manner, a beryllium dome tweeter breaks up very late and often in an uncontrolled manner, but the tweeter shows almost ideal piston behavior up to high frequencies - a silk dome does not.
You can find
more details here and following posts.
As an example, here is an identically constructed tweeter with the cone materials silk, berylium and diamant. The FRs are normalized to the on-axis FR. Each diagram show the measured FR for different angles compared to an ideal piston like behaving tweeter:
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3. Even small changes (small offset or tiny "waveguide") on the front plate of the tweeter lead to changes in the radiation. You can
find more details here.
4. The overall radiation of a loudspeaker plays a major role (apart from direct sound) in how the sound is perceived - especially when the SPL increases.
Speakers with identical on-axis frequency response can sound completely different. Therefore, complete measurements as required by the CTA-2034-A standard are essential to identify potential problems.
For me personally, for example, in the frequency range around 2-4kHz in the sound power, early reflection and PIR, there should be no hump at all (small dip is okay, but can sound "lame", too laid back). Otherwise I perceive the speaker as harsh and aggressive.
View attachment 324515
Since SP, ER and PIR are all "averaged" curves, even the smallest changes can have a major impact on the sound.
A flat on-axis FR (and LW) is usually not enough for a good speaker sound in a normal listening room - except the speaker has perfect radiation, then yes (near field listening in heavy dampened studio is another matter).
Update:
A speaker with design flaws, like below with such radiation can never be fixed by EQ - it will never sound correct. Therefore one needs full measurements of a speaker.
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