Fast driver depends on...?

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Fast is an ill-defined term for loudspeaker quality. Assume being fast equals to high sensitivity. This property in itself is still no guarantee that a loudspeaker will be sounding good. Actually there is very high sensitivity full range drivers out there that play pretty bad on their own. Ragged response throughout the spectrum and shallow bass.

If you care for higher sound quality, decide what sensitivity range you need your speakers for and use the common sense building rules to achieve that.
 
"fast" in loudspeakers seems to me largly in the ability to STOP quickly

sure there are other parameters that may be discribed as "fast", but across the board its the ability to stop moving flexing ringging when the signal says to stop.

There are times when I heard a very light mms sound "slow" and there have been times when I have heard much heavyer mms sound "fast". Sometimes fast has been just a bumped up mid bass but over time its obvious its a fake fast.

Once I had a cheap 15 inch poly woofer with a very low Fs. The thing was worthless and had no control. Very soft suspension and small motor. My daughter wanted a sub in car sooo I just put it on a thick board and mounted to back seat(no box). It was the slowest sounding bass Ive ever heard and it would bottom out easily.
Took woofer out and coated the foam surround with a few layers of silicone calk to stiffen suspension , raise Fs, AND change the mechanical properties to a more spring like charachture. I did not expect the result. It was the fastest sounding bass I have heard to date. I hate to admit this really I do, but it was sooo dynamic and seemed to have zero overhang.
 
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Not again. Try the search facility.

Assuming you are referring to transient response, under notionally pistonic conditions 'fast' = a critically (or an overdamped if you are happy to sacrifice accuracy) response in the rising response ('acceleration') BW up to the mass corner taken as 2Fs/Q' where Q' = effective system Q once any series R &c. is accounted for. Above that in the mass dominated region it is defined by the inherent upper BW of the driver before VC inductance rolls it off.

In drivers designed to exploit the transverse wave & extend their functional BW through resonance of the cone substrate and / or dustcap / sub-cones, then this will further vary depending on the materials, their self-damping and decay time. This is not likely to be revealed in bald T/S data; you need more than that, whatever you 'wanna' do. ;)

Motor linearity and the absense of distortion products also play a role; typically the lower the better, although a higher level of HD3 can make for a subjectively sharper sound which some might consider 'faster'.
 
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