Can we measure "Fast"?

In this blog I pose the following question:

Does a light diaphragm result in a fast sound character? Is lightness a virtue to good sound quality? 

Follow up question number 1:

If it is, then what measurement metric can be used to identify this virtue? 

Follow up question number 2: 

Again, if it is, what measurement metric does not identify this virtue? 

This blog post may be part 1 of multiple blog posts on the topic. It is a refreshing change from my heavy focus on intermodulation distortion. 

Premise 

There exists a polarization of understanding between the audiophile community and the more technical professional community. Generally it is understood that moving mass (mms) has little to do with actual driver performance in terms of what is achievable in terms of sound quality. While the audiophile community empirically observes that supposedly light drivers provide a faster more 'immediate' sound. I began this blog post siding with the professional community wanting to 'educate' the audiophile. However I decided to take a step back and ask myself if perhaps the topic was not so clear cut. Was there actually a test that could reveal a 'fast' driver compared to another? 

If I take two tweeters with obviously different moving mass and I match the frequency response using DSP, will there be any measurable difference between the two tweeters using a standard suite of tests? 

To show this, I decided to take two tweeters and match their response using DSP. The two tweeters are the Dayton AMTPRO4 and the Dayton AMT2-4.

Visually it's quite obvious that the AMTPRO4 has a much heavier diaphragm, although it is not specified by the manufacturer. Below is the overlay after I've matched the responses to the best of my ability using the Hypex DSP software. I've limited the tweeters to the most linear portion of their frequency bandwidth.

Below are the DSP settings for each tweeter to get them matched up with AMT2-4 on the left and AMTPRO4 on the right. Click the image to enlarge.

If we now measure and display the full test data we can see there is little difference between the two tweeters. 

Below are the results between the two tweeters, with the AMT2-4 on the left and the AMTPRO-4 on the right. Click to enlarge each image if needed. 

 

 

 

 

 

 

As you tell from the results, especially the step response, there is no difference in the measurements despite drastically different tweeter sizes. 
We are left then, are there are any other tests that we can do that might bring to light a 'faster' tweeter. I decided to conduct a square wave test to see how each tweeter responds to the abrupt transients found in this type of test signal. Using REW's oscilloscope function I was able to visualize a 5kHz square wave.
Below are the signal generator setup screen within REQ for reference.
Below is the comparison of the square wave performance for the two tweeters. Obviously the square wave shape is not going to be perfect since the tweeter is bandwidth limited (a perfect square wave requires infinite bandwidth), but we want to see if there are significant differences between the two tweeters.
The square wave comparison shows some significant differences starting with the overall spacing of the cycles. The AMTPRO4 has a very non-symmetrical positive/negative relationship as shown below. 
Additionally, the AMT2-4 follows the square shape much better especially in the bottom portion of the cycle as shown below. 
These are just observations at this point. This represents, at the very least, a nice little rabbit hole to investigate further. In the interim, any feedback or suggestions on next steps are certainly welcome.
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