Agreed, from that point of view. I am trying to be cautions in what assumptions we make without stating. For one thing, in visual perception the patterns of light that fall on the retinas is rather different from what subject report seeing. Second, reading @Floyd Toole 's book gives me the impression that we are capable of some similarly clever tricks in hearing.
As already mentioned, everything that has to do with spatial perception and partly tonality is certainly not optimally reproduced via headphone using binaural recordings of a dummy head because HRTF is individually slightly different. So, as is so often the case in science, a small uncertainty remains.
But the question of the OP was "If all things equal measuring on the machine..." and the quoted master thesis shows, even if not all "things" are equal, no more distinction can be made.
What is also missing using binaural recordings via headphone is the sensation that the sound pressure fluctuations trigger on the entire body (but if "all measured values are the equal", this also does not matter).
But perhaps it is the case that the measured sound pressure signal at the simulated eardrum captures everything that a headphone can do to a human listener. It seems likely. Now, assuming that's the case, what is OP's question driving at?
I'll stick with tweeter examples, assuming we have a $1000 diamond tweeter or one with a new fairy dust material and a $20 metal tweeter and all measured values of both tweeters are very similar (on- and off-axis measurement, distortion and decay), will these two tweeters nevertheless be audibly distinguishable?
The scientific research that I know and that deals with it says no, as long as the known parameters (on- and off-axis measurement, distortion and decay) remain very similar.
This is quite plausible, since it is very likely that there are no undiscovered, mysterious sound pressure transmission mechanisms that are not detected by measurements (and would play a major role in auditory perception by human hearing) - so that rules out such things as "telepathic material waves" or the like, except when using fairy dust tweeters
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