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Rupert (Neve) was always of the opinion that extreme HF has a significant effect on perceived ‘quality’; he tried to keep his designs as flat as possible right out to 100KHz.
I have done some experimental work on this but it has usually been inconclusive owing to the difficulty of reproducing these higher frequencies accurately enough….. Ribbon tweeters can sound ever so sweet, but how much of it is harmonic distortion?
My own work has shown clearly that smooth phase response is actually more important than amplitude accuracy; in the bottom end, phase ripples and a ‘kick’ at 75Hz may sound impressive, but is bad for long term listening. At the HF end I tend to not worry if the response is a bit droopy above 10KHz as long as it does not dive off as a result of filters that twist the phase response; The filters used in digital conversions are now so good that they don’t bother me any more! (You could be cynical and say that it’s because I’m 77 years old!)
There’s so much more to listening than just the physics of sound and the biology of the ears, my short answer is that sounds that do not occur in nature tend to be unpleasant, so as designers we should avoid 3rd order distortion and filters that cause sharp phase shifts. My view is that attention to phase is more important than extending amplitude response.
As for telling the difference between sine and square at 12KHz….. That sounds like a simple experiment, but does it lead anywhere? No, hearing is much too complex for that!