I don't know whether to laugh or cry

I've been having a long term discussion with the owner of Audible Arts in Campbell about what makes a good sound system (he says $40k, I say $500). One thing he brought up was the idea of a 'super tweeter'. These are designed to extend the range of the system from says 22kHz to 50kHz. He claims they significantly improve the sound. I have two distinct issues with them. Firstly, it is generally accepted that "the ear can't hear as high as that - still I ought to please any passing bat" to quote the timeless Flanders and Swann. This is possibly just because we aren't measuring the right thing. Perhaps we don't hear sound, but we sense it, like we don't hear a 64foot flute stop, but ooooh boy, we feel it.

My second issue is far more fundamental (or is that harmonic?). We know from digital signal theory that there is no information encoded above half the sample frequency (because we can't distinguish a higher frequency from a lower frequency). For CDs, sampled 44100 times a second, this means 22050Hz is the highest frequency reproducible. A modern CD player enforces this (for technical reasons, a naïve A2D produces aliasing noise with a flipped frequency spectrum above the sampling frequency) by digitally resampling the CD to many times the base frequency (say 176400 samples per second) then digitally cuts off the signal sharply at 22kHz.

Thus, a super tweeter may reproduce exactly the frequencies on its terminals, but a good linear amplifier will have none.

I explained this to the owner and he said 'yes, I'm sure you are right, but I can hear the improvement with the simple experiment of covering the super tweeter with my hand'. It is plausible that covering the ST with a hand would block almost all of the signal, so I agreed to a demonstration.

He set the speakers up with high quality ($4.5k) mono amps (*2) and a good preamp (only $5k) and CD player (a bargain at $1k), cables ($2k each), power conditioning (like a line filter, only $5k) and quality power cables ($10 at walmart?). I sat in the hot seat and closed my eyes. He would then move his hand up and down randomly and I would listen for the dramatic change. He asked if I noticed the airy sound.

I sat there for a while but couldn't hear anything, so I suggested that as he could hear it clearly I would do the covering and he would say when it changed and whether it was good or bad. I did 10 transistions, he was reasonably close to picking the transition times (but he may have heard my clothes rustle). Out of 10 trials he got 2 measurements correct to 8 wrong. Now my stats is a little rusty, but I think this is a binomial trial. That means that the probability of getting 8-2 given no information is 10C0 + 10C1 + 10C2 + 10C8 + 10C9 + 10C10 = 0.109375 (originally I had a one sided test here - but as we're looking for correlation rather than matching 2-sided seems better) so given the poor testing, it is quite believable that he could hear something. However, if he was consistent, he was consistently wrong. Thus, he should consider selling systems with super tweeters that have a human hand covering them.

Code for 2-sided binomial test:

def choose(n,k):
    r = 1
    for i in range(1, k+1):
        r = (r*(n-k+i))/i
    return r

hits = 2 total_trials=10 print sum(choose(total_trials, i) for i in range(hits+1))*0.5**total_trials*2