this music puzzle makes my brain hurt... 

You know what? Credit to you for realizing you'd left off the 's' at about <a href="#" class="seekto" data-time="300">5:00</a> and taking the moment to add it. I appreciate people being self-aware.

Cool reaction video! I was just messing around with Benedetti's puzzle in 34-tone equal temperament which does not temper out the syntonic comma (81/80) as well. By going through 6 iterations, I was a whole tone higher from where I started (6 steps of 34-TET is a whole tone). I tried lowering the A sustained in the treble clef by 81/80 to keep the pitch from drifting, but I ended up creating a wolf 5th (40/27) with the D in the first beat. Comma drifts can be annoying at times but they can be used for interesting modulations that I plan to exploit :)

To break your "tuning brain" on an even higher level, check out Zhea Erose on one of her hour long discussions on tuning systems. Have mercy!

If it were possible to tune a piano perfectly (perfect fifths with a ratio of <a href="#" class="seekto" data-time="123">2:3</a> and major thirds with a ratio of <a href="#" class="seekto" data-time="245">4:5</a> ) then piano tuners would do this. But sadly it's not possible. It's like the leap years. A year does not have exactly 365 days. And an octave does not have exactly 12 fifth (if you play alternating 5th up 4th down 5th up 4th down etc.)

It's actually very interesting. Using the tempered scale, to raise a pitch by a semitone, you would multiply its frequency by the positive twelfth root of two, i.e. the number which when multiplied by itself 12 times becomes two, approx. 1.059. That's an irrational number which means you cannot precisely write out the frequency as a decimal number; there will always be some error involved. I checked this by Wolfram Alpha-ing (like googling but with Wolfram Alpha) "is the 12th root of 2 rational". So if the pitch of A4 is 440 Hz (it's not always), the pitch of A#4 would be 440*2^(1/12) Hz; Wolfram Alpha gives me around 50 digits starting 466.16376 for this number, but that's still not fully precise (but good enough for most purposes of course). Is music that does not use the tempered scale "bad-tempered music?" :D

A440 hertz and a447 hertz various raise or lower for a brighter sound.chromatic harmonica etc.

or maybe a lot of people have unrecognized perfect pitch because it does not match the modern system

I’m not a fan of these types of “reaction” videos where you just sit and show us another person’s work for long periods of time without giving us some useful input. It feels like we’re both learning at the same time, when you’re supposed to be teaching us.