Chinese 12TET - Approximation

>>1. Square the guess => guess sixth root of 2.

This probably should have said:
1. Square the guess of 12th root of two => guess of sixth root of 2.

>>2. Square that => guessed cube root of 2.
>>3. Multiply by result of step 1 => guessed square root of 2.
>>4. Square that => guess of 2.

Johnny K's point here is that it just doesn't take too many iterations of
insightful guessing before you resolve the 12th root of 2 down to as fine
accuracy as you realistically need for musical purposes.

But you don't need any sort of analytical technique to choose your guesses -
just intelligence. I'd probably start by noticing that 1.1, since 1.2 is almost
certainly too high. 1.1 squared is 1.21, and that squared is 1.46, and that is
going to square up to something close to 1.5 squared (2.25). So since 1.1 to
only the eighth power is already larger than two, the twelfth power of two must
be somewhere between 1.1 and 1. So, I would probably then split the difference
and guess 1.05. Doing those four multiplications, I find that 1.05 to the
twelfth power is 1.795..., meaning that the real twelfth root of two must be
somewhat larger than that. Doing those same multiplications on 1.06 gives me
something a lot closer: 2.01.

For the limited purposes of very crude and/or imprecise-pitched instruments,
that's about all of the precision you need. And that in only three iterations
of insightful guessing, and a total of eight multiplications. Not too tough!

From there I'd probably see that I'm so close that I'd probably be best off
shaving just a little bit off the 1.06, and try 1.059. That then lands me at
1.9895...: Low by about the same amount by which I was high at 1.06. So I'd
then try 1.0595. That puts me at 2.00083...

So, after just five iterations - easily less than a day's work with
multiplication on paper - I'm down to an error of less than a cent!


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