I had thought that the '12^{th} root of 2' meant taking the square root of 2 12 times.
But no. It means taking 2 to the 1/12 exponent (2^{1/12})

Taking this value (1.05946309) and multiplying it by say, A_{1} which has an agreed-upon value of 55, we get the following JavaScript output:

We can compare with an existing chart and see that we're pretty close.

Which then leads to the obvious question, what would a musical scale based on the 'golden ratio' (1.61803399) sound like? (If only Pythagorus were here!)

Which results in something that starts with what we would call 'A', then what we would call between 'F' and 'F#", then between 'C#' and 'D', then 'A#', etc.
It's a mess.

(golden.mxb)

There must be some relationship between the golden ratio and musical scales, but I can't find it.

This page explains some of the ideas here.

So now let's write a script that multiplies good ol' A_{1} by the numbers in the Fibonacci sequence:

Which gives us (translated into standard names) A_{1}, A_{2}, E_{3}, C#_{4}, A_{4}, F_{5}/F#_{5}, C#_{6}/D_{6}, A#_{6}, F#_{7}/G_{7}, etc.

This is interesting. The first several values match those of a major A chord, with the values getting sharper to the point where, 6 octaves up, we get A#.

(fib.mxb)

At least part of the mystery may be resolved here. Or maybe not.