talking in circles

I just wrote up this ditty about roots for the Xenharmonic Wiki
<http://xenharmonic.wikispaces.com/roots>. But I worry that it would not elucidate
anything for a novice. Here is the text:

The arithmetical concept of roots is often encountered in discussions about tuning.

How are roots related to equal divisions?

To divide an interval a into b equal parts, that is, to calculate the size of the interval that,
when repeated b times, would add up to a, calculate the bth root of a. The equivalent
expression is to take a to the (1/b)th power.

Why roots and powers? Because intervals are proportions, which you must multiply in
order to "add".

Take a simple example: what's half of an octave? Well, an octave means "twice the
frequency" or "2 times whatever you have" or "2 to 1" or simply "2". (The 2 itself has no
units, because they cancel out: to calculate that octave between A-220 and A-440, we
divide 440 Hertz by 220 Hertz and get... plain ol' 2.) If an octave means "twice", then
what's half of "twice"?

It isn't once...because two onces is just another once!

It's the square root of 2! Try it: The b2 *multiplied* twice is b2*b2 = 2. (Note that b2
*added* twice would be 2b2.)

--

It's confusing and takes a while to get if you were never good at math, yes, but the real
problem is that I don't really know the best way (or any non-slippery language) to talk
about addition turning into multiplication and multiplication turning to powers, these
transforms which bridge the gap between ratio-world and logarithmic-world.

Feel free to add to or modify the growing body of knowledge that is the xenharmonic wiki.
Do note that it does not aspire to be an encyclopedia, and that original research and new
attempts at making sense of the tuning world are kinda the whole point...!

My my,
Jacob