# Root

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

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 *b*-th 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 is 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 A220 and A440, we divide 440 Hertz by 220 Hertz and get… plain ol' 2.) If an octave means "twice", then what is half of "twice"?

It is not once… because two onces is just another once!

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