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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 √2 *multiplied* twice is √2*√2 = 2. (Note that √2 *added* twice would be 2√2.)