Fret Spacing on a Guitar

As you look down the neck of a guitar, the frets get closer and closer. What is the rule for deciding how close the frets should be?

Preliminary Information:

1. A string halved in length vibrates at twice the frequency (physics).
2. An octave is the interval between one musical pitch and another with half or double its frequency (Wikipedia article on octaves).
3. There are 12 notes in an octave (music theory).

To go up one octave, you have to halve your string. To go up another octave you must quarter your string. In general, to go up t octaves you need to have a string of length 

To have our length function, l, go up by twelfths of an octave (notes) rather than octaves, we need to adjust it to be: