Why G# and Ab are (not) the same

If you divide the octave into N notes, the equal tempered interval between notes is the number which when raised to the N-th power, that is, when itis multiplied by itself N times, equals 2. Mathematically this would be called 'the N-th root of 2.' Our normal chromatic scale has N = 12 and the 12-th root of 2 is 1.059463..., an irrational number. Since rational numbers and irrational numbers can never be equal, equal temperament will always be out of tune (i.e. there will be beats). If you want to use an equal tempered scale which closely matches the just intervals above labeled 3rd, 4th, and especially the 5th, then N = 12 happens to work reasonably well. The next higher value of N which works much better is above 30. Having over 30 keys per octave on a keyboard is just too many so we compromise and use 12. A comparison between theequal temperated scale and the just scale can be found here.