In INTERVAL.EXE, you hear two pitches, and you can find out what happens when as they move into various frequency ratios to each other.
The display looks like a slide rule, with the numbers showing the position of the fundamental frequency (1) and its harmonics (2, 3, 4 ...). Here, we see the display of a 3:2 ratio:
It turns out that two pitches with small whole-number frequency ratios, like 3:2 or 5:4, sound better together than those with bigger number ratios like 15:16. Why? Because more of the lower (and louder) harmonics in the two pitches match up (here, 3:2, 6:4, 9:6 ...).
So, assuming we'd like to use whole-number-ratio intervals, what if we'd like to use a scale in which the octave is divided into equal parts? In DIVISION.EXE, the intervals with small whole-number ratios are shown (at their appropriate positions around the outside of the circle), and can be compared to the pitches you get when you divide the octave into equal parts (shown as spokes in the circle). Here we see eleven divisions:
...and here we see twelve:
This shows why we settled on twelve: the 3:2, 4:3, and 5:4 intervals are much closer.
Of course, we don't necessarily have to space the pitches evenly. In Tantrum (Mac software available for download here), we can experiment with the kinds of tradeoffs that can be made in constructing a 12-pitch tuning system.