> I will say that having a base 12 length measuring system (in/ft/yd) is much nicer for carpentry than base 10 (m/cm), since it can be divided evenly by 2 and 3. It's the same reason why having 12 notes in a scale works for music.
When would you want to divide an octave into two or three intervals like that?
The normal way to divide an octave in "half" is at the fifth, which is a ratio of 3:2. If you instead go six half-steps up from the tonic, you'll be a half-step short of a fifth, at fi instead of so, with an irrational ratio of √2:1. That's the opposite of what you want. [From wikipedia: The tritone is a restless interval, classed as a dissonance in Western music from the early Middle Ages through to the end of the common practice period. This interval was frequently avoided in medieval ecclesiastical singing because of its dissonant quality. The first explicit prohibition of it seems to occur with the development of Guido of Arezzo's hexachordal system]
If you divide the octave into "quarters" by going up three half-steps, you get a minor third.
(Going for "thirds" by going up four half-steps will take you to mi and form a third [the name of the chord is not related to the idea of dividing an octave into three equal parts; it's called a "third" because "do re mi" is three notes], which is fine. But it's pretty clearly just a coincidence that that works out; the methodology of "there are twelve half-steps in the octave, so I'll divide that into equal parts and see what I get" just isn't related to the way you compose music.)