So I sat down at the keyboard, deciding that I would figure out major and minor scales, and what triads they contained at each position, and got lost just in the major scale. See, the whole TTsTTTs thing always bothered me -- so asymmetric. Why two tones, then a semitone, followed by three... never made sense. Why not the other way around, or some other pattern. And the whole layout of the piano keyboard contains this pattern. The structure of the major scale is built into the interface. It's mathematically unpleasing, which is one of the reasons why, when extending my interval practice into scales and arpeggios, I often stuck to the totally symmetric ones, built entirely from a single interval: chromatic, whole tone, dim7, augmented.
While diddling, I remembered reading a claim that the scale is in fact symmetric, being built from two identical four-note tetrachords, each consisting of a perfect fourth divided as TTs. That claim confused me at the time, too, since it's clearly not symmetric. Still, it kinda is, since you've got the same note pattern in each half (approximately half, anyway) of the octave, separated by a whole step.
So what happens if you start minor, eg C-D-Eb-F, ie TsT? Does repeating that in the upper half give you one of the minor scales? Well no. You get a Dorian mode, which has a minor feel, from the flatted third, but is not one of the classical ones (natural, harmonic, or melodic). To get melodic minor (going up melodic, TsTTTTs, sometimes called jazz minor if you're planning on playing the same notes going down) you put the major tetrachord on the top, TTs, since those notes are the same as the major scale. Fine, so different scales can get built by combining different tetrachords. That's kind of interesting, since we can therefore use the tetrachords as an organizing concept to relate different scales which have parts which sound the same. To build a natural minor, we use the last tetrachord, sTT, on the top. If you build a scale from two of those, you get the Phyrgian mode, C-Db-Eb-F G-Ab-Bb-C. I guess it's almost obvious that building a scale from two of the same of any these tetrachords (what wikipedia calls diatonic tetrachords, and names the three Lydian, Dorian, and Phrygian) has to give you a mode of the major scale, since the semitones are the same distance apart. But there are only three of them anyway. I suspect pianists learn those shapes, and use those patterns in playing.
The asymmetry of the major scale comes from placing the upper tetrachord a whole step above the lower tetrachord, but not vice versa. So we can distinguish upper and lower. This is not the only choice, and other choices give us other scales. Each tetrachord covers a fourth, so to make a complete octave, we need to add a total of two semitones. We can put them both between lower and upper, giving the major scale, so that the last note of the upper tetrachord, C, is the same as the first note of the lower tetrachord; or both after the upper tetrachord, so that the last note of the lower tetrachord, F, is the first note of the upper tetrachord, giving us the Lydian mode. Conceivably we could divide them equally, with a semitone between lower and upper, and another between upper and lower. This gives the eight note scale, C-D-E-F-F#-G#-A#-B-C, a scale for which I don't know a name. It sounds kinda like a whole tone scale, but with passing tones inserted. Like the whole tone scale, it is symmetrical, or at least, more symmetrical than the major scale. Might be worth looking at what scales are generated by those choices for the other tetrachords.
We could also symmetrize by not demanding we end up with an octave. You lose the idea of the scale as a closed circle, but maybe it could work anyway. Without adding any gaps, just extending the TTs (Lydian) tetrachord, two fourths make a m7, and you start moving through the circle of fifths as you go up the scale, every part sounding kind of in a major key, but mutating as you go: C-D-E-F, F-G-A-Bb, Bb-C-D-Eb, and so on. Or if you put a whole step between both upper and lower, you get something similar, but moving up by fifths instead of by fourths: C-D-E-F, G-A-B-C, D-E-F#-G, A-B-C#-D, ...
You might have noticed I skipped harmonic minor. It's that minor third, which doesn't occur in any permutation of TTs. Wikipedia puts this into a separate class of tetrachords, calling it chromatic, with two semitones and a minor 3rd. Lots more scales can come out of here no doubt, and almost certainly as well from another class of tetrachord that I wouldn't have thought of, because you can't play it on a piano: major 3rd, quartertone, quartertone.
Finally, why a fourth? This, at least, I have a good answer for. It's the smallest perfect interval, and the perfect intervals are unique, not because they have simple integer ratios of frequencies (though they do), and not because we call them a special name (though we do), but rather because it is for perfect intervals alone that we have hardware in our brains to measure. Perfect intervals are innate, all other intervals are learned and cultural. And the fourth is the smallest one.
So I didn't figure out major and minor, but I did manage to clear up a longstanding confusion of mine over the symmetries, and lack thereof, of the basic major scale. Seems like enough for an evening.