I've never liked explaining the scale as a Pythagorean derivation. It's not really correct historically (Greek music didn't have anything approximating a full

I've never liked explaining the scale as a Pythagorean derivation. It's not really correct historically (Greek music didn't have anything approximating a full major scale) or mathematically (it doesn't understand the idea of a "third" interval the way tonal music does, so playing triads with pythagorean tuning sounds awful!). Here's my take: late medieval singers discovered The Major Chord. That's the combination of three (!) notes that is "most consonant" (mathematically: beats in the shortest period). This combines two notes a major fifth apart (ratio 3:2), with a third note that is 5:4 with the low note. You can write some code to prove this if you like. So now take that "best" chord with its three notes, and start moving it around. If you go up a fifth (i.e. by "the most consonent interval", that is the "closest best chord to your first best chord") you can play the same chord, adding two needed notes that weren't in the scale before. You can likewise go down a fifth to add two new notes. Then you compress these seven notes into a single octave, and you get... the major scale! It's just there. All you need is that one "best" three-note chord and an obvious metric for "nearest" (i.e. transpose by a fifth) and you have almost all of modern tonal music. Play the same tunes starting on different notes and you get "modalities", etc... You can transpose up and down to nearby keys and keep playing by "cheating" with your tunings to move a note half way up or down. And the practice of formalizing those transpositional cheats because what we now know as the equitempered scale. But they're still just cheats. And the fact that pow(2, 1.0/12) happens to work is, basically, just dumb luck.