>Not counting the octave, there are seven fairly nice fractions here. Hmm. Seven. What a conspicuous number.

There's seven nice fraction because of how they were rounded.

But there's no explanation of why he rounds 1.682 to 5/3 but doesn't round 1.587 to the even closer 8/5.

It's because scales are a social construction, not a result of vague math.