Perfect Fifths in Equal Tempered Scales (Part 2)

When I listed some equal-tempered scales with good perfect fifths in Part 1, a reader named Sylvain noticed something interesting. The scales with 5, 7, 12, 29, 41 and 53 tones are particularly good, and

29 = 12 + 12 + 5
41 = 12 + 12 + 12 + 5
53 = 12 + 12 + 12 + 12 + 5

I asked if this is a coincidence. And now I know the answer: no, it’s not! There really is a reason for this pattern.

As we’ll see, part of the reason is that

7/12 ≈ 0.58333333333…

is very close to

log(3/2)/log(2) ≈ 0.58496...