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You are using multiplication where you should be using exponentiation. If one fifth is (3/2) then 2 fifths is (3/2) times (3/2) which is (9/4). Your math looked semi-plausible when comparing fifths and octaves, but comparing 3 thirds and an octave it was way off.

Therefore the first comparison should be 7 octaves which is (2/1)^7 = 128, versus 12 perfect fifths which is (3/2)^12 = 531441/4096 = 129+3057/4096 = 129.746337890625.

Similarly for thirds, you're comparing one octave (2/1) = 1 with 3 thirds (5/4)^3 = 125/128 = 1.953125.

As you can see, the ratios are close, but not quite right. Hence the problem.




I think we just lost some carats in the mix. The original comment meant to type exponentials but they didn't come out right on the page.

Someday HTML will support TeX and we'll never have this problem again. ;)




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