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Quick mathematical summary of the problem:

An octave is a 2:1 ratio between frequencies, so 880 hz is one octave above 440 hz. A perfect fifth is a 3:2 ratio between frequencies, so 660 hz is a perfect fifth above 440.

In the modern western system of music, twelve perfect fifths is harmonically equal to seven octaves. In other words,

(2/1)7 == (3/2)12

Unfortunately, we know this is mathematically untrue.

Furthermore, three major thirds is harmonically equal to one octave:

(2/1) == (5/4)3

This also is mathematically untrue.

Hilarity ensues.

I'll also summarize the advantage of equal temperament:

Regardless of what key the song is in, a certain interval is always the same exact ratio. A major third in the key of F is the same as a major third in the key of Bb. This is good for instruments like the guitar and piano, which aren't made or tuned for a single immovable key. Contrast that with harmonicas, for examples, each of which is made for only a certain key.

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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