Really, what this comes down to is that there is no power of 3/2 that is also an exact power of 2. The 3/2 ratio is a perfect fifth; if you apply it twelve times you get 129.746338, which is just a bit higher than 128, the seventh power of 2—that is, twelve fifths are very close to seven octaves. But you can't make it exact; arithmetic won't allow it.
The Pythagorean approach did it all with 3:2 and 2:1 ratios, so their major third was 81:64, just a bit sharp from 80:64 or 5:4, and it apparently sounded awful. Later systems allowed a 5:4 ratio, so you could take a major third (5:4) followed by a minor third (6:5) to get a perfect fifth (6:4 or 3:2). After that it gets really complicated and I've never learned all the weird numerological variants. I'd always been told that the modern tuning used the twelfth root of 2, or 1.05946309, for a semitone. In an odd way that seems like going from monarchy, with one ruling tone, to democracy, with all tones equal. . . .
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Date: 2010-04-25 04:06 am (UTC)The Pythagorean approach did it all with 3:2 and 2:1 ratios, so their major third was 81:64, just a bit sharp from 80:64 or 5:4, and it apparently sounded awful. Later systems allowed a 5:4 ratio, so you could take a major third (5:4) followed by a minor third (6:5) to get a perfect fifth (6:4 or 3:2). After that it gets really complicated and I've never learned all the weird numerological variants. I'd always been told that the modern tuning used the twelfth root of 2, or 1.05946309, for a semitone. In an odd way that seems like going from monarchy, with one ruling tone, to democracy, with all tones equal. . . .