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To approximate the major fifth 3:2 ratio, approximations of log[2](3/2) are needed. The partial quotients of the continuous fraction are

[0, 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2, 1, 1, 55, 1, ... ]

the first few approximants are

0, 1, 1/2, 3/5, 7/12, 24/41, 31/53, 179/306, 389/665, 9126/15601, ...

so the 5, 12, 41, 53, 306, 665 tone scales as good for the 3:2, 4:3, 8:3... harmonics as you can cope for. Especially 12, 53 and 665 tone scales are good, because the next partial quotients 3, 5 or 23 are large. (The devil must be using the 666-scale.)

What about other important ratios?

The continuous fraction for 5:4 is [0, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18, 1, ...]
with the first two approximants 0, 1/3, 9/28, 19/59, 47/146, 207/643, 1289/4004...
Here we have 1/3=4/12 (and there is nothing better until the denominator 28). This luck is a bit of coincidence.

The continuous fraction for 5:3 is [0, 1, 2, 1, 4, 22, 4, 1, 1, 13, 137, 1, 1, ...]
with the first two approximants 0, 1, 2/3, 3/4, 14/19, 311/422, 1258/1707....
Here again, 2/3=8/12 and 3/4=9/12 (and there is nothing better until the denominator 19), which must be very convenient for the major and minor subtonalities...

So the 12 is helped by the fact that it is richly divisible.

by das monde on Wed May 14th, 2008 at 06:44:07 AM EST
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