After thinking about this today, here's how it would work in real life. If you keep the A2-A4 around 1 bps, you will get a near-perfect to perfect 12th. Let's say F2-A4 beats 4.0 bps. F2-A3 would beat a little slower -- maybe 3.5 bps. F2-A2 would beat 3.0 bps, making the A2-A4 double octave beat 1.0 bps. If your fourths are around 1.0 bps, A2-D3 will of course beat 1.0 bps. This means the your F2-D3 sixth will beat 4.0 bps, which is what F2-A4 is. Hence, a pefect twelfth...or near-perfect depending on inharmonicity. JF On 8/22/07, Jason Kanter <jkanter at rollingball.com> wrote: > Double octave, yes. But within this, check all the twelfths -- they should > be as close to pure, beatless as possible and this will guarantee the right > amount of stretch. The test for a perfect 12th is a sixth below the lower > note. That is: to test C4-G5, use Eb3 against the C4 (a sixth that beats at > the frequency of G5) and Eb against G5 - should beat the same. This will > almost always give you an octave stretch that is the sweet spot between 4:2 > and 6:3. > Note - mathematically perfect ET twelfths in a world without inharmonicity > would be narrow. Inharmonicity stretches them. The spot of the perfect 12th > turns out to be a great choice for the stretch because the 3rd partial is > usually very strong. > > Perfect twelfths are also an excellent test up into the high treble.
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