2:1, 4:2, 6:3 octaves

[John Formsma]

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.