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

[John Formsma]

Yes, near-perfect to perfect 12s sound good in the pianos they fit.

Remember we're talking to someone who is *learning* to fit stuff in
aurally.  Maybe also telling Matthew how to aurally construct a P12
tuning would be helpful.

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.