43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets))

------------

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, February 02, 2002 1:50 AM
> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced
bases for ets)
>
> from manuel's page:
> http://www.xs4all.nl/~huygensf/doc/measures.html
>
> >> ... Sauveur ... found 43 to be optimal
> >> because 4 steps is almost exactly a 16/15 minor second
> >> and 7 steps almost exactly the geometric mean of
> >> three 9/8 and two 10/9 whole tones. The chromatic scale
> >> contained in 43-tET is virtually identical to 1/5-comma
> >> meantone tuning.

>
>
>
> [-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone)
> +
> [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone)
> ----------
> [-7 2 2] (= 225:128 "augmented 6th")
>
>
> [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents
>
>
> but what significance does that have? i don't get it
>
> manuel?

the only thing that i think i can see is some kind of
tritone-equivalence in action, because if you ignore
prime-factor 2 you get a mean for the 225:128 of 15:8,
which is 2^(1/2) higher than the above interval, and
which is the interval that is given exactly by 5 generators
of 1/5-comma meantone

but i really don't understand what's going on

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> ------------
>
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Saturday, February 02, 2002 1:50 AM
> > Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT
reduced
> bases for ets)
> >
> > from manuel's page:
> > http://www.xs4all.nl/~huygensf/doc/measures.html
> >
> > >> ... Sauveur ... found 43 to be optimal
> > >> because 4 steps is almost exactly a 16/15 minor second
> > >> and 7 steps almost exactly the geometric mean of
> > >> three 9/8 and two 10/9 whole tones. The chromatic scale
> > >> contained in 43-tET is virtually identical to 1/5-comma
> > >> meantone tuning.
>
>
>
>
>
> >
> >
> >
> > [-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone)
> > +
> > [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone)
> > ----------
> > [-7 2 2] (= 225:128 "augmented 6th")
> >
> >
> > [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents
> >
> >
> > but what significance does that have? i don't get it
> >
> > manuel?
>
>
>
> the only thing that i think i can see is some kind of
> tritone-equivalence in action, because if you ignore
> prime-factor 2 you get a mean for the 225:128 of 15:8,
> which is 2^(1/2) higher than the above interval, and
> which is the interval that is given exactly by 5 generators
> of 1/5-comma meantone
>
> but i really don't understand what's going on

you have to take the weighted mean of three 9/8 and two 10/9 whole
tones. that means three 9/8s "plus" two 10/9s "divided" by five, that
is, ( (9/8)^3 * (10/9)^2 )^(1/5).