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

> 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).