Re: [tuning-math] Re: 43-edo (was: 171-EDO...)

----- Original Message -----
From: paulerlich <paul@stretch-music.com>
To: <tuning-math@yahoogroups.com>
Sent: Saturday, February 02, 2002 2:11 AM
Subject: [tuning-math] Re: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT
reduced bases for ets))

> --- 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).

so that's what manuel means by "geometric mean"? i would
have never understood it that way

ok, i see it now ... but it took a little bit of work to
comprehend what's going on there ... perhaps you'd like to
reword that a bit, manuel?

isn't there a better way to say that, with a quantifiable
numeric? ... perhaps "almost exactly the geometric mean
1/5 part of the 5-tone interval composed of three 9/8 and
two 10/9 whole tones"?

anyway,

1/5-comma meantone whole-tone

= ( (9/8)^3 * (10/9)^2 )^(1/5)

= [2 3 5]^[-7/5 2/5 2/5] (did i write that correctly?)

= ~195.3074859 cents

7 steps of 47-edo

= 2^(7/43)

= ~195.3488372 cents

The difference between them is

[2 3 5]^[336/215 -2/5 -2/5]

~0.041351317 cent = ~1/24 cent = ~1 jot

and paul, there might be a hint of an answer (from
the "new cylindrical meantones lattice" thread) as to
how to draw spirals which compare a meantone-like edo
with a fraction-of-a-comma meantone

we need to include an axis for 2 ... but hmmm ...
this pair of points would be rather far apart, since
the exponents of 3 and 5 have their signs reversed

and of course, what would including 2 mean when
the lattice is wrapped into a cylinder?

my spatial imagination can't handle this ...

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> > 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).
>
>
> so that's what manuel means by "geometric mean"? i would
> have never understood it that way
>
> ok, i see it now ... but it took a little bit of work to
> comprehend what's going on there ... perhaps you'd like to
> reword that a bit, manuel?

manuel worded it just fine. it's the geometric mean of three 9/8
whole tones and two 10/9 whole tones:

( (9/8) * (9/8) * (9/8) * (10/9) * (10/9) )^(1/5).