In praise of 400-EDO (was: Ozan's 159-edo-based tuning)

Hi all,

On Wed, 22 Feb 2006, Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > > What, in a perfect world, would these three sizes of
> > > fifth be?
> >
> > 696, 702, 708 cents or so.
>
> 224-et seems like a good choice then.

200-EDO seems perfect!

With every step being 6 cents, 702 cents is 117 steps,
and the other two are just one step more or less.

Mind you, all the thirds are "horrible"! That is, if you
find a discrepancy of around 3 cents to be too much.

Minor third:
A 6/5 minor third is 315.641287 c.
53 X 6 c = 318 cents, error ~= 2.36 c.

Neutral third(1):
A 49/40 neutral third is 351.3380991 c.
59 X 6 c = 354 cents, error ~= 2.66 c.

Neutral third(2):
A sqrt(3/2) neutral third is 350.9775004 c.
58 X 6 c = 348 cents, error ~= -2.98 c.

Major third:
A 5/4 major third is 386.3137139 c.
64 X 6 c = 384 cents, error ~=2.31 c.

-----

Since these are all within 1 cent of a half-step of
the 200-EDO gamut, I guess it would make sense
to choose a 400-EDO gamut instead.

Then we have in 400-EDO, with errors < 1 cent:
Minor third of 6/5: 105 steps
Neutral third(1) of 49/40: 117 steps
Neutral third(2) of sqrt(3/2): 117 steps
Major third of 5/4: 127 steps
Fifth of 696 cents: 232 steps
Fifth of 702 cents: 234 steps
Fifth of 708 cents: 236 steps

Regards,
Yahya

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Why not up the ante and go all the way to 600? Forget cents, let's use pence
instead.

Is it a schismatic system?

SNIP

>
>
> 200-EDO seems perfect!
>
> With every step being 6 cents, 702 cents is 117 steps,
> and the other two are just one step more or less.
>
> Mind you, all the thirds are "horrible"! That is, if you
> find a discrepancy of around 3 cents to be too much.
>
> Minor third:
> A 6/5 minor third is 315.641287 c.
> 53 X 6 c = 318 cents, error ~= 2.36 c.
>
> Neutral third(1):
> A 49/40 neutral third is 351.3380991 c.
> 59 X 6 c = 354 cents, error ~= 2.66 c.
>
> Neutral third(2):
> A sqrt(3/2) neutral third is 350.9775004 c.
> 58 X 6 c = 348 cents, error ~= -2.98 c.
>
> Major third:
> A 5/4 major third is 386.3137139 c.
> 64 X 6 c = 384 cents, error ~=2.31 c.
>
> -----
>
> Since these are all within 1 cent of a half-step of
> the 200-EDO gamut, I guess it would make sense
> to choose a 400-EDO gamut instead.
>
> Then we have in 400-EDO, with errors < 1 cent:
> Minor third of 6/5: 105 steps
> Neutral third(1) of 49/40: 117 steps
> Neutral third(2) of sqrt(3/2): 117 steps
> Major third of 5/4: 127 steps
> Fifth of 696 cents: 232 steps
> Fifth of 702 cents: 234 steps
> Fifth of 708 cents: 236 steps
>
> Regards,
> Yahya
>