EDO superset containing approximation of Werckmeister III?

could someone please explain how to find an EDO superset
that gives a good approximation of the 12 pitches in
Werckmeister III, with the scale data given here?

/tuning-math/files/dict/werckmeister.htm

-monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> could someone please explain how to find an EDO superset
> that gives a good approximation of the 12 pitches in
> Werckmeister III, with the scale data given here?
>
> /tuning-math/files/dict/werckmeister.htm

I used Manual's scale data rather than trying to figure out where the data was on your page. It turns out that Werckmeister III can be expressed with extreme accuracy in terms of what I call "schismas", steps of the 612 et. In 612-et terms, it is

0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557

hi Gene,

> From: "Gene Ward Smith" <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, October 10, 2002 10:08 PM
> Subject: [tuning-math] Re: EDO superset containing approximation of
Werckmeister III?
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > could someone please explain how to find an EDO superset
> > that gives a good approximation of the 12 pitches in
> > Werckmeister III, with the scale data given here?
> >
> > /tuning-math/files/dict/werckmeister.htm
>
> I used Manual's scale data rather than trying to figure out
> where the data was on your page.

there's a table showing the tunings as a chain of generators.
anyway, i tried it and came up with the same results you did.

> It turns out that Werckmeister III can be expressed with
> extreme accuracy in terms of what I call "schismas", steps
> of the 612 et. In 612-et terms, it is
>
> 0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557

awesome!! i was hoping you'd give some details as to how
you found out that 612edo was the best approximation.

-monz
"all roads lead to n^0"

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

> awesome!! i was hoping you'd give some details as to how
> you found out that 612edo was the best approximation.

I ran a search and 612 came out the best, but other strange-looking
possibilities are out there, such as 200 and 412 (200+412=612, of course.)

> From: "Gene Ward Smith" <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, October 11, 2002 12:37 PM
> Subject: [tuning-math] Re: EDO superset containing approximation of
Werckmeister III?
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > awesome!! i was hoping you'd give some details as to how
> > you found out that 612edo was the best approximation.
>
> I ran a search and 612 came out the best,

well, OK, but ... AARRRGGH! -- *how* did you do that search?

since i'm math-challenged, the only way i know how to do it
is to set up an Excel spreadsheet with the EDO-cardinality
as a variable, but then i have to manually enter each cardinality
and look at the graphs of deviation to see which EDOs are best.

> but other strange-looking possibilities are out there,
> such as 200 and 412 (200+412=612, of course.)

ah, now that's useful! i was hoping to find something smaller
than 612edo which could describe Werckmeister III, and 200
does the trick nicely.

unfortunately, however, neither 200 nor 412 give
integer-divisions for 12edo, so they're not as useful
for comparing Werckmeister III to 12edo as 612edo is.

please, Gene, more info on how your search method works.
do you know how to set it up in an Excel spreadsheet?
if not, then do you have some code that i could run on
my PC? i have Mathematica -- just don't know a lot
about how to use it.

-monz
"all roads lead to n^0"

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

> well, OK, but ... AARRRGGH! -- *how* did you do that search?

Brute force, much like a search for good ets. I totaled up the relative error for each n from 1 to 1000 by running a simple Maple routine, and insisted they be at least as good as 12-et.

> please, Gene, more info on how your search method works.
> do you know how to set it up in an Excel spreadsheet?
> if not, then do you have some code that i could run on
> my PC? i have Mathematica -- just don't know a lot
> about how to use it.

Mathematica is very similar to Maple, but you need to learn how to use it.

Joe and Gene,

I must have told this before but in Scala it's very easy
to do too:

load werck3
fit/mode

This show successively better approximations and stops at
some point. To go beyond that, and show all divisions,
use a negative number:

fit/mode -612

With a positive parameter it only shows that division.

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
>
> Joe and Gene,
>
> I must have told this before but in Scala it's very easy
> to do too:
>
> load werck3
> fit/mode
>
> This show successively better approximations and stops at
> some point. To go beyond that, and show all divisions,
> use a negative number:
>
> fit/mode -612

Nice! Is there a way to go beyond the stop point, and *not* show all divisions? I notice 612 popping up a lot with temperaments I hadn't looked at yet, but the stop point is set too low to easily see it.

Gene wrote:
>Nice! Is there a way to go beyond the stop point, and *not* show all
divisions? I notice 612 popping up a lot >with temperaments I hadn't
looked at yet, but the stop point is set too low to easily see it.

Not at the moment I'm afraid. I'll add another qualifier to the next
version to make this possible.

Manuel