w-optimal meantone generators

hi paul,

these graphs you're making are indeed beautiful,
but i sure wish i knew what the heck they're showing.
i don't get it.

i'm still confused about "p", and now you throw
in "w".

please explain, in as much detail as possible.
i'd like to add this stuff to my Dictionary
"meantone" pages.

thanks.

-monz

> From: <tuning@yahoogroups.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, February 25, 2003 3:31 PM
> Subject: [tuning] New file uploaded to tuning
>

>
> Hello,
>
> This email message is a notification to let you know that
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> group.
>
> File : /perlich/woptimal.gif
> Uploaded by : wallyesterpaulrus <wallyesterpaulrus@yahoo.com>
> Description : w-optimal meantone generators
>
> You can access this file at the URL
>
> /tuning/files/perlich/woptimal.gif
>
> To learn more about file sharing for your group, please visit
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> Regards,
>
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--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> these graphs you're making are indeed beautiful,
> but i sure wish i knew what the heck they're showing.
> i don't get it.
>
> i'm still confused about "p", and now you throw
> in "w".
>
> please explain, in as much detail as possible.
> i'd like to add this stuff to my Dictionary
> "meantone" pages.
>
> thanks.

given p and w, the optimal meantone tuning is the one that minimizes

|w*error in perfect fifth (or perfect fourth)|^p +
|error in major third (or minor sixth)|^p +
|error in major sixth (or minor third)|^p.

if you wish, you can divide through by 2+w^p (to get the weighted
mean error) and/or then take the pth root of the whole expression (to
get back units of cents) -- neither operation affect which meantone
minimizes is optimal for a given p and w, though they may become
necessary when w or p goes to infinity.

when w=1 and p=2, we have the woolhouse case -- equal weighting, and
minimizing sum-of-squared error (or, what amounts to the same thing,
RMS error)

w=3/5 corresponds to limit-weighting, w=5/3 corresponds to inverse-
limit-weighting, and w=0 corresponds to weighting only the ratios of
5. w=infinity obviously gives you pythagorean, since the ratios of 5
get zero weight relative to the ratio of 3.

p=infinity means you're just minimizing the maximum (weighted) error,
p=1 means you're minimizing the sum of (weighted) absolute errors
(or, what amounts to the same thing, MAD error).

thus both of the meantone graphs show the "special" meantones already
in my table on your meantone page . . .

get it?

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

/tuning/topicId_42562.html#42562

> hi paul,
>
>
> these graphs you're making are indeed beautiful,
> but i sure wish i knew what the heck they're showing.
> i don't get it.
>
> i'm still confused about "p", and now you throw
> in "w".
>
> please explain, in as much detail as possible.
> i'd like to add this stuff to my Dictionary
> "meantone" pages.
>

***I agree with Monz here that Paul's chart is indeed beautiful. I
think I'll use it for an "improvisatory" piece... ;)

Since I didn't understand "poptimal" (and Dave Keenan says don't
bother), I'm *certain* not to understand "woptimal..."

Still waiting for "do-woptimal...)

J. Pehrson

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42562.html#42565

<>
> given p and w, the optimal meantone tuning is the one that minimizes
>
> |w*error in perfect fifth (or perfect fourth)|^p +
> |error in major third (or minor sixth)|^p +
> |error in major sixth (or minor third)|^p.
>

***Oh... it kinda looks like "w" includes the optimization for the
perfect fifth that Dave Keenan was "complaining" about...

J. Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
<jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
>
> /tuning/topicId_42562.html#42565
>
> <>
> > given p and w, the optimal meantone tuning is the one that minimizes
> >
> > |w*error in perfect fifth (or perfect fourth)|^p +
> > |error in major third (or minor sixth)|^p +
> > |error in major sixth (or minor third)|^p.
> >
>
> ***Oh... it kinda looks like "w" includes the optimization for the
> perfect fifth that Dave Keenan was "complaining" about...

Hi Joseph,

I don't recall writing anything that could be construed as
"complaining" about including the perfect fifth in the optimisation.
If you can let me know the message number, perhaps I can clear it up.
But it's not important.

I didn't exactly say "don't bother" about p-optimal. It definitely
warrants investigation, and Paul's graphs are certainly interesting to
me. But I don't think folks on tuning(not math) should be bothered by
it until and unless those questions I asked are answered in the
positive. i.e. does it ever go significantly outside the range of RMS
to max absolute optima?

his is a new question: If it does go outside RMS to max absolute, does
it go significantly outside the range of mean absolute, RMS, max absolute?

--- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:

But I don't think folks on tuning(not math) should be bothered by
> it until and unless those questions I asked are answered in the
> positive. i.e. does it ever go significantly outside the range of RMS
> to max absolute optima?

You're making a mountain out of a molehill, it seems to me. People
don't need to understand the definition of popimal unless they want
to, but they don't need to understand RMS, mean absolute, max absolute
either, which you seem to think is obligatory while poptimal isn't. All
anyone really needs to know to follow the conversation is that we are
talking about some kind of optimum.

> his is a new question: If it does go outside RMS to max absolute, does
> it go significantly outside the range of mean absolute, RMS, max
absolute?

Who cares? The question is not even well-defined.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
>
> But I don't think folks on tuning(not math) should be bothered by
> > it until and unless those questions I asked are answered in the
> > positive. i.e. does it ever go significantly outside the range of
RMS
> > to max absolute optima?
>
> You're making a mountain out of a molehill, it seems to me. People
> don't need to understand the definition of popimal unless they want
> to, but they don't need to understand RMS, mean absolute, max
absolute
> either, which you seem to think is obligatory while poptimal isn't.

I believe RMS and max absolute (also known as least squares and
minimax) have been in use on this list and its predecessors for at
least a decade.

> All
> anyone really needs to know to follow the conversation is that we
are
> talking about some kind of optimum.
>
> > This is a new question: If it does go outside RMS to max
absolute, does
> > it go significantly outside the range of mean absolute, RMS, max
> absolute?
>
> Who cares?

I care, because there seems to be an awful lot of number crunching
involved in determining a p-optimal range and I would like to know if
it's warranted. I also got the feeling Carl Lumma cared.

> The question is not even well-defined.

You must have missed where I earlier suggested that 50% outside the
range would be significant.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@u...> wrote:

> I believe RMS and max absolute (also known as least squares and
> minimax) have been in use on this list and its predecessors for at
> least a decade.

Poptimal puts it all in one neat package, saving wear and tear on the
brain. Of the 600 odd (sometimes very odd) people on this list, how
many know what RMS means?

> I care, because there seems to be an awful lot of number crunching
> involved in determining a p-optimal range and I would like to know if
> it's warranted. I also got the feeling Carl Lumma cared.

Paul is the only one who has done that. Why do you think this matters?

--- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"

/tuning/topicId_42562.html#42674
>
> I didn't exactly say "don't bother" about p-optimal. It definitely
> warrants investigation, and Paul's graphs are certainly interesting
to me. But I don't think folks on tuning(not math) should be bothered
by it until and unless those questions I asked are answered in the
> positive. i.e. does it ever go significantly outside the range of
RMS to max absolute optima?
>

***Hi Dave!

Wasn't the RMS method the method that you and Paul used to derive
Blackjack?? Just kinda curious...

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
>
> /tuning/topicId_42562.html#42674
> >
> > I didn't exactly say "don't bother" about p-optimal. It definitely
> > warrants investigation, and Paul's graphs are certainly interesting
> to me. But I don't think folks on tuning(not math) should be bothered
> by it until and unless those questions I asked are answered in the
> > positive. i.e. does it ever go significantly outside the range of
> RMS to max absolute optima?
> >
>
> ***Hi Dave!
>
> Wasn't the RMS method the method that you and Paul used to derive
> Blackjack?? Just kinda curious...
>
> Joseph

Hi Joseph,

No. Optimum generators were not relevant to that at all. We were
looking in 72-ET, because that's where you asked us to look. And so
any generator was bound to be an interval of 72-ET, not necessarily
any kind of optimum. But as it turns out, 7/72 oct is so close to most
kinds of optima for the secor that it barely matters.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@u...> wrote:

/tuning/topicId_42562.html#42809

>
> Hi Joseph,
>
> No. Optimum generators were not relevant to that at all. We were
> looking in 72-ET, because that's where you asked us to look. And so
> any generator was bound to be an interval of 72-ET, not necessarily
> any kind of optimum. But as it turns out, 7/72 oct is so close to
most kinds of optima for the secor that it barely matters.

***Well, that sounds like "good luck" to me! :)

Didn't Paul Erlich do *some* kind of optimization calculations
regarding Blackjack when it was being invented??

Paul, do you remember any such??

J. Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@u...> wrote:
>
> /tuning/topicId_42562.html#42809
>
> >
> > Hi Joseph,
> >
> > No. Optimum generators were not relevant to that at all. We were
> > looking in 72-ET, because that's where you asked us to look. And
so
> > any generator was bound to be an interval of 72-ET, not
necessarily
> > any kind of optimum. But as it turns out, 7/72 oct is so close to
> most kinds of optima for the secor that it barely matters.
>
> ***Well, that sounds like "good luck" to me! :)
>
> Didn't Paul Erlich do *some* kind of optimization calculations
> regarding Blackjack when it was being invented??

sure, after it was invented, and after dave and others had already
done some others.