Meantone generators

Having done this for Miracle and Orwell, I don't want to leave Meantone
out of it, so here is the 120th row of the Farey sequence between 11/19
and 7/12 (which means every rational number with denominator less than
121 between the values of 11/19 and 7/12, inclusive, ordered by size):

[11/19, 62/107, 51/88, 40/69, 69/119, 29/50, 47/81, 65/112, 18/31,
61/105, 43/74,
68/117, 25/43, 57/98, 32/55, 39/67, 46/79, 53/91, 60/103, 67/115,7/12]

"Poptimal" 5-limit generators are 29/50<47/81<65/112, coming nearly to
18/31 but not quite making it. However,
18/31 is poptimal as a 7-limit generator for standard septimal meantone,
for the 11-limit version of this we have
65/112 as a poptimal generator, so 112 has the distinction of being
poptimal for the 5, 7, and 11 limits. Despite all that, 31 equal which is
very close is probably the obvious choice for meantone.

None of this takes into account the use of meantone to temper a fixed
number of notes, such as 12 or 19; in those cases we might want to adjust
towards the equal values, which explains how eg 55-equal gets into the
picture. If we leave that out, and weight all of the consonaces we
propose to approximate the same, then the optimal range is centered
fairly narrowly around 65/112.

--- In tuning@yahoogroups.com, Gene W Smith <genewardsmith@j...>

/tuning/topicId_42350.html#42350

>
> "Poptimal" 5-limit generators are 29/50<47/81<65/112, coming nearly
to
> 18/31 but not quite making it. However,
> 18/31 is poptimal as a 7-limit generator for standard septimal
meantone,
> for the 11-limit version of this we have
> 65/112 as a poptimal generator, so 112 has the distinction of being
> poptimal for the 5, 7, and 11 limits. Despite all that, 31 equal
which is
> very close is probably the obvious choice for meantone.
>

***"Poptimal." Gene, this is cute. What does it mean??

Thanks!

Joe Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
<jpehrson@r...> wrote:

> ***"Poptimal." Gene, this is cute. What does it mean??

Poptimal = p-optimal. For some p, 2 <= p <= infinity, we have that
the generator is optimal for unweighted pth powers of the absolute
values of the errors. Another one for Monzo's dictionary?

hi Gene,

> From: <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, February 18, 2003 9:23 PM
> Subject: [tuning] Re: Meantone generators
>
>
> --- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
> <jpehrson@r...> wrote:
>
> > ***"Poptimal." Gene, this is cute. What does it mean??
>
> Poptimal = p-optimal. For some p, 2 <= p <= infinity, we have that
> the generator is optimal for unweighted pth powers of the absolute
> values of the errors. Another one for Monzo's dictionary?

thanks.

http://sonic-arts.org/dict/poptimal.htm

i know you've posted pages of lists of poptimal
generators, but i didn't know what it meant either,
and never bothered to ask.

how about some nice examples which i can add
to the webpage?

-monz

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"

/tuning/topicId_42350.html#42429

<gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
> <jpehrson@r...> wrote:
>
> > ***"Poptimal." Gene, this is cute. What does it mean??
>
> Poptimal = p-optimal. For some p, 2 <= p <= infinity, we have that
> the generator is optimal for unweighted pth powers of the absolute
> values of the errors. Another one for Monzo's dictionary?

***"p" is between 2 and infinity?? Whyzzat?

jp

--- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
<jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
>
> /tuning/topicId_42350.html#42429
>
> <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
> > <jpehrson@r...> wrote:
> >
> > > ***"Poptimal." Gene, this is cute. What does it mean??
> >
> > Poptimal = p-optimal. For some p, 2 <= p <= infinity, we have that
> > the generator is optimal for unweighted pth powers of the
absolute
> > values of the errors. Another one for Monzo's dictionary?
>
>
> ***"p" is between 2 and infinity?? Whyzzat?
>
> jp

i've been trying to convince gene to lower the 2 down at least to a
1, but to no avail . . .

>Poptimal = p-optimal. For some p, 2 <= p <= infinity, we have that
>the generator is optimal for unweighted pth powers of the absolute
>values of the errors.

I'm having trouble understanding what's it's optimal for minimizing.
The sum of the pth powers of the errors when p is... what? A limit
for a single p as it goes to infinity? The sum of the sums for each
p up to infinity?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Poptimal = p-optimal. For some p, 2 <= p <= infinity, we have that
> >the generator is optimal for unweighted pth powers of the absolute
> >values of the errors.
>
> I'm having trouble understanding what's it's optimal for minimizing.
> The sum of the pth powers of the errors when p is... what?

p is some particular number or numbers 2<=p<="infinity". You don't
actually calculate p, as that would be difficult; you merely show at
least one such p exists.

A limit
> for a single p as it goes to infinity?

The limit of the pth root of the pth powers of the absolute values of
all of the errors has the dimensions--for instance, cents--of the
error. As p goes to infinity this gives us our standard minimax
solution--we use only this unique solution for the definition of
poptimal.

>p is some particular number or numbers 2<=p<="infinity". You don't
>actually calculate p, as that would be difficult; you merely show at
>least one such p exists.

That satisfies what condition?

>The limit of the pth root of the pth powers of the absolute values of
>all of the errors has the dimensions--for instance, cents--of the
>error. As p goes to infinity this gives us our standard minimax
>solution--we use only this unique solution for the definition of
>poptimal.

The way I'm currently reading this, the poptimal generator is
just the optimal minimax generator (which is nothing new).

I wonder, if you showed the calculation in standard math notation,
if it would involve any notation I don't already understand or
could look up on mathworld...

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I wonder, if you showed the calculation in standard math notation,
> if it would involve any notation I don't already understand or
> could look up on mathworld...

Suppose we let

E(x, p)=(|x+1-log2(3)|^p + |4*x-log2(5)|^p + |3*x-1-log2(5/3)|^p)^1/p

and then define

G(p) = minimum over x of E(x, p)

Then if there exists a value of p, p>=2 such that G(p)=g, we call g a
poptimal generator for 5-limit meantone. For instance, there is some
p greater than 4 for which G(p)=47/81, so this is a poptimal
generator for 5-limit meantone.

after generating the graph, i decided to attempt to localize the
minimum, and it appears that it occurs at

p=2.8149

where the optimal generator is

696.1146 cents.

any generator between that and the 1/4-comma meantone generator

696.5784 cents

is therefore poptimal for meantone.

. . . that is, if you weight all the 5-limit consonance equally . . .

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:

> G(p) = minimum over x of E(x, p)

What I meant was the value of x giving such a minimum; sorry for
being unclear.

--- In tuning@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> after generating the graph, i decided to attempt to localize the
> minimum, and it appears that it occurs at
>
> p=2.8149
>
> where the optimal generator is
>
> 696.1146 cents.
>
> any generator between that and the 1/4-comma meantone generator
>
> 696.5784 cents
>
> is therefore poptimal for meantone.

Which means 50 really and truly doesn't quite make it. The best we
can do is still 47/81, though of course there are an infinity of
other possibilities, such as 65/112, 76/131, 83/143, etc.

> . . . that is, if you weight all the 5-limit consonance
equally . . .

There's always that.

hi paul,

> From: <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 21, 2003 1:51 PM
> Subject: [tuning] Re: Meantone generators
>
>
> after generating the graph, i decided to attempt to localize the
> minimum, and it appears that it occurs at
>
> p=2.8149
>
> where the optimal generator is
>
> 696.1146 cents.
>
> any generator between that and the 1/4-comma meantone generator
>
> 696.5784 cents
>
> is therefore poptimal for meantone.
>
> . . . that is, if you weight all the 5-limit consonance equally . . .

i've added it to the dictionary
http://sonic-arts.org/dict/poptimal.htm

now ... can you explain why the size of the 1/4-comma
meantone generator plays such an important role here?

-monz

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

>
> i've added it to the dictionary
> http://sonic-arts.org/dict/poptimal.htm

can't look now, i'm at a freeosk.
>
>
> now ... can you explain why the size of the 1/4-comma
> meantone generator plays such an important role here?
>
>
it's the largest poptimal generator -- that's all.

it (1/4-comma meantone) happens to be optimal for 0<p<1 and also in
the limit as p goes to infinity. in between p=1 and p=infinity, you
get an optimal generator that is smaller (or larger, if you're
considering it a fourth instead of a fifth), as the graph shows.

>
> -monz

hi paul,

> From: <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, February 22, 2003 12:25 PM
> Subject: [tuning] Re: Meantone generators
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> >
> > i've added it to the dictionary
> > http://sonic-arts.org/dict/poptimal.htm
>
> can't look now, i'm at a freeosk.
> >
> >
> > now ... can you explain why the size of the 1/4-comma
> > meantone generator plays such an important role here?
> >
> >
> it's the largest poptimal generator -- that's all.
>
> it (1/4-comma meantone) happens to be optimal for 0<p<1 and also in
> the limit as p goes to infinity. in between p=1 and p=infinity, you
> get an optimal generator that is smaller (or larger, if you're
> considering it a fourth instead of a fifth), as the graph shows.

ah, OK, now i get it ... but i still don't understand
what "p" is. can you or Gene (or someone) explain more?

-monz

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

> ah, OK, now i get it ... but i still don't understand
> what "p" is. can you or Gene (or someone) explain more?

It's the exponent for least pth powers. In other words, for least
squares, p=2, for least fourth powers, p=4, etc.

Hi all, and greetings from miserable Pennsylvania!

I'm here interviewing for an IT job at Lehigh University, so I won't
be able to follow the lists properly until I return on the 28th.
Nevertheless...

>Suppose we let
>
>E(x, p)=(|x+1-log2(3)|^p + |4*x-log2(5)|^p + |3*x-1-log2(5/3)|^p)^1/p
>
>and then define
>
>G(p) = minimum over x of E(x, p)
//
>>G(p) = minimum over x of E(x, p)
>
>What I meant was the value of x giving such a minimum; sorry for
>being unclear.
//
>Then if there exists a value of p, p>=2 such that G(p)=g, we call g a
>poptimal generator for 5-limit meantone.

So check the following:

So for a given p, there may or may not be a single minimum x.
If there is, it's a poptimal generator.
For p=infinity the poptimal generator is the minimax-optimal generator.

>For instance, there is some p greater than 4 for which G(p)=47/81,
>so this is a poptimal generator for 5-limit meantone.

Oh, you have a method for doing this. How many poptimal generators
are there, usually? Is there always an infinite number, is there
ever just one, etc.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> So for a given p, there may or may not be a single minimum x.
> If there is, it's a poptimal generator.

It's not required to be single, just a minimum.

> For p=infinity the poptimal generator is the minimax-optimal
generator.
>
> >For instance, there is some p greater than 4 for which G(p)=47/81,
> >so this is a poptimal generator for 5-limit meantone.
>
> Oh, you have a method for doing this.

Continuity; that was why there was some math in my first posting on
this topic on tuning-math.

How many poptimal generators
> are there, usually? Is there always an infinite number, is there
> ever just one, etc.

Either just one, or uncountably infinite.

>>>For instance, there is some p greater than 4 for which G(p)=47/81,
>>>so this is a poptimal generator for 5-limit meantone.
>>
>>Oh, you have a method for doing this.
>
>Continuity; that was why there was some math in my first posting on
>this topic on tuning-math.

So you just check when the values stop going down as you change p?

>How many poptimal generators are there, usually? Is there always an
>infinite number, is there ever just one, etc.
>
>Either just one, or uncountably infinite.

So what is the significance of this method? 'The next best thing
if we can't agree on an ideal p.'?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>>For instance, there is some p greater than 4 for which G(p)
=47/81,
> >>>so this is a poptimal generator for 5-limit meantone.
> >>
> >>Oh, you have a method for doing this.
> >
> >Continuity; that was why there was some math in my first posting
on
> >this topic on tuning-math.
>
> So you just check when the values stop going down as you change p?

Matlab can do that, but I haven't figured out how to get Maple up to
the job, so I don't even do that much.

> >How many poptimal generators are there, usually? Is there always
an
> >infinite number, is there ever just one, etc.
> >
> >Either just one, or uncountably infinite.
>
> So what is the significance of this method? 'The next best thing
> if we can't agree on an ideal p.'?

It gives you a range of "ideal" values, out of which you can pick
what you like--for instance a closed system or something where the
ratio of the beats is simple, or whatever.

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >How many poptimal generators are there, usually? Is there always
> an
> > >infinite number, is there ever just one, etc.
> > >
> > >Either just one, or uncountably infinite.
> >
> > So what is the significance of this method? 'The next best thing
> > if we can't agree on an ideal p.'?
>
> It gives you a range of "ideal" values, out of which you can pick
> what you like--for instance a closed system or something where the
> ratio of the beats is simple, or whatever.

It doesn't sound like a significant advance on how we used to do it.
Which was simply to quote both the RMS and Max-Absolute (p=2 and
p=infinity) optimal generators and assume that anything in between (or
near-by) will be acceptable.

It's a confusing term, even when you know it is really "p-optimal" and
you know what "p" is. One expects <any-kind-of>-optimal to be a single
value.

I don't see the point of it.

--- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > >How many poptimal generators are there, usually? Is there
always
> > an
> > > >infinite number, is there ever just one, etc.
> > > >
> > > >Either just one, or uncountably infinite.
> > >
> > > So what is the significance of this method? 'The next best
thing
> > > if we can't agree on an ideal p.'?
> >
> > It gives you a range of "ideal" values, out of which you can pick
> > what you like--for instance a closed system or something where
the
> > ratio of the beats is simple, or whatever.
>
> It doesn't sound like a significant advance on how we used to do it.
> Which was simply to quote both the RMS and Max-Absolute (p=2 and
> p=infinity) optimal generators and assume that anything in between
(or
> near-by) will be acceptable.

and don't forget the p=1 or MAD value.

> It's a confusing term, even when you know it is really "p-optimal"
and
> you know what "p" is. One expects <any-kind-of>-optimal to be a
single
> value.
>
> I don't see the point of it.

the point, for me, is that if you're willing to accept p=2 *and*
p=infinity, you should also be willing to accept 2.812, and as we've
seen, this choice results in an optimal meantone generator that is
*not* in the range between the p=2 (or p=1) and the p=infinity
values, so can't be ruled out even if you know you'd like to weight
the 5-limit consonances equally.

by the way, i certainly don't know if i'd like to do the latter --
for me, 3/14-comma meantone might really be the "optimal" one -- i
can't "disprove" this to my ears, especially given the limited tuning
resolution of my ensoniq vfx-sd (i hear these are going for $300
these days) . . .

>>I don't see the point of it.
>
>the point, for me, is that if you're willing to accept p=2 *and*
>p=infinity, you should also be willing to accept 2.812, and as we've
>seen, this choice results in an optimal meantone generator that is
>*not* in the range between the p=2 (or p=1) and the p=infinity
>values, so can't be ruled out even if you know you'd like to weight
>the 5-limit consonances equally.

I think it's somewhat academic, really. The particulars of the
instrument and listening conditions, etc. will have much to say
at this level. In a perfect world the exponent should be chosen
empirically -- it should be the one that best matches judgements
of random mistunings of just chords. . .

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>I don't see the point of it.
> >
> >the point, for me, is that if you're willing to accept p=2 *and*
> >p=infinity, you should also be willing to accept 2.812, and as
we've
> >seen, this choice results in an optimal meantone generator that is
> >*not* in the range between the p=2 (or p=1) and the p=infinity
> >values, so can't be ruled out even if you know you'd like to
weight
> >the 5-limit consonances equally.
>
> I think it's somewhat academic, really.

;) you bet! this is what i was getting at with the last part of my
post, which you snipped . . .

> The particulars of the
> instrument and listening conditions, etc. will have much to say
> at this level. In a perfect world the exponent should be chosen
> empirically -- it should be the one that best matches judgements
> of random mistunings of just chords. . .
>
> -Carl

these judgments will probably show an asymmetry with regard to otonal
vs. utonal, so the whole idea of a single value for p will be out the
window. still, i think the above was a logical reply to what dave was
suggesting . . .

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> > From: <wallyesterpaulrus@y...>
> > To: <tuning@yahoogroups.com>
> > Sent: Saturday, February 22, 2003 12:25 PM
> > Subject: [tuning] Re: Meantone generators
> >
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > >
> > > i've added it to the dictionary
> > > http://sonic-arts.org/dict/poptimal.htm
> >
> > can't look now, i'm at a freeosk.
> > >
> > >
> > > now ... can you explain why the size of the 1/4-comma
> > > meantone generator plays such an important role here?
> > >
> > >
> > it's the largest poptimal generator -- that's all.
> >
> > it (1/4-comma meantone) happens to be optimal for 0<p<1 and also
in
> > the limit as p goes to infinity. in between p=1 and p=infinity,
you
> > get an optimal generator that is smaller (or larger, if you're
> > considering it a fourth instead of a fifth), as the graph shows.
>
>
>
> ah, OK, now i get it ... but i still don't understand
> what "p" is. can you or Gene (or someone) explain more?
>
>
>
> -monz

hi monz,

let me try to link this better with what's already on your meantone
page,

http://sonic-arts.org/dict/meantone.htm

specifically my table of optimal meantone tunings.

the first column corresponds to p=infinity, the second column to p=2,
and the third column to p=1.

i've updated the pop.gif graph to represent all four rows of the
table, but of course on the graph p varies continuously instead of
being restricted to three discrete values:

/tuning/files/perlich/pop.gif

hopefully you can see how the values in the table correspond to
points on the graph . . .

cheers,
paul

>From: "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...>
>--- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
><d.keenan@u...> wrote:
> > It doesn't sound like a significant advance on how we used to do it.
> > Which was simply to quote both the RMS and Max-Absolute (p=2 and
> > p=infinity) optimal generators and assume that anything in between
>(or
> > near-by) will be acceptable.
>
>and don't forget the p=1 or MAD value.
>
> > It's a confusing term, even when you know it is really "p-optimal"
>and
> > you know what "p" is. One expects <any-kind-of>-optimal to be a
>single
> > value.
> >
> > I don't see the point of it.
>
>the point, for me, is that if you're willing to accept p=2 *and*
>p=infinity, you should also be willing to accept 2.812, and as we've
>seen, this choice results in an optimal meantone generator that is
>*not* in the range between the p=2 (or p=1) and the p=infinity
>values, so can't be ruled out even if you know you'd like to weight
>the 5-limit consonances equally.

Of course I had to check it for myself and I found that a power of 2.812 gives an optimal generator that differs by only 0.05 cents from the RMS optimum generator, but yes, it _is_ outside the RMS to Max-Abs range, which is 0.4 cents wide.

That certainly is fascinating from a mathematical perspective, but has anyone found a temperament where a p-optimal value is outside the RMS to Max-Abs range by something that is musically significant? Say by 50% of that range or half a cent?

I also notice that the Mean-Abs (MAD) value is the same as the Max-Abs value in the case of meantone. Has anyone found a case where Mean-Abs is outside the RMS to Max-Abs range by more than 50% of that range, or half a cent?

And as we know, considerations of weighting, or minimising beats or rationalising beat ratios or just plain idiosyncratic personal preference can have a far bigger impact. So, so far, p-optimal doesn't seem worth the trouble to me.
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com