Notating Pajara

This is the system with wedgie [2, -4, -4, 2, 12, -11] which we used to call Paultone. It has [1/2, 5/56] as poptimal in both the 7-limit and the 9-limit, and my recommendation is that the 56-et be used to notate it. The alternative is 22, but with all due respect for Paul's favorite division, 56 is in much better tune. As a way of tuning a 22-tone MOS and playing Decatonic, it's something Paul might try if he hasn't already.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> This is the system with wedgie [2, -4, -4, 2, 12, -11] which we used
to call Paultone. It has [1/2, 5/56] as poptimal in both the 7-limit
and the 9-limit, and my recommendation is that the 56-et be used to
notate it. The alternative is 22, but with all due respect for Paul's
favorite division, 56 is in much better tune. As a way of tuning a
22-tone MOS and playing Decatonic, it's something Paul might try if he
hasn't already.
>

I believe 56-ET would notate a chain of 22 just the same as 22-ET
would, since the only symbol required is the 5-comma symbol /|. But
beyond that, I suspect that the best 7 of 56-ET may not be the 7 of
pajara/paultone.

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

> I believe 56-ET would notate a chain of 22 just the same as 22-ET
> would, since the only symbol required is the 5-comma symbol /|. But
> beyond that, I suspect that the best 7 of 56-ET may not be the 7 of
> pajara/paultone.

Eeek, you're right; the val for Pajara is [56,89,130,158]. This still gives 5-limit harmony quite a bit better than that of the 22-et, but 22 has a better 7/4 and 7/6, and the same 7/5, and should be considered. Does Paul want to weigh in?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:

> This is the system with wedgie [2, -4, -4, 2, 12, -11] which we
>used to call Paultone. It has [1/2, 5/56] as poptimal in both the 7-
>limit and the 9-limit, and my recommendation is that the 56-et be
>used to notate it. The alternative is 22, but with all due respect
>for Paul's favorite division, 56 is in much better tune.

in my paper, i found that the fifth should be in the range 708.8143
to 710.0927 cents, with a preference for the former (equal-weighted
RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while
56-equal is 707.1428, which is not even in the range.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>
> > This is the system with wedgie [2, -4, -4, 2, 12, -11] which we
> >used to call Paultone. It has [1/2, 5/56] as poptimal in both the 7-
> >limit and the 9-limit, and my recommendation is that the 56-et be
> >used to notate it. The alternative is 22, but with all due respect
> >for Paul's favorite division, 56 is in much better tune.
>
> in my paper, i found that the fifth should be in the range 708.8143
> to 710.0927 cents, with a preference for the former (equal-weighted
> RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while
> 56-equal is 707.1428, which is not even in the range.

Hmmm, on what basis? I'd better go look. The advantage of 56 is that it brings us nearer to diaschismic; it gives us better 5-limit harmony without much disadvantage in the 7-limit, which already has the tritone as a 7/5 anyway.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> in my paper, i found that the fifth should be in the range 708.8143
> to 710.0927 cents, with a preference for the former (equal-weighted
> RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while
> 56-equal is 707.1428, which is not even in the range.

Opinions on that might differ. For 7-limit, I get

least squares 708.8143306
least cubes 708.9699583
least fourth powers 709.0364438
minimax 706.8431431

From this we can't prove that 22-equal is poptimal, though it at least comes close and might be. I can't decypher your figures one and two, so I don't know where 710.0927 comes from--this wouldn't me the notorious MAD value, by any chance?

For the 9 limit, we have

least squares 707.2464833
least cubes 707.4361060
least fourth powers 707.5632154
minimax 706.8431431

The 56-et fifth of 707.1429 clearly works better here.

If we intersect the range of known 7 and 9 limit poptimals, we get values between 706.8431 and 707.5632 cents, with 33/56 right in the midst of it. The best choice might still be 22, but it isn't a walk.

Gene wrote:
>minimax 706.8431431

Could this be wrong? I have 709.363 in the scale archive.

Manuel

Gene:
>>minimax 706.8431431

Manuel:
> Could this be wrong? I have 709.363 in the scale archive.

I agree with Manuel

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene:
> >>minimax 706.8431431
>
> Manuel:
> > Could this be wrong? I have 709.363 in the scale archive.
>
> I agree with Manuel
>
>
> Graham

Me too.

I get 709.363 for 7-limit and 708.128 for 9-limit.

Kalle

Kalle Aho wrote:

> I get 709.363 for 7-limit and 708.128 for 9-limit. But I agree with Gene's 706.843 for the 9-limit pajara minimax. It looks like he accidentally copied that for the 7-limit.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Kalle Aho wrote:
>
> > I get 709.363 for 7-limit and 708.128 for 9-limit.
>
> But I agree with Gene's 706.843 for the 9-limit pajara minimax. It
> looks like he accidentally copied that for the 7-limit.
>
>
> Graham

With a generator of 706.843 cents the worst error (7:4) is 17.488
cents.
With a generator of 708.128 cents it's 14.918.

Kalle

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith
> > <genewardsmith@j...>" <genewardsmith@j...> wrote:
> >
> > > This is the system with wedgie [2, -4, -4, 2, 12, -11] which we
> > >used to call Paultone. It has [1/2, 5/56] as poptimal in both
the 7-
> > >limit and the 9-limit, and my recommendation is that the 56-et
be
> > >used to notate it. The alternative is 22, but with all due
respect
> > >for Paul's favorite division, 56 is in much better tune.
> >
> > in my paper, i found that the fifth should be in the range
708.8143
> > to 710.0927 cents, with a preference for the former (equal-
weighted
> > RMS). 22-equal is 709.0909 cents, so it's obviously quite fine,
while
> > 56-equal is 707.1428, which is not even in the range.
>
> Hmmm, on what basis?

i said, equal-weighted RMS. 7-limit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > in my paper, i found that the fifth should be in the range
708.8143
> > to 710.0927 cents, with a preference for the former (equal-
weighted
> > RMS). 22-equal is 709.0909 cents, so it's obviously quite fine,
while
> > 56-equal is 707.1428, which is not even in the range.
>
> Opinions on that might differ. For 7-limit, I get
>
> least squares 708.8143306

i get 708.814329511048; close enough!

> least cubes 708.9699583

i get 708.969922809224; close enough!

> least fourth powers 709.0364438

i get 709.051244958997 -- we have some discrepancy!!
i get a p=4 norm of 0.0162470021271903 for your value, and
0.0162469162102062 for mine, so clearly something's wrong with your
algorithm!

> minimax 706.8431431

i'm not sure, but i do get this same figure for MAD.

> From this we can't prove that 22-equal is poptimal, though it at
>least comes close and might be.

according to my calculations, for p=5 we get 709.112411004975, so 22-
equal is in there!

I can't decypher your figures one >and two, so I don't know where
710.0927 comes from--this wouldn't me >the notorious MAD value, by
any chance?

no, it's least squares with limit weighting (instead of the more
common inverse-limit weighting or equal weighting) -- the
corresponding meantone is 175/634-comma, as you can see in my paper
or here:

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

>
> For the 9 limit, we have
>
> least squares 707.2464833
> least cubes 707.4361060
> least fourth powers 707.5632154
> minimax 706.8431431
>
> The 56-et fifth of 707.1429 clearly works better here.
>
> If we intersect the range of known 7 and 9 limit poptimals, we get
>values between 706.8431 and 707.5632 cents, with 33/56 right in the
>midst of it. The best choice might still be 22, but it isn't a walk.

on my keyboard it is!

--- In tuning-math@yahoogroups.com, "Kalle Aho <kalleaho@m...>"
<kalleaho@m...> wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
> > Kalle Aho wrote:
> >
> > > I get 709.363 for 7-limit and 708.128 for 9-limit.
> >
> > But I agree with Gene's 706.843 for the 9-limit pajara minimax.
It
> > looks like he accidentally copied that for the 7-limit.
> >
> >
> > Graham
>
> With a generator of 706.843 cents the worst error (7:4) is 17.488
> cents.
> With a generator of 708.128 cents it's 14.918.
>
> Kalle

actually there is no single minimax value, since 7:5 is the worst
error, since it's stuck at 600 cents, across a range of generator
values.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Kalle Aho <kalleaho@m...>"
> <kalleaho@m...> wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
> wrote:
> > > Kalle Aho wrote:
> > >
> > > > I get 709.363 for 7-limit and 708.128 for 9-limit.
> > >
> > > But I agree with Gene's 706.843 for the 9-limit pajara
minimax.
> It
> > > looks like he accidentally copied that for the 7-limit.
> > >
> > >
> > > Graham
> >
> > With a generator of 706.843 cents the worst error (7:4) is 17.488
> > cents.
> > With a generator of 708.128 cents it's 14.918.
> >
> > Kalle
>
> actually there is no single minimax value, since 7:5 is the worst
> error, since it's stuck at 600 cents, across a range of generator
> values.

though i suppose if you were taking the limit as p approached
infinity, you'd essentially take the minimax without considering 7:5.

so gene, it looks like you're not getting the right answer for p>3.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Kalle Aho wrote:
>
> > I get 709.363 for 7-limit and 708.128 for 9-limit.
>
> But I agree with Gene's 706.843 for the 9-limit pajara minimax. It
> looks like he accidentally copied that for the 7-limit.

I'm afraid minimax can sometimes involve a range of values, and that's what's happened here. I was simply feeding it to Maple's simplex routine and turning the crank, but I need to redo the code so that the range is returned when there is one. My minimax value has an error on major thirds of zero, and hence 7 and 7/5 with the same error. Your milage may vary, and it seems it does.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
> > Kalle Aho wrote:
> >
> > > I get 709.363 for 7-limit and 708.128 for 9-limit.
> >
> > But I agree with Gene's 706.843 for the 9-limit pajara minimax.
It
> > looks like he accidentally copied that for the 7-limit.
>
> I'm afraid minimax can sometimes involve a range of values, and
>that's what's happened here. I was simply feeding it to Maple's
>simplex routine and turning the crank, but I need to redo the code
>so that the range is returned when there is one. My minimax value
>has an error on major thirds of zero, and hence 7 and 7/5 with the
>same error. Your milage may vary, and it seems it does.

so what's going wrong with p>3?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> i get 709.051244958997 -- we have some discrepancy!!
> i get a p=4 norm of 0.0162470021271903 for your value, and
> 0.0162469162102062 for mine, so clearly something's wrong with your
> algorithm!

Nothing is wrong with my algorithm; I reran it and got your value. I have no idea where mine came from.

> > minimax 706.8431431
>
> i'm not sure, but i do get this same figure for MAD.
>
> > From this we can't prove that 22-equal is poptimal, though it at
> >least comes close and might be.
>
> according to my calculations, for p=5 we get 709.112411004975, so 22-
> equal is in there!

And p=6 gives 709.1585, better yet; it's simply headed off where you suggested, to 709.363.

>The best choice might still be 22, but it isn't a walk.
>
> on my keyboard it is!

Depends on how much you favor triads, I suppose. Compromising between 22 and 34 does seem like an interesting possibility.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> I'm afraid minimax can sometimes involve a range of values, and
that's what's happened here. I was simply feeding it to Maple's
simplex routine and turning the crank, but I need to redo the code so
that the range is returned when there is one.
>

I've been Internet-challenged for the past few days.

I'm glad Paul remembered this common pitfall with max-absolute
calculations.

Standard practice (among tuning folk) when calculating generators
giving the least max-absolute error (minimax) is not to return a range
(musically we don't really care about such ranges) but simply to
eliminate from the optimisation any consonance whose size is
independent of the generator size (like the 5:7 in Pajara). Of course
one must still take the errors of such consonances into account when
quoting the least max-absolute error for the tuning in cents, as
required for comparison with other temperaments.

Previously, in my case at least, the justifaction for this procedure
was purely musical, and so it appeared somewhat ad hoc mathematically.
Thanks Gene for pointing out that least max-absolute corresponds to
least sum-of-p_th-powers-of-absolutes as p -> oo. As Paul pointed out,
this now provides a purely mathematical justification for the standard
procedure.

So Gene, does this now throw into doubt all your previous p-optimal
calculations?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > i get 709.051244958997 -- we have some discrepancy!!
> > i get a p=4 norm of 0.0162470021271903 for your value, and
> > 0.0162469162102062 for mine, so clearly something's wrong with
your
> > algorithm!
>
> Nothing is wrong with my algorithm; I reran it and got your value.
I have no idea where mine came from.
>
> > > minimax 706.8431431
> >
> > i'm not sure, but i do get this same figure for MAD.
> >
> > > From this we can't prove that 22-equal is poptimal, though it
at
> > >least comes close and might be.
> >
> > according to my calculations, for p=5 we get 709.112411004975, so
22-
> > equal is in there!
>
> And p=6 gives 709.1585, better yet; it's simply headed off where
>you suggested, to 709.363.

when did i suggest that? and isn't it true that this value, the limit
as p goes to infinity, is *not* the midpoint of the minimax range, as
you stated it would be?

> >The best choice might still be 22, but it isn't a walk.
> >
> > on my keyboard it is!
>
> Depends on how much you favor triads, I suppose. Compromising
>between 22 and 34 does seem like an interesting possibility.

well, with 22 i have more triads and more of everything, since the
chain closes on itself. plus the guitar, by its nature with frets
across differently-tuned strings, *really* calls strongly for a
closed chain.

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

> Thanks Gene for pointing out that least max-absolute corresponds to
> least sum-of-p_th-powers-of-absolutes as p -> oo.

huh? *i* suggested that this was the case, and gene denied it.

> So Gene, does this now throw into doubt all your previous p-optimal
> calculations?

something else seems to have thrown a large number of them into
doubt -- copying error perhaps, but since the conclusions were wrong
about 22 and pajara, i don't feel too comfortable about any others.

and i think it's insanity not to allow p to go as low as 1, or
perhaps even lower, when talking this poptimal stuff.

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

> Standard practice (among tuning folk) when calculating generators
> giving the least max-absolute error (minimax) is not to return a range
> (musically we don't really care about such ranges)

It seems to me that for this business that musically we might in fact care about them. Of course it would be easy enough to code so that the
p-->infinity limit minimax was returned, but is this what we want?

> So Gene, does this now throw into doubt all your previous p-optimal
> calculations?

Fraid so. :(

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> > And p=6 gives 709.1585, better yet; it's simply headed off where
> >you suggested, to 709.363.
>
> when did i suggest that? and isn't it true that this value, the limit
> as p goes to infinity, is *not* the midpoint of the minimax range, as
> you stated it would be?

I cancel that 10 seconds after posting it and you want to hold me to it. :)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > > And p=6 gives 709.1585, better yet; it's simply headed off
where
> > >you suggested, to 709.363.
> >
> > when did i suggest that? and isn't it true that this value, the
limit
> > as p goes to infinity, is *not* the midpoint of the minimax
range, as
> > you stated it would be?
>
> I cancel that 10 seconds after posting it and you want to hold me
>to it. :)

sorry, i wasn't aware of the cancellation! did you delete the post?
wow, i must have really had my trigger finger on the "next" button!

glad to know my intuition isn't failing me at least on *some*
things . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
>
> > Standard practice (among tuning folk) when calculating generators
> > giving the least max-absolute error (minimax) is not to return a range
> > (musically we don't really care about such ranges)
>
> It seems to me that for this business that musically we might in
fact care about them. Of course it would be easy enough to code so
that the
> p-->infinity limit minimax was returned, but is this what we want?

Gene,

As evidence, I offer the minimax generators for Pajara quoted by a
number of different people earlier in this thread. I believe the same
value was supplied independently by 2 or 3 people, and I think you
will find that it is the p-->infinity value.

I think you'd better include p=1 before Paul starts campaigning for
the limit as p-->0 or p negative. ;-)

Actually I think that the absolutely and ideally perfect optimisation
will probably occur when p = 2*pi. ;-)

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
>
> > Thanks Gene for pointing out that least max-absolute corresponds to
> > least sum-of-p_th-powers-of-absolutes as p -> oo.
>
> huh? *i* suggested that this was the case, and gene denied it.

Paul,

I took your word for it when you wrote in
/tuning-math/message/5353
"... gene suggested using the entire range of p-norms, with p from 2
to infinity ...".

But if it was actually you who first pointed it out, then thanks. I
can't find where Gene denied it, but if he did, I suspect it was only
because his algorithm was apparently producing counterexamples. We now
know it is was the algorithm that was the problem.

Regards,
-- Dave Keenan

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

> As evidence, I offer the minimax generators for Pajara quoted by a
> number of different people earlier in this thread. I believe the same
> value was supplied independently by 2 or 3 people, and I think you
> will find that it is the p-->infinity value.

I've decided to cave, and re-wrote my program to get it to correspond to what other people are doing.

> I think you'd better include p=1 before Paul starts campaigning for
> the limit as p-->0 or p negative. ;-)

So it seems, but I really think we'd better draw a line there.

> Actually I think that the absolutely and ideally perfect optimisation
> will probably occur when p = 2*pi. ;-)

Obviously we need to work pi in somehow.

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

> Paul,
>
> I took your word for it when you wrote in
> /tuning-math/message/5353
> "... gene suggested using the entire range of p-norms, with p from 2
> to infinity ...".

that's right.

> But if it was actually you who first pointed it out, then thanks.

i was the one who pointed out that the limit of the norm as p goes to
infinity is the result that minimizes the maximum error amongst those
errors which are not constant as a function of generator size. i.e.,
the kind of "minimax" value that manuel and kalle quoted.

> I
> can't find where Gene denied it,

i think he said he deleted his claim (which was that you'd get to the
center of the minimax range if you let p go to infinity) 10 seconds
after posting it.