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.