Poptimal generators

"Poptimal" is short for "p-optimal". The p here is a real variable
p>=2, which is what analysts normally use when discussing these Holder
type normed linear spaces.

A pair of generators [1/n, x] for a linear temperament is *poptimal* if there is some p, 2 <= p <= infinity, such that x is precisely optimal according to the sum-of-absolute-values-of-pth-powers definition of optimal, and where p=infinity is shorthand for minimax.

We can show a generator is poptimal by using a continuity argument; the sum of pth powers is a sum

SUM abs(ai/n + bi*x - l2(consonant interval))^p

over the consonances of the tuning. It is twice continuously differentiable with respect to x, and the second derviative is positive, since not all of the error terms can be zero. Hence it has a unique minimum, which is the p-optimized tuning value.

If we denote by F(x, p) the above derivative with respect to x, we can check that it is continuously differentiable with respect to both x and p (it might look as if the p derivative blows up, but
|x|^n log|x| is dominated by x^n near zero for n>0.) We also have that the partial is never zero, since the partial with respect to x,
by the above, is positive. Hence the conditions of the implicit function theorem obtain, and we can not only find x as a continuous function of p, we have p as a continuous function of x. The set of poptimal generators is therefore convex--if x1 and x2 are poptimal, so is any x such that x1<x<x2.

Since p=2 can be computed by least squares, p=infinity by setting it up as a linear programming problem, and p=4 by solving a cubic polynomial with a single real root, they are easy to compute. If x2, x4, xinf are the three values we obtain, we can find the interval
[xa, xb] where xa = min(x2,x4,xinf) and xb = max(x2, x4, xinf).
Usually, this interval will be more than a single point, and so we can find a rational poptimal generator. This is a generator which can reasonably claim to be *precisely* optimal, since it is optimal by some reasonable definition of "optimal", and yet it is an interval of an equal temperament.

In some cases, a rational poptimal generator does not exist, since there is only one unique poptimal generator. This can happen for "funky" temperaments where two of the consonances are identified.
I don't know if it can happen otherwise, and might try to prove that impossible.

Below are poptimal generators for 44 7-limit linear temperaments. In many cases they will be the rational poptimal generator of smallest height, but this is not assured, since they were calculated by the x2, x4, xinf method.

"Dominant seventh" [1, 4, -2, -16, 6, 4] [1, 24/41]

"Diminished" [4, 4, 4, -2, 5, -3] [1/4, 1/16]

"Blackwood" [0, 5, 0, -14, 0, 8] [1/5, 3/25]

"Augmented" [3, 0, 6, 14, -1, -7] [1/3, 8/33]

"Pajara" [2, -4, -4, 2, 12, -11] [1/2, 5/56]

"Hexadecimal" [1, -3, 5, 20, -5, -7] [1, 23/41]

"Tertiathirds" [4, -3, 2, 13, 8, -14] [1, 5/48]

"Kleismic" [6, 5, 3, -7, 12, -6] [1, 14/53]

"Tripletone" [3, 0, -6, -14, 18, -7] [1/3, 2/27]

"Hemifourth" [2, 8, 1, -20, 4, 8] [1, 4/19]

"Meantone" [1, 4, 10, 12, -13, 4] [1, 119/205]

"Injera" [2, 8, 8, -4, -7, 8] [1/2, 1/13]

"Double wide" [8, 6, 6, -3, 13, -9] [1/2, 7/26]

"Porcupine" [3, 5, -6, -28, 18, 1] [1, 8/59]

"Superpythagorean" [1, 9, -2, -30, 6, 12] [1, 45/76]

"Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197]

"Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]

"Flattone" [1, 4, -9, -32, 17, 4] [1, 63/109]

"Magic" [5, 1, 12, 25, -5, -10] [1, 13/41]

"Nonkleismic" [10, 9, 7, -9, 17, -9] [1, 23/89]

"Semisixths" [7, 9, 13, 5, -1, -2] [1, 44/119]

"Orwell" [7, -3, 8, 27, 7, -21] [1, 26/115]

"Miracle" [6, -7, -2, 15, 20, -25] [1, 17/175]

"Quartaminorthirds" [9, 5, -3, -21, 30, -13] [1, 16/247]

"Supermajor seconds" [3, 12, -1, -36, 10, 12] [1, 59/305]

"Schismic" [1, -8, -14, -10, 25, -15] [1, 134/229]

"Superkleismic" [9, 10, -3, -35, 30, -5] [1, 37/138]

"Squares" [4, 16, 9, -24, -3, 16] [1, 93/262]

"Semififth" [2, 8, -11, -48, 23, 8] [1, 119/410]

"Diaschismic" [2, -4, -16, -26, 31, -11] [1/2, 9/104]

"Octafifths" [8, 18, 11, -25, 5, 10] [1, 13/177]

"Tritonic" [5, -11, -12, 3, 33, -29] [1, 163/337]

"Supersupermajor" [3, 17, -1, -50, 10, 20] [1, 25/128]

"Shrutar" [4, -8, 14, 55, -11, -22] [1/2, 4/91]

"Catakleismic" [6, 5, 22, 37, -18, -6] [1, 52/197]

"Hemiwuerschmidt" [16, 2, 5, 6, 37, -34] [1, 37/229]

"Hemikleismic" [12, 10, -9, -49, 48, -12] [1, 25/189]

"Hemithird" [15, -2, -5, -6, 50, -38] [1, 24/149]

"Wizard" [12, -2, 20, 52, 2, -31] [1/2, 23/72]

"Duodecimal" [0, 12, 24, 22, -38, 19] [1/12, 1/76]

"Slender" [13, -10, 6, 42, 27, -46] [1, 5/156]

"Amity" [5, 13, -17, -76, 41, 9] [1, 155/548]

"Hemififth" [2, 25, 13, -40, -15, 35] [1, 70/239]

"Ennealimmal" [18, 27, 18, -34, 22, 1] [1/9, 43/612]

I think I actually followed most of that, and what's more, found it
very interesting. Thanks.

But with regard to sagittal notation for these temperaments, some of
those results looked pretty wild, such as 205-ET for 7-limit Meantone.

What do you mean by rational generator of minimum height? What is
"height" here?

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

> "Injera" [2, 8, 8, -4, -7, 8] [1/2, 1/13]

> "Duodecimal" [0, 12, 24, 22, -38, 19] [1/12, 1/76]

Maple, which I used to print this out, automatically reduces fractions to their lowest terms, so you need to take the lcm of the denomiators to find the et--for Injera, it would be 26, for Duodecimal, 228.

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

> But with regard to sagittal notation for these temperaments, some of
> those results looked pretty wild, such as 205-ET for 7-limit Meantone.

I agree, that is way out there, but I think these are worth considering. They certainly make sense for computer music, at least, if not for notational purposes.

> What do you mean by rational generator of minimum height? What is
> "height" here?

Size of denominator, or numerator times denomiator, etc etc. It doesn't really matter which one you choose. "Height" is what number theorists call a function which measures the complexity of a rational (or algebraic) number.

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

> But with regard to sagittal notation for these temperaments, some of
> those results looked pretty wild, such as 205-ET for 7-limit Meantone.

Incidentally, the convergents for the log base 2 of the fourth root of five, which is the 1/4-comma meantone fifth, are as follows:

1, 1/2, 3/5, 4/7, 7/12, 11/19, 18/31, 101/174, 119/205, 458/789,
1493/2572, 1951/3361, 5395/9294, 7346/12655, 56817/97879 ...

Whether this justifies using 119/205 for notating meantone may be doubted, but better it than than 56817/97879, also poptimal.

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

> Whether this justifies using 119/205 for notating meantone may be doubted, but better it than than 56817/97879, also poptimal.

I used my r3 value of (7+sqrt(11))/38-comma meantone, and found that
47/81 is poptimal for 7-limit meantone. Since it was also what I got for a poptimal 5-limit meantone, it is looking good. Forget
(7+sqrt(11))/38-comma meantone; I think I'll opt for this as my personal meantone. It's about 5/19-comma, or 1/3.8 comma if you prefer, meantone. You could reasonably use it to notate meantone if you wanted. Of course the next theory I expect to hear is that we *should* be looking at weighted poptimal generators, which would probably be enough to bring 31-et back into the fold.

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

> I used my r3 value of (7+sqrt(11))/38-comma meantone, and found that
> 47/81 is poptimal for 7-limit meantone. Since it was also what I got for a poptimal 5-limit meantone, it is looking good. Forget
> (7+sqrt(11))/38-comma meantone; I think I'll opt for this as my personal meantone. It's about 5/19-comma, or 1/3.8 comma if you prefer, meantone.

The same r3 value allows us to determine that Golden Meantone is in the rather small range of values which are poptimal in both the 5 and 7 limits. Notating it is a whole other story, of course; I presume one would use ordinary sharps and flats--and just keep on using them.

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

I think I need to get some sleep; neither 81 nor, sadly, Golden Meantone can be added in the 7-limit by using r3. The 205-et I gave for 7-limit Meantone may be the best that can be done. Of course, I could get some rather different ones by moving on up to the 9-limit.

>"Height" is what number theorists call a function
>which measures the complexity of a rational (or
>algebraic) number.

What common feature would all such functions share?
IOW, how would they define "complexity"?

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> What common feature would all such functions share?

Roughly, the bigger the coefficients of the defining polynomial of the algebraic number, the greater the height; but of course there are different ways to measure it. Common is the sum of absolute values of the coefficients, but there are height functions with nifty properties like Mahler height or Silverman height. I've never thought about musical applications, but we don't usually worry about algebraic numbers anyway.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> "Poptimal" is short for "p-optimal". The p here is a real variable
> p>=2, which is what analysts normally use when discussing these
Holder
> type normed linear spaces.
>
> A pair of generators [1/n, x] for a linear temperament is
>*poptimal* if there is some p, 2 <= p <= infinity,

why not go all the way to 1? MAD, or p-1, error certainly seems most
appropriate for dissonance curves such as vos's or secor's -- which
are in fact even pointier at the local minima (resembling exp
(|error|)) . . .

> "Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197]
>
> "Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]

what did i miss?

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

>MAD, or p-1, error

i meant p=1, not p-1.

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

> why not go all the way to 1? MAD, or p-1, error certainly seems most
> appropriate for dissonance curves such as vos's or secor's -- which
> are in fact even pointier at the local minima (resembling exp
> (|error|)) . . .

I didn't trust a p as low as 1 to give good results, since it tends to ignore outliers, and the argument that we get continuous functions for p as a function of generator size is easier to make if we have p>=2; not that that really matters, I suppose.

Maybe I should give it a shot and see if I can convince 31 to be a poptimal meantone in this way.

> > "Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197]
> >
> > "Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]
>
> what did i miss?

They've been around forever, I just decided to name them. "Muggles" is an inferior sort of Magic, and 19/64 goes with Beatles for those of us who were around back then. It certainly seems better than "Gulf of Tonkin Incident" or "Lyndon Johnson Elected in a Landslide", which are the only other public events I recall from that year.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > "Poptimal" is short for "p-optimal". The p here is a real variable
> > p>=2, which is what analysts normally use when discussing these
> Holder
> > type normed linear spaces.
> >
> > A pair of generators [1/n, x] for a linear temperament is
> >*poptimal* if there is some p, 2 <= p <= infinity,
>
> why not go all the way to 1? MAD, or p=1, error certainly seems
most
> appropriate for dissonance curves such as vos's or secor's -- which
> are in fact even pointier at the local minima (resembling exp
> (|error|)) . . .

Possibly because no one in the history of this endeavour has ever
before now suggested that mean-absolute error corresponds in any way
to the human perception of these things. I think everyone agrees that
the worst errors dominate; either totally as in max-absolute (p=oo)
or partially as in RMS (p=2).

However, I strongly object to this statement of Gene's in the tuning
list:

"A "poptimal" generator can lay claim to being absolutely and ideally
perfect as a generator for a given temperament ..."

When we're talking about human perception, as we are, it should be
obvious that nothing can be absolutely and ideally perfect for
everyone. Even a single person might prefer slightly different
generators for different purposes. To validate such a claim
of "perfection" you would at least need to produce statistics on the
opinions of many listeners.

Gene, you seem to be confusing beautiful mathematics with accurate
modelling of a psychoacoustic property (yet to be established).

> > "Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197]

I think the names "Muggles" and "Wizard" could be swapped for these
two temperaments. The one above, also called "Narrow major thirds",
is significantly better than the Twin major thirds temperament by
anyone's badness. The twin major thirds temperament is so bad (due
mostly to its complexity) that it's barely worth mentioning (at least
at the 7-limit).

> > "Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]

Previously called "Neutral thirds with complex 5s". I guess the
name's based of the "64" denominator?

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> 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:
> > > "Poptimal" is short for "p-optimal". The p here is a real variable
> > > p>=2, which is what analysts normally use when discussing these
> > Holder
> > > type normed linear spaces.
> > >
> > > A pair of generators [1/n, x] for a linear temperament is
> > >*poptimal* if there is some p, 2 <= p <= infinity,
> >
> > why not go all the way to 1? MAD, or p=1, error certainly seems
> most
> > appropriate for dissonance curves such as vos's or secor's -- which
> > are in fact even pointier at the local minima (resembling exp
> > (|error|)) . . .
>
> Possibly because no one in the history of this endeavour has ever
> before now suggested that mean-absolute error corresponds in any way
> to the human perception of these things.

no one in history? you've gotta be kidding me. sum or mean of absolute
errors is a quite common error criterion.

>> "A "poptimal" generator can lay claim to being absolutely and ideally
>> perfect as a generator for a given temperament ..."
>
> When we're talking about human perception, as we are, it should be
> obvious that nothing can be absolutely and ideally perfect for
> everyone. Even a single person might prefer slightly different
> generators for different purposes. To validate such a claim
> of "perfection" you would at least need to produce statistics on the
> opinions of many listeners.

clearly dave missed the clever mockery hidden in gene's statement.

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> 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:
> > > "Poptimal" is short for "p-optimal". The p here is a real variable
> > > p>=2, which is what analysts normally use when discussing these
> > Holder
> > > type normed linear spaces.
> > >
> > > A pair of generators [1/n, x] for a linear temperament is
> > >*poptimal* if there is some p, 2 <= p <= infinity,
> >
> > why not go all the way to 1? MAD, or p=1, error certainly seems
> most
> > appropriate for dissonance curves such as vos's or secor's -- which
> > are in fact even pointier at the local minima (resembling exp
> > (|error|)) . . .
>
> Possibly because no one in the history of this endeavour has ever
> before now suggested that mean-absolute error corresponds in any way
> to the human perception of these things.

no one in history? you've gotta be kidding me. sum or mean of absolute
errors is a quite common error criterion.

>> "A "poptimal" generator can lay claim to being absolutely and ideally
>> perfect as a generator for a given temperament ..."
>
> When we're talking about human perception, as we are, it should be
> obvious that nothing can be absolutely and ideally perfect for
> everyone. Even a single person might prefer slightly different
> generators for different purposes. To validate such a claim
> of "perfection" you would at least need to produce statistics on the
> opinions of many listeners.

clearly dave missed the clever mockery hidden in gene's statement.

--- 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:
> > --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
> > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith
> > > <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > > > "Poptimal" is short for "p-optimal". The p here is a real variable
> > > > p>=2, which is what analysts normally use when discussing these
> > > Holder
> > > > type normed linear spaces.
> > > >
> > > > A pair of generators [1/n, x] for a linear temperament is
> > > >*poptimal* if there is some p, 2 <= p <= infinity,
> > >
> > > why not go all the way to 1? MAD, or p=1, error certainly seems
> > most
> > > appropriate for dissonance curves such as vos's or secor's -- which
> > > are in fact even pointier at the local minima (resembling exp
> > > (|error|)) . . .
> >
> > Possibly because no one in the history of this endeavour has ever
> > before now suggested that mean-absolute error corresponds in any way
> > to the human perception of these things.
>
> no one in history? you've gotta be kidding me. sum or mean of absolute
> errors is a quite common error criterion.

By "this endeavour" I meant specifically the mathematical modelling of
perceptual optimality of generators for musical temperaments, not
mathematic or statistics in general. I also only said "possibly". I'm
happy to be corrected.

> >> "A "poptimal" generator can lay claim to being absolutely and ideally
> >> perfect as a generator for a given temperament ..."
> >
> > When we're talking about human perception, as we are, it should be
> > obvious that nothing can be absolutely and ideally perfect for
> > everyone. Even a single person might prefer slightly different
> > generators for different purposes. To validate such a claim
> > of "perfection" you would at least need to produce statistics on the
> > opinions of many listeners.
>
> clearly dave missed the clever mockery hidden in gene's statement.

Sorry. I must have missed some previous discussion that would have
made it clear that irony was intended. A smiley or winky after it
wouldn't have gone astray. I just read it and thought, hey this is the
sort of talk that gets the non-math folk pissed at us math folk.
Thanks Gene, for being kind enough to ignore my patronising pedantry.

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

> Sorry. I must have missed some previous discussion that would have
> made it clear that irony was intended. A smiley or winky after it
> wouldn't have gone astray.

"Irony" is too strong, but a knowing grin would do excellently.

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- 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:
> > > --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
> > > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith
> > > > <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > > > > "Poptimal" is short for "p-optimal". The p here is a real
variable
> > > > > p>=2, which is what analysts normally use when discussing
these
> > > > Holder
> > > > > type normed linear spaces.
> > > > >
> > > > > A pair of generators [1/n, x] for a linear temperament is
> > > > >*poptimal* if there is some p, 2 <= p <= infinity,
> > > >
> > > > why not go all the way to 1? MAD, or p=1, error certainly
seems
> > > most
> > > > appropriate for dissonance curves such as vos's or secor's --
which
> > > > are in fact even pointier at the local minima (resembling exp
> > > > (|error|)) . . .
> > >
> > > Possibly because no one in the history of this endeavour has
ever
> > > before now suggested that mean-absolute error corresponds in
any way
> > > to the human perception of these things.
> >
> > no one in history? you've gotta be kidding me. sum or mean of
absolute
> > errors is a quite common error criterion.
>
> By "this endeavour" I meant specifically the mathematical modelling
of
> perceptual optimality of generators for musical temperaments, not
> mathematic or statistics in general. I also only said "possibly".
I'm
> happy to be corrected.

consider yourself corrected :)

> > >> "A "poptimal" generator can lay claim to being absolutely and
ideally
> > >> perfect as a generator for a given temperament ..."
> > >
> > > When we're talking about human perception, as we are, it should
be
> > > obvious that nothing can be absolutely and ideally perfect for
> > > everyone. Even a single person might prefer slightly different
> > > generators for different purposes. To validate such a claim
> > > of "perfection" you would at least need to produce statistics
on the
> > > opinions of many listeners.
> >
> > clearly dave missed the clever mockery hidden in gene's statement.
>
> Sorry. I must have missed some previous discussion that would have
> made it clear that irony was intended.

nope. it´s just that an infinite number of tunings, a continuous
range of generators, can be considered poptimal for a given
temperament -- thus "absolutely and ideally perfect" cannot be taken
seriously, and was obviously poking fun of similar claims by people
like lucy.

paul (stuck in mannheim right now due to trains not running on time)

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > By "this endeavour" I meant specifically the mathematical modelling
> of
> > perceptual optimality of generators for musical temperaments, not
> > mathematic or statistics in general. I also only said "possibly".
> I'm
> > happy to be corrected.
>
> consider yourself corrected :)

So who, before you, has championed the use of mean-absolute error to
find optimum generators?

> nope. it´s just that an infinite number of tunings, a continuous
> range of generators, can be considered poptimal for a given
> temperament -- thus "absolutely and ideally perfect" cannot be taken
> seriously, and was obviously poking fun of similar claims by people
> like lucy.

Ah! Well the problem was that I didn't understand that a continuous
range was being referred to. Gene was quoting specific rational
fractions of an octave for the generators. And if I didn't understand
that, I suspect a lot of other people didn't either.

Hope you're having a great trip apart from the trains.

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

> nope. it´s just that an infinite number of tunings, a continuous
> range of generators, can be considered poptimal for a given
> temperament -- thus "absolutely and ideally perfect" cannot be taken
> seriously, and was obviously poking fun of similar claims by people
> like lucy.

Are we still on this? But Paul did get the point, which is that we have a *=
range* of absolutely, completely, 100% perfect systems which everyone must a=
nd shall use. I was saying my 100% perfect systems were are perfect as anyon=
e's, so I should be taken as seriously as anyone else making such a claim. T=
hat means I am just as serious as anyone else for an uncountably infinite nu=
mber of different 100% perfect systems, which is pretty danged serious if yo=
u can count that high.

> paul (stuck in mannheim right now due to trains not running on time)

Have a burger at the Burger King. When I lived in Heidelberg, it had two Ma=
cDonalds but no Burger King. I tried the one in Mannheim. Cold. :(

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus like lucy.
>
> paul (stuck in mannheim right now due to trains not running on time)

Had you considered taking the next steamroller? ;-)

--George

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > > By "this endeavour" I meant specifically the mathematical
modelling
> > of
> > > perceptual optimality of generators for musical temperaments,
not
> > > mathematic or statistics in general. I also only
said "possibly".
> > I'm
> > > happy to be corrected.
> >
> > consider yourself corrected :)
>
> So who, before you, has championed the use of mean-absolute error to
> find optimum generators?

i'm not championing them, but i've seen them suggested several times.
i think carl lumma was one of those people. it seems that carl,
george, and some others feel that even a single just interval can
profoundly affect the sound of a chord -- in which case weighting the
largest errors most isn't pointing you in the right direction.

> > nope. it´s just that an infinite number of tunings, a continuous
> > range of generators, can be considered poptimal for a given
> > temperament -- thus "absolutely and ideally perfect" cannot be
taken
> > seriously, and was obviously poking fun of similar claims by
people
> > like lucy.
>
> Ah! Well the problem was that I didn't understand that a continuous
> range was being referred to. Gene was quoting specific rational
> fractions of an octave for the generators. And if I didn't
understand
> that, I suspect a lot of other people didn't either.

you must have skipped over most of gene's original message. why do
you think we're having this discussion about absolute error criteria
in the first place? it's because gene suggested using the entire
range of p-norms, with p from 2 to infinity, and i suggested moving
the lower limit down to 1.

> Hope you're having a great trip apart from the trains.

tonight i'm stuck in london -- plane screwups this time.

>>So who, before you, has championed the use of mean-absolute
>>error to find optimum generators?
>
>i'm not championing them, but i've seen them suggested several
>times. i think carl lumma was one of those people.

I don't remember it, but I won't deny it. Lately I've been
behind RMS, though I'm interested in the possiblity of a
universal thinger such as the heuristic or the poptimal stuff.

>you must have skipped over most of gene's original message. why
>do you think we're having this discussion about absolute error
>criteria in the first place? it's because gene suggested using
>the entire range of p-norms, with p from 2 to infinity, and i
>suggested moving the lower limit down to 1.

How does this sort of thing compare with the heuristic approach?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> >you must have skipped over most of gene's original message. why
> >do you think we're having this discussion about absolute error
> >criteria in the first place? it's because gene suggested using
> >the entire range of p-norms, with p from 2 to infinity, and i
> >suggested moving the lower limit down to 1.
>
> How does this sort of thing compare with the heuristic approach?

In general, requiring poptimal generators seems to be a little over-precise in its demands; however combining the 7-limit with the 9-limit helped. Paul's idea would relax things some, but I don't really trust p to go all the way down to 1 myself. I think we have good candidates for most of the really significant 7-limit temperaments, but the 11 and 5 limit stuff (which I've looked at but not posted) again seems to be a bit problematic.