Euclidean expressibility tuning

Kees height gives a height function on interval classes, by reducing to lowest terms N/D and taking max(N, D). If we take logarithms translate that in terms of interval space, we get that the Kees expressibility metric is

|| |* w3 w5 ... wp> || = |w3|+|w5|+...+|wp|+|w3+w5+...+wp|

It is, therefore, the L1 norm on points w in interval space projected onto the subspace where J(w)=0, where J = <1 1 1 ... 1| is the JIP.

That suggests using the corresponding L2 norm, which would be simply the ordinary Euclidean norm applied after projecting orthogonally to the subspace orthogonal to J. Dual to that we would have points in tuning space (by duality these can be identified with points in intrval space, so don't let that confuse you.) A tuning map T can be decomposed into a part parallel to J, and a part perpendicular to J. If n is the dimension of the space, the average value of the coordinates of T is a = (J.T)/n, and the part parallel to J is a*J. Subtracting that from T gives T - a*J, the part perpendicular to J. It is this, applied to points on the subspace perpendicular to J regarded as representing interval classes which results in error and which we want to minimize.

Hence we can define a tuning by finding the smallest value for T-a*J for T on some flat representing tunings of a regular temperament. For example, take the line <1 (g+1)/log2(3) 4g/log2(5) (10g-3)/log2(7)| which gives the points of octaves-pure septimal meantone tuning in tuning space. Taking the average of the four coordinates, subtracting it off from each, and finding the value of g which minimizes the Euclidean size of the points thus defined gives us a value of g about 696.5/1200, so that we have a meantone fifth of ~696.5, perfectly reasonable for septimal meantone optimization.

Anyway, the theory seems nice, even if the name doesn't. Comments? I'm looking for a canonical pure-octaves tuning to adopt, and came up with this.

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

> Anyway, the theory seems nice, even if the name doesn't. Comments? I'm looking for a canonical pure-octaves tuning to adopt, and came up with this.

No comments. I was hoping Graham would tell us he's thought of it already and has a name for it.

The tuning assumes pure octaves, and finds the minimum standard deviation for the coordinates of the weighted tuning map given that assumption. (If no such assumption was made, it would return the null tuning, which sends everything to 1/1 and kind of misses the point.) The most obvious alternative is to pick the point nearest the JIP, which would be the Euclidean alternative to NOT, but people seemed pretty cold to NOT.

In any case, given how it works, what about Mistad (MInimal STAndard Deviation) as a name?

On 22 May 2010 05:49, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>> Anyway, the theory seems nice, even if the name doesn't. Comments? I'm looking for a canonical pure-octaves tuning to adopt, and came up with this.
>
> No comments. I was hoping Graham would tell us he's thought of it already and has a name for it.

I think I will, now I'm online.

> The tuning assumes pure octaves, and finds the minimum standard
> deviation for the coordinates of the weighted tuning map given that
> assumption. (If no such assumption was made, it would return the
> null tuning, which sends everything to 1/1 and kind of misses the
> point.) The most obvious alternative is to pick the point nearest the
> JIP, which would be the Euclidean alternative to NOT, but people
> seemed pretty cold to NOT.

It sounds like the STD error, then. It's all explained in the usual

http://x31eq.com/primerr.pdf

> In any case, given how it works, what about Mistad (MInimal
> STAndard Deviation) as a name?

The error is the actual standard deviation, so it's the STD error.
The optimal STD error is the optimal STD error. I've never found the
need to shorten that. You can search through my PDFs for places where
I forgot the "optimal" though.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> It sounds like the STD error, then. It's all explained in the usual
>
> http://x31eq.com/primerr.pdf
>
> > In any case, given how it works, what about Mistad (MInimal
> > STAndard Deviation) as a name?
>
> The error is the actual standard deviation, so it's the STD error.
> The optimal STD error is the optimal STD error. I've never found the
> need to shorten that. You can search through my PDFs for places where
> I forgot the "optimal" though.

I can't see calling a tuning map "the error". The error in the tuning map sounds like what you are calling the STD error, but what about the tuning itself?

On 22 May 2010 11:22, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> I can't see calling a tuning map "the error". The error in the
> tuning map sounds like what you are calling the STD error,
> but what about the tuning itself?

Call it the optimal STD tuning map. I expect I've left out the
"optimal" when talking about tunings because it's obvious.

Graham

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

> I can't see calling a tuning map "the error". The error in the tuning map sounds like what you are calling the STD error, but what about the tuning itself?

There's a discussion of pure octave tunings in there, where you mention this thing and remark that it needs pure octaves to keep from shrinking to zero, as I just got done doing. You even say there's an argument for considering it to be an optimal choice for a pure octave tuning, and point out that it's measuring norms of what you might call zero-size "intervals". So I could have saved myself some trouble, except I think I needed to go through the exercise to get to the theoretical justification.

Anyway, Graham clearly has priority here, but I would prefer a less ugly an unfortunately suggestive name than "STD tuning" and flatly refuse to call a tuning an error.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 22 May 2010 11:22, genewardsmith <genewardsmith@...> wrote:
>
> > I can't see calling a tuning map "the error". The error in the
> > tuning map sounds like what you are calling the STD error,
> > but what about the tuning itself?
>
> Call it the optimal STD tuning map. I expect I've left out the
> "optimal" when talking about tunings because it's obvious.

What about Graham tuning? That has to be better than calling it an especially virulent sexually transmitted disease. Plus, it has vowels in the name.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > On 22 May 2010 11:22, genewardsmith <genewardsmith@> wrote:
> >
> > > I can't see calling a tuning map "the error". The error in the
> > > tuning map sounds like what you are calling the STD error,
> > > but what about the tuning itself?
> >
> > Call it the optimal STD tuning map. I expect I've left out the
> > "optimal" when talking about tunings because it's obvious.
>
> What about Graham tuning? That has to be better than calling it an especially virulent sexually transmitted disease. Plus, it has vowels in the name.

Or there's STOP tuning, I suppose, for STandard OPtimal.
>

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

>> What about Graham tuning? That has to be better than calling
>> it an especially virulent sexually transmitted disease. Plus,
>> it has vowels in the name.

It's not the only tuning I've thought of.

The virulence doesn't enter into it. STD is an abbreviation for
Sexually Transmitted Disease. And also Standard Telephone Dialling,
or something. And, shockingly enough, STandard Deviation.

> Or there's STOP tuning, I suppose, for STandard OPtimal.

If it's a question of marketing, TOP with pure octaves would be TOPPO.
Whether the TOPPO-RMS tuning should be the unstretched TOP-RMS or the
STD is largely academic because they'll sound the same in practice.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> The virulence doesn't enter into it. STD is an abbreviation for
> Sexually Transmitted Disease. And also Standard Telephone Dialling,
> or something. And, shockingly enough, STandard Deviation.

By far the most common meaning for STD is Sexually Transmitted Disease, sorry. Standard deviation is usually called sigma or s, and if a shorthand version is wanted, I think "stdev" is actually more common.

> > Or there's STOP tuning, I suppose, for STandard OPtimal.
>
> If it's a question of marketing, TOP with pure octaves would be TOPPO.
> Whether the TOPPO-RMS tuning should be the unstretched TOP-RMS or the
> STD is largely academic because they'll sound the same in practice.

Toppo has to be unstretched RMS if it's going to be unstretched MAX, for consistency. Since what you rather horribly want to call "STD" is a kind of TOP as applied to interval classes, so Topic would be a name similar to Toppo-RMS for it. Since for other norms the related notion isn't really about interval classes any more, Topic could remain a purely Euclidean notion, needing no qualifier, which would be fine by me.

Is the fact that Topic and Toppo-RMS have very similar results in practive a bug or a feature? You could argue that if you want a standardized pure-octaves tuning, unstretching gives you that for free if you are already using a standardized TOP of some kind. But the theory justifying Topic seems clearer to me. Why should unstretching be regarded as optimal?

FWIW, I think TOPPO sounds stupid. Graham or STOP work for me.
STD is definitely a disease. I always use "sigma" for standard
deviations.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> FWIW, I think TOPPO sounds stupid. Graham or STOP work for me.
> STD is definitely a disease. I always use "sigma" for standard
> deviations.

How does TOPIC grab you as a name?

Also, did you get my email? I'm trying to confirm that Erv Wilson discovered 7-limit "Tonnetz" lattices back in the 60s.

Gene wrote:

>How does TOPIC grab you as a name?

What's the IC stand for?

>Also, did you get my email? I'm trying to confirm that Erv Wilson
>discovered 7-limit "Tonnetz" lattices back in the 60s.

Dude, I replied to that like 24 hours ago.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Gene wrote:
>
> >How does TOPIC grab you as a name?
>
> What's the IC stand for?

"Interval Classes".

> >Also, did you get my email? I'm trying to confirm that Erv Wilson
> >discovered 7-limit "Tonnetz" lattices back in the 60s.
>
> Dude, I replied to that like 24 hours ago.
>

Thanks. Do you have a reference where Fokker uses the lattice of notes and not just the lattice of tetrads?

Gene wrote:

>> >How does TOPIC grab you as a name?
>>
>> What's the IC stand for?
>
>"Interval Classes".

Yeah, that's a decent name.

>Thanks. Do you have a reference where Fokker uses the lattice of notes
>and not just the lattice of tetrads?

That is the lattice of notes. Have a second look.

-Carl

On 22 May 2010 20:00, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> The virulence doesn't enter into it.  STD is an abbreviation for
>> Sexually Transmitted Disease.  And also Standard Telephone Dialling,
>> or something.  And, shockingly enough, STandard Deviation.
>
> By far the most common meaning for STD is Sexually Transmitted
> Disease, sorry. Standard deviation is usually called sigma or s,
> and if a shorthand version is wanted, I think "stdev" is actually
> more common.

It doesn't matter how common another meaning is. Some words have more
than one meaning. The president can still refer to his aides without
collapsing in puritanical horror. Banco Santander seem to be happy
with their stock symbol and visitors to Sao Tome/Principe must somehow
put aside their revulsion and get stds before their journey. (Unless
US dollars are widely accepted there. I'm not an expert on this.)

The letters sigma and s have other meanings in mathematics that are
more likely to be confusing.

You could call it stdev, yes, that's hardly likely to lead to
confusion, although nobody's used it on this list before.

It was April 2004 that I first mentioned this as an error function:

/tuning-math/message/10228

I called it STD then. I've been abbreviating it STD for the past 6
years and nobody's complained. But now you've come up with it
independently so we need to get into a stupid terminology dispute.

> Toppo has to be unstretched RMS if it's going to be unstretched
> MAX, for consistency. Since what you rather horribly want to call
> "STD" is a kind of TOP as applied to interval classes, so Topic
> would be a name similar to Toppo-RMS for it. Since for other
> norms the related notion isn't really about interval classes any
> more, Topic could remain a purely Euclidean notion, needing no
> qualifier, which would be fine by me.

STD and unstretched TOP-RMS errors can both be applied to interval
classes, as you call them. At least, if interval classes are what I
thought I understood them to be: intervals under octave equivalence.
Why on earth wouldn't other norms apply to them?

> Is the fact that Topic and Toppo-RMS have very similar results
> in practive a bug or a feature? You could argue that if you want
> a standardized pure-octaves tuning, unstretching gives you that
> for free if you are already using a standardized TOP of some kind.
> But the theory justifying Topic seems clearer to me. Why should
> unstretching be regarded as optimal?

Of course it's a feature. It means one approximates the other and so
they're both measuring nearly the same thing.

The logic of unstretching TOP is that small intervals are largely
unaffected by the scale stretch. Also, the basic melodic patterns are
preserved. So once you have an optimal tuning, stretching the whole
thing equally does the least violence to the small intervals.
Balancing large and small intervals is impossible with an unoptimal
stretch anyway. This is the whole point of applying projective space,
which you shouldn't need to be convinced of.

As you've noticed in another message, TOPs can be written such that
they don't depend on the scale stretch of what you feed in. Which is
to say, they can be defined in projective space. This also works for
TOP-max: [max(w)-min(w)]/[max(w)+min(w)].

I thought the Kees-max led to an unstretched TOP-max tuning. Or, it
could lead to this, because the TOP-max tuning isn't always uniquely
defined. From what I remember, I thought I'd proved this, and you
thought you'd disproved it. However solid that equivalence is, it's
certainly a feature as well.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> You could call it stdev, yes, that's hardly likely to lead to
> confusion, although nobody's used it on this list before.
>
> It was April 2004 that I first mentioned this as an error function:
>
> /tuning-math/message/10228
>
> I called it STD then. I've been abbreviating it STD for the past 6
> years and nobody's complained. But now you've come up with it
> independently so we need to get into a stupid terminology dispute.

You've been abbreviating standard deviation "STD". You haven't been using anything as a name for the tuning, you are just objecting because someone else wants to give it a name. Has it occurred to you that this might be a good plan if other people want to talk about this tuning?

> The logic of unstretching TOP is that small intervals are largely
> unaffected by the scale stretch.

Which means that's it's really a first-order approximation, which is what I'm objecting to. It works in practice, but the theoretical basis for considering it optimal is another matter--precisely your objection to going with ZMD as I recall.

Also, the basic melodic patterns are
> preserved. So once you have an optimal tuning, stretching the whole
> thing equally does the least violence to the small intervals.
> Balancing large and small intervals is impossible with an unoptimal
> stretch anyway. This is the whole point of applying projective space,
> which you shouldn't need to be convinced of.

Yes, but it isn't a real projective method. You are taking distance in tuning space to define distance to the projective val, which doesn't make a lot of sense except as a first order approximation to something else. You should take distance on the projective plane, for instance by || u ^ J || where u is a unit vector, as I was just suggesting.

> I thought the Kees-max led to an unstretched TOP-max tuning. Or, it
> could lead to this, because the TOP-max tuning isn't always uniquely
> defined. From what I remember, I thought I'd proved this, and you
> thought you'd disproved it. However solid that equivalence is, it's
> certainly a feature as well.

So you claim Kees-RMS equals unstretched TOP-RMS? Because we have uniquelt defined definitions for these. I get 696.49489538 cents for unstretched TOP-RMS for septimal meantone, and previously I got 696.4936084 cents for the Tuning with No Name, and 696.4989638 for the projective metric tuning.

On 24 May 2010 08:03, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> You've been abbreviating standard deviation "STD". You haven't
> been using anything as a name for the tuning, you are just
> objecting because someone else wants to give it a name. Has it
> occurred to you that this might be a good plan if other people
> want to talk about this tuning?

Of course it's occurred to me. I've had six years to think about it.
And I concluded that it isn't a good plan. It's better to call it a
standard deviation tuning because it's a tuning that minimizes a
standard deviation. That's makes it easier for people to remember
which tuning you're talking about. I haven't had enough to say about
such tunings that I needed a cryptic name.

>> The logic of unstretching TOP is that small intervals are largely
>> unaffected by the scale stretch.
>
> Which means that's it's really a first-order approximation, which is
> what I'm objecting to. It works in practice, but the theoretical basis
> for considering it optimal is another matter--precisely your
> objection to going with ZMD as I recall.

First order in what sense? I thought it was at least second order.
It works in practice because it works in theory.

I don't remember objecting to the zero mean deviation (ZMD). I took
it out of my PDF because it didn't look interesting. You still
haven't found a reason for us to be interested in it.

>  Also, the basic melodic patterns are
>> preserved.  So once you have an optimal tuning, stretching the whole
>> thing equally does the least violence to the small intervals.
>> Balancing large and small intervals is impossible with an unoptimal
>> stretch anyway.  This is the whole point of applying projective space,
>> which you shouldn't need to be convinced of.
>
> Yes, but it isn't a real projective method. You are taking distance
> in tuning space to define distance to the projective val, which
> doesn't make a lot of sense except as a first order approximation
> to something else. You should take distance on the projective
> plane, for instance by || u ^ J || where u is a unit vector, as I was
> just suggesting.

What isn't a projective method? Am I taking a distance to a
projective val? I thought I was taking an angle between two lines,
which are both projections. What's a projective plane? Why should I
do that? Where did you suggest it?

>> I thought the Kees-max led to an unstretched TOP-max tuning.  Or, it
>> could lead to this, because the TOP-max tuning isn't always uniquely
>> defined.  From what I remember, I thought I'd proved this, and you
>> thought you'd disproved it.  However solid that equivalence is, it's
>> certainly a feature as well.
>
> So you claim Kees-RMS equals unstretched TOP-RMS? Because
> we have uniquelt defined definitions for these. I get 696.49489538
> cents for unstretched TOP-RMS for septimal meantone, and
> previously I got 696.4936084 cents for the Tuning with No Name,
> and 696.4989638 for the projective metric tuning.

No, I claim Kees-max can equal unstretched TOP-max, like I said.

What's a projective metric tuning? I recognize your other numbers.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 24 May 2010 08:03, genewardsmith <genewardsmith@...> wrote:
>
> > You've been abbreviating standard deviation "STD". You haven't
> > been using anything as a name for the tuning, you are just
> > objecting because someone else wants to give it a name. Has it
> > occurred to you that this might be a good plan if other people
> > want to talk about this tuning?
>
> Of course it's occurred to me. I've had six years to think about it.
> And I concluded that it isn't a good plan. It's better to call it a
> standard deviation tuning because it's a tuning that minimizes a
> standard deviation. That's makes it easier for people to remember
> which tuning you're talking about. I haven't had enough to say about
> such tunings that I needed a cryptic name.

Standard deviation tuning is a name, which you never actually gave before. It's a name, which is good. It connects it to standard deviations, which is both good and bad, but on the whole probably bad. This is because it gives the impression the tuning is really about standard deviations, and square roots of average values of squares minus squares of average values and all that apparatus, which managed to confuse the issue of what the tuning is really all about for so long. Rather than bury the key ideas under statistics terminology which leaves the impression the tuning is some kind of application of statistical ideas, an impression you left me with and which caused me to overlook the whole business as some kind of statistics tangent you had gone off on, a name which brings out that it's connected to interval classes and Kees expressibility and Euclidean norms might be better. But that can be corrected in the exposition, so really "standard deviation tuning" is fine by me if you must have it so.

> What isn't a projective method? Am I taking a distance to a
> projective val? I thought I was taking an angle between two lines,
> which are both projections. What's a projective plane? Why should I
> do that? Where did you suggest it?

If u is a non-zero vector in R^n, the associated point in n-1 dimensional projective space, P^(n-1), can be equated to the line through u and the origin. It can also be taken to be the unit vector u/||u||, provided we equate that with - u/||u||. So, you identify points on opposite sides of a hypersphere as points in projective space. Hyperspheres and hence projective space have a natural metric structure via the angle between the points. If you have a subspace associated to a temperament, it intersects the hypersphere in a projective line, plane, or etc. Finding the closest point on this projective object to J/||J|| gives us a projectivized TOP-like tuning, which is the natural object to stretch. It doesn't make much logical sense to stretch the actual TOP-style tunings, as they aren't defined projectively in the first place.

> What's a projective metric tuning? I recognize your other numbers.

What you get by doing the above, and making octaves pure.

On 25 May 2010 04:35, genewardsmith <genewardsmith@sbcglobal.net> wrote:

>> What isn't a projective method?  Am I taking a distance to a
>> projective val?  I thought I was taking an angle between two lines,
>> which are both projections.  What's a projective plane?  Why should I
>> do that?  Where did you suggest it?
>
> If u is a non-zero vector in R^n, the associated point in
> n-1 dimensional projective space, P^(n-1), can be equated to
> the line through u and the origin. It can also be taken to be the
> unit vector u/||u||, provided we equate that with - u/||u||. So,
> you identify points on opposite sides of a hypersphere as
> points in projective space. Hyperspheres and hence projective
> space have a natural metric structure via the angle between
> the points. If you have a subspace associated to a temperament,
> it intersects the hypersphere in a projective line, plane, or etc.
> Finding the closest point on this projective object to J/||J||
> gives us a projectivized TOP-like tuning, which is the natural
> object to stretch. It doesn't make much logical sense to stretch
> the actual TOP-style tunings, as they aren't defined projectively
> in the first place.

Why aren't the actual TOP tunings defined projectively? I thought I'd
done that and you haven't explained why I'm wrong. What you've
written there looks like a roundabout way of defining it, given a
suitable metric.

>> What's a projective metric tuning?  I recognize your other numbers.
>
> What you get by doing the above, and making octaves pure.

Then why isn't it the same as an unstretched TOP? You'll have to show
your working.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Why aren't the actual TOP tunings defined projectively? I thought I'd
> done that and you haven't explained why I'm wrong. What you've
> written there looks like a roundabout way of defining it, given a
> suitable metric.

TOP tunings are defined in terms of distance to the JIP, which is an undefined concept in projective space. All it knows about are angles to the JIP.

> >> What's a projective metric tuning?  I recognize your other numbers.
> >
> > What you get by doing the above, and making octaves pure.
>
> Then why isn't it the same as an unstretched TOP? You'll have to show
> your working.

(1) Take some linear combination of (weighted) vals with undetermined dcoefficients, which equates to some regular temperament you want to tune. Call that u = c1*v1 + ... + ck*vk.

(2) Take the wedge product u^J and divide by ||u||

(3) Square this, getting (u^J)^2/(u.u), which has no square roots to screw things up.

(4) Take the derivative, equate to 0, and solve for c1 ... ck. This will give two solutions, corresponding to the minimum and maximum possible angles with J.

(5) Substitute back into u, and set u[1] = 1.

All in all, a bother compared to finding TOP-RMS, but it makes more sense than stretched TOP-RMS it seems to me.

Gene wrote:

>All in all, a bother compared to finding TOP-RMS, but it makes more
>sense than stretched TOP-RMS it seems to me.

I take it the preceding was your procedure for finding the
STD tuning? (yes/no)

I'm trying to figure out what "stretched TOP-RMS" is. TOP-RMS
will often have octaves larger than 2. Are you referring to
some procedure whereby the TOP-RMS tuning is computed and then
the scale is stretched (or compressed) until the octaves are
exactly 2?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Gene wrote:
>
> >All in all, a bother compared to finding TOP-RMS, but it makes more
> >sense than stretched TOP-RMS it seems to me.
>
> I take it the preceding was your procedure for finding the
> STD tuning? (yes/no)

No. And must you call it that?

It's a procedure for finding the projective pure-octaves tuning.

> I'm trying to figure out what "stretched TOP-RMS" is.

You find a tuning map in the usual way, and then multiply it by a scalar to make octaves exact.

TOP-RMS
> will often have octaves larger than 2. Are you referring to
> some procedure whereby the TOP-RMS tuning is computed and then
> the scale is stretched (or compressed) until the octaves are
> exactly 2?

As has often been recommended by Paul and Graham, yes. It works, it just has never appealed much to me due to what I think is an inadequate theoretical basis.

At 06:56 PM 5/25/2010, you wrote:

>>> All in all, a bother compared to finding TOP-RMS, but it makes more
>>> sense than stretched TOP-RMS it seems to me.
>>
>> I take it the preceding was your procedure for finding the
>> STD tuning? (yes/no)
>
> No. And must you call it that?

No, but I'm not aware of any agreed-upon alternative.

>It's a procedure for finding the projective pure-octaves tuning.

I wonder if it's "Kees tuning".

>> I'm trying to figure out what "stretched TOP-RMS" is.
>
>You find a tuning map in the usual way, and then multiply it by a
>scalar to make octaves exact.

That's as I thought then.

>> TOP-RMS
>> will often have octaves larger than 2. Are you referring to
>> some procedure whereby the TOP-RMS tuning is computed and then
>> the scale is stretched (or compressed) until the octaves are
>> exactly 2?
>
>As has often been recommended by Paul and Graham, yes. It works,
>it just has never appealed much to me due to what I think is an
>inadequate theoretical basis.

Yes, why bother finding the TOP-RMS tuning if you're going to
modify it capriciously?

I don't recall Paul recommending it. I recall Paul recommending
something like TOP, but based on a Kees norm instead of the
Tenney norm.

-Carl

May I suggest that it is not just "STD" if you are referring to standard
deviation.
It should be STD DEV or sigma - but Carl you should know this.

Chris

On Tue, May 25, 2010 at 10:18 PM, Carl Lumma <carl@lumma.org> wrote:

>
>
> At 06:56 PM 5/25/2010, you wrote:
>
> >>> All in all, a bother compared to finding TOP-RMS, but it makes more
> >>> sense than stretched TOP-RMS it seems to me.
> >>
> >> I take it the preceding was your procedure for finding the
> >> STD tuning? (yes/no)
> >
> > No. And must you call it that?
>
> No, but I'm not aware of any agreed-upon alternative.
>
> >It's a procedure for finding the projective pure-octaves tuning.
>
> I wonder if it's "Kees tuning".
>
> >> I'm trying to figure out what "stretched TOP-RMS" is.
> >
> >You find a tuning map in the usual way, and then multiply it by a
> >scalar to make octaves exact.
>
> That's as I thought then.
>
> >> TOP-RMS
> >> will often have octaves larger than 2. Are you referring to
> >> some procedure whereby the TOP-RMS tuning is computed and then
> >> the scale is stretched (or compressed) until the octaves are
> >> exactly 2?
> >
> >As has often been recommended by Paul and Graham, yes. It works,
> >it just has never appealed much to me due to what I think is an
> >inadequate theoretical basis.
>
> Yes, why bother finding the TOP-RMS tuning if you're going to
> modify it capriciously?
>
> I don't recall Paul recommending it. I recall Paul recommending
> something like TOP, but based on a Kees norm instead of the
> Tenney norm.
>
> -Carl
>
>
>

At 07:27 PM 5/25/2010, you wrote:
>May I suggest that it is not just "STD" if you are referring to standard deviation.
>It should be STD DEV or sigma - but Carl you should know this.
>Chris

Hi Chris, good to see you here. You may have missed the recent
thread where I said I usually use sigma. STD is Graham's term,
which neither Gene nor I like, but which Graham has used in
several papers. -Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> >> I take it the preceding was your procedure for finding the
> >> STD tuning? (yes/no)
> >
> > No. And must you call it that?
>
> No, but I'm not aware of any agreed-upon alternative.

Do you object to "standard deviation tuning"?

> >It's a procedure for finding the projective pure-octaves tuning.
>
> I wonder if it's "Kees tuning".

Naah.

> I don't recall Paul recommending it. I recall Paul recommending
> something like TOP, but based on a Kees norm instead of the
> Tenney norm.

That's the Kees tuning, the analog of "standard deviation tuning" which I've had coded up for years now. Harder to do than standard deviation, or even the projective thing.

I guess then I'm another person who finds STD (meaning in my mind standard)
confusing.
No offense Graham. Within my analytical chemistry background STD and STD DEV
are not the same.
Another common abbreviation for standard deviation that may be useful is
S.D.

And hi everyone. I'm here to try to pick up more knowledge by osmosis.

Chris

On Tue, May 25, 2010 at 11:42 PM, Carl Lumma <carl@lumma.org> wrote:

>
>
> At 07:27 PM 5/25/2010, you wrote:
> >May I suggest that it is not just "STD" if you are referring to standard
> deviation.
> >It should be STD DEV or sigma - but Carl you should know this.
> >Chris
>
> Hi Chris, good to see you here. You may have missed the recent
> thread where I said I usually use sigma. STD is Graham's term,
> which neither Gene nor I like, but which Graham has used in
> several papers. -Carl
>
>
>

On 26 May 2010 08:42, Chris Vaisvil <chrisvaisvil@gmail.com> wrote:
>
>
> I guess then I'm another person who finds STD (meaning in my mind standard) confusing.
> No offense Graham. Within my analytical chemistry background STD and STD DEV are not the same.
> Another common abbreviation for standard deviation that may be useful is S.D.

Please, enough bike shedding. STD is standard deviation. It says so
here, I didn't make it up:

http://en.wiktionary.org/wiki/std

If you want to call it STD DEV or stdev or S.D., then do that. It
doesn't matter.

Graham

>> >> I take it the preceding was your procedure for finding the
>> >> STD tuning? (yes/no)
>> >
>> > No. And must you call it that?
>>
>> No, but I'm not aware of any agreed-upon alternative.
>
>Do you object to "standard deviation tuning"?

No, but my opinion doesn't mean much.

>> I don't recall Paul recommending it. I recall Paul recommending
>> something like TOP, but based on a Kees norm instead of the
>> Tenney norm.
>
>That's the Kees tuning, the analog of "standard deviation tuning"
>which I've had coded up for years now. Harder to do than standard
>deviation, or even the projective thing.

Gad, so many of these. Thanks for your work on the xen wiki
by the way. I think we need a page listing each of these
tuning optimizations...

-Carl

On 25 May 2010 10:52, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> Why aren't the actual TOP tunings defined projectively?  I thought I'd
>> done that and you haven't explained why I'm wrong.  What you've
>> written there looks like a roundabout way of defining it, given a
>> suitable metric.
>
> TOP tunings are defined in terms of distance to the JIP, which
> is an undefined concept in projective space. All it knows about
> are angles to the JIP.

TOP tunings can be defined different ways. The TOP-RMS definition
that works for projective space is the sine of the angle between the
line passing through the weighted tuning map and the line passing
through the JIP. See, angles. It looks fine to me.

With statistical functions, this is stdev(w)/rms(w) where w is the
Tenney weighted tuning map. You can verify that multiplying w by a
constant on the top and bottom doesn't alter the result, so it's
invariant in projective space, or however you phrase that.

With exterior algebra, this is ||w^j||/||w||||j|| where j is the
weighted JI point. That'll be ||u^J||/||u||||J|| with your symbols.
It also generalizes to higher ranks with u as the wedge product of
weighted vals. But in this case we're looking for a tuning so it has
to be a tuning map.

The TOP-max definition is [max(w) - min(w)]/[max(w) + min(w)]. I
don't know how the geometry works out, but it's also invariant if you
multiply w by a scalar.

>> >> What's a projective metric tuning?  I recognize your other numbers.
>> >
>> > What you get by doing the above, and making octaves pure.
>>
>> Then why isn't it the same as an unstretched TOP?  You'll have to show
>> your working.
>
> (1) Take some linear combination of (weighted) vals with undetermined dcoefficients, which equates to some regular temperament you want to tune. Call that u = c1*v1 + ... + ck*vk.
>
> (2) Take the wedge product u^J and divide by ||u||
>
> (3) Square this, getting (u^J)^2/(u.u), which has no square roots to screw things up.

Are you doing this numerically? If so, u.u is prone to rounding error.

> (4) Take the derivative, equate to 0, and solve for c1 ... ck. This will give two solutions, corresponding to the minimum and maximum possible angles with J.
>
> (5) Substitute back into u, and set u[1] = 1.
>
> All in all, a bother compared to finding TOP-RMS, but it makes
> more sense than stretched TOP-RMS it seems to me.

It still looks like TOP-RMS to me. You have ||u^J||/||u|| and I'm
also dividing by ||J||, which is a constant. It shouldn't matter.
Show me a symbolic argument for a step in my reasoning being wrong.
(You have the PDF with the proofs, although some of that's redundant
when you think about the geometry.)

Whether it's a sine, tangent, or raw angle shouldn't affect the minimization.

Are you minimizing something other than stdev(u)/rms(u)? Is ||u^J||
proportional to stdev(u) for Tenney weighting (meaning u^J is
proportional to the variance of u)? I take it we agree rms(u) is
proportional to ||u|| with a Euclidean metric.

Graham

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> Gad, so many of these. Thanks for your work on the xen wiki
> by the way. I think we need a page listing each of these
> tuning optimizations...

Writing a Wiki page is exactly the reason I got marinaded over they STD thing. I wanted to write about it, but didn't want to go ahead and call it something without consulting Graham.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > TOP tunings are defined in terms of distance to the JIP, which
> > is an undefined concept in projective space. All it knows about
> > are angles to the JIP.
>
> TOP tunings can be defined different ways. The TOP-RMS definition
> that works for projective space is the sine of the angle between the
> line passing through the weighted tuning map and the line passing
> through the JIP. See, angles. It looks fine to me.

It's not a TOP tuning. You can call it one, but it's another beast altogether.

> With statistical functions, this is stdev(w)/rms(w) where w is the
> Tenney weighted tuning map. You can verify that multiplying w by a
> constant on the top and bottom doesn't alter the result, so it's
> invariant in projective space, or however you phrase that.

How I phrased it is in terms of geometry, not statistics, but it seems to come to the same thing. Why you insist on terminology like
stdev and rms for this business I wish I knew. It confuses the issue unless you analyze the definitions in geometric terms, which raises the question of why you didn't say it that way to start out with.

> The TOP-max definition is [max(w) - min(w)]/[max(w) + min(w)]. I
> don't know how the geometry works out, but it's also invariant if you
> multiply w by a scalar.

It's the L-infinity version of the same thing, but it isn't as well justified.

> Are you doing this numerically? If so, u.u is prone to rounding error.

I did it symbolically, but I doubt since I was using 20 digits rounding error would have hurt much.

> It still looks like TOP-RMS to me.

It is NOT what we have been calling TOP all these years, so don't know why you think it looks like TOP.

You have ||u^J||/||u|| and I'm
> also dividing by ||J||, which is a constant. It shouldn't matter.
> Show me a symbolic argument for a step in my reasoning being wrong.
> (You have the PDF with the proofs, although some of that's redundant
> when you t

Proof of what? That's it's TOP-RMS? I just *disproved* that by a counterexample, and in any case it's completely obvious it can't be. Did you see what I said about an extraneous solution? Does TOP-RMS have one of those?

> Whether it's a sine, tangent, or raw angle shouldn't affect the minimization.

It doesn't.

> Are you minimizing something other than stdev(u)/rms(u)? Is ||u^J||
> proportional to stdev(u) for Tenney weighting (meaning u^J is
> proportional to the variance of u)? I take it we agree rms(u) is
> proportional to ||u|| with a Euclidean metric.

||u^J||^2 is defined as the sum of squares of all the differences between coordinates of u, and it differs by only an irrelevant constant of proportionality from stdev(u)^2. Hence minimizing ||u^J||/||u|| comes to the same thing as minimizing stdev(u)/rms(u). Whatever that thing is, it's clear what it isn't--it isn't TOP-RMS. It is, however, a tuning which it makes sense to stretch and shrink to fit.

On 26 May 2010 10:53, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> > TOP tunings are defined in terms of distance to the JIP, which
>> > is an undefined concept in projective space. All it knows about
>> > are angles to the JIP.
>>
>> TOP tunings can be defined different ways.  The TOP-RMS definition
>> that works for projective space is the sine of the angle between the
>> line passing through the weighted tuning map and the line passing
>> through the JIP.  See, angles.  It looks fine to me.
>
> It's not a TOP tuning. You can call it one, but it's another beast altogether.

Proof?

>> With statistical functions, this is stdev(w)/rms(w) where w is the
>> Tenney weighted tuning map.  You can verify that multiplying w by a
>> constant on the top and bottom doesn't alter the result, so it's
>> invariant in projective space, or however you phrase that.
>
> How I phrased it is in terms of geometry, not statistics, but it seems to come to the same thing. Why you insist on terminology like
> stdev and rms for this business I wish I knew. It confuses the issue unless you analyze the definitions in geometric terms, which raises the question of why you didn't say it that way to start out with.

I don't insist on the statistics. I phrase things two different ways.
If you want geometry you can have geometry. The statistical
functions are what I use to do the calculations because I can
calculate them.

I didn't use geometry in the first place because I couldn't. I really
couldn't. It wasn't some trick, I really couldn't do it. And when I
asked, on this list, if anybody knew of a geometry involving standard
deviations it was because I really wanted to know, not some strange
debating strategy. Like when I asked you about geometric complexity
it was because I didn't understand it (and still don't). I didn't use
geometric terms, like orthogonal projections, because I didn't know
them back then.

>> Are you doing this numerically?  If so, u.u is prone to rounding error.
>
> I did it symbolically, but I doubt since I was using 20 digits rounding error would have hurt much.

Right, with 20 digits it shouldn't matter. u.u only loses you about 6
digits. So have you checked to make sure it really doesn't matter?

>> It still looks like TOP-RMS to me.
>
> It is NOT what we have been calling TOP all these years, so don't know why you think it looks like TOP.

It is, and shouting me down won't change that. Try offering a proof.

>  You have ||u^J||/||u|| and I'm
>> also dividing by ||J||, which is a constant.  It shouldn't matter.
>> Show me a symbolic argument for a step in my reasoning being wrong.
>> (You have the PDF with the proofs, although some of that's redundant
>> when you t
>
> Proof of what? That's it's TOP-RMS? I just *disproved* that by
> a counterexample, and in any case it's completely obvious it
> can't be. Did you see what I said about an extraneous solution?
> Does TOP-RMS have one of those?

What counterexample? All I have is

"""
So you claim Kees-RMS equals unstretched TOP-RMS? Because we have
uniquelt defined definitions for these. I get 696.49489538 cents for
unstretched TOP-RMS for septimal meantone, and previously I got
696.4936084 cents for the Tuning with No Name, and 696.4989638 for the
projective metric tuning.
"""

Using your projective metric tuning, I get the following weighted
tuning map for septimal meantone (12&19):

[1.0, 0.99713135323238189, 0.99988592143689592, 0.99886124466635151]

Standard deviation: 0.0011502457551365284
RMS: 0.99897029204866306

TOP-RMS error allowing scale stretch as the ratio of these numbers:

0.001151431393197523

My unstretched TOP-RMS generator:

696.49489538329351

Weighted tuning map:

[1.0, 0.99712921416142353, 0.99988008086454949, 0.99884916800575996]

Standard deviation: 0.0011502308567138398
RMS: 0.99896527795885814

Error: 0.0011514222587035816
c/f: 0.001151431393197523

Well, what do you know? The unstretched TOP-RMS tuning is doing a
better job of minimizing this function than your counterexample.

Where am I going wrong?

||u^J||/||u|| is TOP-RMS. It isn't in primerr.pdf because I hadn't
worked it out then but the equivalent formulas using linear algebra
are. And I prove them.

I remember something about an extraneous solution but as it's
extraneous I don't know why I should worry about it.

Here we are: "This will give two solutions, corresponding to the
minimum and maximum possible angles with J." So, no, my TOP-RMS
definition doesn't involve this as it entails a *minimum* angle.

>> Whether it's a sine, tangent, or raw angle shouldn't affect the minimization.
>
> It doesn't.
>
>> Are you minimizing something other than stdev(u)/rms(u)?  Is ||u^J||
>> proportional to stdev(u) for Tenney weighting (meaning u^J is
>> proportional to the variance of u)?  I take it we agree rms(u) is
>> proportional to ||u|| with a Euclidean metric.
>
> ||u^J||^2 is defined as the sum of squares of all the differences
> between coordinates of u, and it differs by only an irrelevant
> constant of proportionality from stdev(u)^2. Hence minimizing
> ||u^J||/||u|| comes to the same thing as minimizing stdev(u)/rms(u).
> Whatever that thing is, it's clear what it isn't--it isn't TOP-RMS.
> It is, however, a tuning which it makes sense to stretch and
> shrink to fit.

It is TOP-RMS. And not only have I proved that symbolically, but I've
refuted your counterexample with my floating point variables. (I can
go further if you like). And I've given a geometric argument here
since you came back.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > It is NOT what we have been calling TOP all these years, so don't know why you think it looks like TOP.
>
> It is, and shouting me down won't change that. Try offering a proof.

What a headache. I was going to write something on the xenwiki to explain all this, and now it seems we never had an agreement on what the word meant in the first place. For the record, I've been assuming "TOP" means take the tuning closest to the JIP. That's what I said on my old web site, and no one complained.

If the tuning closest to the JIP isn't TOP (whether L-infinity or L2) then what is it? Who else has their own personal theory about what a TOP tuning is?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@> wrote:
>
> > > It is NOT what we have been calling TOP all these years, so don't know why you think it looks like TOP.
> >
> > It is, and shouting me down won't change that. Try offering a proof.
>
> What a headache.

By the way, how do you decide how much to stretch octaves using your definition of TOP?

On 26 May 2010 15:44, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> > It is NOT what we have been calling TOP all these years, so don't know why you think it looks like TOP.
>>
>> It is, and shouting me down won't change that.  Try offering a proof.
>
> What a headache. I was going to write something on the xenwiki
> to explain all this, and now it seems we never had an agreement
> on what the word meant in the first place. For the record, I've been
> assuming "TOP" means take the tuning closest to the JIP. That's
> what I said on my old web site, and no one complained.

Yes, the TOP tuning is the one that optimizes some Tenney-weighted
error. The TOP-RMS tuning will be the one closest to the JI point
(JIP) in Tenney-weighted tuning space. The TOP-RMS error is the
distance from this optimal tuning to the JIP. It's also the sine of
the angle between the JI line and the tuning line.

This is simple trigonometry. The JIP is a distance of 1 from the
origin, if the norm measures RMS. The TOP-RMS tuning is the closest
point on the tuning line to the JIP. You can form a right angled
triangle with the JIP position vector on the hypotenuse and the
TOP-RMS error on the far side. Hence the TOP-RMS error is the sine of
the angle between these two lines.

Now, there are other things that might be called "TOP". You could
take a regular temperament with an arbitrary tuning, and only optimize
the scale stretch. I have called that a TOP tuning. It is a tuning
that's Tenney optimized. If you don't want to call it that, fine.
It's what the function of the weighted tuning map gives you.

There's also the unstretched TOP. In that case the tuning is on the
same line (maybe projective space) as the TOP tuning. There's a
function which is invariant along this line, and gives the same
result, which is the TOP error for the TOP tuning. If you don't want
to call that function TOP, fine. It still gives the TOP error if you
allow all parameters to be optimized.

> If the tuning closest to the JIP isn't TOP (whether L-infinity or L2)
> then what is it? Who else has their own personal theory about
> what a TOP tuning is?

The tuning closest to the JIP is TOP.

Graham

On 26 May 2010 15:59, genewardsmith
> By the way, how do you decide how much to stretch octaves using your definition of TOP?

Cut the crap, Gene. Disprove the equation you've been ridiculing or apologize.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 26 May 2010 15:59, genewardsmith
> > By the way, how do you decide how much to stretch octaves using your definition of TOP?
>
> Cut the crap, Gene. Disprove the equation you've been ridiculing or apologize.

What crap? Why should I apologize, and for what? What equation have I allegedly been ridiculing?

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> This is simple trigonometry. The JIP is a distance of 1 from the
> origin, if the norm measures RMS.

I may not know what TOP is, but for damned sure I know what the JIP is, since I defined it. The JIP is <1 1 1 1 ... 1|, which will be at a distance of sqrt(n) from the origin.

The TOP-RMS tuning is the closest
> point on the tuning line to the JIP.

Yes. However, the closest *unit vector* to the unit vector J/||J|| is how I define what I've been calling the projective tuning, which can be stretched in various ways (pure octaves, mean value = 1, closest to the JIP, etc.)

You can form a right angled
> triangle with the JIP position vector on the hypotenuse and the
> TOP-RMS error on the far side. Hence the TOP-RMS error is the sine of
> the angle between these two lines.

The TOP-RMS error is the distance to the JIP, ||T-J||. ||T-J||/||J||
is the sine of the angle between them. However, this isn't treating T and J symmetrically, as J is the hypotenuse. You want to minimize the angle between them, which means minimize || T^J ||/(||T|| ||J||) or || T/||T|| - J/||J|| || or some such thing to minimize the projective distance.

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

> The TOP-RMS tuning is the closest
> > point on the tuning line to the JIP.
>
> Yes. However, the closest *unit vector* to the unit vector J/||J|| is how I define what I've been calling the projective tuning, which can be stretched in various ways (pure octaves, mean value = 1, closest to the JIP, etc.)

While we are defining things, how would you define "standard deviation tuning"?

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

>You want to minimize the angle between them, which means minimize || T^J ||/(||T|| ||J||) or || T/||T|| - J/||J|| || or some such thing to minimize the projective distance.

Sorry, Graham, I was thinking you hadn't defined a metric, which has to be symmetrical. But if u and v are unit vectors, then ||u - (u.v)V|| = ||v - (u.v)u||. You could look at it as maximizing u.v. Now I have to figure out why I was coming up with different numbers...sigh.

Anyway, the upshot is that TOP is projective, which apparently you knew and I didn't. Good show.

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

> Anyway, the upshot is that TOP is projective, which apparently you knew and I didn't. Good show.

I just got done calculating projectively for maximizing u.v,
minimizing u^v, and shrinking TOP to pure octaves and got the same answer with no trouble. Why I got a different answer before I don't know, but it didn't help matters.

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

>I just got done calculating projectively for maximizing u.v,
>minimizing u^v, and shrinking TOP to pure octaves and got the same
>answer with no trouble. Why I got a different answer before I don't
>know, but it didn't help matters.

There's a special place in heaven for he who can describe these
different methods with the same framework, in a way that makes sense.
(preferably a geometric one of which pretty pictures can be drawn)

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> There's a special place in heaven for he who can describe these
> different methods with the same framework, in a way that makes sense.
> (preferably a geometric one of which pretty pictures can be drawn)

Someone else will need to draw the pictures but I think I finally understand the situation, and could explain it on the xenwiki.

On 27 May 2010 09:25, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> Someone else will need to draw the pictures but I think I finally understand the situation, and could explain it on the xenwiki.

That's good, reality's been restored. I take it we have:

TOP-max

TOP-RMS

Unstretched TOP-RMS, which is defined in projective space

Unstretched TOP-max, which also looks like it lives in projective
space, but I don't know the geometry

STD error, which is an approximation to some kind of TOP-RMS

Kees-max, which is probably the same as unstretched TOP-max

Odd limit errors, which are a different category

Probably the STD error is some kind of Kees optimization, at least
related to the RMS. But I've never been able to prove that.

Pictures would be really nice. But I don't know how to make them
either. That's what's been holding up graphics in my PDFs. I've
heard that Inkscape's good but I haven't looked at it.

How are you with complexities and badnesses?

Graham

Graham wrote:

>That's good, reality's been restored. I take it we have:
>
>TOP-max
>
>TOP-RMS
>
>Unstretched TOP-RMS, which is defined in projective space
>
>Unstretched TOP-max, which also looks like it lives in projective
>space, but I don't know the geometry
>
>STD error, which is an approximation to some kind of TOP-RMS
>
>Kees-max, which is probably the same as unstretched TOP-max

Maybe a table would help

norm............L1 L2 Linfty
weighting
Tenney ? TOP-RMS TOP
Kees ? ? Kees-max

I strongly object to calling TOP "TOP-max".

I don't understand "stretching", but presumably one would want
to explain why it produces different results than just omitting
2 from the basis.

>Odd limit errors, which are a different category

Yes. I don't think they should be considered on the same page.

>Pictures would be really nice. But I don't know how to make them
>either. That's what's been holding up graphics in my PDFs. I've
>heard that Inkscape's good but I haven't looked at it.

Presumably maple could produce them. Inkscape's fine but you
have to draw it. Something like maple or mathematica that understood
the geometry would probably be easier.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> How are you with complexities and badnesses?

If W is the wedgie in weighted coordinates, I would recommend ||W|| as the complexity measure and ||W^J|| as the error measure. As usual, error and complexity can be cobbled together to give logflat badness, or not to give it, depending on how you feel like that. Just, please, no moats!

>> How are you with complexities and badnesses?
>
>If W is the wedgie in weighted coordinates, I would recommend ||W|| as
>the complexity measure and ||W^J|| as the error measure. As usual,
>error and complexity can be cobbled together to give logflat badness,
>or not to give it, depending on how you feel like that. Just, please,
>no moats!

In my rank1 explorations, I've found that the logflat exponents do
seem to make the most sense.

Why aren't we using one of the errors we've just been discussing for
the error part -- or is ||W^J|| one?

-Carl

On 28 May 2010 02:02, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> If W is the wedgie in weighted coordinates, I would
> recommend ||W|| as the complexity measure and
> ||W^J|| as the error measure. As usual, error and
> complexity can be cobbled together to give logflat
> badness, or not to give it, depending on how you
> feel like that. Just, please, no moats!

||W|| is the scalar complexity and ||W^J|| is the scalar badness.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 28 May 2010 02:02, genewardsmith <genewardsmith@...> wrote:
>
> > If W is the wedgie in weighted coordinates, I would
> > recommend ||W|| as the complexity measure and
> > ||W^J|| as the error measure. As usual, error and
> > complexity can be cobbled together to give logflat
> > badness, or not to give it, depending on how you
> > feel like that. Just, please, no moats!
>
> ||W|| is the scalar complexity and ||W^J|| is the scalar badness.

Very cryptic. What does it mean?

On 28 May 2010 23:12, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>
>> On 28 May 2010 02:02, genewardsmith <genewardsmith@...> wrote:
>>
>> > If W is the wedgie in weighted coordinates, I would
>> > recommend ||W|| as the complexity measure and
>> > ||W^J|| as the error measure. As usual, error and
>> > complexity can be cobbled together to give logflat
>> > badness, or not to give it, depending on how you
>> > feel like that. Just, please, no moats!
>>
>> ||W|| is the scalar complexity and ||W^J|| is the scalar badness.
>
> Very cryptic. What does it mean?

Given that I copied (cut and paste none the less) your notation, you
should understand it. I did also spend a long time writing a PDF in
the vain hope that I wouldn't have to keep explaining these things.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Given that I copied (cut and paste none the less) your notation, you
> should understand it. I did also spend a long time writing a PDF in
> the vain hope that I wouldn't have to keep explaining these things.

Most of us would prefer something posted here, I suspect. To start out with, what do you mean by "scalar" in this context?

On 29 May 2010 12:22, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> Given that I copied (cut and paste none the less) your notation, you
>> should understand it.  I did also spend a long time writing a PDF in
>> the vain hope that I wouldn't have to keep explaining these things.
>
> Most of us would prefer something posted here, I suspect. To start out with, what do you mean by "scalar" in this context?

Yes, and I'd rather not have to type the same arguments out over and over again.

By "scalar" I mean the kind of complexity you can write as ||W|| and
the kind of badness you can write as ||W^J||. They're scalar
quantities. What isn't clear?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> By "scalar" I mean the kind of complexity you can write as ||W|| and
> the kind of badness you can write as ||W^J||. They're scalar
> quantities. What isn't clear?

What wasn't clear was why you dragged in the term "scalar". All they were were numerical quantities not living in any particular vector space and there were no alternatives to consider in any case, but never mind. All is clear on that point. It seems to me that ||W|| measures complexity, ||W^J|| error, and ||W^J||/||W|| relative error. I don't see why ||W^J|| is a very good badness measure. As a badness measure, a lot of people are going to hate it, since it absolutely adores high complexity, low error temperaments. You've got to punish complexity more for a badness measure. Even relative error isn't going to be enough.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>I don't see why ||W^J|| is a very good badness measure.

I think I see why you are calling it badness--it's relative error, which is ||W^J||/||W||, times complexity, ||W||. Error times complexity! But the point remains that you would need to put more oomph into the complexity part to get a badness measure which will make most people happy.

On 29 May 2010 15:50, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>>I don't see why ||W^J|| is a very good badness measure.
>
> I think I see why you are calling it badness--it's relative error, which
> is ||W^J||/||W||, times complexity, ||W||. Error times complexity!
> But the point remains that you would need to put more oomph
> into the complexity part to get a badness measure which will
> make most people happy.

Right, it's error times complexity. When did the TOP-RMS error become
"relative error"?

I do put more oomph into the complexity part. I have pointed you
toward the PDF I explained all that.

Graham

Graham wrote:

>Yes, and I'd rather not have to type the same arguments out over and
>over again.

The problem with the pdfs is that they are long, rambling, and
generally unreadable trains of somebody talking to himself.
The glossaries at the ends are a possible exception. To test this,
I looked up "scalar" in the glossaries of 3 of your 7 PDFs.
I found it in one, and it referred me to primerr, which is one
of the other 2 I'd checked, that doesn't have an entry for
scalar anything.

>By "scalar" I mean the kind of complexity you can write as ||W|| and
>the kind of badness you can write as ||W^J||. They're scalar
>quantities. What isn't clear?

I assumed you meant it had to do with scales, as this was the
basis of the complexity measure formerly known as "Graham".

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> >By "scalar" I mean the kind of complexity you can write as ||W|| and
> >the kind of badness you can write as ||W^J||. They're scalar
> >quantities. What isn't clear?
>
> I assumed you meant it had to do with scales, as this was the
> basis of the complexity measure formerly known as "Graham".

And I was looking for a vector space, because when a mathematician says "scalar" it means there's a vector space over some field F, meaning an abelian group V and a field F and a scalar product of members of F on members of V and a list of linearity properties. Physicists have their own arcane meaning for the term, and probably people in other fields do too. But hey, I believe the term arose in the first place out of mathematical usage. Why make a big deal out of explaining such things?

Gene wrote:

>And I was looking for a vector space, because when a mathematician
>says "scalar" it means there's a vector space over some field F,
>meaning an abelian group V and a field F and a scalar product of
>members of F on members of V and a list of linearity properties.
>Physicists have their own arcane meaning for the term, and probably
>people in other fields do too.

As far as I know, while physicists may not insist on the precise
algebraic properties above, their definition is basically the same.
Certainly, it implies a vector space.

>Why make a big deal out of explaining such things?

I give up; why?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> >Physicists have their own arcane meaning for the term, and probably
> >people in other fields do too.
>
> As far as I know, while physicists may not insist on the precise
> algebraic properties above, their definition is basically the same.
> Certainly, it implies a vector space.

Not hardly. A scalar property such as mass or temperature does not require a vector space, but does require units.

Gene wrote:

>> As far as I know, while physicists may not insist on the precise
>> algebraic properties above, their definition is basically the same.
>> Certainly, it implies a vector space.
>
>Not hardly. A scalar property such as mass or temperature does not
>require a vector space, but does require units.

I thought temperature is a scalar in phase space. Mass, in Minkowski
space (in special relativity) or, for QM I presume, in Hilbert space.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Right, it's error times complexity. When did the TOP-RMS error become
> "relative error"?

Good point. I was thinking "the error which does not depend on the size of what is giving the error", but that's absolute error. So
E = ||W^J||/||W|| and complexity is C = ||W||, and a logflat badness measure would go E * C^(d/(d-r)) = ||W^J|| ||W||^(r/(d-r)), I guess, where d = pi(p) and r = rank. For 7-limit rank two temperaments, for example, that would be ||W^J|| ||W||, which punishes complexity a lot much more than ||W^J|| does.

> I do put more oomph into the complexity part. I have pointed you
> toward the PDF I explained all that.

Experience has shown I'm much better off trying to get answers from you directly.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> >Not hardly. A scalar property such as mass or temperature does not
> >require a vector space, but does require units.
>
> I thought temperature is a scalar in phase space. Mass, in Minkowski
> space (in special relativity) or, for QM I presume, in Hilbert space.

Scalar properties live at points in space-time. If you transform coordinates, rest mass (which is usually but not always what physicists mean by "mass", and yes that's confusing) and temperature are the same numbers in the same units.

On 30 May 2010 01:06, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> And I was looking for a vector space, because when a
> mathematician says "scalar" it means there's a vector space
> over some field F, meaning an abelian group V and a field F
> and a scalar product of members of F on members of V and
> a list of linearity properties. Physicists have their own arcane
> meaning for the term, and probably people in other fields do too.
> But hey, I believe the term arose in the first place out of
> mathematical usage. Why make a big deal out of explaining
> such things?

In this case, it's because it uses a scalar product. With exterior
algebra, anyway, it looks like a scalar product. I didn't say that
yesterday because I'd forgotten. I don't have a clear memory of
everything I did years ago.

There is another argument about it being a scalar quantity because it
doesn't depend on the representation of the mapping. Some other
complexities do depend on the choice of period. I can't remember
where I got this meaning of "scalar" from.

Graham