Total absolute error, redux

Of course, George is right, in a 'well'-temperament the total absolute
error is just twice the error on the minor third. But if you compare
one triad with one another, you're still just comparing the minor
thirds. Multiplying or dividing by a factor 2 has no effect on the
comparison.

I also agree that it is important for the minor thirds to sound good.
But a triad has two dimensions to it: no single score can reflect its
overall effect.

I invented a couple of weeks ago a two-dimensional diagram which
displays the quality of the triad in a single point. Unfortunately I
don't have the means to display it in an accessible manner just at the
moment.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> Of course, George is right, in a 'well'-temperament the total
absolute
> error is just twice the error on the minor third.

Thank you, but ...

> But if you compare
> one triad with one another, you're still just comparing the minor
> thirds.

You just contradicted what you said in the previous sentence, where
you acknowledged that I was measuring the error of *all 3 intervals*
in the triad.

> Multiplying or dividing by a factor 2 has no effect on the
> comparison.

True, but that's not total absolute error. Comparisons using the
error of the minor 3rd are only valid under certain conditions, such
as those found in a strict well-temperament, whereas total absolute
error can be used to evaluate the error of *any* chord in *any*
tuning, whether triad, tetrad, pentad, etc. A few days ago I was
using it to evaluate _temperaments ordinaires_ (in which total
absolute error of a triad with wide 5th is twice the error of the
*major* 3rd):

/tuning/topicId_59689.html#60205

(I was hoping that Terry would reply to that one; evidently he didn't
stick around long enough to read it.)

> I also agree that it is important for the minor thirds to sound
good.
> But a triad has two dimensions to it:

Actually, there's more to it than that: in addition to the 3
intervals with their separate (but interacting) beat rates, there are
also combinational (sum and difference) tones to consider.

> no single score can reflect its
> overall effect.

Not prefectly, at any rate, but they're quick and convenient in
giving you a general idea of how one tuning compares with another.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> >
> > Of course, George is right, in a 'well'-temperament the total
> absolute
> > error is just twice the error on the minor third.
>
> Thank you, but ...
>
> > But if you compare
> > one triad with one another, you're still just comparing the minor
> > thirds.
>
> You just contradicted what you said in the previous sentence, where
> you acknowledged that I was measuring the error of *all 3 intervals*
> in the triad.

No contradiction. You measure all three, but the way they are added up
means that two cancel against each other, ending up with the same
answer as if you had measured only one error twice. In any case, the
three aren't independent of one another.

Of course this is a peculiarity of triads only.

>
> (...) a triad has two dimensions to it:
>
> Actually, there's more to it than that: in addition to the 3
> intervals with their separate (but interacting) beat rates, there are
> also combinational (sum and difference) tones to consider.

There are at most *two* independent variables, if we don't consider
transpositions as significant. Every beat rate ratio and combination
tone can be written in terms of (say) the two pitches of the 3rd and
5th relative to the root.

This motivates my diagrams which are finally taking some sort of shape.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > >
> > > Of course, George is right, in a 'well'-temperament the total
> > absolute
> > > error is just twice the error on the minor third.
> >
> > Thank you, but ...
> >
> > > But if you compare
> > > one triad with one another, you're still just comparing the
minor
> > > thirds.
> >
> > You just contradicted what you said in the previous sentence,
where
> > you acknowledged that I was measuring the error of *all 3
intervals*
> > in the triad.
>
> No contradiction. You measure all three, but the way they are added
up
> means that two cancel against each other, ending up with the same
> answer as if you had measured only one error twice.

That's not always the case, as George pointed out.

> There are at most *two* independent variables, if we don't consider
> transpositions as significant. Every beat rate ratio and combination
> tone can be written in terms of (say) the two pitches of the 3rd and
> 5th relative to the root.

That's why I use a 2-dimensional graph, with the three axes
corresponding with the three intervals spaced at 120-degree angles
from one another, for these sorts of issues. Neither of the three
intervals is priveledged, and their dependence on one another is
built right in to the geometry of the graph.

Does scoring the total absolute error account for weighting against
extreme distances from the pure, for example a wolf fifth would err
geometrically higher than a tempered fifth?
If the absolute value of the distance from pure is taken, then
combining positive and negative distances would not cancel. However
combining two barely-sharp-from-pure intervals might result in the
same score as two barely flat intervals, and I think the former
would have a different quality than the latter, and deserve a
different score (assuming the score only comes in a one-dimentional
variety).
(Pardon my ignorance.)
-r

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > >
> > > Of course, George is right, in a 'well'-temperament the total
> > absolute
> > > error is just twice the error on the minor third.
> >
> > Thank you, but ...
> >
> > > But if you compare
> > > one triad with one another, you're still just comparing the
minor
> > > thirds.
> >
> > You just contradicted what you said in the previous sentence,
where
> > you acknowledged that I was measuring the error of *all 3
intervals*
> > in the triad.
>
> No contradiction. You measure all three, but the way they are
added up
> means that two cancel against each other, ending up with the same
> answer as if you had measured only one error twice. In any case,
the
> three aren't independent of one another.
>
> Of course this is a peculiarity of triads only.
>
> >
> > (...) a triad has two dimensions to it:
> >
> > Actually, there's more to it than that: in addition to the 3
> > intervals with their separate (but interacting) beat rates,
there are
> > also combinational (sum and difference) tones to consider.
>
> There are at most *two* independent variables, if we don't consider
> transpositions as significant. Every beat rate ratio and
combination
> tone can be written in terms of (say) the two pitches of the 3rd
and
> 5th relative to the root.
>
> This motivates my diagrams which are finally taking some sort of
shape.
>
> ~~~T~~~

> That's why I use a 2-dimensional graph, with the three axes
> corresponding with the three intervals spaced at 120-degree angles
> from one another, for these sorts of issues. Neither of the three
> intervals is priveledged, and their dependence on one another is
> built right in to the geometry of the graph.

Paul! It just occurred to me that a good way to find test
cases for mean vs. max error would be to take a Chalmers-style
triad plot and color each point a shade of red depending on
how much mean and max error differ. Then one could look for
edges and listen triads on either side of an edge.

-Carl

--- In tuning@yahoogroups.com, "Richard Eldon Barber"
<bassooner42@y...> wrote:

> Does scoring the total absolute error account for weighting against
> extreme distances from the pure,

What does that mean?

> for example a wolf fifth would err
> geometrically higher than a tempered fifth?

It would score much worse if its error was much larger, obviously.
Other than that, I may not be catching your drift. Can you elaborate,
perhaps with an example?

> If the absolute value of the distance from pure is taken, then
> combining positive and negative distances would not cancel.
However
> combining two barely-sharp-from-pure intervals

Which two intervals? The chords in question have three intervals, not
all of which can be wide, and not all of which can be narrow.

> might result in the
> same score as two barely flat intervals, and I think the former
> would have a different quality than the latter, and deserve a
> different score

A higher or lower score?

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

> > That's why I use a 2-dimensional graph, with the three axes
> > corresponding with the three intervals spaced at 120-degree
angles
> > from one another, for these sorts of issues. Neither of the three
> > intervals is priveledged, and their dependence on one another is
> > built right in to the geometry of the graph.
>
> Paul! It just occurred to me that a good way to find test
> cases for mean vs. max error would be to take a Chalmers-style
> triad plot and color each point a shade of red depending on
> how much mean and max error differ.

It's easier than that. The max contours and the mean contours are
both regular hexagons, but rotated 60 degrees relative to one another.

> Then one could look for
> edges and listen triads on either side of an edge.

Well, you could take one equally large hexagon from each set,
resulting in a 12-pointed star. Then two triads within two
adjacent "points" of this 12-pointed star would be inside the first
hexagon and outside the second hexagon, and outside the first and
inside the second, respectively. Theoretically this allow you to
determine which of mean or max was more relevant. Unfortunately, the
points of this star are not very pointy, so you're talking about
rather small percentage differences according to either measure -- so
it's likely rather difficult to make confident choices by ear. But a
similar test with many pairs of triads and many listening subjects
might be worthwhile . . . unfortunately, you'd probably want to take
into account other possiblities, such as weighted error measures
(which would be have irregular hexagonal contours on this graph), so
you'd *really* need a lot of data to even hope for a statistically
significant result.

Sum-squared error (or RMS) would correspond to circles here, as you
know.

> > Paul! It just occurred to me that a good way to find test
> > cases for mean vs. max error would be to take a Chalmers-style
> > triad plot and color each point a shade of red depending on
> > how much mean and max error differ.
>
> It's easier than that. The max contours and the mean contours
> are both regular hexagons, but rotated 60 degrees relative to
> one another.

Thanks!!

> > Then one could look for
> > edges and listen triads on either side of an edge.
>
> Well, you could take one equally large hexagon from each set,
> resulting in a 12-pointed star. Then two triads within two
> adjacent "points" of this 12-pointed star would be inside the
> first hexagon and outside the second hexagon, and outside the
> first and inside the second, respectively. Theoretically this
> allow you to determine which of mean or max was more relevant.
> Unfortunately, the points of this star are not very pointy, so
> you're talking about rather small percentage differences
> according to either measure -- so it's likely rather difficult
> to make confident choices by ear.

Hmm... can't one just use a larger pair of hexagons?

Maybe pairs of points farther apart on the star would help too.

> But a similar test with many pairs of triads and many listening
> subjects might be worthwhile . . . unfortunately, you'd probably
> want to take into account other possiblities, such as weighted
> error measures (which would be have irregular hexagonal contours
> on this graph), so you'd *really* need a lot of data to even hope
> for a statistically significant result.

One could test mean vs. max first and see if there's anything
to it. That would be worth knowing.

> Sum-squared error (or RMS) would correspond to circles here,
> as you know.

What about max squared?

-Carl

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Richard Eldon Barber"
> <bassooner42@y...> wrote:
>
> > Does scoring the total absolute error account for weighting against
> > extreme distances from the pure,
>
> What does that mean?
>

I mean intervals close to pure, for instance within n=5 hz, would have
an error on the order of n Hz * 10^0, between 5 and 7.5 hz would be n
Hz * 10^1, between 7.5 hz and 8.75 hz would be n Hz * 10^2, etc.

> > for example a wolf fifth would err
> > geometrically higher than a tempered fifth?
>
> It would score much worse if its error was much larger, obviously.
> Other than that, I may not be catching your drift. Can you elaborate,
> perhaps with an example?
>

Thats what I mean, except as the distance of the error increases, so
increases the weight of the absolute unit of error by some factor or
secondary scale based on psychoacoustic perception as opposed to a
linear relationship.

> > If the absolute value of the distance from pure is taken, then
> > combining positive and negative distances would not cancel.
> However
> > combining two barely-sharp-from-pure intervals
>
> Which two intervals? The chords in question have three intervals, not
> all of which can be wide, and not all of which can be narrow.

Any or all of the intervals included in the score.

>
> > might result in the
> > same score as two barely flat intervals, and I think the former
> > would have a different quality than the latter, and deserve a
> > different score
>
> A higher or lower score?

Well, for example, if the fifth is wide, maybe the absolute score
should be positive, and if the fifth is narrow, the absolute score
should be negative, without regard to the major and minor thirds. Or
vise versa, with narrow major thirds leading to a negative absolute error.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > Paul! It just occurred to me that a good way to find test
> > > cases for mean vs. max error would be to take a Chalmers-style
> > > triad plot and color each point a shade of red depending on
> > > how much mean and max error differ.
> >
> > It's easier than that. The max contours and the mean contours
> > are both regular hexagons, but rotated 60 degrees relative to
> > one another.
>
> Thanks!!
>
> > > Then one could look for
> > > edges and listen triads on either side of an edge.
> >
> > Well, you could take one equally large hexagon from each set,
> > resulting in a 12-pointed star. Then two triads within two
> > adjacent "points" of this 12-pointed star would be inside the
> > first hexagon and outside the second hexagon, and outside the
> > first and inside the second, respectively. Theoretically this
> > allow you to determine which of mean or max was more relevant.
> > Unfortunately, the points of this star are not very pointy, so
> > you're talking about rather small percentage differences
> > according to either measure -- so it's likely rather difficult
> > to make confident choices by ear.
>
> Hmm... can't one just use a larger pair of hexagons?

Sure. But the *percentage* difference, or ratio, of the error
measures will still be just as small, even if their absolute
difference is now large. Other factors such as beating and
combinational tones, which we didn't explicitly take into account,
are likely to be proportionally more important too, and depend
heavily on inversion, voicing, register, timbre, and loudness. So
making reliable judgments "out there" might not be as much easier as
one would have hoped.

> Maybe pairs of points farther apart on the star would help too.

It's only the angular orientation that would affect the ratio of the
two error measures, and it's at a maximum when the angle is any odd
multiple of 60 degrees.

> > But a similar test with many pairs of triads and many listening
> > subjects might be worthwhile . . . unfortunately, you'd probably
> > want to take into account other possiblities, such as weighted
> > error measures (which would be have irregular hexagonal contours
> > on this graph), so you'd *really* need a lot of data to even hope
> > for a statistically significant result.
>
> One could test mean vs. max first and see if there's anything
> to it. That would be worth knowing.

Right, but one would have to be sure to *control* for these other
possibilities -- to make sure they don't unduly influence the result
one way or the other -- if one is going to believe the answer of this
A/B kind of question.

> > Sum-squared error (or RMS) would correspond to circles here,
> > as you know.
>
> What about max squared?

Same hexagon as max-abs, since if you're minimizing the largest
squared error, it's the same as minimizing the largest absolute error.

--- In tuning@yahoogroups.com, "Richard Eldon Barber"
<bassooner42@y...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Richard Eldon Barber"
> > <bassooner42@y...> wrote:
> >
> > > Does scoring the total absolute error account for weighting
against
> > > extreme distances from the pure,
> >
> > What does that mean?
> >
>
> I mean intervals close to pure, for instance within n=5 hz, would
have
> an error on the order of n Hz * 10^0, between 5 and 7.5 hz would be
n
> Hz * 10^1, between 7.5 hz and 8.75 hz would be n Hz * 10^2, etc.

You mean an error of 5.01 cents is 10 times worse than an error of
4.99 cents? Or . . . (?)

> > > for example a wolf fifth would err
> > > geometrically higher than a tempered fifth?
> >
> > It would score much worse if its error was much larger,
obviously.
> > Other than that, I may not be catching your drift. Can you
elaborate,
> > perhaps with an example?
> >
>
> Thats what I mean, except as the distance of the error increases, so
> increases the weight of the absolute unit of error by some factor or
> secondary scale based on psychoacoustic perception as opposed to a
> linear relationship.

Right. For small errors relative to an ideal JI interval, taking
something proportional to the square of the error makes some sense to
me as a way of assessing the 'awfulness' of the resulting sound. But
this levels off for larger errors and even turns around as you begin
to approach other JI ratios.

However, when optimizing tunings, this squaring or whatever you want
to use may be irrelevant. For example, as I just told Carl,
minimizing the maximum *squared* error is the same as minimizing the
maximum *absolute* error.

> > > If the absolute value of the distance from pure is taken, then
> > > combining positive and negative distances would not cancel.
> > However
> > > combining two barely-sharp-from-pure intervals
> >
> > Which two intervals? The chords in question have three intervals,
not
> > all of which can be wide, and not all of which can be narrow.
>
> Any or all of the intervals included in the score.

See above.

> > > might result in the
> > > same score as two barely flat intervals, and I think the former
> > > would have a different quality than the latter, and deserve a
> > > different score
> >
> > A higher or lower score?
>
> Well, for example, if the fifth is wide, maybe the absolute score
> should be positive, and if the fifth is narrow, the absolute score
> should be negative, without regard to the major and minor thirds.

Well, that would imply that wide fifths sound better than JI if the
score was a good thing, or that narrow fifths sound better than JI if
the score is one of 'demerit'. However, I think *both* wide *and*
narrow fifths sound worse than JI.

> Or
> vise versa, with narrow major thirds leading to a negative absolute
>error.

Perhaps the word "absolute" is confusing since we've been using it in
this discussion to mean the absolute value function, |x|. Anyway, I'm
trying my best to follow you, so let's continue.

> > > Well, you could take one equally large hexagon from each set,
> > > resulting in a 12-pointed star. Then two triads within two
> > > adjacent "points" of this 12-pointed star would be inside the
> > > first hexagon and outside the second hexagon, and outside the
> > > first and inside the second, respectively. Theoretically this
> > > allow you to determine which of mean or max was more relevant.
> > > Unfortunately, the points of this star are not very pointy, so
> > > you're talking about rather small percentage differences
> > > according to either measure -- so it's likely rather difficult
> > > to make confident choices by ear.
> >
> > Hmm... can't one just use a larger pair of hexagons?
>
> Sure. But the *percentage* difference, or ratio, of the error
> measures will still be just as small, even if their absolute
> difference is now large. Other factors such as beating and
> combinational tones, which we didn't explicitly take into account,
> are likely to be proportionally more important too, and depend
> heavily on inversion, voicing, register, timbre, and loudness. So
> making reliable judgments "out there" might not be as much easier
> as one would have hoped.

Looks like I don't know what you mean by "percentage differences".
I thought you meant the chords would not sound different enough.
You actually meant their scores would not be different enough?

> > Maybe pairs of points farther apart on the star would help too.
>
> It's only the angular orientation that would affect the ratio of
> the two error measures, and it's at a maximum when the angle is
> any odd multiple of 60 degrees.

Ok, so adjacent points are too close...

> > > But a similar test with many pairs of triads and many listening
> > > subjects might be worthwhile . . . unfortunately, you'd probably
> > > want to take into account other possiblities, such as weighted
> > > error measures (which would be have irregular hexagonal contours
> > > on this graph), so you'd *really* need a lot of data to even
> > > hope for a statistically significant result.
> >
> > One could test mean vs. max first and see if there's anything
> > to it. That would be worth knowing.
>
> Right, but one would have to be sure to *control* for these other
> possibilities -- to make sure they don't unduly influence the
> result one way or the other -- if one is going to believe the
> answer of this A/B kind of question.

If you test enough pairs (and chords other than 4:5:6) and see a
preference it would seem highly unlikely that weighting was to
blame.

> > > Sum-squared error (or RMS) would correspond to circles here,
> > > as you know.
> >
> > What about max squared?
>
> Same hexagon as max-abs, since if you're minimizing the largest
> squared error, it's the same as minimizing the largest absolute
> error.

Check.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > Well, you could take one equally large hexagon from each set,
> > > > resulting in a 12-pointed star. Then two triads within two
> > > > adjacent "points" of this 12-pointed star would be inside the
> > > > first hexagon and outside the second hexagon, and outside the
> > > > first and inside the second, respectively. Theoretically this
> > > > allow you to determine which of mean or max was more relevant.
> > > > Unfortunately, the points of this star are not very pointy, so
> > > > you're talking about rather small percentage differences
> > > > according to either measure -- so it's likely rather difficult
> > > > to make confident choices by ear.
> > >
> > > Hmm... can't one just use a larger pair of hexagons?
> >
> > Sure. But the *percentage* difference, or ratio, of the error
> > measures will still be just as small, even if their absolute
> > difference is now large. Other factors such as beating and
> > combinational tones, which we didn't explicitly take into
account,
> > are likely to be proportionally more important too, and depend
> > heavily on inversion, voicing, register, timbre, and loudness. So
> > making reliable judgments "out there" might not be as much easier
> > as one would have hoped.
>
> Looks like I don't know what you mean by "percentage differences".
> I thought you meant the chords would not sound different enough.
> You actually meant their scores would not be different enough?

Right, like one will only be a few percent larger than the other.

> > > Maybe pairs of points farther apart on the star would help too.
> >
> > It's only the angular orientation that would affect the ratio of
> > the two error measures, and it's at a maximum when the angle is
> > any odd multiple of 60 degrees.
>
> Ok, so adjacent points are too close...

Whoops, I meant any odd multiple of 30 degrees! Sorry. Adjacent
points on the star are 30 degrees apart, so work as well as possible
for this comparison.

> If you test enough pairs (and chords other than 4:5:6) and see a
> preference it would seem highly unlikely that weighting was to
> blame.

Yes, if the pairs are chosen broadly enough.

>> Looks like I don't know what you mean by "percentage differences".
>> I thought you meant the chords would not sound different enough.
>> You actually meant their scores would not be different enough?
>
> Right, like one will only be a few percent larger than the other.

Maybe the test could be bolstered by comparing pairs of chords
with the same scores and seeing if they sound different.

Maybe it means they're not different enough to bother about.

I assume a similar problem exists when comparing RMS to either
of these?

Maybe the thing to do then, is to compare Tenney weighting with
one of these unweighted measures...

-Carl

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

> However, when optimizing tunings, this squaring or whatever you want
> to use may be irrelevant. For example, as I just told Carl,
> minimizing the maximum *squared* error is the same as minimizing the
> maximum *absolute* error.

Eh? For 5-limit meantone the p=1 fifth is 1/4 comma, which is the same
as the minimax (p=infinity) fifth. It isn't the same as the rms (p=2)
fifth, as you well know, and in general this sort of thing works out
in various ways.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Richard Eldon Barber"
> <bassooner42@y...> wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > > --- In tuning@yahoogroups.com, "Richard Eldon Barber"
> > > <bassooner42@y...> wrote:
> > >
> > > > Does scoring the total absolute error account for weighting
> against
> > > > extreme distances from the pure,
> > >
> > > What does that mean?
> > >
> >
> > I mean intervals close to pure, for instance within n=5 hz, would
> have
> > an error on the order of n Hz * 10^0, between 5 and 7.5 hz would be
> n
> > Hz * 10^1, between 7.5 hz and 8.75 hz would be n Hz * 10^2, etc.
>
> You mean an error of 5.01 cents is 10 times worse than an error of
> 4.99 cents? Or . . . (?)
>

Or unless the weighting is curved by floating point exponent.
Curvature may not be desirable also if your error were to show passage
over a particular distance as an order of 10^n.

> > > > for example a wolf fifth would err
> > > > geometrically higher than a tempered fifth?
> > >
> > > It would score much worse if its error was much larger,
> obviously.
> > > Other than that, I may not be catching your drift. Can you
> elaborate,
> > > perhaps with an example?
> > >
> >
> > Thats what I mean, except as the distance of the error increases, so
> > increases the weight of the absolute unit of error by some factor or
> > secondary scale based on psychoacoustic perception as opposed to a
> > linear relationship.
>
> Right. For small errors relative to an ideal JI interval, taking
> something proportional to the square of the error makes some sense to
> me as a way of assessing the 'awfulness' of the resulting sound. But
> this levels off for larger errors and even turns around as you begin
> to approach other JI ratios.
>

If you approach some other interval not traditionally
psychoacoustically associated with the fifth, for instance, the m6,
the weighting might curve factorially approaching the interval
asymptotically. At the "turn around", perhaps the error weigh
intervals calculated from another fundamental freq.

> However, when optimizing tunings, this squaring or whatever you want
> to use may be irrelevant. For example, as I just told Carl,
> minimizing the maximum *squared* error is the same as minimizing the
> maximum *absolute* error.
>

Yes for optimization thats true. For sorting and classification
however it seems the more complex the weighting system, the more
information can be packed into the numerical value itself.

> > > > If the absolute value of the distance from pure is taken, then
> > > > combining positive and negative distances would not cancel.
> > > However
> > > > combining two barely-sharp-from-pure intervals
> > >
> > > Which two intervals? The chords in question have three intervals,
> not
> > > all of which can be wide, and not all of which can be narrow.
> >
> > Any or all of the intervals included in the score.
>
> See above.
>
> > > > might result in the
> > > > same score as two barely flat intervals, and I think the former
> > > > would have a different quality than the latter, and deserve a
> > > > different score
> > >
> > > A higher or lower score?
> >
> > Well, for example, if the fifth is wide, maybe the absolute score
> > should be positive, and if the fifth is narrow, the absolute score
> > should be negative, without regard to the major and minor thirds.
>
> Well, that would imply that wide fifths sound better than JI if the
> score was a good thing, or that narrow fifths sound better than JI if
> the score is one of 'demerit'. However, I think *both* wide *and*
> narrow fifths sound worse than JI.

I dont mean positive and negative to assign perceptual quality, but to
numerically oppose errors generated by narrow fifth scales and wide
fifth scales. A system can thereby retain vector difference according
to the position of the interval relative to JI.

>
> > Or
> > vise versa, with narrow major thirds leading to a negative absolute
> >error.
>
> Perhaps the word "absolute" is confusing since we've been using it in
> this discussion to mean the absolute value function, |x|. Anyway, I'm
> trying my best to follow you, so let's continue.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:

> > There are at most *two* independent variables, if we don't consider
> > transpositions as significant. Every beat rate ratio and combination
> > tone can be written in terms of (say) the two pitches of the 3rd and
> > 5th relative to the root.
>
> That's why I use a 2-dimensional graph, with the three axes
> corresponding with the three intervals spaced at 120-degree angles
> from one another, for these sorts of issues. Neither of the three
> intervals is privileged, and their dependence on one another is
> built right in to the geometry of the graph.

I've looked in your folder in tuning files (Paul Erlich, right?) and
not found an example. Would be nice to see one.

The diagrams you mention are probably equivalent to mine (now in
tuning_files / sphaerenklang) up to a linear transformation and
relabeling in cents etc. As to what scale you use on what axis and at
what angles... I would find triangular graphs much more difficult to
draw properly, and don't personally find it necessary to make the
deviations of each interval look the same.

Coincidentally, the scales (as in measuring units) I put on my graphs
do result in the fifth appearing to be weighted about 3/2 as much as
the M3.

~~~T~~~

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

> Maybe the test could be bolstered by comparing pairs of chords
> with the same scores and seeing if they sound different.

. . . and seeing if one sounds worse than the other.

> Maybe it means they're not different enough to bother about.
>
> I assume a similar problem exists when comparing RMS to either
> of these?

Yes, and that's even harder to begin with, as now it's a circle vs. a
hexagon, rather than two hexagons oriented as differently as possible.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > However, when optimizing tunings, this squaring or whatever you
want
> > to use may be irrelevant. For example, as I just told Carl,
> > minimizing the maximum *squared* error is the same as minimizing
the
> > maximum *absolute* error.
>
> Eh?

Let me repeat: minimizing the maximum *squared* error is the same as
minimizing the maximum *absolute* error.

> For 5-limit meantone the p=1 fifth is 1/4 comma, which is the same
> as the minimax (p=infinity) fifth. It isn't the same as the rms >
(p=2)
> fifth, as you well know, and in general this sort of thing works out
> in various ways.

Gene, you clearly aren't reading my words very carefully. Minimizing
the maximum squared error is *not* the same as minimizing the rms
error, and only the latter corresponds to the p=2 case.

--- In tuning@yahoogroups.com, "Richard Eldon Barber"
<bassooner42@y...> wrote:

> > > I mean intervals close to pure, for instance within n=5 hz,
would
> > have
> > > an error on the order of n Hz * 10^0, between 5 and 7.5 hz
would be
> > n
> > > Hz * 10^1, between 7.5 hz and 8.75 hz would be n Hz * 10^2, etc.
> >
> > You mean an error of 5.01 cents is 10 times worse than an error
of
> > 4.99 cents? Or . . . (?)
> >
>
> Or unless the weighting is curved by floating point exponent.

I don't understand what that means.

> Curvature may not be desirable also if your error were to show
passage
> over a particular distance as an order of 10^n.

Nor that.

> > > > > for example a wolf fifth would err
> > > > > geometrically higher than a tempered fifth?
> > > >
> > > > It would score much worse if its error was much larger,
> > obviously.
> > > > Other than that, I may not be catching your drift. Can you
> > elaborate,
> > > > perhaps with an example?
> > > >
> > >
> > > Thats what I mean, except as the distance of the error
increases, so
> > > increases the weight of the absolute unit of error by some
factor or
> > > secondary scale based on psychoacoustic perception as opposed
to a
> > > linear relationship.
> >
> > Right. For small errors relative to an ideal JI interval, taking
> > something proportional to the square of the error makes some
sense to
> > me as a way of assessing the 'awfulness' of the resulting sound.
But
> > this levels off for larger errors and even turns around as you
begin
> > to approach other JI ratios.
> >
>
> If you approach some other interval not traditionally
> psychoacoustically associated with the fifth, for instance, the m6,
> the weighting might curve factorially approaching the interval
> asymptotically.

Why is there a "weighting" when we're talking about only one
interval? Perhaps we need to understand one another's terminology
first. It seems to me that one can draw a continuous curve as a
function of interval size, that depicts discordance. For example, see:

/harmonic_entropy/?yguid=124315585

> At the "turn around", perhaps the error weigh
> intervals calculated from another fundamental freq.

Another fundamental frequency? I don't understand. Can you illustrate
with an example?

> > However, when optimizing tunings, this squaring or whatever you
want
> > to use may be irrelevant. For example, as I just told Carl,
> > minimizing the maximum *squared* error is the same as minimizing
the
> > maximum *absolute* error.
> >
> Yes for optimization thats true. For sorting and classification
> however it seems the more complex the weighting system, the more
> information can be packed into the numerical value itself.

What is it you're hoping to sort and classify, and what kind of
information are you hoping to pack into a numerical value?

> > > Well, for example, if the fifth is wide, maybe the absolute
score
> > > should be positive, and if the fifth is narrow, the absolute
score
> > > should be negative, without regard to the major and minor
thirds.
> >
> > Well, that would imply that wide fifths sound better than JI if
the
> > score was a good thing, or that narrow fifths sound better than
JI if
> > the score is one of 'demerit'. However, I think *both* wide *and*
> > narrow fifths sound worse than JI.
>
> I dont mean positive and negative to assign perceptual quality, but
to
> numerically oppose errors generated by narrow fifth scales and wide
> fifth scales.

Now why would you want to do that? Errors are errors, scales are
another issue entirely . . . typically, different scale systems have
different mappings to JI, so one can optimize and obtain narrow fifth
scales for one mapping, wide fifth scales for another mapping, and
scales not generated by the fifth at all for still other mappings.
But it sounds like maybe you have something different in mind?

> A system can thereby retain vector difference according
> to the position of the interval relative to JI.

Can you elaborate with an example?

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> > > There are at most *two* independent variables, if we don't
consider
> > > transpositions as significant. Every beat rate ratio and
combination
> > > tone can be written in terms of (say) the two pitches of the
3rd and
> > > 5th relative to the root.
> >
> > That's why I use a 2-dimensional graph, with the three axes
> > corresponding with the three intervals spaced at 120-degree
angles
> > from one another, for these sorts of issues. Neither of the three
> > intervals is privileged, and their dependence on one another is
> > built right in to the geometry of the graph.
>
>
> I've looked in your folder in tuning files (Paul Erlich, right?) and
> not found an example. Would be nice to see one.

The first graph you see here is an example of this kind of geometry:

http://www.tonalsoft.com/enc/e/equal-temperament.aspx

The grid of yellow lines tells you the error in each of the 5-odd-
limit interval classes.

> > Maybe the test could be bolstered by comparing pairs of chords
> > with the same scores and seeing if they sound different.
>
> . . . and seeing if one sounds worse than the other.

Right -- it's just a control. You seed some of these in the
test corpus.

> > Maybe it means they're not different enough to bother about.
> >
> > I assume a similar problem exists when comparing RMS to either
> > of these?
>
> Yes, and that's even harder to begin with, as now it's a
> circle vs. a hexagon, rather than two hexagons oriented as
> differently as possible.

Gotcha.

-Carl

Sorry, Carl -- big brain fart here on my part. The two hexagons are
the same, not rotated relative to one another at all. For triads,
minimizing equally-weighted total (or mean) error amounts to the same
thing as minimizing equally-weighted maximum. This is because of the
constraint we were talking about: the three signed errors are
constrained to add up to zero (when the right sign convention is
used). Therefore the largest error's absolute value is the sum of the
two other errors' absolute values. So the total error is just twice
the maximum error. The two criteria are therefore equivalent for
triads. When different weightings are used, though, the equivalence
disappears.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > > Well, you could take one equally large hexagon from each
set,
> > > > > resulting in a 12-pointed star. Then two triads within two
> > > > > adjacent "points" of this 12-pointed star would be inside
the
> > > > > first hexagon and outside the second hexagon, and outside
the
> > > > > first and inside the second, respectively. Theoretically
this
> > > > > allow you to determine which of mean or max was more
relevant.
> > > > > Unfortunately, the points of this star are not very pointy,
so
> > > > > you're talking about rather small percentage differences
> > > > > according to either measure -- so it's likely rather
difficult
> > > > > to make confident choices by ear.
> > > >
> > > > Hmm... can't one just use a larger pair of hexagons?
> > >
> > > Sure. But the *percentage* difference, or ratio, of the error
> > > measures will still be just as small, even if their absolute
> > > difference is now large. Other factors such as beating and
> > > combinational tones, which we didn't explicitly take into
> account,
> > > are likely to be proportionally more important too, and depend
> > > heavily on inversion, voicing, register, timbre, and loudness.
So
> > > making reliable judgments "out there" might not be as much
easier
> > > as one would have hoped.
> >
> > Looks like I don't know what you mean by "percentage differences".
> > I thought you meant the chords would not sound different enough.
> > You actually meant their scores would not be different enough?
>
> Right, like one will only be a few percent larger than the other.
>
> > > > Maybe pairs of points farther apart on the star would help
too.
> > >
> > > It's only the angular orientation that would affect the ratio of
> > > the two error measures, and it's at a maximum when the angle is
> > > any odd multiple of 60 degrees.
> >
> > Ok, so adjacent points are too close...
>
> Whoops, I meant any odd multiple of 30 degrees! Sorry. Adjacent
> points on the star are 30 degrees apart, so work as well as
possible
> for this comparison.
>
> > If you test enough pairs (and chords other than 4:5:6) and see a
> > preference it would seem highly unlikely that weighting was to
> > blame.
>
> Yes, if the pairs are chosen broadly enough.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > Maybe the test could be bolstered by comparing pairs of chords
> > > with the same scores and seeing if they sound different.
> >
> > . . . and seeing if one sounds worse than the other.
>
> Right -- it's just a control. You seed some of these in the
> test corpus.
>
> > > Maybe it means they're not different enough to bother about.
> > >
> > > I assume a similar problem exists when comparing RMS to either
> > > of these?
> >
> > Yes, and that's even harder to begin with, as now it's a
> > circle vs. a hexagon, rather than two hexagons oriented as
> > differently as possible.
>
> Gotcha.
>
> -Carl

But see the correction I just posted. There never were two different
hexagons to begin with -- just one.

> Sorry, Carl -- big brain fart here on my part. The two hexagons
> are the same, not rotated relative to one another at all. For
> triads, minimizing equally-weighted total (or mean) error
> amounts to the same thing as minimizing equally-weighted maximum.
> This is because of the constraint we were talking about: the
> three signed errors are constrained to add up to zero (when the
> right sign convention is used). Therefore the largest error's
> absolute value is the sum of the two other errors' absolute
> values. So the total error is just twice the maximum error. The
> two criteria are therefore equivalent for triads. When different
> weightings are used, though, the equivalence disappears.

Huhzah! Whatabout tetrads?

-Carl

> Let me repeat: minimizing the maximum *squared* error is the same
> as minimizing the maximum *absolute* error.
>
> > For 5-limit meantone the p=1 fifth is 1/4 comma, which is the
> > same as the minimax (p=infinity) fifth. It isn't the same as the
> > rms (p=2) fifth, as you well know, and in general this sort of
> > thing works out in various ways.
>
> Gene, you clearly aren't reading my words very carefully.
> Minimizing the maximum squared error is *not* the same as
> minimizing the rms error, and only the latter corresponds
> to the p=2 case.

So clearly mean = max does not apply once things are squared?
Is there some elementary example that'll make all this clear?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > Sorry, Carl -- big brain fart here on my part. The two hexagons
> > are the same, not rotated relative to one another at all. For
> > triads, minimizing equally-weighted total (or mean) error
> > amounts to the same thing as minimizing equally-weighted maximum.
> > This is because of the constraint we were talking about: the
> > three signed errors are constrained to add up to zero (when the
> > right sign convention is used). Therefore the largest error's
> > absolute value is the sum of the two other errors' absolute
> > values. So the total error is just twice the maximum error. The
> > two criteria are therefore equivalent for triads. When different
> > weightings are used, though, the equivalence disappears.
>
> Huhzah! Whatabout tetrads?

For tetrads, it's quite possible to construct a situation where two
tetrads have the same maximum absolute error but different total
absolute error. If you start with a JI tetrad w:x:y:z, then

w : x : y : z+e

has maximum absolute error e and total absolute error 3*e, while

w : x: y+e : z+e

also has maximum absolute error e but has total absolute error 4*e.

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

> > Let me repeat: minimizing the maximum *squared* error is the same
> > as minimizing the maximum *absolute* error.
> >
> > > For 5-limit meantone the p=1 fifth is 1/4 comma, which is the
> > > same as the minimax (p=infinity) fifth. It isn't the same as the
> > > rms (p=2) fifth, as you well know, and in general this sort of
> > > thing works out in various ways.
> >
> > Gene, you clearly aren't reading my words very carefully.
> > Minimizing the maximum squared error is *not* the same as
> > minimizing the rms error, and only the latter corresponds
> > to the p=2 case.
>
> So clearly mean = max does not apply once things are squared?

Right, they are distinct criteria when applied to the squared errors
of the triads (or any other chords), and lead in general to different
optima. Even when they (mean and max) are applied to the absolute
errors, they are only equivalent in the special case of *triads* with
the errors equally weighted.

> Is there some elementary example that'll make all this clear?

That would be nice. What did you have in mind?

> > > Gene, you clearly aren't reading my words very carefully.
> > > Minimizing the maximum squared error is *not* the same as
> > > minimizing the rms error, and only the latter corresponds
> > > to the p=2 case.
> >
> > So clearly mean = max does not apply once things are squared?
>
> Right, they are distinct criteria when applied to the squared
> errors of the triads (or any other chords), and lead in general
> to different optima. Even when they (mean and max) are applied
> to the absolute errors, they are only equivalent in the special
> case of *triads* with the errors equally weighted.
>
> > Is there some elementary example that'll make all this clear?
>
> That would be nice. What did you have in mind?

I wish I knew. -Carl