Digital BW, The Print

Austin et al,

I don't know if you are listening to me any more but I'll try.
Here's a light hearted attempt.  Maybe it'll be amusing enough
to bother reading and ease the tension.

Briefly, I'd like to get really simple -- let's get away from
DyR and talk about elementary school rulers.  Dumb question:
How many inches in a 12 inch ruler?  12.   How do you know?
(12 inches/1 inch) = 12 steps.  How many quarter inches in the
same ruler?  48 = (12 / (1/4)).  How many inches in a 36 inch
yard stick?  36.   blah blah blah.  This so obvious that its
painful. --- please no more examples!

This is approximately what you've been feeling in some of the
posts.  Close maybe?  

Austin:  this is so obvious why are we talking about it.
Roy:  I know, I know, but... I have a different ruler ...
Austin:  no, no,  who cares if its metric or time or anything
     just take the last number (12) by the first number (1)
     and you'll know how many units.  it always works.
Roy:  Well it works a lot of the time, but what about ...
Austin: come on, really, this is idiotic.
Roy:  Well I've a couple of rulers that're different. Can
      I tell you about them?
Austin: (you'll have to supply this one!)

Roy: This ruler I have goes from 1 to 16.
Austin: Great (16/1) = 16. It must have 16 steps.

Roy: Well I only see 8 even marks on it, but maybe they just
     missed every other one.

Austin: Yep.

Roy: Hey, I have another that also goes from 1 to 16,
    but it only has 4 steps.

Austin:  Jees, Why don't you just go buy some decent rulers.
    Or at least add the marks they left out and just 
    relabel them with all the numbers.  it'll be good as new.
    its so simple: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16

Roy:  Well, I sure like these rulers, I've used them for
    years and they've always worked perfectly.  Maybe
    I'll start with cleaning them up and polishing.

Austin: Great. Go for it.

Roy: Hey, guess what that first ruler actually goes all
     the way to 32 now.  Cool, clear  from 1 to 32.

Austin: Well that sounds a lot longer. How the heck did
     you miss the whole other half of the ruler?

Roy: I didn't miss half!  It's just a little longer than
     I thought.  In fact there's only 2 more marks on it.
     So its 10 marks long not the 8 marks I thought it was.

Austin:  This is getting nuttier and nuttier.  How are you
     ever going to get all 32 marks you need on that
     dumb ruler.  Buy a new one.

Roy:  Well I'm going to clean some more and see if there's
      any more stuff written under all this gunk.

Austin:  You're wasting your time.

Roy:  Well its really starting to look good now.  In fact
      it turns out that there really are numbers on all
      the marks.  But, they sure don't look like we
      expected.  Here's what's written on the marks:
     
      1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32.
      Boy, it sure is good I didn't put those other
      numbers on it.  It would have confused things royally.

      My other ruler goes: 1, 2, 4, 8, 16
      Its a lot shorter, they didn't bother with the
      intermediate numbers.

----------------------------

Well obviously the rulers are on my lens,  The first one
is the aperture ring and the second one is the shutter
speed.  I wanted to illustate how different an exponential
series is from a linear series.  Linear is so natural to
our thinking many of its properties just seem universal.
Any kid can answer the questions in my first paragraph
without any deep thought.  Exponential series on the
other hand are very non-intuitive.

Quick Question: How many stops are between f5.6 and f64?  
Ask experienced photographers who've known the series for years.  
How many know the answer instantly?
How many can calculate stops = 2*log2(f1/f2) in their head?
How many are reduced to kindergarten counting on fingers?
See my point.  It is not intuitive.

Going back to my two "rulers", we know that the aperture
scale is a series of equivalent intervals.  They don't
really "look" equivalent but they sure work that way.
Similarly, the intervals of the shutter speed series are
all equivalent. In going from 1 to 16 there are 8 intervals
on the aperture scale but only 4 intervals on the shutter
scale.  Are the intervals bigger on the shutters?  No,
as photographers we all know that intervals on both scales are 
also equivalent to each other.  I.e. every interval = one stop.

Notice that the end points of 1 and 16 tell us nothing
about how many steps there are.  For the same overall
ratio of (16/1) we come up with two different numbers of
steps even when we've used the same "photographic" sense
of one step = one stop.  Actually could also have tick
marks on the lens for fractional stops, giving more
possible "number of steps" values.


Mathematically, within an exponential series a ratio
of any two elements only gives you a relative measure
of "how far apart they are" not any sense of "how much
stuff (how many elements) are between them".

The identical analogy in a linear series would be subtracting
two elements.  Using the ruler, I can say 5 - 3 = 2 inch
interval tells me nothing about how many tick marks there
are in the 2 inches.

------------------------------------------

The DynRange formula is always based on a ratio of two
measured values.  The formula doesn't "care" which kind
of underlying scale is used for the two values.
But the meaning of the underlying scale effects the
meaning of the ratio which finally effects the meaning
of DynRange.  

You've stated quite a few times that the "meaning" of
DynRange is number of steps. I entirely agree if the
underlying scale means something as a linear scale.
But I think the meaning of DynRange loses that connotation
when applied to a scale whose meaning is only as an
exponential scale.

Kind of dense wordage so here's an example.
I consider the fstop scale exponential; but is that really
builtin?  After all its based on a measuring the aperture
size in mm and the focal length in mm.  What if it wasn't?
How about a lens that had: f1,f2,f3,f4,f5,f6,f7...f16.
It wouldn't be very useful because photographically
the intervals would all be different sizes.  So if
you were to apply DynRange the "meaning" of the
results would have no notion of "number of steps".

--------------------------------------

So going way back.  I think the DynRange formula for
sound is fairly well documented and agreed on.  Its based
on the ratio of power levels delivered to sounds waves and
to our ears.  The power scale must be exponential to
give an evenly spaced perception of sound.  Therefore
the DynRange of that sound has no notion of number of
sounds levels.  In a much earlier post I showed my analogies
from sound to light and all the same scaling existed
in light as well as sound.  So I keep coming to the
conclusion that number of gray tones in a print can't
be a property measured by DynRange.


Roy

> I don't know if you are listening to me any more but I'll try.
> Here's a light hearted attempt.  Maybe it'll be amusing enough
> to bother reading and ease the tension.

Roy,

I do read all your posts!  I just don't have the time right now to devote to
responding.  I believe I've simply stated my case, and it should stand on
it's own.  I do want to write this up, as I feel it is important enough to
devote the time to it when I have the time.  I have discussed this very
topic with others (far smarter than I am, and also professionals in the
field), and they, too, view life the same way...so it appears the only thing
I'm doing is attempting to change your mind, and it appears you don't want
to change it.

<snip>

> This is approximately what you've been feeling in some of the
> posts.  Close maybe?

Sure.

> Roy:  Well I've a couple of rulers that're different. Can
>       I tell you about them?
> Austin: (you'll have to supply this one!)

OK ;-)  But they're only different if you look at them sideways/with your
eyes closed/underwater and hold the ruler above water/drink a lot of
gin...etc..

<snip>

> Roy: Hey, I have another that also goes from 1 to 16,
>     but it only has 4 steps.
>
> Austin:  Jees, Why don't you just go buy some decent rulers.

GLASSES, I'd say buy some decent GLASSES ;-)

<snip>

> Austin:  This is getting nuttier and nuttier.

Sounds like I was right about the gin ;-)

<snip>

> Roy:  Well its really starting to look good now.  In fact
>       it turns out that there really are numbers on all
>       the marks.  But, they sure don't look like we
>       expected.  Here's what's written on the marks:
>
>       1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32.

Ah, but that has to do with the area of a circle...vs radius...  You could
very well re-mark those anything you want...like relative area, then it
would be marked 1, 2, 4 etc. (as a divisor), or 1, 1/2, 1/4 etc.

Area = pi * r**2

For a 50mm lense, f1 has an opening of 50mm...and the area (which is
directly proportional to the amount of light let in) is 1963 sq mm.  f1.4,
which is 1/2 the amount of light is 1001 sq mm, f2.0 is 490 etc.

> Well obviously the rulers are on my lens,  The first one
> is the aperture ring and the second one is the shutter
> speed.  I wanted to illustate how different an exponential
> series is from a linear series.

But they aren't different...they ARE both linear.  One is merely part of an
equation that represents the actual area of the opening...it's merely a
representative number and could have been the actual area for that matter!

<snip>

> Going back to my two "rulers", we know that the aperture
> scale is a series of equivalent intervals.  They don't
> really "look" equivalent but they sure work that way.

But to me they do look that way, as I understand what, exactly, they mean.
Any way, you are talking about representative numbers on a scale.

> Similarly, the intervals of the shutter speed series are
> all equivalent. In going from 1 to 16 there are 8 intervals
> on the aperture scale

What numbers happen to be marked on the scale is not relevant.  There are,
theoretically, an infinite number of intervals IN the aperture scale
(typically not in the shutter scale though)...the numbers are merely a
marking of a point along the scale.  Scale markings have nothing to do with
dynamic range, unless the interval between markings is the smallest
discernable signal, and in the case of the aperture, they are not.  In the
case of the shutter speed, they typically are, BTW, as there are typically
not intermediate shutter speeds.

<snip>

> In going from 1 to 16 there are 8 intervals
> on the aperture scale but only 4 intervals on the shutter
> scale.

<snip>

> For the same overall
> ratio of (16/1) we come up with two different numbers of
> steps even when we've used the same "photographic" sense
> of one step = one stop.  Actually could also have tick
> marks on the lens for fractional stops, giving more
> possible "number of steps" values.

This is a misunderstanding.  You are assigning 1 as the minimum discernable
"signal", and it is not.  You are not comparing two dynamic ranges here at
all.

It was a nice try, but your example here is erroneous.  I'll read the rest
of it later, but I believe this misunderstanding really shows why you are
having trouble understanding what really dynamic range is.

As a note, the dynamic range of the aperture is FAR larger than that of the
shutter speed, since shutter speeds have fixed intervals, and apertures do
not.  And yes, there IS noise in the aperture system (as well as the shutter
speed system, but the shutter systems noise should be far less than the
minimum discernable change between shutter steps), as it will only hold so
much of a tolerance...and as I've said before, mechanical systems do have
dynamic ranges.

Regards,

Austin

--- In DigitalBlackandWhiteThePrint@y..., "Austin Franklin" <darkroom@i...> wrote:

> Roy,
> 
> I do read all your posts!  I just don't have the time right now to devote to
> responding.  I believe I've simply stated my case, and it should stand on
> it's own.  I do want to write this up, as I feel it is important enough to
> devote the time to it when I have the time.  I have discussed this very
> topic with others (far smarter than I am, and also professionals in the
> field), and they, too, view life the same way...so it appears the only thing
> I'm doing is attempting to change your mind, and it appears you don't want
> to change it.
> 

That last sentence pretty much works equally well either way.  I could
easily have written it also, word for word.  That's the dilemma.

> 
> > Roy:  Well its really starting to look good now.  In fact
> >       it turns out that there really are numbers on all
> >       the marks.  But, they sure don't look like we
> >       expected.  Here's what's written on the marks:
> >
> >       1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32.
> 
> Ah, but that has to do with the area of a circle...vs radius...  You could
> very well re-mark those anything you want...like relative area, then it
> would be marked 1, 2, 4 etc. (as a divisor), or 1, 1/2, 1/4 etc.

Certainly, could have. They would still be exponential. They would look exactly
like the shutter numbers.

> 
> Area = pi * r**2
> 
> For a 50mm lense, f1 has an opening of 50mm...and the area (which is
> directly proportional to the amount of light let in) is 1963 sq mm.  f1.4,
> which is 1/2 the amount of light is 1001 sq mm, f2.0 is 490 etc.
> 
> > Well obviously the rulers are on my lens,  The first one
> > is the aperture ring and the second one is the shutter
> > speed.  I wanted to illustate how different an exponential
> > series is from a linear series.
> 
> But they aren't different...they ARE both linear.  One is merely part of an
> equation that represents the actual area of the opening...it's merely a
> representative number and could have been the actual area for that matter!

Are you really going to call that series of aperture numbers
and the shutter numbers, LINEAR??  That's a serious problem!
I can't imagine how a claim like that is helping your cause.

The numbers themselves are exponentially varying. 
They measure real physical parameters that are exponentially varying.  
The lens is specifically designed for the exponential variation, explicitly
because the response of either film or the eye requires the
light intensity to be exponentially varying to produce evenly spaced
intervals in our perception.

> 
> Regards,
> 
> Austin

Roy

> > But they aren't different...they ARE both linear.  One is
> merely part of an
> > equation that represents the actual area of the opening...it's merely a
> > representative number and could have been the actual area for
> that matter!
>
> Are you really going to call that series of aperture numbers
> and the shutter numbers, LINEAR??

Sigh.  I know exactly what linear means.  If you really believe I don't,
then you obviously have a very low belief of my mathematical skills, and why
are you even bothering discussing this?

I believe what would help you out a lot, is if you would try to understand
what is meant.  Obviously a series of numbers 1, 2, 4, 8, 16, or 1, 1.4, 2,
2.8 etc are not linear.   I was not talking about the "series of aperture
number" or the "series of shutter numbers", but what they represent IS a
linear scale.  Their INDICATION, no matter what representational "set" of
symbols you assign to them, is ONE f-stop, and f-stops ARE linear, whether
they are shutter speed or aperture, as well as film speed.  Someone who
would want to understand that (not misapply it and then claim it was wrong)
would have understood that.

The shutter speed AND the aperture are exactly the same...one doubles/halves
the speed of the shutter, and the other doubles/halves the AREA of the
opening that allows the light in, and each doubling/halving is one
f-stop...no matter what representation you assign to them.

And, again, this is ALL a red herring on your part, and merely confuses the
issue...since none of this has anything to do with dynamic range.  Yes, the
shutter speeds and apertures have a dynamic range, and they are NOT based on
"1" as you erroneous claimed.  In a digital system, dynamic range is based
on 1, if your digital representation is integer, but that is not necessarily
true in an analog system...no matter what the representative symbols used
within that system are.

Linear or exponential is ALSO not relevant to dynamic range, providing both
the range and error are both in same "system".

> I can't imagine how a claim like that is helping your cause.

Roy, I don't have a "cause".  What I've said is merely fact, and I am simply
trying to explain it to you, but you aren't getting it (or don't want to get
it, for what ever reason), you are only looking for things to claim are
wrong, that aren't wrong...and when I point out erroneous
statements/understandings you make/have, you ignore them...  I completely
understand what you are trying to claim, and as I've said, it is simply
wrong.  It is a misunderstanding.  You can cite all the examples you
want...but your misunderstanding is crystal clear to me.  I believe you are
having trouble conceptualizing what the real meaning of dynamic range is,
and obviously how it applies to imaging.  Dynamic range has a very important
meaning/understanding, and it is not what you are claiming it is.  And, yes,
it IS very misused and misunderstood, not only by your.

There is apparently no end to this that can be achieved, so it is foolish on
my part to continue this, at least with respect to discussing this any
further with you.  There is nothing new being added.  I don't really want to
continue this off list, so I suggest we drop it.

Austin

Austin, 

I'm very disappointed that this has degenerated to this name calling.
This forum is made for everyone to contribute information and learn
information -- its a two-way street.  The attitude that "I know everything"
has become "There is nothing that I need to learn".  How unfortunate!

--- In DigitalBlackandWhiteThePrint@y..., "Austin Franklin" <darkroom@i...> wrote:
> > > But they aren't different...they ARE both linear.  One is
> > merely part of an
> > > equation that represents the actual area of the opening...it's merely a
> > > representative number and could have been the actual area for
> > that matter!
> >
> > Are you really going to call that series of aperture numbers
> > and the shutter numbers, LINEAR??
> 
> Sigh.  I know exactly what linear means.  If you really believe I don't,
> then you obviously have a very low belief of my mathematical skills, and why
> are you even bothering discussing this?

I would have preferred to avoid personal attacks, but yes I do believe
your mathematical skills are a little rusty.  Why am I bothering?  Well,
two-fold, first I thought with a few nudges in the right direction, we collectively
could come up with useful model that might help some people communicate
better and help each other get "better" images.  Second, I saw what I
considered a lot of mathematical misinformation that was leading us all
down the wrong path.

There are those on the list who don't care a bit about the math, and frankly
I applaud them the most.  The most important thing about making pictures
IS actually making pictures, not measuring anything about them.  In the
final test the picture is "what you see", no more, no less.

> 
> I believe what would help you out a lot, is if you would try to understand
> what is meant.  Obviously a series of numbers 1, 2, 4, 8, 16, or 1, 1.4, 2,
> 2.8 etc are not linear.   I was not talking about the "series of aperture
> number" or the "series of shutter numbers", but what they represent IS a
> linear scale.  Their INDICATION, no matter what representational "set" of
> symbols you assign to them, is ONE f-stop, and f-stops ARE linear, whether
> they are shutter speed or aperture, as well as film speed.  Someone who
> would want to understand that (not misapply it and then claim it was wrong)
> would have understood that.
> 
> The shutter speed AND the aperture are exactly the same...one doubles/halves
> the speed of the shutter, and the other doubles/halves the AREA of the
> opening that allows the light in, and each doubling/halving is one
> f-stop...no matter what representation you assign to them.

Rethink some of this: every physical parameter you can measure in a lens
goes up/down exponentially as you go up or down f-stops.  That's means the
diameter of the opening, the area of the opening, the amount of light thru
the opening, and the amount of time that the shutter is open.  Notice also
the numbers used (f-numbers and shutter numbers) are actually reciprocals
of the physical parameters.  The beauty of an exponential series is that all
these math operations (square, square root, reciprocal) all maintain the
exponential character of the series.

In the last post you made some deal about AREA versus diameter, you seemed
to be saying that the diameter (f-number) went up linearly but the area
was the important number and that went up exponentially.  Well you take
a linear scale and square it (getting area) you get a quadratic series,
yet another different series.  Exponential series are the only ones that have
invariance with square, square root, reciprocal.

This high school math.  All of this can be seen in those texts.

> 
> And, again, this is ALL a red herring on your part, and merely confuses the
> issue...since none of this has anything to do with dynamic range.  

Well, you could probably spec and calculate DyR values to your heart's content
without understanding or caring anything about linear vs. exponential scales.
But if you then go and try to describe "what is all means" you darn
better know what the underlying stuff "means" before you try to explain
what the outside result "means".

Yes, the
> shutter speeds and apertures have a dynamic range, and they are NOT based on
> "1" as you erroneous claimed.  In a digital system, dynamic range is based
> on 1, if your digital representation is integer, but that is not necessarily
> true in an analog system...no matter what the representative symbols used
> within that system are.
> 
> Linear or exponential is ALSO not relevant to dynamic range, providing both
> the range and error are both in same "system".
> 
> > I can't imagine how a claim like that is helping your cause.
> 
> Roy, I don't have a "cause".  What I've said is merely fact, and I am simply
> trying to explain it to you, but you aren't getting it (or don't want to get
> it, for what ever reason), you are only looking for things to claim are
> wrong, that aren't wrong...and when I point out erroneous
> statements/understandings you make/have, you ignore them...  I completely
> understand what you are trying to claim, and as I've said, it is simply
> wrong.  It is a misunderstanding.  You can cite all the examples you
> want...but your misunderstanding is crystal clear to me.  I believe you are
> having trouble conceptualizing what the real meaning of dynamic range is,
> and obviously how it applies to imaging.  Dynamic range has a very important
> meaning/understanding, and it is not what you are claiming it is.  And, yes,
> it IS very misused and misunderstood, not only by your.
> 
> There is apparently no end to this that can be achieved, so it is foolish on
> my part to continue this, at least with respect to discussing this any
> further with you.  There is nothing new being added.  I don't really want to
> continue this off list, so I suggest we drop it.

I guess you're right about that.  I agree.  Let's drop it.

I might make one other little suggestion,  go talk to those people who
are "far smarter", ask them "how many steps of power" does that
stereo power amp have.  How big is a step.  The only spec you can
tell them is the dynamic range spec in the owner's manual.  

> 
> Austin

We should all get on with our life.

Good luck,
Roy

Roy,

> Austin,
>
> I'm very disappointed that this has degenerated to this name calling.

I don't see where it has.  Why are you claiming this has "degenerated to
[this?] name calling"???  It wouldn't be the first time one of us
misunderstood/misread something.

> This forum is made for everyone to contribute information and learn
> information -- its a two-way street.

Absolutely!

> The attitude that "I know
> everything"
> has become "There is nothing that I need to learn".  How unfortunate!

That is not what I said (and especially not what I meant).  I have a LOT to
learn, and I learn new things most every day, but I also know what I
know...especially if I have a solid foundation on not only the meaning, but
experience with application and use.

> > Sigh.  I know exactly what linear means.  If you really believe I don't,
> > then you obviously have a very low belief of my mathematical
> skills, and why
> > are you even bothering discussing this?
>
> I would have preferred to avoid personal attacks,
> but yes I do believe
> your mathematical skills are a little rusty.

It IS obvious that you believe I'm not "up to par" mathematically, since you
question whether I understand some basic principles, such as linear...,
yet...you have not shown me to be at all mathematically wrong!

As far as what is necessary to understand dynamic range, everything, related
to mathematics that we've discussed/you've brought up, is very basic, at
least for me (and I do believe for you too).....but when I say something
that I figure you would "get", because it is simply just too "simple" to
explain, you don't...

> Why am I bothering?  Well,
> two-fold, first I thought with a few nudges in the right
> direction, we collectively
> could come up with useful model that might help some people communicate
> better and help each other get "better" images.

Well, I believe I HAVE given some very useful "models".  I also believe the
models you've used don't demonstrate an understanding of what dynamic range
is...but demonstrate the "understanding" you keep "petitioning" for.

> Second, I saw what I
> considered a lot of mathematical misinformation that was leading us all
> down the wrong path.

You haven't pointed out any mathematical misinformation that I have written.
I find your statement kind of odd, since I believe you're the one who threw
in things that just weren't relevant, and were "leading [us all] down the
wrong path"!

> There are those on the list who don't care a bit about the math,
> and frankly
> I applaud them the most.

Er, right...but they aren't discussing the dynamic range
**equation**...which IS what you challenged, and IS math...though the
concept can be understood from examples, it is still the equation that must
be translated into those examples...and without understand the equation (and
the concept behind it), one would really have no foundation for deriving a
correct example.

> The most important thing about making pictures
> IS actually making pictures, not measuring anything about them.  In the
> final test the picture is "what you see", no more, no less.

Though true, if you are trying to understand what a characteristic of a
scanner is, or film is, or an image is...that's where something like this
can very well come into play.  Of course, some people can make exceptional
images without ever understanding dynamic range at all...but that doesn't
mean it doesn't exist, or that it isn't "important" to how this stuff
"works", and/or can't improve your images!

> Rethink some of this: every physical parameter you can measure in a lens
> goes up/down exponentially as you go up or down f-stops.

I don't need to rethink it, I understand what you said...you obviously
missed my point.

> In the last post you made some deal about AREA versus diameter, you seemed
> to be saying that the diameter (f-number) went up linearly but the area
> was the important number and that went up exponentially.

I don't see how you could have misunderstood this.  I even included the
formula for area of a circle, as well as cited some examples.  I obviously
know how that works.  One of the point you missed...that it is stop based,
no matter what number symbol system you assign to it.  You also, apparently,
missed the point of why I brought up area into the discussion.

> > And, again, this is ALL a red herring on your part, and merely
> confuses the
> > issue...since none of this has anything to do with dynamic range.
>
> Well, you could probably spec and calculate DyR values to your
> heart's content
> without understanding or caring anything about linear vs.
> exponential scales.

Exactly.  We have agreement ;-)

> But if you then go and try to describe "what is all means" you darn
> better know what the underlying stuff "means" before you try to explain
> what the outside result "means".

I still don't believe that matters much, in two senses.  Cameras are marked
in shutter speeds because one can relate shutter speed to a real live
event...like you can't take a picture of a speeding car perpendicularly at
1/30th of a second.  It brings the numbers into "reality".

As far as aperture, it's merely a relative scale, and really doesn't matter
what it's marked, as far as relating to reality.  Whether the photographer
understand that it's linear or exponential, it's not going to effect their
ability to take a great picture...  People use EV scales quite a lot, in
fact, I really like using EV...but none of my current use cameras use EV,
though one uses Zones, which is very nice (and it IS related to dynamic
range ;-).

Second, You seem to want to make some pertinent distinction between linear
and exponential, but linear or exponential will not effect one's ability to
understand what dynamic range is, and how it applies.  Even if we agreed on
a "core" concept of what dynamic range was, it's application is the same.
Which, is why I believe it's a red herring.

> I guess you're right about that.  I agree.  Let's drop it.

OK...probably no one but you, me and Todd are reading our posts now anyway
;-)

Austin

> OK...probably no one but you, me and Todd are reading our posts now anyway
> ;-)


Nope, even I bailed... ;-)

Todd