Zone system mathematically inconsistent

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alanrockwood

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I don't know why I didn't realize this before, but I think I finally understand part of the reason why I have been having trouble understanding the finer points of the zone system, at least in a quantitative sense. The reason is that it makes no sense in a quantitative sense, assuming we discuss it in terms of powers of two. By this I mean that if we assume that each step in the zone system corresponds to a power of two in exposure then the system is mathematically inconsistent.

Let me explain. Let us assume, for example, that an object that of 100% reflectance is placed in zone 9. Then 50% reflectance (corresponding to a one f-stop difference) would be in zone 8, 25% reflectance would be in zone 7, 12.5% reflectance would be in zone 6, and 6.25% reflectance would be in zone 5. Here is the inconsistency. Zone 5 is commonly said to correspond to 18% reflectance, but the 6.25% number we get by counting down from zone 9 is only about a third of 18%. Therefore, the zone system is not mathematically consistent.

My conclusion is that, while the zone system may be qualitatively/conceptually useful, it is not quantitatively correct.
 

baachitraka

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I understood, it was more empirical at first and fit many other things later to make it look more consistent. So 18% or 12.5% of reflectance for middle gray and rest according to the practitioner.
 

phaedrus

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I don't know why I didn't realize this before, but I think I finally understand part of the reason why I have been having trouble understanding the finer points of the zone system, at least in a quantitative sense. The reason is that it makes no sense in a quantitative sense, assuming we discuss it in terms of powers of two. By this I mean that if we assume that each step in the zone system corresponds to a power of two in exposure then the system is mathematically inconsistent.

Let me explain. Let us assume, for example, that an object that of 100% reflectance is placed in zone 9. Then 50% reflectance (corresponding to a one f-stop difference) would be in zone 8, 25% reflectance would be in zone 7, 12.5% reflectance would be in zone 6, and 6.25% reflectance would be in zone 5. Here is the inconsistency. Zone 5 is commonly said to correspond to 18% reflectance, but the 6.25% number we get by counting down from zone 9 is only about a third of 18%. Therefore, the zone system is not mathematically consistent.

My conclusion is that, while the zone system may be qualitatively/conceptually useful, it is not quantitatively correct.
Don‘t you forget the toe and the shoulder of the density curves of the photographic materials?
 

trendland

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I don't know why I didn't realize this before, but I think I finally understand part of the reason why I have been having trouble understanding the finer points of the zone system, at least in a quantitative sense. The reason is that it makes no sense in a quantitative sense, assuming we discuss it in terms of powers of two. By this I mean that if we assume that each step in the zone system corresponds to a power of two in exposure then the system is mathematically inconsistent.

Let me explain. Let us assume, for example, that an object that of 100% reflectance is placed in zone 9. Then 50% reflectance (corresponding to a one f-stop difference) would be in zone 8, 25% reflectance would be in zone 7, 12.5% reflectance would be in zone 6, and 6.25% reflectance would be in zone 5. Here is the inconsistency. Zone 5 is commonly said to correspond to 18% reflectance, but the 6.25% number we get by counting down from zone 9 is only about a third of 18%. Therefore, the zone system is not mathematically consistent.

My conclusion is that, while the zone system may be qualitatively/conceptually useful, it is not quantitatively correct.


I guess here you are absolute right Alan (I will not state that is it NOT in most other cases also)

[that you are not allmost right:kissing:]

Seriously I agree! Why is it inkonsistent? Comming to the point I first heard about zone - system!
I have thoughts eastern europe ......zones? ADIZ(AIR DEFENSE IDENTIFICATION ZONE )and wonder about this photographer! And for sure I did not unterstand Ansel's zones first!
Alan "the first thought is alltime the best" = my first thought was :
Aha ...a photographer made experience and did find a roule!

THAT IS THE POINT = THE MATH CAME LATER BUT HE WAS JUST A GENIOUS IN PHOTOGRAPHY!

agree with you:wink:
 

zanxion72

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I don't know why I didn't realize this before, but I think I finally understand part of the reason why I have been having trouble understanding the finer points of the zone system, at least in a quantitative sense. The reason is that it makes no sense in a quantitative sense, assuming we discuss it in terms of powers of two. By this I mean that if we assume that each step in the zone system corresponds to a power of two in exposure then the system is mathematically inconsistent.

Let me explain. Let us assume, for example, that an object that of 100% reflectance is placed in zone 9. Then 50% reflectance (corresponding to a one f-stop difference) would be in zone 8, 25% reflectance would be in zone 7, 12.5% reflectance would be in zone 6, and 6.25% reflectance would be in zone 5. Here is the inconsistency. Zone 5 is commonly said to correspond to 18% reflectance, but the 6.25% number we get by counting down from zone 9 is only about a third of 18%. Therefore, the zone system is not mathematically consistent.

My conclusion is that, while the zone system may be qualitatively/conceptually useful, it is not quantitatively correct.

I think it goes the other way, placing 18% at Zone 5 and then the rest above and below that. At both ends it all fits, because 90% and 100% reflectance for example record the same on film.
 

cowanw

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I have always wondered why this discussion always starts at the top.
Let's try starting with zero % reflectance in Zone 1.
In Zone 2, we have two times zero % … Oh, I guess that's why we don't start at the bottom; 2x0 is ….
Any way Adams and Archer first defined Zone 5 as the exposure that reproduces middle grey on the print, and then went from there.
There many many possible doublings of 100% white between paper white and sun light white.
 

Photo Engineer

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Dont forget that the zone system is based on a sensitometric curve of 21 steps. So, a few steps are left out. Maybe that may be contributing to your problems Alan. I can't use the zone system. I am too immersed in the 21 step system to use something that is a (to me) kluge.

PE
 

Bill Burk

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I think it goes the other way, placing 18% at Zone 5 and then the rest above and below that. At both ends it all fits, because 90% and 100% reflectance for example record the same on film.

I'd start reasoning along that line as well.

Zone V covers a range that could include 12.5% and 18% (knock off the woohoo voodoo about 12% or 18%). This gives ample room for doubling a couple times, taking you up to Zone VIII and then anything above that, a mirror's reflection or light source, could provide you with the Zone IX.

On the shady side, you can have something that reflects 18% but since shade might be a quarter of the main light that might be your Zone III object.
 

jim10219

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You're thinking too much. The whole point of the zone system was to minimize thinking. Its designed to get a wide range of tones captured onto film that is repeatable for just about anyone using any equipment.

If you go too far down the rabbit hole, you start to realize that you probably should be taking into account the film and developer you're using, since they will greatly effect the actual range of the negative and what happens near the extremes of that range. At what speed are you shooting the film? Then you realize the age of film will also matter, as well as contrast characteristics of the lens. But what about the reciprocity effects of film? And spectral sensitivity? Blah, blah, blah. It goes on and on.

So rather than calculate all of that and worry about every little detail imaginable to come up with the most comprehensive exposure method possible, the Zone System was devised to strike a good balance between ease of use, accuracy of effects, and repeatability of results. If you're a scientist, it may not have enough accuracy for you. If you're a fly-by-the-seat-of-your-pants type, it may be too cumbersome. If you're a middle of the road guy looking for a middle of the road solution, it's a good place to start.
 
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alanrockwood

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You're thinking too much. The whole point of the zone system was to minimize thinking... If you're a scientist, it may not have enough accuracy for you...
I think that's part of my problem. I am a scientist by training and profession.
 

Luckless

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Alan, it works perfectly with the real 21 step system! :D

And, it is scientific.

PE

Do you have a write-up posted that explains the 21 step system and how/why it improves over the "conventional zone" system? (Or possibly a 'totally unbiased' suggestion of a book in which details of such a system may be found? I've been dragging my feet on ordering books authored by members on this site...)

- I've always just used the zone system as an approximate binning method to ballpark my shots into something that, in theory, should net me a usable negative to work with, and have barely bothered with really in-depth detailed usage of it, but "The more tools you know, the more options you have to work with" has always been a useful mindset.
 
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alanrockwood

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I think it goes the other way, placing 18% at Zone 5 and then the rest above and below that. At both ends it all fits, because 90% and 100% reflectance for example record the same on film.
Thanks for the comment. starting at zone 5 does not solve the problem of logical inconsistency. If you assume 18% diffuse reflectance is so-called "middle gray" and place it in zone 5, then the first doubling of reflectance (i.e. one stop difference) puts you in zone 6 with 36% diffuse reflectance. Another doubling puts you in zone 7 with 72% diffuse reflectance. A third doubling puts you in zone 8 with 144% diffuse reflectance (a physical impossibility by the way). A fourth doubling puts you in zone 9 with with 288% diffuse reflectance (also a physical impossibility).

If we allow specular reflectance into the picture, then the picture changes somewhat and comes closer to mapping the zone system onto physical reality, but it doesn't quite get us there because in the qualitative description of the zones specular reflection doesn't enter the qualitative description until zone 10, whereas in the description based on one stop increments zone 8 is already into the realm of specular reflections (144% diffuse reflection is impossible, but if we allow specular reflection then we can get 144% effective reflection.)

Some posters have mentioned the difference between 18% and 12.5% reflectance as the definition of middle gray. I purposely did not bring up that issue because I didn't want to complicate the discussion at the first introduction. However, using a 12.5% standard gets us closer to physical reality if we want to resolve the zone system with physical reality because a 12.5% standard would place zone 8 at 100% diffuse reflectance, which of course is a physical impossibility, but some objects come close enough to 100% diffuse reflectance. However, according to the qualitative description, specular reflection doesn't enter until zone 10, so we still haven't quite managed to make the zone system consistent between the qualitative description, the numerical description, and physical reality.

Let me repeat something I mentioned in another post. I am not disputing that the zone system is useful from a qualitative/conceptual point of view, but rather that it is hard to square with a quantitative point of view.

I have a few other thoughts, but I don't want to make this post even longer than it already is.
 

MattKing

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The 18% grey card isn't middle grey - unless you want it to be.
It is merely a really useful standard.
 

jim10219

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I think that's part of my problem. I am a scientist by training and profession.
Yeah, it's easy to believe the hype on this one. It gets passed off as a super accurate system. And for those of us who are used to the averaging meters in our old cameras and maybe an incident reading, it is. But in reality its more practical than scientific in it's approach.
 

baachitraka

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That said in BTZS there are five stops from Black to White ;-)
 
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alanrockwood

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The 18% grey card isn't middle grey - unless you want it to be.
It is merely a really useful standard.
Yes, I agree it is basically an arbitrary standard, and that is one topic I wanted to discuss more or less as a separate though related issue.

Discussions of the 18% gray card tend to get mired in a lot of esoteric historical and technical minutia that ends up confusing almost everyone, including me.

Aside from the comments I just posted that a 12.5% card would (arguably) fit a little better into a quantitative description of the zone system, there is a matter of perception. To my eye, depending on the day of the week, the phase of the moon, what I had for breakfast, and my general mood, I don't perceive my 18% gray card to be "middle". To me it seems a bit on the dark side. I think to my eye a 25% reflector would be closer to what I would call as a middle tone. Note that this goes in the opposite direction from calling 12.5% reflectance middle gray.

The whole 18% thing can also get tied up into the issue of light meter calibration. I understand that most manufacturers calibrate their meters more along the lines of ~12% rather than 18%.
 

faberryman

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And yet when you do a zone system film test there is a difference in density in the zone XIII and zone IX negative. I think there is something off about merely doubling reflectance. And dynamic range can exceed ten stops, That's the whole point of the zone system: how to fit the dynamic range of a scene onto the dynamic range of your paper, either by compression or expansion.
 
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Photo Engineer

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Do you have a write-up posted that explains the 21 step system and how/why it improves over the "conventional zone" system? (Or possibly a 'totally unbiased' suggestion of a book in which details of such a system may be found? I've been dragging my feet on ordering books authored by members on this site...)

- I've always just used the zone system as an approximate binning method to ballpark my shots into something that, in theory, should net me a usable negative to work with, and have barely bothered with really in-depth detailed usage of it, but "The more tools you know, the more options you have to work with" has always been a useful mindset.

See any of the major works by Mees or Mees and James, or by Haist. Another good but old book is Practical Sensitometry by G. L. Wakefield. There are so many text books it defies detailed listing here. And no, I have not posted anything because the basic idea is probably too hard to do for most. I did it many times though when developing products at EK.

Best of luck.

PE
 

KN4SMF

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The fatal flaw in the zone system is that it is base on an exposure meter with no equality of color sensitivity. No meter does.
 
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alanrockwood

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The ... zone system was ... designed to get a wide range of tones captured onto film ....
Thanks for making that point, along with some points that I did include in quoting your post.

That points to the topic of how to best specify exposure and film speed.

I am familiar with the concept that, historically speaking, our current iso system has its roots in research done at Kodak where they exposed a series of negatives, made the best print possible from each negative, and then asked a panel of people which prints were excellent. The film speed scale was based on what minimum exposure was required to produce an excellent print. There have been some simplifications/refinements since then, but that is the gist of it.

When it comes to capturing a wide range of tones onto film, I think the "first excellent print" method probably missed the point if the goal is to capture the widest range of tones. Admittedly, capturing the widest range of tones was not the goal of the research that ultimately lead to our current ISO film speed scale.

Here are my thoughts about capturing the widest range of tones. To capture the widest range of tones (rather than determining the minimum exposure that can produce an excellent print) the best approach would be to go back to basing the system on gamma rather than on contrast index or related quantities such as G-bar. If we make the simplifying assumption that a typical characteristic curve is approximately a symmetrical sigmoid, then basing the exposure on the middle of the straight line portion of the characteristic curve will give the widest tonal range captured on film. This implies that the system would be based on the slope of the relatively straight portion of the characteristic curve, i.e. based on gamma. (Sometimes people incorrectly equate gamma with contrast index or G-bar, but those quantities are not the same either by definition or by typical numerical value.)

There are some mathematical reasons for making that argument, but let me try to explain with a mix of mathematical and qualitative comments.

One idea behind my suggestion is that one should usually place the subject of interest so that it's average density lies on the part of the curve that has the highest contrast. (Note: this also means that one meters off from that point.) That point is the center of the relatively straight part of the characteristic curve, and the slope of the curve is at its maximum at that point. Mathematically speaking, that point is defined by the second derivative of the curve (i.e. a value of zero for the second derivative), and the contrast is determined by the slope of the curve at that point, i.e. by gamma (i.e. the first derivative of the curve). That will give you the maximum tonal separation for the subject of interest. The darker and lighter portions of the photograph will then fall where they may, but since they are not of primary interest we accept the compromise that their tonal range will not be as well separated as the tonal range of the subject of interest. That compromise is pretty much inevitable no matter what system one uses. I am just stating it explicitly here for sake of clarity.

Here is an example of how this would work. This figure is from some actual film/developer testing I did.
basing film speed on gamma.jpg




In this case, if I could set my meter at an iso of 54 it would give the same density that the graph is showing at the gamma point (density = 1.145, gamma = 0.820). Therefore, I would classify this film as having an exposure index of 50.

The lower point on the graph marked as slope = gamma/3 has a density of 0.516, which as luck would have it is very close to the ideal speed point of 0.525 (base plus fog was 0.425), and that point is 3.5 exposure stops below the gamma point. The upper point with a slope of gamma/3 has a density of 1.870 and is 4.3 exposure stops above the gamma point. Thus, in this case, the curve is not quite symmetrical, but that detail is not too important from a practical point of view.

Interestingly, the parameters for this curve are very close to the iso definition (average slope of 0.64 for my curve vs. slope of 0.62 at a density of 0.8 units above the iso-defined speed point), and in this case I also get a predicted exposure index close to half way between 50 and 64, which is almost the same as what I get from my proposed method.

One could say, "since they give essentially the same result, why use your proposed method instead of the iso method?" That would be a good point. However, I think that if one is interested in recording the widest tonal range on film then my conceptual starting point and method is more defensible than the iso method, which takes as its conceptual starting point to achieve an adequate level of shadow detail. I guess one could summarize the philosophical difference as saying that my method emphasizes the main subject whereas the iso method emphasizes the shadows. In this particular example they end up at about the same end point, even though their conceptual starting points were different. For other film/developer combinations the two approaches might not give a similar recommended exposure index.

Please note that one advantage of my proposed method is that if two film/developer combinations are set up for the same gamma then, regardless of whether or not the same exposure indexes are recommended for the two film/developer combinations, if the recommended exposure indexes for the two film/developer combinations are followed they will give essentially the same tonal separation for the main subject of the photograph.

By the way, I would call this particular film/developer combination as long-toe. What do you think?

One last point: If one wants to emphasize the shadows a bit more one could shift the effective film speed down a little bit without greatly affecting the tonal separation at of the main subject, provided the one doesn't go too far in that direction. Conversely, one could shift the effective film speed up a little if one wants to give more tonal separation to the highlights without greatly affecting the tonal range of the main subject, provided one doesn't go too far in that direction.
 
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alanrockwood

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One additional point to add to what I wrote in my last post. I have not said what gamma one should shoot for. That is something to be left up to the shooter. This is where one could bring zone system concepts back into the discussion in a slightly different way. If one has a scene with a wide tonal range then one might want to decrease development so as to reduce the gamma, thereby bringing in a wider tonal range from the scene. Conversely, of one has a narrow tonal range then more development might be recommended in order to get better tonal separation.

One could summarize this by changing the old saying "expose for the shadows and develop for the highlights" to something more like "expose for the main subject and develop for the tonal range of the scene" though there might be some subtleties not captured in such a short slogan.
 
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alanrockwood

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And how do you meter a scene for gamma 0.82?

You first do some background work, i.e. testing to set up your film/developer combination for your preferred gamma. That gives you a recommended exposure index for that film/developer combination at the selected gamma. When you are going to shoot a photo (which could be years after you have set up your preferred gamma and your effective film speed) you set your meter to the recommended "iso" for that film/developer combination and meter the main subject of interest in the photo.

There is nothing sacred about a gamma of 0.82. It is simply the value I got when testing this film developer combination.
 
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