The mystery of different values of C in incident light exposure meters

[Doug Kerr]

I have been brooding here lately over the matter of exposure meters that, for the incident light exposure metering mode, offer two confirmations, one with a hemispherical (dome) light collector (which we expect would give the meter a cardiod directivity pattern) and one with a flat (disk) collector, which we would expect to give the meter a cosine directive pattern. Almost invariably, the manufacturers of those meters state quite different values of C, the incident light metering calibration constant, for the two configurations.

Why is that?

Here is one possible explanation (admittedly a bit far-fetched, but that ls the best I have just now).

I note that there are an infinity of different illumination situations that can deliver the same illuminance on a certain surface. Here are two:

• A single beam of light with a certain luminous flux density where it strikes the target surface, striking the target surface "head on".

• Two beams of light each with half that luminous flux density where they strike the target surface, both striking the target surface at an angle of 60° from "head on".

If we have two meters, one with cardiod directivity and one with cosine directivity, which have the same "calibration constant" when tested with "head on" light (in exposure meter times, they would have the same value of "C", which is to be determined by testing with "head-on" light), they will read differently in those two lighting situations. The "cardioid" meter will read, in terms of "illuminance", 0.87 times the reading of the cosine meter.

Now consider this new test lighting situation:

• A single beam of light with a certain luminous flux density where it strikes the target surface, striking the target surface "head on".

• Two beams of light each with half that luminous flux density where they strike the target surface, both striking the target surface at an angle of 76.66° from "head on".

It turns out that if now the "cardiod" (dome) exposure meter had a value of C that was 340/250 the value of C for the "cosine" (disk) meter, the two would read the same for this lighting situation.

If in fact exposure meter developers chose, for some reason, to use that "contrived" lighting situation as "representative" of the range of situations to be encountered in actual use, this could have led to the choice of values of C with the ratio 340/250 (such as 340 and 250).

Best regards,

Doug

 

[Ted Cousins]

Another wrinkle is that the value of 'C' is directly proportional to the value of ISO 'S'.

So ... how accurate is the value of S set into your meter?

Well, for the popular methods SOS and REI, entering 100 for example means that the value measured per I.S.O. was anything between 89.09 and 112.2 !!

ergo a presumed 'C' of 320 is anything between 285 and 359.

It gets worse ... in the literature we often see that ISO S=10/Hm ... so, for Hm=0.1 lx.s ... ISO = 100?

Nope ... ISO actually = 9.8/Hm:

1) S = Constant / Hm |The starting point

2) Hm = pi/4 * (Lm*t/N^2) lx-s |Generic formula for mean exposure on a surface

3) (Lm*t/N^2) = K/S |Reflected light meter equation with calibration constant K

Next plug the right hand side of 2) into 3) and then do the same for 2) into 1). Result:

4) Constant = K*pi/4

It says at the light meter link above that Nikon, Canon and Sekonic use K around 12.5 for their light meters. In that case:

5) Constant = 12.5*pi/4 = 9.8

Perhaps they should use K = 12.7 !!

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

Constant = S*Hm

S=K/(Lm*t/N^2)

Hm = pi/4 * (Lm*t/N^2) lx-s

Constant = K/(Lm*t/N^2)*pi/4 * (Lm*t/N^2) lx-s

Constant = K*pi/4 lx-s

If 10 = K*pi/4 then K = 40/pi = 12.7324

12.7 looks rather familiar, ho ho.

best,

Ted.

 

[Doug Kerr]

Hi, Ted,

Another wrinkle is that the value of 'C' is directly proportional to the value of ISO 'S'.

So ... how accurate is the value of S set into your meter?

Well, for the popular methods SOS and REI, entering 100 for example means that the value measured per I.S.O. was anything between 89.09 and 112.2 !!

ergo a presumed 'C' of 320 is anything between 285 and 359.

It gets worse ... in the literature we often see that ISO S=10/Hm ... so, for Hm=0.1 lx.s ... ISO = 100?

Nope ... ISO actually = 9.8/Hm:



12.7 looks rather familiar, ho ho.

best,

Ted.

Yes, all fascinating.

I suspect that the intent in the L398A is that when the sensitivity is set to the detent point marked ISO 100 that this is the actual value.

The ISO 12232 table of assigning "handy" values to S for ranges of the calculated value does not, in my opinion, fit into this conundrum. I can't imagine some engineer at Sekonic saying, "Now when the ISO dial is set to the ISO 100 point, we should make that actually be ISO 85.

But then ya never know!

The detent values for ISO in the L-398A are apparently at 1/3 stop intervals, and so are probably intended to follow the preferred values of ISO speed (arithmetic form) in ISO 12232. But in fact except for the series 25-50-100-200-400 etc, those preferred values are not at precisely 1/3 stop intervals.

So if the ideal intent in the L-398A is for the "100" detent point to be ISO 100, then one detent click higher would theoretically be ISO 125.99, and one click higher yet would theoretically be ISO 158.74 (which if labeled, would no doubt be labeled "160").

Best regards,

Doug

 

[Doug Kerr]

The "dome collector" configuration of photographic exposure meters is attributed to Donald Norwood. His work was seemingly initially motivated by the quest for a technique that could, with a single measurement, indicate the appropriate photographic exposure in the specific use case of photography of a human face illuminated by the "key-fill" lighting technique. Norwood's seminal paper supporting his measurement concept (with a hemispherical light collector on the meter, giving ideally a cardiod directivity) was wholly predicated on that case.

But in subsequent years, it was seemingly determined that this meter configuration was advantageous (compared to a meter with a disk collector) in a wide range of use cases, including various kinds of outdoor photography. Analytical review of that class of situations is certainty beyond my capability.

Although various tasks are suggested in various manuals for the alternative "disk collector" configuration of various exposure meters, the task that seems most relevant is the photography of flat objects, such as manuscripts and paintings. There, it can fairly readily be shown analytically that the appropriate photographic exposure will be indicated by a meter with a cosine directivity.

But what about the "best" value of C for these two meter configurations? If we adopt some exposure objective, such as, "a scene element with 100% reflectivity should give a phtometric exposure on the digital sensor of Hsat, the saturation phtometric exposure, then we can neatly determine by theoretical calculation the value of C that will bring that about.

[In doing that we must assume some value for the lens transmissivity, a parameter that fits into various parts of this story but which is rarely mentioned.]

But as to the panoply of use situations for which the cadioid meter directivity (with the dome collector) is recommended, there is no tidy phtometric model for which we can analytically determine the "best" value of of C.

Almost certainly, the values of C adopted, for the two configurations, by exposure meter manufacturers, we based on numerous empirical tests over a wide range of scene situations. We cannot "second guess" these by logical or mathematical analysis.

Best regards,

Doug

 

[Doug Kerr]

In my prior note, I mentioned that, if we were to assume a certain exposure objective we can readily determine the value of C an incident light exposure meter would have to have to attain that.

I will assume a scene in which all elements were in a plane "facing the camera", for which the use of an exposure meter with a cosine directivity meter (disk collector) is recommended.

Suppose we assume as our exposure objective that a scene element (actual or hypothetical) with a reflectance of 100% (the brightest possible natural element that coiuld be in the scene) would give to the sensor a phtometric exposure of 0.707 Hsat (1/2 stop below the saturation photometric exposure - the infamous "half-stop headroom" paradigm).

Suppose we also assume a lens transmittance of 0.88. [Perfectly reasonable, but yes, I chose that exact value to make this example work out prettily.]

Then for an incident light exposure meter to attain our exposure objective, in theory it would have to have a C of 250. [I will spare the reader the details of the calculation.]

Fancy that!

Best regards,

Doug

 

[Ted Cousins]

The detent values for ISO in the L-398A are apparently at 1/3 stop intervals, and so are probably intended to follow the preferred values of ISO speed (arithmetic form) in ISO 12232. But in fact except for the series 25-50-100-200-400 etc, those preferred values are not at precisely 1/3 stop intervals.

So if the ideal intent in the L-398A is for the "100" detent point to be ISO 100, then one detent click higher would theoretically be ISO 125.99, and one click higher yet would theoretically be ISO 158.74 (which if labeled, would no doubt be labeled "160").

Yes, no coincidence that the cube root of two = 1.259921 and that the aforesaid ISO "reported" values ascend by that ratio but rounded to integers.

 

[Doug Kerr]

I said earlier that the "theoretical" bases for the preferred ISO speed values were separated by "1/3 stop" (that is, by a ratio of the cube root of 2).

To be absolutely precise, the step ratio between the theoretical values is the 10th root of 10.

These two step ratio values only differ by less than one part in 1000, so the difference is of no practical consequence.

If we look at the manual for the Sekonic L-398, the last one to show the "as would be labeled" values of all steps in the sensitivity scale, we see that (once we get out of the small values, where fractional parts screw up the deal) 10 steps up always leads to a value exactly 10 times as great.

Best regards,

Doug

 

[Ted Cousins]

I said earlier that the "theoretical" bases for the preferred ISO speed values were separated by "1/3 stop" (that is, by a ratio of the cube root of 2).

To be absolutely precise, the step ratio between the theoretical values is the 10th root of 10.

Sorry, Doug, I don't get that at all. see below.

These two step ratio values only differ by less than one part in 1000, so the difference is of no practical consequence.

If we look at the manual for the Sekonic L-398, the last one to show the "as would be labeled" values of all steps in the sensitivity scale, we see that (once we get out of the small values, where fractional parts screw up the deal) 10 steps up always leads to a value exactly 10 times as great.

What page?

If we have n steps in a logarithmic range of x₁ to x₂ then x₂ is always exactly equal to x₁ times the ratioⁿ.

Let there be 5 steps in the range 40 to 80 fc. The ratio for that is the fifth root of 2 (because x₂/x₁=2), about 1.148698355 on my Casio.

ergo, 40 x 1.148698355 ..⁵ = exactly 80 fc.

 

[Doug Kerr]

Hi, Ted,

Sorry, Doug. What page?

Page 10.

This figure:

This of course uses the older terminology of "ASA speed" and "DIN degree".

We note that (once we get out of the smaller values), 10 little steps is exactly a ratio of 10. (Note from 100 to "1M" [1000].)

If we have n steps in a logarithmic range of x₁ to x₂ then x₂ is always exactly equal to x₁ times the ratioⁿ.

Let there be 5 steps in the range 40 to 80 fc. The ratio for that is the fifth root of 2 (because x₂/x₁=2), about 1.148698355 on my Casio.

ergo, 40 x 1.148698355 ..⁵ = exactly 80 fc.

What I say only directly pertains only to the sensitivity scale.

Note that per the figure above there are 10 little steps from ASA 100 to ASA 1000. Thus theoretically each little step is a ratio of exactly the 10th root of 10.

Formally, as to sensitivity, a distance of 10 "little steps" steps is intended to be a change in the value of exactly 10:1.

I can't put my hands on the reference to that just now. But we see it recognized in this table from ISO 1223:

The ration between items that are separated by 10 entries ("little steps") is 10; the ratio between items that are separated by 20 entries is 100.

Thus the ratio between adjacent entries ("little steps") is the 10th root of 10.

That is so nearly the cube root of 2 that we generally speak of three little steps as being a ratio of 2:1.

But the theoretical ratio of one big step is theoretically (to be very scrupulous) about 1.9953 (not 2). That ratio is 10^(3/10).

In contrast, on the exposure time scale, one big step is in theory exactly a ratio of 2. Thus the ratio for a little step there is in theory exactly the cube root of 2

Best regards,

Doug

 

[Doug Kerr]

Hi, Ted,

The fact that when it comes to the sensitivity property, the 10th root of 10 is so very nearly the same as the cube root of 2 that we often interchange the two as the ratio of a "little step" without really even knowing we are doing that. Any"error" by interchaging those is negligible.

Where we might see the difference is in the now obsolete" "logarithmic" expression of film speed , which was carried forward into the ISO scheme from the scheme used in the German ("DIN) system. There, the relationship of the "arithmetic" and "logarithmic" expressions was quite precise: an increase in 10 steps of the list of "valid" logarithmic values (which were all integers) was defined as exactly equaivelent to a ratio of 10 in the arithmetic values. Thus a logarithmic value ("S°") of ISO 21° corresponded exactly to the arithmetic value ("S") of ISO 100; a logarithmic value of of ISO 31° corresponded exactly to the arithmetic value ISO 1000.

If indeed 3 little steps would be a ratio of 2, then we would expect that an increase of 9 little steps in the logarithmic expression would be a ratio of exactly 8 in the actual value, but is is not quite.

The relationship is explicitly:

Interestingly enough, this relationship is involved in some of the math in ISO 2720. but the authors (in one place) misbelieved the relationship to be:

which led to some really weird stuff, but that is another story altogether.

Best regards,

Doug

 

[Doug Kerr]

Hi, Ted,

Notwithstanding the fast that the idealized values of successive "preferred" values of S increases by the 10th root of 10 per "step", I do not mean to suggest that in the L-398A calculator the intended spacing between "little steps" on the S scale is the 10th root of 10. Those steps are I'm sure intended to be at the ratio of the cube root of 2, just as on all the other scales (except of course the N scale).

But the labeling is based on 10 steps being a ratio of 10 in the value (a very tiny discrepancy).

Best regards,

Doug

 

[Doug Kerr]

Hi, Ted,

It seems clear that in the exposure calculator of the Sekonic L-398A, 12 steps on the S dial is exactly 4 "one-stop" steps in value on any other scale.

And this is consistent with the labeling of the steps on the S scale, which doubles every 3 clicks.

So we can certainly think in terms of the little steps in S as being "1/3 stop".

Best regards,

Doug

 

[Doug Kerr]

Hi, Ted,

To recap, on the exposure calculator of the Sekonic L-398A, on the "ISO" scale:

• As they affect the calculations, the detent clicks are separated by the ratio of the cube root of 2.

• The markings are as if the clicks were separated by the ratio of the 10th root of 10.

Best regards,

Doug

 

[Ted Cousins]

Hi, Ted,

To recap, on the exposure calculator of the Sekonic L-398A, on the "ISO" scale:

• As they affect the calculations, the detent clicks are separated by the ratio of the cube root of 2.

yes, the cube root of two raised to the power of three is exactly two.

• The markings are as if the clicks were separated by the ratio of the 10th root of 10. [?]

Doug, there seems to be a fundamental misunderstanding of the ISO markings. The 10th root of 10 raised to the power of three is close (1.995) but no cigar. On my 398A, the markings go ... 50, 100, 200, 400, 800 ... increasing by one step each time, each step divided into thirds where one click = one third. Since each marking is the previous one times two, each click must be the cube root of two (as indeed you said), such that three clicks equals one step. No different to the EV scale except that EV has no detents.

In other words, (∛2)³= exactly 2, whereas (tenth√10)³≠2 and 50 x (tenth√10)^12 ≠ 800 ... etc.


best,

Ted.

 

[Doug Kerr]

Hi, Ted,

It is possible to have a series of numbers (not longer than 30) that obey both these:

• If we move ahead by 3 steps, the value changes by exactly 2.

• If we move ahead by 10 steps, the value changes by exactly 10.

How can this be so? The answer is that, in a sequence of numbers not longer than 30, only one number will be a member ofboth:

• The set of every 3rd number.

• The set of every 10th number.

But at the 31st number, this fall apart. That number would need to be:

• 2^ (1024) times the first number, and

• 10^3 (1000) times the first number

and of course it can't be both. So that number must be arbitrarily chosen to be one or the other.

In the ISO dial of the Sekonic L-398A exposure meter, only every 3rd tick is labeled. These are the subset where the value change by exactly 2 from one marked tick to the other (except for the last such jump, from 6400 to 12000). Here, the "12000" number is the 31st of a "season", where the discrepancy has to be dealt with.

For the ticks whose labels are not marked, we find that their "assigned" values (as seen on that figure from the L3-98 manual) are all such that an advance from one number to one 10 ticks further has a value change of 10.

But on the actual dial, the only labels are at every third thick. Then if we ignore the last one (12k) and ignore those below 25 (where the fractional part screws up the deal), we see only a progression of an increase by a factor of exactly 2 from marked tick to marked tick.

And so we think that the whole story is a progression of values by the cube root of two from tick to tick.

Now, can we operate on the belief that the ticks on the ISO dial must theoretically be the cube root of 2 apart? We have to, to make the calculator work as we know it must.

But the underlying story is not quite that.

We see this subtle story play out in the sequence of "preferred values" for the ISO speeds given by that table in ISO 12232. Over a certain range:

• Values that are 3 steps apart have a value ratio of exactly 2.

• Values that are 10 steps apart have a value ratio of exactly 10.

Except that at certain places that fails, for the reason I described earlier. (An example is 640 to 1250. !250 fits the 10 step pattern. while for the 3 step pattern the value would have to be 1280.

I think.

Best regards,

Doug

 

[Doug Kerr]

Hi, Will,

This is my guess as the overall story.

******

In the original German (DIN) "film speed" rating standard, the "index" by which speeds were indicated was logarithmically-related to the actual speed parameter. Only integer value were to be used to mark film packages.

The underlying plan was that a 10-"step" difference in the index represented exactly a 10:1 difference on film speed. (Someting that a bunch of wonks could surely enjoy!)

But since the 10th root of 10 is almost the same as the cube root of 2, photographers were taught that a three-step change in the DIN index meant a 2:1 change ("one stop") in film speed. This was so close to the exact situation that it served well (especially in the context in which such things were spoken of).

When the DIN system and various other film speed systems were consolidated into an ISO standard, out of respect for its wide use, the DIN logarithmnic index system was included as one of the forms in which film speed could be expressed. Thus the "logarithmic" ISO speed system has permitted values that matched the DIN values: a 10-step change in the index meant a 2:1 change in film speed.

But as the film speed standard matured, it was apparently decided (for ISO 12232, for example) that the list of "permitted" film speeds for product marking would follow the "3 steps = 2X" plan, but would also follow the "10 steps=10X" plan. (As I described a little earlier, those two differnt schemes can coexist without conflict for a set of up to 30 values.)

Eventually, the logarithmic form of expressing film speed became obsolete. But the shadow of the DIN scheme remained.

So the ISO 12232 table for quantizing the "speed" property of a sensor into "permitted values" followed both increment schemes, until after a certain number of steps the two would clash. Then one of the two contenders was chosen as the "official" value of that step. Which one? Well, the one that followed the "10 steps=10X" plan. We see that in the ISO 12232 table for the entry after 1000, which is 1250 (10 times the ten steps earlier value, 125), not twice the three steps earlier value, 640.

Why do we not see such a "bump" elsewhere in the table? Well, that table only has 30 entries, so we wouod expect to find only one "bump" in it.

Maybe.

******

Now, what effect does this have on the marking of the ISO dial in the L-298A exposure meter? Not really anything. The detent clicks that are labeled (every third one) are labeled with the precise theoretical value of the ISO speed for that click (excpt for the topmost one,, which exhibits a "bump").

Best regards,

Doug

 

[Ted Cousins]

But the underlying story is not quite that.

We see this subtle story play out in the sequence of "preferred values" for the ISO speeds given by that table in ISO 12232. Over a certain range:

• Values that are 3 steps apart have a value ratio of exactly 2.

• Values that are 10 steps apart have a value ratio of exactly 10.

Not really subtle, Doug, sorry.

Rule: for any number of "steps", the "value ratio" will be equal to the number of steps.

Your If we move ahead by 10 steps, the value changes by exactly 10. obeys that rule, as it must.

Example: for 7 steps the step ratio will be the seventh root of 7 and the ratio of that, raised to the power of seven, equals 7 ...
... the rule above even works for a fractional number of steps, so it's just simple arithmetic, nothing to do with Sekonic ISO at all.

By "step" here I do not mean stops ... can we call them something else?

Sorry to disagree,

best,

Ted.

 

[Doug Kerr]

Hi, Ted.

By "step" here I do not mean stops ... can we call them something else?

I never use "step" to mean stop", notwithstanding the practice of ISO. So here there is no ambiguity.

But I would be glad to adopt a different term for the consecutive "entries" along some discrete scale

******

As to the overall topic, I point out that in the "official" list of the values assigned to all detent points on the L-398A exposure "ISO" dial, if we start with "25" and go up 3 ticks (formerly known as steps), we get to 50 (a ratio for each tick of exactly the 3rd root of 2 (about 1.25499).

If we start with 20 (not marked on the actual scale)and go up 10 ticks, we reach 200. That is a ratio for each tick of the 10th root of 10 (about 1.2598).

If we limit ourselves to the detent points labeled on the actual calculator, and ignore the bottom few and the top one., then it will seem to us the distance between consecutive detent points as marked is uniformly the 3rd root of 2.

But, still limiting ourselves to the marked detent points, if we go from 6400 up 3 ticks, we reach 12000. That would suggest that the ratio between ticks (in that neighborhood) was about 1.2331.

But it is certainly true that the theoretical distance between ticks as actually affects the working of the calculator is uniformly the 3rd root of 2.

Best regards,

Doug

 

[Ted Cousins]

Hi, Ted.


I never use "step" to mean stop", notwithstanding the practice of ISO. So here there is no ambiguity.

But I would be glad to adopt a different term for the consecutive "entries" along some discrete scale

******

As to the overall topic, I point out that in the "official" list of the values assigned to all detent points on the L-398A exposure "ISO" dial, if we start with "25" and go up 3 ticks (formerly known as steps), we get to 50 (a ratio for each tick of exactly the 3rd root of 2 (about 1.25499).

If we start with 20 (not marked on the actual scale)and go up 10 ticks, we reach 200. That is a ratio for each tick of the 10th root of 10 (about 1.2598).

If we limit ourselves to the detent points labeled on the actual calculator, and ignore the bottom few and the top one., then it will seem to us the distance between consecutive detent points as marked is uniformly the 3rd root of 2.

But, still limiting ourselves to the marked detent points, if we go from 6400 up 3 ticks, we reach 12000. That would suggest that the ratio between ticks (in that neighborhood) was about 1.2331.

But it is certainly true that the theoretical distance between ticks as actually affects the working of the calculator is uniformly the 3rd root of 2.

Best regards,

Doug

So,

Amended rule: for any number of [equal] intervals, the "value ratio" will be equal to the number of intervals.

e.g. (the 5.6th root of 5.6)^5.6 = 5.6 not surprisingly.

... and, obviously, (the 10th root of 10)^10 = 10 ... not uniquely to DIN or ISO ...

 

[Doug Kerr]

Hi, Ted,

The relationship between the ISO linear expression of film speed (S) and the ISO logarithmic expression of film speed (S°) is given as:

I understand that was taken directly from the definition of the DIN film speed index.

That of course means that an advance of 10 units in S° gives a factor of 10 in the speed.

I think that in turn led to the ISO table of "accepted" values of film, speed for labeling film packages including, interlaced, both "3 ticks = x2" and "10 ticks=10" patterns.

I think.

Best regards,

Doug

 

[Doug Kerr]

Hi, Ted,

So,

Amended rule: for any number of [equal] ticks, the "value ratio" will be equal to the number of ticks.

So, for a change of 5 ticks, the value ratio wouild be 5?

I would think that for a change of n ticks, the value ratio would be k^n, where k is the value ratio for a change of 1 tick.

Best regards,

Doug

 

[Doug Kerr]

Hi, Ted,

Of course what is actually important is that the markings on the ISO scale of the calculator in the L-398A are such that the calculator will apparently perform the intended calculation (with a very tiny discrepancy for the topmost marked position and the two marked positions below "25".

******

As a related matter, further photogrammetric examination of the calculator suggests that the intent of the scales is that a "one stop" difference in any factor corresponds to an angle of 12° (1/30 of a circle).

Best regards,

Doug

 

[Ted Cousins]

I never use "step" to mean "stop", notwithstanding the practice of ISO. So here there is no ambiguity.

so, on a sekonic l-398a ISO dial, each click is a "step" of value 1/3 "stop"?

Yes, exactly! On the Sekonic L-398A film sensitivity dial, each mechanical click (or marked increment) represents precisely 1/3 of a stop.

Observe the term "precisely" ... and from ISO 50 to 800 is 12 clicks

ergo (cube root of 2)^12 = 800

but (tenth root of ten)^12 is about 792 tsk!

Just sayin' ...

best,

Ted.

 

[Doug Kerr]

Hi, Ted,

so, on a sekonic l-398a ISO dial, each click is a step of value 1/3 stop



Observe the term "precisely" ... and from ISO 50 to 800 is 12 clicks

(cube root of 2)^12=800

but (tenth root of ten)^12=729 tsk!

Each distance between successive ticks, as a matter of slide role mechanics, is intended to represent an interval of precisely 1/3 stop. (I say "intended" to recognize the fact of manufacturing tolerance.) That is true over the entire scale of S.

But because of the situation I have been discussing, where the "preferred values" of S follow two different "meters" (in musical terms), the assigned values of the ticks (only some actually labeled) are not necessarily all separated by the cube root of two.

For example, on the S scale, the distance from 25 (labeled) to 250 (not labeled) is 10 ticks, with an assigned value ratio of 10:1. The distance from 64 (not labeled) to 640 (not labeled) is 10 ticks, a value ratio of 10. The distance from 80 (not labeled) tom 800 (labeled) is 10 ticks, with a value ratio of 10.

But we can't see that on the actual scale, where only every third tick is labeled. For most of the scale, the difference in the marked value of the labeled ticks between one labeled tick and the next labeled tick is indeed a ratio of exactly 2:1.

There is slightly similar matter for the N scale (f-numbers). If we start with, say, f/128, then every 6 steps we have half the f-number. These are the precise f-numbers that follow what I said about "clicks".

But in between, for example, f/16 and f/8 we have "f/11". The theoretical value of that step (from its position on the slide rule scale) should be about f/11.3137. But the "preferred values" for N are all integers, and the preferred value between f/16 and f/8 is "f/11".

But as to the marked value, the ratio between the f/16 marked tick and the f/11 marked tick labels is about 0.541 stops, whereas in terms of slide rule mechanics the angular distance represents exactly 0.5 stops.

Best regards,

Doug

 

[Doug Kerr]

[This should have been made a different thread. My apologies.]

The table to follow shows how we can construct a list of numbers that obeys both:

• If we move 3 steps up the value increases by 2.
• If we move 10 steps up, the value increases by 10.
:

In this example, the "root" number is fairly large (80). That is so all the derived values will be integers. Actual usage of this principle in tables in various photographic standards will involve rounding of some of the final values, but I did not want to introduce that here (as it tends to muddy what is really going on).

The procedure was this. In column 2, starting with the root value (80), I added values every three steps, that value doubling each time.

In column 3, starting with the root value (80) (not shown in that column), I added values every 10 steps, that value increasing by 10 each time.

In column 4, I started with the first "by 10s" value (value number 11). I then went in both directions from it in steps of 3 values, each time increasing or deceasing the value by 2.

In column 5, I started with the second "by 10s" value (value number 21). I then went in both directions from it in steps of 3 values, each time increasing or deceasing the value by 2.

By this time, each of the value positions from 1 through 30 had been populated with a single value. These are gathered in the rightmost column.

Note however that at value number 31, that position gets a value from two processes (the "by 2s" sequence and the "by 10s" sequence, those values being different. Thus the scheme has a fatal conflict at value 31.

In most actual usages of this principle, the list is limited to 30 values, so this conflict does not visit.

Best regards,

Doug

 

[Doug Kerr]

I return here to the original topic of this thread, which is essentially this;

In incident light exposure meters that offer both "dome collector" (cardiod) and "flat collector" (cosine) directivity patterns, why is typically a significantly-different value of C (the incident light exposure metering calibration constant) stated for the two configurations.?

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The use of exposure meters with a cardiod directivity is attributed to Donald Norwood. His original, quest was for a way to, with a single measurement, determine the most appropriate photographic exposure for a shot of a human face illuminated by the "key-fill" lighting technique.

Subjective tests reported in his seminal paper on this matter suggested that if the meter had a certain directivity pattern, it would in fact do that. It turned out the the "cardiod" directivity pattern theoretically exhibited by a meter with a hemispherical ("dome") collector was very near to that ideal directivity pattern.

A bit of analysis of that paper suggests that the ideal calibration of such a meter (today quantified by the factor "C") would be the same as was considered appropriate for a meter with "cosine" response (which responded proportionally to the actual illuminance on the plane of the meter's light collector).

But in modern times, most incident light photographic exposure meters in their "normal" configuration have a dome collector. Bur we do not find that advocated for the photography of a human face lit by the "key-fill" technique. Rather, we find that advocated for use for the incident light metering of most photographic situations.

I can only assume that over the years, that meter behavior had been empirically determined to give the "best" photographic exposure recommendation for the preponderance of photographic scenes.

******

If we consider exposure metering with a "cosine directivity" meter, which responds to the actual illuminance on its collector, and adopt a certain objective exposure "objective" (namely, that an element in the scene with a reflectance of 100% would have a photometric exposure on a digital sensor equal to the saturation phtometric exposure of the sensor system), and with an ideal orientation of the plane of the subject, we can analytically find the value of C needed to bring that about. And that "ideal" value of C is very nearly that often stated for incident light exposure meters in their cosine (flat) collector configuration.

Now it may well be that exposure meter manufacturers had determined empirically that, in the use of "dome" meters over a wide range of general photographic situations (not specifically for the situation on which Norwood originally focused), the best result overall would be given with a value of "C" somewhat greater that the value of C for the use of the flat collector.

Maybe.

Best regards,

Doug

 

[Ted Cousins]

[This should have been made a different thread. My apologies.]

The table to follow shows how we can construct a list of numbers that obeys both:

• If we move 3 steps up the value increases by 2.
• If we move 10 steps up, the value increases by 10.
:

View attachment 14429

In this example, the "root" number is fairly large (80). That is so all the derived values will be integers. Actual usage of this principle in tables in various photographic standards will involve rounding of some of the final values, but I did not want to introduce that here (as it tends to muddy what is really going on).

Doug, with all due respect, I don't see how a starting value of 80 produces only integers, unless "derived" means manipulated so to do; and I am neither talking about rounding, nor am I talking about de facto standard photographic sequences such as shutter speed as found in cameras or even light-meters.

So, on my Casio, and assuming a step ratio of the cube root of two, three steps up from 80 indeed get me 160 but ten steps do not give me exactly 8000, no matter how light-meter mfgs mark their dials.

[edit]written a day or two but forgot to post it[/edit]

Please help us by understanding my view of the matter; it seems to me that I should maybe not be posting mathematical points in this thread.

 

[Doug Kerr]

Hi, Ted,

Doug, with all due respect, I don't see how a starting value of 80 produces only integers, unless "derived" means manipulated so to do;

Well, the algorithm I used generates exactly those values, which all turn out to be integers (in the case of a "root" value of 80) without any manipulation. "Derived" means "by construction from the root value, 80".

You might wish to follow my description of the algorithm I used and see what list of values it gives you.

Remember, that table was not predicated on some uniform ratio between consecutive values. It was solely predicated on satisfying these criteria:

• From any value to one 3 steps further up, the value ratio will be exactly 2.

• From any value to one 10 steps further up, the value ratio will be exactly 10.

which, although it might at first seem otherwise, can in fact be achieved (for not over 30 values).

And this seems to be the predicates of the list of "preferred values" of S, upon which the assignment of values to the 34 clicks of the L-398 sensitivity scale seems to be based.

Yes, these "preferred values" are not separated by a uniform ratio.

Indeed, as to the effective value of the clicks (how they affect the working of the slide rule) ten steps up from 80 does not give 800 (I assume that is what you meant to say, not 8000). It would be 806.349_.

But the value assigned to that click (as it would be labeled, if it were one that received a label) would be 800.

Again it is important that we keep separate:

• The ideal "effective" values of the clicks (as the affect the working of the slide rule)

• The series of "assigned" values (some of which are actually labeled on the associated clicks.

Best regards,

Doug

 

[Doug Kerr]

Hi, Teed,

As to what follows, please at first do not think in terms of clicks on an exposure meter dial, or camera sensitivity ("speed) values. This is just a list of numbers:

80-100-125-160-200-250-320-400-500-640-800-1000-1280-1600-2000-2500

In it:

• If we start from any number and move to the right by 3 numbers, the value changes by exactly a factor of 2.

• If we start from any number and move to the right by 10 numbers, the value changes by exactly a factor of 10.

Note that the numbers as we see them are all integers (and follow those "rules"exactly).

Now back to photography. This might be part of a list of "allowed" values of, say, the ISO film speed rating. In that case, an exposure meter manufacturer might choose to place every 3rd one of them as a "click" label on the sensitivity scale.

I do not mean to suggest that the actual ratio between successive numbers on this list is uniform. In fact, the first several ratios are:

1.25, 1.25, 1.28, 1.25, 1.25, 1.28 (and that sequence in fact continues through 30 numbers)

and in fact, 1.25×1.25×1.28=2.00

and 1.27^7×1.28^3=10.00 (those being the factors involved in an advance of 10 numbers).

Now we know that the intended effective values of the clicks on the L-398 sensitivity scale are separated (ideally) by the uniform ratio of the 3rd root of 2 (1.2599_).

But the assigned values (only every third of them actually labeled) follow the sequence shown above.

Best regards,

Doug

 

[Ted Cousins]

Hi, Ted,

Remember, that table was not predicated on some uniform ratio between consecutive values. It was solely predicated on satisfying these criteria:

• From any value to one 3 steps further up, the value ratio will be exactly 2.

• From any value to one 10 steps further up, the value ratio will be exactly 10.

Thanks Doug,

Now it is clear how you got that table. Interesting approach, assigning different ratios to individual steps ... personally, I'll stick to geometric progressions as fundamental bases for relevant hardware properties and Standards.

best,

Ted.