How to compute half an f-stop

[owen_farmer]

I recently went through an exercise of computing half an f-stop

between f2.8 and f4.0 (trying to determine exact guide numbers for

electronic flashes). The number I arrived at is f3.4 (f3.36, to be

precise), not f3.5, which I had expected. The following summarizes

the calculation.

 

First, exact f-stops are: 1.414, 2.0 , 2.828, 4.0, 5.656, 8.0, etc.

 

Halfway between 2 f-stops might mean arithmetic mean or geometric

mean. For instance, the arithmetic mean of 2 and 4 is 3, while the

geometric mean of 2 and 4 is the square root of 2X4=8, or 2.828. I

assume that geometric mean should be used, since it gives the f-stop

between f2 and f4.

 

Another example: we know that f4.0 is halfway between f2.0 and f8.0.

The geometric mean of 2 and 8 is the square root of 2X8=16, which is 4.

 

The geometric mean of f2.828 and f4.0 is the square root of

2.828X4.0=11.312, which is f3.36.

 

Similarly, half an f-stop between f1.414 and f2.0 is f1.7 (not f1.8).

 

Does anyone disagree with this? If so, how did f3.5 get established

as half an f-stop? Thanks,

Owen

[mark_sirota1]

The actual progression on half-stops (rounded to two digits) is

<p>

1.00<br>

1.19<br>

1.41<br>

1.68<br>

2.00<br>

2.38<br>

2.83<br>

3.36<br>

4.00<br>

4.76<br>

5.66<br>

6.73<br>

8.00<br>

<p>

You get this by multiplying the previous by sqrt(sqrt(2)).

<p>

Never mind 3.5 being wrong... Whole stops are wrong too. 5.6 should be 5.7, and 22 should be 23.

[miha_steinb_cher]

Owen,

 

Your math is right, it's the geometric mean that should be used.

 

Yu're also right with 1.7 being half stop between 1.4 and 2. 1.8 is 2/3 stop from 1.4. It's the same thing with 3.5 - 2/3 stop from 2.8 (or 1/3 from 4 for that matter).

 

Best regards,

 

Miha

 

PS (for computing half stops multiply by 1.19 and by 1.12 for 1/3 stops)

[mark_sirota1]

By the way, who claims that f/3.5 is a half-stop? It's pretty close to a third-stop (just a hair more than a third stop faster than f/4).

 

The third-stop progression (rounding again) is 1, 1.1, 1.3, 1.4, 1.6, 1.8, 2, 2.2, 2.5, 2.8, 3.2, 3.6, 4, 4.5, 5.0, 5.7, 6.3, 7.1, 8

[richard_cochran]

Actually, third-stops are more commonly used than half stops. Note that the film speed scale moves in third stops. f/3.5 is a rounded-off third-stop. The common lens aperture f/4.5 is also a rounded off third-stop.

<p>

To two decimal places, the 1/3 stop sequence goes in part:

<p>

2.00, 2.24, 2.52, 2.83, 3.17, 3.56, 4.0, 4.49, 5.04, 5.66

<p>

Each number is the sixth root of two times the previous number.

<p>

While the half-stop sequence goes

<p>

2.00, 2.38, 2.82, 3.36, 4.0 4.76, 5.66

<p>

Each number is the fourth root of two times the previous number.

 

<p>

There's a convention to aperture scales that doesn't exactly follow the normal rounding rules -- sometimes they truncate to two significant figures instead of properly rounding. f/5.6 should really be f/5.7, but nobody writes it that way. Similarly, f/3.5 should probably be rounded to f/3.6, but nobody does that. It's been done that way for a long, long, time, and photographers don't demand that kind of mathematical precision, at least they don't demand it strongly enough to get the lens makers to defy tradition.

<p>

Don't put too much significance in the precise numbers. Internal reflections and the differences between f-stops and t-stops puts a lot more slop in things than a slight error in the second significant figure.

[charles_stobbs3]

A lot of older camera f stops were based on 3.5 3.5, 4.5, 6.3, 9, 13.5, 18, etc. An f stop system can be based on whatever you want.

[mag_miksch]

I think I have to copy this page of wisdom^^

[jcs56]

Mag:

I will do too!

[stephen hazelton]

Whole stops: Each number is previous number times the square root of 2 (1.4142)

 

Half stops: Each number is previous number times the fourth root of two (1.1892).

 

Third stops: Each number is previous number times the sixth root of two (1.12246).

[dave_young3]

I think someone needs to get a job. Way to much time on your hands

[erin.e]

Or maybe they could go take some pics instead Dave!

[awahlster]

Or just down load the free program Fcalc and have all this math done for you is an instant.

[pvp]

I'm sure f/3.5 was chosen because it was (a) easy to remember and (b) close enough.

 

Someone who has the time to sit around examining your film with a densitometer might be able to tell the difference between f/3.5 and f/3.4 or f/3.36.

 

Nobody else will notice. Or care.

 

You'll get more variation in the light output from a strobe if you let it charge a few seconds more or less between shots.

[owen_farmer]

This is more an exercise in playing with numbers than anything important to a photographer. I am retired and, at least yesterday, didn't have anything better to do. (It was a cold day.) Thanks for the info. Owen

[r_miller1]

Owen,

 

If you are at f4 and want to open-up a half stop, i.e., allow 50% more light in than at f4, then I think you will find that you need to open up to f3.2, not the numbers suggested above.

 

If you start with the definition of the f value, then determine the aperture required to yield an area that is 50% larger, and finally use that aperture to determine the f value, I think you will find that it equates to about f3.2.

 

Kind regards,

[shiver_me_timbrrrre]

R. Miller, you are wrong.

 

The trouble with your assumption, R. Miller, is that you equate opening up a half-stop to letting in 50% more light. It does NOT. Opening up a half-stop actually lets in close to 41% more light i.e. 1.4142...X It has to do with the geometry of a circle. A half-stop is 2 raised to the power of 1/2. You may argue, 50%, 41%, what's the difference? Using your reasoning, the F-stop numbers will come out very differently and your exposures will be all over the place.

 

Do the math for yourself. Start with F1.0 and then try to derive the rest of the series.

[r_miller1]

Shiv,

 

Glad that at least you are awake. And yes, quite a few of us have worked through the sequence in the past and are aware that the area of a circle varies with respect to the square of another dimension. You have simply chosen a different method of pointing out that "half a stop" or "half a step" is not half of an f-stop and that those who simply focus on the progression in an adopted convention, without understanding the underlying rationale, may never understand that half of an f-stop is less than half of step (and for most - "half a stop").

 

Clearly you could not argue with my prior post if the word stop was replaced with step... or would you?

 

Without suggesting that you are wrong, I would guess that other posters may disagree with "[a] half-stop is 2 raised to the power of 1/2."

 

Kind regards,

[jcs56]

If We took a square of light each f Stop will be equal to the half or double of each one--- If We have a Circle then (Pi=3.1416)... is enveloped.

 

Now we can do it 'by had' using paper or a calculator. ;=)

[shiver_me_timbrrrre]

R. Miller, you clearly do not know how the F-stop numbers are derived, do you? And neither do you know what you are talking about. I am awake and it is clear to me that you are not.

 

It has nothing to do with whether you wish to call it step or stop but to do with the geometry of the circle which if you do know what you are talking about would see that it is fact and not a matter of opinion.

 

Yours is another illustration of why the USA is lagging the rest of the world in mathematics and science.

 

I suggest you think about it before you embarrass yourself further.

[r_miller1]

It was a meager attempt to help foster an understanding that an f number is really a function of just two factors, focal length and aperture [and yes, I do understand that the true definition is more sophisticated and involved than simply "aperture"), and that the traditional/common "f stop" scale is simply a convention, albeit one that has basically universal usage, notwithstanding that Zeiss, with their Luminars, and others have used different conventions.

 

As Mr. Stobbs had already correctly pointed out, "[a]n f stop system can be based on whatever you want."

Thus, one could select f5.0 as a base point and you would be perfectly capable of calculating an f stop scale based on a starting point of f5.0, and on the traditional doubling or halving of the amount of light to be transmitted. Or, be original and base your system on a tripling or "thirding" of the amount of light and call it the "Shiv Scale" eh? It would take only a few moments what with calculators and spreadsheets - wouldnﾒt have to resort to those pesky log tables.

 

Even though you are obviously not concerned with embarrassing yourself, you may want to learn an additional phrase beyond "t has to do with the geometry of a circle" - something along the lines of the area of a circle varying with the square of its radius may add at least a little something at least cosmetically even though you have nothing of substance to say.

 

I have wasted enough time. Based on your lack of civility and insulting nature (yikes - based on the preceding paragraph it must be contagious) and obvious personality (quite easy to understand why you wouldn't use your name), you will surely have to have the last word. So be it.

[shiver_me_timbrrrre]

R. Miller. That is sheer and utter nonsense compounded by your chutzpah in pointing out the 'errors' of those who provided correct answers before you.

 

Answer me this: two half-stops or half-steps as you prefer to call it make 1-step. Based on your claim, one half-step is 1.5X of the previous step. So you can make one step to get from 1 to 2. You can also make two half-steps to get from 1 to 2 i.e. 1 to 1 plus one half-step and then 1 plus one half-step plus one half-step to 2. Thus, 1x1.5X= 1.5; 1.5X1.5= 2.25 which is NOT equal to 2, not matter how you slice and dice it. This is plain and simple mathematics. The world does not function on addition and subtraction only. The simple mathematics of f-stops are consistent. Yours isn't.

 

Not only do you not know that you are wrong, you insist on being wrong and spreading this nonsense around.

 

Go back to school. Get some sense knocked into you. You can make fun of my moniker all you like but you are still WRONG.

 

That half-stops or half-steps move in steps of sqrt2 is borne out by photographic practice and by simple mathematical proof.

 

Prove that you are right and that we are wrong. Running away is a coward's game.

[shiver_me_timbrrrre]

So, let's us use F5.0 as your base point. What is the 1 f-stop above and below F5.0? What is one half-stop above and below F5.0?

 

Don't let me have the last word. Shut me up by proving that you know what you are talking about.

[shiver_me_timbrrrre]

Answer: using F5.0 as the base point, 1 stop above is F7.07. 1 stop under is F3.54. One half-stop above is F5.95; one half-stop under is F4.2.

 

An F-stop scale is anything you wish it to be? Such stuff and utter nonsense.

 

I am waiting for your apology after your dim bulb lights up a little, R. Miller. Took you a whole damn day to try to slag me and you still cannot prove that you're right.

 

So, I am a nasty piece of work but you're still WRONG. By the way, hiding behind R. Miller is just as nice a move too. Saves you from further embarrassment.

[shiver_me_timbrrrre]

"an f number is really a function of just two factors, focal length and aperture"

 

Not in a million years. F-stops can also be applied to time.

 

And since you have provided an erudite explanation of my simplistic 'geometry of the circle', you should know better. Despite your smarmy reply, what amazes me is that faced with the facts, you still don't know what it is you are reading. I guess parroting a textbook answer with no real comprehension is all you are capable of.

[shiver_me_timbrrrre]

R. Miller, I would also like to see you construct an f-stop sequence in 1/3, 1/4, 1/5, 1/6 stops. If you truly know what you're doing, it should be no problem at all. Please check your answers for consistency before you make a bloody oaf out of yourself again.

 

I certainly won't be 'tripling or thirding the amount of light' to be original for that seems exactly like something that you would do.

 

Your understanding is minimal and you should not attempt to correct who have provided the right explanations before you.

 

As an addendum, fractional f-stops add so 6 x 1/6 stop = 5 x 1/5 stop = 4 x 1/4 stop = 3 x 1/3 stop = 2 x 1/2 stop = 1 x 1 stop = 2^1. -1 stop is 2^-1.

 

Nothing original in it at all. I am not so smart as to invent it. It is just facts.