A moderately Technical Discussion about f stops.

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This is just a little help for newbies on confusing f-stops that don't add up.

 

One stop has a value of two. That is two times the amount of light. Here is an easy way to write it mathematically; 2 ^1 = 2. Which is 2 to the power of 1 equals 2. A half of a stop is 2 ^0.50 = 1.4. A third of a stop is 2 ^0.33 = 1.3.

 

So here is a lens with a top speed of f/1.4 and the *slowest speed of f/16. Now, as we all know, each stop is double the amount of light added or subtracted, depending on which direction one is going. Please note that on this lens the number immediately to the left of f/1.4 is f/2. I used to think this was one stop, but of course I was wrong. Going from f/1.4 to f/2 is a half of a stop and the full stop is f/2.8. This sequence of ½ to full continues all the way to f/16 on this particular lens. It is not too complicated, but it does get confusing when you find a lens like the following one.

 

 

Note that the top speed of this lens is f/3.5 and it goes down to f/22. Where is the half stop next to f/3.5?! There is no f/7 and f/5.6 is not a half stop from f/3.5! Fortunately, order returns after f/5.6., where the regular sequence of a half stop followed by one stop returns and continues to f/22. So what gives? The answer is that manufacturers want to keep it simple. Most of the numbers follow the "normal" convention with the only one exception.

 

The fact is, it really doesn’t hurt to use f/3.5 to f/5.6 as a half a stop or to use f/8 to f/3.5 as a full stop. It’s close enough for the latitude of most, if not all films. That said, I think it is important to know this is a manufacturer fudge. So next time you come across what looks like a mathematical error on your lens, it’s okay. It’s close enough.

 

I welcome comments.

 

 

 

*Each f-stop represents a fraction of the diameter of the lens. That means f/2 is ½ the diameter of the lens. That is why ½ allows more light to enter than 1/16.

 

 

Edited January 28, 2019 by William Michael

The f/numbers explained:

 

The camera lens gets its name from the lentil seed it resembles. The lens projects an image of the outside world, focusing at the focal plane which is the position occupied by the surface of film or electronic chip. This image is allowed to play briefly on these surfaces by the shutters which acts as a gate. A lens is much like a funnel in that it gathers light. The larger its diameter the more light it gathers. Thus the diameter of the lens is a key controlling factor regarding how bright the projected image will be. In addition to diameter, image brightness is a product of scene brightness and lens focal length.

 

 

Needed is a precision way to control image brightness. The optical solution is adjustable control over the working diameter of the lens. This takes the form of a washer shaped restriction or stop called an aperture. This mechanism is technically known as the Iris, so named after the Greek god of the rainbow (colored portion of the human eye). Sometimes its called a stop, so named after a set of thin metal slides, with different size holes, invented by John Waterhouse in 1858, they were inserted into the lens barrel, each stopped a different amount of light allowing repeatable adjustment of image brightness.

 

 

Photo scientists concluded the logical sequence of adjustment should be in 2 x increments i.e. twofold brightness change stop-to-stop. To accomplish, the diameter of the aperture hole can be enlarged or reduced by a calculated amount thus causing a shift in the area (square measure) of the hole in twofold increments. Scientists recognized that when dealing with a circle you must vary the diameter multiplying it by the square root of 2 which is 1.4142. This math calculates a revised diameter that generates a circle with a twofold area increase. Conversely if you multiply the diameter of a circle by 0.7071, you calculated a revised circle with a twofold area decrease.

 

 

Using 1.4142 (rounded to 1.4) a number set emerged:

 

This is set is called the full stop set or the f/stops:

 

1 – 1.4 – 2 – 2.8 – 4 – 5.6 - 8 – 11 – 16 – 22 – 32 – 45 - 64

 

Note each number is its neighbor on its left multiplied by 1.4

 

Note each number is its neighbor on the right multiplied by 0.7

 

 

As time passed a twofold adjustment increment proved to be too course. A revised set was calculated that generate changes half again as fine. The number set that satisfies this requirement is based on the fourth root of 2 which is 1.1892

 

Thus the finer number set is in ½ stop increments:

 

1 – 1.2 – 1.4 – 1.7 – 2 – 2.4 – 2.8 – 3.5 – 4 – 4.5 – 5.6 – 6.3 – 8 – 9.5 – 11 – 13.5 – 16 – 19 – 22 – 26.9 - 32

 

 

As more time passed and light meters came into common usage, the f/number set was again made finer incrementing in 1/3 f/stop progression. This 1/3 f/stop set is based on the sixth root of 2 which is 1.1225

 

 

This set completely satisfies photo scientists as shutters accuracy, mechanical apertures gear backlash and ISO values are marginally able to hold this tolerance.

 

 

The 1/3 f/number set is:

 

1 – 1.1 – 1.2 – 1.4 – 1.6 – 1.8 – 2 – 2.2 – 2.5 – 2.8 – 3.2 – 3.5 – 4 – 5 – 5.6 – 6.3 – 7 – 8 – 10 – 11 – 12.6 – 14 – 16 – 18 – 20 – 22 – 25 – 40 – 45 – 50 – 57 – 64

 

 

As to image brightness at the film/chip plane (focal plane):

 

Within the camera, the two most significant factors are the working lens diameter and the lens focal length. Image brightness decreases as focal length increases. Conversely as focal length decreased image brightness increases. Thus both diameter and focal length must be taken into account.

 

 

With thousands or perhaps millions of camera designs that bring into play a hodgepodge of lens dimensional combinations, each will presents a different brightness at the focal plane. You guessed it – disorder results. Without a solution, most pictures you take will would be under or over exposed.

 

 

Ratio to the rescue: The ratio of two numbers is pure because it is devoid of dimension. In the case of the camera lens we divide the focal length by the working diameter to get a ratio which we call the focal ratio or in other words the f/number. The beauty of the f/number system is, when any lens is set to a specific f/number it delivers the same image brightness at the image plane as any other lens positioned to the same f/number regardless of differences in dimension.. Simply stated, the focal ratio or f/number gets rid of the chaos revolving around setting the lens aperture. Other systems have been proposed and tried but we still elect to keep the one that has works best. (You’re free to invent a better system).

 

 

Sorry to report that we photographers must cherish the lowly focal ratio (f/number) system until you think up something better.

 

 

Technical gobbledygook from Alan Marcus

the number immediately to the left of f/1.4 is f/2. I used to think this was one stop, but of course I was wrong. Going from f/1.4 to f/2 is a half of a stop and the full stop is f/2.8.

You weren't wrong but are now. For a circular area to let through half as much or twice as much light, the diameters for the two circles must differ by a factor of sqrt(2) = 1.4142... Hence the relation between full f-stops by that factor.

Edited January 23, 2019 by Dieter Schaefer

I used to sweat the f'ing stops, but now I simply set my camera to advance or decrease by 1/3 stop per click of the command dial, and all's good. I can reference the numbers as needed to compare settings, but this is one more place where digital has made life easier by the application of technology. One can also select other settings to fit one's preferences. This is way more flexibility and granularity of choice than I ever knew with my old, manual lenses back in the day.

Some of the earlier cameras with adjustable stops just numbered them 1, 2, 3, 4. No relation to the diameter or area.

(I believe some Brownie box cameras use this system, and also some folding cameras.)

 

Not so long before the f/ system, there was the Uniform System (called US) which uses numbers in inverse proportion to area.

(2, 4, 8, 16, 32, 64). Interesting, the Sunny 16 rule still works, as f/16 is the same as US 16, but otherwise the numbers

double or halve for each stop. I am not so sure why the f/ system caught on over the US system.

Not so long before the f/ system, there was the Uniform System (called US) which uses numbers in inverse proportion to area.

Never even heard of it - which is not surprising since it dates back to the 19th century (and lasted probably into the 1920s). There were apparently five different aperture numbering schemes in effect in the early 1900s.

Not long after I got interested in darkroom photography, and in 1968, visiting my grandparents

I found an Autographic Junior 1A in the attic. My grandfather gave it to me, I found a roll of

VP116 at a nearby store, and started using it. It was pretty obvious that the aperture

wasn't numbered in f/stops, but I didn't know how it was numbered. I don't think

I had a light meter, so it might not have mattered much.

 

I suspect it is common on many of the Kodak models from the 1910's or 1920's.

 

Looking at it and guessing a little, my dad and I thought that US8 was about f/8,

which seems to be one stop off.

 

It wasn't until recently, from Wikipedia, that I found the table explaining the

different scales, and that US16 is f/16.

 

I had another roll for it in 1975, found in the half price bin at a nearby store.

 

I had a Kodak developing tank with aprons, as my reel tank didn't go to 116.

The only time I ever used a tank with aprons.

The f/numbers explained:

. . . . .

 

. . . . Technical gobbledygook from Alan Marcus

 

That's good gobbledygook right there! Loved it.

 

Best,

Henry

You weren't wrong but are now. For a circular area to let through half as much or twice as much light, the diameters for the two circles must differ by a factor of sqrt(2) = 1.4142... Hence the relation between full f-stops by that factor.

 

So you're saying that going from f/1.4 to f/2 is a full stop?

From f/3.5 to f/5.6 is 1 and 1/3 stops not half stop.

I pity the poor beginners who try to follow this.

So you're saying that going from f/1.4 to f/2 is a full stop?

 

That's right.

Here's me thinking that F-stops were simple to understand. Apparently not.

That's right.

Going from 1.4 to 2 is a reduction by double of the light? I want to understand what I am not understanding. Edited January 24, 2019 by Henricvs

A whole stop is a doubling of the light, yes? Going from 1.4 to 2 is a doubling of the light? I want to understand what I am not understanding.

Other way around

I want to understand what I am not understanding.

Then ask yourself the following: how much does the diameter of a circle (aka aperture) has to change so that the circle has half the area (aka lets through half as much light)? The answer is in my first response above. How much is the aperture diameter at f/1.4 (of a 50mm lens, for example)? How much is the aperture diameter at f/2? What are the aperture areas for those two diameters (assume perfect circles here)?

Then ask yourself the following: how much does the diameter of a circle (aka aperture) has to change so that the circle has half the area (aka lets through half as much light)? The answer is in my first response above. How much is the aperture diameter at f/1.4 (of a 50mm lens, for example)? How much is the aperture diameter at f/2? What are the aperture areas for those two diameters (assume perfect circles here)?

 

Yes, I see. Okay, that is helpful. Thanks.

Now a story I often told my students to better explain the f/# system.

 

You are the captain of Cavalry “A” Troop. One hundred men with horses are marching through the American Southwestern Desert. Water is a problem but the forecast is for rain tonight. You order your troops to bivouac and direct that a circular pit 8 feet in diameter be constructed and lined it with canvas tent material. It rains as expected and the pit collects rainwater. Due to your West Point training, you know an 8 foot diameter pit is adequate to collect enough water for your needs. Unexpectedly a lookout spots “B” Troop approaching -- another 100 men with horses. Now you order your men to expand the diameter of the circular pit. The volume of water collected must now accommodate 200 men and horses.

 

How big must the revised pit be to double the amount of collected rain water?

 

Answer: You multiply the pit diameter (8 feet) by 1.4. This works out to a revised pit diameter of 11 feet. Surprise, this enlarged pit collects twice as much water as before. Why? The surface area (catch basin) now has double the surface area; thus it can capture twice the amount of rain.

 

The lens opening or aperture is also a circular geometric figure. The area of any circle (thus its ability to collect rain or light) is doubled if you multiply its diameter by 1.4. Using this factor a number set emerges:

 

1 – 1.4 – 2 – 2.8 – 4 - 5.6 – 8 – 11 – 16 – 22 – 32 – 45 – 64

 

Note each number to the right is its neighbor on the left multiplied by 1.4 and then rounded. Each number to the left is its neighbor on the right divided by 1.4 and then rounded.

 

These are the mysterious values engraved on the lens barrel. With geometric precision they allow the adjustment of the working diameter of a lens, making it smaller or larger. We need this number set because it allows even and logical and predictable changes to be made to image brightness in an increment that either doubles or halves the amount of light allowed to play on the film or digital chip.

Alan, Just tell the cavalry guys to dig the hole twice as deep. That makes sense.;) (I bet you liked having students like that.)

@ James G. Dainis - The key to all this is in the geometry of circles. Multiply the diameter of any circle by 1.4 calculates a revised circle diameter with twice the surface area. Multiple the diameter of any circle by 0.707 calculates a revised diameter of a circle with 1/2 the surface area. Thus the f-number sequence allows the construction of a series of circles each twice or half the surface area. I these circles are lenses, these lenses project an image that is twice as bright or half as bright thus the f-number is based on a 2X incremental change.

The Pizza Pi rule is easier - an 11" pie is twice as much pizza as an 8" pie, a 16" pie is twice as much as an 11" pie.

The Pizza Pi rule is easier - an 11" pie is twice as much pizza as an 8" pie, a 16" pie is twice as much as an 11" pie.

A tip of the hat to chuck909 -- therebefore this Pizza rule was unknown to me! You get an A+

Personally I think the easiest way to "remember" the whole f-stop number sequence is coming to the realization that the f-numbers double for every second stop. It's not so easy to muliply a number by 1.4 in your head, but... It's pretty easy to double (or halve) the f-numbers.

 

So if you can just remember two f-numbers next to each other... say f/1.0 and f/1.4, follow the sequence like so: double the 1.0 to get f/2.0, and double the 1.4 to get f/2.8. Then double the 2.0 to get f/4.0, and double the 2.8 to get f/5.6, etc.

 

If this is not real obvious, look at it this way: start with the numbers f/1.0 and f/1.4. From f/1.0, write down the sequence of doubling numbers: 1, 2, 4, 8, 16, 32, etc. Then, in between these numbers, write down the other doubling sequence: 1.4, 2.8, 5.6, 11.2 (round it to f/11), 22, etc.

 

This will give you the f-number sequences as far as you want to go, aside from those little idiosyncrasies in rounding off.

I'm sure all of this makes it crystal clear to a beginner exactly what the effect on the picture is if they change the aperture number from 2 to 8.

 

Getting bogged down in theory and technical nitty-gritty is the enemy of making good pictures. Nobody needs to know the area of the aperture 'hole' to apply f-numbers practically. The numbers are what they are, and could just as easily be marked 1, 2, 4, 8, 16 etc. like they are on some old enlarging lenses.

 

The point is to make the association of a particular number with its visual effect, and secondarily how it affects the exposure. And nobody needs to know how the numbers were arrived at to do that.

 

They just need to know that f/4 lets in twice as much light as f/5.6, and that f/2.8 lets in twice as much light again. A simple relationship like - 'If you make the aperture number bigger by a step, you need to make the shutter speed smaller by a step' is about all that's needed. Apart from by the terminally curious.

 

We're not living in the stone age. We can (mostly) leave the tedious calculations to the camera now, and concentrate on the important bit of what's in front of the camera and seen through the viewfinder.

Edited January 24, 2019 by rodeo_joe|1

Thanks to everyone who contributed to correct my screwup. Especially, Big Al Marcus. I would have loved your class!

 

Best,

Henry