Why are full fstops the numbers they are?

It's so weird that full fstops are like 2, 4, and 8 but also include 2.8 and 5.6. Can anyone explain why?

 

If you would like, you can sit down and do the math as to how the numbers ended up where they are.

Basically, the aperture RATIO is arrived at by dividing the focal length of the lens by the diameter of the aperture. Thus, a 50mm f/1 lens has an aperture of 50mm.

The AREA of the aperture, however, dictates the amount of light let through. A 50mm aperture has an area of 1963.5mm^2. Half of that is 981.7mm^2, or a diameter of 35.4mm. 50mm/35.4mm gives ~1.4.

Cut that area in half again, and you are at an area of 490.9mm^2 or a diameter of 25mm. That, of course, gives f/2.

I won't work through the whole sequence, but if you wish to the numbers customarily marked on the scale make sense.

BTW, older lenses often didn't use the "standard" stop we are familiar with. My late 1930s Leitz Elmar, for example, has a maximum aperture of f/3.5 and it's marked in full stop increments down from that(I think the smallest is f/18). Of course, it doesn't even have click stops, but still it can take a bit of mental gymnastics if reading off a light meter.

 

Thanks!

 

Just for fun, this also applies to Pizza. If you order an 11 inch pizza, you get twice as much as an 8 inch pizza. Or on a different scale, order a 28cm pizza and you'll get twice as much if you order a 40cm pizza.

You might want to call this the Pizza "Pi" Rule.

 

Wiki is your friend (unless your name is Hillary):

f-number - Wikipedia

 

Before the current f/ system was standardized, there were a few other systems.

One such, Uniform System, popularly named U.S. (but not for U.S.A) goes in multiples of two, or proportional to the area.

The f/ system uses diameter ratios, so there is a square root of two involved.

I suppose that the f/ system has the convenience of the numbers not getting so large, but then they do for shutter speeds.

The other system is E.V., which numbers both shutter speed and aperture with consecutive numbers, increasing by 1.

I have seen cameras with E.V. numbers on them.

 

I think Hasselblad "C" lenses are probably the best known examples that use EVs.

When I first got my Hasselblad, I found it somewhat annoying. Once I got use to going straight to setting EVs, I saw how wonderful the system is. On these lenses, the shutter speed and aperture rings are locked together, so once an EV is set the exposure stays constant regardless of the specific shutter speed and aperture used.

 

For the more mathematically inclined, the key number is the square root of 2 (which is 1,4142...). So, starting with 1, the next full stop is square root of 2 times 1, which is 1.4. Next one is 1.4 times 1.4 again, gives 2. Next 2 times 1.4 gives 2.8 etc.

Ben's explanation contains all the maths needed to figure out why you need this square root of 2. The area is pi times r squared, and the diameter is 2 times r. Some good old high school maths later, you end up with the square root of 2.

 

My Ricoh 500 bodies from the mid 1950s have EV designations as well as traditional stop markings.

 

A lens lets light through in proportion to its area. The area of a circle = pi * r^2.

This means if you double the diameter of the lens or its aperture, then you quadruple the amount of light it lets through.

F-Stops are a doubling or halving of the amount of light, and the aperture diameter therefore has to vary by a power of root 2. The square root of 2 is 1.4 to a near approximation. So if you start with f/1 and multiply by root 2, you get f/1.4. Multiply by root 2 again and you get f/2; and so on. Simple!

 

On the other hand, the actual rounding used isn't so obvious.

4*sqrt(2) rounds to 5.7, and not 5.6.

 

Indeed, but 5.7 isn't 2 * 2.8 and 11 not 2 * 5.7. It should also be 23 instead of 22.

Similar issues with the shutter speed.

 

Oh my, if we're gonna worry about f5.6 v f5.7, we might as well go whole hog and bring "T" stops into the discussion

 

Even though I've "converted" to Nikon, I bought a Canon EF last summer. I THINK this was the first shutter priority camera on the market, and is a bit of an interesting oddball for a couple of reasons.

In any case, shutter speeds longer than the flash sync speed(1/125 thanks to the Copal Square) are electronically timed, and it's common for a broken circuit board to cause the slow speeds to fail.

One came into the local camera store, and I was checking it out for purchase. I set it to 30 seconds and then started counting off-much to my surprise "30 seconds" actually went to 32 seconds. "15 seconds" went to 16 seconds.

I have a couple of Rolleiflexes with the Pronto-Compur shutter. I think that speeds between 1/50 and 1/250 are infinitely variable, but it's marked in what one might call non-conventional markings like 1/50, 1/100 and so forth. It DOES go to 1/500, but that's accomplished with a separate spring from the others so it steps directly from 1/250 to 1/500.

Come to think of it, I think my Leica IIIc has a similar oddball arrangement of speeds, although it goes to 1/1000.

 

By the way, at some point I saw a nice little shortcut for remembering the full stops of aperture if one isn't just used to them. If you start with the most common pair of wide open ones, 1.4 and 2, the subsequent ones are alternate doubles (or close when rounded). A, B, 2A, 2B, 2(2A), 2(2B) and so forth.

By the way in case Ben Hutcherson doesn't have an old Leica to hand, the answer is that yes, it has an odd set of both shutter speeds and apertures, at least on my old IIIb with a Summar lens. The shutter speeds go 1; 2; 4; 8; 20; 30; 40; 60; 100; 200; 500; 1000. The apertures go 2; 2.2; unmarked around 2.8; 3.2; 4.5; 6.3; 9; and 12.5. The shutter speeds are pretty odd, but the apertures, though a bit messy at the open end, settle down to alternate doubles, just a different set.

 

Dang - I do remember a lens that had an f6.3. I'd like to look at my father's old Sekonic ? and see if it had those now non-standard f-stops on it. To bad the meter and my father are both long gone. (At least I think it was a Sekonic - or maybe a Weston)

 

My Polaroid 150 only has EV setting. There is no shutter speed or aperture.

 

It is desirable to make aperture changes in 2X increment i.e. a doubling or halving. To accomplish we increase or decrease the diameter (aperture diameter). Actually we need to change the surface area of this circular opening. If we multiply the diameter of any circle by 1.414 (square root of 2) we calculate a revised diameter that has a 2X greater surface area. Conversely, if we divide the diameter of any circle by 1.414, we compute a revised circle with half the surface area. Thus the f-number set is --- 1 - 1.4 - 2 - 2.8 - 4 - 5.6 - 8 - 11 - 16 - 22 - 32 - 45 - 64. Note each number going right is its neighbor on the left multiplied by 1.4. Conversely each number going left is its neighbor on right divided by 1.4. .

 

I seem to have misplaced some of my old meters, and some have just died and gone to the barn, I think. But of those I found, the Sekonic L28C has half stops but they're not numbered, except for F3.5 and 4.5. The others you need to guess. The Weston Master IV has unlabeled half stops, but the Master V is divided into 1/3 stops for both aperture and shutter speed, and they're all numbered. Most other light meter dials I've seen just don't have the room for that many divisions but the old weston was pretty big (and pretty busy).

 

@ Matthew Currie -- The multiplying factor for 1/2 f-stop increment is the fourth root of 2 = 1.189 ----- For 1/3 f-stop increments the multiplier is the 6th root of 2 = 1.1225 ---- These multipliers create the f-number sets now in use. Again, they calculate a revised circle with the required area delta.