# Why are full fstops the numbers they are?

Discussion in 'Beginner Questions' started by http://www.photo.net/martinw, Mar 27, 2018.

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1. ### rodeo_joe|1

- Ken, sometimes these threads, like Topsy, just grow.

Andrew: what's the point in worrying about f-numbers to such accuracy?
Unless you're about to drill a box of Waterhouse stops from sheet brass.

As I hinted, multi-leaf irises can't possibly be assembled, linearised or controlled to an accuracy that makes it worth bothering about more than one decimal place of calculation. And even then....

A theoretical f/1.4(1416) lens actually transmits more like a perfect f/1.8 lens, and so on. We work in values that double or halve the amount of light, for goodness sake! That's a sloppy tolerance by anyone's standard; so there's absolutely no point getting hung up on details of nomenclature that make two-tenths of bugger all difference to the end result!

And don't even get me started on studio strobes that pretend to control the flash output to 1/10th of a stop.

Last edited: May 13, 2018
2. ### Andrew Garrard

No argument, Joe. Tolerances are certainly limited, and T-stops aren't f-stops (though the latter is still relevant for depth of field).

Still, Nikon report the alleged f-stop numbers in their user interface, and do so at the accuracy I quoted (irrespective of the lens, which suggests basing it on theory). And if they're going to do that, it does appear that they're rounding incorrectly.

I'm certainly not suggesting that instead of f/5.6 Nikon should report f/5.6568542494923801952067548968388... - but it doesn't take many maths lessons to realise the actual denominator is closer to 5.7 than 5.6. So it's a curio that "5.6" is what we always say. Similarly we say "f/22", not "f/23" when talking about f/22.6274... although the lens I was testing didn't allow me to select that.

I'm assuming this is just convention, and people start with exactly 1.4 rather than sqrt(2) as a number to multiply. This doesn't quite give an excuse for why Nikon apply odd rounding to some intermediate numbers like 3.5 - to get to that number you already need to be doing some exponentiation, and may as well start with exactly 2.

But the topic of this thread does give us some leeway to discuss both the mathematical origin of the numbers, and then why the numbers don't quite line up with the maths. We're already in the land of theory with any of this.

In the real world, factors like vignetting come into play, which is why I assume Nikon have a slight discontinuity in the way the AI ring reports aperture once you've got fast lenses (or it might relate to how the viewfinder screen transmits brightness to the TTL meter, if your meter is in the prism). It's complicated - but that makes it interesting.

3. ### BeBu Lamar

To clear the question. I asked Andrew how Nikon did because I know how it should be mathematically but sometimes they use number that is not mathematically correct.

4. ### Andrew Garrard

Thanks for clarifying, BeBu.

I suspect "5.6" is just a historical approximation caused by multiplying the approximation "2.8" by 2 (and never working with more precision). If someone once said "you multiply the number two stops before by 2" and everything was to two significant figures, you'd get here without having to explain fractional powers to people - although high school maths should cover it easily.

The 3.3/3.4 and 3.5/3.6 errors are harder for me to justify. I guess you can round either way, and especially for maximum apertures the marketing team would rather you rounded to the larger aperture - although there are rumours that some 200mm/2 lenses are actually a bit faster than specified.

5. ### rodeo_joe|1

Actually, I think the theoretical series (logically) starts with f/1, and meanders on from there.

Oddball apertures like f/3.3, f/3.5, f/4.8 and f/6.3 are 'traditional' from before the series was standardised.

Whether the geometric aperture of a nominal f/3.5 lens is closer to f/3.5 or to f/3.6 is totally irrelevant in practise. It is what it is. The approximate 5.8% difference in brightness is of no practical consequence.

I mean, why did many early shutters use the crazy sequence of 1, 2, 5, 10, 25, 50, 100, 200, 400 fractional seconds? Not everything in photography follows maths or logic... to state the bleedin' obvious.

IMO we've now gone too far in the opposite direction, by trying to use needless accuracy. Sorry to bang on about them, but decimal stops?
A ridiculous complication and irrelevance that nobody needs or can even implement on any camera or lens.

6. ### Andrew Garrard

True, Joe, "1" is the start of the system (and arguably "1.4" then defines the step size) - although obviously you can go negative. Not that the "f/0.95" lenses are terribly meaningful (or, arguably, useful) - even the Barry Lyndon f/0.7 lenses were only a stop ahead of f/1.

And yes, it's all down to rounding of mostly irrelevant numbers. But it's still odd that they don't obviously align to sensible fractions of an f-stop. Maybe the f/3.3 lenses really are f/3.3, but as an electronic partial stop on a faster lens, it's weird. Not important, but weird.

As for fractions for shutters, I have to assume that's because we're writing the numbers in decimal rather than binary, and the fractions would confuse people otherwise - though I've not checked how close to accurate the stated numbers there are. Presumably, from other discussions on this thread, the actual timings are closer to binary fractions. Again, it would all be simpler talking in powers of two, but that may not help people wanting to do, for example, the 1/focal length calculation for shutter speeds. Someone would no doubt complain that 1/256s is needlessly complicated and can be "simplified" to 1/200s (and I'd guess not 1/300s).

But you've lost me with the last argument. With decimal stops, are you complaining about having control at the fraction-of-a-stop level, or that apertures are written to two significant figures (ish), and therefore the larger apertures get a decimal point? If the former, I think 1/3 stops are still relevant. If the latter, I suspect any other way of distinguishing f/1.0, f/1.2, f/1.4, f/1.8 and f/2 would be more verbose, possibly excluding the "powers of the square root of a half" representation.

7. ### rodeo_joe|1

"But you've lost me with the last argument. With decimal stops, are you complaining about having control at the fraction-of-a-stop level,..."

- Basically, yes.
I have an otherwise excellent Minolta meter that displays its aperture readings in 1/10th of a stop increments. However the display for, say, 2/10th stops below f/8 is displayed as f/5.6 subscript 8; an absolutely ridiculous way to show an aperture!

The Minolta meter isn't alone in using this infuriating way of showing apertures. It's used by Sekonic and the 'power' display on the back of many studio strobes as well.

I worked out that 1/10th stop represents only a 3.5% change in image brightness. So not only is the display format unfathomable and impossible to implement, it's also totally irrelevant in terms of a practical exposure.

Pretending to be able to control a flash output to 0.1 stop is even more misleading. The full-output pulse rises with a fast exponential curve, and decays at a much slower exponential rate. Therefore attempting to control the integrated amount of light by controlling the pulse duration requires an extremely complex calculation and precision of pulse-width. I very much doubt the designers of such strobes have the expertise to tailor the pulse-width to the required precision, nor the knowledge of the pulse shape to begin to compute such.

You only have to measure how imprecise the so-called 1/3rd stop decrements on speedlights are, to realise that accurate 0.1 stop control of a flash output is pure fantasy.

8. ### BeBu Lamar

1/10 of a stop is about 7% rather than 3.5%. I would want my light meter to be more accurate than that whether or not I can control the light to that accuracy. Knowing accurate light level is important.That's why I wish I could afford the Minolta light meter and not the exposure meter. But put that aside for now.
Regarding the f/3.5 or f/3.6 the difference is small but I want to know which number Nikon uses so that I won't tell someone to set the aperture to f/3.6 which they can't do on their camera.
According to the service manual for the Nikon F3 the aim point for the shutter speeds are exact doubling (I.E. 1/256 and not 1/250) however if the speeds match the marked speed they are well within their tolerance.

9. ### Andrew Garrard

Hmm. I might argue about the merits of accuracy when we're talking 1/10 of a stop, but 1/3 of a stop is absolutely visible - and it put me off camera interfaces (hello Fuji) where the aperture is constrained to a whole stop. It's one reason I'm a bit annoyed that AI-S lenses can't let the camera control the aperture, even though I'm aware of no reason why not. It may be less relevant for older cameras with less precision - but the shutter has been electronically calibrated since, I believe, at least the F5, and I'd hope we have at least a bit more relative accuracy these days. Specifying accurate decimal places is pushing the accuracy a bit far (except below f/1.2), but thirds of a stop are going to have to be described efficiently somehow unless we go with the logarithmic system and start having custom LCD elements for +/- 1/3 and 1/2.

We seem to be stuck with it, anyway - as with using ISO as a sensitivity measure and focal length for field of view (both of which have some disadvantages, although also advantages) the cost of changing would outweigh any benefit. But as BeBu says, it's helpful if we're all using the same terminology, even if it's wrong. If I start describing a lens as f/5.7, people will give me very funny looks.

10. ### alan_marcus|2

The job of the lens is to project an image of the outside world onto the film or digital chip. Thus the camera system acts like a slide projector backwards, the film/chip being the screen. Now, how bright the image on the screen will be is a function of several factors. For this discussion we are only interested in the lens’s working diameter which is defined as the lens’s aperture. We need the ability to change the working diameter to make the screen image brighter or dimmer. Years ago it was determined that the best way to do this was to use an increment that either doubles or halves the screen brightness.

Now changing the subject (maybe):

You are the Captain of Cavalry “A” Troop, one hundred men with horses marching through the desert. Water is a problem. You bivouac for the night and you expect rain. You order the men to dig a circular pit 8 feet in diameter and line it with their canvas tent material. It rains as expected and the pit begins to collect rainwater. By your experience, you know an 8 foot diameter pit is adequate to collect rain water for your needs. Unexpectedly a lookout spots “B” Troop approaching --another 100 men with horses. You order your men to expand the diameter of the circular pit to accumulate water for 200 men and horses.

How big must the revised pit be to double the amount of collected rainwater? Answer: You multiply the pit diameter (8 feet) by 1.4142. This value is the square root of 2. The answer is 11.3 (rounded it’s 11 feet). You order the pit expanded to 11 feet diameter. Surprise, this new value causes the pit to accumulate 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 (1.4142 rounded). 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. The f-numbers are ratios. A ratio is dimensionless. Using a ratio allows us to compare a lens with all other lenses, as to its light transmission, without regard to the actual dimensions of the lens. f-number is short for “focal ratio

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11. ### Gary Naka

Thanks guys.
This finally allowed me to get a handle on some numbers; are they half or third stops?

12. ### rodeo_joe|1

"1/10 of a stop is about 7% rather than 3.5%."

- Nope. Afraid not Bebu. The 20th root of 2 is 1.035264924, slightly over 3.5% as an increment or - 3.4% as a decrement.

7% is 1/5th of a stop. Looks like you've used the 10th root of 2, not the 20th.

But even 1/5th of a stop is beyond the capability of most cameras' accuracy of setting.

Anyway, I wasn't really complaining of bogus accuracy, rather the use of tenths of a stop as a display device on meters. Who the heck can put a stop number on f/8 point 4, for example? OK it's close to a half stop between f/8 and f/11, but it's actually closer to f/8 plus 1/3rd stop. Whatever. It's an absolutely crazy way to display an aperture, and requires unnecessary 'stop and think' time before you can translate the meter reading to the camera.

13. ### rodeo_joe|1

"- Nope. Afraid not Bebu. The 20th root of 2 is 1.035264924, slightly over 3.5% as an increment or - 3.4% as a decrement."

Doh! I was thinking in stop numbers rather than a linear brightness increase.

You're right, it is 7 percent, but it's still insignificant in exposure terms.

"The lens opening or aperture is also a circular geometric figure."

- On most modern lenses, it's far from it.
7, 9, or even 5 straight-edged iris blades are the norm these days. Which kind of throws a spanner in the works of calculating a theoretical aperture diameter. And also puts the kibosh on neatly geometrical depth-of-field calculations!

Last edited: May 15, 2018
14. ### William MichaelModeratorStaff Member

An aside comment because this was previously quoted and commented upon:

I think that this is probably a case of a word used in error and I further think that the sentence was intended to read/mean:

"this is due to the geometric parameters of the lens and so on. And f/ is just a system of notation, for how much (blades) the aperture opens. Read on Wikipedia about this"

WW

15. ### BeBu Lamar

Not the 20th root because you're talking about image brightness and not aperture.

16. ### rodeo_joe|1

Yes, I realised that about 2 minutes after it was too late to edit.
Apologies!

I blame too many years of thinking about exposure as a geometrical progression..... or just too many years!

17. ### Andrew Garrard

Which "they"? Here's the output of a short computer program (which was quicker to write than typing the numbers manually):

Full stops (power of sqrt(2) on the left, increasing in steps of one, focal ratio on the right):
• 0 1.000000
• 1 1.414214
• 2 2.000000
• 3 2.828427
• 4 4.000000
• 5 5.656854 (note: only kind of "f/5.6")
• 6 8.000000
• 7 11.313708
• 8 16.000000
• 9 22.627417 (note: only kind of "f/22")
• 10 32.000000
Half stops (power of sqrt(2) on the left, increasing in steps of a half, focal ratio on the right):
• 0.000000 1.000000
• 0.500000 1.189207 ("f/1.2")
• 1.000000 1.414214
• 1.500000 1.681793
• 2.000000 2.000000
• 2.500000 2.378414
• 3.000000 2.828427
• 3.500000 3.363586 ("f/3.3", kind of)
• 4.000000 4.000000
• 4.500000 4.756828
• 5.000000 5.656854
• 5.500000 6.727171
• 6.000000 8.000000
• 6.500000 9.513657
• 7.000000 11.313708
• 7.500000 13.454343
• 8.000000 16.000000
• 8.500000 19.027314
• 9.000000 22.627417
• 9.500000 26.908685
• 10.000000 32.000000
Third stops (power of sqrt(2) on the left, increasing in steps of 1/3, focal ratio on the right):
• 0.000000 1.000000
• 0.333333 1.122462
• 0.666667 1.259921
• 1.000000 1.414214
• 1.333333 1.587401
• 1.666667 1.781797
• 2.000000 2.000000
• 2.333333 2.244924
• 2.666667 2.519842
• 3.000000 2.828427
• 3.333333 3.174802
• 3.666667 3.563595 ("f/3.5", kind of)
• 4.000000 4.000000
• 4.333333 4.489848
• 4.666667 5.039684
• 5.000000 5.656854
• 5.333333 6.349604 ("f/6.3")
• 5.666667 7.127190
• 6.000000 8.000000
• 6.333333 8.979696
• 6.666667 10.079368
• 7.000000 11.313708
• 7.333333 12.699208
• 7.666667 14.254379
• 8.000000 16.000000
• 8.333333 17.959393
• 8.666667 20.158737
• 9.000000 22.627417
• 9.333333 25.398417
• 9.666667 28.508759
• 10.000000 32.000000
You can set the exposure changes on a (Nikon, at least) dSLR to work in whole, half or third stops. The meter, as I reported with a variable-aperture lens, appears to work in sixths of a stop, which means it can represent both halves and thirds - although it uses the traditional (oddly) rounded numbers.

So you're suggesting that the meter should round its results to the nearest whole/half/third/sixth of a stop depending on what someone's using? Given that the meter has to have a decimal place to show f/1.4 and f/5.6 anyway, I think I'd rather it just gave the most accurate number it could within reason, and left us to spin a dial until it's near the specified number. YMMV, but I'd rather be given the right number and approximate it myself than have it try to work out what rounding I want.

Alan: I approve of the metaphor. I've been going with a swimming pool and a hosepipe, with being down a well as an explanation of why focal ratio can be more relevant than focal length, but I like your rain version. Although focal ratio is dimensionless (and convenient because of that), I do think the dimensioned numbers that produce it are also often relevant and worth teaching - though to depth of field rather than exposure. I may just be confusing people - I'm a specialist in unnecessary complication.

I actually took it as an understanding that different aperture blades in the iris moved from fully open to fully closed independently depending on the selected aperture. Technically, this would work, although it would produce a weird bokeh; to the best of my knowledge, any practical (and not broken) iris moves all of its blades concurrently and to a partial position to produce an approximation to a circular aperture, though leaf shutters could sometimes be a bit more complicated than that. That said, I just discovered that my new Laowa 20mm 4-4.5x macro has three aperture blades, producing a (slightly curved) triangular bokeh - which is just weird; I think they still all move together, though.

18. ### BeBu Lamar

Well first I do want the meter to measure to 1/10 of a stop and more although I can't set that on the camera. If you try to display aperture in 1/10 of a stop using the aperture number you would need more than 2 digits for the larger stops. For me it doesn't bother me if the meter displays in Cd/m^2 in spot mode, in Lux in incident mode and in Lux.Second in flash mode.

19. ### paul ronNYC

its all nonsense... its like tennis scoring. They did that to keep people confused, job security. Otherwise it would have been numbered 1 2 3 4 5 6 7 8 9... etc.

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20. ### Dustin McAmeraYorkshire

My little Sekonic has the display with tenths of the next stop; I agree that tenths are a bit daft. I would like to be able to set the meter to round that to thirds for me, since I'll be doing that in my head anyway. I think any more is precision for the sake of it, which has no effect on my picture (unless maybe to delay me getting it). However, I'm glad they at least left the display starting with a familiar aperture value, rather than multiplying the tenths into it.

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