This article is about the theory of the camera "numbers", but it is NOT about using those camera settings. If any mystery about using the settings to take better pictures, I would suggest starting with this inexpensive book for beginners: Bryan Peterson's Understanding Exposure, 3rd Edition: How to Shoot Great Photographs with Any Camera (an easy fun read, and surely the best $17 a beginner can spend on photography. It is in the public libraries here).
That is math and physics, but still, the very useful purpose of f/stop numbers is the grand concept that f/8 will the same exposure in any lens, regardless of focal length, or physical size of construction.
And while we are momentarily distracted, the marked Focal Length is when focused at infinity, but it changes as we focus closer (focal length normally becomes longer if front element is extended, but internal focus lenses probably become shorter). The actual focal length is measured to the rear Principle Point, H', as shown above. The Principle Point is the designer's apparent plane where the image appears to be. Design of curved lens elements can move this point, and this H' point is in fact often literally outside the actual lens, either in front or behind. In telephoto lenses, this H' point is always slightly in front of the front lens element, because, the actual optical technical definition of "telephoto" is that the lens is physically shorter than its focal length (which is a practical way to build the long lenses that show distant objects enlarged). Wide angle lenses are often retro-focus, which means the rear node H' is well behind the rear element. This allows the short lens to be mounted well forward, leaving space for the SLR camera mirror to be raised. Otherwise for example, an 18 mm lens would block raising a mirror 24 mm tall. FWIW, regarding this H' distance, the ratio of subject size to image size is called Magnification, and when equal sizes (at 1:1 reproduction ratio), then also equal distances - the distance in front of the lens is necessarily equal to the distance behind the lens, at 1:1 size (similar triangles, etc.) Seems a cute fact, which aids understanding.
Is f/stop written f/stop or f-stop or fstop? The lens manufacturers properly write f/8. The term f-stop has become popular on-line, so we see both. But I learned to write f/stop, because we also write f/8, to be remindful of the division defining it:
f/stop number = focal length / aperture diameter
f/8 is an aperture diameter, literally = focal length / 8.
Why do all lenses expose equally if all are set to the same f/stop? For two lenses at the same f/8, the lens with 3x longer focal length has an aperture diameter 3x larger. Tricky, but the 3x focal length magnifies the subject 3x, and so it sees 1/3x width and 1/3x height, which is 1/9 area, which only reflects 1/9 the light the wider lens sees. But the longer 3x lens also has aperture 3x larger, which is 9x area, and so now admits 9x more light, which before was 1/9 as much, from a 1/9 area field... so the 9x times 1/9 result is the same exposure in both f/8 lenses. Another argument is the Inverse Square Law over the 3x longer focal length is 1/9 the light, when the image reproduction reaches the sensor plane (which is just repeating the first explanation again). This is why the f/stop system is used. It's good stuff.
So, f/8 denotes (focal length / 8), which represents the aperture of the lens, and this exposure value can be compared with other lenses in this way. A series of multiple f/stop steps is designed, called "stops". Stop originally denoted the notched detent which marked the 2x area multiples. Today in photography, the word stop is used to mean any step of double or half value of exposure, also in regard to shutter speed and ISO. Each full stop towards larger f/stop numbers gives half the light exposure of the previous step (called stopping down, which also increases depth of field).
The tables below are the computed f/stop numbers of the camera aperture. The first table is the fractional steps in tenth stops. These charts show the actual numbers, and the relationships, and one purpose could be to aid determining span in stops between two values.
The cameras and light meters are marked f/11, and we say it as f/11, but f/11.3 is the necessary correct actual calculated value. This is less than 1/10 stop difference, and any difference exists only in our mind, since the camera will do it right anyway. Most other values are closer, but Guide Number calculations for speedlights can use f/11.3 instead of f/11.
To make this fact be obvious, note that f/stop numbering is the sequence of √ 2 intervals, (which is 1.414) - so every other stop number is a multiple of 2. The sequence progressions, when arranged into rows of every-other doubled aperture values, are also defined as:
Shutter speed marking numbers are also approximated. For example, the standard shutter speed chart (3rd long table below) shows 1/20 second and 1/10 second (and 10 and 20 seconds) to be both third stop values and half stop values. Same value cannot be both, and the camera does compute the actual value closer (half stop 20 seconds will be 22.6 seconds, and full stop 30 seconds will be 32 seconds), but we humans are frequently shown easier rounded or even "equivalents".
First option is the precise f/stop values, in exact third or half stop values that the camera uses (better accuracy than the nominal marked values).
Green is Full stops, Blue is Third stops, Red is Half stops).
Select two f/stop values to compare difference.
Calculate Difference of Any Two Exposures How many stops?
Field of View Calculator for lens and sensor
Handheld light meters typically can also be set to read tenth stops (for metering multiple flash). If you set your light meter to read in tenth stops, the format of the result value we see is (for example):
This is NOT f/8.7. It is 7/10 of the way between f/8 and f/11 - or about f/10, but read as "f/8 plus 7/10 stop".
By definition, both of these equivalent values are simply two third-clicks past f/8, or one third-click below f/11 (easy to set). The camera dial will indicate f/10 there, but we can instead meter and work in tenth-stop differences from full stops.
Fractions: 1/10 is 0.1 stop. The fraction 1/3 stop is 0.33 stops, and 2/3 stop is 0.67 stops, so a reading around 0.3 is one third stop, and one around 0.7 is two third stops. The lens can only be set to third stops, so just pick the nearest third stop: 0, 1/3, 2/3, or 1 stop.
There would seem no point of 1/10 stop meter readings for daylight, since we can only set the camera to third stops. However there are two good reasons to use tenth stops for multiple flash. One is for greater precision in adjusting the power level of individual flash units - the actual difference between two lights could be controlled more closely. But the overwhelming advantage is when pondering fill level for that lighting ratio - how much is one and a third stop less than f/10? It is about f/6.3, but who knows that? But if we read these two as f/5.6 plus 3/10 stop vs. f/8 plus 6/10 stop, then we easily know 1.3 stops difference, in our heads, immediately (in use, that is really big).
Notes: (√ 2 is 1.414). f/stop number = 1.414 (stop number + fraction)
Two tenth stops past f/11 (stop number 7) is 1.414 7.2 = f/12.126
The focal length factor is about the magnification of the field of view.
A short lens (wide angle) gathers a lot of light from a very wide view, and concentrates all of that light onto the camera sensor area.
A long lens (telephoto) gathers much less light from a much smaller view, onto same sensor area. Less light collected from a smaller area.
Exposure is about Illumination per unit of area.
But fstop = focal length / aperture diameter equalizes these scenes, giving constant exposure at equal f/stop numbers. f/8 is always f/8, on any lens. That's why we bother with f/stop numbers, the benefit is great.
Aperture is circular, and the area of a circle is defined as Pi r². Double area is twice the light, or one stop.
For double area: 2 Pi r² = Pi (1.414 x r)² , so 1.414x radius gives one stop. (√ 2 is 1.414).
Since f/stop = focal length / aperture diameter, then f/stop numbers increase in 1.414x steps (or 1/1.414 is 0.707x decreasing steps).
Inversely, when the diameter and area are made larger, the f/number from the ratio f/d becomes a smaller number.
Full f/stop numbers advance in 1.414x multiples. From any f/stop number, in all cases, double or half of that number is two stops (for example, f/6.2 is two stops above f/3.1). Every second stop is the doubled f/number. Or one stop is x1.414 or /1.414.
Third f/stop numbers advance in multiples of the cube root of √ 2, or 1.12246x the previous. Half stops are the square root of 1.414 apart. Less (number) is More (light).
Lens manufacturers seem to truncate instead of round off. For example, f/5.6 is actually 5.66, and f/3.5 is 3.56. We see the same f/1.2 marking for the half stop (f/1.189) and third stop (f/1.260).
The values of shutter speed and ISO are linear scales, meaning that 2x the number is a 2x difference, and 2x is one stop. The very important thing to the definition of our exposure system is that any span of three third stop steps (or any two half stop steps) must come out exactly 1.0 stop of 2.0x exposure difference. To force this, cube root (and square root) steps are the proper values to create and number step intervals.
The next third-stop increment is cube root of 2 (1.26992) greater than the previous value. (but for f/stop, see above)
Every three third-stop steps (from any point) is exactly a 2.0x change of the light.
The next half-stop increment is square root of 2 (1.41421) greater than the previous value.
Every two half-stop steps (from any point) is exactly a 2.0x change of the light.
The next full-stop value is 2x greater than the previous value. Doubling any numeric value is one stop (speaking of shutter speed or ISO, but 2x number is two stops for f/stops, see above.)
The nominally marked numbers may not be exact, but the camera knows exactly what to do. For example, set ISO 250 or ISO 2000 in the Nikon D300 camera. Then near the top of the Exif data will show the ISO 250 or 2000 values, but down in the manufacturers data, it shows the precise values used, ISO 252 or ISO 2016. (The ISO base appears to be 100 instead of 1... 100, 200, 400, 800 instead of 1, 2, 4, 8. This makes third stops of 252 and 2016 instead of full stops 256 and 2048 - which we will call 250 and 2000.) The point is, the numbers we see are just convenient nominal numbers, which really does not much matter to us humans. We just want one stop to always be a 2x light value.
|Shutter Speed||Marked As:|
Shutter speed is of course the time duration when the shutter is open, exposing the sensor or film to the light from the aperture. On many cameras, numerical values for shutter speed are marked on the camera using two methods with different meanings - for example, marked as either 4 or 4". Just the number alone, like 4, is an implied fraction (1 over the number), meaning 1/4 second. The same number written 4" means four whole seconds, not a fraction. A slow shutter is a longer duration, and a fast shutter is a shorter duration.
A flash, especially a speedlight flash, is typically a much shorter duration than the shutter. The flash simply must occur while the shutter is open (sync), but the faster flash exposure is not affected by the slower shutter speed. Keeping the shutter open longer does increase the continuous ambient light seen, but shutter speed does not change what the fast flash does.
Values of 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 are special, each exactly double the previous. That is how our number system works. The few shutter speeds and f/stops and ISO of those values or multiples are exact values, but many/most values are marked with rounded nominal values (easy approximations, even 1/64 is marked 1/60, and 1/1024 is marked 1/1000). The camera uses the exact equivalents internally, so that full, half and third stops always actually are exact half or third stops. For example, both third stop and half stop systems have a shutter speed marked 1/20 second, but the camera knows thirds is 1/20.2 and half is 1/22.6.
ISO speed was a film sensitivity concept. Digital speed is a gain factor, multiplied after the digital sensor native sensitivity does what it does (i.e., around ISO 100), but the same ISO numbering scheme is used, still an apparent "sensitivity" indication, still seems the same to us. We think of ISO 100 as a base, as a full stop, but if we divide 100 by 2 a few times, we end up at 1.5625 or 0.78125, etc - instead of 1. Technically, ISO 100 is a third stop, 2/3 stop above ISO 64. Which is all relative, a round number to be easy for humans.
Technically, the ISO speed specification must start at base value "1" too, and must advance as 1,2,4,8,16, etc, same as f/stop and shutter speed. And it does, Wikipedia shows the ISO and old ASA specs starting at 1. So technically, this makes ISO 128 be the full stop (called 125). And ISO 100 is actually 101.6, a third stop. But the cameras still marks 100, 200, 400, 800 as nice even full stops, and I did it here too (to match the cameras). But there is not much difference either way, and the camera does it right. Technically, when every stop is an exact numerical third stop, full stops are not special to us. So long as all third stop clicks are exactly one third stop (exactly cube root of 2 apart numerically), relationships don't matter where we start the markings. The exact number only matters to humans in precise calculations, for example if we compute EV or a third stop difference. But the camera knows, and always uses the correct values. It is all relative to the user, so long as every three third stop clicks add up to be exactly one full stop at 2x.
You've seen f/stops above, and there are two shutter speed charts (3rd and 4th long table below), the camera's normal rounded marked values, and the theoretical computed value goals that are actually used.
Each "Theoretical Actual" shutter speed (4th long table below) is just the simple progression (starting at one second), showing third stop times in sequential multiples of cube root of 2 (1.2599), and half stop times as sequential multiples of square root of 2 (1.4142). This insures that every interval of three third-stops, or two half-stops, is exactly 2x or 1/2x value (i.e., exactly one stop). These are the "Actual" shutter speed goals the camera uses. I am not implying practical accuracy is within a microsecond, my goal was merely to show four significant digits for 1/1000 to 1/8000 second values, and to show that any and every third value of third stops is exactly double or half value (one stop).
The other "Shutter Speed" chart is the camera's normal marked numbers for same shutter speeds. These marked values take liberties to show even or round values for convenient human use. This is just a marking, which does not affect what the shutter does. It is difficult to verify the fast numbers, but at the 30 second end, we can easily measure and confirm the camera shutter in fact does use the computed theoretical numbers (32 seconds actual instead of the marked 30 seconds). It must do that, because the basis of the system is that one stop is 2x the light.
As an example of nominal settings, users might plan to use the cameras interval timer to record multiple 30 second shots. They set the interval timer to 31 second intervals, so it can fit. Sounds reasonable, but this cannot work, because the camera 30 second setting actually does 32 second exposures (the sequence 1,2,4,8,16,32 seconds must be 2x full stops). You can time your shutter yourself. So the interval timer requires 33 second intervals for a so-called 30 second shutter setting. The difference between nominal and precise does exist.
Most markings have no more than about 2% or 6% numeric discrepancy. Which is a tiny difference, not more than 1/10 stop, but f/stop and shutter together can combine to add (the shutter half-stop markings of 10 and 20 are 13%, near 2/10 stop). Do realize of course, that any such error is Not real, it exists only in our own minds, since the camera is designed to use the right numbers instead of the nominal markings for humans.
So the actual shutter speed sequences 1/2, 1/4, 1/8 second do not suddenly shift to 1/15, 1/30, 1/60, and then suddenly shift again to 1/125, 1/250, 1/500 second. The camera does it right, and only the markings change, thought more helpful for humans. 64 may seem a nice round number today, but it was not always the case. :) This nomenclature was adopted 100 years ago, before the computer era, but if invented today, we would probably have no issue with the real 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 numbers (however the third stop markings, like 1/323 or 1/406 second may still look odd to us, see 4th long table below). But we are used to this old system now, and it is convenient. Nominal does have a certain beauty, and it serves our purpose. The exact markings we see are not very important, the important need is for each full stop (and each three third stops) to be exactly 2x the light from previous stop - easy work for today's crystal timed shutter.
So (unless humans are doing precise calculations), the point is NOT that there is a marking discrepancy, but is instead that we need not be concerned about it. The shutter does the right thing, and it is a rather neat system.
|Shutter Speed Stops|
|1.3 sec||1.5 sec|
|2.5 sec||3 sec|
|5 sec||6 sec|
|10 sec||10 sec|
|20 sec||20 sec|
|Theoretical Actual Shutter Speed|
|Third Stops||Half Stops|
Combining multiple lights (techie stuff, a use of f/stop)
Two lights will add to be brighter than the brightest. A light meter is a good way to meter multiple lights, but the math is like this:
We are assuming lights are ganged to light the scene area the same way.
Multiple Equal lights ganged = (one lights exposure fstop) x square root (of number of equal lights)
Assuming each light is f/8:
2 lights: f/8 x square root (2) = f/11.3 one stop brighter - remember, f/11 is actually f/11.3
3 lights: f/8 x square root (3) = f/13.85 1.58 stops brighter than one
4 lights: f/8 x square root (4) = f/16 2 stops brighter than one
5 lights: f/8 x square root (5) = f/17.9 2.3 stops brighter than one
Diminishing returns. We must double the number of lights to gain one stop.
The same formula is used for combined Guide Number of N equal flashes. GN of N equal lights = GN of one x square root (N).
Unequal lights ganged:
Two lights at f/8 and f/4 : (two stops difference)
Square root of (8² + 4²) = f/8.9 (about 1/3 stop more than brightest)
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