Aperture f-stop, Shutter Speed, ISO

This article is about "Understanding the camera Numbers". It is NOT a primer about using those numbers to take photos (for that, see for example).

That is math and physics, but still** the very useful purpose of f/stop numbers is the grand concept that the same f/stop, like say f/8, will be the same exposure in any lens**, regardless of focal length, or physical size of construction.

Is f/stop written f/stop or f-stop or fstop? The lens manufacturers properly write f/8. The internet changes things, and the term f-stop has become very popular on-line, but we also still see f/stop. 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 of literally = focal length / 8. This is a common and close and useful approximation, the actual physics is deeper (look up "numerical aperture" and "conservation of etendue"). The aperture diameter is the diameter of the entrance pupil from in front of the lens. But what photographers need to know is that the the purpose of using the f/stop system is so that the same numerical f/stop on any lens is expected to produce the same exposure.

OK, yes, there can be slight minor variations, especially in the old days in lenses without modern coatings (losing a lot of light), and even today in fancy lenses containing 15 or 20 glass elements (30 or 40 surfaces), each surface losing a slight amount of light even if with best coatings (lens coatings allow the light to pass through the lens, instead of reflecting it away). There are normally only slight variations today, but which are still important in professional movie cameras, when switching lenses on the same scene.

T-stops: So the professional movie lenses use T-stops, with markings which are calibrated to the actual amount of light the lens transmits, instead of the theoretical amount (a T2 lens actually transmits the light that a perfect f/2 lens should, matching the light meter). Fancy zooms with many elements suffer a little more, and the coatings are to help it. Todays improved modern lens coatings improve this tremendously today. So now, this is relatively solved for still cameras, since the camera meters through the lens anyway, automatically accounting for any possible variance in the lens losses.

Focal length: And while we are momentarily distracted, the marked Focal Length number *applies when focused at infinity*. Focal length changes as we focus closer (focal length normally becomes longer if front element is extended to focus closer). This change is relatively minor if at focus distances of a few feet or more, but at 1:1 macro, the focal length becomes equal to the working distance in front of the lens. The actual focal length is measured to the rear Principle Point, H', as shown above. The Principle Points are the designer's apparent planes where the subject and sensors images appear to be. Design of lens elements can move these points, and this H' point is often inside the lens, but in fact often literally outside the actual lens, either in front or behind the lens. Moved with convex lens elements that converge, and concave lenses diverge.

In **telephoto** lenses, this H' point is designed slightly in front of the front lens element, because, the actual optical technical definition of "telephoto" is that the lens is made to be 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 normally retro-focus (for SLR, DSLR), which means the rear node H' is designed 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 20 mm lens would block raising a mirror 24 mm tall. Here are diagrams of two (old) Nikon lenses.

Telephoto - H' is moved slightly in front of lens, a shorter lens

Retro-focus wide angle, lens is moved forward of H'

Magnification: The ratio of real subject size to image size on the sensor is called Magnification. At 1:1 reproduction ratio, then equal sizes, and the "working distance" in front of the lens (in front of point H) is necessarily equal to the distance behind the lens (at 1:1, the focal length, behind point H', is equal to the subject distance in front of H, due to similar triangles, etc.)

Seems a cute fact, which aids understanding, however (today, with zoom lenses and internal focusing which shifts thing inside), we probably are not told the exact locations of H and H1 (which would probably only apply to just one zoom at one focal length). Panoramic photos can be an issue, because best perspective (of close subjects) is when the camera panoramic axis is rotated around the H point. Panoramic articles describe ways to determine this pivot point they call Nodal point, or sometimes Perspective point.

Field of View: The field of view of the lens is the angle from the rear principle point H' (focal length) back to the sensor size dimensions (sensor size is also a factor of field of view). Then the same angle from the subject principle point H determines the field of view at the focus distance.

Back to f/stops:

Lenses are designed to expose equally if set to the same f/stop. That's what f/stop is. 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 view 3x, and then crops it to 1/3, so only 1/3x width and 1/3x height is seen, 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.

There are two concepts of camera numbers here. "Nominal" numbers (existing in name only) are the numbers actually marked on the camera, which are just simpler approximations of what I call the "Precise" numbers that the camera actually uses. The camera knows to actually do it right, but the marked numbers are made easier for humans. Much more on next page.

Nominal f/stop numerical values might be rounded, but might be truncated, or even approximated into a friendly ballpark number.

The cameras and light meters are marked f/11, and we say it as f/11, but f/11.31 is the necessary correct actual calculated value. This is only about 0.08 stop difference, and any difference exists only in our mind, since the camera will try to do it right anyway. Most other values are closer, but Guide Number calculations for speedlights can use f/11.31 instead of f/11.

As one way to make this fact be obviously true (that f/11 is actually f/11.31), note that f/stop numbering is the sequence of √2 intervals, (which is 1.414 numeric intervals) - making **every other stop number** be a multiple of 2. This sequence of progressions, when arranged into rows of every other doubled aperture values, are:

f/ | 1 | 2 | 4 | 8 | 16 | 32 | |||||

f/ | 1.414 | 2.828 | 5.657 | 11.314 | 22.627 |

It may be handy to realize that doubling any f/stop number (for example, f/5 to f/10) is exactly two stops.

Shutter speed marking numbers are also approximated. For example, the camera nominal markings show 1/20 second and 1/10 second (and 10 and 20 seconds) to be both third stop values and half stop values. But the same value cannot be both values, 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, see standard shutter speed charts on next page). The camera does it right, but we humans are frequently shown easier approximated numbers.

It is not obvious that the difference between f/4 and f/5 is 2/3 stop, and f/9 to f/10 is 1/3 stop, so the calculator purpose is to help with the math.

- Option 1 selects the camera nominal numbers, but like the camera, it uses the corresponding precise values that the camera actually uses (better accuracy than the nominal marked numbers). The next page has a chart of the actual precise values that the camera uses, and the calculator matches that here.
- Option 2 can be any numbers. Not sure there is great use for that, since the only marked numbers we can select on the camera are nominal numbers (hence Option 1 is more useful and accurate).
- Option 3 assumes that adding +EV exposure adds more shutter time, or causes a smaller and wider f/stop number (like the camera does). The calculator range is large, but not quite infinite. f/stop range here is f/0.5 to f/520 (f/0.5 is said to be a theoretical limit for the lens to still focus). Shutter speed range is 512 seconds to 1/32000 second (seems adequate). Range here meaning, it will compute further, but the suggested nearest third holds at those limits. The EV increment can be any value, like even -2.192 EV, but of course 1/3 stop is 0.3333 EV, 2/3 stop is 0.6667 EV, and 1/2 stop is 0.5 EV.
Option 3 could be used to add 10 EV to shutter speed for a 10 stop ND filter (or you could divide it among both shutter and f/stop). Or, there is also an Exposure calculator to compare two exposures.

- Option 4 tenth f/stop values are also detailed in the tenths chart just below, but this option is the difference from tenth stop values to a nominal value we can set on the camera. Difference will not exceed 0.17 EV from the nearest third.

But 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):

f/8 plus 7/10 stop

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, the equivalent value of f/8 plus 7/10 stop is 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 0.333 stops, and 2/3 stop is 0.667 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 not seem much point of 1/10 stop meter readings for daylight (IMO), since we can only set the camera to the nearest third stop. 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 values 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: f/stop = √2 ^{^(stop number + fraction)} (√2 is 1.4142)

e.g., 2/10 stops past f/11 (stop number 7) is √2 ^{7.2} = f/12.126

Or 1/3 stop past f/11 is √2 ^{7.3333} = f/12.698

If interested, here is a one page printable PDF file of this tenths chart.

Nominal and precise values of all (full and third and half) stops are on the next page.

- The reason to invent and use the f/stop method of numbering aperture is so that any two lenses (of any different sizes or focal lengths), will give the same exposure if at the same numerical f/stop. The entire idea is that f/4 exposure is f/4 exposure, in any lens. Prior to about the mid-1920s, Kodak used the "U.S. system" (Uniform System, from Britain in 1880s) of aperture markings, where stops were just numbered 1,2,4,8, with 1 starting at todays f/4.
- Values of f/stop intervals are selected to provide useful standard steps, called stops, each stop providing 2x or /2 of the previous step's light. An aperture with double or half area will pass double or half the light. Cameras today also have 1/3 stop or 1/2 stop increments.

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 wide view, and concentrates that light onto the camera sensor area.

A long lens (telephoto) gathers less light from a smaller view, onto the same sensor area.

But fstop = focal length / aperture diameter equalizes these, a larger aperture in a longer lens, giving equal exposure at equal f/stop numbers. Exposure is about Illumination per unit of scene area, which stays the same. That's why we bother with f/stop numbers, the benefit is great. f/8 is always f/8, on any lens. Our light meter works the same for any lens.

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.

See the note under the tenth stop table above for a formula computing f/stop for partial stops.

**Full f/stop numbers** advance in steps of 1.414x numeric multiples (f/1, f/1.4, f/2, f/2.8, f/4 ...) From any f/stop number, in all cases, double or half of that number is two stops (for example, f/10.2 is two stops above f/5.1).
Every second stop is the doubled f/number. Or one stop is x1.414 (or /1.414 which is x0.707).

**Third f/stop numbers** advance in multiples of the cube root of √2, or 1.12246x the previous (speaking of f/stops).

Every three third-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

**Half f/stop numbers** advance in multiples of the square root of √2, or 1.1892x the previous (speaking of f/stops).

Every two half-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

Less is more, Less f/stop number is More light.

Lens manufacturers seem to truncate numbers instead of round off. For example, f/5.6 is actually 5.66, and f/3.5 is 3.56. Except we see the same f/1.2 marking for the half stop (f/1.189) and third stop (f/1.260). Point is, the markings are just nominal numbers to show us humans. The lens and camera know to try to do it right.

The values of shutter speed and ISO are linear scales (square root of 2 is Not involved), 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 shutter step** 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 1.0 stop and a 2.0x change of the light.

The next **half-stop shutter step** is square root of 2 (1.4142) greater than the previous value.

Every two half-stop steps (from any point) is exactly 1.0 stop and 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 precise, but the camera knows exactly what to do. For example, set ISO 250 or ISO 2000 in the Nikon DSLR camera. Then near the top of the Exif data will show the ISO 250 or 2000 values, but further 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 call 250 and 2000.) Auto ISO is probably using 1/6 stops, but which will still be the precise sixth root of 2 steps. The numbers we see are just convenient nominal numbers, which the number really does not much matter to us humans. We just want one stop to always be a 2x light value. The point here is that the camera typically uses numbers a little different than the numbers we see. The only time that actually matters is if we try calculating ourself, using the nominal numbers instead of the actual precise numbers.

Shutter Speed | Marked As: |

30 seconds | 30" |

4 seconds | 4" |

1/4 second | 4 |

1/30 second | 30 |

1/4000 second | 4000 |

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.

Continued, Nominal and Precise Camera Settings, and charts.

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