Camera settings are marked with Nominal values, but they use their actual precise values
The first page here mentioned how cameras mark their settings dials with approximate numbers (that I call Nominal numbers, existing in name only, close, but not actual or real). The marked numbers are made easy for humans to handle, who care about the 2x intervals, but really don't much care much about the knowing the precise actual numbers. And the marked nominal numbers are pretty close, certainly close enough for human interest.
Charts of all the camera Nominal and Precise settings are below.
In the camera, the overwhelming goal for photography is that each full stop must of course be exactly 2x or 1/2 exposure of the previous stop (necessarily and explicitly powers of two, which I call Precise numbers, meaning the perfect theoretical numbers which the camera actually tries to do. Every three third stops, or every two half stops, must align precisely with full stops, etc.
There is no error caused by the Nominal values, because the cameras already use the correct precise values internally, regardless of the easier approximate nominal numbers they show humans. Even if it were an error, it's small, would have little effect anyway. So this is not info that humans must know, except when doing calculations such as EV or f/stops or Guide Number, we get much better (prettier) precise results if we know the right numbers to use.
And knowing will explain these sequences of shutter speed full stops...
Nominal looks like three sequence methods. We know that can't be exactly right, but at least, they are nice round numbers (often rounded to nearest 1, or 5, or 10, or 100 or something). These markings are just human approximations of the actual precise numbers the digital camera uses. We are used to these numbers.
Old time mechanical shutters (timed with springs and gears) were not so precise, but we have precise digital timing today, and what the digital camera actually does is these precise steps, each are exactly 2x or 0.5x steps, called stops. And the numbers are not actually much different than human notions, at least for these full stops.
Numeric values of 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 are a special system, powers of 2, each exactly double the previous. That 2x is the basis of "binary", and it is how our cameras work. All of those values are exact 2x multiples, critical to our concept of "stops". But many of these on the camera are "marked" with rounded nominal values (easy approximations, for example 1/32 is marked 1/30, and 1/1024 is marked 1/1000). The digital camera uses the exact values internally, so that stops are always exact 2x or 1/2x steps. But the camera always uses the exact precise goal.
This is simple indeed, and the least complicated and yet most precise way it could be. The actual marked shutter speed sequences 1/2, 1/4, 1/8 don't suddenly shift to 1/15, 1/30, 1/60, and then suddenly shift again to 1/125, 1/250, 1/500. The camera does it right, and only the nominal markings change, years ago thought to be more helpful for humans to handle. 64 and 128 may seem nice round numbers today, but it may not have always been obvious. :) This nomenclature was adopted maybe 100 years ago, before the computer era, and before mechanical shutters with springs and gears could be very accurate anyway. But if invented today, we would probably have no issue with seeing 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 may still look odd to us. We humans like rounded numbers, and we are used to this old system now, and it is convenient for humans. Nominal does have a certain beauty, and it serves our purpose. The exact markings we see are not very important (to us), 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.
Cameras control the exposure, and their design is very concerned with exact values. Humans normally don't much care about exact specifics. Even if we want 1/125 second shutter speed, then 0.0078125 second is not a number we want to think about. So unless we're doing precise math calculations, we just specify things in terms of nominal third stops, and the camera tends to it, correctly. We could easily compute the percentage difference between nominal and the actual precise goals, but it seems quite unimportant, since nominal speeds are not real, do not exist, and are not used. Nominal is just a rough abbreviation for the actual precise goal. The point is NOT that there is a marking discrepancy, but that it's not important, that we need not be concerned about it. The important thing is that stops are all exactly 2x steps. Any "error" exists only in our own mind, the camera knows to do the right thing. It's a neat system. But if involved in computing numbers, you will be interested in the precise values.
Same thing in the full-second range:
That's of course why f/stops are f/16 and f/32. Mechanical shutter speeds could not be longer than one second until relatively recent years (of digital electronics), but actual shutter speeds are also 16 and 32 seconds.
It's difficult to verify the fast shutter 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). The basis of "stops" in photography is that one stop is 2x the light, so it is very important that cameras honor the 1, 2, 4, 8, 16, 32 numbers.
As an example of nominal settings, users using a Nikon cameras interval timer to record multiple 30 second shots will have a problem if they set the interval timer to 31 second intervals, so it can fit the 30 second shutter speed. Intervals must be longer than the shutter duration they contain, so this sounds very reasonable, but this cannot work, because the camera 30 second setting actually does 32 second exposures (because the sequence 1, 2, 4, 8, 16, 32 seconds must each be 2x full stops). The timer will miss many exposures if mistimed. But 33 second intervals would work well for "30 second" shutters. Similarly, 20 seconds requires 21 second timer intervals, and 10 seconds requires 11 seconds.
If using an external interval timer with camera Bulb shutter mode, it likely does exactly time the shutter duration you specify, but the interval to hold that duration must still be a bit longer. Many timer instructions say interval one second longer, some say two seconds longer. Run a test in a dark room at home to be sure your numbers will work before you go out on your trip. But if using the cameras 30 second shutter, that shutter will be 32 seconds, and the interval must be a bit longer.
The difference between nominal and precise does exist. The difference in 30 and 32 seconds is only about a 1/10 stop, not very important to us even if it were real, but when doing math, the numbers come out right for 32 seconds. In the old days of mechanically timed shutters, camera shutter speeds could not do more than one second (if that), so it was not much issue then.
The 2x stop concept is quite sacred. Nikon DSLR do 32 seconds for 30, which makes the stops be correct. A Canon compact does 16 seconds for 15. And a Sekonic meter reading tenth stops will show exactly 2.0 EV difference between 8 seconds and "30" seconds, which is computing for 32 seconds, which is of course the expected right thing to do. You can easily verify your camera by timing your shutter yourself (at 15 or 30 seconds).
In some cases, adjacent half stops and third stops are both marked with the same nominal number, but of course, these precise values cannot be equal. The shutter half and third stop markings of 10 and 20 seconds and the marked 1/10 and 1/20 seconds would seem 13% off, near 2/10 stop. And a few f/stops are marked the same for halfs and thirds (f/12.7 and f/13.5 are normally both marked f/13). Most markings have no more than about 2% or 6% or 10% numeric discrepancy. Which is a tiny difference, not more than 1/10 stop. And do realize of course, that any such error is Not real, it exists only in our own minds, since the digital camera is designed to use the right precise numbers instead of the nominal markings shown to make it easy for humans.
Nominal stops cannot be calculated. Rounding is often close, but not always exact (for example, 1/30 second or f/11). Nominal instead has to be a known list based on past convention. Nominal settings are just arbitrary approximations that have been marked over the years to look nice. But to create the 2x stops, the camera must actually use the precise value.
The charts below are calculating with full significant digit precision. Stop Number is merely "printed" here with only two decimal places, but the actual values 1/3 and 2/3 are used for full precision.
In the results, I do NOT imply hardware accuracy is microsecond precision, but computing with five or six significant digits is needed to show close numbers for 1/16384 second, or ISO 102400.
These charts show the camera's usual nominal marked values, and their fractional relationships. The main point is that also shown are the corresponding actual precise computed values that cameras must strive to perform accurately. The red values in the charts are those flagged as "These are Not third stops", which are the 1/2 stops, but also ISO 1/6 stops are shown between ISO 50 and 6400.
There is a two page printable PDF file of these next charts (fits Letter or A4 paper).
When computing, the precise stop value is a small formula to compute the precise value. Value = Base Stop Number. This ensures exact 2x steps of intensity.
The concept is there are "Stop Numbers", the stops are simply numbered 0, 1, 2, 3, 4, 5. We did not just arbitrarily number them, but instead these are the math exponents, to create powers of 2. Any base number to these exponents is exact powers of that base, base 2 gives powers of 2 for example. Base Stop Number is an exact power multiple of Base. Stop number 0 is a good starting point, because any number (except 0) to the power of 0 is value 1, which starts the all-important sequence 1, 2, 4, 8, 16, 32, etc. (powers of two). And negative stop numbers gives the sequence 1, 1/2, 1/4, 1/8, 1/16, 1/32, etc. The numbers are what they are because of this math of the powers of two. Short concept example shown at right, but Stop Number is also shown in the long charts just above. The Stop Numbers are not shown in the camera, but they are used to compute the camera setting stops that are used.
Shutter speed and ISO values are base 2 to power of stop number (is the powers of 2, vales are 1, 2, 4, 8, 16, 32, etc). Each full stop value number is exactly 2.0x apart, and their exposures are exactly 2.0x apart. Base 2 to power of Stop Number 3 is value 8 (seconds).
Aperture f/stop values are base √2 to power of stop number (is the powers of √2, values are 1, 1.414, 2, 2.828, 4, 5.657, etc). Each full stop value number is exactly √2 apart (1.4142x apart), and their exposures are exactly 2.0x apart. Base √2 to power of Stop Number 3 is value 2.828. Or aperture can optionally be computed as sqrt(base 2 to power of Stop Number), same thing, same math.
All full stops are precisely 1.0 EV apart (2.0x intensity). This is a sacred rule, and powers of 2 ensure it.
Fractional values like 1/4 second or f/0.5 are Negative stop numbers, -2 in this example. 2 -2 = 0.25 = 1/4 second.
Fractional stops, like + 1/3 EV, is base to power of (stop number + fraction). We can add 1/3 or 2/3 or 1/2 or 1/10 stop to Stop Number for precise fractional stops. For example, 2 -(6 + 0.3333) = 0.0124 second, and 1 / 0.0124 = 1/80.6 second.
This computes the theoretical "precise" goals actually used, which are often slightly different than the nominal numbers we see marked. The nominal marked numbers are just a guide for humans, but the digital camera always necessarily uses the precise values.
Stop Number creates full stops (of base 2) in a binary 1, 2, 4, 8, 16, 32 ... sequence. The need for 2x exposure steps seems obvious. Stop Number itself is 0, 1, 2, 3, 4, 5 ... for full stops (stop number is the exponent calculating either powers of 2 for shutter speed, or powers of √2 for f/stop). Any number (except 0) to exponent 0 is 1.
f/stop is focal length / aperture diameter, so aperture diameter is focal length / f/stop Number.
Circular aperture Area is Pi x r². Doubled area is 2 (Pi x r²) = Pi x (√2 x r)². Of course doubled areas increment in EV steps of 1.0 (2x), but the f/stop Number to do it increments in steps of √2 Stop Number. Stop Number is the exponent, and f/stop number is the result. The base is √2 0 = f/1. So each full f/stop Number is 1.414 x the previous f/stop number, which is a 2x stop of EV exposure change.
Stop Numbers that are negative compute fractional numbers (but f/0.5 is considered a limit to still be able to focus colors well enough).
The value 2 Stop Number computes the 1, 2, 4, 8, 16, 32 ... shutter speed full stops. 2x shutter speed duration is 2x exposure (linear), which is one stop. The base is 2 0 = 1 second.
The value 2 - Stop Number computes the fractional 1/2, 1/4, 1/8, 1/16, 1/32 ... shutter speed full stops.
ISO is similar to shutter speed (linear powers of 2), except ISO is special and harder today. In history, it was the same, and ISO values also initially began at IS0 1 (from 2 0). The European DIN rating began at DIN 1° = ASA 1, because both used 2 0 as a base. Using the standard Stop Number rule that 2 0 is value ISO 1, and then 2 6.667 = ISO 101.59 which is a third stop less than full stop 125 (like the nominal 1/125 second shutter speed number, both actually 128). Which was OK in film days then, we just bought the film and didn't care if ISO 128 was a full stop.
However modern cameras have adopted the goal for ISO 100 and its 2x multiples to conveniently be full even stops. So, to make the ISO numbers precisely match current practice, we must choose to start our base for stop numbers for example at exactly an even full stop multiple under ISO 100. If choosing 5 stops under, then 2 5 is 32, so ISO 100/32 = ISO 3.125, exactly 5 stops under 100. Then log₂(3.125) = 1.643856, and the inverse means that 2 1.643856 = 3.125. So (if this starting point with this stop number offset to reach ISO 3.125), and 5 stops more at 2 (5 + 1.643856) is exactly ISO 100 at an even full stop.
There's always the concern if repetition adds clarity or confusion, but what that is saying is:
If stop number 0 is conventionally to be ISO 1 (like is used for f/stops and shutter speeds), then nominal ISO 100 is stop number 6.6667, which is precise value ISO 101.6, which is a 2/3 stop between full stops 64 and 128.
ISO (called ASA then) was done that way in say around 1950, when we didn't really know or care about this detail, we just loaded a roll of film. But then in the late 1950s came notions of adding semiconductor light meters into cameras, involving computation, and then ISO 100 seemed like it should instead be considered as a nice round number.
So today, we consider ISO 100 to be exactly 5.0 stops above ISO 3.125 (because 100 / 2^5 = 3.125, exactly 5 stops below 100). Then stop number 1.643856, which is log₂(3.125) is ISO 3.125. Then if adding this ISO offset to all stop numbers, then ISO 100 is stop number 6.643956 (stop 5 plus this offset), which is exactly ISO 100.000, which is now a full stop itself (if considered as stop 5). Or, the math works the same if using base ISO 1.5625 which is 6 stops under ISO 100, so offset in that system is 0.643856. and 100 is stop 6 (same actual 6.643856 stop number).
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