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 two sequences of shutter speed full stops...
Nominal: 1, 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, 1/500, etc. Looks like three sequences. We know that can't be exactly right, but 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 camera uses. We are used to these numbers.
Precise: 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, etc. Pretty simple. We have precise digital timing now, and what the 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.
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 mere 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. The point is NOT that there is a marking discrepancy, but that it's not important, that we need not be concerned about it. 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.
Numeric values of 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 are a special system, 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 camera uses the exact values internally, so that stops are always exact 2x or 1/2x steps. The camera always tries to use the exact precise value.
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 may have a problem if they set the interval timer to 31 second intervals, so it can fit the 30 second shutters. 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). Similarly, 20 seconds requires 21 second timer intervals, and 10 seconds requires 11 seconds. 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.
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.
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 conventions over the years have used to look nice.
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". 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).
Stop Number starts numbering at zero, and is shown in the charts to indicate the necessity for full stops to be in a 1, 2, 4, 8, 16 sequence. The need for 2x exposure steps seems obvious, and shutter speed using value 2Stop Number calculates that same sequence (concept example at right, and f/stop uses powers of √2). Stop Number itself is 0, 1, 2, 3, 4, 5 ... for full stops (calculating powers of 2 or powers of √2), and any number to exponent 0 is 1. We can add 1/3 or 1/2 or 1/10 stop to Stop Number for fractional stops.
f/stop aperture area depends on radius squared, so doubled area increments in steps of square root of 2. The value √2Stop Number computes f/stop number. Stop Number is the exponent, and f/stop number is the result. The base is √20 = f/1. So each full f/stop number is 1.414 x the previous f/stop number, which is one stop of 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).
Shutter Speed: The value 2Stop 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 20 = 1 second.
The value 2negative 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 today. In history, ISO values also initially began at IS0 1 (from 20). The European DIN rating began at DIN 1° = ASA 1, both = 20 as a base. Using the standard Stop Number rule that 20 is value ISO 1, and then 26.667 = 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 125 was a full stop.
However modern and digital cameras have adopted the goal for ISO 100 and 2x multiples to be full even stops. So, to make the ISO numbers precisely match current practice, we can choose to instead start our base for stop numbers for example at exactly 5 full stops under ISO 100. If choosing 5 stops under, then 25 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 21.643856 = 3.125. So (if this starting point with this offset to reach ISO 3.125), and 5 stops more at 2(5 + 1.643856) is exactly ISO 100 at an even full stop. The big advantage is that this plan also now computes precise third and half stops, relative to exactly ISO 100.
For empirical example of today's need of this offset, we see a Nikon DSLR if set to ISO 1250 is 1/3 stop less than 1600. The camera then reports ISO 1250 nominal in the menu and Exif, but deep into the extended Exif (Maker Notes section), it also reports ISO 1270 there, that it actually uses. Users don't much care about the exact numerical value, they just want it to be 1/3 stop less than ISO 1600.
The Nikon DSLR also uses 1/6 stops for ISO in Auto ISO mode, so values like ISO 449 or 566 can be seen in that way (nominal 450 or 560). 566 is a 1/2 stop, 3/6, but 449 is 1/6. This 100/32 = ISO 3.125 starting from 21.643856 creates those specific numbers like 1270 or 449 or 566. But starting at ISO 1 does not.
I did arbitrarily start the ISO chart at Sv = log2(100/32) = 1.643856 to be exactly five stops under ISO 100 in order to match SV charts starting at ISO 3.125 that you might see in Apex documents. Note though, it's different, they refer to the ISO 100 "fifth" stop Value as 5, which worked for their additive system of EV, but the math of converting back to specific ISO number requires the exponent of log2(5 + 1.643856). But the chart here could have started 4 or 6 stops under to simply match ISO values you may see in cameras today (with identical results as 5 stops).
Anyway, the ISO chart above places ISO 100 as a precise full stop, and computes correct values for 1/3 and 1/2 stops (correct relative to ISO 100), and also shows 1/6 stops for ISO too (shown only if between ISO 50 and 6400, to save a few lines).
If curiosity might want to see the ISO table computed with old film origin starting at ISO 1, then:
That was simpler, but it's just for explanation why it isn't that way now. Those numbers won't always match what you see in today's digital camera.
Nominal values cannot be computed, they're just a list of what the third stops have been called. Nominals shown in this second chart are from the first ISO chart above, probably one reason it changed. Rounding sometimes matches conventional nominal, and sometimes does not. Generally for shutter speed and ISO, one stop greater than any nominal third should be 2x the numerical value (2x number is two stops for f/stop).
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