The previous page 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 about the precise actual value. And the marked nominal numbers are pretty close, certainly close enough for human interest.
But 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 (the charts are below). 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 do use the correct precise values internally, regardless of the easier approximate nominal numbers they show humans. This is not info that humans must know, except when doing calculations such as EV or f/stops or Guide Number, we get better 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. They have been marked that way for nearly 100 years, since mechanical shutters with gears and springs. We know that can't be exactly right, but they do seem like nice round numbers. These markings are just 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. We have precise digital timing now, and what the camera actually does is precise steps, each are 2x or 0.5x steps, called stops. And the numbers are not so drastically different than human notions, at least for these full stops.
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. The camera knows to do the right thing, and it is a rather neat system. But if interested in computing actual precise values, there are two methods shown below (the two results should be exactly the same).
Numeric values of 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 are a special system, each exactly double the previous. That is the basis of "binary", and it is how our cameras work. All of 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 full, half and third stops are always actually exact half or third stops. But to make it confusing, also for example, both third stop and half stop systems have a shutter speed marked 1/10, 1/20, 10, and 20 seconds. Both halves and thirds cannot be the same equal values, which is a worse case error of about 13%, but the only error is the marked nominal value. The camera always tries to use the exact precise value.
ISO sensitivity is the equivalent of film speed. Digital sensitivity is a gain factor, results are multiplied after the digital sensor native sensitivity does whatever it does (i.e., around ISO 100). But the same film ISO numbering scheme is still 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. The ISO speed specification ought to start at base value "1" too, and advance as 1,2,4,8,16, etc. And Wikipedia does show the ISO and old ASA specs as starting at 1. However, that would make ISO 128 be the full stop (called 125). And ISO 100 would be a nominal third stop, near but not exactly 1/3 stop less. But digital cameras today instead treat ISO 100 as a full even stop, and compute the other numbers from it. Which if we still started at 1 now, it skews all of the numbers a little (example chart near page bottom here). We humans don't really need to care what the exact number is, but it is important to the math.
We do favor ISO 100 being the base, and the cameras mark 100, 200, 400, 800 as nice even full stops, and I did it here too (to match the cameras). The camera does this too. This works perfectly fine, and technically, full stops are not special to us. All that really matters is that all third stop clicks are exactly one third stop. Shutter intervals are exactly cube root of 2 apart numerically, so that three thirds are 1.0 full stop (however f/stop thirds are cube root of √2 apart, but three are 1.0 stop.) The math makes one second and f/1 be exact full stops, and these are convenient numbers on those scales. However, for ISO today, ISO 1 is not of interest. We must align ISO to compute the precise numbers used (detail below). 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. What's important to the user is that every three third stop clicks add up to be exactly one full stop at exactly 2x.
|Example of Concept|
The nominal chart values below are the camera's normal marked numbers, which take liberties to show nice rounded or even approximate values for convenient human use, numbers we won't struggle with. This is just a marking, which does not precisely affect what the shutter does - the camera knows to do the right thing.
It is 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). I've been told that some digital cameras actually do the 30 seconds (really hard to believe), but of course, most do the correct 32 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 (star trails, etc) may have a problem, if they set the interval timer to 31 second intervals, so it can fit the 30 second 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). Also 20 seconds requires 21 second 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, 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 an issue.
The 2x stop concept does seem sacred, but camera brands may vary. Nikon DSLR do 32 seconds, which makes the stops be correct. A Canon compact does 16 seconds for 15. And a Sekonic meter reading tenth stops will show exactly two 2.0 EV difference between 8 seconds and "30" seconds, which has to be computing for 32 seconds, which IMO, is of course the expected right thing to do. It's hard for me to believe some cameras would actually do 30 seconds, but you can easily verify which way your camera works by timing your shutter yourself.
In some cases, half stops and third stops are 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 1/10 and 1/20 seconds are 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 camera is designed to use the right precise numbers instead of the nominal markings shown to make it easy for humans.
So the actual shutter speed sequences 1/2, 1/4, 1/8 do not 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 markings change, thought more helpful for mere humans. 64 may seem a nice round number today, but it may not have always been obvious. :) This nomenclature was adopted maybe 100 years ago, before the computer era, and before the light meter era, and before mechanical shutters could be very accurate. 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, 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.
Marked nominal values cannot be calculated of course, they are just arbitrary usual approximations established over the years by convention. A few values might vary slightly in different cameras.
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 chart are those flagged as "These are Not third stops"
There is a two page printable PDF file of these next charts (fits Letter or A4 paper).
Stop Number is calculating above with full significant digit precision. Stop Number is merely "printed" here with only two decimal places, but the values 1/3 and 2/3 are used for full precision. I do NOT imply hardware accuracy is microsecond precision, but about six significant digits are needed down at 1/16384 second, or for ISO 182456.
For f/stops, start multiplying with third stop steps, which (for f/stop) is cube root of √2, or 1.12246x. Each third is this factor greater than the previous. For stops f/1, 2, 4, 8, 16, etc (power of 2) nominal is same value as precise. Or Half f/stop numbers advance in multiples of the 4th root of 2, or 1.18921x.
From f/8 (an even starting point, of 1, 2, 4, 8, 16, etc)
8 x 1.12246 = f/8.9797
x 1.12246 = f/10.079
x 1.12246 = f/11.314
x 1.12246 = f/12.699
x 1.12246 = f/14.254
x 1.12246 = f/16
For shutter speeds in third stops, start dividing at 1 with cube root of 2 or 1.26992. Each third is this factor less than the previous. Multiply from 1 to advance non-fractional seconds, greater than 1. Or half-stop shutter steps are square root of 2 (1.4142).
From 1/64 (an even starting point, of 1, 2, 4, 8, 16, 32, 64, etc)
(1/64) / 1.26992 = 0.01240157 = 1/81
/ 1.26992 = 0.00984313 = 1/102
/ 1.26992 = 0.0078125 1/128
/ 1.26992 = 0.00620079 = 1/161
/ 1.26992 = 0.00492157 = 1/203
For ISO, multiply by 1.26992 (cube root of 2 for thirds, linear like shutter speed), but today, start Not at 1, but at 25, 50, or 100. Or 100/128 (0.78125) will make ISO 50, 100, 200, 400, etc all become Full stops at exactly x.000 values. I think the ISO base used to be ISO 1, however digital cameras are using ISO 100 offset to be a full stop instead of an inexact third stop.
Each three thirds exactly hits each full stop (if using sufficient significant digits for precision).
|Fractions use a negative exponent. Additional Stop Numbers are shown in long charts above.|
Shutter speed is powers of 2 from 1 second (stop number 0 is 1 second).
Shutter Speed = 2 (stop number + fraction)
This is 2 "to the power of" (stop number + fraction)
The precise values use the values "2" for shutter speed, or the "square root of 2" for f/stops. Both are raised to the power of the stop number" (0,1,2,3,4...) plus the fractional stop. See chart at right.
Negative exponents give fractions, positive exponents give whole numbers. 22 is 4 seconds, but many shutter speeds are 1/ fractions, using a negative exponent.
Nominal 1/30 second is 2-5 = 0.3125 = 1/32 second precise.
f/stops are powers of √2 from f/1 ( √2 is 1.4142).
f/stop = √2 (stop number + fraction)
e.g., 2/10 stops more than f/2.8 (stop number 3) is 1.414 3.2 = f/3.031
Or 1/3 stop more than f/11 is 1.414 7.333 = f/12.698
Or 1/2 stop below f/11 is 1.414 6.5 = f/9.514
Three facts about Method 2:
Most shutter speeds are fractional, produced by a negative exponent.
1/100 is 6 2/3 stops "below" 1 second, or 2 -6.667 = 0.00984 second second,, which we call 0.01 or 1/100.
1/125 is 7 stops "below" 1 second, or 2 -7 = 0.00781 second, which we call 1/125 (but 1/0.00781 = 128).
The nominal shutter speed 1/1000 is 2 (-10) = 0.00097656, and the reciprocal is 2 (+10) = 1024 (to show 1/1024 seconds).
Negative (stop number + fraction) computes 1/(same numeric value) - e.g., for f/stops, f/0.707 is f/(1/1.414)
The idea is that adding fractional increments to the exponent stop number computes fractional EV stops. For example: Using √2 for f/stop, and the integer stop number for the full stop, then:
√2 (stop number) = full stop
√2 (stop number + 0.1) = tenth stop more
√2 (stop number + 1/6) = 1/6 stop more
√2 (stop number + 1/3) = third stop
√2 (stop number + 0.5) = half stop
√2 (stop number + 2/3) = 2/3 stop
√2 (stop number + 1 ) = Next full stop
ISO uses the same 2 (stop number + fraction) as shutter speeds use. Shutter speed and f/stop are very straight forward to compute, but digital ISO 100 has made ISO become tricky.
One way to compute these values (Method 1) is by starting at ISO 100, and keep continually dividing ISO by 1.122462, which is the 6th root of 2, which subtracts exactly a 1/6 stop from ISO. Repeat 42 times for seven stops, and you will reach ISO 0.78125. Restart again at ISO 100 and continually multiply by 1.122462 to add 1/6 stops to the other end of the chart. Or from ISO 100, you can divide by 0.259921 for 21 third stops to the same 0.78125. Or, you can start at ISO 0.78125, and multiply all the way. You need at least six significant digits for ISO values like 104032 (a third stop).
The second ISO method above (Method 2, used here) is mathematically the same but procedurally different, and it produces exactly identical numbers. ISO 3.125 is 100/32, which is 5 full stops below 100 (2 5 = 32). Stop Number 0 at ISO 1 no longer works as a base for ISO (100 is actually the base today, a round number for human convenience I suppose, but more difficult to work into a binary system).
Seven stops below ISO 100 (100/128, ISO 0.781) does include the IS0 1 range, but IS0 101.6 is a third stop. To hit ISO 100 cleanly, you can instead start at any value of 100/(2 N), N = 0, 1, 2, 3, 4, 5, 6, or 7 (stops below ISO 100). For example: (compare in the top ISO chart above)
N=5, 25 =32, 100/32 =ISO 3.125 at Stop Num =log2(100/32) =1.6439
N=4, 24 =16, 100/16 =ISO 6.25 at Stop Num =log2(100/16) =2.6439
N=3, 23 =8, 100/8 =ISO 12.5 at Stop Num =log2(100/8) =3.6439
N=2, 22 =4, 100/4 =ISO 25 at Stop Num =log2(100/4) =4.6439
N=1, 21 =2, 100/2 =ISO 50 at Stop Num =log2(100/2) =5.6439
N=0, 20 =1, 100/1 =ISO 100 at Stop Num =log2(100/1) =6.6439
Math: A log function is the inverse of an exponential function, for example, log2(100) = 6.6439, and inversely, 26.6439 = 100. So in this case, log2(100) is the exponent (the stop number) of 2 that will produce 100.
This will still use 2 (stop number + fraction), but the exponent 0.6439 is computed to exactly offset ISO 100 to become a full even stop.
There is one more level for aperture, so we either have to use base √2, or else we could use 2 and then take the square root of the result.
I used log2(100/32) to start at ISO 3.125 above, for a shorter chart. The omitted stops of very small numbers seemed unimportant, not useful today.
So it's a shorter chart, but it still creates these full stops, specifically so they (and ISO 100) become an exact even full stop (even if the stop number is not an integer.) But since the camera uses base 100 now, this offsets 100 evenly, and by adding 0.3333 and 0.6667 will also compute all the third stop numbers to come out as the known recognizable values (which the markings will then approximate with Nominal values).
The purpose of showing the Hypothetical chart example at right (it instead conventionally starts at Stop Number 0) was to show that Stop Number 0 was no longer an accurate choice for ISO now. It is a proper math plan, and it is more neat, with values suitable for historic lower film speeds (IMO), but is not how digital cameras number ISO today. It will compute numbers skewed from actual camera usage, because 100 is simply not a binary power of two (128 is). So we need heroics to get 100 aligned right, at least according to usage. This offset ISO numbering (to make ISO 100 be a full even stop) is the only procedure that agrees with my Nikon DSLR cameras. The cameras today really do like ISO 100.
A Canon compact camera also shows up with values like ISO 233 or 261. Canon hardware is said to use only full ISO stops, and software does the rest? I don't know yet how all their values come to be, none of my guessed tries computes those numbers. I suspect they are computed exact numbers? But digital cameras now do appear to NOT start ISO at base 1. I am interested in hearing other ideas about ISO numbering?
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