Is there a sane reason why ¹⁄₁₂₅ is not, instead, exactly half of ¹⁄₆₀?

Here are some common shutter speeds you will find on most DSLR cameras:

1⁄15, 1⁄30, 1⁄60, 1⁄125, 1⁄250, 1⁄500, 1⁄1000, 1⁄2000, 1⁄4000

As you move from left to right, or as you increase the shutter speed, you are halving the amount of light that hits the sensor. In other words, you are decreasing the amount of light by one stop for each step. So 1⁄30 is half of 1⁄15, and 1⁄60 is half of 1⁄30. But then you come to 1⁄125, which is not half of 1⁄60 — half of 1⁄60 is 1⁄120. This is basic math.

So you break the sequence or pattern. But as you continue, it starts to make sense again. So 1⁄250 is in fact half of 1⁄125, and 1⁄500 is in fact half of 1⁄250, and 1⁄1000 is in fact half of 1⁄500, so on and so forth.

So there appear to be two distinct sequences here.

One: 1⁄15, 1⁄30, 1⁄60

Two: 1⁄125, 1⁄250, 1⁄500, 1⁄1000, 1⁄2000, 1⁄4000

Is there a sane reason for this?

I know that people sometimes talk about half stops or even thirds of a full stop. But then what is 1⁄125 the half stop, or third stop of? If you increase 1⁄60 by a third you get 1⁄180. This setting does not exist in the standard sequence; the closest you will get is 1⁄160. If you increase 1⁄60 by a half, you get 1⁄120 and it doesn't exist either.

Is this all arbitrarily set by the camera manufacturers, or is there perhaps some reason and history behind this?

-
Yes, it's a evil conspiracy between camera manufacturers. They are shaving a little off of shutter speeds to save money and hope you won't notice. Don't let them get away with it. Stand up and demand your milliseconds back! – Olin Lathrop Apr 27 '14 at 12:47
Though that is what the label says, is it in fact true that the shutter speeds are those fractions? I would not be in the least surprised if the shutter speeds were actually 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, 1/1024 and so on, and that the numbers were rounded to "nice" multiples of five for people not accustomed to thinking in powers of two. – Eric Lippert Apr 27 '14 at 14:47
You might like this question: What is an easy way to remember the full stop scale? – dpollitt Apr 27 '14 at 15:02
@Eric: The point is it doesn't matter. Given various other sources of error in getting the exact exposure, the 0.034 f-stop difference between 1/1000 second and 1/1024 second is irrelevant. You'd need carefully calibrated sensitive equipment to be able to detect that difference. The error in the actual aperture, error in the real ISO, error in the shutter speed, and light absorbed in the lens are all going to swamp that. Besides, in modern cameras the auto-exposure will compensate for some of these. Is your auto-exposure calibrated to within .034 f-stops? I didn't think so. – Olin Lathrop Apr 27 '14 at 15:12
Zero: n/1, 1/1, 1/2, 1/4, 1/8, not 1/16. – Michael Kjörling Apr 28 '14 at 12:06

Actually 1/125 is half of 1/60, ±0.06 f-stop.

It should be obvious by looking at shutter speeds that they were chosen to be the reciprocal of nice round numbers. Start with 1 second and keep dividing it by 2. Note that you missed the discrepancy between 1/16 s and 1/15 s. If you kept going in strict mathematical multiples of 2, then 1/60 s should really be 1/64 s, 1/1000 s should really be 1/1024 s, etc.

The basic problem is that in photography we are used to dealing with factors of 2, but a sequence of factors of 2 doesn't work out to nice numbers in our decimal numbering system. So we observe that 103 is close to 210, and realize that the 0.034 f-stop error is inconsequential.

Adding slight shifts in the factor of 2 progression to the shutter speed sequence in order to keep them as round numbers in our decimal system allows people to do mental math on them more easily.

-
@KRyan Probably. I wish that the higher ISO numbers would also follow this rule — terms like ISO 51,200 are kind of silly and suffer from the same excess precision. We should just call it ISO 50k. – mattdm Apr 27 '14 at 13:31
@mattdm Especially since the actual sensitivity versus specified ISO for most digital cameras is off by more than the difference between 51,200 and 50K at that setting. – Michael Clark Apr 28 '14 at 6:28
@bdesham: I don't know and it doesn't matter. You'd be unlikely to notice a shutter being off by 1/4 f-stop. Even if the shutter is accurate to 1/10 f-stop (exact for practical photography purposes), that is still three times the difference between 1/1000 s and 1/1024 s. So even if a high-quality shutter is targeting 1/1000 s, it could easily be off my more than the difference to 1/1024 s. 1/1000 s with .1 f-stop error is 1/1072 to 1/933 seconds. – Olin Lathrop Apr 28 '14 at 15:08
@MaxRied In which case you should be using equipment capable of such accurate exposure times, not consumer or even pro grade DSLRs. – Michael Clark Apr 29 '14 at 1:37
@Max: First, Michael is right. It is unreasonable to expect even a pro camera to be that accurate, since it does nothing for the pro photography use. It would add cost to get a feature 99.999% of customer don't care about. Second, really accurate scientific "shutters" are usually done by controlling the light. A bank of LEDs, for example, can be turned on and off with accuracies well below a microsecond. You can easily turn on such LEDs for 1/1000 s (1 ms) +-0.01% (within 100 ns). – Olin Lathrop Apr 29 '14 at 13:49

The difference between the "actual" shutter speeds at powers of 2 (1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, 1/1024, etc.) and the rounded numbers we use (1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000, etc.) is so trivial as to be beyond the limits of the vast majority of cameras in existence to accurately differentiate. Most consumer and pro grade DSLRs are not accurate to within the 0.034 stop difference between 1/1000 and 1/1024 seconds, or even the 0.06 stop difference between 1/125 and 1/120 seconds.

The same is true with alternating whole f-stops. f/1.4 is a rounded version of √2 and so are all of the other f-stops that include the √2: f/2.8, 5.6, 11, 22, etc. are actually (carried out to 16 significant digits) f/2.828427124746919, 5.65685424949238, 11.31370849898476, 22.62741699796952, 45.25483399593904, 90.50966799187808, etc. Notice that f/22 actually rounds closer to f/23 and f/90 actually rounds closer to f/91. This is totally insignificant because all but the most precise laboratory grade lenses can't control the aperture precisely enough to create that small of a difference anyway.

The largest variation in the exposure triangle between actual and theoretical numbers with most DSLRs is ISO sensitivity. Many manufacturers will fudge this number, some by as much as 2/3 stop, to make their performance at "ISO 1600" look better because in actuality the measurements were taken at the actual sensitivity of, say, ISO 1057 when the camera is set to ISO 1600! That is about 20 times more inaccurate than the difference between 1/1000 and 1/1024 seconds. The following graphic demonstrates the actual sensitivities of three top of the line DSLRs at various full stop ISO settings as measured by DxO Labs. When set to ISO 1600 the following cameras are actually sensitive at the ISO value in parenthesis: Canon EOS 1D X (1222), Nikon D4 (1192), Sony SLT ALpha A99 (913). Many other DSLRs are similar.

-
Do you have any citation for the claim that manufacturers use such... dubious tactics with ISO? – David Richerby Apr 28 '14 at 15:26
When photographing things like old-fashioned television sets, the differences between 1/59 second, 1/60, and 1/61, or between 1/29, 1/30 and 1/31, may be very visible (the shorter times--especially those faster than 1/60--being quite objectionable). I don't know if 60Hz power-line has anything to do with the choice of 1/60 as a time, but it can sometimes be much more useful than 1/64. – supercat Apr 28 '14 at 17:40
@supercat I doubt 1/60 is exactly 1/60.. it might 1/64 or 1/58 for what we know - I doubt camera companies apply such strong tolerances. Anyway, just to nitpick, NTSC (and HD in those countries who had NTSC) television frequency is not the powerline 60Hz anymore, it is 59.94Hz. – Marco Mp Apr 28 '14 at 20:31
@DavidRicherby See the following link. Click 'Measurements-->ISO Sensitivity' When set to ISO 1600 the Canon 1D X, The Nikon D4, and the Sony A99 are at ISO 1222, ISO 1192, and ISO 913 respectively. dxomark.com/Cameras/Compare/Side-by-side/… – Michael Clark Apr 29 '14 at 1:45
@supercat Most of the world uses 50Hz so I doubt it has anything to do with that. – David Richerby Apr 29 '14 at 7:28

The rule isn't exact with slower shutter speeds either: 1/15, 1/8, 1/4, 1/2, etc.

I think the only reason for this is that it the basic full stop series (which 1/125 is part of) has been agreed to at some point as a standard so that exposure calculation is easier when working together with the full stop aperture series. I don't think the small "errors" have a meaningful effect regarding doubling or halving the amount of light for your exposure.

-

Those numbers date from a century ago, when everything was mechanic on a camera. There was no way to build a shutter so accurate that there would have been a difference between 1/120 and 1/125... And 1000 is the human readable for 1024...

-

I'm a little surprised that nobody knows this, but the shutter speeds shown on cameras are simply the result of convention. There were two different conventions until about 1939, but that is beside the point.

Back in the days of mechanical cameras, repairmen had a simple device that could be used to determine the actual shutter speed of a camera. They discovered that cameras made by different manufacturers had brand specific biases, for example Leica shutter speeds were 1/10, 1/20, 1/40, 1/80, 1/200, 1/400, 1/800. Hasselblad leaf shutters tended towards the high side, as I seem to recall. It was of little consequence, since the speed rating of the film combined with the temperature, pH, and agitation vagaries of the processing solutions would easily vary by +/- 50%.

I should also mention that most mechanical cameras had two separate timing adjustments for the slow speeds and high speeds. In fact, very early focal-plane cameras had only one "high" speed, with various fixed width shutter openings chosen from a roll of shutter cloth to change the actual exposure without varying the time at all. Slow speeds were created by having a full aperture shutter "dwell" for a certain amount of time.

-
This is interesting, but doesn't seem to have much to do with why we now have 1/60s and 1/125s on cameras. – Philip Kendall Apr 29 '14 at 20:38
Why do you think nobody knows this? – mattdm May 3 '14 at 11:53
The earliest Speed Graphics used at the turn of the 20th century used a combination of slit width and spring tension to cover all of the available shutter speeds. See piercevaubel.com/cam/catalogs/… – Michael Clark May 4 '14 at 2:47

The actual shutter speeds we use are necessarily powers of two, but the nominal shutter speeds MARKED are just convenient rounded approximations. We DO NOT USE THE MARKED number values, the camera knows to substitute and use the actual powers of two. 1/30 is 1/32, 1/60 is 1/64, and 1/125 is 1/128 second. The markings might show three sequences, but the camera only uses the ONE power of two sequence.

For example, time the 30 second shutter on your camera, and the actual will be 32 seconds. Necessarily so, because of 1,2,4,8,16,32 seconds, and we need each stop to be exactly double for the concept to work. But the markings think its convenient to show the same numbers for 30 seconds and 1/30 second, neither of which are actually precise actual values actually used by the camera. It is a very old system, from back before binary was common (1/128 sounded strange), and things didn't matter so much. More detail at http://www.scantips.com/lights/fstop.html

For example, the camera has both third and half stops of 1/10 and 1/20 second, but the same marked value cannot be both a third stop and a half stop, which are 1/6 stop apart. And it isn't. The camera knows to do it right. Stops have to be powers of two.

For example, what we call f/11 is actually f/11.31. This is just rounded convenience in the nominal markings, but the camera knows to do it right.

The only time we might be concerned with this small difference (which only exists in our mind and in the markings) is when doing numerical calculations ourself. We calculate small inconsistencies with the nominal numbers, but the actual numbers used show precise relationships.

And when the camera does it, it comes out right. The camera uses precise values, but marks nominal rounded approximations.

-

In simple terms, the "baseline" exposure in cameras in the "olden days" was 1/125 shutter at f/8 aperture on ISO100 film. This would a give good in bright daylight. I expect camera shutters were actually calibrated for that 1/125 of a second. Going faster by a stop just meant doubling to 1/250, 1/500 and do on. Going slower, you didn't want to express it to the user on some horrible decimal fraction, so 1/60, 1/30,1/15, 1/8 and so on was "good enough" even if the camera might as good as exactly double the exposure each stop up from 1/125.

-