Stopping (and starting) and Powers of Two
What is all this talk about stops and what power does it have?
When you first enter the world of photography you have to reconcile yourself to this term of stops, at least I did. It was a confusing term at first. That and the idea of fast or slow. Eventually I suppose all of us grasp the meaning in whatever way we find useful.
Photography expresses all aspects of an exposure whether on film or paper in f-stops. Though it did not always and even today not everyone everywhere expresses exposure as stops. The term derives from the settings on the aperture of a lens. That apparatus now almost exclusively defined as a movable diaphragm (though not always as some cameras use neutral density filters and some old cameras had plates with different sized holes drilled in them.). Indeed our current system was not so universal. I came across this when I obtained an old Kodak rectilinear lens and was confuse by the aperture markings. This article here explains in more detail this history. I find the table below enough to make the point however.
They are most all concerned with a doubling of light. The current consensus derives from the area that light falls and thus must incorporate the square root of 2. So the modern aperture system uses square root of 2 intervals. Hence that strange number sequence we end up memorizing. (1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64 etc) (Note there is some rounding going on.) Square root of 2 is 1.414... You just keep multiplying the square root of 2 with itself. This of course is an exponent. So rather than say 1.414x1.414x1.414=2.8 we say 1.414 to the power of 3 (1.414^3 in other notation).
We also adopt shutter speeds as doubling or halving. Why? Well certainly it is convenient as the two become interchangeable. If I open the aperture one stop and can increase the shutter speed one setting to achieve the equivalent amount of light on the film or paper.
If I want to reduce the exposure by 2 I can either move to the next higher aperture or increase the shutter speed to the next higher speed. (These numbers tend to be in the sequence 1/2, 1/4, 1/8, (and now a small shift) /1/15, 1/30, 1.60, (another shift) 1/125, 1/250, 1/500 etc. The shifts help align to a more decimal notation and the differences are small enough to to ignore. (Other wise the sequence would be 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512.)
So the other aspect of this square root and powers of two is this is the basis of the binary number system and digital computers and then finally digital photography. So what is this connection? If we have a digital camera sensor that supports 8-bit data. That is 8 bits in each color Red, Green, and Blue (RBG) then we can say the dynamic range is 2^8 levels or 8-stops. If it supports 12 bits then 12 stops. My scanner supports 16-bit scans per color (RGB) or 48-bit (3x16) as a format. The sensor itself is probably no better than 12 bits however.
This article came about as I often have need to manipulate these numbers; particular in the darkroom. I moved some years ago to a stop-based approach to test strips for instance. I, like many people, started by using a simple timing sequence for this work. 2,4,6,8,10, 12 seconds etc. This system works well enough.
I began to understand how a stop based system might work better. Gene Nokon was an advocate of this method. (I bought a used copy of his book Photographic Printing. Interestingly he taught Prince Andrew so as a warning some of his photos are included.) It allows one to move readily between apertures on the enlarger for instance. Increasingly, as a result, I can think in terms of burns and dodges in stops or fractions there-of. In some cases I need a really long burn so I might make the burn at a different aperture confident the result will be the same.
I had then to compute this sequence of times, 8, 11, 16, 22, 32, 45, 64. These are half stop times. A full stop sequence is a simple doubling of times. 8, 16, 32, 64. To find half stops I need a little more math (US) (maths UK). We return to the idea of powers of 2 so for the full stop times the exponents are computed with a base 2 logarithm which is easily available on spreadsheets or some scientific calculators. The normal log function is base 10 and so cannot be used directly. However we remember from high school algebra we can get to a log in any number base (10 in this example) by taking log10(n)/log10(base). In the case of base 2 log10(2) = .30. So if you have the normal log base 10 function use it but divide the log10(n) by .30 to get the log2 result.
Here are the full stop times and their log2 counterparts.
Now we can intersperse 1/2 stop intervals easily as below. Just adding .5 to each log2 interval.
Then take 2^n of these ...
When you first enter the world of photography you have to reconcile yourself to this term of stops, at least I did. It was a confusing term at first. That and the idea of fast or slow. Eventually I suppose all of us grasp the meaning in whatever way we find useful.
Photography expresses all aspects of an exposure whether on film or paper in f-stops. Though it did not always and even today not everyone everywhere expresses exposure as stops. The term derives from the settings on the aperture of a lens. That apparatus now almost exclusively defined as a movable diaphragm (though not always as some cameras use neutral density filters and some old cameras had plates with different sized holes drilled in them.). Indeed our current system was not so universal. I came across this when I obtained an old Kodak rectilinear lens and was confuse by the aperture markings. This article here explains in more detail this history. I find the table below enough to make the point however.
 |
| A Forest of Apertures! |
We also adopt shutter speeds as doubling or halving. Why? Well certainly it is convenient as the two become interchangeable. If I open the aperture one stop and can increase the shutter speed one setting to achieve the equivalent amount of light on the film or paper.
There are exceptions; digital cameras can dispense with this if in aperture priority mode and compute the exact time the shutter should be open. The Yashica GT(S)N series of rangefinders uses an analog shutter which is held open based on the brightness of light falling on the light meter and the aperture size selected. The shutter speed can change across a linear continuum.)
If I want to reduce the exposure by 2 I can either move to the next higher aperture or increase the shutter speed to the next higher speed. (These numbers tend to be in the sequence 1/2, 1/4, 1/8, (and now a small shift) /1/15, 1/30, 1.60, (another shift) 1/125, 1/250, 1/500 etc. The shifts help align to a more decimal notation and the differences are small enough to to ignore. (Other wise the sequence would be 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512.)
So the other aspect of this square root and powers of two is this is the basis of the binary number system and digital computers and then finally digital photography. So what is this connection? If we have a digital camera sensor that supports 8-bit data. That is 8 bits in each color Red, Green, and Blue (RBG) then we can say the dynamic range is 2^8 levels or 8-stops. If it supports 12 bits then 12 stops. My scanner supports 16-bit scans per color (RGB) or 48-bit (3x16) as a format. The sensor itself is probably no better than 12 bits however.
This article came about as I often have need to manipulate these numbers; particular in the darkroom. I moved some years ago to a stop-based approach to test strips for instance. I, like many people, started by using a simple timing sequence for this work. 2,4,6,8,10, 12 seconds etc. This system works well enough.
I began to understand how a stop based system might work better. Gene Nokon was an advocate of this method. (I bought a used copy of his book Photographic Printing. Interestingly he taught Prince Andrew so as a warning some of his photos are included.) It allows one to move readily between apertures on the enlarger for instance. Increasingly, as a result, I can think in terms of burns and dodges in stops or fractions there-of. In some cases I need a really long burn so I might make the burn at a different aperture confident the result will be the same.
I had then to compute this sequence of times, 8, 11, 16, 22, 32, 45, 64. These are half stop times. A full stop sequence is a simple doubling of times. 8, 16, 32, 64. To find half stops I need a little more math (US) (maths UK). We return to the idea of powers of 2 so for the full stop times the exponents are computed with a base 2 logarithm which is easily available on spreadsheets or some scientific calculators. The normal log function is base 10 and so cannot be used directly. However we remember from high school algebra we can get to a log in any number base (10 in this example) by taking log10(n)/log10(base). In the case of base 2 log10(2) = .30. So if you have the normal log base 10 function use it but divide the log10(n) by .30 to get the log2 result.
Here are the full stop times and their log2 counterparts.
| Time (seconds) | 8 | 16 | 32 | 64 |
| log2(time) | 3 | 4 | 5 | 6 |
Now we can intersperse 1/2 stop intervals easily as below. Just adding .5 to each log2 interval.
| Time (seconds) | 8 | 16 | 32 | 64 | |||
| log2(time) | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 | 6 |
Then take 2^n of these ...
| Time (seconds) | 8 | 11 | 16 | 23 | 32 | 45 | 64 |
| log2(time) | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 | 6 |
| Countdown | 56 | 53 | 48 | 41 | 32 | 19 | 0 |
These are rounded to the nearest second which if you are watching a clock is all you need. I also have a Paterson enlarger timer for 'The Beast' 4x5 enlarger. The Paterson timer only counts down. So I have added the countdown row so I can move the test strip at the right times from its display.
Marking quarter stop time is not so easy it must be said.
Around this time I was aware of a product from RH Designs for timing exposures on the basis of f-stops. They also have a eloquent argument in favor of f-stop timing. Their products seem extremely well designed and thought out. They retail for over £300 ($400 US) and when they come up for auction they seem to retain their value so they must be good products. I was also thinking at this time of automating my split filter printing. Having been a programmer in my past I decided it would make a fun and useful project. I document the project in a series of posts beginning here.
As part of the project I wanted to implement f-stop timing and taking a page from RH Designs playbook. I accepted their argument that 1/4 stop timing would be useful (Experience shows them to be right when dialing in the last details in a fine print. At least as fine as mine get.). I note in looking up RH Designs again they have upped the spec to 1/24th stop! Not sure I will be chasing that anytime soon however.
For this problem I took my range of exposures to be 4 seconds to 64 seconds which is 4 stops of range (log2(64)-log2(4)=6-2=4). I then populate the table by 1/4 (.25) intervals and compute the times. Since a computer is timing the lamp I can use more accurate numbers. (The table wraps to the next line below.)
One can see getting finer stops if desired easily.
I resolved this by measuring on the baseboard the brightness of the highlights in each case. I use my Sekonic light-meter in reflected light mode and place it under the highlight area I want to control to. I set the meter to ASA 100 and then use the Ev mode to measure the exposure value as a simple number to determine brightness. Now the nice thing here is again this is calibrated in stops. (If the meter balks at a measurement I can open the aperture 1 or more stops so long as I remember to subtract this from the resulting Ev to get it back to the same reference point.
I also have another complication. On 'The Beast' I often use a Kodak Exposure Scale like the one below. It works simply by overlaying onto of a piece of photo paper and then the exposure timer is set for 60 seconds. Each wedge when developed shows the exposure for the time in seconds. Trouble is this is based on 2 and 3 second times. So the '1/2’ stops are actually multiples of 3.
We can still use the same techniques. Log2 is still our friend.
I determined the best unmasked exposure for the 00 filter was f45 at 48 seconds. I measured the unmasked exposure that I like the highlights in and got an Ev of 4.2 and on the masked version I measured Ev 3.9 (which is darker my .3 stops). This means I need to increase my exposure by .3 stops. The aperture on the enlarger lens only shows 1 stop increments so the only variable I can use is time.
What is .3 stops added to 48 seconds? We can take log2(48) and get 5.58 stops. Add .3 to get 5.88 then take 2^5.88 and we get 59 seconds. (Alternatively I can take 2^.3 = 1.23 and multiply this by 48 to get the same results. Adding logarithms we all recall is the same as multiplying.) Set the time and run the exposure. Another mask I made differed by 0.7 stops and this gave a 78 second exposure by the same calculations. The comparisons are shown below and are closely matched in the highlights on the tree trunk.
Hopefully I have illustrated the usefulness and methods of the f-stop system .
Marking quarter stop time is not so easy it must be said.
Around this time I was aware of a product from RH Designs for timing exposures on the basis of f-stops. They also have a eloquent argument in favor of f-stop timing. Their products seem extremely well designed and thought out. They retail for over £300 ($400 US) and when they come up for auction they seem to retain their value so they must be good products. I was also thinking at this time of automating my split filter printing. Having been a programmer in my past I decided it would make a fun and useful project. I document the project in a series of posts beginning here.
As part of the project I wanted to implement f-stop timing and taking a page from RH Designs playbook. I accepted their argument that 1/4 stop timing would be useful (Experience shows them to be right when dialing in the last details in a fine print. At least as fine as mine get.). I note in looking up RH Designs again they have upped the spec to 1/24th stop! Not sure I will be chasing that anytime soon however.
For this problem I took my range of exposures to be 4 seconds to 64 seconds which is 4 stops of range (log2(64)-log2(4)=6-2=4). I then populate the table by 1/4 (.25) intervals and compute the times. Since a computer is timing the lamp I can use more accurate numbers. (The table wraps to the next line below.)
| Time (seconds) | 4.0 | 4.8 | 5.7 | 6.7 | 8.0 | 9.5 | 11.3 | 13.5 | 16.0 | ||||||||
| log2(time) | 2 | 2.25 | 2.5 | 2.75 | 3 | 3.25 | 3.5 | 3.75 | 4 |
| Time (seconds) | 16.0 | 19.0 | 22.6 | 26.9 | 32.0 | 38.1 | 45.3 | 53.8 | 64.0 |
| log2(time) | 4 | 4.25 | 4.5 | 4.75 | 5 | 5.25 | 5.5 | 5.75 | 6 |
One can see getting finer stops if desired easily.
Other Uses
Recently I have been exploring masks for dodging shadows. Getting a mask to the right darkness and contrast I have found a challenge. But to evaluate the results in a print objectively is a struggle as well. The mask usually impacts the highlights as well as the shadows. This means when I add the mask I need to add some more exposure to get the same highlight values from the un-masked print.I resolved this by measuring on the baseboard the brightness of the highlights in each case. I use my Sekonic light-meter in reflected light mode and place it under the highlight area I want to control to. I set the meter to ASA 100 and then use the Ev mode to measure the exposure value as a simple number to determine brightness. Now the nice thing here is again this is calibrated in stops. (If the meter balks at a measurement I can open the aperture 1 or more stops so long as I remember to subtract this from the resulting Ev to get it back to the same reference point.
I also have another complication. On 'The Beast' I often use a Kodak Exposure Scale like the one below. It works simply by overlaying onto of a piece of photo paper and then the exposure timer is set for 60 seconds. Each wedge when developed shows the exposure for the time in seconds. Trouble is this is based on 2 and 3 second times. So the '1/2’ stops are actually multiples of 3.
We can still use the same techniques. Log2 is still our friend.
I determined the best unmasked exposure for the 00 filter was f45 at 48 seconds. I measured the unmasked exposure that I like the highlights in and got an Ev of 4.2 and on the masked version I measured Ev 3.9 (which is darker my .3 stops). This means I need to increase my exposure by .3 stops. The aperture on the enlarger lens only shows 1 stop increments so the only variable I can use is time.
What is .3 stops added to 48 seconds? We can take log2(48) and get 5.58 stops. Add .3 to get 5.88 then take 2^5.88 and we get 59 seconds. (Alternatively I can take 2^.3 = 1.23 and multiply this by 48 to get the same results. Adding logarithms we all recall is the same as multiplying.) Set the time and run the exposure. Another mask I made differed by 0.7 stops and this gave a 78 second exposure by the same calculations. The comparisons are shown below and are closely matched in the highlights on the tree trunk.
Hopefully I have illustrated the usefulness and methods of the f-stop system .
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