By Ed Buffaloe

When making test exposures for prints, most people just do a series of 3 or 5 second exposures across the paper, as recommended by Fred Picker, among others. This linear series of exposures does not take into account the fact that exposure progressions are inherently geometric in nature. To grasp this easily, imagine the difference between adding 3 seconds to a 3 second exposure and adding 3 seconds to a 30 second exposure. In the first case the exposure is doubled (i.e., one full stop more exposure), whereas in the second case the increase is almost negligible (i.e., about 1/8 stop more exposure).

In practice, if your print exposures are less than 30 seconds in duration, you will do fine with the three-second method. It is faster.  But above 30 seconds you may find it useful to know how to calculate 1/4 stop intervals.

Since 1/4 stop is about the minimum exposure change the unaided eye can easily differentiate, it would be most convenient if we could make our test strips in exact 1/4 stop increments. But how can this be done? The key lies in the logarithms of base 2. You don't need to understand logarithms to use this exposure technique, so if you wish you can skip to the bottom of the article and simply utilize the exposure progressions and multipliers outlined there. [It is also perfectly feasible to use this information and a calculator to come up with values for 1/8 or 1/10 stop intervals.]

First it is necessary to understand the logarithmic nature of full stops. Photographic exposure progressions are based on numbers that are factors of 2, such as 1, 2, 4, 8, 16, 32, 64, etc.

2⁰ = 1       0 = Log₂ of 1
2¹ = 2      1 = Log₂ of 2
2² = 4      2 = Log₂ of 4
2³ = 8      3 = Log₂ of 8
2⁴ = 16      4 = Log₂ of 16
2⁵ = 32      5 = Log₂ of 32
etc.

Any exposure multiplied by 1 gives plus 0 stops (log2 of 1) more exposure.
Any exposure multiplied by 2 gives plus 1 stops (log2 of 2) more exposure.
Any exposure multiplied by 4 gives plus 2 stops (log2 of 4) more exposure.
Any exposure multiplied by 8 gives plus 3 stops (log2 of 8) more exposure.

Most of us just understand intuitively that these numbers are 2, 4, 8, 16, etc., without knowing they are the logarithms of 2.

Now for the 1/4 stop increments. Let us fill in the blanks.

2⁰ = 1.00        0.00 = Log₂ of 1.00
2^.25 = 1.19      0.25 = Log₂ of 1.19
2^.50 = 1.41      0.50 = Log₂ of 1.41
2^.75 = 1.68      0.75 = Log₂ of 1.68
2¹ = 2.00        1.00 = Log₂ of 2.00
etc.

To give +1/4 stop, multiply by 1.19.      To give -1/4 stop, divide by 1.19.
To give +1/2 stop, multiply by 1.41.      To give -1/2 stop, divide by 1.41.
To give +3/4 stop, multiply by 1.68.      To give -3/4 stop, divide by 1.68.

Here is a handy chart: 

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