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I'm curious if there is a formula to determine my exposure in the dark room.
Suppose my print comes out perfectly with the following:
Is there a formula to determine what I need if I want to make the print bigger (for example move to a height of 13)?
I'm sure there is a relationship, but does anyone know what it is?
By the way, you can assume a 50mm enlarging lens and I'd be using the same brand and type of paper, just a bigger size.
The amount of light required to achieve the same exposure is proportional to the print area. That is the same as saying it goes with the square of the linear dimension. This is a approximation that is reasonably valid once the print is a few times larger than the negative.
Let's say your example with a height of 8 results in a 8 x 12 print area. Now you raise the enlarger so that the same image results in a 12 x 18 print area. 8x12 is a area of 96, and 12x18 a area of 216. At first approximation, roughly the same light will be spread over 216/96 = 2.25 as much area. That means the larger print will be exposed Log2(2.25) = 1.2 f-stops less if all else is kept the same.
In this example, you compensate by using f/8, or 18 seconds exposure, or some combination to make up for the larger area.
As I said earlier, this is a approximation that is close enough as long as the print is a few times larger in linear dimension than the negative. Most of the time that is the case, so I won't go into detail about how to correct for that.
A long time ago when I was still using a enlarger, I made myself a cardboard slide rule that showed me exposure times as a function of enlarger height (mine had a numbered scale), f-stop, which lens (I only had two), which attenuator, and general -2,-1,0,+1,+2 overall exposure. Instead of doing the math, I derived these from direct measurements with a light meter. It was quite handy.
To work this problem, you need to know the degree of enlargement before and after. Keep two rulers handy. Both scaled in millimeters. One used to measure the projected image on the baseboard and a thin transparent ruler to slip into the negative carrier.
Before making the height change, slip the transparent ruler into the negative carrier and project an image of the ruler on the baseboard. Using the other millimeter scale; measure the actual length, distance between two adjacent projected millimeter markings. Say one millimeter space = 8 actual millimeters. Now you know the first magnification is 8X. You also note the exposure time, in this case 8 seconds.
Now you compose the image to a new magnification. Again, slip the transparent ruler into the negative gate and measure the length of a projected milometer scaling. Suppose it measures 12, the revised magnification is 12X.
The key formula is called “bellows factor” = magnification (m+1)^2
For original magnification m= 8 thus (8+1)^2 = 81
For the revised magnification m = 12 thus (12+1)^2 = 169
Now we calculate the time difference factor 169/81 = 2.09
The revised exposure time = 8 x 2.09 = 16.7 seconds.
Best would be to use an enlarging light meter or an ordinary hand-held light meter. Remove the negative from the carrier and measure with the meter the center of the projected rectangle of light. Repeat at the new enlarger height.
You can use the inverse square law here!
You conveniently[1] started with a height of 8.
When you raise the enlarger to 11 (1 stop more distance), you need to either add 2 stops of time (increase to 32 seconds) or open up 2 stops of aperture (open up to f/4). If you raise the enlarger to 16 (2 stops), you need to either add 4 stops of time (128 seconds), or open up 4 stops of aperture (f/2).
Why? The inverse square law. As the distance increases, the light falls off (spreads out) by the square of the distance. When you increase the distance by 2 (8 to 16) you need to increase the light by 4 (4 times the exposure time, or 4 stops wider aperture).
This is especially useful when you want to make your next image a "power of two" larger. If your image is currently an 8x10 and you want a 16x20, that's a power of two larger.
[1] (Why is this convenient? It's an f-stop! It's a lot easier to work this out in your head when using familiar f-stop increments.)
Also see:
http://photo.net/black-and-white-photo-printing-finishing-forum/00E2LR
As you can see from the above suggestions, all made in the best faith, working this out using the Inverse Square Law is a bit complicated, and in actual fact you’ll never get it completely right to the point where your two differently sized enlargements will match each other in their tonality perfectly. This is because the Inverse Square Law does not really solve the problem in the first place, because an enlarger is not a ‘point’ source of illumination but is instead a much more complex arrangement of lamp, lamphouse reflector, possibly a diffuser or one or more condenser lenses, and an enlarging lens. But there is a relationship between any two (or more) different print magnifications, in terms of the equivalent exposure time required to expose them, and it has been caught and bundled in a revolutionary new app (June 2017) available on the Apple AppStore for iPhone, iPod touch and iPad. It’s called enLARGE and it lets you easily and simply compute the required equivalent exposure time for all magnifications above 2X. Because each enlarger behaves differently in this respect, thus with a slightly different relationship between print magnification and equivalent exposure time, you need to calibrate enLARGE to work specifically with your enlarger (or enlargers, as it stores profiles for an unlimited number of different enlargers or enlarger assemblies) first, and then it’s 100% accurate, dead on! It’s a breakthrough after 160+ years of darkroom printing in which enlarging has always basically been a ‘one print size only’ affair. And it’s cheap, priced to pay for itself in saved paper alone in the first one or two print sessions!
