How to calculate a new print exposure time for a change in enlarger head height
Hi! For years I thought about this problem of 'What exposure factor do I need - or more practically, what new exposure time -for a change in enlarger head height?' I was running a photo studio at the time and was wanting to display 20x24" b&w SGFB framed prints in my front window - and I was typically using about 5 (five!) whole sheets of paper just to get to my first 'excellent' or 'perfect' display print, going through the usual series of numerous test strips and whole sheet prints, pretty wasteful! And of course, not only was I chewing through my expensive stock of paper, but also all my chemistry was getting pretty quickly worn out as well, mostly on rubbish! So I decided that there must be a way to work out correctly - with total, utter accuracy - a new exposure time for a new enlarger head height, and I did work it out.
It's a mathematical thing, and what I came up with, as a perfectly functioning solution, was complicated! A spreadsheet or computer app runs 'the formula' in a snap, but if you had to work it all out every time you went to a new print using, say, a simple pocket calculator, you'd go nuts and probably make mistakes.
Forget about using the simple, pure 'inverse square law' as it applies only to a 'point light source', whereas our enlargers are not 'point light sources', but rather they are much more complicated multi-element optical systems comprising head casings or reflectors, a lamp, a condenser or diffusor; the negative; the enlarging lens - and finally, the print paper. For these reasons, the use of the Inverse Square Law (ISL) to calculate a new exposure time gives only a very rough approximate result, and our photographically sensitive print papers being what they are, you will always end up simply exposing a print that looks noticeably too dark or too light.
The ISL is the starting point for the calculation, but that's all. Then there's the second problem: As the enlarger head is re-positioned at different heights to make prints of different sizes, the relative 'error-to-ISL' changes. Why is this? Theoretically the enlarger head's internal reflector, lamp, condenser (or diffusor), negative, lens - and of course the print paper - should all change their corresponding positions relative to each other in order to to maintain the same equivalently focused 'conjugate' setup, or if you like, to simply maintain 'the same degree of divergence-from-ISL'. But they don't: our enlargers, even the best and most expensive, are relatively simple 'fixed boxes' in this respect, usually only the lens shifts relative to its distance between the neg and paper (correctly so), but everything above the neg tends to stay the same (incorrect operation) so there's a constantly changing 'degree of error' over the print magnification range.
In fact the initial degree of divergence-from-ISL doesn't matter, but 'the degree of change-through-the print-magnification range' is everything; and all we need to do is identify this 'degree of change through the range' over, say, a continuous range of 2.5X to 20X magnifications, and use this as the controlling value in a slightly modified ISL calc program.
When we do this, the enlarging process goes as follows (this is how I have been printing for the last 20+ years, since I first solved this math):
1) I start off by making a small (c.63x88mm or 'wallet size') print of my entire desired neg, or the part of it (the 'crop') that I'm interested in. I keep working on this tiny (wallet-sized) print until I think it's just "perfect". Typically I might get such a print, after several tries using different exposure times and contrast filters, at an exposure time of, say, 3.4 secs, with my lens set to its best-performing printing aperture of, say, f14.
Because I'm making a tiny print, my enlarger head is set fairly close to the print paper; we're in 'tiny print land' here, using very short exposure times and not using much paper or chemistry. Whilst doing this, I always notice how great it is to work with a complete image, rather than just a test-strip slice, and often, in the course of perfecting this first tiny print, I may realize that it's not such a great photo after all and I can abandon it early and move onto another neg before wasting large amounts of paper and associated chemistry on it. Finally, when I get my first tiny print right, I use a simple steel hardware-store measuring tape to measure the straight-line (eg. vertical) distance from the print paper (in practice, the surface of the print easel) up to the enlarger's negative plane: lets say this distance is 452 mm. I now enter this distance ('452') and my tiny print's perfected basic exposure time ('3.8') into the two waiting fields in my (pocket) computer screen app.
2) Next, I raise or lower the enlarger head to any distance I like (in practice, somewhere between 1X and 20X) to get the degree of print magnification that I want for my second (usually larger, but it could be smaller, program works both ways) print. I lock the enlarger head in this desired new position and refocus the lens sharply for the print, taking care not to touch or change the setting of the lens's aperture ring. I now measure the new straightline (eg. vertical) easel-to-negative distance once again with my tape measure - let's say it now measures 1125 mm - and I enter this new distance ('1125') into a third waiting field in my app. I hit the 'calc' button and my app instantly computes and displays the new exposure time; eg, '38.5 secs'. I now set my exposure timer to this new exposure time and expose the print. After processing for the same period of time as for the earlier perfect tiny print, the 2nd print looks absolutely identical to the first tiny print, and I'm talking no error, but a perfect match. Sounds too good to be true? You just need the right computing app.
And that's it. I have since bundled this calculating program up into a proper app that runs on practically any PalmOS organizer (Zire22, TX, T5), sorry, it isn't available in any other format. You do need to calibrate it to your specific enlarger rig (thus inform it of your rig's 'slope') before you print, not hard, takes 20 minutes of your time and a few tiny scraps of paper in the wet darkroom. The app stores several different rig profiles and has extra features: it can automatically compensate for print dry-down, it can handle up to four different burn/dodge pockets; it can handles up to 9 split exposures with automatic reciprocity compensation; and it can help you to execute a (focal-length) lens-change mid-print. Great for making up sets of differently sized but otherwise identical prints! Send me your PalmOS device user name (eg 'thomasK' or whatever you call it) and I'll send you a full unlocked version on the house. I originally packaged this app, which I call 'enLARGE for PalmOS', about 5 years ago but nobody was interested, everyone too interested in going digital. Well, who says you can't use a computer in the darkroom? As they say, "Add a little computer to your darkroom!"