I am frankly puzzled.
I know what an f/stop is...the numerical value which represents the denominator in the ratio of the diameter of the aperture to the FL of the lens.
I know what a timer is.
The OP is speaking of his stops with reference to a base exposure. What he calls f/stops should probably be called simply "stops" so that they are divorced from apertures.
One stop more light is accomplished by twice as much time. Two stops more light requires four times as much time. Thus it is a power (exponential) function. Logs are the opposite of exponents.
Half a stop more light is had by multiplying time by 1.414 (the square root of 2), since you can take your base time, multiply by 1.414 to get a half stop more, and multiply that by 1.414 to get yet another half stop, a whole stop from the base time. By application of logarithms you can get any intermediate stop in terms of time from the base time.
The method got it's name before there were such things as f-stop timers. The method uses exposure times in seconds that follow the f-stop numbering sequence: 1.0, 1.4, 2.0, 2.8, 4.0, 5.6 seconds ... When used as times each is 1/2 a stop more exposure.
This timer and the Darkroom Automation system use decimal stops. 1.0 stops = 2.00 seconds; 1.5 stops = 2.83 seconds; 2.0 stops = 4.00 seconds ...
The method got it's name before there were such things as f-stop timers. The method uses exposure times in seconds that follow the f-stop numbering sequence: 1.0, 1.4, 2.0, 2.8, 4.0, 5.6 seconds ... When used as times each is 1/2 a stop more exposure.
This timer and the Darkroom Automation system use decimal stops. 1.0 stops = 2.00 seconds; 1.5 stops = 2.83 seconds; 2.0 stops = 4.00 seconds ...
The term 'f-stop timer/timing' probably comes from Gene Nocon, who described his method for printing in his 1987 book Photographic Printing, and I believe the timer his company produced (for a short time) was called an 'f-stop timer'. Those who followed on with similar concepts for enlarging timing kept Nocon's terminology.
Yes. Arduinos are readily available for about $15-18 shipped; if I do a PCB it would probably be a "shield" that plugs into the top of an Arduino to handle the keypad, LCD and driver transistors for backlight and output relay. If one were mass-producing this they'd do it on a single PCB but for people putting this together at home, using a premade Arduino makes a lot of sense because it provides the power regulation and FTDI USB interface, the latter requiring SMD components. Not to mention that it would be quite difficult for an individual to even buy the components+PCB for an Arduino for the price they're sold at.
I've also simplified things a bit so that the external wiring is now trivial. An off-the-shelf solder-prototyping shield is quite sufficient to build this thing as long as it has at least 16 pins in a row to hold the LCD header (or just a 2x8 block if you want to use a ribbon) and most of them seem to. A custom PCB really isn't required at all, though the box-insides would look neater with it.