Fibonacci sequence for f-stop test strips

whojammyflip · Mar 20, 2022

Just noticed a couple of nice features of the Fibonacci sequence applied to f-stop printing. a) gamma of 0.6 and the Fibonacci step of 1.62 equate to approximately 1 stop increase for each increment on the Fib test strip sequence. b) convenient calculation on the fly of increments on the test strip, as the increment is the preceding number in the Fib sequence c) for a 6 step test strip, with 5 increments, the step size is nearly 10% (10.1) which is easier to calculate on the fly.

Here is a script of the steps taken this evening, in my darkroom:

# 1st test strip (6 steps to the test strip)
- use the following exposure range in seconds 3, 5, 8, 13, 21, 34, 55, (likely that the correct exposure lands in this range)
- the difference between steps is also a Fibonacci sequence 2, 3, 5, 8, 13, 21, 34, making incremental test strips easy

# 2nd test strip
- pick a new narrower range of neighbouring times from the above test strip (for example, 13 and 21), which are the upper and lower bounds of appropriate exposure time
- due to the fact that this range is 1.6 (Fibonacci ratio), for 6 stop range, there are 5 increments, which equates to 10% increase per stop

Rinse and repeat if contrast is too high or low, with an increase or decrease of a grade in paper.

Hope this helps. I've been reading Harry Fearn's book, "Better in black and white" and something in there is probably responsible for triggering this idea.

MattKing · Mar 20, 2022

The only problem with that is that the step sequence for 1/2 stop increments is actually 1.414 (the square root of two) rather than 1.6.
Approximating very slightly:
2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64 ....
Recognize the progression?

whojammyflip · Mar 20, 2022

Yeah, its not really f-stops, its Fib-stops. Have a try with the calculations. For me, the size of the steps was convenient to work with, as they are integers and borrow from lag 1 in the series. Plus, there is the gamma aspect, which I thought was also neat.

For example, with the 2, 2.8, 4, 5.6, 8, 11 progression, the increments in time are 0.8, 1.2, 1.6, 2.4, 3....not really so useful. And they are approximations. 5.6 * 2 != 11, for example.

MattKing · Mar 20, 2022

And I wish you were correct, because the Fibonnacci sequence is absolutely fascinating.
Some light reading for everybody: https://www.fq.math.ca/

MattKing · Mar 20, 2022

My test strip guide:

MattKing · Mar 20, 2022

If you aren't using stops, it sort of throws out the ability to make adjustments using the lens aperture as well.

AgX · Mar 20, 2022

MattKing said:
The only problem with that is that the step sequence for 1/2 stop increments is actually 1.414 (the square root of two) rather than 1.6.
Approximating very slightly:
2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64 ....
Recognize the progression?

You mean: 1 stop increments

MattKing · Mar 20, 2022

AgX said:
You mean: 1 stop increments

Not when it comes to time - those are seconds.

albada · Mar 20, 2022

MattKing said:
If you aren't using stops, it sort of throws out the ability to make adjustments using the lens aperture as well.

Yes!
Also, my LED-head controller expresses LED power-levels (brightness) in stops, as well as time. Consequently, I can freely interchange time, aperture, and brightness. I love it. For example, every step in my test strips are exposed for the same time (I vary brightness instead of time). This way, there can be no reciprocity effects in a test strip. But such interchanging of settings requires that time be expressed in stops.

By the way, you can also measure the height of the enlarger lens in stops. Then, when you change height, the numeric height-change is the number of stops to compensate the exposure.

Back to the topic: Using Fibonacci numbers for test strip times is a fascinating idea. But the average step-factor of 1.6 is too large. Even a 1.414 step (0.5 stop) is large. A step of 0.3 stops will extend 1.2 stops from each side of the guess on a 9-step strip, which will be fine if your guess of exposure is within 1.2 stops. Can you think of a modification of the Fibonacci sequence to reduce that 1.6 factor to around 1.25?

Mark Overton

albada · Mar 21, 2022

Thinking about this some more, here's an easy way to get a geometric sequence of times for a typical additive (incremental) test strip:

Set the exposure of the next step to 1/4 of the total exposure time of the prior steps, rounding off the division.

Example: You guess that the exposure won't be under 15 seconds, so start there. For a 7-step strip, we get:

15 plus 4 is 19 plus 5 is 24 plus 6 is 30 plus 7 is 37 plus 9 is 46 plus 11 is 57.

This means the time of a step is about 1.25 times that of the prior step, which is about 0.3 stops.
The above example covers about a two-stop range of time, but if your guess might be farther off, try using 1/2 or 1/3 of the total time so far, instead of 1/4.

Mark Overton

Sirius Glass · Mar 21, 2022

Rather than Fibinacci series, use powers of 2.

faberryman · Mar 21, 2022

Does doing your test strips with a Fibonacci sequence make your final prints better? My experience is that the correct exposure always falls somewhere between the increments of my test strips no matter how they are calculated.

radiant · Mar 21, 2022

Sirius Glass said:
Rather than Fibinacci series, use powers of 2.

Is that you, Mr. Nocon?

gone · Mar 21, 2022

faberryman said:
My experience is that the correct exposure always falls somewhere between the increments of my test strips no matter how they are calculated.

How true. It's just life, I've come to terms w/ this one.

whojammyflip · Mar 21, 2022

Another observation, if you look at the sequence of say 4, 6, 8, 11, 16, 22, the ratio between steps is 1.50, 1.33, 1.38, 1.45, 1.38, with a standard deviation of 0.068.

In comparison, the sequence of 3, 5, 8, 13, 21, 34 has a ratio of 1.67, 1.60, 1.63, 1.62, 1.62, with a standard deviation of 0.025.

And another observation: with gamma of 0.62, step multipliers of 2**0.62, would be 1.54. Fib ratio of 1.62 is closer to this than 2**0.5, or 1.41, would be.

Really, the big advantage is knowing on the second test strip that 10% increments in time are equivalent to 1/5 of a step, and can be programmed in easily. I'm using a digital Durst timer, which allows for this kind of timing. I find I cannot really see the difference between steps less than 10%.

For example, the steps would then go: 13, 14.3, 15.7, 17.3, 19, 20.9, if you take 10% of the previous step, accurate to 0.1s. If you knew the correct exposure was between these steps.

Of course, I am going to push this idea, as I dreamt it up and hadnt seen it discussed. But I do think its pretty useful.

ic-racer · Mar 21, 2022

Sirius Glass said:
Rather than Fibinacci series, use powers of 2.

I got flamed from the Like button thread... how about a Correct Answer button?

MattKing · Mar 21, 2022

hoojammyflip said:
Another observation, if you look at the sequence of say 4, 6, 8, 11, 16, 22, the ratio between steps is 1.50, 1.33, 1.38, 1.45, 1.38, with a standard deviation of 0.068.

The progression should be 4, 5.6, 8, 11, 16, 22 . My in-darkroom table just uses slightly more practical numbers, and as for this progression, the ratio between steps is 1.40, 1.42, 1.375, 1.45, 1.375 - standard deviation of 0.032 I believe.

mrosenlof · Mar 21, 2022

I use powers of two also. 2,4,8,16,32,64

Yes, the exposure I end up with is often between one of those pairs, but I can guestimate where in that gap I want to go next.

Sirius Glass · Mar 21, 2022

Sirius Glass said:
Rather than Fibinacci series, use powers of 2.

mrosenlof said:
I use powers of two also. 2,4,8,16,32,64

Yes, the exposure I end up with is often between one of those pairs, but I can guestimate where in that gap I want to go next.

Also 4, 5.6, 8, 11, 16, 22, 32, 45, 64, 90, 128. 180 ...

Fibonacci sequence for f-stop test strips

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