Archived posting to the Leica Users Group, 2000/03/07

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Subject: [Leica] OT : Stop fractions - how long and tedious can you get?......very
From: "Garrelfs, R (Rick)" <Rick.Garrelfs@utc.rabobank.com>
Date: Tue, 7 Mar 2000 14:45:11 +0100

> 
> Marc - I'm curious...How does one calculate these "stop fractions?"
> 
> B. D.
> 

I have just thought about it a bit, and I think it is a little easier than
we might think. As someone already stated, it is a logaritm thing, and we
don't need to calculate any actual lens diameters. Here is what I think may
work:

we know that subsequent f-stop numbers are calculated by multiplying the
previous number by the square root (sqrt) of 2. That has to do with the
pi*r*r that has been mentioned before, and that is how we get the f-stop
number sequence:
1.0; 1.0*(sqrt 2)= 1.414; 1.414*(sqrt 2)= 2; 2*(sqrt 2)= 2.818.....etc.

or, for instance: 2.8xxxx = 1.0 * (sqrt 2) * (sqrt 2) * (sqrt 2) = 1.0 *
(sqrt 2)^3
(^3 meaning: to the 3rd order).

a difference of 3 f-stops equals: the original f-stop * (sqrt 2)^3
a difference of n f-stops equals: the original f-stop * (sqrt 2)^n

example: 16 = (sqrt 2)^3 * 5.6, or: a difference of 3 stops

so, if one f-stop is Y, the other X:  Y = (sqrt 2)^N * X

The thing is: we want to know N for every for every Y and X we can think of.
So, fiddling around a bit, we can get (and here comes the logaritm  (Ln is
natural logaritm)):

Y = (sqrt 2)^N * X
Y/X = (sqrt 2)^N

Ln(Y/X) = Ln{(sqrt 2)^N} 
Ln(Y) - Ln(X) = N * Ln(sqrt 2)


N = {Ln(Y) - Ln(X)} / Ln(sqrt 2)

Tataa. Flash out the pocket calculator, and:

What is the difference between a 1.4 summilux and a 1.5 Noktor?

Y = 1.5
X = 1.4

Ln(1.5) - Ln(1.4) = 0.06899;  Ln (sqrt 2) = 0,34657

difference is:   0.06899/0,34657= 0,19906, or about 0,2 f-stop.


Other example: what is the difference in f-stops in a zoom between its 4.5
and 6.3 setting?

{Ln(6.3) - Ln(4.5)}/0,34657 = 0,97085 , almost a full stop. Sounds about
right, doesn't it?

Good luck using it. If you have actually read this post up to here: I admire
you patience, hope it was worth it.


Rick Garrelfs

































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