Math behind the scale

By Masayoshi Hayashi for The Large Format Page

It's simple.

Define 
	delta = delta on rail
	delta' = delta on ring
	D = delta'/delta
	N = min. f-number required
	c = circle of confusion
	M = magnification
	Av = Apeture value in APEX notation
	
	
	Givens					Goal

N = delta/2/c/(1+M)		Conversion from delta to delta'
D = delta'/delta	
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delta' = D*N*2*c*(1+M)

Basically that's it.
Because I want to calibrate scales in 1/3 steps, using APEX notation, 

Av = log(N^2)/log(2)
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delta' = D*N*2*c*(1+M)
N = 10^(1/2*log(2))*Av
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delta' = D*10^(1/2*log(2))*Av*2*c*(1+M)

For non-macro work, M is much less than 1, so making M ~ 0, giving:

delta' = D*10^(1/2*log(2))*Av*2*c where Av = { 1/3 stop increments }

This equation is used in making the scale. 
It is strictly correct at infinity focus (M = 0).

For arbitrary N and non-zero M, 

N(M) = delta/2/c/(1+M)

while

N(M=0) = delta/2/c

so

N(M) = 1/(1+M)*N(M=0)	---- (1)

This equation accounts for the facts explained in Good things to know section:

The min. f-numbers calibrated are correct at magnification M = 0, infinity focus. However, at any magnification, the min. f-number 
required for a given delta will be less than that at infinity focus. Therefore you're always guaranteed to get min. dof for a given
delta at any magnification.
At magnification M = 1 (life-size, 1:1), the min. f-numbers N on scale are half. At any magnification in between 0 and 1 (infinity 
focus), the min. f-number required will fall in between the f-number scales and their corresponding halves. For an arbitrary N, use 
N(M) = 1/(1+M)*N(M=0) to figure out exact min. f-number required at M. For example, at M = 0.5, N(M=0.5) = 2/3*N(M=0). So multiply the 
f-numbers on scale by 2/3.

To prove backing off 2 stops on the scale is equivalent to delta/2,

	Givens				Goal

(1)				backing off 2 stops on the scale is equivalent to delta/2	
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				N(M)/2 is equivalent to delta/2
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Using (1) and dividing both sides by two,

N(M)/2 = 1/(1+M)*N(M=0)/2 = 1/(1+M)*delta/2/c/2 = const*delta/2