Q:

Calculating the "35mm equivalent" circle of confusion/DoF

I'm gonna introduce my concrete example, but I'd like to know how do do this calculation in general. Say I have a middle-format camera with film height 58mm and a 90mm/3.5 lens. What is the equivalent aperture on a 24mm film with 50mm lens that has the same circle of confusion/DoF?
Edit for clarification:
Suppose I have two cameras with lenses. One has frame height of 24mm and lens with focal length of 50mm. The second camera has frame height 56mm and a lens focal length of 50mm full frame equivalent (which should be around 117mm physical, I think).
Now suppose I take a photo with the full frame camera with its aperture wide open at let's say f/1.8. A subject at focal plane will be sharp and a background at some distnce will be blurred to a disc of size which can be represented as a fraction of a frame.
Now suppose I take the exact scene with the medium format camera. But if I open the aperture to f/1.8 on that one, the blur disc of the background will be relatively larger. So what should I step my aperture down to to have the same relative blur size.
In even simpler terms, how do I calculate bokeh equivalency? If I have some bokeh on a medium camera at some particular fstop, what would be the fstop to have an equivalent bokeh on a full frame, ceteris paribus?

A:

Calculation of the equivalent lens range
Classically, to calculate the equivalent lens focal range f, you need to use the diagonals (D1, D2) of the two films/sensors to calculate the 'crop factor' k = D1 / D2.
The equivalent lens focal range is then given by the formula :f2 = f1 / k
In your case (from the clarification part of your question):
For 24x36mm film/sensor, D1 = sqrt(24²+36²) = 43.27mm
For 6x9 films, D2 = sqrt(56²+84²) = 100.96
Hence k = 0.429
So for a 6x9, the equivalent focal range to a the 50mm lens on the 24x36, is 50/0.429 = 116.67mm. This confirm your own calculation.
Calculation of the equivalent aperture
With the risk of simplification(*) : source Wikipedia:
DoF(N,D) = 2 N c D² / f²
with :
DoF: depth of field
N: aperture
c: circle of confusion in mm.
D: distance to the subject (focus distance)
f: focal length in mm
At the equivalent aperture, if it exists, the spots of the dots on both films must have diamerers (c1, c2) such as c2 = c1 / k
So DoF2(N,D) = 2 N2 c2 D² / f2² = 2 N2 c1/k D² / (f1/k)² = 2 k N2 c1 D² / f1² =  DoF1(k N2,D) = DoF1(N1,D) when N1 = k N2
This confirms, on the one hand, that there is an equivalent aperture, and gives us the formula to calculate it.
Formula giving the equivalent aperture N2 = N1/k
In your case
N1 = 1.8 and  k = 0.429 so the equivalent aperture is 1.8 /0.429 hence 4.2
Note:
*) D needs to be much greater than f.

