Wouldn’t these frames also look somewhat different when projected, accounting for the warmth of the bulb etc.? Can you really just look at the frames themselves?
There are conflicting reports on the temperatures of these lamps:
correlated color temperature of a white flame carbon arc lamp is 5800K
Xenon arc lamps are approximately 5400K
Which doesn’t make sense since Xenons are supposed to be bluer than the carbon arc lamps.
Here, David Mullen says this about it:
“They (Carbon Arc and Xenon projectors) should be similar (daylight-balanced, like around 5500K) but Xenons are slightly bluer I think. But since carbon arc projection is extremely rare these days, most people color-time for Xenon lamp projection. It’s a bit of an issue when projecting old Technicolor dye transfer prints with modern Xenon lamps since the print looks slightly cooler than it used to, since it was timed for carbon arc projection.”
Also mentioned somewhere is the fact that carbon arc light was notoriously inconsistent since the ends of the carbon would burn at a variable rate.
Here’s a source that gives strangely low numbers, but at least it shows that the carbon arc lamp is yellower: http://inspectapedia.com/electric/Bulb_Color_Temperatures.php
Projector bulbs, carbon arc (traditional movie) 3,900K
Xenon arclamps 4,100K
This image shows just how similar the lights are, if the 200K difference can be believed:
The difference is even less than the difference between two of those panels.
So to answer your question, it appears that you would need to adjust the color of technicolor film about 200K yellower to counteract the grading for carbon arc lamps.
The Brightness of the bulb has very little to do with how the film should look more to do with how far you need to project an image through the air. The Bigger the cinema and a larger sized screen the brighter the bulb you would need. If you were in a small pokey cinema you might have a a less bright bulb but end up with a better brighter image than a very large cinema.
Think about distance and size of projected image vs the brightness of the bulb. but projected images will never look the same as anything on a TV screen ever, because projected images pass through air via light and TV screens display straight on to the display surface without any or very little atmospheric interference.
The temperature would be relative to how far the throw is of the projected image vs the temperature of the lamp before it hits the screen material (drop off) which is either reflective (slight gain) or permeable (greater loss).