Just ran some simple math, and "just intonation" seems to be highly self-conflicting.
The most fundamental frequency relationships under 'just intonation" are: C 1/1; E 5/4; G 3/2; in other words, a major 3rd interval and a perfect 5th interval.
D, F, A and B are then derived based on these two fundamental intervals from C, E and G: D = G x (3/2) / 2 = 9/8, F = 2 x C / (3/2) = 4/3, A = 2 x E / (3/2) = 5/3, B = G x 5/4 = 15/8
Now here's the problem: you can't even play a 'just intoned" D chord in the key of C. Why? the A in a D chord is supposed to be D x 3/2 = 9/8 * 3/2 = 27/16 ≈ 1.69, and our "just intoned" A of 5/3 ≈ 1.66 would simply sound flat.
To me, the core issue with "just intonation" is the arbitrary designation of the 5th harmonic to E. It just don't gel with the Pythagorean "circle of 5ths" (based on the 3rd harmonic) mathematically.
And we are not even talking about key changes yet.