William Lynch wrote:How would these work?

For 10 EDO I know major/minor and perfect assume a chain of 720c fifths or 5 EDO and Gai and Gao would be the other chain of fifths. What's odd here is that it seems a Gai second and a Gao second would be enharmonically equal??

P1 , g2, M2/m3, g3, M3/p4, g4/g5, P5, g5/g6, M6/m7, g7/g8, P8

Is this correct?

This looks like Cam's system, so I'll leave the details to him, but I don't see how a gai second could ever be the same as a gao second. Makes no sense whatsoever. Did you mean a gai second is the same as a gao third (in 10edo)? Quite possibly.

It seems like you've used lowercase "g" to stand for both gai and gao. That would be very confusing.

In a context like this, you could use uppercase G for gai, but in general it will be confused with the nominal G. In which case I suggest just spelling them in full, since they are only 3 letters, same as aug and dim.

An extreme fifth size and short chains like this really shows up the difference between Cam's system (pitch notation based) and mine (sound+symmetry based).

If, as in Cam's system, you base major/minor aug/dim on a single chain of fifths as follows:

... d4 d8 d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 A4 A1 A5 A2 A6 A3 A7 ... then you have the following being equal in all 5n edos

m2 = P1 = 0 c

d4 = m3 = M2 = 240 c

d5 = P4 = M3 = A2 = 480 c

m6 = P5 = A4 = A3 = 720 c

d8 = m7 = M6 = A5 = 960 c

P8 = M7 = A6 = 1200 c

P9 = A8 = A7 = 1440 c

If instead you notated the pitches as a subset of 60 edo (or as a subset of 50 edo in the case of 25edo) then the names of the intervals will all change.