I did a search of possible transformations among the commas in the list 25/24, 28/27, 36/35, 49/48, 50/49, 64/63, 81/80, 2048/2025, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024, 225/224, 3136/3125, 5120/5103, 6144/6125, 2401/2400, 4375/4374. There is nothing special about this list, it's just one I had handy. I checked to see which pairs had the same full octahedral group invariants, where for 3^x 5^y 7^z the invariants I used were

Degree 2 x^2+y^2+z^2+x*y+x*z+y*z

Degree 4 y*x^2*z+x*y*z^2+x*y^2*z

Degree 6 y^4*z^2+y^4*x^2+2*y^3*z^3+2*y^3*z^2*x+2*y^3*z*x^2+2*y^3*x^3+y^2*z^4+2*y^2*z^3*x+4*y^2*z^2*x^2+2*y^2*z*x^3+y^2*x^4+2*y*z^3*x^2+2*y*z^2*x^3+z^4*x^2+2*z^3*x^3+z^2*x^4

This gave me the following possibilites, of pairs of commas which had the same values for all three invariants:

25/24 36/35 3 0 4

25/24 49/48 3 0 4

28/27 64/63 7 0 36

28/27 126/125 7 0 36

36/35 49/48 3 0 4

64/63 126/125 7 0 36

81/80 1029/1024 13 0 144

The last one was unexpected and particularly intriging; on checking it, I found it associated to the 7-limit transformation (of order 4)

given by 2->2, 3->7/2, 5->14/3, 7->28/5. The orbit of 81/80 under this transformation is 81/80->1029/512->2401/2000->378/625->81/80.

Hence a piece in meantone, as a 7-limit planar temperament, can be sent to something in the 1029/1024 temperament. The comma, instead of converting into another comma and then being tempered out, converts to 1029/512, which tempers to 1/2. Hence, 7-limit harmony *is* preserved! We now have two new, albeit related, kinds of transformations.

We might also note the larger groups of transformations arising from

the {28/27, 64/63, 126/125} and {25/24, 36/25, 49/48} sets of commas.