Nope, it can get anywhere in the western musical system.
A triad 1 3 5 can become 7 3 5, then 7 2 5, which is a 5th. Once you can move a 5th you can move anywhere because it's a move of 7 in the integers modulo 12 (numbering all the notes chromatically) which will get you back round to 0 in 12 moves.
Maj -> Min is only one note change, as is moving inversions or adding or removing a 4th note.
Comments
Nope, it can get anywhere in the western musical system.
A triad 1 3 5 can become 7 3 5, then 7 2 5, which is a 5th. Once you can move a 5th you can move anywhere because it's a move of 7 in the integers modulo 12 (numbering all the notes chromatically) which will get you back round to 0 in 12 moves.
Maj -> Min is only one note change, as is moving inversions or adding or removing a 4th note.