Like 3EDO, 4EDO is already familiar as a chord of 12EDO. Again, however, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit tuning map, or val, for 4EDO goes <4 6 9 11|, all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3EDO goes with triadic harmony, mapping the 7-limit consistently, and sending 15/14, 21/20, 25/24, and 36/35 to the unison. Somewhat confusingly, 9/8 is mapped to the unison also.
By putting together the triples of integers which uniquely represent 7-limit tetrads in the 7-limit cubic lattice of tetrads with the number of 4EDO steps returned by the val <4 6 9 11| we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4EDO skeletons of another.
We can also add more kinds of chords, for instance the subminor (1-7/6-3/2-5/3) and supermajor (1-9/7-3/2-9/5) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4EDO skeletal position.