A linear temperament is fully described by a generator and periodicity interval. The mapping from generators to consonances is fixed.
For example, meantones are defined by the generator approximating 3:2 and four generators approximating 5:4. The generator can have different sizes, giving 1/3-comma meantone, 1/4-comma meantone, 1/6-comma meantone, [etc.,] but they're all the same temperament class. 53-equal, however, isn't a meantone because the best 5:4 is 8 fourths rather than 4 fifths.
The difference between a generator and an equivalence interval is only a matter of formalism.