Harmonic class (HC) classifies JI ratios based on the highest prime they contain in either the numerator or denominator. HC tells us that the ratio must contain the prime of whatever class it falls into and will contain no higher prime.
Harmonic limit refers to the highest prime allowed in the ratios and includes all simpler ratios of lower limit, but HC only contains those which contain that prime. For example, while 5/4 falls into the 7-limit, it is not considered a HC7 interval because the highest prime it contains is 5 not 7. Therefore, HC7 must contain a 7 and no higher prime. 9/7 however would be considered HC7 because 9 is not prime but rather a multiple of 3. Therefore, HC9 does not exist.
It has been criticized by some schools that the sound of JI is not well characterized by this classification system. Specifically, it is believed that each harmonic class lacks a consistent sound quality. Rather, primodality classifies intervals by their common denominator, and meanwhile, the 2.3-equivalent class may be used as an enhancement suitable for traditional JI and/or regular temperament theory.