Harmonic limit
In just intonation, the p-limit or p-prime-limit consists of ratios of integers whose prime factors are no larger than p.
A positive rational number q belongs to the p-limit for a given prime number p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. In math, such a number is known as a p-smooth number. An interval does not need to contain p as a factor to be considered within the p-limit. For instance, 3/2 is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a p in it is not necessarily within the p-limit. 23/13 is not within the 13-limit, since 23 is a prime number higher than 13.
All prime limits are infinite sets, and except for the 2-limit, all prime limits are still infinite even if we restrict consideration to a single octave.
Prime limits as subgroups
Prime limits are essentially special cases of subgroups that include all primes up to the limit rather than skipping any. For any prime number p, the set of all rational numbers in the p-limit defines a finitely generated free abelian group The rank of this group is equal to π (p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the 7-limit is 4, as it is generated by 2, 3, 5 and 7. In many cases, it is often more useful to speak of subgroups of the prime-limit, rather than the full limit, and this becomes increasingly true for higher limits as the number of useful temperaments with a good approximation of full limits dwindles, and for that purpose, the term "p-horizon" can be used to refer to an entire umbrella of subgroups encompassed by the p-limit.
Harmonic class
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.