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 just intonation subgroups that do not skip any primes. For any prime number p, the p-limit creates a well-defined mathematical structure. This structure has a dimension (or rank) equal to the number of prime numbers less than or equal to p. For example, the 7-limit works with intervals built from the primes 2, 3, 5, and 7, so it has 4 dimensions.
Often, composers and theorists find it more practical to work with smaller subsets of a prime limit rather than using all possible intervals within that limit. This becomes increasingly important for higher limits, as the number of practical tuning systems that can reasonably approximate the full set of intervals diminishes. For discussing these subsets that exist within a larger prime limit, some theorists use the term "p-horizon" to refer to the collection of all possible subsets within a p-limit.
Harmonic class
While harmonic limit encompasses all ratios up to a given prime, harmonic class (HC) classifies JI ratios based only based on the highest prime they contain in either the numerator or denominator. Equivalently, it is all of the intervals of a prime limit that are not found in a lower prime limit.
A ratio belongs to harmonic class n (HCn) if and only if n is the highest prime number found in its factorization. For example:
- 7/4 is HC7 because 7 is the highest prime in its factorization.
- 5/4 is HC5, not HC7, even though it's within the 7-limit.
- 9/7 is HC7 because the highest prime is 7 (since 9 = 32).
This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime. Unlike harmonic limits, harmonic classes are mutually exclusive categories.
Alternative classification systems
Prime limits contain infinitely many intervals, so they cannot truly function as limits on complexity. Rather, they serve as compositional constraints that help manage choices when working with just intonation or regular temperaments.
Various alternative classification systems exist for characterizing intervals, such as:
- Odd-limit classifies intervals based on the complexity of the ratio itself.
- Pythagorean interval naming characterizes just intervals by their deviation from basic Pythagorean intervals.
- Overtone scales and primodality classify scales and chords based on their relative position in the harmonic series.