# Harmonic limit

In just intonation, the ** p-limit** or

**consists of ratios of integers whose prime factors are no larger than**

*p*-prime-limit*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.

## 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.