# Harmonic limit

(Redirected from Prime limit)

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

**consists of the ratios of**

*p*-prime-limit*p*-smooth numbers, where a

*p*-smooth number is an integer with prime factors 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*. 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.