Harmonic limit: Difference between revisions
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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 [[Wikipedia: Free abelian group|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. | 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 [[Wikipedia: Free abelian group|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. | ||
== | == Individual pages of ''p''-limit JI == | ||
{| class="wikitable center-all" | |||
|- | |||
| [[2-limit]] || [[3-limit]] || [[5-limit]] || [[7-limit]] || [[11-limit]] || [[13-limit]] | |||
|- | |||
| [[17-limit]] || [[19-limit]] || [[23-limit]] || [[29-limit]] || [[31-limit]] || [[37-limit]] | |||
|- | |||
| [[41-limit]] || [[43-limit]] || [[47-limit]] || [[53-limit]] || [[59-limit]] || [[61-limit]] | |||
|- | |||
| [[67-limit]] || [[71-limit]] || [[73-limit]] || [[79-limit]] || [[83-limit]] || [[89-limit]] | |||
|} | |||
== See also == | == See also == | ||
Revision as of 09:19, 2 January 2023
In just intonation, the p-limit or p-prime-limit consists of the ratios of 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.