Harmonic limit: Difference between revisions

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In [[just intonation]], the '''''p''-limit''' or '''''p''-prime-limit''' consists of ~~the ratios~~ of [[~~Wikipedia: Smooth number~~|~~''p''-smooth numbers~~]]~~, where a ''p''-smooth number is an integer with prime factors~~ no larger than ''p''.

In [[just intonation]], the '''''p''-limit''' or '''''p''-prime-limit''' consists of [[ratio]]s of integers whose [[Prime factorization|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''. 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''. In math, such a number is known as a {{w|Smooth number|''p''-smooth number}}.

For any prime number ''p'', the set of all rational numbers in the ''p''-limit defines a {{w|Free abelian group|finitely generated free abelian group}}. The [[rank]] of this group is equal to π (''p''), the {{w|Prime-counting function|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 ==

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 {{Wikipedia|Limit (music)}}
-In [[just intonation]], the '''''p''-limit''' or '''''p''-prime-limit''' consists of the ratios of [[Wikipedia: Smooth number|''p''-smooth numbers]], where a ''p''-smooth number is an integer with prime factors no larger than ''p''.
+In [[just intonation]], the '''''p''-limit''' or '''''p''-prime-limit''' consists of [[ratio]]s of integers whose [[Prime factorization|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''. 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''. In math, such a number is known as a {{w|Smooth number|''p''-smooth number}}.
+For any prime number ''p'', the set of all rational numbers in the ''p''-limit defines a {{w|Free abelian group|finitely generated free abelian group}}. The [[rank]] of this group is equal to π (''p''), the {{w|Prime-counting function|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 ==