Harmonic limit: Difference between revisions
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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''. | 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''. In math, such a number is known as a {{w|Smooth number|''p''-smooth number}}. | 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}}. 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. | ||
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. | 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. | ||
Revision as of 10:22, 29 September 2023
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.
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.