Harmonic limit: Difference between revisions
Jump to navigation
Jump to search
m Categories |
Rework intro |
||
| Line 5: | Line 5: | ||
{{Wikipedia|Limit (music)}} | {{Wikipedia|Limit (music)}} | ||
In [[just intonation]], the '''''p''-limit''' or '''''p''-prime-limit''' consists of | 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 | 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 == | == Individual pages of ''p''-limit JI == | ||
Revision as of 06:40, 10 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.
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.