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''') refers to the set of [[frequency ratio]]s that can be expressed using only [[prime numbers]] less than or equal to ''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~~}}. 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.

A frequency ratio belongs to the ''p''-limit if and only if both its numerator and denominator can be [[prime factorization|factored]] completely into prime numbers no larger than ''p'' (with positive or negative integer exponents). In mathematics, such numbers are known as {{w|Smooth number|''p''-smooth numbers}}.

~~All~~ prime ~~limits are infinite sets~~, ~~and except for~~ the [[2-limit]]~~, all~~ prime limits ~~are still infinite even~~ if we restrict consideration to a single octave.

An interval doesn't need to contain the prime ''p'' itself to be within the ''p''-limit. For example, [[3/2]] belongs to the [[13-limit]] because both 2 and 3 are smaller than 13.

Conversely, containing the prime ''p'' doesn't guarantee membership in the ''p''-limit. For instance, [[23/13]] is not within the 13-limit because 23 is a prime number larger than 13.

All prime limits contain infinitely many intervals. Even if we [[octave reduction|restrict]] our consideration to intervals within a single octave, all prime limits except the [[2-limit]] still contain infinitely many distinct ratios.

== Prime limits as subgroups ==

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 {{Wikipedia|Limit (music)}}
-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''') refers to the set of [[frequency ratio]]s that can be expressed using only [[prime numbers]] less than or equal to ''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}}. 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.
+A frequency ratio belongs to the ''p''-limit if and only if both its numerator and denominator can be [[prime factorization|factored]] completely into prime numbers no larger than ''p'' (with positive or negative integer exponents). In mathematics, such numbers are known as  {{w|Smooth number|''p''-smooth numbers}}.
-All prime limits are infinite sets, and except for the [[2-limit]], all prime limits are still infinite even if we restrict consideration to a single octave.
+An interval doesn't need to contain the prime ''p'' itself to be within the ''p''-limit. For example, [[3/2]] belongs to the [[13-limit]] because both 2 and 3 are smaller than 13.
+Conversely, containing the prime ''p'' doesn't guarantee membership in the ''p''-limit. For instance, [[23/13]] is not within the 13-limit because 23 is a prime number larger than 13.
+All prime limits contain infinitely many intervals. Even if we [[octave reduction|restrict]] our consideration to intervals within a single octave, all prime limits except the [[2-limit]] still contain infinitely many distinct ratios.
 == Prime limits as subgroups ==