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{{Wikipedia|Limit (music)}}
{{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''') is 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 does not 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'' does not 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 ==
== Prime limits as subgroups ==
Prime limits are essentially special cases of [[just intonation subgroup|subgroups]] that include all primes up to the limit rather than skipping any. For any prime number ''p'', the set of all rational numbers in the ''p''-limit defines a [[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. In many cases, it is often more useful to speak of subgroups of the prime-limit, rather than the full limit, and this becomes increasingly true for higher limits as the number of useful temperaments with a good approximation of full limits dwindles, and for that purpose, the term "''p''-horizon" can be used to refer to an entire umbrella of subgroups encompassed by the ''p''-limit.
Prime limits are essentially [[just intonation subgroups]] that do not skip any primes. For any prime number ''p'', the ''p''-limit creates a well-defined mathematical structure, called ''{{w|free abelian group}}''. This structure has a dimension (or rank) equal to the number of prime numbers less than or equal to ''p''. For example, the [[7-limit]] works with intervals built from the primes 2, 3, 5, and 7, so it has 4 dimensions.


== Harmonic class ==
Often, composers and theorists find it more practical to work with smaller subsets of a prime limit rather than using all possible intervals within that limit. This becomes increasingly important for higher limits, as the number of practical tuning systems that can reasonably approximate the full set of intervals diminishes. For discussing these subsets that exist within a larger prime limit, some theorists use the term "''p''-horizon" to refer to the collection of all possible subsets within a ''p''-limit.
While harmonic limit encompasses all ratios up to a given prime, '''harmonic class''' ('''HC''') classifies JI ratios based only based on the ''highest'' prime they contain in either the numerator or denominator. Equivalently, it is all of the intervals of a prime limit that are not found in a lower prime limit.


A ratio belongs to harmonic class n (HCn) if and only if n is the highest prime number found in its factorization. For example:
== Proper harmonic limit ==
* [[7/4]] is HC7 because 7 is the highest prime in its factorization.
While harmonic limit encompasses all ratios up to a given prime, '''proper harmonic limit''' classifies JI ratios based only based on the ''highest'' prime they contain in either the numerator or denominator. Equivalently, it is all of the intervals of a prime limit that are not found in a lower prime limit. It has been called '''harmonic class''' or '''HC'''.
* [[5/4]] is HC5, not HC7, even though it's within the 7-limit.
* [[9/7]] is HC7 because the highest prime is 7 (since 9 = 3<sup>2</sup>).


This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime. Unlike harmonic limits, harmonic classes are mutually exclusive categories.
A ratio belongs to the proper ''p''-prime limit if and only if ''p'' is the highest prime number found in its factorization. For example:
* [[7/4]] is proper 7-limit because 7 is the highest prime in its factorization.
* [[5/4]] is proper 5-limit, not proper 7-limit, even though it's within the 7-limit.
* [[9/7]] is proper 7-limit because the highest prime is 7 (since {{nowrap| 9 {{=}} 3<sup>2</sup> }}).
 
This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime. Unlike regular harmonic limits, proper harmonic limits are mutually exclusive categories.


== Alternative classification systems ==
== Alternative classification systems ==
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Various alternative classification systems exist for characterizing intervals, such as:
Various alternative classification systems exist for characterizing intervals, such as:
* [[Odd-limit]] classifies intervals based on the complexity of the ratio itself.
* [[Odd-limit]] classifies intervals based on the complexity of the ratio itself.
* [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean interval naming]] characterizes just intervals by their deviation from basic Pythagorean intervals.
* [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming]] characterizes just intervals by their deviation from basic Pythagorean intervals.
* [[Overtone scales]] and [[primodality]] classify scales and chords based on their relative position in the harmonic series.
* [[Overtone scales]] and [[primodality]] classify scales and chords based on their relative position in the harmonic series.


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== See also ==
== See also ==
* [[Harmonic class]]
* [[Odd limit]]
* [[Odd limit]]
* [[Cubic and octahedral limits]]
* [[Prime minimum]]
* [[Prime minimum]]
* [[Harmonic class]]
* [[Wikipedia: Størmer's theorem]]
* [[Wikipedia: Størmer's theorem]]


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