利用者:Dummy index/リミットのソースを表示

<!--{{interwiki
| de = P-Limit
| en = Harmonic limit
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{{Wikipedia|限界 (音楽)}}

[[純正律]]において、'''''p''-リミット''' または '''''p''-素数限界''' は、''p'' 以下の素因数のみを持つ分母と分子からなる分数の集合である。

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. 

== Individual pages of ''p''-limit JI ==
{| class="wikitable center-all"
|-
| [[2-limit]] || [[3-limit]] || [[5-limit]] || [[7-limit]] || [[11-limit]] || [[13-limit]]
|-
| [[17-limit]] || [[19-limit]] || [[23-limit]] || [[29-limit]] || [[31-limit]] || [[37-limit]]
|-
| [[41-limit]] || [[43-limit]] || [[47-limit]] || [[53-limit]] || [[59-limit]] || [[61-limit]]
|-
| [[67-limit]] || [[71-limit]] || [[73-limit]] || [[79-limit]] || [[83-limit]] || [[89-limit]]
|}

== See also ==
* [[Harmonic class]]
* [[アドリミット]] (奇数限界)
* [[Prime minimum]]
* [[Wikipedia: Størmer's theorem]]

== External links ==
* [http://tonalsoft.com/enc/l/limit.aspx Tonalsoft Encyclopedia | ''Limit'']