Harmonic limit: Difference between revisions

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| en = Harmonic limit

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A positive rational number q belongs to the '''p-limit''', called the '''p harmonic''' or '''prime limit''', for a given [[~~prime_number|~~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 [~~http~~:~~//en.wikipedia.org/wiki/Free_abelian_group~~ finitely generated free abelian group]. The rank of this group is equal to pi(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. Another way to express the p-limit is that it consists of the ratios of [~~http~~:~~//en.wikipedia.org/wiki/Smooth_number~~ p-smooth numbers], where a p-smooth number is an integer with prime factors no larger than p.

A positive rational number ''q'' belongs to the '''''p''-limit''', called the '''''p'' harmonic''' or '''prime 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 [[Wikipedia: Freeabeliangroup|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. Another way to express the ''p''-limit is that it consists of the ratios of [[Wikipedia: Smooth number|''p''-smooth numbers]], where a ''p''-smooth number is an integer with prime factors no larger than ''p''.

== List of small p-limits ==

== List of small ''p''-limits ==

With increasing limits, the tonal space becomes more dense.

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== See also ==

* [[Odd_limit|Odd limit]]

* [[~~Harmonic_Class_(HC)~~|Harmonic Class (HC)]]

* [[Harmonic Class|Harmonic Class (HC)]]

* [[~~consistency|consistency~~]]

* [[Consistency]]

* [~~http~~:~~//en.wikipedia.org/wiki/Limit_%28music%29~~ Limit (music) ~~- Wikipedia~~] (covers also the distinction between odd-limit and prime-limit)

* [[Wikipedia: Limit (music)]] (covers also the distinction between odd-limit and prime-limit)

* [~~http~~:~~//en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem~~ Størmer's theorem ~~- Wikipedia~~]

* [[Wikipedia: Størmer's theorem]]

[[Category:~~Theory]]~~

[[Category:Regular temperament theory]]

~~[[Category:Definition~~]]

[[Category:Prime limit| ]]

[[Category:Limit]]

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-A positive rational number q belongs to the '''p-limit''', called the '''p harmonic''' or '''prime limit''', for a given [[prime_number|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 [http://en.wikipedia.org/wiki/Free_abelian_group finitely generated free abelian group]. The rank of this group is equal to pi(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. Another way to express the p-limit is that it consists of the ratios of [http://en.wikipedia.org/wiki/Smooth_number p-smooth numbers], where a p-smooth number is an integer with prime factors no larger than p.
+A positive rational number ''q'' belongs to the '''''p''-limit''', called the '''''p'' harmonic''' or '''prime 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 [[Wikipedia: Freeabeliangroup|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. Another way to express the ''p''-limit is that it consists of the ratios of [[Wikipedia: Smooth number|''p''-smooth numbers]], where a ''p''-smooth number is an integer with prime factors no larger than ''p''.
-== List of small p-limits ==
+== List of small ''p''-limits ==
 With increasing limits, the tonal space becomes more dense.
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 == See also ==
 * [[Odd_limit|Odd limit]]
-* [[Harmonic_Class_(HC)|Harmonic Class (HC)]]
+* [[Harmonic Class|Harmonic Class (HC)]]
-* [[consistency|consistency]]
+* [[Consistency]]
-* [http://en.wikipedia.org/wiki/Limit_%28music%29 Limit (music) - Wikipedia] (covers also the distinction between odd-limit and prime-limit)
+* [[Wikipedia: Limit (music)]] (covers also the distinction between odd-limit and prime-limit)
-* [http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem Størmer's theorem - Wikipedia]
+* [[Wikipedia: Størmer's theorem]]
-[[Category:Theory]]
+[[Category:Regular temperament theory]]
-[[Category:Definition]]
 [[Category:Prime limit| ]] <!-- main article -->
 [[Category:Limit]]