Harmonic limit: Difference between revisions
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{{Wikipedia|Limit (music)}} | {{Wikipedia|Limit (music)}} | ||
In [[just intonation]], the '''''p''-limit''' or '''''p''-prime-limit''' 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''. | |||
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 [[Wikipedia: Free abelian group|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. | |||
== Examples of ''p''-limits == | == Examples of ''p''-limits == | ||
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== See also == | == See also == | ||
* [[Odd limit]] | * [[Odd limit]] | ||
* [[Harmonic | * [[Harmonic class]] | ||
* [[Wikipedia: Størmer's theorem]] | * [[Wikipedia: Størmer's theorem]] | ||
== External links == | == External links == | ||
* [http://tonalsoft.com/enc/l/limit.aspx | * [http://tonalsoft.com/enc/l/limit.aspx Tonalsoft Encyclopedia | ''Limit''] | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Prime limit| ]] <!-- main article --> | [[Category:Prime limit| ]] <!-- main article --> | ||
[[Category:Limit]] | [[Category:Limit]] | ||
Revision as of 08:45, 2 January 2023
In just intonation, the p-limit or p-prime-limit consists of the ratios of 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 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 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.
Examples of p-limits
With increasing limits, the tonal space becomes more dense.
- 2-limit contains only multiples of the octave (2/1), see 1edo
- 3-limit contains 3/2, the just perfect fifth
- 5-limit contains 5/4, the perfect major third
- 7-limit contains 7/4, the harmonic seventh or septimal subminor seventh
- 11-limit contains 11/8, the undecimal tritone or "Alphorn-Fa"
- 13-limit
- 17-limit
- 19-limit
- 23-limit
- 29-limit
- 31-limit
- 41-limit
- 47-limit
- 61-limit