Harmonic limit: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
reworked: wiki lists, links simplified, re-added some interval names for 7-limit and 11-limit examples
Line 1: Line 1:
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|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|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.


==List of small p-limits==
== List of small p-limits ==
With increasing limits, the tonal space becomes more dense.
With increasing limits, the tonal space becomes more dense.


<ul><li>[[2-limit|2-limit]] contains only multiples of the [[Octave|octave]] (2/1), see [[1edo|1edo]]</li><li>[[3-limit|3-limit]] contains [[3/2|3/2]], the [[just_perfect_fifth|perfect fifth]]</li><li>[[5-limit|5-limit]] contains [[5/4|5/4]], the perfect major third</li><li>[[7-limit|7-limit]] contains [[7/4|7/4]], the septimal subminor seventh</li><li>[[11-limit|11-limit]] contains [[11/8|11/8]], the undecimal tritone</li><li>[[13-limit|13-limit]]</li><li>[[17-limit|17-limit]]</li><li>[[19-limit|19-limit]]</li><li>[[23-limit|23-limit]]</li><li>[[29-limit|29-limit]]</li><li>[[31-limit|31-limit]]</li><li>[[37-limit|37-limit]]</li><li>[[41-limit|41-limit]]</li><li>[[43-limit|43-limit]]</li><li>[[47-limit|47-limit]]</li><li>[[53-limit|53-limit]]</li><li>[[59-limit|59-limit]]</li><li>[[61-limit|61-limit]]</li></ul>
* [[2-limit]] contains only multiples of the [[Octave|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]]
* [[37-limit]]
* [[41-limit]]
* [[43-limit]]
* [[47-limit]]
* [[53-limit]]
* [[59-limit]]
* [[61-limit]]


==See also==
== See also ==
<ul><li>[[Odd_limit|Odd limit]]</li><li>[[Harmonic_Class_(HC)|Harmonic Class (HC)]]</li><li>[[consistency|consistency]]</li><li>[http://en.wikipedia.org/wiki/Limit_%28music%29 Limit (music) - Wikipedia] (covers also the distinction between odd-limit and prime-limit)</li><li>[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem Størmer's theorem - Wikipedia]</li></ul>      [[Category:definition]]
* [[Odd_limit|Odd limit]]
* [[Harmonic_Class_(HC)|Harmonic Class (HC)]]
* [[consistency|consistency]]
* [http://en.wikipedia.org/wiki/Limit_%28music%29 Limit (music) - Wikipedia] (covers also the distinction between odd-limit and prime-limit)
* [http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem Størmer's theorem - Wikipedia]
 
[[Category:definition]]
[[Category:limit]]
[[Category:limit]]
[[Category:theory]]
[[Category:theory]]

Revision as of 22:33, 10 October 2018

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 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 p-smooth numbers, where a p-smooth number is an integer with prime factors no larger than p.

List of small p-limits

With increasing limits, the tonal space becomes more dense.

See also

Retrieved from "https://en.xen.wiki/index.php?title=Harmonic_limit&oldid=36532"