Harmonic limit: Difference between revisions

Line 9:

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~~ ==

== Individual pages of ''p''-limit JI ==

~~With increasing limits, the tonal space becomes more dense.~~

{| class="wikitable center-all"

|-

* [[2-limit]] ~~contains only multiples of the~~ [[~~octave~~]] ~~(2/1), see~~ [[~~1edo~~]]

| [[2-limit]] || [[3-limit]] || [[5-limit]] || [[7-limit]] || [[11-limit]] || [[13-limit]]

* [[3-limit]] ~~contains~~ [[~~3/2~~]]~~, the~~ [[~~just perfect fifth~~]]

|-

* [[5-limit]] ~~contains~~ [[~~5/4~~]]~~, the perfect major third~~

| [[17-limit]] || [[19-limit]] || [[23-limit]] || [[29-limit]] || [[31-limit]] || [[37-limit]]

* [[7-limit]] ~~contains~~ [[~~7/4~~]]~~, the harmonic seventh or septimal subminor seventh~~

|-

* [[11-limit]] ~~contains~~ [[~~11/8~~]]~~, the undecimal tritone or "Alphorn~~-~~Fa"~~

| [[41-limit]] || [[43-limit]] || [[47-limit]] || [[53-limit]] || [[59-limit]] || [[61-limit]]

* [[13-limit]]

|-

* [[17-limit]]

| [[67-limit]] || [[71-limit]] || [[73-limit]] || [[79-limit]] || [[83-limit]] || [[89-limit]]

* [[19-limit]]

|}

* [[23-limit]]

* [[29-limit]]

* [[31-limit]]

* [[41-limit]]

* [[47-limit]]

* [[61-limit]]

== See also ==

@@ Line 9: / Line 9: @@
 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 ==
+== Individual pages of ''p''-limit JI ==
-With increasing limits, the tonal space becomes more dense.
+{| class="wikitable center-all"
+|-
-* [[2-limit]] contains only multiples of the [[octave]] (2/1), see [[1edo]]
+| [[2-limit]] || [[3-limit]] || [[5-limit]] || [[7-limit]] || [[11-limit]] || [[13-limit]]
-* [[3-limit]] contains [[3/2]], the [[just perfect fifth]]
+|-
-* [[5-limit]] contains [[5/4]], the perfect major third
+| [[17-limit]] || [[19-limit]] || [[23-limit]] || [[29-limit]] || [[31-limit]] || [[37-limit]]
-* [[7-limit]] contains [[7/4]], the harmonic seventh or septimal subminor seventh
+|-
-* [[11-limit]] contains [[11/8]], the undecimal tritone or "Alphorn-Fa"
+| [[41-limit]] || [[43-limit]] || [[47-limit]] || [[53-limit]] || [[59-limit]] || [[61-limit]]
-* [[13-limit]]
+|-
-* [[17-limit]]
+| [[67-limit]] || [[71-limit]] || [[73-limit]] || [[79-limit]] || [[83-limit]] || [[89-limit]]
-* [[19-limit]]
+|}
-* [[23-limit]]
-* [[29-limit]]
-* [[31-limit]]
-* [[41-limit]]
-* [[47-limit]]
-* [[61-limit]]
 == See also ==