Generator: Difference between revisions

Line 1:

~~The term~~ '''generator''' ~~has multiple senses~~.

A '''generator''' is an interval which is stacked repeatedly to create pitches in a [[tuning system]] or a [[scale]].

~~== Generators in MOSes ==~~

~~{{Main|Periods and generators}}~~

In [[MOS]] and [[rank]]-2 temperament contexts, the '''generator''' of a MOS or a rank-2 temperament is an interval that you stack up and reduce by the [[period]] of the MOS to construct the MOS pattern within each period. Along with the [[period]], it is one of two defining intervals of the [[MOS]]. For example:

* In diatonic (LLLsLLs), the perfect fifth is a generator: stacking 6 fifths up from the tonic and reducing by the octave produces the pattern LLLsLLs, the Lydian mode. Note that the perfect fourth and the perfect twelfth are also generators.

* One example for a MOS with multiple periods per octave: for [[pajara]][10] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period.

== ~~Generators in math and JI subgroups~~ ==

See [[Periods and generators]] for a beginner-level introduction.

A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] (such as a [[JI subgroup]], a [[regular temperament]] based on ~~a JI subgroup, or any [[MOS]]-based harmony~~) is a subset of the elements of the group which is not contained in any [[Wikipedia: Subgroup|proper subgroup]], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a '''generator'''.

== Mathematical definition ==

A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] (such as a [[JI subgroup]] or a [[regular temperament]] based on it) is a subset of the elements of the group which is not contained in any [[Wikipedia: Subgroup|proper subgroup]], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a '''generator'''.

A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2, 3, 5/3} are minimal generating sets for the JI subgroup 2.3.5. However, {2, 3, 5, 15} is not a minimal generating set: 15 = 3 · 5 so we can take out 15 from this generating set.

@@ Line 1: / Line 1: @@
-The term '''generator''' has multiple senses.
+A '''generator''' is an interval which is stacked repeatedly to create pitches in a [[tuning system]] or a [[scale]].
-== Generators in MOSes ==
-{{Main|Periods and generators}}
 In [[MOS]] and [[rank]]-2 temperament contexts, the '''generator''' of a MOS or a rank-2 temperament is an interval that you stack up and reduce by the [[period]] of the MOS to construct the MOS pattern within each period. Along with the [[period]], it is one of two defining intervals of the [[MOS]]. For example:
 * In diatonic (LLLsLLs), the perfect fifth is a generator: stacking 6 fifths up from the tonic and reducing by the octave produces the pattern LLLsLLs, the Lydian mode. Note that the perfect fourth and the perfect twelfth are also generators.
 * One example for a MOS with multiple periods per octave: for [[pajara]][10] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period.
-== Generators in math and JI subgroups ==
+See [[Periods and generators]] for a beginner-level introduction.
-A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] (such as a [[JI subgroup]], a [[regular temperament]] based on a JI subgroup, or any [[MOS]]-based harmony) is a subset of the elements of the group which is not contained in any [[Wikipedia: Subgroup|proper subgroup]], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a '''generator'''.
+== Mathematical definition ==
+A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] (such as a [[JI subgroup]] or a [[regular temperament]] based on it) is a subset of the elements of the group which is not contained in any [[Wikipedia: Subgroup|proper subgroup]], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a '''generator'''.
 A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2, 3, 5/3} are minimal generating sets for the JI subgroup 2.3.5. However, {2, 3, 5, 15} is not a minimal generating set: 15 = 3 · 5 so we can take out 15 from this generating set.