Generator: Difference between revisions

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== Generators in MOS ==

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 [[meantone]], the (flattened) perfect fifth is a generator. Note that the perfect fourth and the perfect twelfth are also generators.

* In [[meantone]], the (flattened) 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]], the perfect fifth (a tempered [[3/2]]) is a generator and the half-octave is the period.

== Generators in math ==

A [http://en.wikipedia.org/wiki/Generating_set_of_a_group set of '''generators'''] for a [http://en.wikipedia.org/wiki/Group_%28mathematics%29 group] is a subset of the elements of the group which is not contained in any [http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Abelian_group abelian group], it is called a [http://en.wikipedia.org/wiki/Finitely_generated_abelian_group finitely generated abelian group].

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 == Generators in MOS ==
 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 [[meantone]], the (flattened) perfect fifth is a generator. Note that the perfect fourth and the perfect twelfth are also generators.
+* In [[meantone]], the (flattened) 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]], the perfect fifth (a tempered [[3/2]]) is a generator and the half-octave is the period.
 == Generators in math ==
 A [http://en.wikipedia.org/wiki/Generating_set_of_a_group set of '''generators'''] for a [http://en.wikipedia.org/wiki/Group_%28mathematics%29 group] is a subset of the elements of the group which is not contained in any [http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Abelian_group abelian group], it is called a [http://en.wikipedia.org/wiki/Finitely_generated_abelian_group finitely generated abelian group].