Generator: Difference between revisions

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

== Generators in MOS ==

In MOS ~~theory~~, the '''generator''' of a MOS is an interval that you stack up and reduce by the [[period]] of the MOS to construct the MOS pattern within each period. For example:

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. For example:

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

* One example for a ~~mode of limited transposition~~: for [[pajara]] ~~(half-octave temperament)~~, the perfect fifth (a tempered [[3/2]]) is a generator and the half-octave is the period.

* 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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 The term '''generator''' has multiple senses.
 == Generators in MOS ==
-In MOS theory, the '''generator''' of a MOS is an interval that you stack up and reduce by the [[period]] of the MOS to construct the MOS pattern within each period. For example:
+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. For example:
 * In [[meantone]], the (flattened) perfect fifth is a generator. Note that the perfect fourth and the perfect twelfth are also generators.
-* One example for a mode of limited transposition: for [[pajara]] (half-octave temperament), the perfect fifth (a tempered [[3/2]]) is a generator and the half-octave is the period.
+* 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].