Generator: Difference between revisions

Line 5:

* 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.

== 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].

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].

If the abelian group is written additively, then if {g1, g2, ... gk} is the generating set, every element g of the group can be written

@@ Line 5: / Line 5: @@
 * 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.
 == 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].
+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].
 If the abelian group is written additively, then if {g1, g2, ... gk} is the generating set, every element g of the group can be written