Generator: Difference between revisions

Line 1:

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 [[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.

== 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].

Line 9:

Line 15:

g = g1^n1 g2^n2 ... gk^nk

== Relation to music ==

=== Relation to music ===

An important example is provided by [[regular temperaments]], where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by [[just intonation subgroups]], where the generators are a finite set of positive rational numbers. These two example converge when we seek generators for the [[abstract regular temperament|abstract temperament]] rather than any particular tuning of it.

@@ Line 1: / Line 1: @@
+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 [[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.
+== 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].
@@ Line 9: / Line 15: @@
 g = g1^n1 g2^n2 ... gk^nk
-== Relation to music ==
+=== Relation to music ===
 An important example is provided by [[regular temperaments]], where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by [[just intonation subgroups]], where the generators are a finite set of positive rational numbers. These two example converge when we seek generators for the [[abstract regular temperament|abstract temperament]] rather than any particular tuning of it.