Generator: Difference between revisions
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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 [[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. | * 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 (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]][10] (ssLssssLss), the perfect fifth (a tempered [[3/2]]) is a generator and the half-octave is the period. | ||
== Generators in math and JI subgroups == | == Generators in math and JI subgroups == | ||
A [http://en.wikipedia.org/wiki/Generating_set_of_a_group set of '''generators'''], or '''generating set''', 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'''], or '''generating set''', 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 '''basis''' (plural ''bases'') is a minimal generating set, i.e. a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2 | A '''basis''' (plural ''bases'') is a minimal generating set, i.e. a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2, 3, 5/3} are bases for the JI subgroup 2.3.5. However, {2, 3, 5, 15} is not a basis: 15 = 3*5 so we can take out 15 from this generating set. | ||
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 | 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 | ||