Generator: Difference between revisions
Use Expert template, add "formal fifth" as alternate term |
"formal fifth" has 12 results on the xen discord server, several of which are referring to something completely different ("a fifth tuned differently from 3/2") Not a useful term to include |
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{{Expert|Periods and generators}} | {{Expert|Periods and generators}} | ||
A '''generator''' | A '''generator''' is an interval which is [[Stacking|stacked]] repeatedly to create pitches in a [[tuning system]] or a [[scale]]. | ||
In [[ | In [[MOS scale]]s, the generator 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 can also work as 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 can also work as generators. | ||
* In [[ | * In [[2L 8s|jaric]] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period. | ||
== Mathematical definition == | == Mathematical definition == | ||
A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] is a subset of the elements of the group which is not contained in any [[Wikipedia: 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 [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a generator. | A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] is a subset of the elements of the group which is not contained in any [[Wikipedia: 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 [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia:Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a generator. | ||
A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. In [[Wikipedia: Free abelian group|free abelian groups]] such as [[just intonation subgroup]]s or its [[regular temperament]]s, this is the same thing as a [[basis]]. 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 and the set will remain a generating set. | A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. In [[Wikipedia: Free abelian group|free abelian groups]] such as [[just intonation subgroup]]s or its [[regular temperament]]s, this is the same thing as a [[basis]]. 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 and the set will remain a generating set. | ||
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== References == | == References == | ||
<references/> | <references /> | ||
[[Category:Generator| ]] <!-- | [[Category:Generator| ]] <!-- Main article --> | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:MOS scale]] | [[Category:MOS scale]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||