Generator: Difference between revisions
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A '''generator''' is an interval which is stacked repeatedly to create pitches in a [[tuning system]] or a [[scale]]. | {{Expert|Periods and generators}} | ||
A '''generator''', also called '''formal fifth'''<ref>Op de Coul, E.F. [https://www.huygens-fokker.org/scala/help.htm Scala help.]</ref>, is an interval which is stacked repeatedly to create pitches in a [[tuning system]] or a [[scale]]. | |||
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 [[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 [[8L 2s|jaric]] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period. | * In [[8L 2s|jaric]] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period. | ||
== Mathematical definition == | == Mathematical definition == | ||
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=== 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, forming a [[basis]], which are typically the literal prime numbers up to a given prime limit. These two example converge when we seek generators for the [[abstract regular temperament|abstract temperament]] rather than any particular tuning of it. | 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, forming a [[basis]], which are typically the literal prime numbers up to a given prime limit. These two example converge when we seek generators for the [[abstract regular temperament|abstract temperament]] rather than any particular tuning of it. | ||
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== See also == | == See also == | ||
* [[Wikipedia: Generating set of a group]] | * [[Wikipedia: Generating set of a group]] | ||
== References == | |||
<references/> | |||
[[Category:Generator| ]] <!-- main article --> | [[Category:Generator| ]] <!-- main article --> | ||