Generator - Revision history

Fredg999: Interwiki

2026-07-17T04:44:18Z

Interwiki

← Older revision		Revision as of 04:44, 17 July 2026
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			{{Interwiki
			\| de =
			\| en = Generator
			\| es =
			\| ja = ジェネレーター
			}}
	{{Expert\|Periods and generators}}		{{Expert\|Periods and generators}}
	A '''generator''' is an interval which is [[Stacking\|stacked]] repeatedly to create pitches in a [[tuning system]] or a [[scale]].		A '''generator''' is an interval which is [[Stacking\|stacked]] repeatedly to create pitches in a [[tuning system]] or a [[scale]].

Dummy index: make more connection to

2025-11-17T14:18:15Z

make more connection to

← Older revision		Revision as of 14:18, 17 November 2025
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	A '''generator''' is an interval which is [[Stacking\|stacked]] repeatedly to create pitches in a [[tuning system]] or a [[scale]].		A '''generator''' is an interval which is [[Stacking\|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. (Although this explanation sounds like it is a quotient group, it is more consistent with RTT to consider it as a rank-2 system that includes the period in the generating set without taking remainders.) 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 [[2L 8s\|jaric]] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period.		* In [[2L 8s\|jaric]] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period.

VectorGraphics: "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

2025-03-04T01:26:16Z

"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

← Older revision		Revision as of 01:26, 4 March 2025
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	{{Expert\|Periods and generators}}		{{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 [[Stacking\|stacked]] repeatedly to create pitches in a [[tuning system]] or a [[scale]].		A '''generator''' is an interval which is [[Stacking\|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:

VectorGraphics at 03:37, 21 February 2025

2025-02-21T03:37:41Z

← Older revision		Revision as of 03:37, 21 February 2025
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	{{Expert\|Periods and generators}}		{{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]].		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 [[Stacking\|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:

Zhenlige: fix typo

2025-01-17T08:11:51Z

fix typo

← Older revision		Revision as of 08:11, 17 January 2025
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	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 [[2L 8s\|jaric]] (ssLssssLss), the perfect fifth ([[~]][[3/2]]) is a generator and the half-octave is the period.

	== Mathematical definition ==		== Mathematical definition ==

ArrowHead294 at 17:45, 8 October 2024

2024-10-08T17:45:42Z

← Older revision		Revision as of 17:45, 8 October 2024
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	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]].		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 ==
	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\| ]] <!-- ~~main~~ article -->		[[Category:Generator\| ]] <!-- Main article -->
	[[Category:Math]]		[[Category:Math]]
	[[Category:MOS scale]]		[[Category:MOS scale]]
	[[Category:Terms]]		[[Category:Terms]]

Fredg999: Move period outside of link

2024-03-07T04:50:05Z

Move period outside of link

← Older revision		Revision as of 04:50, 7 March 2024
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	{{Expert\|Periods and generators}}		{{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]].		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:

Fredg999: Use Expert template, add "formal fifth" as alternate term

2024-03-07T04:48:42Z

Use Expert template, add "formal fifth" as alternate term

← Older revision		Revision as of 04:48, 7 March 2024
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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.

	~~See [[Periods and generators]] for a beginner-level introduction.~~

	== 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 -->

FloraC: /* Convention */ rewrite in a more encyclopedian tone

2023-08-05T10:49:02Z

Convention: rewrite in a more encyclopedian tone

← Older revision		Revision as of 10:49, 5 August 2023
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	In [[rank\|multirank]] systems, it is customary that generators are said as opposed to the period. Specifically, the first generator is called the period, and only the rest are called the generators.		In [[rank\|multirank]] systems, it is customary that generators are said as opposed to the period. Specifically, the first generator is called the period, and only the rest are called the generators.

	Combined with another convention~~, canonicalizing the~~ [[~~Domain basis #Canonical form~~\|~~domain basis~~]] ~~and~~ the [[~~Normal lists\|val list~~]], ~~the result is usually~~ a period of ~~2/1~~ or fraction ~~of 2/1~~. ~~If you want a nonoctave (3/1~~, ~~3/2, 7/4, ...) scale~~, ~~let's break~~ one of ~~the conventions. Not canonicalize the domain basis, or do~~ [[generator form manipulation]]~~, or directly make nonoctave scale from canonical form mapping~~.		Combined with another convention that both [[JI subgroup]]s and [[Mapping\|temperament mappings]] are documented in the [[canonical form]], temperaments commonly have a period of the [[octave]] or a fraction thereof. That, however, does not stop one from creating non-octave scales or expressing the same system in terms of other bases through [[generator form manipulation]].

	== See also ==		== See also ==

Dummy index: /* Convention */

2023-08-05T10:04:11Z

Convention

← Older revision		Revision as of 10:04, 5 August 2023
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	In [[rank\|multirank]] systems, it is customary that generators are said as opposed to the period. Specifically, the first generator is called the period, and only the rest are called the generators.		In [[rank\|multirank]] systems, it is customary that generators are said as opposed to the period. Specifically, the first generator is called the period, and only the rest are called the generators.

	Combined with another convention, canonicalizing the domain basis and the val list, the result is usually a period of 2/1 or fraction of 2/1. If you want a nonoctave (3/1, 3/2, 7/4, ...) scale, let's break one of the conventions. Not ~~canonicalizing~~ the domain basis, or directly make nonoctave scale from canonical form mapping.		Combined with another convention, canonicalizing the [[Domain basis #Canonical form\|domain basis]] and the [[Normal lists\|val list]], the result is usually a period of 2/1 or fraction of 2/1. If you want a nonoctave (3/1, 3/2, 7/4, ...) scale, let's break one of the conventions. Not canonicalize the domain basis, or do [[generator form manipulation]], or directly make nonoctave scale from canonical form mapping.

	== See also ==		== See also ==