Tuning system - Revision history

Overthink: /* Open and closed systems */ mention regular temperaments as open systems

2026-02-15T05:05:04Z

Open and closed systems: mention regular temperaments as open systems

← Older revision		Revision as of 05:05, 15 February 2026
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	== Open and closed systems ==		== Open and closed systems ==
	Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all ~~2097151~~ notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which ~~puts~~ no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.		Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2,097,151 notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which places no limit on how high or low the range of tones extends. From a practical point of view, the MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.

	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI (also known as [[Pythagorean tuning]]), a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of [[Erv Wilson]].		Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI (also known as [[Pythagorean tuning]]), a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Regular temperaments (other than equal temperaments) are also open systems with infinitely many pitches in a finite range. Another example would be any of the golden horagrams of [[Erv Wilson]].

	Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.		Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.

Overthink: /* Open and closed systems */ metion pythagorean tuning

2026-01-17T05:44:37Z

Open and closed systems: metion pythagorean tuning

← Older revision		Revision as of 05:44, 17 January 2026
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	Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.		Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.

	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of [[Erv Wilson]].		Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI (also known as [[Pythagorean tuning]]), a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of [[Erv Wilson]].

	Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.		Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.

Lériendil: Protected "Tuning system": deemed priority ([Edit=Allow only autoconfirmed users] (indefinite) [Move=Allow only autoconfirmed users] (indefinite))

2025-04-22T04:51:59Z

Protected "Tuning system": deemed priority ([Edit=Allow only autoconfirmed users] (indefinite) [Move=Allow only autoconfirmed users] (indefinite))

← Older revision	Revision as of 04:51, 22 April 2025
(No difference)

FloraC: /* Concrete and abstract systems */ rework and focus on the principles since the examples were too specific and anecdotal

2025-03-24T15:12:58Z

Concrete and abstract systems: rework and focus on the principles since the examples were too specific and anecdotal

← Older revision		Revision as of 15:12, 24 March 2025
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	== Concrete and abstract systems ==		== Concrete and abstract systems ==
	A very basic distinction among tuning systems is between concrete and abstract systems. A concrete system defines exact intervals between all of its possible notes. Examples include untempered just intonation and equal tunings. An abstract system is a set of characteristics or rules ~~that a concrete tuning can have~~. ~~For example~~, ~~any regular tuning that follows the rule that ~(3/2)<sup>4</sup> = ~5/1 is~~ a tuning of ~~[[meantone]] temperament; if you change the rule to ~(3/2)<sup>4</sup> = ~36/7, then the same tuning can be treated as a~~ tuning of ~~[[archy]] temperament~~.		A very basic distinction among tuning systems is between concrete and abstract systems. A concrete tuning system defines exact intervals between all of its possible notes. Examples include untempered just intonation and equal tunings. An abstract tuning system has at least one variable interval, often constrained to preserve certain important properties. In other words, an abstract system is a set of concrete systems that have some characteristics or follow some rules. Depending on the characteristics or rules being considered, a concrete tuning system may be part of multiple abstract tuning systems. The most notable class of abstract systems is the regular temperaments.

	Analogous definitions exist for [[scale]]s.		Analogous definitions exist for [[scale]]s.

FloraC: /* Concrete and abstract systems */ cleanup

2025-03-24T14:37:18Z

Concrete and abstract systems: cleanup

← Older revision		Revision as of 14:37, 24 March 2025
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	== Concrete and abstract systems ==		== Concrete and abstract systems ==
	A very basic distinction among tuning systems is between concrete and abstract systems. A ~~'''~~concrete system~~'''~~ defines exact intervals between all of its possible notes. Examples include untempered just intonation and equal tunings. An "abstract ~~tuning~~ system~~" doesn't refer to a concrete tuning, but to~~ a set of "characteristics" or "rules" that a concrete tuning can have. For example, any regular tuning that follows the rule that ~3/2^4 = ~5/1 is a tuning of [[meantone]] temperament; if you change the rule to ~3/2^4 = ~36/7, then the same tuning can be treated as a tuning of [[archy]] temperament.		A very basic distinction among tuning systems is between concrete and abstract systems. A concrete system defines exact intervals between all of its possible notes. Examples include untempered just intonation and equal tunings. An abstract system is a set of characteristics or rules that a concrete tuning can have. For example, any regular tuning that follows the rule that ~(3/2)<sup>4</sup> = ~5/1 is a tuning of [[meantone]] temperament; if you change the rule to ~(3/2)<sup>4</sup> = ~36/7, then the same tuning can be treated as a tuning of [[archy]] temperament.

	Analogous definitions exist for [[scale]]s.		Analogous definitions exist for [[scale]]s.

	== Open and closed systems ==		== Open and closed systems ==

FloraC: +concrete and abstract systems (copypasted VectorGraphic's addition in the regular temperament article)

2025-03-24T14:34:26Z

+concrete and abstract systems (copypasted VectorGraphic's addition in the regular temperament article)

← Older revision		Revision as of 14:34, 24 March 2025
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	There are many schools of thought regarding what tuning systems are most useful and how they should be generated. With each of those perspectives comes a different method to generate them. Some of the most common systems are [[just intonation]], [[regular temperament]]s, [[circulating temperament]]s, and [[equal-step tuning\|equal divisions]]. Another source of tunings is those used historically and by various cultures throughout the world. Some of those tunings include Indonesian [[Wikipedia: Pelog\|Pelog]] and {{w\|Slendro}}, the {{w\|Shruti (music)\|Indian Shruti}}, [[Arabic, Turkish, Persian music\|Middle Eastern maqamat]], and [[meantone\|historical meantone tunings]].		There are many schools of thought regarding what tuning systems are most useful and how they should be generated. With each of those perspectives comes a different method to generate them. Some of the most common systems are [[just intonation]], [[regular temperament]]s, [[circulating temperament]]s, and [[equal-step tuning\|equal divisions]]. Another source of tunings is those used historically and by various cultures throughout the world. Some of those tunings include Indonesian [[Wikipedia: Pelog\|Pelog]] and {{w\|Slendro}}, the {{w\|Shruti (music)\|Indian Shruti}}, [[Arabic, Turkish, Persian music\|Middle Eastern maqamat]], and [[meantone\|historical meantone tunings]].

			== Concrete and abstract systems ==
			A very basic distinction among tuning systems is between concrete and abstract systems. A '''concrete system''' defines exact intervals between all of its possible notes. Examples include untempered just intonation and equal tunings. An "abstract tuning system" doesn't refer to a concrete tuning, but to a set of "characteristics" or "rules" that a concrete tuning can have. For example, any regular tuning that follows the rule that ~3/2^4 = ~5/1 is a tuning of [[meantone]] temperament; if you change the rule to ~3/2^4 = ~36/7, then the same tuning can be treated as a tuning of [[archy]] temperament.

			Analogous definitions exist for [[scale]]s.

	== Open and closed systems ==		== Open and closed systems ==
	[[Gene Ward Smith]] ~~considered the most basic of distinctions among tuning systems to be~~ between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.		Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.

	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of [[Erv Wilson]].		Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of [[Erv Wilson]].

	Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a [[regular temperament]].		Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.

	== Where to next ==		== Where to next ==

AraMax: Added a new language

2024-10-27T11:08:21Z

Added a new language

← Older revision		Revision as of 11:08, 27 October 2024
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	\| es =		\| es =
	\| ja = 音律		\| ja = 音律
			\| ro = Sistem de acordare
	}}		}}
	{{Wikipedia\|Musical tuning}}		{{Wikipedia\|Musical tuning}}

BudjarnLambeth: /* See also */

2024-10-15T10:28:53Z

Dummy index: add ja interwiki

2024-10-06T14:28:19Z

add ja interwiki

← Older revision		Revision as of 14:28, 6 October 2024
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			{{interwiki
			\| de =
			\| en = Tuning system
			\| es =
			\| ja = 音律
			}}
	{{Wikipedia\|Musical tuning}}		{{Wikipedia\|Musical tuning}}
	A musical '''tuning system''' or '''interval system''' (commonly referred to as a '''tuning''' or '''system''') is a set of rules or algorithms – including enumeration – that gives all the notes or [[interval]]s theoretically available to a composer. The qualification ''theoretically'' is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system, though it could involve the decision of what fundamental frequencies the notes of instruments will be "tuned to."		A musical '''tuning system''' or '''interval system''' (commonly referred to as a '''tuning''' or '''system''') is a set of rules or algorithms – including enumeration – that gives all the notes or [[interval]]s theoretically available to a composer. The qualification ''theoretically'' is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system, though it could involve the decision of what fundamental frequencies the notes of instruments will be "tuned to."

BudjarnLambeth: /* See also */ added link to general theory page

2024-01-07T09:30:30Z

See also: added link to general theory page

← Older revision		Revision as of 09:30, 7 January 2024
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	* [[List of approaches to musical tuning]]		* [[List of approaches to musical tuning]]
	* [[Taxonomies of xen approaches]]		* [[Taxonomies of xen approaches]]
			* [[General theory]]

	[[Category:Tuning]]		[[Category:Tuning]]

← Older revision		Revision as of 05:44, 17 January 2026
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	Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.		Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the {{w\|MIDI tuning standard}}. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.

	Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of [[Erv Wilson]].		Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is [[3-limit\|3-prime-limit]] JI (also known as [[Pythagorean tuning]]), a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of [[Erv Wilson]].

	Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.		Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.

← Older revision		Revision as of 10:28, 15 October 2024
Line 19:		Line 19:
	Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a [[regular temperament]].		Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w\|Abelian group\|abelian groups}}, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen [[1/1]]; all rational numbers in a given [[harmonic limit]]; all intervals in a [[just intonation subgroup]]; and all intervals in a [[regular temperament]].

	== ~~See also~~ ==		== Where to next ==
			* [[Xen concepts for beginners]]
	* [[List of approaches to musical tuning]]		* [[List of approaches to musical tuning]]
	* [[Taxonomies of xen approaches]]
	* [[General theory]]		* [[General theory]]

	[[Category:Tuning]]		[[Category:Tuning]]