Inthar at 21:44, 27 December 2025

2025-12-27T21:44:30Z

← Older revision		Revision as of 21:44, 27 December 2025
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	== Finite temperaments ==		== Finite temperaments ==
	Any ~~periodic~~ scale other than an equal tuning, if treated as a complete set of notes, is an irregular temperament. They may be produced by ''detempering'' a tempered tuning system such as an edo, or adjusting the notes in the edo to fit some non-equal structure. One way is to choose a MOS that the edo supports, and another is to choose one or more (but not infinite) JI interpretations of each note in the edo. Detemperaments do not allow free transposition, because there is always at least one interval category that has two different "qualities". An example of a detemperament is the detempering of [[Equiheptatonic\|7edo]] to the equable [[diatonic]], 18:20:22:24:27:30:33:36.		Any scale other than an equal tuning, if treated as a complete set of notes, is an irregular temperament. They may be produced by ''detempering'' a tempered tuning system such as an edo, or adjusting the notes in the edo to fit some non-equal structure. One way is to choose a MOS that the edo supports, and another is to choose one or more (but not infinite) JI interpretations of each note in the edo. Detemperaments do not allow free transposition, because there is always at least one interval category that has two different "qualities". An example of a detemperament is the detempering of [[Equiheptatonic\|7edo]] to the equable [[diatonic]], 18:20:22:24:27:30:33:36.

	=== Well temperament ===		=== Well temperament ===
	A well temperament is a scale that closely approximates an edo. They can be considered detemperaments or detunings of the edo in question, and were the focus of much of historical European tuning theory, as opposed to the modern concept of a [[regular temperament]]. An example of a well temperament might be constructed using six slightly narrow fifths of 698c and six slightly wide fifths of 702c, so that there are 12 notes in an octave that are slightly unequally spaced. Common [[equipentatonic]] and [[equiheptatonic]] scales may be considered 5- and 7-well temperaments, and thus a 12-well temperament may be called "[[12edo\|equidodecatonic]]".		A well temperament is a scale that closely approximates an edo. They can be considered detemperaments or detunings of the edo in question, and were the focus of much of historical European tuning theory, as opposed to the modern concept of a [[regular temperament]]. An example of a well temperament might be constructed using six slightly narrow fifths of 698c and six slightly wide fifths of 702c, so that there are 12 notes in an octave that are slightly unequally spaced. Common [[equipentatonic]] and [[equiheptatonic]] scales may be considered 5- and 7-well temperaments, and thus a 12-well temperament may be called "[[12edo\|equidodecatonic]]".

Vector: Created page with "An '''irregular temperament''' is a temperament that is not a regular temperament. Specifically, an irregular temperament either does not allow free transposition, or its JI interpretation is otherwise not consistent (for example, a temperament where stacking 3/2 twice does not result in 9/8). Any temperament with a finite set of notes is an irregular temperament, and other than the equal temperaments, any temperament with a finite set of notes ''within the octav..."

2025-12-13T02:29:46Z

Created page with "An '''irregular temperament''' is a temperament that is not a regular temperament. Specifically, an irregular temperament either does not allow free transposition, or its JI interpretation is otherwise not consistent (for example, a temperament where stacking 3/2 twice does not result in 9/8). Any temperament with a finite set of notes is an irregular temperament, and other than the equal temperaments, any temperament with a finite set of notes ''within the octav..."

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An '''irregular temperament''' is a temperament that is not a [[regular temperament]]. Specifically, an irregular temperament either does not allow free transposition, or its JI interpretation is otherwise not consistent (for example, a temperament where stacking 3/2 twice does not result in [[9/8]]). Any temperament with a finite set of notes is an irregular temperament, and other than the equal temperaments, any temperament with a finite set of notes ''within the octave'' is an irregular temperament.

== Direct approximation ==
Using direct approximation to temper intervals to an [[EDO|edo]] creates an irregular temperament. For example, in [[12edo]], the closest step to [[Classical major third|5/4]] is 4 steps, and the closest step to [[Magic|3125/2048]] = ((5/4)^5)/2 is 7 steps. If these approximations are used, then the temperament is irregular; stacking ~5/4 5 times and octave-reducing yields 8 steps, which is not ~3125/2048. The use of an edo as a [[regular temperament]] (that is, an [[equal temperament]]) is facilitated by vals.

== Finite temperaments ==
Any periodic scale other than an equal tuning, if treated as a complete set of notes, is an irregular temperament. They may be produced by ''detempering'' a tempered tuning system such as an edo, or adjusting the notes in the edo to fit some non-equal structure. One way is to choose a MOS that the edo supports, and another is to choose one or more (but not infinite) JI interpretations of each note in the edo. Detemperaments do not allow free transposition, because there is always at least one interval category that has two different "qualities". An example of a detemperament is the detempering of [[Equiheptatonic|7edo]] to the equable [[diatonic]], 18:20:22:24:27:30:33:36.

=== Well temperament ===
A well temperament is a scale that closely approximates an edo. They can be considered detemperaments or detunings of the edo in question, and were the focus of much of historical European tuning theory, as opposed to the modern concept of a [[regular temperament]]. An example of a well temperament might be constructed using six slightly narrow fifths of 698c and six slightly wide fifths of 702c, so that there are 12 notes in an octave that are slightly unequally spaced. Common [[equipentatonic]] and [[equiheptatonic]] scales may be considered 5- and 7-well temperaments, and thus a 12-well temperament may be called "[[12edo|equidodecatonic]]".

Irregular temperament - Revision history

Inthar at 21:44, 27 December 2025