12edo - Revision history

Sintel: /* Scales */ typo'd

2026-06-07T00:01:20Z

Scales: typo'd

← Older revision		Revision as of 00:01, 7 June 2026
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	The diminished and augmented scales are also MOS scales.		The diminished and augmented scales are also MOS scales.
	* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)		* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
	* Augmented: [[3L 3s]] – ~~13131313~~ (generator = 1\12, period = 4\12)		* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)

	Other widely used scales include:		Other widely used scales include:

Sintel: /* Scales */ Wording, give more obvious examples, remove pdf link

2026-06-06T20:37:01Z

Scales: Wording, give more obvious examples, remove pdf link

← Older revision		Revision as of 20:37, 6 June 2026
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	== Scales ==		== Scales ==
	{{~~Main~~\| List of MOS scales in 12edo }}		{{See also\| List of MOS scales in 12edo }}

	The two most common 12edo ~~mos~~ scales are meantone[5] and meantone[7].		The two most common 12edo MOS scales are meantone[5] and meantone[7].
	* Diatonic ~~(meantone) 5L2s~~ 2221221 (generator = 7\12)		* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
	* Pentatonic ~~(meantone) 2L3s~~ 22323 (generator = 7~~\12)~~		* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)
	* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

	~~=== Non-mos~~ scales ===		The diminished and augmented scales are also MOS scales.
	~~Due to 12edo's dominance~~, ~~some non-mos scales are also widely used in many musical practices around the world.~~		* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
			* Augmented: [[3L 3s]] – 13131313 (generator = 1\12, period = 4\12)

	* Harmonic ~~major~~ – ~~2212132~~		Other widely used scales include:
	* ~~Melodic~~ major – ~~2212122~~		* Melodic minor – 2122221
			* Harmonic minor – 2122131
			* Harmonic major – 2212131
	* Hungarian minor – 2131131		* Hungarian minor – 2131131
	* Maqam hijaz / double harmonic major – 1312131		* Maqam hijaz / double harmonic major – 1312131
	* 5-odd-limit tonality diamond – 3112113
	~~[[File:12edo modes.pdf\|thumb]]~~

	== Well temperaments ==		== Well temperaments ==

Overthink: /* Theory */ various additions

2026-06-06T05:27:59Z

Theory: various additions

← Older revision		Revision as of 05:27, 6 June 2026
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	== Theory ==		== Theory ==
	12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.		12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2\|perfect fifths]] with the [[5/1\|5th harmonic]].

	It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2\|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4\|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5\|minor third]] is even less accurate, being 15.6 cents flat of just.		It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2\|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4\|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5\|minor third]] is even less accurate, being 15.6 cents flat of just.

	Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w\|serialism}} and much of {{w\|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}		Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w\|serialism}} and much of {{w\|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}
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	The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony\|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64\|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val\| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1\|11]] or [[13/1\|13]] at all, though it does a quite credible [[17/1\|17]] and an even better [[19/1\|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].		The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony\|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64\|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val\| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1\|11]] or [[13/1\|13]] at all, though it does a quite credible [[17/1\|17]] and an even better [[19/1\|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

	Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma\|3<sup>12</sup>/2<sup>19</sup>]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out\|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.		Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma\|3<sup>12</sup>/2<sup>19</sup>]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering\|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out\|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

	12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.		12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

	=== Prime harmonics ===		=== Prime harmonics ===

Overthink: /* Intervals */ swap these

2026-06-06T05:11:51Z

Intervals: swap these

← Older revision		Revision as of 05:11, 6 June 2026
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	\| 100		\| 100
	\| Minor second		\| Minor second
	\| [[256/243]] (+9.775)<br>[[25/24]] (~~+29~~.~~328~~)<br>[[16/15]] (~~−11~~.~~731~~)		\| [[256/243]] (+9.775)<br>[[16/15]] (−11.731)<br>[[25/24]] (+29.328)
	\| [[File:piano_1_12edo.mp3]]		\| [[File:piano_1_12edo.mp3]]
	\| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)<br>[[17/16]] (−4.955), [[18/17]] (+1.045)<br>[[19/18]] (+6.397), [[20/19]] (+11.199)		\| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)<br>[[17/16]] (−4.955), [[18/17]] (+1.045)<br>[[19/18]] (+6.397), [[20/19]] (+11.199)

FloraC: /* Regular temperament properties */ add rank-3 back, after discussion with Eufalesio

2026-06-03T13:22:12Z

Regular temperament properties: add rank-3 back, after discussion with Eufalesio

← Older revision		Revision as of 13:22, 3 June 2026
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	<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). For more comprehensive lists, see:		Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[schismic]]/[[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see:
	* [[List of 12et rank two temperaments by badness]]		* [[List of 12et rank two temperaments by badness]]
	* [[List of 12et rank two temperaments by complexity]]		* [[List of 12et rank two temperaments by complexity]]

FloraC: No one cares about atomic

2026-06-03T09:16:52Z

No one cares about atomic

← Older revision		Revision as of 09:16, 3 June 2026
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	<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	Rank-2 ~~'''de'''~~temperaments to which 12et can be [[detempering\|detempered]] include:		Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). For more comprehensive lists, see:

	* [[compton]] (12 & 72)
	* [[garibaldi]] (41 & 53)
	* [[diaschismic]] (46 & 58)
	* [[atomic]] (12 & 612)

	For more comprehensive lists, see:
	* [[List of 12et rank two temperaments by badness]]		* [[List of 12et rank two temperaments by badness]]
	* [[List of 12et rank two temperaments by complexity]]		* [[List of 12et rank two temperaments by complexity]]

Eufalesio: Readded and reorganized

2026-06-03T08:39:54Z

Readded and reorganized

← Older revision		Revision as of 08:39, 3 June 2026
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	<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[garibaldi]] (41 & 53)~~, and~~ [[diaschismic]] (46 & 58). For more comprehensive lists, see:		Rank-2 '''de'''temperaments to which 12et can be [[detempering\|detempered]] include:

			* [[compton]] (12 & 72)
			* [[garibaldi]] (41 & 53)
			* [[diaschismic]] (46 & 58)
			* [[atomic]] (12 & 612)

			For more comprehensive lists, see:
	* [[List of 12et rank two temperaments by badness]]		* [[List of 12et rank two temperaments by badness]]
	* [[List of 12et rank two temperaments by complexity]]		* [[List of 12et rank two temperaments by complexity]]

FloraC: /* Regular temperament properties */ re-organize to make the connection more obvious

2026-06-02T21:31:54Z

Regular temperament properties: re-organize to make the connection more obvious

← Older revision		Revision as of 21:31, 2 June 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	* [[List of 12et rank two temperaments by badness]]
	* [[List of 12et rank two temperaments by complexity]]
	* [[List of edo-distinct 12f rank two temperaments]]

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|-		\|-
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	<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). ~~See~~ [[Schismic–commatic equivalence continuum]] ~~for a more comprehensive list.~~		Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). For more comprehensive lists, see:
			* [[List of 12et rank two temperaments by badness]]
			* [[List of 12et rank two temperaments by complexity]]
			* [[List of edo-distinct 12f rank two temperaments]]
			* [[Schismic–commatic equivalence continuum]]

	== Octave stretch or compression ==		== Octave stretch or compression ==

FloraC: /* Regular temperament properties */ no more srutal. Replace lithium with august, which is simpler

2026-06-02T17:29:10Z

Regular temperament properties: no more srutal. Replace lithium with august, which is simpler

← Older revision		Revision as of 17:29, 2 June 2026
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	\| 1\12		\| 1\12
	\| (P8, P4/5)		\| (P8, P4/5)
	\| [[Ripple]] / [[passion]]		\| [[Ripple]], [[passion]]
	\|-		\|-
	\| 1		\| 1
	\| 5\12		\| 5\12
	\| (P8, P5)		\| (P8, P5)
	\| [[Meantone]] / [[~~Dominant~~ (temperament)\|dominant]]		\| [[Meantone]] / [[dominant (temperament)\|dominant]]
	\|-		\|-
	\| 2		\| 2
	\| 5\12 (1\12)		\| 5\12 (1\12)
	\| (P8/2, P5)		\| (P8/2, P5)
	\| [[~~Srutal~~]] ~~/ [[pajara]] /~~ [[injera]]		\| [[Pajara]], [[injera]]
	\|-		\|-
	\| 3		\| 3
	\| 5\12 (1\12)		\| 5\12 (1\12)
	\| (P8/3, P5)		\| (P8/3, P5)
	\| [[Augmented (temperament)\|Augmented]] / [[~~lithium~~]]		\| [[Augmented (temperament)\|Augmented]] / [[august]]
	\|-		\|-
	\| 4		\| 4

FloraC: /* Regular temperament properties */ schismic -> garibaldi

2026-06-02T17:23:54Z

Regular temperament properties: schismic -> garibaldi

← Older revision		Revision as of 17:23, 2 June 2026
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	<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[~~schismic~~]] (12 & 53), and [[diaschismic]] (46 & 58). See [[Schismic–commatic equivalence continuum]] for a more comprehensive list.		Rank-2 temperaments to which 12et can be [[detempering\|detempered]] include [[compton]] (12 & 72), [[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). See [[Schismic–commatic equivalence continuum]] for a more comprehensive list.

	== Octave stretch or compression ==		== Octave stretch or compression ==