12edo - Revision history

Inthar at 02:02, 22 October 2026

2026-10-22T02:02:29Z

← Older revision		Revision as of 02:02, 22 October 2026
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	[[File:Circle of fifths.png\|thumb\|The circle of fifths in 12edo. Source: Wikipedia]]		[[File:Circle of fifths.png\|thumb\|The circle of fifths in 12edo. Source: Wikipedia]]
	'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by ~~the Xenharmonic Reference~~ for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory describes conventional musical terminology.		'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by XenReference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory describes conventional musical terminology.

	12edo has its dominant position due to its approximation of the just perfect fifth, a frequency ratio of [[3/2]], at 7 steps. This approximation is less than 2% of a 12edo step (2 cents) flat of just.		12edo has its dominant position due to its approximation of the just perfect fifth, a frequency ratio of [[3/2]], at 7 steps. This approximation is less than 2% of a 12edo step (2 cents) flat of just.

Vector at 03:43, 21 October 2026

2026-10-21T03:43:33Z

← Older revision		Revision as of 03:43, 21 October 2026
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	'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory describes conventional musical terminology.		'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory describes conventional musical terminology.

	12edo~~'s most important feature is~~ its approximation of [[3/2]] at 7 steps~~, which~~ is less than 2~~{{c}} flat~~ of ~~just, and which, when stacked against the octave, produces the basic [[diatonic]] scale,~~ 2~~-2-1-2-2-2-1. As the fifth is still slightly~~ flat~~, this has allowed 12edo to be conventionally interpreted as a tuning~~ of [[Meantone]] and hence a [[5-limit]] system, where the minor third of 3 steps and the major third of 4 steps represent [[6/5]] and [[5/4]] respectively. Also important to 12edo's structure is the fact that these thirds derive from [[4edo]] and [[3edo]], allowing for the construction of scales that repeat at fractions of the octave.		12edo has its dominant position due to its approximation of the just perfect fifth, a frequency ratio of [[3/2]], at 7 steps. This approximation is less than 2% of a 12edo step (2 cents) flat of just.

	However, as the 12edo thirds are 16{{c}} and 14{{c}} out of tune with ~~the~~ 5~~-limit~~, another important interpretation is the 2.3.17.19 subgroup, which is tuned more accurately than prime 5. In particular, the minor third ~~closely approximates [[~~19/16]], and the ~~100{{c}}~~ semitone ~~can be thought of as [[~~17/16~~]]~[[18/17]]~~.		The [[diatonic]] scale used in conventional music theory is, in xenharmonic theory, considered to be generated by repeatedly stacking this perfect fifth (i.e. by going around the circle of fifths). It is the moment-of-symmetry scale 5L 2s, meaning that it has 5 large steps and 2 small steps, as equally spaced as possible. In xenharmonic theory, it is common to consider diatonic as a whole as a single scale, being agnostic to rotation and to transposition. This is because xenharmonic theory often deals with scales that don't follow the standard pattern of whole tones and semitones. In 12edo, the diatonic scale has a step ratio of 2:1 (whole steps to half steps, or whole tones to semitones).

			The major third of 12edo is flat of the Pythagorean major third (the one generated by the just 3/2 fifth, at about 408 cents), due to the fifth itself being slightly flat. Because of this, it is generally also interpreted as the simple just intonation major third [[5/4]], which is not generated by a fifth in just intonation. This 'double interpretation' is a temperament, and that particular temperament is called [[meantone]]. Because of meantone, the major triad itself represents the just intonation triad 4:5:6, one of the most basic just intonation triads. As a result (due to the rules of [[regular temperament]]), the minor third is interpreted as [[6/5]]. Also important to 12edo's structure is the fact that these thirds derive from [[4edo]] and [[3edo]], allowing for the construction of scales that repeat at fractions of the octave. This means that 12edo can represent any interval built from combinations of 3/2, 6/5, and 5/4, such as 9/8 (the whole tone) or 25/24 (the semitone between 6/5 and 5/4 themselves).

			However, as the 12edo thirds are 16{{c}} and 14{{c}} out of tune with 6/5 and 5/4 respectively, another important interpretation is the 2.3.17.19 subgroup, which is tuned more accurately than prime 5. A subgroup is a set of intervals that you can multiply and divide to get other intervals in the same subgroup - which makes it a group in mathematical terms - and is generally defined as all the intervals that you can get by multiplying or dividing (moving up or down by) a certain set of frequency ratios. In the 2.3.17.19 subgroup, these ratios In particular are the 19-limit minor third (19/16, about 298c, and approximated by 12edo's minor third), the perfect fifth (as previously discussed), and the 17-limit semitone (17/16, about 105c, and approximated by 12edo's semitone), alongside the octave.

	== Theory ==		== Theory ==
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	* 25/24 (the difference between 6/5 and 5/4)		* 25/24 (the difference between 6/5 and 5/4)
	* 27/25 (the difference between 10/9 and 6/5)		* 27/25 (the difference between 10/9 and 6/5)

			This means that when looking at just intonation interpretations of 12edo music or scales, these intervals show up wherever 12edo has a single edostep.

	When primes 17 and 19 are included, it also serves as 17/16, 18/17, [[19/18]], and [[20/19]].		When primes 17 and 19 are included, it also serves as 17/16, 18/17, [[19/18]], and [[20/19]].

Vector at 03:24, 21 October 2026

2026-10-21T03:24:40Z

← Older revision		Revision as of 03:24, 21 October 2026
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	[[File:Circle of fifths.png\|thumb\|The circle of fifths in 12edo. Source: Wikipedia]]		[[File:Circle of fifths.png\|thumb\|The circle of fifths in 12edo. Source: Wikipedia]]
	'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory ~~descirbes~~ conventional musical terminology.		'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory describes conventional musical terminology.

	12edo's most important feature is its approximation of [[3/2]] at 7 steps, which is less than 2{{c}} flat of just, and which, when stacked against the octave, produces the basic [[diatonic]] scale, 2-2-1-2-2-2-1. As the fifth is still slightly flat, this has allowed 12edo to be conventionally interpreted as a tuning of [[Meantone]] and hence a [[5-limit]] system, where the minor third of 3 steps and the major third of 4 steps represent [[6/5]] and [[5/4]] respectively. Also important to 12edo's structure is the fact that these thirds derive from [[4edo]] and [[3edo]], allowing for the construction of scales that repeat at fractions of the octave.		12edo's most important feature is its approximation of [[3/2]] at 7 steps, which is less than 2{{c}} flat of just, and which, when stacked against the octave, produces the basic [[diatonic]] scale, 2-2-1-2-2-2-1. As the fifth is still slightly flat, this has allowed 12edo to be conventionally interpreted as a tuning of [[Meantone]] and hence a [[5-limit]] system, where the minor third of 3 steps and the major third of 4 steps represent [[6/5]] and [[5/4]] respectively. Also important to 12edo's structure is the fact that these thirds derive from [[4edo]] and [[3edo]], allowing for the construction of scales that repeat at fractions of the octave.

Vector at 03:24, 21 October 2026

2026-10-21T03:24:32Z

← Older revision		Revision as of 03:24, 21 October 2026
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	[[File:Circle of fifths.png\|thumb\|The circle of fifths in 12edo. Source: Wikipedia]]		[[File:Circle of fifths.png\|thumb\|The circle of fifths in 12edo. Source: Wikipedia]]
	'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic".		'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave\|2/1]]. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory descirbes conventional musical terminology.

	12edo's most important feature is its approximation of [[3/2]] at 7 steps, which is less than 2{{c}} flat of just, and which, when stacked against the octave, produces the basic [[diatonic]] scale, 2-2-1-2-2-2-1. As the fifth is still slightly flat, this has allowed 12edo to be conventionally interpreted as a tuning of [[Meantone]] and hence a [[5-limit]] system, where the minor third of 3 steps and the major third of 4 steps represent [[6/5]] and [[5/4]] respectively. Also important to 12edo's structure is the fact that these thirds derive from [[4edo]] and [[3edo]], allowing for the construction of scales that repeat at fractions of the octave.		12edo's most important feature is its approximation of [[3/2]] at 7 steps, which is less than 2{{c}} flat of just, and which, when stacked against the octave, produces the basic [[diatonic]] scale, 2-2-1-2-2-2-1. As the fifth is still slightly flat, this has allowed 12edo to be conventionally interpreted as a tuning of [[Meantone]] and hence a [[5-limit]] system, where the minor third of 3 steps and the major third of 4 steps represent [[6/5]] and [[5/4]] respectively. Also important to 12edo's structure is the fact that these thirds derive from [[4edo]] and [[3edo]], allowing for the construction of scales that repeat at fractions of the octave.

Vector at 14:38, 4 April 2026

2026-04-04T14:38:19Z

← Older revision		Revision as of 14:38, 4 April 2026
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	=== Notation ===		=== Notation ===
	12edo has a standard notation system, consistent with classical theory. As a result, [[ups and downs]] notation, [[KISS notation]] for diatonic, [[Pythagorean notation]], and [[sagittal]] notation all converge on 12edo.		12edo has a standard notation system, consistent with classical theory. As a result, [[ups and downs]] notation, [[KISS notation]] for diatonic, [[Pythagorean notation]], and [[sagittal]] notation all converge on 12edo.

			=== Derivation ===
			12edo is derived, as in historical theory, by setting the diatonic and chromatic semitone equal to avoid the problem of "wolf fifths".

	== Compositional theory ==		== Compositional theory ==

Overthink: /* Scales */ add 3L 3s

2026-04-04T03:34:04Z

Scales: add 3L 3s

← Older revision		Revision as of 03:34, 4 April 2026
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	=== Scales ===		=== Scales ===
	12edo, due to its large number of factors, contains many MOS scales with periods that are some fraction of the octave. One well-known example is [[4L 4s]], which is known as the octatonic scale. Additionally, it contains [[6edo]] as a subset, which is the whole tone scale.		12edo, due to its large number of factors, contains many MOS scales with periods that are some fraction of the octave. One well-known example is [[4L 4s]], which is known as the octatonic scale. Another example is the augmented scale, or [[3L 3s]]. Additionally, it contains [[6edo]] as a subset, which is the whole tone scale.

	12edo is small enough that the EDO itself functions as a chromatic scale.		12edo is small enough that the EDO itself functions as a chromatic scale.

Overthink: /* JI approximation */ Add more explanation

2026-04-04T03:33:18Z

JI approximation: Add more explanation

← Older revision		Revision as of 03:33, 4 April 2026
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	12edo is conventionally seen as a [[5-limit\|2.3.5]] edo, though perhaps more salient to conventional Western musical practice is the fact that it contains the basic [[diatonic]] scale, 2-2-1-2-2-2-1.		12edo is conventionally seen as a [[5-limit\|2.3.5]] edo, though perhaps more salient to conventional Western musical practice is the fact that it contains the basic [[diatonic]] scale, 2-2-1-2-2-2-1.
	{{Harmonics in ED\|12\|31\|0}}		{{Harmonics in ED\|12\|31\|0}}

			12edo has two qualities of thirds when not including augmented and diminished ones, those being the major third and minor third, and stacking them in either order produces a triad bounded by a perfect fifth.

	{\| class="wikitable"		{\| class="wikitable"
	\|+Thirds in 12edo		\|+Thirds in 12edo
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	\|'''5/4'''		\|'''5/4'''
	\|}		\|}
	Diatonic thirds are bolded.
			Diatonic thirds are bolded. Note that there are only two third types, both of which are found in the diatonic scale.

	=== Notation ===		=== Notation ===

Overthink: /* Compton temperament */ add 7-limit extension

2026-04-04T03:28:36Z

Compton temperament: add 7-limit extension

← Older revision		Revision as of 03:28, 4 April 2026
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	=== Compton temperament ===		=== Compton temperament ===
	If 12edo is taken as a temperament of [[Pythagorean tuning]] instead of a 2.3.5 temperament, an independent dimension for any other prime (conventionally 5) may be added. This temperament, called Compton, takes advantage of the fact that due to 12edo's extraordinary accuracy in the 3-limit, higher-limit intervals tend to be off by roughly consistent offsets from 12edo ones. Compton temperament tempers out the Pythagorean comma (531441/524288, the ~24c difference between the Pythagorean chroma and minor second), and equates any mosdiatonic interval with its enharmonic counterpart. However, intervals of 5 are distinct, with 5/4 usually being tuned around 384 cents. As such, this is well-tuned in [[72edo]], and is supported by any multiple of 12 up to and including 300edo.		If 12edo is taken as a temperament of [[Pythagorean tuning]] instead of a 2.3.5 temperament, an independent dimension for any other prime (conventionally 5) may be added. This temperament, called Compton, takes advantage of the fact that due to 12edo's extraordinary accuracy in the 3-limit, higher-limit intervals tend to be off by roughly consistent offsets from 12edo ones. Compton temperament tempers out the Pythagorean comma (531441/524288, the ~24c difference between the Pythagorean chroma and minor second), and equates any mosdiatonic interval with its enharmonic counterpart. However, intervals of 5 are distinct, with 5/4 usually being tuned around 384 cents (or 81/80 tuned to 16 cents). As such, this is well-tuned in [[72edo]], and is supported by any multiple of 12 up to and including 300edo.

			The [[canonical extension]] of Compton to the [[7-limit]] tempers out [[225/224]], so that 7/4 is mapped to 10 steps of 12edo minus 2 comma steps.

	{{Cat\|edos}}{{Navbox EDO}}		{{Cat\|edos}}{{Navbox EDO}}

Hotcrystal0: 120edo

2026-03-09T00:13:23Z

120edo

← Older revision		Revision as of 00:13, 9 March 2026
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	===72edo===		===72edo===
	''Main article: [[72edo]]''		''Main article: [[72edo]]''

			===120edo===

			''See [[40edo#120edo]].''


	===612edo===		===612edo===

Inthar: /* 36edo */

2026-03-03T15:24:19Z

36edo

← Older revision		Revision as of 15:24, 3 March 2026
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	===36edo===		===36edo===

	36edo is another way to extend 12edo while maintaining a manageable amount of notes, correcting the 7th and 13th harmonics (in a way distinct from 24edo). Its 5 is still inherited from 12edo, although it is now about halfway between the steps. It is a good tuning for the 2.3.7.13.17.19 subgroup, and its 8/7 makes it a good tuning for [[~~slendric~~]] temperament.		36edo is another way to extend 12edo while maintaining a manageable amount of notes, correcting the 7th and 13th harmonics (in a way distinct from 24edo). Its 5 is still inherited from 12edo, although it is now about halfway between the steps. It is a good tuning for the 2.3.7.13.17.19 subgroup, and its 8/7 makes it a good tuning for [[Slendric]] temperament.

	{{Harmonics in ED\|36\|31\|0}}		{{Harmonics in ED\|36\|31\|0}}