22edo - Revision history

Vector: Restore moved theory sections

2026-07-19T06:41:07Z

Restore moved theory sections

Ground: /* Superpyth scales */

2026-07-07T23:21:58Z

Superpyth scales

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	\|P-chromatic		\|P-chromatic
	\|{{Interval ruler\|22\|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}		\|{{Interval ruler\|22\|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
	\|		\|The most important scale of Superpyth, Superpyth[12]. Replacing ss with m yields pental blackdye, the intersection of Superpyth and Porcupine.
	\|}		\|}

Hotcrystal0: One decimal place cent values

2026-06-17T16:52:13Z

One decimal place cent values

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	\|-		\|-
	!Cents		!Cents
	\|'''~~273~~'''		\|'''272.7'''
	\|327		\|327.3
	\|~~382~~		\|381.8
	\|'''436'''		\|'''436.4'''
	\|-		\|-
	!Just interpretation		!Just interpretation

Hotcrystal0: fixes in the comparison section

2026-05-22T20:51:56Z

fixes in the comparison section

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	* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.		* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
	* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.		* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
	* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks ~~(e.g.~~ compressed 5-limit thirds ~~and~~ 7/5~10/7 being mapped to the semioctave).		* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks, such as compressed 5-limit thirds, 7/5~10/7 being mapped to the semioctave (supporting jubilismic), and four versions of every 12edo interval quality.
	* [[27edo]] shares Superpyth, ~~and 32edo, also an Archy tuning, shares Pajara~~ with a ~~particularly sharp~~ tuning.		* [[27edo]] shares Superpyth, with a sharper tuning than 22edo where 5/4 is tuned to 400¢.
	* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.		* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
			* [[32edo]] shares Pajara with a particularly sharp tuning.
	* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.		* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

Vector: the 22edo diatonic harmony is on the porcupine page

2026-05-22T17:29:49Z

the 22edo diatonic harmony is on the porcupine page

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	==== Modal and functional harmony. ====		==== Modal and functional harmony. ====
	There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on ~~their respective~~ pages), each with tonal and modal approaches.		There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on the Pajara and Porcupine pages respectively), each with tonal and modal approaches.

	In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.		In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

Lériendil at 06:04, 21 May 2026

2026-05-21T06:04:41Z

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	'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.		'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

	22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup\|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (~~including~~ in the [[ADIN]] system for melodic qualities, which will be used in ~~the remainder~~ of this article).		22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup\|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (as in the [[ADIN]] system for melodic qualities, which will be used in much of this article).

	As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.		As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
	[[File:24edo 22edo comparison.png\|thumb\|The interval qualities found in 22edo vs. those found in 24edo.]]		[[File:24edo 22edo comparison.png\|thumb\|The interval qualities found in 22edo vs. those found in 24edo.]]
	22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" ~~thirds in ADIN~~). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8\|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.		22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (which can be called "nearminor" and "nearmajor" to disambiguate). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8\|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
	[[File:22edo 1.mp3\|thumb\|22edo pajara scale and chords (0-3-6-8 major and minor)]]		[[File:22edo 1.mp3\|thumb\|22edo pajara scale and chords (0-3-6-8 major and minor)]]
	[[File:22edo 62.mp3\|thumb\|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]		[[File:22edo 62.mp3\|thumb\|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]

Lériendil: /* Tables of scales */

2026-05-21T06:01:12Z

Tables of scales

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	\|Mosh		\|Mosh
	\|{{Interval ruler\|22\|0, 330, 380, 700, 760, 1090, 1150, 1200}}		\|{{Interval ruler\|22\|0, 330, 380, 700, 760, 1090, 1150, 1200}}
	\|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better ~~conceptualizes~~ as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.		\|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualized as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
	\|-		\|-
	\|Sephiroid		\|Sephiroid
	\|{{Interval ruler\|22\|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}		\|{{Interval ruler\|22\|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
	\|		\|Magic may be conceptualized decatonically as well; however, 4/3 and 3/2 are placed on the same degree, unlike in a standard 10-form.
	\|-		\|-
	\|Antiluachoid		\|Antiluachoid

Lériendil: /* Building scales from tetrachords */

2026-05-21T05:59:52Z

Building scales from tetrachords

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	A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].		A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

	There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remembering that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", ~~equable~~ diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).		There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remembering that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equitetrachordal diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

	===== Other polychordal structures =====		===== Other polychordal structures =====

Lériendil: /* Building scales from tetrachords */

2026-05-21T05:59:25Z

Building scales from tetrachords

← Older revision		Revision as of 05:59, 21 May 2026
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	A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].		A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

	There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (~~remember~~ that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).		There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remembering that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

	===== Other polychordal structures =====		===== Other polychordal structures =====

Lériendil: /* 11edo temperaments */

2026-05-21T05:59:03Z

11edo temperaments

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	11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.		11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

	Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''~~sentry~~''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "~~father~~" temperament, although that is not supported by 11edo patent.)		Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''Sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "Father" temperament, although that is not supported by 11edo patent.)

	Another temperament that resides in 11edo is called ''~~orgone~~'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.		Another temperament that resides in 11edo is called ''Orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

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

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	'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.		'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

	22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup\|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (~~including~~ in the [[ADIN]] system for melodic qualities, which will be used in ~~the remainder~~ of this article).		22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup\|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (as in the [[ADIN]] system for melodic qualities, which will be used in much of this article).

	As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.		As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
	[[File:24edo 22edo comparison.png\|thumb\|The interval qualities found in 22edo vs. those found in 24edo.]]		[[File:24edo 22edo comparison.png\|thumb\|The interval qualities found in 22edo vs. those found in 24edo.]]
	22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" ~~thirds in ADIN~~). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8\|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.		22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (which can be called "nearminor" and "nearmajor" to disambiguate). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8\|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
	[[File:22edo 1.mp3\|thumb\|22edo pajara scale and chords (0-3-6-8 major and minor)]]		[[File:22edo 1.mp3\|thumb\|22edo pajara scale and chords (0-3-6-8 major and minor)]]
	[[File:22edo 62.mp3\|thumb\|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]		[[File:22edo 62.mp3\|thumb\|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]