22edo: Difference between revisions

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The 22edo system is the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4{{c}} per octave. Moreover, there is more to it than just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent]]ly.

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out [[81/80]] (S9, syntonic), instead ~~mapping~~ it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out [[81/80]] ([[S-expression|S9]], the syntonic comma), and instead maps it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

22edo's approximation to the 7th harmonic is about 12 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. In that manner, 22edo can be thought of as widening the gap of [[49/48]] ~~(S7, semaphore)~~ between septimal intervals to a full quarter-tone. However, the opposite effect ~~is consequently had on~~ the 5-limit: 5/4 is narrow, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] ~~(S5, dicot)~~ narrowed to a quarter tone. Reasonably, [[36/35]] ~~(S6, mint)~~ is tempered to the same 1-step interval.

22edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and supporting [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: 5/4 is narrow, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], equating the [[7/5]] and [[10/7]] tritones. Reasonably, [[36/35]] is tempered to the same 1-step interval.

22edo's 11-limit ~~status~~ is somewhat contentious: while it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit ~~an inaccurate one~~), it lacks a [[neutral third]] (unless the sharp 6/5 is considered to be one), meaning that 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp.

22edo's approximation of the 11-limit is somewhat contentious: while it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] (unless the sharp ~6/5 is considered to be one), meaning that 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp, but also because 22edo's step is too large to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 31edo all includes).

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].

22edo is also the third-smallest edo (after [[10edo]] and [[15edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

=== Prime harmonics ===

=== As a tuning of other temperaments ===

==== Observance of 81/80 ====

22edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to [[5L 2s|mosdiatonic]] as in meantone systems. Instead, it is a ternary scale.

22edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to [[5L 2s|mosdiatonic]] as in meantone systems. Instead, it is a ternary scale, with patterns including [[nicetone]].

==== Superpyth temperament ====

The 5L 2s diatonic (LLsLLLs) in 22edo is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic MOS has subminor and supermajor thirds of 7/6 and 9/7, rather than classical minor and major thirds of 6/5 and 5/4. This means that the septimal comma 64/63 (S8, archytas) is tempered out, rather than the syntonic comma of 81/80, which one of 22et's core features.

Superpyth as a temperament equates the Pythagorean sevenths (such as A-G, C–Bb in chain-of-fifths notation) to ''harmonic'' sevenths instead of 5-limit minor sevenths (~~approximately~~ [[7/4]] instead of [[9/5]]). In addition to the more uneven diatonic scale as compared to meantone systems and 12edo, 22edo has a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size). The step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively. In superpyth (and thus in 22edo and technically 12edo) the dominant seventh chord and an otonal tetrad are represented by the same chord.

Superpyth as a temperament equates the Pythagorean sevenths (such as A-G, C–Bb in chain-of-fifths notation) to ''harmonic'' sevenths instead of 5-limit minor sevenths (approximating [[7/4]] instead of [[9/5]]). In addition to the more uneven diatonic scale as compared to meantone systems and 12edo, 22edo has a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size). The step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively. In superpyth (and thus in 22edo and technically 12edo), the [[36:45:54:64|1–5/4–3/2–16/9]] dominant seventh chord and an otonal tetrad are represented by the same chord.

==== Porcupine temperament ====

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==== Pajara temperament ====

A third important temperament that 22edo supports is pajara. In the 5-limit, [[2048/2025]] (S16<sup>2</sup> × S17, [[diaschismic]]) is tempered out, meaning that the 5-limit tritones are equated to one another and to the [[semioctave]]. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, [[50/49]] (S5 / S7, [[jubilismic]]) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are also merged to the semioctave, and consequently 64/63 is tempered out as in superpyth - 5/4 is a semioctave away from 7/4 ~~(which~~ is ~~why in 22edo~~, ~~their tunings~~ are ~~inaccurate~~ in the ~~opposite directions~~, and ~~why both semitwelfths~~ and ~~thirds have one particularly accurate~~ interval ~~(12~~/7, 5/4) and ~~the opposite being rather sharp (~~7/4, 6/~~5))~~. Pajara temperament is also supported by [[12edo]], as that system also tempers out 50/49 and 64/63.

A third important temperament that 22edo supports is pajara. In the 5-limit, [[2048/2025]] (S16<sup>2</sup> × S17, [[diaschismic]]) is tempered out, meaning that the 5-limit tritones are equated to one another and to the [[semioctave]]. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, [[50/49]] (S5 / S7, [[jubilismic]]) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are also merged to the semioctave, and consequently 64/63 is tempered out as in superpyth - 5/4 is a semioctave away from 7/4. Since 50/49 is tempered out, the 25/24 and 49/48 intervals are equated to a single interval, and it functions as a chroma in the [[2L 8s]] MOS. This suggests the use of a decatonic notation system, where 7/6 and 8/7 are the same number of scale degrees, and 7/4 is a major interval. Thus the [[4:5:6:7|1–5/4–3/2–7/4]] major tetrad has 5/4 and 7/4 as major intervals, and replacing them with the corresponding minor intervals gives us the [[70:84:105:120|1–6/5–3/2–12/7]] harmonic sixth chord or minor tetrad. Pajara temperament is also supported by [[12edo]], as that system also tempers out 50/49 and 64/63.

==== Additional commas ====

Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} (S15, [[marvel]]), so that the ~~septimal kleisma~~ augmented triad is a chord of 22et, as it also is of any meantone tuning. A ~~septimal~~ comma not tempered out by 12et which 22et does temper out is 1728/1715, the ~~[[1728/1715|~~orwell comma]]; ~~and~~ the [[orwell tetrad]] is also a chord of 22et. The [[orwell]] temperament uses the septimal subminor third ~~as a generator~~ (5 degrees) and forms mos scales with step patterns {{dash|2, 3, 2, 3, 2, 3, 2, 3, 2|med}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2|med}}. Harmonically, orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]]. ~~But~~ 22edo has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish.

Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} (S15, marvel comma), so that the [[marvel]] augmented triad is a chord of 22et, as it also is of any meantone tuning. A 7-limit comma not tempered out by 12et which 22et does temper out is [[1728/1715]], the orwell comma; therefore, the [[orwell tetrad]] is also a chord of 22et. The [[orwell]] temperament uses the septimal subminor third (5 degrees) as a generator, and forms mos scales with step patterns {{dash|2, 3, 2, 3, 2, 3, 2, 3, 2|med}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2|med}}. Harmonically, orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]]. However, 22edo has a leg-up on the others melodically, as the large and small steps of [[Orwell[9]]] are easier to distinguish.

=== Subsets, supersets, and inheritances ===

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=== Other features ===

The 163.6{{c}} "flat minor whole tone" is a key interval in 22edo, in part because it functions as no less than three different consonant ratios in the [[11-limit]]: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22edo can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.

The 163.6{{c}} "flat minor whole tone" is a key interval in 22edo, in part because it functions as no less than three different consonant ratios in the 11-limit: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22edo can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third, but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.

=== Higher-limit interpretations ===

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. ~~Let us also mind~~ its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31.

22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Also note that its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with the 2.3.5.7.11.17.29.31 subgroup.

== Intervals ==

@@ Line 15: / Line 15: @@
 The 22edo system is the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4{{c}} per octave. Moreover, there is more to it than just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents/oct of error. While [[31edo]] does much better, 22edo still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division of the octave to represent the [[11-odd-limit]] [[consistent]]ly.
-Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out [[81/80]] (S9, syntonic), instead mapping it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.
+Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out [[81/80]] ([[S-expression|S9]], the syntonic comma), and instead maps it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.
-edo's approximation to the 7th harmonic is about 12 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. In that manner, 22edo can be thought of as widening the gap of [[49/48]] (S7, semaphore) between septimal intervals to a full quarter-tone. However, the opposite effect is consequently had on the 5-limit: 5/4 is narrow, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] (S5, dicot) narrowed to a quarter tone. Reasonably, [[36/35]] (S6, mint) is tempered to the same 1-step interval.
+edo's approximation to the 7th harmonic is about 13 cents sharp, somewhat similar to 12edo's approximation to the 5th harmonic. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and supporting [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: 5/4 is narrow, and as a result, the interval of 6/5 is significantly sharp of just intonation, with [[25/24]] narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], equating the [[7/5]] and [[10/7]] tritones. Reasonably, [[36/35]] is tempered to the same 1-step interval.
-edo's 11-limit status is somewhat contentious: while it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit an inaccurate one), it lacks a [[neutral third]] (unless the sharp 6/5 is considered to be one), meaning that 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp.
+edo's approximation of the 11-limit is somewhat contentious: while it represents 11/8 well (about 5-6 cents flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] (unless the sharp ~6/5 is considered to be one), meaning that 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7 cents sharp, but also because 22edo's step is too large to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which 31edo all includes).
-Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma [[superpyth]]", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].
+Since 22edo's fifth is sharp of just by approximately one-quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].
 edo is also the third-smallest edo (after [[10edo]] and [[15edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].
 === Prime harmonics ===
-{{Harmonics in equal|22|columns=11}}
+{{Harmonics in equal|22}}
 === As a tuning of other temperaments ===
 ==== Observance of 81/80 ====
-edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to [[5L 2s|mosdiatonic]] as in meantone systems. Instead, it is a ternary scale.
+edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to [[5L 2s|mosdiatonic]] as in meantone systems. Instead, it is a ternary scale, with patterns including [[nicetone]].
 ==== Superpyth temperament ====
 The 5L 2s diatonic (LLsLLLs) in 22edo is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic MOS has subminor and supermajor thirds of 7/6 and 9/7, rather than classical minor and major thirds of 6/5 and 5/4. This means that the septimal comma 64/63 (S8, archytas) is tempered out, rather than the syntonic comma of 81/80, which one of 22et's core features.
-Superpyth as a temperament equates the Pythagorean sevenths (such as A-G, C&ndash;Bb in chain-of-fifths notation) to ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]). In addition to the more uneven diatonic scale as compared to meantone systems and 12edo, 22edo has a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size). The step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively. In superpyth (and thus in 22edo and technically 12edo) the dominant seventh chord and an otonal tetrad are represented by the same chord.
+Superpyth as a temperament equates the Pythagorean sevenths (such as A-G, C&ndash;Bb in chain-of-fifths notation) to ''harmonic'' sevenths instead of 5-limit minor sevenths (approximating [[7/4]] instead of [[9/5]]). In addition to the more uneven diatonic scale as compared to meantone systems and 12edo, 22edo has a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size). The step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively. In superpyth (and thus in 22edo and technically 12edo), the [[36:45:54:64|1–5/4–3/2–16/9]] dominant seventh chord and an otonal tetrad are represented by the same chord.
 ==== Porcupine temperament ====
@@ Line 45: / Line 45: @@
 ==== Pajara temperament ====
-A third important temperament that 22edo supports is pajara. In the 5-limit, [[2048/2025]] (S16<sup>2</sup> × S17, [[diaschismic]]) is tempered out, meaning that the 5-limit tritones are equated to one another and to the [[semioctave]]. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, [[50/49]] (S5 / S7, [[jubilismic]]) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are also merged to the semioctave, and consequently 64/63 is tempered out as in superpyth - 5/4 is a semioctave away from 7/4 (which is why in 22edo, their tunings are inaccurate in the opposite directions, and why both semitwelfths and thirds have one particularly accurate interval (12/7, 5/4) and the opposite being rather sharp (7/4, 6/5)). Pajara temperament is also supported by [[12edo]], as that system also tempers out 50/49 and 64/63.
+A third important temperament that 22edo supports is pajara. In the 5-limit, [[2048/2025]] (S16<sup>2</sup> × S17, [[diaschismic]]) is tempered out, meaning that the 5-limit tritones are equated to one another and to the [[semioctave]]. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, [[50/49]] (S5 / S7, [[jubilismic]]) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are also merged to the semioctave, and consequently 64/63 is tempered out as in superpyth - 5/4 is a semioctave away from 7/4. Since 50/49 is tempered out, the 25/24 and 49/48 intervals are equated to a single interval, and it functions as a chroma in the [[2L 8s]] MOS. This suggests the use of a decatonic notation system, where 7/6 and 8/7 are the same number of scale degrees, and 7/4 is a major interval. Thus the [[4:5:6:7|1–5/4–3/2–7/4]] major tetrad has 5/4 and 7/4 as major intervals, and replacing them with the corresponding minor intervals gives us the [[70:84:105:120|1–6/5–3/2–12/7]] harmonic sixth chord or minor tetrad. Pajara temperament is also supported by [[12edo]], as that system also tempers out 50/49 and 64/63.
 ==== Additional commas ====
-Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} (S15, [[marvel]]), so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et. The [[orwell]] temperament uses the septimal subminor third as a generator (5 degrees) and forms mos scales with step patterns {{dash|2, 3, 2, 3, 2, 3, 2, 3, 2|med}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2|med}}. Harmonically, orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]]. But 22edo has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish.
+Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} (S15, marvel comma), so that the [[marvel]] augmented triad is a chord of 22et, as it also is of any meantone tuning. A 7-limit comma not tempered out by 12et which 22et does temper out is [[1728/1715]], the orwell comma; therefore, the [[orwell tetrad]] is also a chord of 22et. The [[orwell]] temperament uses the septimal subminor third (5 degrees) as a generator, and forms mos scales with step patterns {{dash|2, 3, 2, 3, 2, 3, 2, 3, 2|med}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2|med}}. Harmonically, orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]]. However, 22edo has a leg-up on the others melodically, as the large and small steps of [[Orwell[9]]] are easier to distinguish.
 === Subsets, supersets, and inheritances ===
@@ Line 56: / Line 56: @@
 === Other features ===
-The 163.6{{c}} "flat minor whole tone" is a key interval in 22edo, in part because it functions as no less than three different consonant ratios in the [[11-limit]]: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22edo can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.
+The 163.6{{c}} "flat minor whole tone" is a key interval in 22edo, in part because it functions as no less than three different consonant ratios in the 11-limit: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22edo can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third, but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.
 === Higher-limit interpretations ===
-edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31.
+edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Also note that its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with the 2.3.5.7.11.17.29.31 subgroup.
 == Intervals ==