22edo: Difference between revisions

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{{ED intro}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.

{{ED intro}} Because it distinguishes [[10/9]] and [[9/8]], it is not a [[meantone]] system.

== Theory ==

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=== Overview to JI approximation quality ===

The 22edo system is in fact 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~~. While not an [[zeta integral edo|integral]] or [[zeta gap edo|gap edo]] it at least qualifies as a [[zeta peak edo|zeta peak]]~~. 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. Furthermore, 22edo, unlike 12 and 19, is not a [[meantone]] system. 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.

The 22edo system is in fact 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. Furthermore, 22edo, unlike 12 and 19, is not a [[meantone]] system. 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.

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.

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=== Subsets and supersets ===

As 22 is divisible by 11, a 22edo instrument can play any music in 11edo, in the same way that [[12edo]] can play [[6edo]] (the whole tone scale). 11edo is interesting for sounding melodically very similar to 12edo (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.

== Defining features ==

=== Septimal vs. syntonic comma ===

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80. This means that 22 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 and [[27edo]] 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.

The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A–G♮ and C–B♭ are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.

=== Porcupine comma ===

It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a very flat minor whole tone of ~[[10/9]] (usually tuned slightly flat of [[11/10]]), two of which is a sharp ~[[6/5]], and three of which is a slightly flat ~[[4/3]], implying the existence of an equal-step tetrachord, which is characteristic of porcupine. It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3|med}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3|med}} (and their respective modes).

Porcupine temperament also allows the [[zarlino]] scale, present as 4-3-2-4-3-4-2 and tuned particularly accurately in 22edo, to be notated with only 1 set of accidentals (conventionally sharps and flats) representing both the syntonic comma and the classical chromatic semitone.

=== 5-limit commas ===

Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

=== 7-limit commas ===

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. [[50/49]] (the jubilisma), and 64/63 (the septimal comma) are tempered out in both systems, so they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 the dominant seventh chord and an otonal tetrad are represented by the same chord. Hence both also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}}, the [[septimal kleisma]], 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.

=== 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.

22edo also supports the [[orwell]] temperament, which 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.

22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.

== Intervals ==

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! Cents

! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref>

! colspan="3" | [[Ups and ~~Downs Notation~~|Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and ^^d2)

! colspan="3" | [[Ups and downs notation|Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and ^^d2)

! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}

! Audio

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! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! colspan="2" | [[Ups and downs notation|Ups and ~~Downs Notation~~]]

! colspan="2" | [[Ups and downs notation|Ups and downs notation]]

|-

! [[5L 2s|Diatonic Interval Names]]

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|}

Treating [[Ups and ~~Downs Notation~~|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

Treating [[Ups and downs notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:

[[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]]

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The keyboard runs {{nowrap|D * * E * * F * * G * * * A * * B * * C * * D}}.

A score video demonstrating this type of notation using redefined sharp and flat symbols is available: [https://www.youtube.com/watch?v=se79rdp705Y ''Study #1 in Porcupine Temperament: "Flying Straight Down" (Microtonal/Xenharmonic)''] (2020) by [[John Moriarty]]. Note that the sharp of one note is lower than the flat of the next note, in contrast to sharps and flats in the diatonic notation with ups and downs described above.

=== Pentatonic notation ===

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! [[Cent]]s

! colspan="2" | Superpyth/Porcupine

! colspan="3" | Porcupine

! colspan="3" | Porcupine (Onyx)

! colspan="3" |Porcupine (Zarlino)

! colspan="3" | Pentatonic

! colspan="3" | Decatonic

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Line 558:

| P1

| D

|perfect unison

|P1

|C

| perfect unison

| P1

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Line 581:

| A1

| D#

|augmented unison

|A1

|C#

| aug unison

| A1

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| d2

| Eb

|minor second

|m2

|Db

| double-aug unison, double-dim sub3rd

| AA1, dds3

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| P2

| E

|narrow major second

|nM2

|D

| dim sub3rd

| ds3

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| A2

| E#

|wide major second

|WM2

|D#

| minor sub3rd

| ms3

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| d3

| Fb

|wolf third

|w3

|Ebb

| major sub3rd

| Ms3

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| m3

| F

|minor third

|m3

|Eb

| aug sub3rd

| As3

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Line 719:

| M3

| F#

|major third

|M3

|E

| double-aug sub3rd, double-dim 4thoid

| AAs3, dd4d

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| A3, d4

| Fx, Gb

|augmented third

|A3

|E#

| dim 4thoid

| d4d

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| m4

| G

|perfect fourth

|P4

|F

| perfect 4thoid

| P4d

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| M4

| G#

|wolf fourth

|w4

|F#

| aug 4thoid

| A4d

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| A4, d5

| Gx, Abb

|augmented fourth, diminished fifth

|A4, d5

|F##, Gbb

| double-aug 4thoid, double-dim 5thoid

| AA4d, dd5d

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| m5

| Ab

|wolf fifth

|w5

|Gb

| dim 5thoid

| d5d

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| M5

| A

|perfect fifth

|P5

|G

| perfect 5thoid

| P5d

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| A5, d6

| A#, Bbb

|diminished sixth

|d6

|Abb

| aug 5thoid

| A5d

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| m6

| Bb

|minor sixth

|m6

|Ab

| double-aug 5thoid, double-dim sub7th

| AA5d, dds7

Line 851:

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| M6

| B

|major sixth

|M6

|A

| dim sub7th

| ds7

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| A6

| B#

|wolf sixth

|w6

|A#

| minor sub7th

| ms7

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| d7

| Cb

|narrow minor seventh

|nm7

|Bbb

| major sub7th

| Ms7

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| P7

| C

|wide minor seventh

|Wm7

|Bb

| aug sub7th

| As7

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| A7

| C#

|major seventh

|M7

|B

| double-aug sub7th, double-dim octave

| AAs7, dd8

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| d8

| Db

|diminished octave

|d8

|Cb

| dim octave

| d8

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| P8

| D

|perfect octave

|P8

|C

| perfect octave

| P8

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~~=== Zeta peak index ===~~

== Regular temperament properties ==

~~{{ZPI~~

{| class="wikitable center-4 center-5 center-6"

~~| zpi = 80~~

~~| steps = 22.0251467420146~~

~~| step size = 54.4831784348982~~

~~| tempered height = 6.062600~~

~~| pure height = 5.857510~~

~~| integral = 1.258178~~

~~| gap = 16.213941~~

~~| octave = 1198.62992556776~~

~~| consistent = 12~~

~~| distinct = 8~~

}}

~~== Defining features ==~~

~~=== Septimal vs. syntonic comma ===~~

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80. This means that 22 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 and [[27edo]] 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.

The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A–G♮ and C–B♭ are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.

~~=== Porcupine comma ===~~

It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a very flat minor whole tone of ~[[10/9]] (usually tuned slightly flat of [[11/10]]), two of which is a sharp ~[[6/5]], and three of which is a slightly flat ~[[4/3]], implying the existence of an equal-step tetrachord, which is characteristic of porcupine. It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3|med}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3|med}} (and their respective modes).

~~=== 5-limit commas ===~~

Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

~~=== 7-limit commas ===~~

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. [[50/49]] (the jubilisma), and 64/63 (the septimal comma) are tempered out in both systems, so they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 the dominant seventh chord and an otonal tetrad are represented by the same chord. Hence both also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}}, the [[septimal kleisma]], 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.

~~=== 11-limit commas===~~

In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property—although it should be noted that the related [[159edo]] ''does''.

~~=== 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.

22edo also supports the [[orwell]] temperament, which 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.

22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

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| [[Undeka]] [[Hendecatonic]]

|}

== Octave stretch or compression ==

22edo can benefit from slightly compressing the octave, especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11.

; 22edo

* Step size: 54.545{{c}}, octave size: 1200.000{{c}}

Pure-octaves 22edo approximates all harmonics up to 16 but 13 within 14.3{{c}}.

{{Harmonics in equal|22|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22edo}}

{{Harmonics in equal|22|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edo (continued)}}

; [[WE|22et, 11-limit WE tuning]]

* Step size: 54.494{{c}}, octave size: 1198.859{{c}}

Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3, 7, and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. Both 11-limit TE and WE tunings do this.

{{Harmonics in cet|54.493592|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning}}

{{Harmonics in cet|54.493592|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}

; [[ZPI|80zpi]]

* Step size: 54.483{{c}}, octave size: 1198.630{{c}}

Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3, 7 and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. The tuning 80zpi does this.

{{Harmonics in cet|54.483|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80zpi}}

{{Harmonics in cet|54.483|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 80zpi (continued)}}

; [[57ed6]]

* Step size: 54.420{{c}}, octave size: 1197.246{{c}}

Compressing the octave of 22edo by around 2.8{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 but 13 within 15.4{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7 subgroup|2.3.7-subgroup]] tuning, e.g. for [[archy]] (2.3.7-subgroup superpyth) temperament. The tuning 57ed6 does this.

{{Harmonics in equal|57|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed6}}

{{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}

; [[35edt]]

* Step size: 54.342{{c}}, octave size: 1195.515{{c}}

Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[62ed7]] both do this. This extends 57ed6's 2.3.7-subgroup tuning into a [[2.3.7.13 subgroup|2.3.7.13-subgroup]] tuning.

{{Harmonics in equal|35|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edt}}

{{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}}

== Scales ==

''See [[22edo ~~modes]]''.~~

== Tetrachords ==

~~''See [[~~22edo tetrachords~~]].''~~

== ~~Chord names~~ ==

== Chords ==

Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:

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|-

| fourthward wa

| {{monzo| a b }} where b < ~~−1~~

| {{monzo| a b }} where {{nowrap|b < −1}}

| 32/27, 16/9

|-

| upminor

| gu

| {{monzo| a b -1 }}

| {{monzo| a b −1 }}

| 6/5, 9/5

|-

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Line 1,519:

| rowspan="2" | major

| fifthward wa

| {{monzo| a b }} where b > 1

| {{monzo| a b }} where {{nowrap|b > 1}}

| 9/8, 27/16

|-

| ru

| {{monzo| a b 0 -1 }}

| {{monzo| a b 0 −1 }}

| 9/7, 12/7

|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo, and ru triads:

{| class="wikitable center-all"

Line 1,483:

Line 1,575:

* 0-5-11 = C Eb ^Gb = Cd(^5)

* 0-5-12 = C Eb vG = Cm(v5)

~~Further discussion of 22edo chord naming:~~

* [[22edo Chord Names]]

* [[22 EDO Chords]]

* [[Ups and Downs Notation #Chords and Chord Progressions]]

* [[Chords of orwell]]

== Instruments ==

=== Keyboards ===

[[File:22-tone halberstadt layout~~.png|link=https://en.xen.wiki/w/File:22-tone%20halberstadt%20layout~~.png|alt=|frameless]]

[[File:22-tone halberstadt layout.png|alt=|frameless]]

A potential layout for a 22edo keyboard with both split black and white keys.

[[Lumatone mapping for 22edo|Lumatone mappings for 22edo]] are available.

== Music ==

== ~~Related pages ==~~

== See also ==

* [[Lumatone mapping for 22edo]]

* [[List of MOS scales in 22edo]]

~~=== Approaches =~~==

* [[User:Unque/22edo Composition Theory|Unque's approach]]

* [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch's approach]]

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Line 1,610:

[[Category:Alpharabian]]

[[Category:Superpyth]]

[[Category:Orwell]]

[[Category:Porcupine]]

[[Category:Magic]]

[[Category:Quartismic]]

[[Category:Todo:complete table]]

@@ Line 7: / Line 7: @@
 {{Infobox ET}}
 {{Wikipedia|22 equal temperament}}
-{{ED intro}} Because it distinguishes [[10/9]] and [[9/8]], it is not a meantone system.
+{{ED intro}} Because it distinguishes [[10/9]] and [[9/8]], it is not a [[meantone]] system.
 == Theory ==
@@ Line 14: / Line 14: @@
 === Overview to JI approximation quality ===
-The 22edo system is in fact 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. While not an [[zeta integral edo|integral]] or [[zeta gap edo|gap edo]] it at least qualifies as a [[zeta peak edo|zeta peak]]. 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. Furthermore, 22edo, unlike 12 and 19, is not a [[meantone]] system. 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.
+The 22edo system is in fact 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. Furthermore, 22edo, unlike 12 and 19, is not a [[meantone]] system. 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.
 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.
@@ Line 27: / Line 27: @@
 === Subsets and supersets ===
 As 22 is divisible by 11, a 22edo instrument can play any music in 11edo, in the same way that [[12edo]] can play [[6edo]] (the whole tone scale). 11edo is interesting for sounding melodically very similar to 12edo (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.
+== Defining features ==
+=== Septimal vs. syntonic comma ===
+Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80. This means that 22 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 and [[27edo]] 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.
+The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L&nbsp;2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A&ndash;G&#x266E; and C&ndash;B&#x266D; are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.
+=== Porcupine comma ===
+It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a very flat minor whole tone of ~[[10/9]] (usually tuned slightly flat of [[11/10]]), two of which is a sharp ~[[6/5]], and three of which is a slightly flat ~[[4/3]], implying the existence of an equal-step tetrachord, which is characteristic of porcupine. It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3|med}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3|med}} (and their respective modes).
+Porcupine temperament also allows the [[zarlino]] scale, present as 4-3-2-4-3-4-2 and tuned particularly accurately in 22edo, to be notated with only 1 set of accidentals (conventionally sharps and flats) representing both the syntonic comma and the classical chromatic semitone.
+=== 5-limit commas ===
+Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.
+=== 7-limit commas ===
+In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. [[50/49]] (the jubilisma), and 64/63 (the septimal comma) are tempered out in both systems, so they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 the dominant seventh chord and an otonal tetrad are represented by the same chord. Hence both also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}}, the [[septimal kleisma]], 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.
+=== 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.
+edo also supports the [[orwell]] temperament, which 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.
+edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.
 == Intervals ==
@@ Line 37: / Line 61: @@
 ! Cents
 ! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref>
-! colspan="3" | [[Ups and Downs Notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^^d2)
+! colspan="3" | [[Ups and downs notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^^d2)
 ! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}
 ! Audio
@@ Line 307: / Line 331: @@
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
-! colspan="2" | [[Ups and downs notation|Ups and Downs Notation]]
+! colspan="2" | [[Ups and downs notation|Ups and downs notation]]
 |-
 ! [[5L 2s|Diatonic Interval Names]]
@@ Line 428: / Line 452: @@
 |}
-Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:
+Treating [[Ups and downs notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:
 [[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]]
@@ Line 495: / Line 519: @@
 The keyboard runs {{nowrap|D * * E * * F * * G * * * A * * B * * C * * D}}.
+A score video demonstrating this type of notation using redefined sharp and flat symbols is available:  [https://www.youtube.com/watch?v=se79rdp705Y ''Study #1 in Porcupine Temperament: "Flying Straight Down" (Microtonal/Xenharmonic)''] (2020) by [[John Moriarty]]. Note that the sharp of one note is lower than the flat of the next note, in contrast to sharps and flats in the diatonic notation with ups and downs described above.
 === Pentatonic notation ===
@@ Line 518: / Line 544: @@
 ! [[Cent]]s
 ! colspan="2" | Superpyth/Porcupine
-! colspan="3" | Porcupine
+! colspan="3" | Porcupine (Onyx)
+! colspan="3" |Porcupine (Zarlino)
 ! colspan="3" | Pentatonic
 ! colspan="3" | Decatonic
@@ Line 531: / Line 558: @@
 | P1
 | D
+|perfect unison
+|P1
+|C
 | perfect unison
 | P1
@@ Line 551: / Line 581: @@
 | A1
 | D#
+|augmented unison
+|A1
+|C#
 | aug unison
 | A1
@@ Line 571: / Line 604: @@
 | d2
 | Eb
+|minor second
+|m2
+|Db
 | double-aug unison,<br />double-dim sub3rd
 | AA1,<br />dds3
@@ Line 591: / Line 627: @@
 | P2
 | E
+|narrow major second
+|nM2
+|D
 | dim sub3rd
 | ds3
@@ Line 611: / Line 650: @@
 | A2
 | E#
+|wide major second
+|WM2
+|D#
 | minor sub3rd
 | ms3
@@ Line 631: / Line 673: @@
 | d3
 | Fb
+|wolf third
+|w3
+|Ebb
 | major sub3rd
 | Ms3
@@ Line 651: / Line 696: @@
 | m3
 | F
+|minor third
+|m3
+|Eb
 | aug sub3rd
 | As3
@@ Line 671: / Line 719: @@
 | M3
 | F#
+|major third
+|M3
+|E
 | double-aug sub3rd,<br />double-dim 4thoid
 | AAs3,<br />dd4d
@@ Line 691: / Line 742: @@
 | A3, d4
 | Fx, Gb
+|augmented third
+|A3
+|E#
 | dim 4thoid
 | d4d
@@ Line 711: / Line 765: @@
 | m4
 | G
+|perfect fourth
+|P4
+|F
 | perfect 4thoid
 | P4d
@@ Line 731: / Line 788: @@
 | M4
 | G#
+|wolf fourth
+|w4
+|F#
 | aug 4thoid
 | A4d
@@ Line 751: / Line 811: @@
 | A4, d5
 | Gx, <br />Abb
+|augmented fourth, diminished fifth
+|A4, d5
+|F##, Gbb
 | double-aug 4thoid,<br />double-dim 5thoid
 | AA4d, <br />dd5d
@@ Line 771: / Line 834: @@
 | m5
 | Ab
+|wolf fifth
+|w5
+|Gb
 | dim 5thoid
 | d5d
@@ Line 791: / Line 857: @@
 | M5
 | A
+|perfect fifth
+|P5
+|G
 | perfect 5thoid
 | P5d
@@ Line 811: / Line 880: @@
 | A5, d6
 | A#, Bbb
+|diminished sixth
+|d6
+|Abb
 | aug 5thoid
 | A5d
@@ Line 831: / Line 903: @@
 | m6
 | Bb
+|minor sixth
+|m6
+|Ab
 | double-aug 5thoid,<br />double-dim sub7th
 | AA5d,<br />dds7
@@ Line 851: / Line 926: @@
 | M6
 | B
+|major sixth
+|M6
+|A
 | dim sub7th
 | ds7
@@ Line 871: / Line 949: @@
 | A6
 | B#
+|wolf sixth
+|w6
+|A#
 | minor sub7th
 | ms7
@@ Line 891: / Line 972: @@
 | d7
 | Cb
+|narrow minor seventh
+|nm7
+|Bbb
 | major sub7th
 | Ms7
@@ Line 911: / Line 995: @@
 | P7
 | C
+|wide minor seventh
+|Wm7
+|Bb
 | aug sub7th
 | As7
@@ Line 931: / Line 1,018: @@
 | A7
 | C#
+|major seventh
+|M7
+|B
 | double-aug sub7th,<br />double-dim octave
 | AAs7,<br />dd8
@@ Line 951: / Line 1,041: @@
 | d8
 | Db
+|diminished octave
+|d8
+|Cb
 | dim octave
 | d8
@@ Line 971: / Line 1,064: @@
 | P8
 | D
+|perfect octave
+|P8
+|C
 | perfect octave
 | P8
@@ Line 991: / Line 1,087: @@
 {{Q-odd-limit intervals|22}}
-=== Zeta peak index ===
+== Regular temperament properties ==
-{{ZPI
+{| class="wikitable center-4 center-5 center-6"
-| zpi = 80
-| steps = 22.0251467420146
-| step size = 54.4831784348982
-| tempered height = 6.062600
-| pure height = 5.857510
-| integral = 1.258178
-| gap = 16.213941
-| octave = 1198.62992556776
-| consistent = 12
-| distinct = 8
-}}
-== Defining features ==
-=== Septimal vs. syntonic comma ===
-Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80. This means that 22 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 and [[27edo]] 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.
-The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A&ndash;G&#x266E; and C&ndash;B&#x266D; are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.
-=== Porcupine comma ===
-It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a very flat minor whole tone of ~[[10/9]] (usually tuned slightly flat of [[11/10]]), two of which is a sharp ~[[6/5]], and three of which is a slightly flat ~[[4/3]], implying the existence of an equal-step tetrachord, which is characteristic of porcupine. It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. It forms [[mos scale]]s of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3|med}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3|med}} (and their respective modes).
-=== 5-limit commas ===
-Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.
-=== 7-limit commas ===
-In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. [[50/49]] (the jubilisma), and 64/63 (the septimal comma) are tempered out in both systems, so they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 the dominant seventh chord and an otonal tetrad are represented by the same chord. Hence both also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}}, the [[septimal kleisma]], 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.
-=== 11-limit commas===
-In the 11-limit, 22edo tempers out the [[quartisma]], leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with [[24edo]], is surprisingly ''not'' shared with a number of other relatively small edos such as [[17edo]], [[26edo]] and [[34edo]]. In fact, not even the famous [[53edo]] has this property—although it should be noted that the related [[159edo]] ''does''.
-=== 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.
-edo also supports the [[orwell]] temperament, which 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.
-edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.
-== Regular temperament properties ==
-{| class="wikitable center-4 center-5 center-6"
 |-
 ! rowspan="2" | [[Subgroup]]
@@ Line 1,389: / Line 1,446: @@
 | [[Undeka]]<br>[[Hendecatonic]]
 |}
+== Octave stretch or compression ==
+edo can benefit from slightly compressing the octave, especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11.
+; 22edo
+* Step size: 54.545{{c}}, octave size: 1200.000{{c}}
+Pure-octaves 22edo approximates all harmonics up to 16 but 13 within 14.3{{c}}.
+{{Harmonics in equal|22|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22edo}}
+{{Harmonics in equal|22|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edo (continued)}}
+; [[WE|22et, 11-limit WE tuning]]
+* Step size: 54.494{{c}}, octave size: 1198.859{{c}}
+Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3, 7, and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. Both 11-limit TE and WE tunings do this.
+{{Harmonics in cet|54.493592|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
+{{Harmonics in cet|54.493592|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
+; [[ZPI|80zpi]]
+* Step size: 54.483{{c}}, octave size: 1198.630{{c}}
+Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3, 7 and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. The tuning 80zpi does this.
+{{Harmonics in cet|54.483|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80zpi}}
+{{Harmonics in cet|54.483|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 80zpi (continued)}}
+; [[57ed6]]
+* Step size: 54.420{{c}}, octave size: 1197.246{{c}}
+Compressing the octave of 22edo by around 2.8{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 but 13 within 15.4{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7 subgroup|2.3.7-subgroup]] tuning, e.g. for [[archy]] (2.3.7-subgroup superpyth) temperament. The tuning 57ed6 does this.
+{{Harmonics in equal|57|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed6}}
+{{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}
+; [[35edt]]
+* Step size: 54.342{{c}}, octave size: 1195.515{{c}}
+Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[62ed7]] both do this. This extends 57ed6's 2.3.7-subgroup tuning into a [[2.3.7.13 subgroup|2.3.7.13-subgroup]] tuning.
+{{Harmonics in equal|35|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edt}}
+{{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}}
 == Scales ==
-''See [[22edo modes]]''.
+{{Main|22edo modes}}
+{{See also|List of MOS scales in 22edo}}
 == Tetrachords ==
-''See [[22edo tetrachords]].''
+{{Main|22edo tetrachords}}
-== Chord names ==
+== Chords ==
+{{Main|22edo chords}}
 Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
@@ Line 1,412: / Line 1,504: @@
 |-
 | fourthward wa
-| {{monzo| a b }} where b &lt; &minus;1
+| {{monzo| a b }} where {{nowrap|b &lt; −1}}
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| {{monzo| a b -1 }}
+| {{monzo| a b −1 }}
 | 6/5, 9/5
 |-
@@ Line 1,427: / Line 1,519: @@
 | rowspan="2" | major
 | fifthward wa
-| {{monzo| a b }} where b &gt; 1
+| {{monzo| a b }} where {{nowrap|b &gt; 1}}
 | 9/8, 27/16
 |-
 | ru
-| {{monzo| a b 0 -1 }}
+| {{monzo| a b 0 −1 }}
 | 9/7, 12/7
 |}
-All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:
+All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo, and ru triads:
 {| class="wikitable center-all"
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 * 0-5-11 = C Eb ^Gb = Cd(^5)
 * 0-5-12 = C Eb vG = Cm(v5)
-Further discussion of 22edo chord naming:
-* [[22edo Chord Names]]
-* [[22 EDO Chords]]
-* [[Ups and Downs Notation #Chords and Chord Progressions]]
-* [[Chords of orwell]]
 == Instruments ==
 === Keyboards ===
-[[File:22-tone halberstadt layout.png|link=https://en.xen.wiki/w/File:22-tone%20halberstadt%20layout.png|alt=|frameless]]
+[[File:22-tone halberstadt layout.png|alt=|frameless]]
 A potential layout for a 22edo keyboard with both split black and white keys.
+[[Lumatone mapping for 22edo|Lumatone mappings for 22edo]] are available.
 == Music ==
 {{Main| 22edo/Music }}
 {{Catrel|22edo tracks}}
-== Related pages ==
+== See also ==
-* [[Lumatone mapping for 22edo]]
-* [[List of MOS scales in 22edo]]
-=== Approaches ===
 * [[User:Unque/22edo Composition Theory|Unque's approach]]
 * [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch's approach]]
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 [[Category:Alpharabian]]
 [[Category:Superpyth]]
+[[Category:Orwell]]
 [[Category:Porcupine]]
 [[Category:Magic]]
 [[Category:Quartismic]]
 [[Category:Todo:complete table]]