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{{interwiki
{{Interwiki
| en = 22edo
| de = 22-EDO
| de = 22-EDO
| en = 22edo
| es = 22 EDO
| es = 22 EDO
| ja = 22平均律
| ja = 22平均律
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{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|22 equal temperament}}
{{Wikipedia|22 equal temperament}}
{{EDO intro|22}} 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.
 
== History ==
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the supposed division of the octave into 22 unequal parts in the [[Indian music|music theory of India]], Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo]], and {{w|James Murray Barbour|J. Murray Barbour}} in his classic survey of tuning history, ''Tuning and Temperament''.


== Theory ==
== Theory ==
=== History ===
22edo is the third edo, 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 cents. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents of error, and in fact 22 is the smallest edo to represent the [[11-odd-limit]] [[consistent]]ly, though [[31edo]] is considerably more accurate.
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the division of the octave into 22 unequal parts in the [[Indian music|music theory of India]], Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo]], and {{w|James Murray Barbour|J. Murray Barbour}} in his classic survey of tuning history, ''Tuning and Temperament''.
 
Possibly the most striking characteristic of 22edo to those not used to it is that it does ''not'' [[tempering out|temper out]] [[81/80]] (the syntonic comma), and instead maps it to one step. Additionally, it is a superset of 11edo and is close to [[24edo]], having only 2 fewer steps than it, and thus behaves like [[11edo]] and [[13edo]] in that melodic movements similar to 12edo can quickly arrive at an unfamiliar place. 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 [[7/1|7th harmonic]] is about 13 cents sharp, somewhat similar to 12edo's approximation to the [[5/1|5th harmonic]]. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and [[support]]ing [[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: while 5/4 and 6/5 are closer to JI than in 12edo, 5/4 is flat and 6/5 is sharp, resulting in [[25/24]] being narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], so the [[7/5]] and [[10/7]] are equated to the 600{{c}} half-octave tritone, and 5/4 and 7/4 are separated by a semioctave, as well as 6/5 and [[12/7]]. Reasonably, [[36/35]] is also tempered to 1 step just like 25/24 and 49/48.


=== Overview to JI approximation quality ===
22edo's approximation of the 11-limit is somewhat contentious: While it represents 11/8 well (about 5–6{{c}} flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] dividing the perfect fifth in two, which means 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{{c}} sharp, but also because 22edo's step is just short of being small enough to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which [[24edo]], [[27edo]], and 31edo all include fully). Because 22edo does not contain "neutral" intervals, [[11/9]] is mapped to the same interval as 6/5 and [[12/11]] is mapped to the submajor second, inflating [[243/242]] to a full step.
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 [[TE error]] of 4 cents/oct. 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|consistently]]. 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.
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 ===
=== Prime harmonics ===
{{Harmonics in equal|22|columns=11}}
{{Harmonics in equal|22}}
 
=== 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 these intervals. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to the [[5L 2s]] [[mos]] as in meantone systems. Instead, it is a ternary scale, having the [[nicetone]] pattern.


=== Subsets and supersets ===
==== Superpyth temperament ====
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.
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 is tempered out rather than the syntonic comma of 81/80, which one of 22et's core features.
 
Superpyth temperament equates the Pythagorean sevenths (such as A–G and C–B♭ in [[chain-of-fifths notation]]) to ''harmonic'' sevenths instead of 5-limit minor sevenths (approximating [[7/4]] instead of [[9/5]]). Due to the sharper fifths, the diatonic scale is more uneven than in meantone systems and 12edo. In addition to the more uneven diatonic scale, 22edo has a quasi-equal pentatonic scale (the major whole tone and subminor third are rather close in size). The step patterns of the pentatonic and diatonic scales in 22et are {{dash|4, 4, 5, 4, 5}} and {{dash|4, 4, 1, 4, 4, 4, 1}} 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 ====
22edo additionally tempers out the porcupine comma or maximal diesis of [[250/243]] ([[S-expression|S10<sup>2</sup>⋅S11]]), 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.
 
Porcupine temperament allows the 5-limit diatonic scale (the [[zarlino]] scale), present as {{nowrap|{{dash|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, as the difference between them (250/243) is tempered out.
 
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. Porcupine's generator forms mos scales of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3}} (and their respective modes).
 
==== Pajara temperament ====
A third important temperament that 22edo supports is [[pajara]]. In the 5-limit, [[2048/2025]] (diaschisma) 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]] (jubilisma) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are equated 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&nbsp;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]] subharmonic sixth chord or minor tetrad. Pajara temperament is also supported by [[12edo]], as it also tempers out 50/49 and 64/63.
 
The decatonic scales of pajara have been considered by many to be a system in the 7-limit analogous to the diatonic scale of meantone temperament in the 5-limit, as described in Paul Erlich's paper [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf Tuning, Tonality and 22-Tone Temperament].
 
==== Additional commas ====
Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} ({{S|15}}, [[marvel comma]]), so that the marvel augmented triad is a chord of 22et. 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}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2}}. While orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]], 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 ===
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 quartertones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In particular, 22edo can be roughly conceptualized as 24 but with only two types of thirds rather than three. In [[Sagittal notation]], 11 can be notated as every other note of 22.
 
22 inherits 11edo's [[11/8]] and [[7/4]], and inherits [[2edo]]'s tritone, which is mapped in both systems to [[7/5]] and [[10/7]].
 
=== Other features ===
The 163.6{{c}} "flat minor whole tone" or "submajor second" 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. 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 ==
== Intervals ==
Line 34: Line 71:
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<ref group="note">{{sg|2.3.5.7.11.17 subgroup}}</ref>
! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref>
! colspan="3" | [[Ups and Downs Notation]]<br>([[Enharmonic intervals in ups and downs notation|EIs]]: v<sup>3</sup>A1 and ^^d2)
! Audio
! 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)}}
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}
! Audio
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| [[File:0-0.000c_P1.mp3]]
| perfect unison
| perfect unison
| P1
| P1
Line 48: Line 86:
| P1
| P1
| D
| D
| [[File:0-0.000c_P1.mp3]]
|-
|-
| 1
| 1
| 54.545
| 54.5
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
| [[File:0-54.545c_22edo.mp3]]
| up-unison, minor 2nd
| up-unison, minor 2nd
| ^1, m2
| ^1, m2
Line 59: Line 97:
| K1, m2
| K1, m2
| KD, Eb
| KD, Eb
| [[File:0-54.545c_22edo.mp3]]
|-
|-
| 2
| 2
| 109.091
| 109.1
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
| [[File:0-109.091c_11edo.mp3]]
| downaug 1sn, upminor 2nd
| downaug 1sn, upminor 2nd
| vA1, ^m2
| vA1, ^m2
Line 70: Line 108:
| Km2
| Km2
| KEb
| KEb
| [[File:0-109.091c_11edo.mp3]]
|-
|-
| 3
| 3
| 163.636
| 163.6
| [[12/11]], [[11/10]], [[10/9]]
| [[12/11]], [[11/10]], [[10/9]]
| [[File:0-163.636c_22edo.mp3]]
| aug 1sn, downmajor 2nd
| aug 1sn, downmajor 2nd
| A1, vM2
| A1, vM2
Line 81: Line 119:
| kM2
| kM2
| kE
| kE
| [[File:0-163.636c_22edo.mp3]]
|-
|-
| 4
| 4
| 218.182
| 218.2
| [[9/8]], [[17/15]], [[8/7]]
| [[9/8]], [[17/15]], [[8/7]]
| [[File:0-218.182c_11edo.mp3]]
| major 2nd
| major 2nd
| M2
| M2
Line 92: Line 130:
| M2
| M2
| E
| E
| [[File:0-218.182c_11edo.mp3]]
|-
|-
| 5
| 5
| 272.727
| 272.7
| [[20/17]], [[7/6]]
| [[20/17]], [[7/6]]
| [[File:0-272.727c_22edo.mp3]]
| minor 3rd
| minor 3rd
| m3
| m3
Line 103: Line 141:
| m3
| m3
| F
| F
| [[File:0-272.727c_22edo.mp3]]
|-
|-
| 6
| 6
| 327.273
| 327.3
| [[6/5]], [[17/14]], [[11/9]]
| [[6/5]], [[17/14]], [[11/9]]
| [[File:0-327.273c_11edo.mp3]]
| upminor 3rd
| upminor 3rd
| ^m3
| ^m3
Line 114: Line 152:
| Km3
| Km3
| KF
| KF
| [[File:0-327.273c_11edo.mp3]]
|-
|-
| 7
| 7
| 381.818
| 381.8
| [[5/4]], [[96/77]]
| [[5/4]], [[96/77]]
| [[File:0-381.818c_22edo.mp3]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 125: Line 163:
| kM3
| kM3
| kF#
| kF#
| [[File:0-381.818c_22edo.mp3]]
|-
|-
| 8
| 8
| 436.364
| 436.4
| [[14/11]], [[9/7]], [[22/17]]
| [[14/11]], [[9/7]], [[22/17]]
| [[File:0-436.364c_11edo.mp3]]
| major 3rd
| major 3rd
| M3
| M3
Line 136: Line 174:
| M3
| M3
| F#
| F#
| [[File:0-436.364c_11edo.mp3]]
|-
|-
| 9
| 9
| 490.909
| 490.9
| [[4/3]]
| [[4/3]]
| [[File:0-490.909c_22edo.mp3]]
| perfect 4th
| perfect 4th
| P4
| P4
Line 147: Line 185:
| P4
| P4
| G
| G
| [[File:0-490.909c_22edo.mp3]]
|-
|-
| 10
| 10
| 545.455
| 545.5
| [[15/11]], [[11/8]]
| [[15/11]], [[11/8]]
| [[File:0-545.455c_11edo.mp3]]
| up-4th, dim 5th
| up-4th, dim 5th
| ^4, d5
| ^4, d5
Line 158: Line 196:
| K4
| K4
| KG
| KG
| [[File:0-545.455c_11edo.mp3]]
|-
|-
| 11
| 11
| 600.000
| 600.0
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
| [[File:0-600.000c_2edo.mp3]]
| downaug 4th, updim 5th
| downaug 4th, updim 5th
| vA4, ^d5
| vA4, ^d5
Line 169: Line 207:
| kA4<br />Kd5
| kA4<br />Kd5
| kG#, KAb
| kG#, KAb
| [[File:0-600.000c_2edo.mp3]]
|-
|-
| 12
| 12
| 654.545
| 654.5
| [[16/11]], [[22/15]]
| [[16/11]], [[22/15]]
| [[File:0-654.545c_11edo.mp3]]
| aug 4th, down-5th
| aug 4th, down-5th
| A4, v5
| A4, v5
Line 180: Line 218:
| k5
| k5
| kA
| kA
| [[File:0-654.545c_11edo.mp3]]
|-
|-
| 13
| 13
| 709.091
| 709.1
| [[3/2]]
| [[3/2]]
| [[File:0-709.091c_22edo.mp3]]
| perfect 5th
| perfect 5th
| P5
| P5
Line 191: Line 229:
| P5
| P5
| A
| A
| [[File:0-709.091c_22edo.mp3]]
|-
|-
| 14
| 14
| 763.636
| 763.6
| [[17/11]], [[14/9]], [[11/7]]
| [[17/11]], [[14/9]], [[11/7]]
| [[File:0-763.636c_11edo.mp3]]
| minor 6th
| minor 6th
| m6
| m6
Line 202: Line 240:
| m6
| m6
| Bb
| Bb
| [[File:0-763.636c_11edo.mp3]]
|-
|-
| 15
| 15
| 818.182
| 818.2
| [[8/5]], [[77/48]]
| [[8/5]], [[77/48]]
| [[File:0-818.182c_22edo.mp3]]
| upminor 6th
| upminor 6th
| ^m6
| ^m6
Line 213: Line 251:
| Km6
| Km6
| KBb
| KBb
| [[File:0-818.182c_22edo.mp3]]
|-
|-
| 16
| 16
| 872.727
| 872.7
| [[18/11]], [[28/17]], [[5/3]]
| [[18/11]], [[28/17]], [[5/3]]
| [[File:0-872.727c_11edo.mp3]]
| downmajor 6th
| downmajor 6th
| vM6
| vM6
Line 224: Line 262:
| kM6
| kM6
| kB
| kB
| [[File:0-872.727c_11edo.mp3]]
|-
|-
| 17
| 17
| 927.273
| 927.3
| [[17/10]], [[12/7]]
| [[17/10]], [[12/7]]
| [[File:0-927.273c_22edo.mp3]]
| major 6th
| major 6th
| M6
| M6
Line 235: Line 273:
| M6
| M6
| B
| B
| [[File:0-927.273c_22edo.mp3]]
|-
|-
| 18
| 18
| 981.818
| 981.8
| [[7/4]], [[30/17]], [[16/9]]
| [[7/4]], [[30/17]], [[16/9]]
| [[File:0-981.818c_11edo.mp3]]
| minor 7th
| minor 7th
| m7
| m7
Line 246: Line 284:
| m7
| m7
| C
| C
| [[File:0-981.818c_11edo.mp3]]
|-
|-
| 19
| 19
| 1036.364
| 1036.4
| [[9/5]], [[11/6]], [[20/11]]
| [[9/5]], [[11/6]], [[20/11]]
| [[File:0-1036.364c_22edo.mp3]]
| upminor 7th, dim 8ve
| upminor 7th, dim 8ve
| ^m7, d8
| ^m7, d8
Line 257: Line 295:
| Km7
| Km7
| kC
| kC
| [[File:0-1036.364c_22edo.mp3]]
|-
|-
| 20
| 20
| 1090.909
| 1090.9
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
| [[File:0-1090.909c_11edo.mp3]]
| downmajor 7th, updim 8ve
| downmajor 7th, updim 8ve
| vM7, ^d8
| vM7, ^d8
Line 268: Line 306:
| kM7
| kM7
| kC#
| kC#
| [[File:0-1090.909c_11edo.mp3]]
|-
|-
| 21
| 21
| 1145.455
| 1145.5
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
| [[File:0-1145.455c_22edo.mp3]]
| major 7th, down 8ve
| major 7th, down 8ve
| M7, v8
| M7, v8
Line 279: Line 317:
| M7 / k8
| M7 / k8
| C#, kD
| C#, kD
| [[File:0-1145.455c_22edo.mp3]]
|-
|-
| 22
| 22
| 1200.000
| 1200.0
| [[2/1]]
| [[2/1]]
| [[File:0-1200.000c_P8.mp3]]
| perfect octave
| perfect octave
| P8
| P8
Line 290: Line 328:
| P8
| P8
| D
| D
| [[File:0-1200.000c_P8.mp3]]
|}
|}


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that E&#x266D; and D&#x266F; are different notes and that E&#x266D; is significantly lower in pitch than D&#x266F;.
Since a sharp raises by three steps, 22edo is a good candidate for [[Stein–Zimmermann–Gould notation]], using sharps and flats with arrows similar to 29edo:
{{Sharpness-sharp3-szg}}
 
If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with double arrows.
 
=== Kite's ups and downs notation ===
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{sharpness-sharp3a}}
 
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 edostep (1\22). Note that E♭ and D♯ are different notes and that E♭ is significantly lower in pitch than D♯.


{| class="wikitable right-1 right-2 center-3 center-4"
{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 22edo
|+ style="font-size: 105%;" | Notation of 22edo
|-
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Degree|#]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="2" | [[Kite's ups and downs notation]]
|-
|-
! [[5L 2s|Diatonic Interval Names]]
! [[5L 2s|Diatonic interval names]]
! Note Names<br />on D
! Note names
|-
|-
| 0
| 0
| 0.00
| 0.0
| '''Perfect unison (P1)'''
| '''Perfect unison (P1)'''
| '''D'''
| '''D'''
|-
|-
| 1
| 1
| 54.545
| 54.5
| Minor second (m2)<br />Up-unison (^1)
| Minor second (m2)<br>Up unison (^1)
| Eb<br />^D
| Eb<br>^D
|-
|-
| 2
| 2
| 109.091
| 109.1
| Upminor 2nd (^m2)<br />Down-augmented unison (vA1)<br />Diminished third (d3)
| Upminor second (^m2)<br>Downaugmented unison (vA1)<br>Diminished third (d3)
| ^Eb<br />vD#<br />Fb
| ^Eb<br>vD#<br>Fb
|-
|-
| 3
| 3
| 163.636
| 163.6
| Downmajor second (vM2)<br />Augmented unison (A1)
| Downmajor second (vM2)<br>Augmented unison (A1)
| vE<br />D#
| vE<br>D#
|-
|-
| 4
| 4
| 218.182
| 218.2
| '''Major second (M2)'''<br />Up-augmented unison (^A1)<br />Downminor third (vm3)
| '''Major second (M2)'''<br>Upaugmented unison (^A1)<br>Downminor third (vm3)
| '''E'''<br />^D#<br />vF
| '''E'''<br>^D#<br />vF
|-
|-
| 5
| 5
| 272.727
| 272.7
| Upmajor second (^M2)<br />'''Minor third (m3)'''
| Upmajor second (^M2)<br>'''Minor third (m3)'''
| ^E<br />'''F'''
| ^E<br>'''F'''
|-
|-
| 6
| 6
| 327.273
| 327.3
| '''Upminor third (^m3)'''<br />Diminished fourth (d4)
| '''Upminor third (^m3)'''<br>Diminished fourth (d4)
| '''^F'''<br />Gb
| '''^F'''<br>Gb
|-
|-
| 7
| 7
| 381.818
| 381.8
| '''Downmajor third (vM3)'''<br />Augmented second (A2)<br />Up-diminished fourth (^d4)
| '''Downmajor third (vM3)'''<br>Augmented second (A2)<br>Updiminished fourth (^d4)
| '''vF#'''<br />E#<br />^Gb
| '''vF#'''<br>E#<br>^Gb
|-
|-
| 8
| 8
| 436.364
| 436.4
| '''Major third (M3)'''<br />Up-augmented second (^A2)<br />Down-fourth (v4)
| '''Major third (M3)'''<br>Upaugmented second (^A2)<br>Down fourth (v4)
| '''F#'''<br />^E#<br />vG
| '''F#'''<br>^E#<br>vG
|-
|-
| 9
| 9
| 490.909
| 490.9
| '''Perfect fourth (P4)'''
| '''Perfect fourth (P4)'''
| '''G'''
| '''G'''
|-
|-
| 10
| 10
| 545.455
| 545.5
| Up-fourth (^4)<br />Diminished fifth (d5)
| Up fourth (^4)<br>Diminished fifth (d5)
| ^G<br />Ab
| ^G<br>Ab
|-
|-
| 11
| 11
| 600.000
| 600.0
| Down-augmented fourth (vA4)<br />Up-diminished fifth (^d5)
| Downaugmented fourth (vA4)<br>Updiminished fifth (^d5)
| vG#<br />^Ab
| vG#<br>^Ab
|-
|-
| 12
| 12
| 654.545
| 654.5
| Augmented fourth (A4)<br />Down-fifth (v5)
| Augmented fourth (A4)<br>Down fifth (v5)
| G#<br />vA
| G#<br>vA
|-
|-
| 13
| 13
| 709.091
| 709.1
| '''Perfect fifth (P5)'''
| '''Perfect fifth (P5)'''
| '''A'''
| '''A'''
|-
|-
| 14
| 14
| 763.636
| 763.6
| Up-fifth (^5)<br />Minor sixth (m6)
| Up fifth (^5)<br>Minor sixth (m6)
| ^A<br />Bb
| ^A<br>Bb
|-
|-
| 15
| 15
| 818.182
| 818.2
| Down-augmented fifth (vA5)<br />Upminor sixth (^m6)
| Downaugmented fifth (vA5)<br>Upminor sixth (^m6)
| vA#<br />^Bb
| vA#<br>^Bb
|-
|-
| 16
| 16
| 872.727
| 872.7
| Augmented fifth (A5)<br />'''Downmajor sixth (vM6)'''
| Augmented fifth (A5)<br>'''Downmajor sixth (vM6)'''
| A#<br />'''vB'''
| A#<br>'''vB'''
|-
|-
| 17
| 17
| 927.273
| 927.3
| '''Major sixth (M6)'''<br />Up-augmented fifth (^A5)<br />Downminor seventh (vm7)
| '''Major sixth (M6)'''<br>Upaugmented fifth (^A5)<br>Downminor seventh (vm7)
| '''B'''<br />^A#<br />vC
| '''B'''<br>^A#<br />vC
|-
|-
| 18
| 18
| 981.818
| 981.8
| '''Minor seventh (m7)'''<br />Upmajor sixth (^M6)<br />Down-diminished octave (vd8)
| '''Minor seventh (m7)'''<br>Upmajor sixth (^M6)<br>Downdiminished octave (vd8)
| '''C'''<br />^B<br />vDb
| '''C'''<br>^B<br>vDb
|-
|-
| 19
| 19
| 1036.364
| 1036.4
| '''Upminor seventh (^m7)'''<br />Diminished octave (d8)
| '''Upminor seventh (^m7)'''<br>Diminished octave (d8)
| '''^C'''<br />Db
| '''^C'''<br>Db
|-
|-
| 20
| 20
| 1090.909
| 1090.9
| Downmajor seventh (vM7)<br />Up-diminished octave (^d8)<br />Augmented sixth (A6)
| Downmajor seventh (vM7)<br>Updiminished octave (^d8)<br>Augmented sixth (A6)
| vC#<br />^Db<br />B#
| vC#<br>^Db<br>B#
|-
|-
| 21
| 21
| 1145.455
| 1145.5
| Major seventh (M7)<br />Down-octave (v8)
| Major seventh (M7)<br>Down octave (v8)
| C#<br />vD
| C#<br>vD
|-
|-
| 22
| 22
| 1200.000
| 1200.0
| '''Perfect octave (P8)'''
| '''Perfect octave (P8)'''
| '''D'''
| '''D'''
|}
|}


Treating [[Ups and Downs Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:
Treating 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]]
[[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 434: Line 480:


[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]
[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]
Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:
{{Sharpness-sharp3}}


Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.
Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.
Line 446: Line 488:
</gallery>
</gallery>


=== Superpyth/Porcupine Notation ===
=== Sagittal notation ===
Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.
This notation uses the same sagittal sequence as edos [[15edo #Sagittal notation|15]] and [[29edo #Sagittal notation|29]], is a subset of the notations for edos [[44edo #Sagittal notation|44]] and [[66edo #Sagittal notation|66]], and is a superset of the notation for [[11edo #Sagittal notation|11edo]].
 
==== Evo flavor ====
{{Sagittal chart|Evo}}
 
==== Revo flavor ====
{{Sagittal chart}}
 
When 22edo is treated as generated by a cycle of its fifths, the natural notes {{nowrap|F C G D A E B}} represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (Pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:
 
[[File:22edo.png|alt=22edo.png|22edo.png]]
 
This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.
 
The division of the apotome into three syntonic commas also indicates 22's tempering out of the [[250/243|porcupine comma]] (which is equivalent to three syntonic commas minus a Pythagorean apotome).
 
We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:


=== Porcupine Notation ===
[[File:22edo Sagittal.png|800px]]
Porcupine Notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. The natural notes represent a chain of 2nds ABCDEFG. This is the only way to use a heptatonic notation without additional accidentals.
 
=== Superpyth/porcupine notation ===
Superpyth/porcupine notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.
 
=== Porcupine notation ===
Porcupine notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. The natural notes represent a chain of 2nds ABCDEFG. This is the only way to use a heptatonic notation without additional accidentals.


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


=== Pentatonic Notation ===
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.
In Pentatonic Notation, the degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The natural notes represent a chain of 5ths FCGDA. This is the only way to use a chain-of-fifths notation without additional accidentals.  
 
=== Pentatonic notation ===
In Pentatonic notation, the degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The natural notes represent a chain of 5ths FCGDA. This is the only way to use a chain-of-fifths notation without additional accidentals.  


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


=== Decatonic Notation ===
=== Decatonic notation ===
The Decatonic Notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.
The Decatonic notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.


Chain 1: {{nowrap|C G D A E}}
Chain 1: {{nowrap|C G D A E}}
Line 469: Line 534:


In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G&ndash;D is a fifth, and so is γ&ndash;δ.
In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G&ndash;D is a fifth, and so is γ&ndash;δ.
=== Sagittal Notation ===
When 22edo is treated as generated by a cycle of its fifths, the natural notes {{nowrap|F C G D A E B}} represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (Pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:
[[File:22edo.png|alt=22edo.png|22edo.png]]
This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.
The division of the apotome into three syntonic commas also indicates 22's tempering out of the [[250/243|porcupine comma]] (which is equivalent to three syntonic commas minus a Pythagorean apotome).
We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:
[[File:22edo Sagittal.png|800px]]


=== Comparison of 22edo notation systems ===
=== Comparison of 22edo notation systems ===
{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2 mw-collapsible mw-collapsed"
|-
|-
! [[Degree]]
! [[Degree]]
! [[Cent]]s
! [[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="2" | Superpyth/porcupine
! colspan="3" | Porcupine
! colspan="3" | Porcupine (Onyx)
! colspan="3" | Porcupine (Zarlino)
! colspan="3" | Pentatonic
! colspan="3" | Pentatonic
! colspan="3" | Decatonic
! colspan="3" | Decatonic
! colspan="3" | [[Ups and downs notation|Ups and Downs]]
! colspan="3" | [[Ups and downs notation|Ups and downs]]
! colspan="3" | [[SKULO interval names]]
! colspan="3" | [[SKULO interval names]]
|-
|-
| 0
| 0
| 0
| 0
| Natural Unison
| Natural unison
| 1
| 1
| perfect unison
| perfect unison
| P1
| P1
| D
| D
| perfect unison
| P1
| C
| perfect unison
| perfect unison
| P1
| P1
Line 522: Line 578:
| A1
| A1
| D#
| D#
| augmented unison
| A1
| C#
| aug unison
| aug unison
| A1
| A1
Line 542: Line 601:
| d2
| d2
| Eb
| Eb
| double-aug unison,<br />double-dim sub3rd
| minor second
| AA1,<br />dds3
| m2
| Dx,<br />Fb<span style="vertical-align: super;">3</span>
| Db
| double-aug unison,<br>double-dim sub3rd
| AA1,<br>dds3
| Dx,<br>Fb<span style="vertical-align: super;">3</span>
| natural 2nd
| natural 2nd
| N2
| N2
Line 562: Line 624:
| P2
| P2
| E
| E
| narrow major second
| nM2
| D
| dim sub3rd
| dim sub3rd
| ds3
| ds3
Line 577: Line 642:
| 4
| 4
| 218
| 218
| (s/p) Major second
| (s/p) major second
| M2
| M2
| aug 2nd
| aug 2nd
| A2
| A2
| E#
| E#
| wide major second
| WM2
| D#
| minor sub3rd
| minor sub3rd
| ms3
| ms3
Line 602: Line 670:
| d3
| d3
| Fb
| Fb
| wolf third
| w3
| Ebb
| major sub3rd
| major sub3rd
| Ms3
| Ms3
Line 622: Line 693:
| m3
| m3
| F
| F
| minor third
| m3
| Eb
| aug sub3rd
| aug sub3rd
| As3
| As3
Line 637: Line 711:
| 7
| 7
| 382
| 382
| p-Major third
| p-major third
| pM3
| pM3
| major 3rd
| major 3rd
| M3
| M3
| F#
| F#
| double-aug sub3rd,<br />double-dim 4thoid
| major third
| AAs3,<br />dd4d
| M3
| Fx,<br />Gbb
| E
| double-aug sub3rd,<br>double-dim 4thoid
| AAs3,<br>dd4d
| Fx,<br>Gbb
| natural 4th
| natural 4th
| N4
| N4
Line 657: Line 734:
| 8
| 8
| 436
| 436
| s-Major third
| s-major third
| sM3
| sM3
| aug 3rd, dim 4th
| aug 3rd, dim 4th
| A3, d4
| A3, d4
| Fx, Gb
| Fx, Gb
| augmented third
| A3
| E#
| dim 4thoid
| dim 4thoid
| d4d
| d4d
Line 677: Line 757:
| 9
| 9
| 491
| 491
| Natural Fourth
| Natural fourth
| 4, N4
| 4, N4
| minor 4th
| minor 4th
| m4
| m4
| G
| G
| perfect fourth
| P4
| F
| perfect 4thoid
| perfect 4thoid
| P4d
| P4d
Line 697: Line 780:
| 10
| 10
| 545
| 545
| p-Major fourth, s-dim fifth
| p-major fourth, s-dim fifth
| pM4, sd5
| pM4, sd5
| major 4th
| major 4th
| M4
| M4
| G#
| G#
| wolf fourth
| w4
| F#
| aug 4thoid
| aug 4thoid
| A4d
| A4d
Line 717: Line 803:
| 11
| 11
| 600
| 600
| p-Augmented Fourth,<br />p-diminished Fifth,<br />Half-Octave
| p-augmented fourth,<br>p-diminished fifth,<br>half-octave
| A4, HO
| A4, HO
| aug 4th, <br />dim 5th
| aug 4th, <br>dim 5th
| A4, d5
| A4, d5
| Gx, <br />Abb
| Gx, <br>Abb
| double-aug 4thoid,<br />double-dim 5thoid
| augmented fourth, diminished fifth
| AA4d, <br />dd5d
| A4, d5
| Gx, <br />Abb
| F##, Gbb
| double-aug 4thoid,<br>double-dim 5thoid
| AA4d, <br>dd5d
| Gx, <br>Abb
| natural 6th
| natural 6th
| N6
| N6
Line 731: Line 820:
| vA4, ^d5
| vA4, ^d5
| vG#, ^Ab
| vG#, ^Ab
| comma-narrow augmented 4th<br />comma-wide diminished 5th
| comma-narrow augmented 4th<br>comma-wide diminished 5th
| kA4<br />Kd5
| kA4<br>Kd5
| kG#, KAb
| kG#, KAb
|-
|-
| 12
| 12
| 655
| 655
| p-minor Fifth, s-aug Fourth
| p-minor fifth, s-aug fourth
| pm5, sA4
| pm5, sA4
| minor 5th
| minor 5th
| m5
| m5
| Ab
| Ab
| wolf fifth
| w5
| Gb
| dim 5thoid
| dim 5thoid
| d5d
| d5d
Line 757: Line 849:
| 13
| 13
| 709
| 709
| Natural Fifth
| Natural fifth
| 5, N5
| 5, N5
| major 5th
| major 5th
| M5
| M5
| A
| A
| perfect fifth
| P5
| G
| perfect 5thoid
| perfect 5thoid
| P5d
| P5d
Line 782: Line 877:
| A5, d6
| A5, d6
| A#, Bbb
| A#, Bbb
| diminished sixth
| d6
| Abb
| aug 5thoid
| aug 5thoid
| A5d
| A5d
Line 802: Line 900:
| m6
| m6
| Bb
| Bb
| double-aug 5thoid,<br />double-dim sub7th
| minor sixth
| AA5d,<br />dds7
| m6
| Ax,<br />Cb<span style="vertical-align: super;">3</span>
| Ab
| double-aug 5thoid,<br>double-dim sub7th
| AA5d,<br>dds7
| Ax,<br>Cb<span style="vertical-align: super;">3</span>
| flat 8th
| flat 8th
| f8
| f8
Line 817: Line 918:
| 16
| 16
| 873
| 873
| p-Major sixth
| p-major sixth
| pM6
| pM6
| major 6th
| major 6th
| M6
| M6
| B
| B
| major sixth
| M6
| A
| dim sub7th
| dim sub7th
| ds7
| ds7
Line 837: Line 941:
| 17
| 17
| 927
| 927
| s-Major sixth
| s-major sixth
| sM6
| sM6
| aug 6th
| aug 6th
| A6
| A6
| B#
| B#
| wolf sixth
| w6
| A#
| minor sub7th
| minor sub7th
| ms7
| ms7
Line 862: Line 969:
| d7
| d7
| Cb
| Cb
| narrow minor seventh
| nm7
| Bbb
| major sub7th
| major sub7th
| Ms7
| Ms7
Line 877: Line 987:
| 19
| 19
| 1036
| 1036
| p-Major seventh
| p-major seventh
| pM7
| pM7
| perfect 7th
| perfect 7th
| P7
| P7
| C
| C
| wide minor seventh
| Wm7
| Bb
| aug sub7th
| aug sub7th
| As7
| As7
Line 897: Line 1,010:
| 20
| 20
| 1091
| 1091
| p-Augmented seventh
| p-augmented seventh
| pA7
| pA7
| aug 7th
| aug 7th
| A7
| A7
| C#
| C#
| double-aug sub7th,<br />double-dim octave
| major seventh
| AAs7,<br />dd8
| M7
| Cx,<br />Dbb
| B
| double-aug sub7th,<br>double-dim octave
| AAs7,<br>dd8
| Cx,<br>Dbb
| natural 10th
| natural 10th
| N10
| N10
Line 917: Line 1,033:
| 21
| 21
| 1145
| 1145
| s-Major seventh
| s-major seventh
| sM7
| sM7
| dim 8ve
| dim 8ve
| d8
| d8
| Db
| Db
| diminished octave
| d8
| Cb
| dim octave
| dim octave
| d8
| d8
Line 942: Line 1,061:
| P8
| P8
| D
| D
| perfect octave
| P8
| C
| perfect octave
| perfect octave
| P8
| P8
Line 958: Line 1,080:
== Approximation to JI ==
== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
=== Interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals|22}}
{{Q-odd-limit intervals|22}}
 
{{Q-odd-limit intervals|22.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 22f val mapping}}
== 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. Both [[50/49]], (jubilee comma), and 64/63, (septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. 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¢ "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|3, 2, 3, 2, 3, 2, 3, 2, 2|med}} and {{dash|1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2|med}}. Harmonically, orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]] and [[84edo]]. But 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.
 
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 ==
== Regular temperament properties ==
Line 1,001: Line 1,100:
| {{monzo| 35 -22 }}
| {{monzo| 35 -22 }}
| {{mapping| 22 35 }}
| {{mapping| 22 35 }}
| &minus;2.25
| −2.25
| 2.25
| 2.25
| 4.12
| 4.12
Line 1,008: Line 1,107:
| 250/243, 2048/2025
| 250/243, 2048/2025
| {{mapping| 22 35 51 }}
| {{mapping| 22 35 51 }}
| &minus;0.86
| −0.86
| 2.70
| 2.70
| 4.94
| 4.94
Line 1,015: Line 1,114:
| 50/49, 64/63, 245/243
| 50/49, 64/63, 245/243
| {{mapping| 22 35 51 62 }}
| {{mapping| 22 35 51 62 }}
| &minus;1.80
| −1.80
| 2.85
| 2.85
| 5.23
| 5.23
Line 1,022: Line 1,121:
| 50/49, 55/54, 64/63, 99/98
| 50/49, 55/54, 64/63, 99/98
| {{mapping| 22 35 51 62 76 }}
| {{mapping| 22 35 51 62 76 }}
| &minus;1.11
| −1.11
| 2.90
| 2.90
| 5.33
| 5.33
Line 1,029: Line 1,128:
| 50/49, 55/54, 64/63, 85/84, 99/98
| 50/49, 55/54, 64/63, 85/84, 99/98
| {{mapping| 22 35 51 62 76 90 }}
| {{mapping| 22 35 51 62 76 90 }}
| &minus;1.09
| −1.09
| 2.65
| 2.65
| 4.87
| 4.87
Line 1,037: Line 1,136:


=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|21.5|22.5}}
{{Uniform map|edo=22}}


=== Commas ===
=== Commas ===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)
22et [[tempering out|tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
Line 1,053: Line 1,152:
| 3
| 3
| <abbr title="34359738368/31381059609">(22 digits)</abbr>
| <abbr title="34359738368/31381059609">(22 digits)</abbr>
| {{monzo| 35 -22 }}
| {{Monzo| 35 -22 }}
| 156.98
| 156.98
| Trisawa
| Trisawa
Line 1,060: Line 1,159:
| 5
| 5
| [[20480/19683]]
| [[20480/19683]]
| {{monzo| 12 -9 1 }}
| {{Monzo| 12 -9 1 }}
| 68.72
| 68.72
| Sayo
| Sayo
| Superpyth comma, retroptolematic minor second
| Superpyth comma
|-
|-
| 5
| 5
| [[250/243]]
| [[250/243]]
| {{monzo| 1 -5 3 }}
| {{Monzo| 1 -5 3 }}
| 49.17
| 49.17
| Triyo
| Triyo
| Porcupine comma, maximal diesis
| Porcupine comma
|-
|-
| 5
| 5
| [[3125/3072]]
| [[3125/3072]]
| {{monzo|-10 -1 5 }}
| {{Monzo| -10 -1 5 }}
| 29.61
| 29.61
| Laquinyo
| Laquinyo
Line 1,081: Line 1,180:
| 5
| 5
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| {{Monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Sagugu
Line 1,088: Line 1,187:
| 5
| 5
| [[2109375/2097152| (14 digits)]]
| [[2109375/2097152| (14 digits)]]
| {{monzo|-21 3 7 }}
| {{Monzo| -21 3 7 }}
| 10.06
| 10.06
| Lasepyo
| Lasepyo
Line 1,095: Line 1,194:
| 5
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{monzo| 32 -7 -9 }}
| {{Monzo| 32 -7 -9 }}
| 9.49
| 9.49
| Sasa-tritrigu
| Sasa-tritrigu
Line 1,102: Line 1,201:
| 5
| 5
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
| {{monzo|-53 10 16 }}
| {{Monzo| -53 10 16 }}
| 0.57
| 0.57
| Quadla-quadquadyo
| Quadla-quadquadyo
Line 1,109: Line 1,208:
| 7
| 7
| [[50/49]]
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| {{Monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyo
Line 1,116: Line 1,215:
| 7
| 7
| [[64/63]]
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ru
Line 1,123: Line 1,222:
| 7
| 7
| [[875/864]]
| [[875/864]]
| {{monzo|-5 -3 3 1 }}
| {{Monzo|-5 -3 3 1 }}
| 21.90
| 21.90
| Zotriyo
| Zotriyo
Line 1,130: Line 1,229:
| 7
| 7
| [[2430/2401]]
| [[2430/2401]]
| {{monzo| 1 5 1 -4 }}
| {{Monzo| 1 5 1 -4 }}
| 20.79
| 20.79
| Quadru-ayo
| Quadru-ayo
Line 1,137: Line 1,236:
| 7
| 7
| [[245/243]]
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| {{Monzo| 0 -5 1 2 }}
| 14.19
| 14.19
| Zozoyo
| Zozoyo
Line 1,144: Line 1,243:
| 7
| 7
| [[1728/1715]]
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| {{Monzo| 6 3 -1 -3 }}
| 13.07
| 13.07
| Triru-agu
| Triru-agu
Line 1,151: Line 1,250:
| 7
| 7
| [[225/224]]
| [[225/224]]
| {{monzo|-5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
Line 1,158: Line 1,257:
| 7
| 7
| [[10976/10935]]
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| {{Monzo| 5 -7 -1 3 }}
| 6.48
| 6.48
| Trizo-agu
| Trizo-agu
Line 1,165: Line 1,264:
| 7
| 7
| [[6144/6125]]
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| {{Monzo| 11 1 -3 -2 }}
| 5.36
| 5.36
| Saruru-atrigu
| Saruru-atrigu
Line 1,172: Line 1,271:
| 7
| 7
| [[65625/65536]]
| [[65625/65536]]
| {{monzo|-16 1 5 1 }}
| {{Monzo| -16 1 5 1 }}
| 2.35
| 2.35
| Lazoquinyo
| Lazoquinyo
Line 1,179: Line 1,278:
| 7
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| <abbr title="420175/419904">(12 digits)</abbr>
| {{monzo|-6 -8 2 5 }}
| {{Monzo| -6 -8 2 5 }}
| 1.12
| 1.12
| Quinzo-ayoyo
| Quinzo-ayoyo
Line 1,186: Line 1,285:
| 11
| 11
| [[99/98]]
| [[99/98]]
| {{monzo|-1 2 0 -2 1 }}
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruru
Line 1,193: Line 1,292:
| 11
| 11
| [[100/99]]
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyo
Line 1,200: Line 1,299:
| 11
| 11
| [[121/120]]
| [[121/120]]
| {{monzo|-3 -1 -1 0 2 }}
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
Line 1,207: Line 1,306:
| 11
| 11
| [[176/175]]
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
Line 1,214: Line 1,313:
| 11
| 11
| [[896/891]]
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| 9.69
| Saluzo
| Saluzo
Line 1,221: Line 1,320:
| 11
| 11
| [[65536/65219]]
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| 8.39
| Satrilu-aruru
| Satrilu-aruru
Line 1,228: Line 1,327:
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{monzo|-7 -1 1 1 1 }}
| {{Monzo|-7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
Line 1,235: Line 1,334:
| 11
| 11
| [[540/539]]
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| 3.21
| Lururuyo
| Lururuyo
Line 1,242: Line 1,341:
| 11
| 11
| [[4000/3993]]
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| {{Monzo| 5 -1 3 0 -3 }}
| 3.03
| 3.03
| Triluyo
| Triluyo
Line 1,249: Line 1,348:
| 11
| 11
| [[9801/9800]]
| [[9801/9800]]
| {{monzo|-3 4 -2 -2 2 }}
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Bilorugu
Line 1,256: Line 1,355:
| 13
| 13
| [[65/64]]
| [[65/64]]
| {{monzo|-6 0 1 0 0 1 }}
| {{Monzo| -6 0 1 0 0 1 }}
| 26.84
| 26.84
| Thoyo
| Thoyo
Line 1,263: Line 1,362:
| 13
| 13
| [[78/77]]
| [[78/77]]
| {{monzo| 1 1 0 -1 -1 1 }}
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.34
| 22.34
| Tholuru
| Tholuru
Line 1,270: Line 1,369:
| 13
| 13
| [[91/90]]
| [[91/90]]
| {{monzo|-1 -2 -1 1 0 1 }}
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozogu
Line 1,277: Line 1,376:
| 13
| 13
| [[31213/31104]]
| [[31213/31104]]
| {{monzo|-7 -5 0 4 0 1 }}
| {{Monzo| -7 -5 0 4 0 1 }}
| 6.06
| 6.06
| Thoquadzo
| Thoquadzo
Line 1,284: Line 1,383:
| 31
| 31
| [[125/124]]
| [[125/124]]
| {{monzo|-2 0 3 0 0 0 0 0 0 0 -1 }}
| {{Monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| 13.91
| Thiwutriyo
| Thiwutriyo
Line 1,343: Line 1,442:
| 11
| 11
| 1\22
| 1\22
| [[Undeka]]<br>[[Hendecatonic]]
| [[Undeka]]<br>[[Hendecatonic (temperament)|Hendecatonic]]
|}
|}
== Octave stretch or compression ==
22edo can benefit from slightly compressing the octave, especially when using it as an 7-limit equal temperament. With the right amount of compression 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.
Good compressed-22 options include: [[ZPI|80zpi]] or [[57ed6]].


== Scales ==
== Scales ==
''See [[22edo modes]]''.
{{Main|22edo modes}}
{{See also|List of MOS scales in 22edo}}


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


Line 1,368: Line 1,474:
|-
|-
| fourthward wa
| fourthward wa
| {{monzo| a b }} where b &lt; &minus;1
| {{monzo| a b }} where {{nowrap|b &lt; −1}}
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
| gu
| gu
| {{monzo| a b -1 }}
| {{monzo| a b −1 }}
| 6/5, 9/5
| 6/5, 9/5
|-
|-
Line 1,383: Line 1,489:
| rowspan="2" | major
| rowspan="2" | major
| fifthward wa
| fifthward wa
| {{monzo| a b }} where b &gt; 1
| {{monzo| a b }} where {{nowrap|b &gt; 1}}
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| ru
| ru
| {{monzo| a b 0 -1 }}
| {{monzo| a b 0 −1 }}
| 9/7, 12/7
| 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"
{| class="wikitable center-all"
Line 1,440: Line 1,546:
* 0-5-12 = C Eb vG = Cm(v5)
* 0-5-12 = C Eb vG = Cm(v5)


Further discussion of 22edo chord naming:
== Instruments ==
== Scordatura piano ==
Although it does not allow for much in the way of modulation, it is possible to make some music using a piano tuned to a 12 note subset of 22edo, as shown by [[Juhani Nuorvala]]'s [https://www.youtube.com/watch?v=raRiTvogBBA ''Improvisations on a piano tuned to 22edo''] (2026).


* [[22edo Chord Names]]
* [[22 EDO Chords]]
* [[Ups and Downs Notation #Chords and Chord Progressions]]
* [[Chords of orwell]]
== Instruments ==
=== Keyboards ===
=== 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.
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 ==
== Music ==
{{Main| 22edo/Music }}
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}
{{Catrel|22edo tracks}}


== Related pages ==
== See also ==
* [[Lumatone mapping for 22edo]]
* [[User:Unque/22edo Composition Theory|Unque's approach]]
* [[William Lynch's thoughts on septimal harmony and 22edo]]
* [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch's approach]]
* [[22edo/Eliora's approach|22edo/Eliora's Approach]]
* [[22edo/Eliora's approach|Eliora's approach]]
* [[List of MOS scales in 22edo]]


== Further reading ==
== Further reading ==
Line 1,479: Line 1,583:
[[Category:Alpharabian]]
[[Category:Alpharabian]]
[[Category:Superpyth]]
[[Category:Superpyth]]
[[Category:Pajara]]
[[Category:Orwell]]
[[Category:Porcupine]]
[[Category:Porcupine]]
[[Category:Magic]]
[[Category:Magic]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Todo:complete table]]
[[Category:Todo:complete table]]
Retrieved from "https://en.xen.wiki/w/22edo"