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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 ==
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.
 
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.
 
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.
 
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 ===
{{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.
 
==== Superpyth temperament ====
The 5L 2s diatonic (LLsLLLs) in 22edo is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic mos has subminor and supermajor thirds of 7/6 and 9/7, rather than classical minor and major thirds of 6/5 and 5/4. This means that the septimal comma 64/63 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).


==Theory==
==== Pajara temperament ====
=== Prime harmonics===
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.
{{Harmonics in equal|22|columns=11}}


===History===
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].
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 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 J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.


===Overview to JI approximation quality===
==== Additional commas ====
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 integral or gap [[EDO]] it at least qualifies as a [[The Riemann zeta function and tuning #Zeta EDO lists|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.
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.


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 fairly 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.
=== 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.


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]].
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]].


===Subsets and supersets===
=== Other features ===
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 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.


==Intervals==
=== 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 ==
{{See also|22edo solfege}}
{{See also|22edo solfege}}
{{See also|SKULO interval names#Alternatives}}
{{See also|SKULO interval names#Alternatives}}
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{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
!Degree
! Degree
!Cents
! Cents
!Approximate Ratios*
! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref>
! colspan="3" | [[Ups and Downs Notation]]
! Audio
! colspan="3" |[[SKULO interval names|SKULO notation]] (K = 1)
! 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)
!Audio
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}
|-
|-
|0
| 0
|0.000
| 0.0
|[[1/1]]
| [[1/1]]
|perfect unison
| [[File:0-0.000c_P1.mp3]]
|P1
| perfect unison
|D
| P1
|perfect unison
| D
|P1
| perfect unison
|D
| P1
|[[File:0-0.000c_P1.mp3]]
| D
|-
|-
|1
| 1
|54.545
| 54.5
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
| [[File:0-54.545c_22edo.mp3]]
|^1, m2
| up-unison, minor 2nd
|^D, Eb
| ^1, m2
|comma-wide unison, minor 2nd
| ^D, Eb
|K1, m2
| comma-wide unison, minor 2nd
|KD, Eb
| K1, m2
|[[File:0-54.545c_22edo.mp3]]
| KD, Eb
|-
|-
|2
| 2
|109.091
| 109.1
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
| [[File:0-109.091c_11edo.mp3]]
|vA1, ^m2
| downaug 1sn, upminor 2nd
|vD#, ^Eb
| vA1, ^m2
|classic minor 2nd
| vD#, ^Eb
| classic minor 2nd
| 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]]
|aug 1sn, downmajor 2nd
| [[File:0-163.636c_22edo.mp3]]
|A1, vM2
| aug 1sn, downmajor 2nd
|D#, vE
| A1, vM2
|classic/comma-narrow major 2nd
| D#, vE
| classic/comma-narrow major 2nd
| 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
|E
| E
| major 2nd
| major 2nd
|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
|F
| F
| minor 3rd
| minor 3rd
|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]]
|upminor 3rd
| [[File:0-327.273c_11edo.mp3]]
| upminor 3rd
| ^m3
| ^m3
|^F
| ^F
|classic minor 3rd
| classic minor 3rd
| Km3
| Km3
|KF
| KF
|[[File:0-327.273c_11edo.mp3]]
|-
|-
|7
| 7
|381.818
| 381.8
|[[5/4]], [[96/77]]
| [[5/4]], [[96/77]]
|downmajor 3rd
| [[File:0-381.818c_22edo.mp3]]
| downmajor 3rd
| vM3
| vM3
| vF#
| vF#
|classic major 3rd
| classic major 3rd
| 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
|F#
| F#
| major 3rd
| major 3rd
|M3
| M3
|F#
| F#
|[[File:0-436.364c_11edo.mp3]]
|-
|-
|9
| 9
|490.909
| 490.9
|[[4/3]]
| [[4/3]]
|perfect 4th
| [[File:0-490.909c_22edo.mp3]]
|P4
| perfect 4th
|G
| P4
|perfect 4th
| G
|P4
| perfect 4th
|G
| P4
|[[File:0-490.909c_22edo.mp3]]
| G
|-
|-
|10
| 10
|545.455
| 545.5
|[[15/11]], [[11/8]]
| [[15/11]], [[11/8]]
|up-4th, dim 5th
| [[File:0-545.455c_11edo.mp3]]
|^4, d5
| up-4th, dim 5th
|^G, Ab
| ^4, d5
|comma-wide 4th
| ^G, Ab
|K4
| comma-wide 4th
|KG
| K4
|[[File:0-545.455c_11edo.mp3]]
| KG
|-
|-
|11
| 11
|600.000
| 600.0
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
| [[File:0-600.000c_2edo.mp3]]
|vA4, ^d5
| downaug 4th, updim 5th
|vG#, ^Ab
| vA4, ^d5
|comma-narrow augmented 4th<br>comma-wide diminished 5th
| vG#, ^Ab
|kA4<br>Kd5
| comma-narrow augmented 4th<br />comma-wide diminished 5th
|kG#, KAb
| kA4<br />Kd5
|[[File:0-600.000c_2edo.mp3]]
| kG#, KAb
|-
|-
|12
| 12
|654.545
| 654.5
|[[16/11]], [[22/15]]
| [[16/11]], [[22/15]]
|aug 4th, down-5th
| [[File:0-654.545c_11edo.mp3]]
|A4, v5
| aug 4th, down-5th
|G#, vA
| A4, v5
|comma-narrow 5th
| G#, vA
|k5
| comma-narrow 5th
|kA
| k5
|[[File:0-654.545c_11edo.mp3]]
| kA
|-
|-
|13
| 13
|709.091
| 709.1
|[[3/2]]
| [[3/2]]
|perfect 5th
| [[File:0-709.091c_22edo.mp3]]
|P5
| perfect 5th
|A
| P5
|perfect 5th
| A
|P5
| perfect 5th
|A
| P5
|[[File:0-709.091c_22edo.mp3]]
| A
|-
|-
|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
|Bb
| Bb
| minor 6th
| minor 6th
|m6
| m6
|Bb
| Bb
|[[File:0-763.636c_11edo.mp3]]
|-
|-
|15
| 15
|818.182
| 818.2
|[[8/5]], [[77/48]]
| [[8/5]], [[77/48]]
|upminor 6th
| [[File:0-818.182c_22edo.mp3]]
| upminor 6th
| ^m6
| ^m6
| ^Bb
| ^Bb
|classic minor 6th
| classic minor 6th
| 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]]
|downmajor 6th
| [[File:0-872.727c_11edo.mp3]]
| downmajor 6th
| vM6
| vM6
|vB
| vB
|classic major 6th
| classic major 6th
| 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
|B
| B
| major 6th
| major 6th
|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
|C
| C
| minor 7th
| minor 7th
|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]]
|upminor 7th, dim 8ve
| [[File:0-1036.364c_22edo.mp3]]
|^m7, d8
| upminor 7th, dim 8ve
|^C, Db
| ^m7, d8
|classic minor 7th
| ^C, Db
| classic minor 7th
| 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]]
|downmajor 7th, updim 8ve
| [[File:0-1090.909c_11edo.mp3]]
|vM7, ^d8
| downmajor 7th, updim 8ve
|vC#, ^Db
| vM7, ^d8
|classic major 7th
| vC#, ^Db
| classic major 7th
| 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
|C#, vD
| C#, vD
| major 7th / comma-narrow 8ve
| major 7th / comma-narrow 8ve
|M7 / k8
| M7 / k8
|C#, kD
| C#, kD
|[[File:0-1145.455c_22edo.mp3]]
|-
|-
|22
| 22
|1200.000
| 1200.0
|[[2/1]]
| [[2/1]]
|perfect octave
| [[File:0-1200.000c_P8.mp3]]
|P8
| perfect octave
|D
| P8
|perfect 8ve
| D
|P8
| perfect 8ve
|D
| P8
|[[File:0-1200.000c_P8.mp3]]
| D
|}
|}
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.


==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 Eb and D# are different notes and that Eb is lower in pitch than D#.
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}}


{| class="wikitable right-1 right-2 center-3 center-4"
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.
|+Notation of 22edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Ups and downs notation|Ups and Downs Notation]]
|-
![[5L 2s|Diatonic Interval Names]]
!Note Names<br>on D
|-
|0
|0.00
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 54.545
|Minor second (m2)<br>Up-unison (^1)
|Eb<br>^D
|-
|2
|109.091
|Upminor 2nd (^m2)<br>Down-augmented unison (vA1)
Diminished third (d3)
|^Eb<br>vD#
Fb
|-
|3
|163.636
|Downmajor second (vM2)<br>Augmented unison (A1)
|vE<br>D#
|-
|4
|218.182
| '''Major second (M2)'''
Up-augmented unison (^A1)<br>Downminor third (vm3)
|'''E'''
^D#<br>vF
|-
|5
|272.727
|Upmajor second (^M2)<br>'''Minor third (m3)'''
|^E<br>'''F'''
|-
| 6
|327.273
|'''Upminor third (^m3)'''<br>Diminished fourth (d4)
|'''^F'''<br>Gb
|-
|7
|381.818
|'''Downmajor third (vM3)'''<br>Augmented second (A2)
Up-diminished fourth (^d4)
|'''vF#'''<br>E#
^Gb
|-
|8
| 436.364
|'''Major third (M3)'''
Up-augmented second (^A2)


Down-fourth (v4)
=== Kite's ups and downs notation ===
|'''F#'''
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
^E#
{{sharpness-sharp3a}}
 
vG
|-
|9
|490.909
| '''Perfect fourth (P4)'''
|'''G'''
|-
|10
|545.455
|Up-fourth (^4)
Diminished fifth (d5)
|^G<br>Ab
|-
|11
| 600.000
|Down-augmented fourth (vA4)
Up-diminished fifth (^d5)
| vG#
^Ab
|-
|12
|654.545
|Augmented fourth (A5)<br>Down-fifth (v5)
|G#<br>vA
|-
|13
|709.091
|'''Perfect fifth (P5)'''
|'''A'''
|-
|14
| 763.636
|Up-fifth (^5)<br>Minor sixth (m6)
|^A
Bb
|-
|15
| 818.182
|Down-augmented fifth (vA5)
Upminor sixth (^m6)
|vA#
^Bb
|-
|16
|872.727
|Augmented fifth (A5)
'''Downmajor sixth (vM6)'''
|A#<br>'''vB'''
|-
| 17
|927.273
|'''Major sixth (M6)'''
Up-augmented fifth (^A5)


Downminor seventh (vm7)
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♯.
|'''B'''
^A#


vC
{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 22edo
|-
|-
|18
! rowspan="2" | [[Degree|#]]
|981.818
! rowspan="2" | [[Cent]]s
|'''Minor seventh (m7)'''
! colspan="2" | [[Kite's ups and downs notation]]
Upmajor sixth (^M6)
 
Down-diminished octave (vd8)
|'''C'''
^B
 
vDb
|-
|-
|19
! [[5L 2s|Diatonic interval names]]
|1036.364
! Note names
|'''Upminor seventh (^m7)'''
Diminished octave (d8)
|'''^C'''
Db
|-
|-
|20
| 0
|1090.909
| 0.0
|Downmajor seventh (vM7)
| '''Perfect unison (P1)'''
Up-diminished octave (^d8)
| '''D'''
 
Augmented sixth (A6)
|vC#
^Db
 
B#
|-
|-
|21
| 1
|1145.455
| 54.5
|Major seventh (M7)
| Minor second (m2)<br>Up unison (^1)
Down-octave (v8)
| Eb<br>^D
|C#
vD
|-
|-
|22
| 2
|1200.000
| 109.1
|'''Perfect octave (P8)'''
| Upminor second (^m2)<br>Downaugmented unison (vA1)<br>Diminished third (d3)
|'''D'''
| ^Eb<br>vD#<br>Fb
|}
 
== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
===Interval mappings===
{{Q-odd-limit intervals|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, and therefore is not a system of [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]] and [[53edo]].
 
The diatonic scale it produces is instead derived from [[superpyth]] temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22et. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, 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 flat minor whole tone of [[10/9]], two of which is a slightly 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 is notable for being the 5-limit temperament lowest in [[badness]] which is ''not'' approximated by the familiar 12-tone equal temperament, 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 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (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 (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 164¢ "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 3 2 3 2 3 2 3 2 2 and 1 2 2 1 2 2 1 2 2 1 2 2 2. 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==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
| 3
![[TE simple badness|Relative]] (%)
| 163.6
| Downmajor second (vM2)<br>Augmented unison (A1)
| vE<br>D#
|-
|-
|2.3
| 4
|{{monzo| 35 -22 }}
| 218.2
|[{{val| 22 35 }}]
| '''Major second (M2)'''<br>Upaugmented unison (^A1)<br>Downminor third (vm3)
|−2.25
| '''E'''<br>^D#<br />vF
|2.25
|4.12
|-
|-
|2.3.5
| 5
|250/243, 2048/2025
| 272.7
|[{{val| 22 35 51 }}]
| Upmajor second (^M2)<br>'''Minor third (m3)'''
| −0.86
| ^E<br>'''F'''
|2.70
|4.94
|-
|-
|2.3.5.7
| 6
| 50/49, 64/63, 245/243
| 327.3
|[{{val| 22 35 51 62 }}]
| '''Upminor third (^m3)'''<br>Diminished fourth (d4)
|−1.80
| '''^F'''<br>Gb
|2.85
|5.23
|-
|2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
|−1.11
|2.90
|5.33
|-
|2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
|−1.09
| 2.65
|4.87
|}
 
22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].
 
===Uniform maps ===
{{Uniform map|13|21.5|22.5}}
 
===Commas===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)
 
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime <br>limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
![[Cents]]
![[Color name]]
!Name
|-
|3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
|{{monzo| 35 -22 }}
|156.98
|Trisawa
|22-comma
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
|49.17
|Triyo
|Porcupine comma, maximal diesis
|-
|5
|[[3125/3072]]
|{{monzo| -10 -1 5 }}
|29.61
| Laquinyo
|Magic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
| Sagugu
|Diaschisma
|-
|5
|[[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
|10.06
|Lasepyo
|[[Semicomma]]
|-
|5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
|9.49
|Sasa-tritrigu
|[[Escapade comma]]
|-
|5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
|0.57
|Quadla-quadquadyo
|[[Kwazy comma]]
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.26
| Ru
|Septimal comma
|-
|7
|[[875/864]]
|{{monzo| -5 -3 3 1 }}
| 21.90
|Zotriyo
|Keema
|-
|7
|[[2430/2401]]
|{{monzo| 1 5 1 -4 }}
|20.79
|Quadru-ayo
|Nuwell comma
|-
|7
|[[245/243]]
|{{monzo| 0 -5 1 2 }}
|14.19
|Zozoyo
|Sensamagic comma
|-
|7
|[[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
|13.07
|Triru-agu
|Orwellisma
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
| 7.71
|Ruyoyo
|Marvel comma
|-
|-
| 7
| 7
|[[10976/10935]]
| 381.8
|{{monzo| 5 -7 -1 3 }}
| '''Downmajor third (vM3)'''<br>Augmented second (A2)<br>Updiminished fourth (^d4)
|6.48
| '''vF#'''<br>E#<br>^Gb
|Trizo-agu
| Hemimage comma
|-
|-
|7
| 8
|[[6144/6125]]
| 436.4
|{{monzo| 11 1 -3 -2 }}
| '''Major third (M3)'''<br>Upaugmented second (^A2)<br>Down fourth (v4)
|5.36
| '''F#'''<br>^E#<br>vG
|Saruru-atrigu
|Porwell comma
|-
|-
|7
| 9
|[[65625/65536]]
| 490.9
|{{monzo| -16 1 5 1 }}
| '''Perfect fourth (P4)'''
|2.35
| '''G'''
|Lazoquinyo
|Horwell comma
|-
|7
|<abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
|1.12
|Quinzo-ayoyo
|[[Wizma]]
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
| Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
|Luyoyo
|Ptolemisma
|-
|11
|[[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
|14.37
|Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle comma
|-
|-
|11
| 10
|[[65536/65219]]
| 545.5
|{{monzo| 16 0 0 -2 -3 }}
| Up fourth (^4)<br>Diminished fifth (d5)
|8.39
| ^G<br>Ab
|Satrilu-aruru
|Orgonisma
|-
|-
| 11
| 11
|[[385/384]]
| 600.0
|{{monzo| -7 -1 1 1 1 }}
| Downaugmented fourth (vA4)<br>Updiminished fifth (^d5)
|4.50
| vG#<br>^Ab
|Lozoyo
|Keenanisma
|-
|-
|11
| 12
|[[540/539]]
| 654.5
|{{monzo| 2 3 1 -2 -1 }}
| Augmented fourth (A4)<br>Down fifth (v5)
|3.21
| G#<br>vA
|Lururuyo
|Swetisma
|-
|-
|11
| 13
|[[4000/3993]]
| 709.1
|{{monzo| 5 -1 3 0 -3 }}
| '''Perfect fifth (P5)'''
|3.03
| '''A'''
|Triluyo
|Wizardharry comma
|-
|-
|11
| 14
|[[9801/9800]]
| 763.6
|{{monzo| -3 4 -2 -2 2 }}
| Up fifth (^5)<br>Minor sixth (m6)
|0.18
| ^A<br>Bb
|Bilorugu
| Kalisma
|-
|-
|13
| 15
|[[65/64]]
| 818.2
|{{monzo| -6 0 1 0 0 1 }}
| Downaugmented fifth (vA5)<br>Upminor sixth (^m6)
|26.84
| vA#<br>^Bb
|Thoyo
|Wilsorma
|-
|-
|13
| 16
|[[78/77]]
| 872.7
|{{monzo| 1 1 0 -1 -1 1 }}
| Augmented fifth (A5)<br>'''Downmajor sixth (vM6)'''
|22.34
| A#<br>'''vB'''
|Tholuru
|Negustma
|-
|-
|13
| 17
|[[91/90]]
| 927.3
|{{monzo| -1 -2 -1 1 0 1 }}
| '''Major sixth (M6)'''<br>Upaugmented fifth (^A5)<br>Downminor seventh (vm7)
| 19.13
| '''B'''<br>^A#<br />vC
| Thozogu
| Superleap comma, biome comma
|-
|-
|13
| 18
|[[31213/31104]]
| 981.8
|{{monzo| -7 -5 0 4 0 1 }}
| '''Minor seventh (m7)'''<br>Upmajor sixth (^M6)<br>Downdiminished octave (vd8)
|6.06
| '''C'''<br>^B<br>vDb
|Thoquadzo
|Praveensma
|-
|-
|31
| 19
|[[125/124]]
| 1036.4
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| '''Upminor seventh (^m7)'''<br>Diminished octave (d8)
| 13.91
| '''^C'''<br>Db
| Thiwutriyo
|Twizzler comma
|}
<references />
 
===Rank-2 temperaments===
*[[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]
 
{| class="wikitable center-1 center-2"
|-
|-
!Periods <br> per octave
| 20
!Generator
| 1090.9
!Temperaments
| Downmajor seventh (vM7)<br>Updiminished octave (^d8)<br>Augmented sixth (A6)
| vC#<br>^Db<br>B#
|-
|-
|1
| 21
|1\22
| 1145.5
|[[Sensamagic clan #Sensa|Sensa]]<br>[[Chromo]]<br>[[Ceratitid]]
| Major seventh (M7)<br>Down octave (v8)
| C#<br>vD
|-
|-
| 1
| 22
| 3\22
| 1200.0
|[[Porcupine]]
| '''Perfect octave (P8)'''
|-
| '''D'''
|1
| 5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
|-
|1
|7\22
|[[Magic]] / telepathy
|-
|1
| 9\22
|[[Superpyth]] / [[suprapyth]]
|-
|2
|1\22
|[[Shrutar]] / hemipaj<br>[[Comic]]
|-
|2
| 2\22
|[[Srutal]] / [[pajara]] / pajarous
|-
|2
|3\22
|[[Hedgehog]] / [[echidna]]
|-
|2
|4\22
|[[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]
|-
|2
|5\22
|[[Doublewide]] / fleetwood
|-
|11
|1\22
|[[Undeka]]<br>[[Hendecatonic]]
|}
|}


==Scales==
Treating ups and downs as "fused" with sharps and flats, and never appearing separately:
''See [[22edo modes]]''.


==Tetrachords ==
[[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]]
''See [[22edo tetrachords]].''


==Notation==
Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:
===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===
[[File:Tibia_22edo_ups_and_downs_guide_2.png|alt=Tibia 22edo ups and downs guide 2.png|800x150px|Tibia 22edo ups and downs guide 2.png]]
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 D * * E * * F * * G * * * A * * B * * C * * D.  
A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.


=== Pentatonic Notation===
[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]
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 D * * * * F * * * G * * * A * * * * C * * * D.  
Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.


===Decatonic Notation ===
<gallery mode="slideshow">
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.
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
 
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
Chain 1: C G D A E
</gallery>


Chain 2: γ δ α ε β
=== Sagittal notation ===
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]].


The alphabet is, in ascending order: C δ D ε E γ G α A β C
==== Evo flavor ====
{{Sagittal chart|Evo}}


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-D is a fifth, and so is γ-δ.
==== Revo flavor ====
{{Sagittal chart}}


===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:
When 22edo is treated as generated by a cycle of its fifths, the naturals 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]]
[[File:22edo.png|alt=22edo.png|22edo.png]]
Line 892: Line 509:
[[File:22edo Sagittal.png|800px]]
[[File:22edo Sagittal.png|800px]]


===Ups and Downs Notation===
=== 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.


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


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


Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:
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.


[[File:Tibia_22edo_ups_and_downs_guide_2.png|alt=Tibia 22edo ups and downs guide 2.png|800x150px|Tibia 22edo ups and downs guide 2.png]]
=== 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.  


A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.
The keyboard runs {{nowrap|D * * * * F * * * G * * * A * * * * C * * * D}}.
 
[[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:
=== 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.


{{Sharpness-sharp3}}
Chain 1: {{nowrap|C G D A E}}


Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.
Chain 2: {{nowrap|γ δ α ε β}}


<gallery mode="slideshow">
The alphabet is, in ascending order: {{nowrap|C δ D ε E γ G α A β C}}
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)
</gallery>


===Comparison of 22edo notation systems===
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.


{| class="wikitable center-all right-2"
=== Comparison of 22edo notation systems ===
{| 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" |Pentatonic
! colspan="3" | Porcupine (Zarlino)
! colspan="3" |Decatonic
! colspan="3" | Pentatonic
! colspan="3" |[[Ups and downs notation|Ups and Downs]]
! colspan="3" | Decatonic
! colspan="3" |[[SKULO interval names]]
! colspan="3" | [[Ups and downs notation|Ups and downs]]
! 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
| perfect unison
|P1
| P1
|D
|natural 1st
|N1
| C
| C
|perfect unison
| perfect unison
|P1
| P1
|D
| D
|perfect unison
| natural 1st
|P1
| N1
|D
| C
| perfect unison
| P1
| D
| perfect unison
| P1
| D
|-
|-
|1
| 1
|55
| 55
|s-minor second
| s-minor second
|sm2
| sm2
|aug unison
| aug unison
|A1
| A1
|D#
| D#
|aug unison
| augmented unison
|A1
| A1
|D#
| C#
|flat 2nd
| aug unison
|f2
| A1
|C#, δb
| D#
|up-unison, minor 2nd
| flat 2nd
| f2
| C#, δb
| up-unison, minor 2nd
| ^1, m2
| ^1, m2
|^D, Eb
| ^D, Eb
|comma-wide unison, minor 2nd
| comma-wide unison, minor 2nd
|K1, m2
| K1, m2
|KD, Eb
| KD, Eb
|-
|-
| 2
| 2
|109
| 109
|p-diminished second
| p-diminished second
|pd2
| pd2
|dim 2nd
| dim 2nd
|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
|natural 2nd
| double-aug unison,<br>double-dim sub3rd
|N2
| AA1,<br>dds3
|δ
| Dx,<br>Fb<span style="vertical-align: super;">3</span>
|downaug 1sn, upminor 2nd
| natural 2nd
|vA1, ^m2
| N2
|vD#, ^Eb
| δ
|classic minor 2nd
| downaug 1sn, upminor 2nd
|Km2
| vA1, ^m2
|KEb
| vD#, ^Eb
| classic minor 2nd
| Km2
| KEb
|-
|-
|3
| 3
| 164
| 164
| p-minor second
| p-minor second
|pm2
| pm2
|perfect 2nd
| perfect 2nd
|P2
| P2
|E
| E
|dim sub3rd
| narrow major second
|ds3
| nM2
|Fbb
| D
|sharp 2nd, flat 3rd
| dim sub3rd
|s2, f3
| ds3
|δ#, Db
| Fbb
| sharp 2nd, flat 3rd
| s2, f3
| δ#, Db
| aug 1sn, downmajor 2nd
| aug 1sn, downmajor 2nd
|A1, vM2
| A1, vM2
|D#, vE
| D#, vE
|classic/comma-narrow major 2nd
| classic/comma-narrow major 2nd
|kM2
| kM2
|kE
| kE
|-
|-
| 4
| 4
|218
| 218
|(s/p) Major second
| (s/p) major second
|M2
| M2
|aug 2nd
| aug 2nd
|A2
| A2
|E#
| E#
|minor sub3rd
| wide major second
|ms3
| WM2
|Fb
| D#
|natural 3rd
| minor sub3rd
|N3
| ms3
|D
| Fb
|major 2nd
| natural 3rd
|M2
| N3
|E
| D
|major 2nd
| major 2nd
|M2
| M2
|E
| E
| major 2nd
| M2
| E
|-
|-
|5
| 5
|273
| 273
|s-minor third
| s-minor third
|sm3
| sm3
|dim 3rd
| dim 3rd
|d3
| d3
|Fb
| Fb
|major sub3rd
| wolf third
| w3
| Ebb
| major sub3rd
| Ms3
| Ms3
|F
| F
|sharp 3rd
| sharp 3rd
| s3
| s3
|D#
| D#
|minor 3rd
| minor 3rd
|m3
| m3
|F
| F
|minor 3rd
| minor 3rd
|m3
| m3
| F
| F
|-
|-
|6
| 6
|327
| 327
|p-minor third
| p-minor third
|pm3
| pm3
|minor 3rd
| minor 3rd
|m3
| m3
|F
| F
|aug sub3rd
| minor third
|As3
| m3
|F#
| Eb
|flat 4th
| aug sub3rd
|f4
| As3
|εb
| F#
|upminor 3rd
| flat 4th
| f4
| εb
| upminor 3rd
| ^m3
| ^m3
| ^F
| ^F
| classic minor 3rd
| classic minor 3rd
|Km3
| Km3
|KF
| KF
|-
|-
| 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
|natural 4th
| double-aug sub3rd,<br>double-dim 4thoid
|N4
| AAs3,<br>dd4d
| Fx,<br>Gbb
| natural 4th
| N4
| ε
| ε
|downmajor 3rd
| downmajor 3rd
|vM3
| vM3
| vF#
| vF#
| classic major 3rd
| classic major 3rd
|kM3
| kM3
|kF#
| kF#
|-
|-
|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
|Gb
| Gb
|sharp 4th, flat 5th
| sharp 4th, flat 5th
|s4, f5
| s4, f5
|ε#, Eb
| ε#, Eb
|major 3rd
| major 3rd
|M3
| M3
|F#
| F#
|major 3rd
| major 3rd
|M3
| M3
|F#
| F#
|-
|-
| 9
| 9
| 491
| 491
|Natural Fourth
| Natural fourth
|4, N4
| 4, N4
|minor 4th
| minor 4th
|m4
| m4
| G
| perfect fourth
| P4
| F
| perfect 4thoid
| P4d
| G
| natural 5th
| N5
| E
| perfect 4th
| P4
| G
| perfect 4th
| P4
| G
| G
|perfect 4thoid
|P4d
|G
|natural 5th
|N5
|E
|perfect 4th
|P4
|G
|perfect 4th
|P4
|G
|-
|-
|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
|G#
| G#
|sharp 5th, flat 6th
| sharp 5th, flat 6th
|s5, f6
| s5, f6
|E#, γb
| E#, γb
|up-4th, dim 5th
| up-4th, dim 5th
|^4, d5
| ^4, d5
|^G, Ab
| ^G, Ab
|comma-wide 4th
| comma-wide 4th
|K4
| K4
|KG
| KG
|-
|-
| 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
| A4, d5
| F##, Gbb
| double-aug 4thoid,<br>double-dim 5thoid
| AA4d, <br>dd5d
| AA4d, <br>dd5d
|Gx, <br>Abb
| Gx, <br>Abb
|natural 6th
| natural 6th
| N6
| N6
|γ
| γ
| downaug 4th, updim 5th
| downaug 4th, updim 5th
|vA4, ^d5
| vA4, ^d5
|vG#, ^Ab
| vG#, ^Ab
|comma-narrow augmented 4th
| comma-narrow augmented 4th<br>comma-wide diminished 5th
comma-wide diminished 5th
| kA4<br>Kd5
|kA4
| kG#, KAb
Kd5
|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
|dim 5thoid
| wolf fifth
|d5d
| w5
|Ab
| Gb
| dim 5thoid
| d5d
| Ab
| sharp 6th, flat 7th
| sharp 6th, flat 7th
|s6, f7
| s6, f7
|γ#, Gb
| γ#, Gb
|aug 4th, down-5th
| aug 4th, down-5th
|A4, v5
| A4, v5
|G#, vA
| G#, vA
| comma-narrow 5th
| comma-narrow 5th
|k5
| k5
|kA
| kA
|-
|-
|13
| 13
| 709
| 709
|Natural Fifth
| Natural fifth
|5, N5
| 5, N5
|major 5th
| major 5th
|M5
| M5
|A
| A
|perfect 5thoid
| perfect fifth
|P5d
| P5
|A
| G
|natural 7th
| perfect 5thoid
|N7
| P5d
|G
| A
|perfect 5th
| natural 7th
|P5
| N7
|A
| G
|perfect 5th
| perfect 5th
|P5
| P5
|A
| A
| perfect 5th
| P5
| A
|-
|-
|14
| 14
|764
| 764
| s-minor sixth
| s-minor sixth
|sm6
| sm6
|aug 5th, dim 6th
| aug 5th, dim 6th
|A5, d6
| A5, d6
|A#, Bbb
| A#, Bbb
|aug 5thoid
| diminished sixth
|A5d
| d6
|A#
| Abb
|sharp 7th
| aug 5thoid
|s7
| A5d
|G#
| A#
| sharp 7th
| s7
| G#
| minor 6th
| m6
| Bb
| minor 6th
| minor 6th
|m6
|Bb
|minor 6th
| m6
| m6
| Bb
| Bb
|-
|-
| 15
| 15
|818
| 818
|p-minor sixth
| p-minor sixth
|pm6
| pm6
|minor 6th
| minor 6th
|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
|flat 8th
| double-aug 5thoid,<br>double-dim sub7th
|f8
| AA5d,<br>dds7
|αb
| Ax,<br>Cb<span style="vertical-align: super;">3</span>
|upminor 6th
| flat 8th
|^m6
| f8
|^Bb
| αb
| upminor 6th
| ^m6
| ^Bb
| classic minor 6th
| classic minor 6th
| Km6
| Km6
|KBb
| KBb
|-
|-
|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
|Cbb
| Cbb
|natural 8th
| natural 8th
|N8
| N8
|α
| α
| downmajor 6th
| downmajor 6th
|vM6
| vM6
|vB
| vB
|classic major 6th
| classic major 6th
|kM6
| kM6
|kB
| kB
|-
|-
| 17
| 17
|927
| 927
| s-Major sixth
| s-major sixth
|sM6
| sM6
|aug 6th
| aug 6th
|A6
| A6
|B#
| B#
|minor sub7th
| wolf sixth
|ms7
| w6
|Cb
| A#
| minor sub7th
| ms7
| Cb
| sharp 8th, flat 9th
| sharp 8th, flat 9th
|s8, f9
| s8, f9
|α#, Ab
| α#, Ab
|major 6th
| major 6th
|M6
| M6
|B
| B
|major 6th
| major 6th
|M6
| M6
|B
| B
|-
|-
|18
| 18
|982
| 982
|(s/p) minor seventh
| (s/p) minor seventh
|m7
| m7
| dim 7th
| dim 7th
|d7
| d7
|Cb
| Cb
|major sub7th
| narrow minor seventh
| nm7
| Bbb
| major sub7th
| Ms7
| Ms7
|C
| C
|natural 9th
| natural 9th
| N9
| N9
|A
| A
|minor 7th
| minor 7th
|m7
| m7
| C
| C
| minor 7th
| minor 7th
| m7
| m7
|C
| C
|-
|-
|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
|C#
| C#
|sharp 9th, flat 10th
| sharp 9th, flat 10th
|s9, f10
| s9, f10
|A#, βb
| A#, βb
|upminor 7th, dim 8ve
| upminor 7th, dim 8ve
|^m7, d8
| ^m7, d8
|^C, Db
| ^C, Db
|classic minor 7th
| classic minor 7th
|Km7
| Km7
|kC
| kC
|-
|-
| 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
|natural 10th
| double-aug sub7th,<br>double-dim octave
|N10
| AAs7,<br>dd8
| Cx,<br>Dbb
| natural 10th
| N10
| β
| β
|downmajor 7th, updim 8ve
| downmajor 7th, updim 8ve
|vM7, ^d8
| vM7, ^d8
|vC#, ^Db
| vC#, ^Db
|classic major 7th
| classic major 7th
|kM7
| kM7
|kC#
| kC#
|-
|-
|21
| 21
|1145
| 1145
|s-Major seventh
| s-major seventh
|sM7
| sM7
|dim 8ve
| dim 8ve
|d8
| d8
|Db
| Db
|dim octave
| diminished octave
|d8
| d8
|Db
| Cb
| dim octave
| d8
| Db
| sharp 10th
| sharp 10th
|s10
| s10
|β#, Cb
| β#, Cb
|major 7th, down 8ve
| major 7th, down 8ve
|M7, v8
| M7, v8
|C#, vD
| C#, vD
|major 7th / comma-narrow 8ve
| major 7th / comma-narrow 8ve
|M7 / k8
| M7 / k8
|C#, kD
| C#, kD
|-
|-
|22
| 22
|1200
| 1200
| Octave
| Octave
|8
| 8
|perfect octave
| perfect octave
| P8
| P8
|D
| D
|perfect octave
| perfect octave
|P8
| P8
|D
| C
|natural 11th
| perfect octave
|N11
| P8
|C
| D
|perfect octave
| natural 11th
|P8
| N11
|D
| C
|perfect 8ve
| perfect octave
|P8
| P8
|D
| D
| perfect 8ve
| P8
| D
|}
 
== Approximation to JI ==
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]
 
=== Interval mappings ===
{{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}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 35 -22 }}
| {{mapping| 22 35 }}
| −2.25
| 2.25
| 4.12
|-
| 2.3.5
| 250/243, 2048/2025
| {{mapping| 22 35 51 }}
| −0.86
| 2.70
| 4.94
|-
| 2.3.5.7
| 50/49, 64/63, 245/243
| {{mapping| 22 35 51 62 }}
| −1.80
| 2.85
| 5.23
|-
| 2.3.5.7.11
| 50/49, 55/54, 64/63, 99/98
| {{mapping| 22 35 51 62 76 }}
| −1.11
| 2.90
| 5.33
|-
| 2.3.5.7.11.17
| 50/49, 55/54, 64/63, 85/84, 99/98
| {{mapping| 22 35 51 62 76 90 }}
| −1.09
| 2.65
| 4.87
|}
* 22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]].
* 22et does best in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]].
 
=== Uniform maps ===
{{Uniform map|edo=22}}
 
=== Commas ===
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"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name
|-
| 3
| <abbr title="34359738368/31381059609">(22 digits)</abbr>
| {{Monzo| 35 -22 }}
| 156.98
| Trisawa
| 22-comma
|-
| 5
| [[20480/19683]]
| {{Monzo| 12 -9 1 }}
| 68.72
| Sayo
| Superpyth comma
|-
| 5
| [[250/243]]
| {{Monzo| 1 -5 3 }}
| 49.17
| Triyo
| Porcupine comma
|-
| 5
| [[3125/3072]]
| {{Monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[2109375/2097152| (14 digits)]]
| {{Monzo| -21 3 7 }}
| 10.06
| Lasepyo
| [[Semicomma]]
|-
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{Monzo| 32 -7 -9 }}
| 9.49
| Sasa-tritrigu
| [[Escapade comma]]
|-
| 5
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
| {{Monzo| -53 10 16 }}
| 0.57
| Quadla-quadquadyo
| [[Kwazy comma]]
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[875/864]]
| {{Monzo|-5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[2430/2401]]
| {{Monzo| 1 5 1 -4 }}
| 20.79
| Quadru-ayo
| Nuwell comma
|-
| 7
| [[245/243]]
| {{Monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[1728/1715]]
| {{Monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[10976/10935]]
| {{Monzo| 5 -7 -1 3 }}
| 6.48
| Trizo-agu
| Hemimage comma
|-
| 7
| [[6144/6125]]
| {{Monzo| 11 1 -3 -2 }}
| 5.36
| Saruru-atrigu
| Porwell comma
|-
| 7
| [[65625/65536]]
| {{Monzo| -16 1 5 1 }}
| 2.35
| Lazoquinyo
| Horwell comma
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo| -6 -8 2 5 }}
| 1.12
| Quinzo-ayoyo
| [[Wizma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{Monzo|-7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3 }}
| 3.03
| Triluyo
| Wizardharry comma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{Monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[78/77]]
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.34
| Tholuru
| Negustma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[31213/31104]]
| {{Monzo| -7 -5 0 4 0 1 }}
| 6.06
| Thoquadzo
| Praveensma
|-
| 31
| [[125/124]]
| {{Monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
| Twizzler comma
|}
 
=== Rank-2 temperaments ===
* [[List of 22et rank two temperaments by badness]]
* [[List of 22et rank two temperaments by complexity]]
* [[List of edo-distinct 22et rank two temperaments]]
 
{| class="wikitable center-1 center-2"
|-
! Periods<br>per 8ve
! Generator
! Temperaments
|-
| 1
| 1\22
| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]
|-
| 1
| 3\22
| [[Porcupine]]
|-
| 1
| 5\22
| [[Orwell]] (22) / blair (22) / winston (22f)
|-
| 1
| 7\22
| [[Magic]] / telepathy
|-
| 1
| 9\22
| [[Superpyth]] / [[suprapyth]]
|-
| 2
| 1\22
| [[Shrutar]] / hemipaj<br>[[Comic]]
|-
| 2
| 2\22
| [[Srutal]] / [[pajara]] / pajarous
|-
| 2
| 3\22
| [[Hedgehog]] / [[echidna]]
|-
| 2
| 4\22
| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]
|-
| 2
| 5\22
| [[Doublewide]] / fleetwood
|-
| 11
| 1\22
| [[Undeka]]<br>[[Hendecatonic (temperament)|Hendecatonic]]
|}
|}


==Chord names==
== 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 ==
{{Main|22edo modes}}
{{See also|List of MOS scales in 22edo}}
 
== Tetrachords ==
{{Main|22edo tetrachords}}
 
== 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:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
!Quality
! Quality
![[Color name]]
! [[Color name]]
![[Monzo]] Format
! [[Monzo]] Format
!Examples
! Examples
|-
|-
| rowspan="2" |minor
| rowspan="2" | minor
|zo
| zo
|[a b 0 1>
| {{monzo| a b 0 1 }}
|7/6, 7/4
| 7/6, 7/4
|-
|-
|fourthward wa
| fourthward wa
|[a b> where b &lt; -1
| {{monzo| a b }} where {{nowrap|b &lt; −1}}
|32/27, 16/9
| 32/27, 16/9
|-
|-
|upminor
| upminor
|gu
| gu
|[a b -1>
| {{monzo| a b −1 }}
|6/5, 9/5
| 6/5, 9/5
|-
|-
|downmajor
| downmajor
|yo
| yo
|[a b 1>
| {{monzo| a b 1 }}
|5/4, 5/3
| 5/4, 5/3
|-
|-
| rowspan="2" |major
| rowspan="2" | major
|fifthward wa
| fifthward wa
|[a b> where b &gt; 1
| {{monzo| a b }} where {{nowrap|b &gt; 1}}
|9/8, 27/16
| 9/8, 27/16
|-
|-
|ru
| ru
|[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"
|-
|-
![[Kite's color notation|Color of the 3rd]]
! [[Kite's color notation|Color of the 3rd]]
!JI Chord
! JI Chord
!Notes as edosteps
! Notes as edosteps
!Notes of C chord
! Notes of C chord
!Written name
! Written name
!Spoken name
! Spoken name
|-
|-
|zo
| zo
|6:7:9
| 6:7:9
|0-5-13
| 0-5-13
|C Eb G
| C Eb G
|Cm
| Cm
|C minor
| C minor
|-
|-
|gu
| gu
|10:12:15
| 10:12:15
|0-6-13
| 0-6-13
|C ^Eb G
| C ^Eb G
|C^m
| C^m
|C upminor
| C upminor
|-
|-
|yo
| yo
|4:5:6
| 4:5:6
|0-7-13
| 0-7-13
|C vE G
| C vE G
|Cv
| Cv
|C downmajor or C down
| C downmajor or C down
|-
|-
|ru
| ru
|14:18:21
| 14:18:21
|0-8-13
| 0-8-13
|C E G
| C E G
|C
| C
|C major or C
| C major or C
|}
|}


Examples:
Examples:


*0-4-13 = C D G = C2
* 0-4-13 = C D G = C2
*0-9-13 = C F G = C4
* 0-9-13 = C F G = C4
*0-10-13 = C ^F G = C^4 or C(^4)
* 0-10-13 = C ^F G = C^4 or C(^4)
*0-5-10 = C Eb Gb = Cd = Cdim
* 0-5-10 = C Eb Gb = Cd = Cdim
*0-5-11 = C Eb ^Gb = Cd(^5)
* 0-5-11 = C Eb ^Gb = Cd(^5)
*0-5-12 = C Eb vG = Cm(v5)
* 0-5-12 = C Eb vG = Cm(v5)
 
== 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).


Further discussion of 22edo chord naming:
=== Keyboards ===
[[File:22-tone halberstadt layout.png|alt=|frameless]]


*[[22edo Chord Names]]
A potential layout for a 22edo keyboard with both split black and white keys.
*[[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]


==Music==
[[Lumatone mapping for 22edo|Lumatone mappings for 22edo]] are available.
 
== 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 22 EDO]]
* [[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]]
 
== Further reading ==
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
* [http://lumma.org/tuning/erlich/erlich-decatonic.pdf Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']
* [http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]


==Further reading==
== Notes ==
*[[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
<references group="note" />
*[http://lumma.org/tuning/erlich/erlich-decatonic.pdf Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']
*[http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music]
*[https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
*[https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]


==References==
== References ==
#Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
# Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]
#Bosanquet, R.H.M. [https://www.webcitation.org/5kjJcrhEx ''On the Hindoo division of the octave, with additions to the theory of higher orders''], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965
# Bosanquet, R.H.M. [https://www.webcitation.org/5kjJcrhEx ''On the Hindoo division of the octave, with additions to the theory of higher orders''], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965


[[Category:Twentuning]]
[[Category:Twentuning]]
[[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]]
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