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<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.


== Notations ==
== JI approximation ==
=== 15-odd-limit interval mappings ===
The following tables show how [[15-odd-limit intervals]] are represented in 22edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[9/7]], [[14/9]]
| 1.280
| 2.3
|-
| [[11/10]], [[20/11]]
| 1.368
| 2.5
|-
| [[15/8]], [[16/15]]
| 2.640
| 4.8
|-
| '''[[5/4]], [[8/5]]'''
| '''4.496'''
| '''8.2'''
|-
| [[7/6]], [[12/7]]
| 5.856
| 10.7
|-
| '''[[11/8]], [[16/11]]'''
| '''5.863'''
| '''10.7'''
|-
| '''[[3/2]], [[4/3]]'''
| '''7.136'''
| '''13.1'''
|-
| [[15/11]], [[22/15]]
| 8.504
| 15.6
|-
| [[15/14]], [[28/15]]
| 10.352
| 19.0
|-
| [[5/3]], [[6/5]]
| 11.631
| 21.3
|-
| '''[[7/4]], [[8/7]]'''
| '''12.992'''
| '''23.8'''
|-
| [[11/6]], [[12/11]]
| 12.999
| 23.8
|-
| [[9/8]], [[16/9]]
| 14.272
| 26.2
|-
| [[13/11]], [[22/13]]
| 16.482
| 30.2
|-
| [[7/5]], [[10/7]]
| 17.488
| 32.1
|-
| [[13/10]], [[20/13]]
| 17.850
| 32.7
|-
| ''[[13/9]], [[18/13]]''
| ''17.928''
| ''32.9''
|-
| [[9/5]], [[10/9]]
| 18.767
| 34.4
|-
| [[11/7]], [[14/11]]
| 18.856
| 34.6
|-
| ''[[13/7]], [[14/13]]''
| ''19.207''
| ''35.2''
|-
| [[11/9]], [[18/11]]
| 20.135
| 36.9
|-
| '''[[13/8]], [[16/13]]'''
| '''22.346'''
| '''41.0'''
|-
| [[15/13]], [[26/15]]
| 24.986
| 45.8
|-
| ''[[13/12]], [[24/13]]''
| ''25.064''
| ''46.0''
|}
{{15-odd-limit|22}}
 
=== Selected 17-limit intervals ===
[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]
 
== 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]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 35 -22 }}
| [{{val| 22 35 }}]
| -2.25
| 2.25
| 4.12
|-
| 2.3.5
| 250/243, 2048/2025
| [{{val| 22 35 51 }}]
| -0.86
| 2.70
| 4.94
|-
| 2.3.5.7
| 50/49, 64/63, 245/243
| [{{val| 22 35 51 62 }}]
| -1.80
| 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]].
 
=== 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
|-
| 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]]
|-
| 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
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic
|-
| 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
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Saruru-atrigu
| Porwell
|-
| 7
| [[65625/65536]]
| {{monzo| -16 1 5 1 }}
| 2.35
| Lazoquinyo
| Horwell
|-
| 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
|-
| 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
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 31
| [[125/124]]
| {{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
| Twizzler
|}
<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
! Generator
! Temperaments
|-
| 1
| 1\22
| [[Sensamagic clan #Sensa|Sensa]]<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]]
|}
 
== Scales ==
''See [[22edo modes]]''.
 
== Tetrachords ==
''See [[22edo tetrachords]].''
 
== Approaches to notation ==
=== Superpyth/Porcupine 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.
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.
Line 283: Line 722:
{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
|-
|-
! [[Degree]]
![[Degree]]
! [[cent|Cents]]
![[cent|Cents]]
! colspan="2" | Superpyth/Porcupine Notation
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" | Porcupine
Line 763: Line 1,202:
! quality
! quality
![[color name]]
![[color name]]
! [[monzo]] format
![[monzo]] format
! examples
! examples
|-
|-
Line 799: Line 1,238:
{| 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
Line 848: Line 1,287:


For a more complete list, see [[22edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].
For a more complete list, see [[22edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].
== JI approximation ==
=== 15-odd-limit interval mappings ===
The following tables show how [[15-odd-limit intervals]] are represented in 22edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[9/7]], [[14/9]]
| 1.280
| 2.3
|-
| [[11/10]], [[20/11]]
| 1.368
| 2.5
|-
| [[15/8]], [[16/15]]
| 2.640
| 4.8
|-
| '''[[5/4]], [[8/5]]'''
| '''4.496'''
| '''8.2'''
|-
| [[7/6]], [[12/7]]
| 5.856
| 10.7
|-
| '''[[11/8]], [[16/11]]'''
| '''5.863'''
| '''10.7'''
|-
| '''[[3/2]], [[4/3]]'''
| '''7.136'''
| '''13.1'''
|-
| [[15/11]], [[22/15]]
| 8.504
| 15.6
|-
| [[15/14]], [[28/15]]
| 10.352
| 19.0
|-
| [[5/3]], [[6/5]]
| 11.631
| 21.3
|-
| '''[[7/4]], [[8/7]]'''
| '''12.992'''
| '''23.8'''
|-
| [[11/6]], [[12/11]]
| 12.999
| 23.8
|-
| [[9/8]], [[16/9]]
| 14.272
| 26.2
|-
| [[13/11]], [[22/13]]
| 16.482
| 30.2
|-
| [[7/5]], [[10/7]]
| 17.488
| 32.1
|-
| [[13/10]], [[20/13]]
| 17.850
| 32.7
|-
| ''[[13/9]], [[18/13]]''
| ''17.928''
| ''32.9''
|-
| [[9/5]], [[10/9]]
| 18.767
| 34.4
|-
| [[11/7]], [[14/11]]
| 18.856
| 34.6
|-
| ''[[13/7]], [[14/13]]''
| ''19.207''
| ''35.2''
|-
| [[11/9]], [[18/11]]
| 20.135
| 36.9
|-
| '''[[13/8]], [[16/13]]'''
| '''22.346'''
| '''41.0'''
|-
| [[15/13]], [[26/15]]
| 24.986
| 45.8
|-
| ''[[13/12]], [[24/13]]''
| ''25.064''
| ''46.0''
|}
{{15-odd-limit|22}}
=== Selected 17-limit intervals ===
[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]
== 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]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 35 -22 }}
| [{{val| 22 35 }}]
| -2.25
| 2.25
| 4.12
|-
| 2.3.5
| 250/243, 2048/2025
| [{{val| 22 35 51 }}]
| -0.86
| 2.70
| 4.94
|-
| 2.3.5.7
| 50/49, 64/63, 245/243
| [{{val| 22 35 51 62 }}]
| -1.80
| 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]].
=== 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
|-
| 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]]
|-
| 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
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic
|-
| 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
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Saruru-atrigu
| Porwell
|-
| 7
| [[65625/65536]]
| {{monzo| -16 1 5 1 }}
| 2.35
| Lazoquinyo
| Horwell
|-
| 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
|-
| 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
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 31
| [[125/124]]
| {{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| Thiwutriyo
| Twizzler
|}
<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
! Generator
! Temperaments
|-
| 1
| 1\22
| [[Sensamagic clan #Sensa|Sensa]]<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]]
|}
== Scales ==
''See [[22edo modes]]''.
== Tetrachords ==
''See [[22edo tetrachords]].''


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/22edo"