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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.
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.


==Chord Names==
== Chord names ==


See also [[22 EDO Chords]], [[Chords of orwell]].
See also [[22 EDO Chords]], [[Chords of orwell]].
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|}
|}


All 31edo 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"
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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]].


== Just approximation ==
== JI approximation ==
 
=== 15-odd-limit interval mappings ===
=== Selected just intervals by error ===
 
==== 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''.  
The following tables show how [[15-odd-limit intervals]] are represented in 22edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
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|}
|}


==== Selected 17-limit intervals ====
=== Selected 17-limit intervals ===


[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]
[[File:22ed2-001e.svg|alt=alt : Your browser has no SVG support.]]
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''See also: [[22edo Solfege]], [[22edo tetrachords]], [[22 EDO Chords]], [[22edo Modes]]''
''See also: [[22edo Solfege]], [[22edo tetrachords]], [[22 EDO Chords]], [[22edo Modes]]''


=== Temperament measures ===
== Regular temperament properties ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 22et.
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-all"
! rowspan="2" | Subgroup
! colspan="2" |
! rowspan="2" | [[Comma list]]
! 3-limit
! rowspan="2" | [[Mapping]]
! 5-limit
! rowspan="2" | Optimal<br>8ve stretch (¢)
! 7-limit
! colspan="2" | Tuning error
! 11-limit
|-
! 2.3.5.7.11.17
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3
| {{monzo| 35 -22 }}
| [{{val| 22 35 }}]
| -2.25
| -2.25
| 2.25
| 4.12
|-
| 2.3.5
| 250/243, 2048/2025
| [{{val| 22 35 51 }}]
| -0.86
| -0.86
| 2.70
| 4.94
|-
| 2.3.5.7
| 50/49, 64/63, 245/243
| [{{val| 22 35 51 62 }}]
| -1.80
| -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
| -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
| -1.09
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 2.25
| 2.70
| 2.85
| 2.90
| 2.65
| 2.65
|-
! [[TE simple badness|relative]] (%)
| 4.12
| 4.94
| 5.23
| 5.33
| 4.87
| 4.87
|}
|}


* 22et has a lower relative error than any previous ET in the 11-limit. The next ET that does better in this subgroup is 31.  
22et is lower in relative error than any previous equal temperaments in the 11-limit. The next ET that does better in this subgroup is 31.  
* 22et is most prominent in the 2.3.5.7.11.17 subgroup, and the next ET that does better in this is 46.
 
== Rank Two 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]]
 
Important MOSes include:
 
* [[superpyth]] pentatonic 2L3s 44545 (13\22, 1\1)
* [[superpyth]] diatonic 5L2s 1441444 (13\22, 1\1)
* [[superpyth]] chromatic 5L7s 113131131313 (13\22, 1\1)
* [[superpyth]] hyperchromatic 5L12s 11121121112112112 (13\22, 1\1)
* [[porcupine]] 7L1s 13333333 (3\22, 1\1)
* [[porcupine]] 7L8s 112121212121212 (3\22, 1\1)
* [[pajara]] 2L8s 2232222322 (2\22, 1\2)
* [[pajara]] 10L2s 221222221222 (2\22, 1\2)
* [[orwell]] pentatonic 4L1s 55552 (5\22, 1\1)
* [[orwell]] diatonic 4L5s 323232322 (5\22, 1\1)
* [[orwell]] chromatic 9L4s 2122122122122 (5\22, 1\1)
* [[magic]] diatonic 3L4s 1616161 (7\22, 1\1)
* [[magic]] superdiatonic 3L7s 1511511511 (7\22, 1\1)
* [[magic]] chromatic 3L10s 1411141114111 (7\22, 1\1)
* [[magic]] mega chromatic 3L13s 1131111311113111 (7\22, 1\1)
* Pathological [[magic]] enharmonic 3L16s 1112111112111112111 (7\22, 1\1)
* [[hedgehog]] hexatonic 2L4s 353353 (3\22, 1\2)
* [[hedgehog]] symmetric octatonic 6L2s 33233323 (3\22, 1\2)
* [[hedgehog]] symmetric chromatic 8L6s 21212212121221 (3\22, 1\2)
* [[astrology]] hexatonic 4L2s 434434 (4\22, 1\2)
* [[astrology]] symmetric decatonic 6L4s 3133131331 (4\22, 1\2)
* [[astrology]] symmetric hexadecatonic 6L10s 2112121121121211 (4\22, 1\2)
* [[doublewide]] tetrad 2L2s 6565 (5\22, 1\2)
* [[doublewide]] hexatonic 4L2s 515515 (5\22, 1\2)
* [[doublewide]] symmetric decatonic 4L6s 4114141141 (5\22, 1\2)
* [[Astrology|doublewide]] symmetric tetradecatonic 4L10s 31113113111311 (5\22, 1\2)
* Pathological [[doublewide]] symmetric octokaidecatonic 4L14s 211112111211112111 (5\22, 1\2)
 
{| class="wikitable"
|-
! Periods <br> per octave
! Period
! Generator
! Temperaments
|-
| 1
| 22\22
| 1\22
| [[Sensamagic clan#Sensa|Sensa]]/chromo/ceratitid
|-
| 1
| 22\22
| 3\22
| [[Porcupine]]
|-
| 1
| 22\22
| 5\22
| [[Orwell]]/blair/orson
|-
| 1
| 22\22
| 7\22
| [[Magic]]/telepathy
|-
| 1
| 22\22
| 9\22
| [[Superpyth]]/[[Suprapyth]]
|-
| 2
| 11\22
| 1\22
| [[Shrutar]]/hemipaj/comic
|-
| 2
| 11\22
| 2\22
| [[Srutal]]/[[pajara]]/pajarous
|-
| 2
| 11\22
| 3\22
| [[Porcupine family#Hedgehog|Hedgehog]]/[[echidna]]
|-
| 2
| 11\22
| 4\22
| [[Astrology]]/[[wizard]]/[[antikythera]]
|-
| 2
| 11\22
| 5\22
| [[Doublewide]]/fleetwood
|-
| 11
| 2\22
| 1\22
| [[Hendecatonic]]/undeka
|}
 
== Scales ==
Scales are written be steps in degrees of 22edo. [[MOS scale]]s are listed in their symmetric mode if one exists, and otherwise in the "brightest" mode - the mode with the highest average pitch height / the lexicographically highest mode
 
=== MOS scales ===


:''See also [[22edo Modes]], [[22edo tetrachords]]
22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next ET that does better in this is 46.
 
* Porcupine[7] - 3334333
* Porcupine[8] - 33333331
* Porcupine[15] - 121212121212121
* Orwell[5] - 55255
* Orwell[9] - 232323232
* Orwell[13] - 2122122212212
* Magic[7] - 1616161
* Magic[10] - 5115115111
* Magic[13] - 1141114111411
* Magic[16] - 3111131111311111
* Magic[19] - 1112111112111112111
* Superpyth[5] - pentatonic - 45454
* Superpyth[7] - diatonic - 4144414
* Superpyth[12] - chromatic - 313131131311
* Superpyth[17] - hyperchromatic - 12111211211211121
* Pajara[10] - symmetric decatonic - 2232222322
* Pajara[12] - 222221222221
* Hedgehog[6] - 353353
* Hedgehog[8] - 33323332
* Hedgehog[14] - 21212122121212
* Astrology[6] - 434434
* Astrology[10] - 3131331313
* Astrology[16] - 2121121121211211
* Doublewide[4] - 5656
* Doublewide[6] - 551551
* Doublewide[10] - 4141141411
* Doublewide[14] - 31131113113111
* Doublewide[18] - 211121111211121111
 
=== Other Scales ===
 
* Pentachordal decatonic - Pajara[10] 4|4(2) #8 - 2232223222
* Zarlino/Ptolemy diatonic, "just" major, Ma grama - 4324342
* inverse of Zarlino/Ptolemy diatonic, natural minor - 4234243
* tetrachordal major, Sa grama - 4324432
* inverse of tetrachordal major, "just"/tetrachordal minor - 4234234
* Porcupine bright major #7 - Porcupine[7] 6|0 #7 - 4333342
* Porcupine bright major #6 #7 - Porcupine[7] 6|0 #6 #7 - 4333432
* Porcupine bright minor #2 - Porcupine[7] 4|2 #2 4243333 (mode of bright major #7)
* Porcupine dark minor #2 - Porcupine[7] 3|3 #2 4234333 (inverse of bright major #6 #7)
* Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7 4333324
* Superpyth harmonic minor - Superpyth[7] 2|4 #7 - 4144171
* Superpyth harmonic major - Superpyth[7] 5|1 b6 - 4414171 (inverse of harmonic minor)
* Superpyth melodic minor - Superpyth[7] 5|1 b3 - 4144441
* Superpyth double harmonic - Superpyth[7] 5|1 b2 b6 - 1714171
* "just" harmonic minor - 4234252
* "just" harmonic major - 4324252
* "just" melodic minor - 4234342
* "just" double harmonic - 2524252


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


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<references/>
<references/>


== Staff Notation ==
=== 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]]
 
Important MOSes include:
 
* [[superpyth]] pentatonic 2L3s 44545 (13\22, 1\1)
* [[superpyth]] diatonic 5L2s 1441444 (13\22, 1\1)
* [[superpyth]] chromatic 5L7s 113131131313 (13\22, 1\1)
* [[superpyth]] hyperchromatic 5L12s 11121121112112112 (13\22, 1\1)
* [[porcupine]] 7L1s 13333333 (3\22, 1\1)
* [[porcupine]] 7L8s 112121212121212 (3\22, 1\1)
* [[pajara]] 2L8s 2232222322 (2\22, 1\2)
* [[pajara]] 10L2s 221222221222 (2\22, 1\2)
* [[orwell]] pentatonic 4L1s 55552 (5\22, 1\1)
* [[orwell]] diatonic 4L5s 323232322 (5\22, 1\1)
* [[orwell]] chromatic 9L4s 2122122122122 (5\22, 1\1)
* [[magic]] diatonic 3L4s 1616161 (7\22, 1\1)
* [[magic]] superdiatonic 3L7s 1511511511 (7\22, 1\1)
* [[magic]] chromatic 3L10s 1411141114111 (7\22, 1\1)
* [[magic]] mega chromatic 3L13s 1131111311113111 (7\22, 1\1)
* Pathological [[magic]] enharmonic 3L16s 1112111112111112111 (7\22, 1\1)
* [[hedgehog]] hexatonic 2L4s 353353 (3\22, 1\2)
* [[hedgehog]] symmetric octatonic 6L2s 33233323 (3\22, 1\2)
* [[hedgehog]] symmetric chromatic 8L6s 21212212121221 (3\22, 1\2)
* [[astrology]] hexatonic 4L2s 434434 (4\22, 1\2)
* [[astrology]] symmetric decatonic 6L4s 3133131331 (4\22, 1\2)
* [[astrology]] symmetric hexadecatonic 6L10s 2112121121121211 (4\22, 1\2)
* [[doublewide]] tetrad 2L2s 6565 (5\22, 1\2)
* [[doublewide]] hexatonic 4L2s 515515 (5\22, 1\2)
* [[doublewide]] symmetric decatonic 4L6s 4114141141 (5\22, 1\2)
* [[Astrology|doublewide]] symmetric tetradecatonic 4L10s 31113113111311 (5\22, 1\2)
* Pathological [[doublewide]] symmetric octokaidecatonic 4L14s 211112111211112111 (5\22, 1\2)
 
{| class="wikitable"
|-
! Periods <br> per octave
! Period
! Generator
! Temperaments
|-
| 1
| 22\22
| 1\22
| [[Sensamagic clan#Sensa|Sensa]]/chromo/ceratitid
|-
| 1
| 22\22
| 3\22
| [[Porcupine]]
|-
| 1
| 22\22
| 5\22
| [[Orwell]]/blair/orson
|-
| 1
| 22\22
| 7\22
| [[Magic]]/telepathy
|-
| 1
| 22\22
| 9\22
| [[Superpyth]]/[[Suprapyth]]
|-
| 2
| 11\22
| 1\22
| [[Shrutar]]/hemipaj/comic
|-
| 2
| 11\22
| 2\22
| [[Srutal]]/[[pajara]]/pajarous
|-
| 2
| 11\22
| 3\22
| [[Porcupine family#Hedgehog|Hedgehog]]/[[echidna]]
|-
| 2
| 11\22
| 4\22
| [[Astrology]]/[[wizard]]/[[antikythera]]
|-
| 2
| 11\22
| 5\22
| [[Doublewide]]/fleetwood
|-
| 11
| 2\22
| 1\22
| [[Hendecatonic]]/undeka
|}
 
== Scales ==
Scales are written be steps in degrees of 22edo. [[MOS scale]]s are listed in their symmetric mode if one exists, and otherwise in the "brightest" mode - the mode with the highest average pitch height / the lexicographically highest mode
 
=== MOS scales ===
 
:''See also [[22edo Modes]], [[22edo tetrachords]]
 
* Porcupine[7] - 3334333
* Porcupine[8] - 33333331
* Porcupine[15] - 121212121212121
* Orwell[5] - 55255
* Orwell[9] - 232323232
* Orwell[13] - 2122122212212
* Magic[7] - 1616161
* Magic[10] - 5115115111
* Magic[13] - 1141114111411
* Magic[16] - 3111131111311111
* Magic[19] - 1112111112111112111
* Superpyth[5] - pentatonic - 45454
* Superpyth[7] - diatonic - 4144414
* Superpyth[12] - chromatic - 313131131311
* Superpyth[17] - hyperchromatic - 12111211211211121
* Pajara[10] - symmetric decatonic - 2232222322
* Pajara[12] - 222221222221
* Hedgehog[6] - 353353
* Hedgehog[8] - 33323332
* Hedgehog[14] - 21212122121212
* Astrology[6] - 434434
* Astrology[10] - 3131331313
* Astrology[16] - 2121121121211211
* Doublewide[4] - 5656
* Doublewide[6] - 551551
* Doublewide[10] - 4141141411
* Doublewide[14] - 31131113113111
* Doublewide[18] - 211121111211121111
 
=== Other scales ===


===How to Notate 22edo in Sagittal===
* Pentachordal decatonic - Pajara[10] 4|4(2) #8 - 2232223222
* Zarlino/Ptolemy diatonic, "just" major, Ma grama - 4324342
* inverse of Zarlino/Ptolemy diatonic, natural minor - 4234243
* tetrachordal major, Sa grama - 4324432
* inverse of tetrachordal major, "just"/tetrachordal minor - 4234234
* Porcupine bright major #7 - Porcupine[7] 6|0 #7 - 4333342
* Porcupine bright major #6 #7 - Porcupine[7] 6|0 #6 #7 - 4333432
* Porcupine bright minor #2 - Porcupine[7] 4|2 #2 4243333 (mode of bright major #7)
* Porcupine dark minor #2 - Porcupine[7] 3|3 #2 4234333 (inverse of bright major #6 #7)
* Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7 4333324
* Superpyth harmonic minor - Superpyth[7] 2|4 #7 - 4144171
* Superpyth harmonic major - Superpyth[7] 5|1 b6 - 4414171 (inverse of harmonic minor)
* Superpyth melodic minor - Superpyth[7] 5|1 b3 - 4144441
* Superpyth double harmonic - Superpyth[7] 5|1 b2 b6 - 1714171
* "just" harmonic minor - 4234252
* "just" harmonic major - 4324252
* "just" melodic minor - 4234342
* "just" double harmonic - 2524252
 
== Staff notation ==
=== Sagittal Notation ===


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:
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:
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[[File:22edo Sagittal.png|800px]]
[[File:22edo Sagittal.png|800px]]


===How to Notate 22edo with Ups and Downs===
=== Ups and Downs Notation ===


Treating [[Ups_and_Downs_Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:
Treating [[Ups_and_Downs_Notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:
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* [http://www.archive.org/download/HumanAstronomy/03Sevish-Ambrosia.mp3 Ambrosia]
* [http://www.archive.org/download/HumanAstronomy/03Sevish-Ambrosia.mp3 Ambrosia]
* [https://youtu.be/l9wINwlgxRU "Gleam"] (from his 2017 album "Harmony Hacker")
* [https://youtu.be/l9wINwlgxRU "Gleam"] (from his 2017 album "Harmony Hacker")
[[Category:22edo| ]]  
 
[[Category:22edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Listen]]
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[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Zeta]]
[[Category:Zeta]]
{{todo| cleanup }}
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