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== Theory ==
== 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 more accurate.
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. 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.
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 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.
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== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
Since a sharp raises by three steps, 22edo is a good candidate for [[Stein–Zimmermann–Gould notation]], using sharps and flats with arrows similar to 29edo:
{{Sharpness-sharp3-szg}}
 
If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with double arrows.
 
=== Kite's ups and downs notation ===
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{sharpness-sharp3a}}
{{sharpness-sharp3a}}


Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that E♭ and D♯ are different notes and that E♭ is significantly lower in pitch than D♯.
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 edostep (1\22). Note that E♭ and D♯ are different notes and that E♭ is significantly lower in pitch than D♯.


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


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


[[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]]
[[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]]
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[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]
[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]
Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:
{{Sharpness-sharp3}}
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.


Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.
Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.
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=== Sagittal notation ===
=== 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|11-EDO]].
This notation uses the same sagittal sequence as edos [[15edo #Sagittal notation|15]] and [[29edo #Sagittal notation|29]], is a subset of the notations for edos [[44edo #Sagittal notation|44]] and [[66edo #Sagittal notation|66]], and is a superset of the notation for [[11edo #Sagittal notation|11edo]].


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


=== 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.


=== Porcupine notation ===
=== Porcupine notation ===
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! [[Degree]]
! [[Degree]]
! [[Cent]]s
! [[Cent]]s
! colspan="2" | Superpyth/Porcupine
! colspan="2" | Superpyth/porcupine
! colspan="3" | Porcupine (Onyx)
! colspan="3" | Porcupine (Onyx)
! colspan="3" |Porcupine (Zarlino)
! colspan="3" | Porcupine (Zarlino)
! colspan="3" | Pentatonic
! colspan="3" | Pentatonic
! colspan="3" | Decatonic
! colspan="3" | Decatonic
! colspan="3" | [[Ups and downs notation|Ups and Downs]]
! colspan="3" | [[Ups and downs notation|Ups and downs]]
! colspan="3" | [[SKULO interval names]]
! colspan="3" | [[SKULO interval names]]
|-
|-
| 0
| 0
| 0
| 0
| Natural Unison
| Natural unison
| 1
| 1
| perfect unison
| perfect unison
| P1
| P1
| D
| D
|perfect unison
| perfect unison
|P1
| P1
|C
| C
| perfect unison
| perfect unison
| P1
| P1
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| A1
| A1
| D#
| D#
|augmented unison
| augmented unison
|A1
| A1
|C#
| C#
| aug unison
| aug unison
| A1
| A1
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| d2
| d2
| Eb
| Eb
|minor second
| minor second
|m2
| m2
|Db
| Db
| double-aug unison,<br />double-dim sub3rd
| double-aug unison,<br>double-dim sub3rd
| AA1,<br />dds3
| AA1,<br>dds3
| Dx,<br />Fb<span style="vertical-align: super;">3</span>
| Dx,<br>Fb<span style="vertical-align: super;">3</span>
| natural 2nd
| natural 2nd
| N2
| N2
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| P2
| P2
| E
| E
|narrow major second
| narrow major second
|nM2
| nM2
|D
| D
| dim sub3rd
| dim sub3rd
| ds3
| ds3
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| 4
| 4
| 218
| 218
| (s/p) Major second
| (s/p) major second
| M2
| M2
| aug 2nd
| aug 2nd
| A2
| A2
| E#
| E#
|wide major second
| wide major second
|WM2
| WM2
|D#
| D#
| minor sub3rd
| minor sub3rd
| ms3
| ms3
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| d3
| d3
| Fb
| Fb
|wolf third
| wolf third
|w3
| w3
|Ebb
| Ebb
| major sub3rd
| major sub3rd
| Ms3
| Ms3
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| m3
| m3
| F
| F
|minor third
| minor third
|m3
| m3
|Eb
| Eb
| aug sub3rd
| aug sub3rd
| As3
| As3
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| 7
| 7
| 382
| 382
| p-Major third
| p-major third
| pM3
| pM3
| major 3rd
| major 3rd
| M3
| M3
| F#
| F#
|major third
| major third
|M3
| M3
|E
| E
| double-aug sub3rd,<br />double-dim 4thoid
| double-aug sub3rd,<br>double-dim 4thoid
| AAs3,<br />dd4d
| AAs3,<br>dd4d
| Fx,<br />Gbb
| Fx,<br>Gbb
| natural 4th
| natural 4th
| N4
| N4
Line 734: Line 734:
| 8
| 8
| 436
| 436
| s-Major third
| s-major third
| sM3
| sM3
| aug 3rd, dim 4th
| aug 3rd, dim 4th
| A3, d4
| A3, d4
| Fx, Gb
| Fx, Gb
|augmented third
| augmented third
|A3
| A3
|E#
| E#
| dim 4thoid
| dim 4thoid
| d4d
| d4d
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| 9
| 9
| 491
| 491
| Natural Fourth
| Natural fourth
| 4, N4
| 4, N4
| minor 4th
| minor 4th
| m4
| m4
| G
| G
|perfect fourth
| perfect fourth
|P4
| P4
|F
| F
| perfect 4thoid
| perfect 4thoid
| P4d
| P4d
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| 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
| wolf fourth
|w4
| w4
|F#
| F#
| aug 4thoid
| aug 4thoid
| A4d
| A4d
Line 803: Line 803:
| 11
| 11
| 600
| 600
| p-Augmented Fourth,<br />p-diminished Fifth,<br />Half-Octave
| p-augmented fourth,<br>p-diminished fifth,<br>half-octave
| A4, HO
| A4, HO
| aug 4th, <br />dim 5th
| aug 4th, <br>dim 5th
| A4, d5
| Gx, <br>Abb
| augmented fourth, diminished fifth
| A4, d5
| A4, d5
| Gx, <br />Abb
| F##, Gbb
|augmented fourth, diminished fifth
| double-aug 4thoid,<br>double-dim 5thoid
|A4, d5
| AA4d, <br>dd5d
|F##, Gbb
| Gx, <br>Abb
| double-aug 4thoid,<br />double-dim 5thoid
| AA4d, <br />dd5d
| Gx, <br />Abb
| natural 6th
| natural 6th
| N6
| N6
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| vA4, ^d5
| vA4, ^d5
| vG#, ^Ab
| vG#, ^Ab
| comma-narrow augmented 4th<br />comma-wide diminished 5th
| comma-narrow augmented 4th<br>comma-wide diminished 5th
| kA4<br />Kd5
| kA4<br>Kd5
| kG#, KAb
| kG#, KAb
|-
|-
| 12
| 12
| 655
| 655
| p-minor Fifth, s-aug Fourth
| p-minor fifth, s-aug fourth
| pm5, sA4
| pm5, sA4
| minor 5th
| minor 5th
| m5
| m5
| Ab
| Ab
|wolf fifth
| wolf fifth
|w5
| w5
|Gb
| Gb
| dim 5thoid
| dim 5thoid
| d5d
| d5d
Line 849: Line 849:
| 13
| 13
| 709
| 709
| Natural Fifth
| Natural fifth
| 5, N5
| 5, N5
| major 5th
| major 5th
| M5
| M5
| A
| A
|perfect fifth
| perfect fifth
|P5
| P5
|G
| G
| perfect 5thoid
| perfect 5thoid
| P5d
| P5d
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| A5, d6
| A5, d6
| A#, Bbb
| A#, Bbb
|diminished sixth
| diminished sixth
|d6
| d6
|Abb
| Abb
| aug 5thoid
| aug 5thoid
| A5d
| A5d
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| m6
| m6
| Bb
| Bb
|minor sixth
| minor sixth
|m6
| m6
|Ab
| Ab
| double-aug 5thoid,<br />double-dim sub7th
| double-aug 5thoid,<br>double-dim sub7th
| AA5d,<br />dds7
| AA5d,<br>dds7
| Ax,<br />Cb<span style="vertical-align: super;">3</span>
| Ax,<br>Cb<span style="vertical-align: super;">3</span>
| flat 8th
| flat 8th
| f8
| f8
Line 918: Line 918:
| 16
| 16
| 873
| 873
| p-Major sixth
| p-major sixth
| pM6
| pM6
| major 6th
| major 6th
| M6
| M6
| B
| B
|major sixth
| major sixth
|M6
| M6
|A
| A
| dim sub7th
| dim sub7th
| ds7
| ds7
Line 941: Line 941:
| 17
| 17
| 927
| 927
| s-Major sixth
| s-major sixth
| sM6
| sM6
| aug 6th
| aug 6th
| A6
| A6
| B#
| B#
|wolf sixth
| wolf sixth
|w6
| w6
|A#
| A#
| minor sub7th
| minor sub7th
| ms7
| ms7
Line 969: Line 969:
| d7
| d7
| Cb
| Cb
|narrow minor seventh
| narrow minor seventh
|nm7
| nm7
|Bbb
| Bbb
| major sub7th
| major sub7th
| Ms7
| Ms7
Line 987: Line 987:
| 19
| 19
| 1036
| 1036
| p-Major seventh
| p-major seventh
| pM7
| pM7
| perfect 7th
| perfect 7th
| P7
| P7
| C
| C
|wide minor seventh
| wide minor seventh
|Wm7
| Wm7
|Bb
| Bb
| aug sub7th
| aug sub7th
| As7
| As7
Line 1,010: Line 1,010:
| 20
| 20
| 1091
| 1091
| p-Augmented seventh
| p-augmented seventh
| pA7
| pA7
| aug 7th
| aug 7th
| A7
| A7
| C#
| C#
|major seventh
| major seventh
|M7
| M7
|B
| B
| double-aug sub7th,<br />double-dim octave
| double-aug sub7th,<br>double-dim octave
| AAs7,<br />dd8
| AAs7,<br>dd8
| Cx,<br />Dbb
| Cx,<br>Dbb
| natural 10th
| natural 10th
| N10
| N10
Line 1,033: Line 1,033:
| 21
| 21
| 1145
| 1145
| s-Major seventh
| s-major seventh
| sM7
| sM7
| dim 8ve
| dim 8ve
| d8
| d8
| Db
| Db
|diminished octave
| diminished octave
|d8
| d8
|Cb
| Cb
| dim octave
| dim octave
| d8
| d8
Line 1,061: Line 1,061:
| P8
| P8
| D
| D
|perfect octave
| perfect octave
|P8
| P8
|C
| C
| perfect octave
| perfect octave
| P8
| P8
Line 1,547: Line 1,547:


== Instruments ==
== 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).
=== Keyboards ===
=== Keyboards ===
[[File:22-tone halberstadt layout.png|alt=|frameless]]
[[File:22-tone halberstadt layout.png|alt=|frameless]]
Retrieved from "https://en.xen.wiki/w/22edo"