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== Theory ==
== Theory ==
=== History ===
=== History ===
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist [https://en.wikipedia.org/wiki/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|19 equal temperament]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and 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|19 equal temperament]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.


=== Overview to JI approximation quality ===
=== Overview to JI approximation quality ===
The 22et 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|31 equal temperament]] does much better, 22et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22et, unlike 12 and [[19edo|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, such as 22-tone guitars and the like.
The 22et 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|31 equal temperament]] does much better, 22et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the [[11-odd-limit]] [[consistent|consistently]]. Furthermore, 22et, unlike 12 and [[19edo|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, such as 22-tone guitars and the like.


22et can also be treated as adding harmonics 3 and 5 to 11edo's 2.7.9.11.15.17 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.
22et 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.


22et is very close to an extended "quarter-comma archy", a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma [[64/63]] instead of the syntonic comma [[81/80]]. Because of this it has nearly pure septimal major thirds ([[9/7]]).
22et is very close to an extended "quarter-comma archy", a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma [[64/63]] instead of the syntonic comma [[81/80]]. Because of this it has nearly pure septimal major thirds ([[9/7]]).
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! Cents
! Cents
! Approximate Ratios*
! Approximate Ratios*
! colspan="3" |[[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! Audio
! Audio
|-
|-
|0
| 0
|0.000
| 0.000
|[[1/1]]
| [[1/1]]
|perfect unison
| perfect unison
|P1
| P1
|D
| D
|[[File:0-0.000c_P1.mp3]]
| [[File:0-0.000c_P1.mp3]]
|-
|-
|1
| 1
|54.545
| 54.545
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
|up-unison, minor 2nd
| up-unison, minor 2nd
|^1, m2
| ^1, m2
|^D, Eb
| ^D, Eb
|[[File:0-54.545c_22edo.mp3]]
| [[File:0-54.545c_22edo.mp3]]
|-
|-
|2
| 2
|109.091
| 109.091
|[[18/17]], [[17/16]], [[16/15]], [[15/14]]
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
|downaug 1sn, upminor 2nd
| downaug 1sn, upminor 2nd
|vA1, ^m2
| vA1, ^m2
|vD#, ^Eb
| vD#, ^Eb
|[[File:0-109.091c_11edo.mp3]]
| [[File:0-109.091c_11edo.mp3]]
|-
|-
|3
| 3
|163.636
| 163.636
|[[12/11]], [[11/10]], [[10/9]]
| [[12/11]], [[11/10]], [[10/9]]
| aug 1sn, downmajor 2nd
| aug 1sn, downmajor 2nd
|A1, vM2
| A1, vM2
|D#, vE
| D#, vE
|[[File:0-163.636c_22edo.mp3]]
| [[File:0-163.636c_22edo.mp3]]
|-
|-
|4
| 4
|218.182
| 218.182
|[[9/8]], [[17/15]], [[8/7]]
| [[9/8]], [[17/15]], [[8/7]]
|major 2nd
| major 2nd
|M2
| M2
|E
| E
|[[File:0-218.182c_11edo.mp3]]
| [[File:0-218.182c_11edo.mp3]]
|-
|-
|5
| 5
|272.727
| 272.727
|[[20/17]], [[7/6]]
| [[20/17]], [[7/6]]
|minor 3rd
| minor 3rd
|m3
| m3
|F
| F
|[[File:0-272.727c_22edo.mp3]]
| [[File:0-272.727c_22edo.mp3]]
|-
|-
|6
| 6
|327.273
| 327.273
|[[6/5]], [[17/14]], [[11/9]]
| [[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
| upminor 3rd
|^m3
| ^m3
|^F
| ^F
|[[File:0-327.273c_11edo.mp3]]
| [[File:0-327.273c_11edo.mp3]]
|-
|-
|7
| 7
|381.818
| 381.818
|[[5/4]], [[96/77]]
| [[5/4]], [[96/77]]
|downmajor 3rd
| downmajor 3rd
|vM3
| vM3
|vF#
| vF#
|[[File:0-381.818c_22edo.mp3]]
| [[File:0-381.818c_22edo.mp3]]
|-
|-
| 8
| 8
|436.364
| 436.364
|[[14/11]], [[9/7]], [[22/17]]
| [[14/11]], [[9/7]], [[22/17]]
|major 3rd
| major 3rd
|M3
| M3
|F#
| F#
|[[File:0-436.364c_11edo.mp3]]
| [[File:0-436.364c_11edo.mp3]]
|-
|-
|9
| 9
|490.909
| 490.909
|[[4/3]]
| [[4/3]]
|perfect 4th
| perfect 4th
| P4
| P4
|G
| G
|[[File:0-490.909c_22edo.mp3]]
| [[File:0-490.909c_22edo.mp3]]
|-
|-
|10
| 10
|545.455
| 545.455
|[[15/11]], [[11/8]]
| [[15/11]], [[11/8]]
|up-4th, dim 5th
| up-4th, dim 5th
|^4, d5
| ^4, d5
|^G, Ab
| ^G, Ab
|[[File:0-545.455c_11edo.mp3]]
| [[File:0-545.455c_11edo.mp3]]
|-
|-
| 11
| 11
|600.000
| 600.000
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
| downaug 4th, updim 5th
|vA4, ^d5
| vA4, ^d5
| vG#, ^Ab
| vG#, ^Ab
|[[File:0-600.000c_2edo.mp3]]
| [[File:0-600.000c_2edo.mp3]]
|-
|-
| 12
| 12
|654.545
| 654.545
|
| [[16/11]], [[22/15]]
[[16/11]], [[22/15]]
| aug 4th, down-5th
|aug 4th, down-5th
| A4, v5
|A4, v5
| G#, vA
|G#, vA
| [[File:0-654.545c_11edo.mp3]]
|[[File:0-654.545c_11edo.mp3]]
|-
|-
|13
| 13
|709.091
| 709.091
|[[3/2]]
| [[3/2]]
|perfect 5th
| perfect 5th
|P5
| P5
|A
| A
|[[File:0-709.091c_22edo.mp3]]
| [[File:0-709.091c_22edo.mp3]]
|-
|-
|14
| 14
|763.636
| 763.636
|[[17/11]], [[14/9]], [[11/7]]
| [[17/11]], [[14/9]], [[11/7]]
|minor 6th
| minor 6th
|m6
| m6
|Bb
| Bb
|[[File:0-763.636c_11edo.mp3]]
| [[File:0-763.636c_11edo.mp3]]
|-
|-
|15
| 15
|818.182
| 818.182
|[[8/5]], [[77/48]]
| [[8/5]], [[77/48]]
|upminor 6th
| upminor 6th
|^m6
| ^m6
| ^Bb
| ^Bb
|[[File:0-818.182c_22edo.mp3]]
| [[File:0-818.182c_22edo.mp3]]
|-
|-
| 16
| 16
|872.727
| 872.727
|[[18/11]], [[28/17]], [[5/3]]
| [[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
| downmajor 6th
|vM6
| vM6
|vB
| vB
|[[File:0-872.727c_11edo.mp3]]
| [[File:0-872.727c_11edo.mp3]]
|-
|-
|17
| 17
|927.273
| 927.273
|[[17/10]], [[12/7]]
| [[17/10]], [[12/7]]
|major 6th
| major 6th
|M6
| M6
|B
| B
|[[File:0-927.273c_22edo.mp3]]
| [[File:0-927.273c_22edo.mp3]]
|-
|-
|18
| 18
|981.818
| 981.818
|[[7/4]], [[30/17]], [[16/9]]
| [[7/4]], [[30/17]], [[16/9]]
|minor 7th
| minor 7th
|m7
| m7
|C
| C
|[[File:0-981.818c_11edo.mp3]]
| [[File:0-981.818c_11edo.mp3]]
|-
|-
|19
| 19
|1036.364
| 1036.364
|[[9/5]], [[11/6]], [[20/11]]
| [[9/5]], [[11/6]], [[20/11]]
|upminor 7th, dim 8ve
| upminor 7th, dim 8ve
|^m7, d8
| ^m7, d8
|^C, Db
| ^C, Db
|[[File:0-1036.364c_22edo.mp3]]
| [[File:0-1036.364c_22edo.mp3]]
|-
|-
|20
| 20
|1090.909
| 1090.909
|[[28/15]], [[15/8]], [[32/17]], [[17/9]]
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th, updim 8ve
| downmajor 7th, updim 8ve
|vM7, ^d8
| vM7, ^d8
|vC#, ^Db
| vC#, ^Db
|[[File:0-1090.909c_11edo.mp3]]
| [[File:0-1090.909c_11edo.mp3]]
|-
|-
|21
| 21
|1145.455
| 1145.455
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
|major 7th, down 8ve
| major 7th, down 8ve
|M7, v8
| M7, v8
|C#, vD
| C#, vD
|[[File:0-1145.455c_22edo.mp3]]
| [[File:0-1145.455c_22edo.mp3]]
|-
|-
|22
| 22
|1200.000
| 1200.000
|[[2/1]]
| [[2/1]]
|perfect octave
| perfect octave
|P8
| P8
|D
| D
|[[File:0-1200.000c_P8.mp3]]
| [[File:0-1200.000c_P8.mp3]]
|}
|}
<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.
<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.


==JI approximation==
== Approximation to JI ==
===15-odd-limit interval mappings===
=== 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''.  


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" |15-odd-limit intervals by direct approximation (even if inconsistent)
|+ style="white-space:nowrap" | 15-odd-limit intervals by direct approximation (even if inconsistent)
!Interval, complement
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (abs, [[Cent|¢]])
!Error (rel, [[Relative cent|%]])
! Error (rel, [[Relative cent|%]])
|-
|-
|[[9/7]], [[14/9]]
| [[9/7]], [[14/9]]
|1.280
| 1.280
| 2.3
| 2.3
|-
|-
|[[11/10]], [[20/11]]
| [[11/10]], [[20/11]]
|1.368
| 1.368
| 2.5
| 2.5
|-
|-
|[[15/8]], [[16/15]]
| [[15/8]], [[16/15]]
|2.640
| 2.640
|4.8
| 4.8
|-
|-
|'''[[5/4]], [[8/5]]'''
| '''[[5/4]], [[8/5]]'''
|'''4.496'''
| '''4.496'''
|'''8.2'''
| '''8.2'''
|-
|-
|[[7/6]], [[12/7]]
| [[7/6]], [[12/7]]
|5.856
| 5.856
|10.7
| 10.7
|-
|-
|'''[[11/8]], [[16/11]]'''
| '''[[11/8]], [[16/11]]'''
|'''5.863'''
| '''5.863'''
|'''10.7'''
| '''10.7'''
|-
|-
|
| '''[[3/2]], [[4/3]]'''
'''[[3/2]], [[4/3]]'''
| '''7.136'''
|'''7.136'''
| '''13.1'''
|'''13.1'''
|-
|-
|[[15/11]], [[22/15]]
| [[15/11]], [[22/15]]
|8.504
| 8.504
|15.6
| 15.6
|-
|-
|[[15/14]], [[28/15]]
| [[15/14]], [[28/15]]
|10.352
| 10.352
|19.0
| 19.0
|-
|-
|[[5/3]], [[6/5]]
| [[5/3]], [[6/5]]
|11.631
| 11.631
|21.3
| 21.3
|-
|-
|'''[[7/4]], [[8/7]]'''
| '''[[7/4]], [[8/7]]'''
|'''12.992'''
| '''12.992'''
|
| '''23.8'''
'''23.8'''
|-
|-
|[[11/6]], [[12/11]]
| [[11/6]], [[12/11]]
|12.999
| 12.999
|23.8
| 23.8
|-
|-
|[[9/8]], [[16/9]]
| [[9/8]], [[16/9]]
|14.272
| 14.272
|26.2
| 26.2
|-
|-
|[[13/11]], [[22/13]]
| [[13/11]], [[22/13]]
|16.482
| 16.482
|30.2
| 30.2
|-
|-
|[[7/5]], [[10/7]]
| [[7/5]], [[10/7]]
|17.488
| 17.488
|32.1
| 32.1
|-
|-
|[[13/10]], [[20/13]]
| [[13/10]], [[20/13]]
|17.850
| 17.850
|32.7
| 32.7
|-
|-
|''[[13/9]], [[18/13]]''
| ''[[13/9]], [[18/13]]''
|''17.928''
| ''17.928''
|''32.9''
| ''32.9''
|-
|-
|[[9/5]], [[10/9]]
| [[9/5]], [[10/9]]
|18.767
| 18.767
|34.4
| 34.4
|-
|-
|[[11/7]], [[14/11]]
| [[11/7]], [[14/11]]
| 18.856
| 18.856
|34.6
| 34.6
|-
|-
|''[[13/7]], [[14/13]]''
| ''[[13/7]], [[14/13]]''
|''19.207''
| ''19.207''
|''35.2''
| ''35.2''
|-
|-
|[[11/9]], [[18/11]]
| [[11/9]], [[18/11]]
|20.135
| 20.135
|36.9
| 36.9
|-
|-
|'''[[13/8]], [[16/13]]'''
| '''[[13/8]], [[16/13]]'''
|'''22.346'''
| '''22.346'''
|'''41.0'''
| '''41.0'''
|-
|-
|[[15/13]], [[26/15]]
| [[15/13]], [[26/15]]
|24.986
| 24.986
|45.8
| 45.8
|-
|-
|''[[13/12]], [[24/13]]''
| ''[[13/12]], [[24/13]]''
|''25.064''
| ''25.064''
|''46.0''
| ''46.0''
|}
|}
{{15-odd-limit|22}}
{{15-odd-limit|22}}


===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.]]


==Defining features==
== Defining features ==


===Septimal vs syntonic comma===
=== 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]].
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.
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===
=== 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).
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===
=== 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.
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 ===
=== 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.
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===
=== 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''.
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 ===
=== 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.
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.


Line 364: Line 360:
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.
22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.


==Regular temperament properties==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo| 35 -22 }}
| {{monzo| 35 -22 }}
|[{{val| 22 35 }}]
| [{{val| 22 35 }}]
| -2.25
| -2.25
|2.25
| 2.25
|4.12
| 4.12
|-
|-
|2.3.5
| 2.3.5
|250/243, 2048/2025
| 250/243, 2048/2025
|[{{val| 22 35 51 }}]
| [{{val| 22 35 51 }}]
| -0.86
| -0.86
| 2.70
| 2.70
|4.94
| 4.94
|-
|-
|2.3.5.7
| 2.3.5.7
|50/49, 64/63, 245/243
| 50/49, 64/63, 245/243
|[{{val| 22 35 51 62 }}]
| [{{val| 22 35 51 62 }}]
| -1.80
| -1.80
|2.85
| 2.85
|5.23
| 5.23
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|50/49, 55/54, 64/63, 99/98
| 50/49, 55/54, 64/63, 99/98
| [{{val| 22 35 51 62 76 }}]
| [{{val| 22 35 51 62 76 }}]
| -1.11
| -1.11
|2.90
| 2.90
|5.33
| 5.33
|-
|-
|2.3.5.7.11.17
| 2.3.5.7.11.17
|50/49, 55/54, 64/63, 85/84, 99/98
| 50/49, 55/54, 64/63, 85/84, 99/98
|[{{val| 22 35 51 62 76 90 }}]
| [{{val| 22 35 51 62 76 90 }}]
| -1.09
| -1.09
|2.65
| 2.65
|4.87
| 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]].  
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 maps ===
{{Uniform map|13|21.5|22.5}}
{{Uniform map|13|21.5|22.5}}


===Commas===
=== Commas ===
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.)
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"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
![[Harmonic limit|Prime <br>limit]]
! [[Harmonic limit|Prime <br>limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
! [[Monzo]]
![[Cents]]
! [[Cents]]
![[Color name]]
! [[Color name]]
!Name
! Name
|-
|-
|3
| 3
|<abbr title="34359738368/31381059609">(22 digits)</abbr>
| <abbr title="34359738368/31381059609">(22 digits)</abbr>
|
| {{monzo| 35 -22 }}
{{monzo| 35 -22 }}
| 156.98
|156.98
|  
|
|  
|
|-
|-
|5
| 5
|[[250/243]]
| [[250/243]]
|{{monzo| 1 -5 3 }}
| {{monzo| 1 -5 3 }}
|49.17
| 49.17
|Triyo
| Triyo
|Porcupine comma
| Porcupine comma
|-
|-
|5
| 5
|[[3125/3072]]
| [[3125/3072]]
|{{monzo| -10 -1 5 }}
| {{monzo| -10 -1 5 }}
|29.61
| 29.61
|Laquinyo
| Laquinyo
|Magic comma
| Magic comma
|-
|-
|5
| 5
|[[2048/2025]]
| [[2048/2025]]
|{{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
|19.55
| 19.55
| Sagugu
| Sagugu
|Diaschisma
| Diaschisma
|-
|-
| 5
| 5
|[[2109375/2097152|(14 digits)]]
| [[2109375/2097152|(14 digits)]]
|{{monzo| -21 3 7 }}
| {{monzo| -21 3 7 }}
|10.06
| 10.06
|Lasepyo
| Lasepyo
|[[Semicomma]]
| [[Semicomma]]
|-
|-
|5
| 5
|<abbr title="4294967296/4271484375">(20 digits)</abbr>
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
|{{monzo| 32 -7 -9 }}
| {{monzo| 32 -7 -9 }}
|9.49
| 9.49
| Sasa-tritrigu
| Sasa-tritrigu
|[[Escapade comma]]
| [[Escapade comma]]
|-
|-
|5
| 5
|<abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr>
|{{monzo| -53 10 16 }}
| {{monzo| -53 10 16 }}
|0.57
| 0.57
|Quadla-quadquadyo
| Quadla-quadquadyo
|
| [[Kwazy]]
[[Kwazy]]
|-
|-
|7
| 7
|[[50/49]]
| [[50/49]]
|{{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
|34.98
| 34.98
|Biruyo
| Biruyo
|Jubilisma
| Jubilisma
|-
|-
|7
| 7
|[[64/63]]
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
|27.26
| 27.26
|Ru
| Ru
|Septimal comma
| Septimal comma
|-
|-
|7
| 7
|[[875/864]]
| [[875/864]]
|{{monzo| -5 -3 3 1 }}
| {{monzo| -5 -3 3 1 }}
|21.90
| 21.90
|Zotriyo
| Zotriyo
|Keema
| Keema
|-
|-
|7
| 7
|[[2430/2401]]
| [[2430/2401]]
|{{monzo| 1 5 1 -4 }}
| {{monzo| 1 5 1 -4 }}
|20.79
| 20.79
| Quadru-ayo
| Quadru-ayo
|Nuwell
| Nuwell
|-
|-
|7
| 7
|[[245/243]]
| [[245/243]]
|{{monzo| 0 -5 1 2 }}
| {{monzo| 0 -5 1 2 }}
|14.19
| 14.19
|Zozoyo
| Zozoyo
|Sensamagic
| Sensamagic
|-
|-
|7
| 7
|[[1728/1715]]
| [[1728/1715]]
|{{monzo| 6 3 -1 -3 }}
| {{monzo| 6 3 -1 -3 }}
|13.07
| 13.07
| Triru-agu
| Triru-agu
|Orwellisma
| Orwellisma
|-
|-
|7
| 7
|[[225/224]]
| [[225/224]]
|{{monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
|7.71
| 7.71
|Ruyoyo
| Ruyoyo
|Marvel comma
| Marvel comma
|-
|-
|7
| 7
|[[10976/10935]]
| [[10976/10935]]
|{{monzo| 5 -7 -1 3 }}
| {{monzo| 5 -7 -1 3 }}
|6.48
| 6.48
| Trizo-agu
| Trizo-agu
|Hemimage
| Hemimage
|-
|-
|7
| 7
|[[6144/6125]]
| [[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
| {{monzo| 11 1 -3 -2 }}
|5.36
| 5.36
|Saruru-atrigu
| Saruru-atrigu
|Porwell
| Porwell
|-
|-
| 7
| 7
|[[65625/65536]]
| [[65625/65536]]
|{{monzo| -16 1 5 1 }}
| {{monzo| -16 1 5 1 }}
|2.35
| 2.35
|Lazoquinyo
| Lazoquinyo
|Horwell
| Horwell
|-
|-
|7
| 7
|<abbr title="420175/419904">(12 digits)</abbr>
| <abbr title="420175/419904">(12 digits)</abbr>
|{{monzo| -6 -8 2 5 }}
| {{monzo| -6 -8 2 5 }}
|1.12
| 1.12
|Quinzo-ayoyo
| Quinzo-ayoyo
|[[Wizma]]
| [[Wizma]]
|-
|-
|11
| 11
|[[99/98]]
| [[99/98]]
|{{monzo| -1 2 0 -2 1 }}
| {{monzo| -1 2 0 -2 1 }}
|17.58
| 17.58
|Loruru
| Loruru
|Mothwellsma
| Mothwellsma
|-
|-
|11
| 11
|[[100/99]]
| [[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
|17.40
| 17.40
| Luyoyo
| Luyoyo
|Ptolemisma
| Ptolemisma
|-
|-
|11
| 11
|[[121/120]]
| [[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
| {{monzo| -3 -1 -1 0 2 }}
|14.37
| 14.37
|Lologu
| Lologu
|Biyatisma
| Biyatisma
|-
|-
|11
| 11
|[[176/175]]
| [[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
|9.86
| 9.86
|Lorugugu
| Lorugugu
|Valinorsma
| Valinorsma
|-
|-
|11
| 11
|[[896/891]]
| [[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
| {{monzo| 7 -4 0 1 -1 }}
|9.69
| 9.69
|Saluzo
| Saluzo
|Pentacircle
| Pentacircle
|-
|-
|11
| 11
|[[65536/65219]]
| [[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
| {{monzo| 16 0 0 -2 -3 }}
|8.39
| 8.39
|Satrilu-aruru
| Satrilu-aruru
|Orgonisma
| Orgonisma
|-
|-
|11
| 11
|[[385/384]]
| [[385/384]]
|{{monzo| -7 -1 1 1 1 }}
| {{monzo| -7 -1 1 1 1 }}
|4.50
| 4.50
|Lozoyo
| Lozoyo
|Keenanisma
| Keenanisma
|-
|-
|11
| 11
|[[540/539]]
| [[540/539]]
|{{monzo| 2 3 1 -2 -1 }}
| {{monzo| 2 3 1 -2 -1 }}
|3.21
| 3.21
|Lururuyo
| Lururuyo
|Swetisma
| Swetisma
|-
|-
|11
| 11
|[[4000/3993]]
| [[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
| {{monzo| 5 -1 3 0 -3 }}
|3.03
| 3.03
|Triluyo
| Triluyo
|Wizardharry
| Wizardharry
|-
|-
|11
| 11
|[[9801/9800]]
| [[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
|0.18
| 0.18
|Bilorugu
| Bilorugu
|Kalisma
| Kalisma
|-
|-
|13
| 13
|
| [[65/64]]
[[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
|{{monzo| -6 0 1 0 0 1 }}
| 26.84
| 26.84
| Thoyo
| Thoyo
| Wilsorma
| Wilsorma
|-
|-
|13
| 13
|[[78/77]]
| [[78/77]]
|{{monzo| 1 1 0 -1 -1 1 }}
| {{monzo| 1 1 0 -1 -1 1 }}
|22.34
| 22.34
| Tholuru
| Tholuru
|Negustma
| Negustma
|-
|-
| 13
| 13
|[[91/90]]
| [[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
| {{monzo| -1 -2 -1 1 0 1 }}
|19.13
| 19.13
|Thozogu
| Thozogu
|Superleap
| Superleap
|-
|-
|31
| 31
|[[125/124]]
| [[125/124]]
|{{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| {{monzo| -2 0 3 0 0 0 0 0 0 0 -1 }}
| 13.91
| 13.91
|Thiwutriyo
| Thiwutriyo
|Twizzler
| Twizzler
|}
|}
<references />
<references />


===Rank-2 temperaments===
=== Rank-2 temperaments ===
*[[List of 22et rank two temperaments by badness]]
* [[List of 22et rank two temperaments by badness]]
*[[List of 22et rank two temperaments by complexity]]
* [[List of 22et rank two temperaments by complexity]]
*[[List of edo-distinct 22et rank two temperaments]]
* [[List of edo-distinct 22et rank two temperaments]]


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
!Periods <br> per octave
! Periods <br> per octave
!Generator
! Generator
!Temperaments
! Temperaments
|-
|-
| 1
| 1
|1\22
| 1\22
|[[Sensamagic clan #Sensa|Sensa]]<br>[[Chromo]]<br>[[Ceratitid]]
| [[Sensamagic clan #Sensa|Sensa]]<br>[[Chromo]]<br>[[Ceratitid]]
|-
|-
|1
| 1
|3\22
| 3\22
|[[Porcupine]]
| [[Porcupine]]
|-
|-
| 1
| 1
|5\22
| 5\22
|[[Orwell]] (22) / blair (22) / winston (22f)
| [[Orwell]] (22) / blair (22) / winston (22f)
|-
|-
|1
| 1
|7\22
| 7\22
|
| [[Magic]] / telepathy
[[Magic]] / telepathy
|-
|-
|1
| 1
| 9\22
| 9\22
|
| [[Superpyth]] / [[suprapyth]]
[[Superpyth]] / [[suprapyth]]
|-
|-
|2
| 2
|1\22
| 1\22
|[[Shrutar]] / hemipaj<br>[[Comic]]
| [[Shrutar]] / hemipaj<br>[[Comic]]
|-
|-
|2
| 2
|2\22
| 2\22
|[[Srutal]] / [[pajara]] / pajarous
| [[Srutal]] / [[pajara]] / pajarous
|-
|-
|2
| 2
|3\22
| 3\22
|[[Hedgehog]] / [[echidna]]
| [[Hedgehog]] / [[echidna]]
|-
|-
|2
| 2
|4\22
| 4\22
|[[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]
| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]
|-
|-
|2
| 2
|5\22
| 5\22
|[[Doublewide]] / fleetwood
| [[Doublewide]] / fleetwood
|-
|-
|11
| 11
|1\22
| 1\22
|[[Undeka]]<br>[[Hendecatonic]]
| [[Undeka]]<br>[[Hendecatonic]]
|}
|}


==Scales==
== Scales ==
''See [[22edo modes]]''.
''See [[22edo modes]]''.


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


==Notation==
== 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.


===Porcupine Notation===
=== 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.
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.  
The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.  


===Pentatonic Notation===
=== 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.  
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.  
The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.  


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


Line 746: Line 737:
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.


===Sagittal 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:


Line 759: Line 750:
[[File:22edo Sagittal.png|800px]]
[[File:22edo Sagittal.png|800px]]


===Ups and Downs Notation===
=== 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:
Line 780: Line 771:
</gallery>
</gallery>


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


{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
|-
|-
![[Degree]]
! [[Degree]]
![[cent|Cents]]
! [[Cent]]s
! colspan="2" | Superpyth/Porcupine Notation
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" |Porcupine
! colspan="3" | Porcupine
! colspan="3" |Pentatonic
! colspan="3" | Pentatonic
! colspan="3" |Decatonic
! colspan="3" | Decatonic
! colspan="3" |Sagittal
! colspan="3" | Sagittal
! colspan="3" |Ups and Downs
! colspan="3" | Ups and Downs
|-
|-
|0
| 0
| 0
|Natural Unison
| 0
|1
| Natural Unison
|perfect unison
| 1
|P1
| perfect unison
|D
| P1
|perfect unison
| D
|P1
| perfect unison
|D
| P1
|natural 1st
| D
|N1
| natural 1st
|C
| N1
|
| C
|
|  
|
|  
|perfect unison
|  
|P1
| perfect unison
|D
| P1
| D
|-
|-
|1
| 1
|55
| 55
| s-minor second
| s-minor second
|sm2
| sm2
|aug unison
|A1
|D#
| aug unison
| aug unison
|A1
| A1
|D#
| D#
|flat 2nd
| aug unison
|f2
| A1
|C#, δb
| D#
|
| flat 2nd
|
| f2
|
| C#, δb
|minor 2nd
|  
|m2
|  
|Eb
|  
| minor 2nd
| m2
| 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
| 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
|δ
| δ
|
|  
|
|  
|
|  
|upminor 2nd
| upminor 2nd
| ^m2
| ^m2
| ^Eb
| ^Eb
|-
|-
|3
| 3
|164
| 164
|p-minor second
| p-minor second
|pm2
| pm2
|perfect 2nd
| perfect 2nd
|P2
| P2
|E
| E
|dim sub3rd
| dim sub3rd
|ds3
| ds3
|Fbb
| Fbb
|sharp 2nd, flat 3rd
| sharp 2nd, flat 3rd
|s2, f3
| s2, f3
|δ#, Db
| δ#, Db
|
|  
|
|  
|
|  
|downmajor 2nd
| downmajor 2nd
| vM2
| vM2
|vE
| vE
|-
|-
|4
| 4
|218
| 218
|(s/p) Major second
| (s/p) Major second
|M2
| M2
|aug 2nd
| aug 2nd
|A2
| A2
|E#
| E#
|minor sub3rd
| minor sub3rd
|ms3
| ms3
|Fb
| Fb
|natural 3rd
| natural 3rd
|N3
| N3
|D
| D
|
|  
|
|  
|
|  
| major 2nd
| major 2nd
| M2
| M2
|E
| E
|-
|-
|5
| 5
|273
| 273
|s-minor third
| s-minor third
|sm3
| sm3
| dim 3rd
| dim 3rd
| d3
| d3
|Fb
| Fb
|major sub3rd
| major sub3rd
|Ms3
| Ms3
|F
| F
|sharp 3rd
| sharp 3rd
|s3
| s3
|D#
| D#
|
|  
|
|  
|
|  
|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
| aug sub3rd
|As3
| As3
|F#
| F#
|flat 4th
| flat 4th
| f4
| f4
|εb
| εb
|
|  
|
|  
|
|  
|upminor 3rd
| upminor 3rd
| ^m3
| ^m3
|^F
| ^F
|-
|-
| 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
| 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
|ε
| ε
|
|  
|
|  
|
|  
|downmajor 3rd
| downmajor 3rd
|vM3
| vM3
|vF#
| vF#
|-
|-
|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
|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#
|-
|-
|9
| 9
|491
| 491
|Natural Fourth
| Natural Fourth
|4, N4
| 4, N4
|minor 4th
| minor 4th
| m4
| m4
|G
| G
|perfect 4thoid
| perfect 4thoid
|P4d
| P4d
|G
| G
|natural 5th
| natural 5th
|N5
| N5
| E
| E
|
|  
|
|  
|
|  
|perfect fourth
| perfect fourth
|P4
| P4
|G
| 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#
|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
|-
|-
|11
| 11
|600
| 600
|p-Augmented Fourth,  
| p-Augmented Fourth, <br>p-diminished Fifth, <br>Half-Octave
p-diminished Fifth  
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
| 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
|-
|-
|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
| dim 5thoid
|d5d
| d5d
|Ab
| 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
|-
|-
|13
| 13
| 709
| 709
|Natural Fifth
| Natural Fifth
|5, N5
| 5, N5
| major 5th
| major 5th
|M5
| M5
|A
| A
|perfect 5thoid
| perfect 5thoid
| P5d
| P5d
|A
| A
|natural 7th
| natural 7th
|N7
| N7
|G
| G
|
|  
|
|  
|
|  
|perfect 5th
| perfect 5th
|P5
| P5
|A
| 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
| aug 5thoid
|A5d
| A5d
|A#
| A#
|sharp 7th
| sharp 7th
|s7
| s7
| G#
| G#
|
|  
|
|  
|
|  
|minor 6th
| 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
| 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
|αb
| αb
|
|  
|
|  
|
|  
|upminor 6th
| upminor 6th
|^m6
| ^m6
|^Bb
| ^Bb
|-
|-
| 16
| 16
|873
| 873
|p-Major sixth
| p-Major sixth
|pM6
| pM6
|major 6th
| major 6th
| M6
| M6
|B
| B
|dim sub7th
| dim sub7th
|ds7
| ds7
|Cbb
| Cbb
|natural 8th
| natural 8th
|N8
| N8
|α
| α
|
|  
|
|  
|
|  
| downmajor 6th
| downmajor 6th
| vM6
| vM6
|vB
| vB
|-
|-
|17
| 17
|927
| 927
|s-Major sixth
| s-Major sixth
|sM6
| sM6
|aug 6th
| aug 6th
|A6
| A6
|B#
| B#
| minor sub7th
| minor sub7th
| ms7
| ms7
|Cb
| 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
|-
|-
|18
| 18
|982
| 982
|(s/p) minor seventh
| (s/p) minor seventh
| m7
| m7
|dim 7th
| dim 7th
|d7
| d7
|Cb
| Cb
| major sub7th
| major sub7th
| Ms7
| Ms7
|C
| C
| natural 9th
| natural 9th
|N9
| N9
|A
| A
|
|  
|
|  
|
|  
|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
|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
| upminor 7th
|^m7
| ^m7
|^C
| ^C
|-
|-
|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
| 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
| β
| β
|
|  
|
|  
|
|  
|downmajor 7th
| downmajor 7th
|vM7
| vM7
|vC#
| vC#
|-
|-
|21
| 21
| 1145
| 1145
|s-Major seventh
| s-Major seventh
|sM7
| sM7
|dim 8ve
| dim 8ve
|d8
| d8
|Db
| Db
|dim octave
| dim octave
|d8
| d8
|Db
| Db
|sharp 10th
| sharp 10th
|s10
| s10
|β#, Cb
| β#, Cb
|
|  
|
|  
|
|  
| major 7th
| major 7th
|M7
| M7
|C#
| C#
|-
|-
|22
| 22
|1200
| 1200
| Octave
| Octave
|8
| 8
|perfect octave
| perfect octave
|P8
| P8
|D
| D
|perfect octave
| perfect octave
| P8
| D
| natural 11th
| N11
| C
|
|
|
| perfect octave
| P8
| P8
|D
| D
|natural 11th
|N11
|C
|
|
|
|perfect octave
|P8
|D
|}
|}


===Chord names===
=== Chord names ===
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>
| [a b 0 1>
| 7/6, 7/4
| 7/6, 7/4
|-
|-
|fourthward wa
| fourthward wa
|[a b> where b &lt; -1
| [a b> where b &lt; -1
|32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
|gu
| gu
|[a b -1>
| [a b -1>
|6/5, 9/5
| 6/5, 9/5
|-
|-
|downmajor
| downmajor
|yo
| yo
|[a b 1>
| [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
| [a b> where b &gt; 1
|9/8, 27/16
| 9/8, 27/16
|-
|-
|ru
| ru
|[a b 0 -1>
| [a b 0 -1>
|9/7, 12/7
| 9/7, 12/7
|}
|}


Line 1,299: Line 1,288:
{| 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)


Further discussion of 22edo chord naming:
Further discussion of 22edo chord naming:


*[[22edo Chord Names]]
* [[22edo Chord Names]]
*[[22 EDO Chords]]
* [[22 EDO Chords]]
*[[Ups and Downs Notation #Chords and Chord Progressions]]
* [[Ups and Downs Notation #Chords and Chord Progressions]]
*[[Chords of orwell]]
* [[Chords of orwell]]


==Music ==
== Music ==
{{Main| 22edo/Music }}
{{Main| 22edo/Music }}
{{Catrel|22edo tracks}}
{{Catrel|22edo tracks}}


==Related pages==
== Related pages ==
*[[Lumatone mapping for 22edo]]
* [[Lumatone mapping for 22edo]]
*[[William_Lynch's_Thoughts_on_Septimal_Harmony_and_22_EDO|William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
* [[William Lynch's Thoughts on Septimal Harmony and 22 EDO]]
*[[22edo/Eliora's approach|22edo/Eliora's Approach]]
* [[22edo/Eliora's approach|22edo/Eliora's Approach]]


== Further reading==
== 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.
* [[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://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]
* [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://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]
* [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]]
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