22edo: Difference between revisions

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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. This means that 22 is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably 12edo, 19edo, 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo and [[27edo]] in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony.

The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A–G♮ and C–B♭ are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.

The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A–G♮ and C–B♭ are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.

=== Porcupine comma ===

Line 61:

! Cents

! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref>

! colspan="3" | [[Ups and ~~Downs Notation~~|Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and ^^d2)

! colspan="3" | [[Ups and downs notation|Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and ^^d2)

! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}

! Audio

Line 331:

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! colspan="2" | [[Ups and downs notation|Ups and ~~Downs Notation~~]]

! colspan="2" | [[Ups and downs notation|Ups and downs notation]]

|-

! [[5L 2s|Diatonic Interval Names]]

Line 452:

|}

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:

[[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]]

Line 519:

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

A score video demonstrating this type of notation using redefined sharp and flat symbols is available: [https://www.youtube.com/watch?v=se79rdp705Y ''Study #1 in Porcupine Temperament: "Flying Straight Down" (Microtonal/Xenharmonic)''] (2020) by [[John Moriarty]]. Note that the sharp of one note is lower than the flat of the next note, in contrast to sharps and flats in the diatonic notation with ups and downs described above.

=== Pentatonic notation ===

Line 1,084:

Line 1,086:

=== Interval mappings ===

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 80~~

~~| steps = 22.0251467420146~~

~~| step size = 54.4831784348982~~

~~| tempered height = 6.062600~~

~~| pure height = 5.857510~~

~~| integral = 1.258178~~

~~| gap = 16.213941~~

~~| octave = 1198.62992556776~~

~~| consistent = 12~~

~~| distinct = 8~~

}}

== Regular temperament properties ==

Line 1,458:

Line 1,446:

| [[Undeka]] [[Hendecatonic]]

|}

== Octave compression ==

What follows is a comparison of compressed-octave 22edo tunings.

; 22edo

* Step size: 54.545{{c}}, octave size: 1200.0{{c}}

Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal. It is a good 13-limit tuning for its size.

{{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}}

{{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}

; [[WE|22et, 11-limit WE tuning]]

* Step size: 54.494{{c}}, octave size: 1198.9{{c}}

Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.

{{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}}

{{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}

; [[zpi|80zpi]]

* Step size: 54.483{{c}}, octave size: 1198.6{{c}}

Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.

{{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}}

{{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}}

; [[57ed6]]

* Step size: 54.420{{c}}, octave size: 1197.2{{c}}

Compressing the octave of 22edo by around 3{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7]] tuning, eg for [[archy]] (2.3.7 superpyth) temperament. The tuning 57ed6 does this.

{{Harmonics in equal|57|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}}

{{Harmonics in equal|57|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}

; [[35edt]]

* Step size: 54.342{{c}}, octave size: 1195.5{{c}}

Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[equal tuning|62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.

{{Harmonics in equal|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt}}

{{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt (continued)}}

== Scales ==

''See [[22edo ~~modes]]''.~~

== Tetrachords ==

~~''See [[~~22edo tetrachords~~]].''~~

== ~~Chord names~~ ==

== Chords ==

Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:

Line 1,481:

Line 1,504:

|-

| fourthward wa

| {{monzo| a b }} where b < ~~−1~~

| {{monzo| a b }} where {{nowrap|b < −1}}

| 32/27, 16/9

|-

| upminor

| gu

| {{monzo| a b -1 }}

| {{monzo| a b −1 }}

| 6/5, 9/5

|-

Line 1,496:

Line 1,519:

| rowspan="2" | major

| fifthward wa

| {{monzo| a b }} where b > 1

| {{monzo| a b }} where {{nowrap|b > 1}}

| 9/8, 27/16

|-

| ru

| {{monzo| a b 0 -1 }}

| {{monzo| a b 0 −1 }}

| 9/7, 12/7

|}

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo, and ru triads:

{| class="wikitable center-all"

Line 1,552:

Line 1,575:

* 0-5-11 = C Eb ^Gb = Cd(^5)

* 0-5-12 = C Eb vG = Cm(v5)

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

* [[22edo Chord Names]]

* [[22 EDO Chords]]

* [[Ups and Downs Notation #Chords and Chord Progressions]]

* [[Chords of orwell]]

== Instruments ==

Line 1,565:

Line 1,581:

A potential layout for a 22edo keyboard with both split black and white keys.

[[Lumatone mapping for 22edo|Lumatone mappings for 22edo]] are available.

== Music ==

== ~~Related pages ==~~

== See also ==

* [[Lumatone mapping for 22edo]]

* [[List of MOS scales in 22edo]]

~~=== Approaches =~~==

* [[User:Unque/22edo Composition Theory|Unque's approach]]

* [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch's approach]]

@@ Line 32: / Line 32: @@
 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. This means that 22 is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably 12edo, 19edo, 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo and [[27edo]] in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony.
-The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A&ndash;G&#x266E; and C&ndash;B&#x266D; are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.
+The diatonic scale it produces is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L&nbsp;2s]]), superpyth's diatonic scale has subminor and supermajor thirds of 7/6 and 9/7, rather than minor and major thirds of 6/5 and 5/4. This means that the septimal comma of 64/63 is tempered out, rather than the syntonic comma of 81/80, one of 22et's core features. Superpyth is melodically interesting in that intervals such as A&ndash;G&#x266E; and C&ndash;B&#x266D; are ''harmonic'' sevenths instead of 5-limit minor sevenths (approximately [[7/4]] instead of [[9/5]]), in addition to having a quasi-equal pentatonic scale (as the major whole tone and subminor third are rather close in size) and more uneven diatonic scale, as compared with 12et and other meantone systems; the step patterns in 22et are {{dash|4, 4, 5, 4, 5|med}} and {{dash|4, 4, 1, 4, 4, 4, 1|med}}, respectively.
 === Porcupine comma ===
@@ Line 61: / Line 61: @@
 ! Cents
 ! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref>
-! colspan="3" | [[Ups and Downs Notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^^d2)
+! colspan="3" | [[Ups and downs notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^^d2)
 ! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}
 ! Audio
@@ Line 331: / Line 331: @@
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
-! colspan="2" | [[Ups and downs notation|Ups and Downs Notation]]
+! colspan="2" | [[Ups and downs notation|Ups and downs notation]]
 |-
 ! [[5L 2s|Diatonic Interval Names]]
@@ Line 452: / Line 452: @@
 |}
-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:
 [[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]]
@@ Line 519: / Line 519: @@
 The keyboard runs {{nowrap|D * * E * * F * * G * * * A * * B * * C * * D}}.
+A score video demonstrating this type of notation using redefined sharp and flat symbols is available:  [https://www.youtube.com/watch?v=se79rdp705Y ''Study #1 in Porcupine Temperament: "Flying Straight Down" (Microtonal/Xenharmonic)''] (2020) by [[John Moriarty]]. Note that the sharp of one note is lower than the flat of the next note, in contrast to sharps and flats in the diatonic notation with ups and downs described above.
 === Pentatonic notation ===
@@ Line 1,084: / Line 1,086: @@
 === Interval mappings ===
 {{Q-odd-limit intervals|22}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 80
-| steps = 22.0251467420146
-| step size = 54.4831784348982
-| tempered height = 6.062600
-| pure height = 5.857510
-| integral = 1.258178
-| gap = 16.213941
-| octave = 1198.62992556776
-| consistent = 12
-| distinct = 8
-}}
 == Regular temperament properties ==
@@ Line 1,458: / Line 1,446: @@
 | [[Undeka]]<br>[[Hendecatonic]]
 |}
+== Octave compression ==
+What follows is a comparison of compressed-octave 22edo tunings.
+; 22edo
+* Step size: 54.545{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal. It is a good 13-limit tuning for its size.
+{{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}}
+{{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}
+; [[WE|22et, 11-limit WE tuning]]
+* Step size: 54.494{{c}}, octave size: 1198.9{{c}}
+Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.
+{{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
+{{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
+; [[zpi|80zpi]]
+* Step size: 54.483{{c}}, octave size: 1198.6{{c}}
+Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.
+{{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}}
+{{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}}
+; [[57ed6]]
+* Step size: 54.420{{c}}, octave size: 1197.2{{c}}
+Compressing the octave of 22edo by around 3{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7]] tuning, eg for [[archy]] (2.3.7 superpyth) temperament. The tuning 57ed6 does this.
+{{Harmonics in equal|57|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}}
+{{Harmonics in equal|57|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}
+; [[35edt]]
+* Step size: 54.342{{c}}, octave size: 1195.5{{c}}
+Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[equal tuning|62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.
+{{Harmonics in equal|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt}}
+{{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt (continued)}}
 == Scales ==
-''See [[22edo modes]]''.
+{{Main|22edo modes}}
+{{See also|List of MOS scales in 22edo}}
 == Tetrachords ==
-''See [[22edo tetrachords]].''
+{{Main|22edo tetrachords}}
-== Chord names ==
+== Chords ==
+{{Main|22edo chords}}
 Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
@@ Line 1,481: / Line 1,504: @@
 |-
 | fourthward wa
-| {{monzo| a b }} where b &lt; &minus;1
+| {{monzo| a b }} where {{nowrap|b &lt; −1}}
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| {{monzo| a b -1 }}
+| {{monzo| a b −1 }}
 | 6/5, 9/5
 |-
@@ Line 1,496: / Line 1,519: @@
 | rowspan="2" | major
 | fifthward wa
-| {{monzo| a b }} where b &gt; 1
+| {{monzo| a b }} where {{nowrap|b &gt; 1}}
 | 9/8, 27/16
 |-
 | ru
-| {{monzo| a b 0 -1 }}
+| {{monzo| a b 0 −1 }}
 | 9/7, 12/7
 |}
-All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:
+All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo, and ru triads:
 {| class="wikitable center-all"
@@ Line 1,552: / Line 1,575: @@
 * 0-5-11 = C Eb ^Gb = Cd(^5)
 * 0-5-12 = C Eb vG = Cm(v5)
-Further discussion of 22edo chord naming:
-* [[22edo Chord Names]]
-* [[22 EDO Chords]]
-* [[Ups and Downs Notation #Chords and Chord Progressions]]
-* [[Chords of orwell]]
 == Instruments ==
@@ Line 1,565: / Line 1,581: @@
 A potential layout for a 22edo keyboard with both split black and white keys.
+[[Lumatone mapping for 22edo|Lumatone mappings for 22edo]] are available.
 == Music ==
 {{Main| 22edo/Music }}
 {{Catrel|22edo tracks}}
-== Related pages ==
+== See also ==
-* [[Lumatone mapping for 22edo]]
-* [[List of MOS scales in 22edo]]
-=== Approaches ===
 * [[User:Unque/22edo Composition Theory|Unque's approach]]
 * [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch's approach]]