224edo: Difference between revisions

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== Theory ==

224edo is a very strong [[13-limit]] system, [[tempering out]] [[32805/32768]] in the [[5-limit]]; [[4375/4374]], [[16875/16807]] and [[65625/65536]] in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned [[essentially tempered chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. It is the twelfth [[~~The~~ Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] and is the second-smallest ~~EDO~~ after [[87edo|87]] to approximate all of the first 16 harmonics of the harmonic series with [[minimal consistent EDOs|no greater than 25%]] relative error.

224edo is a very strong [[13-limit]] system. It is the twelfth [[the Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] and is the second-smallest edo after [[87edo|87]] to approximate all of the first 16 harmonics of the harmonic series with [[minimal consistent EDOs|no greater than 25%]] relative error.

224edo tempers the [[syntonic comma]] to 1/56th of the octave (4 steps) and as a corollary supports the [[barium]] temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for [[meantone]] and is very close (0.15 cents) to the [[quarter-comma meantone]] fifth. The generator however reduces to [[112edo]], being 65\112.

As an equal temperament, 224et [[tempering out]] [[32805/32768]] in the [[5-limit]]; [[4375/4374]], [[16875/16807]] and [[65625/65536]] in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned [[essentially tempered chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the [[15-odd-limit]]. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. [[217edo]], only a bit smaller, has a worse 13-limit, but it achieves a much higher [[consistency limit]], almost [[31-odd-limit|31-odd]].

224edo tempers the [[syntonic comma]] to 1/56th of the octave (4 steps) and as a corollary supports the [[barium]] temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for [[meantone]] and is very close (0.15 cents) to the [[quarter-comma meantone]] fifth. The generator however reduces to [[112edo]], being 65\112; that said, the use of both types of fifth enables creation of a closed circle of 24 notes per octave, generated as 16 patent fifths plus 8 bb fifths (as in [[quadrant]] temperament)<ref>[http://www.youtube.com/@Xen-p6p @Xen-p6p] (2026), YouTube post on [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024).</ref>, although a different distribution than the quarter-octave distribution specified by quadrant might be desired for a well-tempered 24 note tuning system.

=== Prime harmonics ===

{{Harmonics in equal|224|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 224edo (continued)}}

=== Subsets and supersets ===

Since 224 factors into {{~~factorisation~~|~~224~~}}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.

Since 224 factors into primes as {{nowrap| 25 × 7 }}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.

== Intervals ==

== Notation ==

=== Sagittal ===

224edo can be written in Sagittal using ''almost'' the entire Athenian extension, by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>[https://sagittal.org/sagittal.pdf Sagittal – A Microtonal Notation System] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>

~~=== Sagittal ===~~

224edo can be written in Sagittal using almost the entire Athenian extension (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian<ref name=":0">[[Ragismic microtemperaments#Brahmagupta]]</ref>. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>https://sagittal.org/sagittal.pdf p. 11</ref>

{| class="wikitable"

|+Sagittal notation

|+ Sagittal notation

!~~224edosteps~~

! colspan="2" |Steps

!0

! 0

!1

! 1

!2

! 2

!3

! 3

!4

! 4

!5

! 5

!6

! 6

!7

! 7

!8

! 8

!9

! 9

!10

! 10

!11

! 11

!12

! 12

!13

! 13

!14

! 14

!15

! 15

!16

! 16

!17

! 17

!18

! 18

!19

! 19

!20

! 20

!21

! 21

|-

|~~Revo~~

! rowspan="2" |Symbol

| rowspan="2" |{{sagittal||//|}}

! Evo

| rowspan="2" |{{sagittal||(}}

| rowspan="2" | <big>{{sagittal||//|}}</big>

| rowspan="2" |{{sagittal|)|(}}

| rowspan="2" | <big>{{sagittal||(}}</big>

| rowspan="2" |{{sagittal|~|(}}

| rowspan="2" | <big>{{sagittal|)|(}}</big>

| rowspan="2" |{{sagittal|/|}}

| rowspan="2" | <big>{{sagittal|~|(}}</big>

| rowspan="2" |{{sagittal||)}}

| rowspan="2" | <big>{{sagittal|/|}}</big>

| rowspan="2" |{{sagittal||\}}

| rowspan="2" | <big>{{sagittal||)}}</big>

| rowspan="2" |{{sagittal|(|(}}

| rowspan="2" | <big>{{sagittal||\}}</big>

| rowspan="2" |{{sagittal|//|}}

| rowspan="2" | <big>{{sagittal|(|(}}</big>

| rowspan="2" |{{sagittal|/|)}}

| rowspan="2" | <big>{{sagittal|//|}}</big>

| rowspan="2" |{{sagittal|/|\}}

| rowspan="2" | <big>{{sagittal|/|)}}</big>

|{{sagittal|(|)}}

| rowspan="2" | <big>{{sagittal|/|\}}</big>

|{{sagittal|(|\}}

| {{sagittal|#}}{{sagittal|\!/}}

|{{sagittal|)||(}}

| {{sagittal|#}}{{sagittal|\!)}}

|{{sagittal|~||(}}

| {{sagittal|#}}{{sagittal|\\!}}

|{{sagittal|/||}}

| {{sagittal|#}}{{sagittal|(!(}}

|{{sagittal|||)}}

| {{sagittal|#}}{{sagittal|!/}}

|{{sagittal|||\}}

| {{sagittal|#}}{{sagittal|!)}}

|{{sagittal|(||(}}

| {{sagittal|#}}{{sagittal|\!}}

|{{sagittal|~~//|~~|}}

| {{sagittal|#}}{{sagittal|~!(}}

|{{sagittal|/||)}}

| {{sagittal|#}}{{sagittal|)!(}}

|{{sagittal|~~/||\~~}}

| {{sagittal|#}}{{sagittal|!(}}

| {{sagittal|#}}

|-

~~|Evo~~

! Revo

|{{sagittal|#}}~~{{sagittal|\!~~/}}

| <big>{{sagittal|(|)}}</big>

|{{sagittal|~~#}}{{sagittal~~|\!)}}

| <big>{{sagittal|(|\}}</big>

|{{sagittal|~~#}}{{sagittal~~|~~\\!~~}}

| <big>{{sagittal|)||(}}</big>

|{{sagittal|~~#}}{{sagittal~~|(!(}}

| <big>{{sagittal|~||(}}</big>

|{{sagittal|#}}~~{{sagittal|!~~/}}

| <big>{{sagittal|/||}}</big>

|{{sagittal|~~#}}{{sagittal~~|!)}}

| <big>{{sagittal|||)}}</big>

|{{sagittal|~~#}}{{sagittal~~|\!}}

| <big>{{sagittal|||\}}</big>

|{{sagittal|~~#}}{{sagittal~~|~!(}}

| <big>{{sagittal|(||(}}</big>

|{{sagittal|~~#}}{{sagittal~~|~~)!(~~}}

| <big>{{sagittal|//||}}</big>

|{{sagittal|~~#}}{{sagittal~~|!(}}

| <big>{{sagittal|/||)}}</big>

|{{sagittal|#}}

| <big>{{sagittal|/||\}}</big>

|}

Because it uses the entire Athenian system (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), it allows no accidental enharmonic respellings

=== Ups-and-downs notation ===

The 4-up (quup) alteration maps to the pythagorean/syntonic comma.

{| class="wikitable" style="text-align:center;"

|+Ups-and-downs notation

|+ Ups-and-downs notation

! rowspan="6" |224edosteps

! rowspan="6" | 224edosteps

!0

! 0

!1

! 1

!2

! 2

!3

! 3

!4

! 4

!5

! 5

!6

! 6

!7

! 7

!8

! 8

!9

! 9

!10

! 10

|-

| rowspan="2" |h

| rowspan="2" | h

|^

| ^

|^^

| ^^

|^^^

| ^^^

|v>

| v>

|>

| >

|^>

| ^>

|^^>

| ^^>

|^^^>

| ^^^>

|v>>

| v>>

|>>

| >>

|-

|<<<<#

| <<<<#

|^<<<<#

| ^<<<<#

|vvv<<<#

| vvv<<<#

|vv<<<#

| vv<<<#

|v<<<#

| v<<<#

|<<<#

| <<<#

|^<<<#

| ^<<<#

|vvv<<#

| vvv<<#

|vv<<#

| vv<<#

|v<<#

| v<<#

|-

!11

! 11

!12

! 12

!13

! 13

!14

! 14

!15

! 15

!16

! 16

!17

! 17

!18

! 18

!19

! 19

!20

! 20

!21

! 21

|-

|^>>

| ^>>

|^^>>

| ^^>>

|^^^>>

| ^^^>>

|v>>>

| v>>>

|>>>

| >>>

|^>>>

| ^>>>

|^^>>>

| ^^>>>

|^^^>>>

| ^^^>>>

|v>>>>

| v>>>>

|>>>>

| >>>>

| rowspan="2" |#

|-

|<<#

| <<#

|^<<#

| ^<<#

|vvv<#

| vvv<#

|vv<#

| vv<#

|v<#

| v<#

|<#

| <#

|^<#

| ^<#

|vvv#

| vvv#

|vv#

| vv#

|v#

| v#

|}

== Approximation to JI ==

=== Interval mappings ===

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

~~{{ZPI~~

~~| zpi = 1546~~

~~| steps = 224.002551156014~~

~~| step size = 5.35708184485908~~

~~| tempered height = 11.730463~~

~~| pure height = 11.721612~~

~~| integral = 1.700865~~

~~| gap = 19.715639~~

~~| octave = 1199.98633324843~~

~~| consistent = 16~~

~~| distinct = 16~~

}}

== Regular temperament properties ==

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|-

| 2.3

| {{~~monzo~~| -355 224 }}

| {{Monzo| -355 224 }}

| {{~~mapping~~| 224 355 }}

| {{Mapping| 224 355 }}

| +0.053

| 0.0534

Line 201:

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| 2.3.5

| 32805/32768, {{monzo| -5 -32 24 }}

| {{~~mapping~~| 224 355 520 }}

| {{Mapping| 224 355 520 }}

| +0.122

| 0.1059

Line 208:

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| 2.3.5.7

| 4375/4374, 16875/16807, 32805/32768

| {{~~mapping~~| 224 355 520 629 }}

| {{Mapping| 224 355 520 629 }}

| +0.018

| 0.2009

Line 215:

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| 2.3.5.7.11

| 540/539, 1375/1372, 4000/3993, 32805/32768

| {{~~mapping~~| 224 355 520 629 775 }}

| {{Mapping| 224 355 520 629 775 }}

| −0.012

| 0.1899

Line 222:

Line 213:

| 2.3.5.7.11.13

| 540/539, 625/624, 729/728, 1375/1372, 2200/2197

| {{~~mapping~~| 224 355 520 629 775 829 }}

| {{Mapping| 224 355 520 629 775 829 }}

| −0.035

| 0.1805

Line 229:

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| 2.3.5.7.11.13.17

| 375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197

| {{~~mapping~~| 224 355 520 629 775 829 916 }}

| {{Mapping| 224 355 520 629 775 829 916 }}

| −0.106

| 0.2420

Line 391:

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| [[Barium]]

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==

; [[Mercury Amalgam]]

* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning

; [[Gene Ward Smith]]

* ''Dreyfus'' (archived 2010) – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – ~~octoid~~[72] in 224edo tuning

* ''Dreyfus'' (archived 2010) – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning

~~; [[Mercury Amalgam]]~~

== References ==

* [https://www.youtube.com/watch?v=~~iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – octant[24], hemigamera[26], oquatonic[56], bezique[64] in 224edo tuning~~

[[Category:Listen]]

[[Category:Canopic]]

[[Category:Indra]]

[[Category:~~Listen]]~~

[[Category:Shibi]]

~~[[Category:Mirkwai~~]]

[[Category:Octoid]]

~~[[Category:Quartismic]]~~

~~[[Category:Shibi]]~~

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a very strong [[13-limit]] system, [[tempering out]] [[32805/32768]] in the [[5-limit]]; [[4375/4374]], [[16875/16807]] and [[65625/65536]] in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned [[essentially tempered chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. It is the twelfth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] and is the second-smallest EDO after [[87edo|87]] to approximate all of the first 16 harmonics of the harmonic series with [[minimal consistent EDOs|no greater than 25%]] relative error.
+edo is a very strong [[13-limit]] system. It is the twelfth [[the Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] and is the second-smallest edo after [[87edo|87]] to approximate all of the first 16 harmonics of the harmonic series with [[minimal consistent EDOs|no greater than 25%]] relative error.
-edo tempers the [[syntonic comma]] to 1/56th of the octave (4 steps) and as a corollary supports the [[barium]] temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for [[meantone]] and is very close (0.15 cents) to the [[quarter-comma meantone]] fifth. The generator however reduces to [[112edo]], being 65\112.
+As an equal temperament, 224et [[tempering out]] [[32805/32768]] in the [[5-limit]]; [[4375/4374]], [[16875/16807]] and [[65625/65536]] in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned [[essentially tempered chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the [[15-odd-limit]]. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. [[217edo]], only a bit smaller, has a worse 13-limit, but it achieves a much higher [[consistency limit]], almost [[31-odd-limit|31-odd]].
+edo tempers the [[syntonic comma]] to 1/56th of the octave (4 steps) and as a corollary supports the [[barium]] temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for [[meantone]] and is very close (0.15 cents) to the [[quarter-comma meantone]] fifth. The generator however reduces to [[112edo]], being 65\112; that said, the use of both types of fifth enables creation of a closed circle of 24 notes per octave, generated as 16 patent fifths plus 8 bb fifths (as in [[quadrant]] temperament)<ref>[http://www.youtube.com/@Xen-p6p @Xen-p6p] (2026), YouTube post on [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024).</ref>, although a different distribution than the quarter-octave distribution specified by quadrant might be desired for a well-tempered 24 note tuning system.
 === Prime harmonics ===
-{{Harmonics in equal|224|columns=14}}
+{{Harmonics in equal|224|columns=11}}
+{{Harmonics in equal|224|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 224edo (continued)}}
 === Subsets and supersets ===
-Since 224 factors into {{factorisation|224}}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
+Since 224 factors into primes as {{nowrap| 2<sup>5</sup> × 7 }}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
 == Intervals ==
-{{todo|create page|comment=Table of 224edo intervals}}
+{{Todo|create page|comment=Table of 224edo intervals}}
 {{Interval table}}
 == Notation ==
+=== Sagittal ===
+edo can be written in Sagittal using ''almost'' the entire Athenian extension, by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>[https://sagittal.org/sagittal.pdf Sagittal – A Microtonal Notation System] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>
-=== Sagittal ===
-edo can be written in Sagittal using almost the entire Athenian extension (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian<ref name=":0">[[Ragismic microtemperaments#Brahmagupta]]</ref>. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>https://sagittal.org/sagittal.pdf p. 11</ref>
 {| class="wikitable"
-|+Sagittal notation
+|+ Sagittal notation
-!224edosteps
+! colspan="2" |Steps
-!0
+! 0
-!1
+! 1
-!2
+! 2
-!3
+! 3
-!4
+! 4
-!5
+! 5
-!6
+! 6
-!7
+! 7
-!8
+! 8
-!9
+! 9
-!10
+! 10
-!11
+! 11
-!12
+! 12
-!13
+! 13
-!14
+! 14
-!15
+! 15
-!16
+! 16
-!17
+! 17
-!18
+! 18
-!19
+! 19
-!20
+! 20
-!21
+! 21
 |-
-|Revo
+! rowspan="2" |Symbol
-| rowspan="2" |{{sagittal||//|}}
+! Evo
-| rowspan="2" |{{sagittal||(}}
+| rowspan="2" | <big>{{sagittal||//|}}</big>
-| rowspan="2" |{{sagittal|)|(}}
+| rowspan="2" | <big>{{sagittal||(}}</big>
-| rowspan="2" |{{sagittal|~|(}}
+| rowspan="2" | <big>{{sagittal|)|(}}</big>
-| rowspan="2" |{{sagittal|/|}}
+| rowspan="2" | <big>{{sagittal|~|(}}</big>
-| rowspan="2" |{{sagittal||)}}
+| rowspan="2" | <big>{{sagittal|/|}}</big>
-| rowspan="2" |{{sagittal||\}}
+| rowspan="2" | <big>{{sagittal||)}}</big>
-| rowspan="2" |{{sagittal|(|(}}
+| rowspan="2" | <big>{{sagittal||\}}</big>
-| rowspan="2" |{{sagittal|//|}}
+| rowspan="2" | <big>{{sagittal|(|(}}</big>
-| rowspan="2" |{{sagittal|/|)}}
+| rowspan="2" | <big>{{sagittal|//|}}</big>
-| rowspan="2" |{{sagittal|/|\}}
+| rowspan="2" | <big>{{sagittal|/|)}}</big>
-|{{sagittal|(|)}}
+| rowspan="2" | <big>{{sagittal|/|\}}</big>
-|{{sagittal|(|\}}
+| <small>{{sagittal|#}}{{sagittal|\!/}}</small>
-|{{sagittal|)||(}}
+| <small>{{sagittal|#}}{{sagittal|\!)}}</small>
-|{{sagittal|~||(}}
+| <small>{{sagittal|#}}{{sagittal|\\!}}</small>
-|{{sagittal|/||}}
+| <small>{{sagittal|#}}{{sagittal|(!(}}</small>
-|{{sagittal|||)}}
+| <small>{{sagittal|#}}{{sagittal|!/}}</small>
-|{{sagittal|||\}}
+| <small>{{sagittal|#}}{{sagittal|!)}}</small>
-|{{sagittal|(||(}}
+| <small>{{sagittal|#}}{{sagittal|\!}}</small>
-|{{sagittal|//||}}
+| <small>{{sagittal|#}}{{sagittal|~!(}}</small>
-|{{sagittal|/||)}}
+| <small>{{sagittal|#}}{{sagittal|)!(}}</small>
-|{{sagittal|/||\}}
+| <small>{{sagittal|#}}{{sagittal|!(}}</small>
+| <small>{{sagittal|#}}</small>
 |-
-|Evo
+! Revo
-|{{sagittal|#}}{{sagittal|\!/}}
+| <big>{{sagittal|(|)}}</big>
-|{{sagittal|#}}{{sagittal|\!)}}
+| <big>{{sagittal|(|\}}</big>
-|{{sagittal|#}}{{sagittal|\\!}}
+| <big>{{sagittal|)||(}}</big>
-|{{sagittal|#}}{{sagittal|(!(}}
+| <big>{{sagittal|~||(}}</big>
-|{{sagittal|#}}{{sagittal|!/}}
+| <big>{{sagittal|/||}}</big>
-|{{sagittal|#}}{{sagittal|!)}}
+| <big>{{sagittal|||)}}</big>
-|{{sagittal|#}}{{sagittal|\!}}
+| <big>{{sagittal|||\}}</big>
-|{{sagittal|#}}{{sagittal|~!(}}
+| <big>{{sagittal|(||(}}</big>
-|{{sagittal|#}}{{sagittal|)!(}}
+| <big>{{sagittal|//||}}</big>
-|{{sagittal|#}}{{sagittal|!(}}
+| <big>{{sagittal|/||)}}</big>
-|{{sagittal|#}}
+| <big>{{sagittal|/||\}}</big>
 |}
+Because it uses the entire Athenian system (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), it allows no accidental enharmonic respellings
 === Ups-and-downs notation ===
 The 4-up (quup) alteration maps to the pythagorean/syntonic comma.
 {| class="wikitable" style="text-align:center;"
-|+Ups-and-downs notation
+|+ Ups-and-downs notation
-! rowspan="6" |224edosteps
+! rowspan="6" | 224edosteps
-!0
+! 0
-!1
+! 1
-!2
+! 2
-!3
+! 3
-!4
+! 4
-!5
+! 5
-!6
+! 6
-!7
+! 7
-!8
+! 8
-!9
+! 9
-!10
+! 10
 |-
-| rowspan="2" |h
+| rowspan="2" | h
-|^
+| ^
-|^^
+| ^^
-|^^^
+| ^^^
-|v>
+| v>
-|>
+| >
-|^>
+| ^>
-|^^>
+| ^^>
-|^^^>
+| ^^^>
-|v>>
+| v>>
-|>>
+| >>
 |-
-|<<<<#
+| <<<<#
-|^<<<<#
+| ^<<<<#
-|vvv<<<#
+| vvv<<<#
-|vv<<<#
+| vv<<<#
-|v<<<#
+| v<<<#
-|<<<#
+| <<<#
-|^<<<#
+| ^<<<#
-|vvv<<#
+| vvv<<#
-|vv<<#
+| vv<<#
-|v<<#
+| v<<#
 |-
-!11
+! 11
-!12
+! 12
-!13
+! 13
-!14
+! 14
-!15
+! 15
-!16
+! 16
-!17
+! 17
-!18
+! 18
-!19
+! 19
-!20
+! 20
-!21
+! 21
 |-
-|^>>
+| ^>>
-|^^>>
+| ^^>>
-|^^^>>
+| ^^^>>
-|v>>>
+| v>>>
-|>>>
+| >>>
-|^>>>
+| ^>>>
-|^^>>>
+| ^^>>>
-|^^^>>>
+| ^^^>>>
-|v>>>>
+| v>>>>
-|>>>>
+| >>>>
 | rowspan="2" |#
 |-
-|<<#
+| <<#
-|^<<#
+| ^<<#
-|vvv<#
+| vvv<#
-|vv<#
+| vv<#
-|v<#
+| v<#
-|<#
+| <#
-|^<#
+| ^<#
-|vvv#
+| vvv#
-|vv#
+| vv#
-|v#
+| v#
 |}
 == Approximation to JI ==
 === Interval mappings ===
 {{Q-odd-limit intervals}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 1546
-| steps = 224.002551156014
-| step size = 5.35708184485908
-| tempered height = 11.730463
-| pure height = 11.721612
-| integral = 1.700865
-| gap = 19.715639
-| octave = 1199.98633324843
-| consistent = 16
-| distinct = 16
-}}
 == Regular temperament properties ==
@@ Line 193: / Line 184: @@
 |-
 | 2.3
-| {{monzo| -355 224 }}
+| {{Monzo| -355 224 }}
-| {{mapping| 224 355 }}
+| {{Mapping| 224 355 }}
 | +0.053
 | 0.0534
@@ Line 201: / Line 192: @@
 | 2.3.5
 | 32805/32768, {{monzo| -5 -32 24 }}
-| {{mapping| 224 355 520 }}
+| {{Mapping| 224 355 520 }}
 | +0.122
 | 0.1059
@@ Line 208: / Line 199: @@
 | 2.3.5.7
 | 4375/4374, 16875/16807, 32805/32768
-| {{mapping| 224 355 520 629 }}
+| {{Mapping| 224 355 520 629 }}
 | +0.018
 | 0.2009
@@ Line 215: / Line 206: @@
 | 2.3.5.7.11
 | 540/539, 1375/1372, 4000/3993, 32805/32768
-| {{mapping| 224 355 520 629 775 }}
+| {{Mapping| 224 355 520 629 775 }}
 | −0.012
 | 0.1899
@@ Line 222: / Line 213: @@
 | 2.3.5.7.11.13
 | 540/539, 625/624, 729/728, 1375/1372, 2200/2197
-| {{mapping| 224 355 520 629 775 829 }}
+| {{Mapping| 224 355 520 629 775 829 }}
 | −0.035
 | 0.1805
@@ Line 229: / Line 220: @@
 | 2.3.5.7.11.13.17
 | 375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197
-| {{mapping| 224 355 520 629 775 829 916 }}
+| {{Mapping| 224 355 520 629 775 829 916 }}
 | −0.106
 | 0.2420
@@ Line 391: / Line 382: @@
 | [[Barium]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Music ==
+; [[Mercury Amalgam]]
+* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning
 ; [[Gene Ward Smith]]
-* ''Dreyfus'' (archived 2010) – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning
+* ''Dreyfus'' (archived 2010) – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning
-; [[Mercury Amalgam]]
+== References ==
-* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – octant[24], hemigamera[26], oquatonic[56], bezique[64] in 224edo tuning
+[[Category:Listen]]
+[[Category:Canopic]]
 [[Category:Indra]]
-[[Category:Listen]]
+[[Category:Shibi]]
-[[Category:Mirkwai]]
 [[Category:Octoid]]
-[[Category:Quartismic]]
-[[Category:Shibi]]