224edo - Revision history

FloraC: Recategorize (name change followup)

2026-06-06T10:28:19Z

Recategorize (name change followup)

← Older revision		Revision as of 10:28, 6 June 2026
Line 393:		Line 393:
	== References ==		== References ==

			[[Category:Listen]]
			[[Category:Canopic]]
	[[Category:Indra]]		[[Category:Indra]]
	[[Category:~~Listen]]~~		[[Category:Shibi]]
	~~[[Category:Mirkwai~~]]
	[[Category:Octoid]]		[[Category:Octoid]]
	~~[[Category:Quartismic]]~~
	~~[[Category:Shibi]]~~

Lucius Chiaraviglio: /* Theory */ Closed circle of 16 patent fifths + 8 bb fifths

2026-05-10T11:46:13Z

Theory: Closed circle of 16 patent fifths + 8 bb fifths

← Older revision		Revision as of 11:46, 10 May 2026
Line 7:		Line 7:
	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]].		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.		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 ===		=== Prime harmonics ===

FloraC: Don't reference the wiki itself lol

2026-03-28T14:44:41Z

Don't reference the wiki itself lol

← Older revision		Revision as of 14:44, 28 March 2026
Line 14:		Line 14:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 224 factors into {{nowrap\| 2<sup>5</sup> × 7 }}, 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 ==		== Intervals ==
Line 22:		Line 22:
	== Notation ==		== Notation ==
	=== Sagittal ===		=== 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~~<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>		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>

	{\| class="wikitable"		{\| class="wikitable"
Line 385:		Line 385:

	== Music ==		== 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]]		; [[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:Indra]]		[[Category:Indra]]

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:30:02Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:30, 13 March 2026
Line 382:		Line 382:
	\| [[Barium]]		\| [[Barium]]
	\|}		\|}
	<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|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 ==		== Music ==

Eufalesio: Standardize format Sagittal

2026-03-07T12:01:39Z

Standardize format Sagittal

← Older revision		Revision as of 12:01, 7 March 2026
Line 22:		Line 22:
	== Notation ==		== Notation ==
	=== Sagittal ===		=== 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>		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<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"		{\| class="wikitable"
	\|+ Sagittal notation		\|+ Sagittal notation
	! ~~224edosteps~~		! colspan="2" \|Steps
	! 0		! 0
	! 1		! 1
Line 50:		Line 50:
	! 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 ===		=== Ups-and-downs notation ===

Eufalesio: Connection to 217edo

2026-01-21T13:55:37Z

Connection to 217edo

← Older revision		Revision as of 13:55, 21 January 2026
Line 5:		Line 5:
	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 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.

	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.		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.		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.

FloraC: Cleanup

2026-01-18T15:53:27Z

Cleanup

Show changes

FloraC: /* Approximation to JI */ - zeta table

2026-01-18T15:50:07Z

Approximation to JI: - zeta table

@@ Line 165: / Line 165: @@
 == Approximation to JI ==
 === Interval mappings ===
 {{Q-odd-limit intervals}}
 == Regular temperament properties ==

FloraC: /* Theory */ re-organize and wrap prime harmonics table

2026-01-18T15:49:26Z

Theory: re-organize and wrap prime harmonics table

← Older revision		Revision as of 15:49, 18 January 2026
Line 3:		Line 3:

	== Theory ==		== 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.

			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.

	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.		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.

	=== Prime harmonics ===		=== 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 ===		=== 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 {{nowrap\| 2<sup>5</sup> × 7 }}, 224edo has subset edos {{EDOs\| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.

	== Intervals ==		== Intervals ==

Overthink: /* Intervals */ oops, should be comment

2025-12-27T11:14:13Z

Intervals: oops, should be comment

← Older revision		Revision as of 11:14, 27 December 2025
Line 14:		Line 14:

	== Intervals ==		== Intervals ==
	{{todo\|create page\|Table of 224edo intervals}}		{{todo\|create page\|comment=Table of 224edo intervals}}
	{{Interval table}}		{{Interval table}}

← Older revision		Revision as of 12:01, 7 March 2026
Line 22:		Line 22:
	== Notation ==		== Notation ==
	=== Sagittal ===		=== 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>		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<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"		{\| class="wikitable"
	\|+ Sagittal notation		\|+ Sagittal notation
	! ~~224edosteps~~		! colspan="2" \|Steps
	! 0		! 0
	! 1		! 1
Line 50:		Line 50:
	! 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 ===		=== Ups-and-downs notation ===