270edo - Revision history

FloraC: /* Rank-2 temperaments */ + trivish. Also convert the table to minimal form since we're already reducing it to the first semi-octave

2026-06-20T15:12:07Z

Rank-2 temperaments: + trivish. Also convert the table to minimal form since we're already reducing it to the first semi-octave

← Older revision		Revision as of 15:12, 20 June 2026
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	\| 25/19		\| 25/19
	\| [[Vulture]]		\| [[Vulture]]
			\|-
			\| 2
			\| 4\270
			\| 17.{{overline\|7}}
			\| 99/98
			\| [[Quarvish]]
	\|-		\|-
	\| 2		\| 2
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	\| 71.{{overline\|1}}		\| 71.{{overline\|1}}
	\| 25/24		\| 25/24
	\| [[Vishnu]] / ananta ~~/ acyuta~~		\| [[Vishnu]] / acyuta / ananta
	\|-		\|-
	\| 2		\| 2
	\| ~~112\270<br>(~~23\270)		\| 23\270
	\| ~~497.{{overline\|7}}<br>(~~102.{{overline\|2}})		\| 102.{{overline\|2}}
	\| ~~4/3<br>(~~35/33)		\| 35/33
	\| [[Gariwizmic]]		\| [[Gariwizmic]]
	\|-		\|-
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	\| 8/7		\| 8/7
	\| [[Orga]]		\| [[Orga]]
	\|-
	~~\| 2~~
	~~\| 131\270<br>(4\270)~~
	~~\| 582.{{overline\|2}}<br>(17.{{overline\|7}})~~
	~~\| 7/5<br>(99/98)~~
	~~\| [[Quarvish]]~~
	\|-		\|-
	\| 3		\| 3
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	\|-		\|-
	\| 5		\| 5
	\| ~~83\270<br>(~~25\270)		\| 25\270
	\| ~~368.{{overline\|8}}<br>(~~111.{{overline\|1}})		\| 111.{{overline\|1}}
	\| ~~1024/891<br>(~~16/15)		\| 16/15
	\| [[Quintosec]]		\| [[Quintosec]]
	\|-		\|-
	\| 6		\| 6
	\| ~~112~~\270~~<br>(4\270)~~		\| 16\270
	\| ~~497~~.{{overline\|7}}~~<br>(~~97.{{overline\|7}})		\| 71.{{overline\|1}}
	\| ~~4/3<br>(~~128/121)		\| 25/24
			\| [[Trivish]]
			\|-
			\| 6
			\| 22\270
			\| 97.{{overline\|7}}
			\| 128/121
	\| [[Sextile]]		\| [[Sextile]]
	\|-		\|-
	\| 9		\| 9
	\| ~~71\270<br>(~~11\270)		\| 11\270
	\| ~~315.{{overline\|5}}<br>(~~48.{{overline\|8}})		\| 48.{{overline\|8}}
	\| ~~6/5<br>(~~36/35)		\| 36/35
	\| [[Ennealimmal]] / enneabiotic / ennealympic		\| [[Ennealimmal]] / enneabiotic / ennealympic
	\|-		\|-
	\| 10		\| 10
	\| 16\270~~<br>(11\270)~~		\| 2\270
	\| ~~71.{{overline\|1}}<br>(48~~.{{overline\|8}})		\| 8.{{overline\|8}}
	\| ~~25/24<br>(36~~/~~35)~~		\| 176/175
	\| [[~~Decavish~~]]		\| [[Decoid]]
	\|-		\|-
	\| 10		\| 10
	\| 56\270~~<br>(2\270)~~		\| 10\270
	\| ~~248.{{overline\|8}}<br>(8~~.{{overline\|8}})		\| 44.{{overline\|4}}
	\| ~~15/13<br>(176~~/~~175)~~		\| 40/39
	\| [[~~Decoid~~]]		\| [[Deca]]
	\|-		\|-
	\| 10		\| 10
	\| 71\270~~<br>(10\270)~~		\| 11\270
	\| ~~315.{{overline\|5}}<br>(44~~.{{overline\|4}})		\| 48.{{overline\|8}}
	\| ~~6/5<br>(40~~/~~39)~~		\| 36/35
	\| [[~~Deca~~]]		\| [[Decavish]]
	\|-		\|-
	\| 18		\| 18
	\| 71\270~~<br>(4\270)~~		\| 2\270
	\| ~~248~~.{{overline\|8}}~~<br>(17.{{overline\|7}})~~		\| 8.{{overline\|8}}
	\| ~~15/13<br>(99~~/~~98)~~		\| 1287/1280
	\| [[~~Hemiennealimmal~~]]		\| [[Semihemiennealimmal]]
	\|-		\|-
	\| 18		\| 18
	\| 71\270~~<br>(2\270)~~		\| 4\270
	\| ~~475.{{overline\|5}}<br>(8~~.{{overline\|8}})		\| 17.{{overline\|7}}
	\| ~~1053/800<br>(1287~~/~~1280)~~		\| 99/98
	\| [[~~Semihemiennealimmal~~]]		\| [[Hemiennealimmal]]
	\|-		\|-
	\| 27		\| 27
	\| ~~61\270<br>(~~1\270)		\| 1\270
	\| ~~271.{{overline\|1}}<br>(~~4.{{overline\|4}})		\| 4.{{overline\|4}}
	\| ~~1375/1176<br>(~~385/384)		\| 385/384
	\| [[Trinealimmal]]		\| [[Trinealimmal]]
	\|-		\|-
	\| 30		\| 30
	\| ~~82\270<br>(~~1\270)		\| 1\270
	~~\| 364.{{overline~~\|~~4}}<br>(~~4.{{overline\|4}})		\| 4.{{overline\|4}}
	\| ~~216/175<br>(~~385/384)		\| 385/384
	\| [[Zinc]]		\| [[Zinc]]
	\|-		\|-
	\| 45		\| 45
	\| ~~59\270<br>(~~1\270)		\| 1\270
	\| ~~262.{{overline\|2}}<br>(~~4.{{overline\|4}})		\| 4.{{overline\|4}}
	\| ~~64/55<br>(~~385/384)		\| 385/384
	\| [[Rhodium]]		\| [[Rhodium]]
	\|}		\|}
	<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~~		<nowiki/>* In [[normal forms #Minimal-generator form\|minimal-generator form]]

	== Scales ==		== Scales ==

Dave Keenan: /* Sagittal notation */

2026-06-02T03:12:02Z

Sagittal notation

← Older revision		Revision as of 03:12, 2 June 2026
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	\|<big>{{Sagittal\|/\|\|\}}</big>		\|<big>{{Sagittal\|/\|\|\}}</big>
	\|}		\|}
	Note that the Revo notation has matching flag sequences between the double shaft symbols and a subsequence of the single-shaft symbols.		Note that the Revo notation has matching flag sequences between the double-shaft symbols and a subsequence of the single-shaft symbols.

	<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span>		<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span>

Dave Keenan: /* Sagittal notation */ Corrected some Revo symbols and added note about matching flag sequences.

2026-06-02T02:59:07Z

Sagittal notation: Corrected some Revo symbols and added note about matching flag sequences.

← Older revision		Revision as of 02:59, 2 June 2026
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	\|<big>{{Sagittal\|\|\|)}}</big>		\|<big>{{Sagittal\|\|\|)}}</big>
	\|<big>{{Sagittal\|\|\|\}}</big>		\|<big>{{Sagittal\|\|\|\}}</big>
			\|<big>{{sagittal\|~\|\|)}}</big>
	\|<big>{{sagittal\|(\|\|(}}</big>		\|<big>{{sagittal\|(\|\|(}}</big>
	~~\|<big>{{sagittal\|~\|\|\}}</big>~~
	\|<big>{{sagittal\|//\|\|}}</big>		\|<big>{{sagittal\|//\|\|}}</big>
	\|<big>{{sagittal\|/\|\|)}}</big>		\|<big>{{sagittal\|/\|\|)}}</big>
	\|<big>{{Sagittal\|/\|\|\}}</big>		\|<big>{{Sagittal\|/\|\|\}}</big>
	\|}		\|}
			Note that the Revo notation has matching flag sequences between the double shaft symbols and a subsequence of the single-shaft symbols.

	<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span>		<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span>

Dave Keenan: /* Sagittal notation */

2026-06-02T02:45:18Z

Sagittal notation

← Older revision		Revision as of 02:45, 2 June 2026
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	=== Sagittal notation ===		=== Sagittal notation ===
	<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses symbols from the Promethean set. Since the apotome can be split in two, the Stein-Zimmermann half-sharp and a half-flat may be used.</span>		<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses symbols from the Promethean set. Since the apotome can be split in two, the Stein-Zimmermann half-sharp and half-flat may be used.</span>
	{\| class="wikitable center-all" data-darkreader-inline-color=""		{\| class="wikitable center-all" data-darkreader-inline-color=""
	! colspan="2" \|+ edosteps		! colspan="2" \|+ edosteps

Dave Keenan: /* Sagittal notation */ Corrected the table to agree with George Secor's notation as given in the Sagittal Forum.

2026-06-02T02:42:56Z

Sagittal notation: Corrected the table to agree with George Secor's notation as given in the Sagittal Forum.

← Older revision		Revision as of 02:42, 2 June 2026
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	=== Sagittal notation ===		=== Sagittal notation ===
	<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses ~~alterations of~~ the ~~Promethian~~ set. Since the apotome can be split in two, ~~a SZ~~ half-sharp and a half-flat may be used.</span>		<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses symbols from the Promethean set. Since the apotome can be split in two, the Stein-Zimmermann half-sharp and a half-flat may be used.</span>
	{\| class="wikitable center-all" data-darkreader-inline-color=""		{\| class="wikitable center-all" data-darkreader-inline-color=""
	! colspan="2" \|+ edosteps		! colspan="2" \|+ edosteps
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	\|-		\|-
	! rowspan="3" \|Symbol		! rowspan="3" \|Symbol
	!SZ		!Evo-SZ
	\| rowspan="3" \|<big>{{sagittal\|\|(}}</big>		\| rowspan="3" \|<big>{{sagittal\|\|(}}</big>
	\| rowspan="3" \|<big>{{sagittal\|)\|(}}</big>		\| rowspan="3" \|<big>{{sagittal\|)\|(}}</big>
	~~\| rowspan="3" \|<big>{{sagittal\|)~\|}}</big>~~
	\| rowspan="3" \|<big>{{Sagittal\|~\|(}}</big>		\| rowspan="3" \|<big>{{Sagittal\|~\|(}}</big>
			\| rowspan="3" \|<big>{{Sagittal\|~~\|}}</big>
	\| rowspan="3" \|<big>{{Sagittal\|/\|}}</big>		\| rowspan="3" \|<big>{{Sagittal\|/\|}}</big>
	\| rowspan="3" \|<big>{{Sagittal\|\|)}}</big>		\| rowspan="3" \|<big>{{Sagittal\|\|)}}</big>
	\| rowspan="3" \|<big>{{sagittal\|\|\}}</big>		\| rowspan="3" \|<big>{{sagittal\|\|\}}</big>
	\| rowspan="3" \|<big>{{sagittal\|(\|}}</big>		\| rowspan="3" \|<big>{{sagittal\|~\|)}}</big>
	\| rowspan="3" \|<big>{{sagittal\|(\|(}}</big>		\| rowspan="3" \|<big>{{sagittal\|(\|(}}</big>
	\| rowspan="3" \|<big>{{sagittal\|//\|}}</big>		\| rowspan="3" \|<big>{{sagittal\|//\|}}</big>
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	\| rowspan="3" \|<big>{{Sagittal\|/\|\}}</big>		\| rowspan="3" \|<big>{{Sagittal\|/\|\}}</big>
	\|<big>{{Sagittal\|t}}</big>		\|<big>{{Sagittal\|t}}</big>
	\|~~<small>~~{{Sagittal\|\|(}}{{sagittal\|t}}~~</small>~~		\|{{Sagittal\|\|(}}{{sagittal\|t}}
	\|~~<small>~~{{Sagittal\|)\|(}}{{sagittal\|t}}~~</small>~~		\|{{Sagittal\|)\|(}}{{sagittal\|t}}
	\|~~<small>~~{{~~Sagittal\|)~~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|\\!}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|~~~\|(}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|(!(}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|/~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|~!)}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|\|)}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|!/}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|\|\}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|!)}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|(~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|\!}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|~~(\|(~~}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|~~!}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|//~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|~!(}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|/~~\|)}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|)!(}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|~~/\|\~~}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|!(}}{{sagittal\|#}}
	\| rowspan="2" \|<big>{{Sagittal\|#}}</big>		\| rowspan="2" \|<big>{{Sagittal\|#}}</big>
	\|-		\|-
	!Evo		!Evo
	\| rowspan="2" \|<big>{{sagittal\|)/\|\}}</big>		\| rowspan="2" \|<big>{{sagittal\|)/\|\}}</big>
	\|~~<small>{{sagittal\|\!/}}{{sagittal\|#}}</small>~~		\| rowspan="2" \|<big>{{Sagittal\|(\|)}}</big>
	~~\|<small>{{sagittal\|\!)}}{{sagittal\|#}}</small>~~		\| rowspan="2" \|<big>{{sagittal\|(\|\}}</big>
	~~\|<small>{{sagittal\|\\!}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|(!(}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|(!}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|!/}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|!)}}{{sagittal\|#}}</small>~~
	\|<~~small~~>{{~~sagittal~~\|~~\!}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|~!~~(~~}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal~~\|)~~~!}}{{sagittal\|#~~}}</~~small~~>
	\|~~<small>{{sagittal\|)!(}}{{sagittal~~\|~~#}}~~<~~/small>~~
	~~\|<small~~>{{sagittal\|!(~~}}{{sagittal~~\|#}}</~~small~~>
	\|-		\|-
	!Revo		!Revo
	~~\|<big>{{Sagittal\|(\|)}}</big>~~
	~~\|<big>{{sagittal\|(\|\}}</big>~~
	\|<big>{{sagittal\|)\|\|(}}</big>		\|<big>{{sagittal\|)\|\|(}}</big>
	\|<big>{{sagittal\|~\|\|(}}</big>		\|<big>{{sagittal\|~\|\|(}}</big>

FloraC: /* Regular temperament properties */ update

2026-05-19T09:45:00Z

Regular temperament properties: update

← Older revision		Revision as of 09:45, 19 May 2026
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	\| 315.{{overline\|5}}<br>(48.{{overline\|8}})		\| 315.{{overline\|5}}<br>(48.{{overline\|8}})
	\| 6/5<br>(36/35)		\| 6/5<br>(36/35)
	\| [[Ennealimmal]] / ~~ennealimmia~~		\| [[Ennealimmal]] / enneabiotic / ennealympic
	\|-		\|-
	\| 10		\| 10

TallKite: /* Ups and downs notation */ added the spoken terms

2026-05-10T00:58:04Z

Ups and downs notation: added the spoken terms

← Older revision		Revision as of 00:58, 10 May 2026
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	== Notation ==		== Notation ==
	=== Ups and downs notation ===		=== Ups and downs notation ===
	270edo can be notated using [[Kite's ups and downs notation\|ups and downs]] with Stein-Zimmerman quarter-tone accidentals representing half-~~apotomes:~~		270edo can be notated using [[Kite's ups and downs notation\|ups and downs]] with Stein-Zimmerman quarter-tone accidentals representing half-sharps and half-flats. These can be spoken as ''sha'' and ''fla''. For example, the note 12\270 above C is C downsha, and the note 39\270 above C is C shasharp.
	{{Ups and downs sharpness\|270\|true}}		{{Ups and downs sharpness\|270\|true}}

Eufalesio: /* Regular temperament properties */

2026-04-24T17:06:57Z

Regular temperament properties

← Older revision		Revision as of 17:06, 24 April 2026
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	\| 4.58		\| 4.58
	\|}		\|}
	* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo\|72]] in the 11-limit, [[224edo\|224]] in the 13-limit, and [[217edo\|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo\|342]], in the 13-limit [[494edo\|494]], in the 23-limit [[282edo\|282]]; and in the 19-limit, [[311edo\|311]] for absolute error and [[581edo\|581]] for relative error. It is also a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit and 13-limit up until [[342edo]] and [[~~96478edo~~]] respectively.		* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo\|72]] in the 11-limit, [[224edo\|224]] in the 13-limit, and [[217edo\|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo\|342]], in the 13-limit [[494edo\|494]], in the 23-limit [[282edo\|282]]; and in the 19-limit, [[311edo\|311]] for absolute error and [[581edo\|581]] for relative error. It is also a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit, 13-limit and 19-limit up until [[342edo]], [[96478edo]] and [[3395edo]] respectively.
	* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.		* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.
	* It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo\|552]], and the least relative error until [[2190edo\|2190]].		* It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo\|552]], and the least relative error until [[2190edo\|2190]].

FloraC: At least don't put it in the theory section. Also remove semantic inflation

2026-04-18T14:51:40Z

At least don't put it in the theory section. Also remove semantic inflation

← Older revision		Revision as of 14:51, 18 April 2026
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	== Theory ==		== Theory ==
	270edo is an extremely strong [[13-limit]] ~~and~~ system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency#Consistency to distance d\|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit ''and'' 13-limit up until [[342edo]] and [[96478edo]] respectively.		270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency #Consistency to distance d\|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]]). It is the 11th [[zeta gap edo]], the 13th [[zeta integral edo]], the 23rd [[zeta peak edo]], and the 18th [[zeta peak integer edo]], making it a [[strict zeta edo]].

	It is [[~~The Riemann zeta function and tuning #Zeta EDO lists\|the 11th~~ zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a [[strict zeta edo]].

	In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo\| 1 -27 18 }}, the [[vulture comma]], {{monzo\| 24 -21 4 }}, and the [[vishnuzma]], {{monzo\| 23 6 -14 }}.		In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo\| 1 -27 18 }}, the [[vulture comma]], {{monzo\| 24 -21 4 }}, and the [[vishnuzma]], {{monzo\| 23 6 -14 }}.
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	The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit, and [[island chords]] in the 15-odd-limit.		The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit, and [[island chords]] in the 15-odd-limit.

	Beyond the 13-limit, the approximated [[17/1\|harmonic 17]] is more than 1/3-edostep, but the [[19/1\|harmonic 19]] is ~~extremely~~ accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1\|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. It tempers out [[715/714]], [[936/935]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.		Beyond the 13-limit, the approximated [[17/1\|harmonic 17]] is more than 1/3-edostep, but the [[19/1\|harmonic 19]] is very accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1\|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. It tempers out [[715/714]], [[936/935]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.

	The harmonics [[29/1\|29]] and [[31/1\|31]] are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency ([[29/26]] and [[31/26]] are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 [[35-odd-limit]]. Notably, it tempers out [[784/783]], [[900/899]], and [[1024/1023]], while inflating [[841/840]] and [[961/960]].		The harmonics [[29/1\|29]] and [[31/1\|31]] are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency ([[29/26]] and [[31/26]] are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 [[35-odd-limit]]. Notably, it tempers out [[784/783]], [[900/899]], and [[1024/1023]], while inflating [[841/840]] and [[961/960]].
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	\| 4.58		\| 4.58
	\|}		\|}
	* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo\|72]] in the 11-limit, [[224edo\|224]] in the 13-limit, and [[217edo\|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo\|342]], in the 13-limit [[494edo\|494]], in the 23-limit [[282edo\|282]]; and in the 19-limit, [[311edo\|311]] for absolute error and [[581edo\|581]] for relative error.		* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo\|72]] in the 11-limit, [[224edo\|224]] in the 13-limit, and [[217edo\|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo\|342]], in the 13-limit [[494edo\|494]], in the 23-limit [[282edo\|282]]; and in the 19-limit, [[311edo\|311]] for absolute error and [[581edo\|581]] for relative error. It is also a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit and 13-limit up until [[342edo]] and [[96478edo]] respectively.
	* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.		* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.
	* It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo\|552]], and the least relative error until [[2190edo\|2190]].		* It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo\|552]], and the least relative error until [[2190edo\|2190]].

Eufalesio: Added back TE logflat badness mention (Come on there's nothing like it until the tens or hundreds of thousands!)

2026-04-18T13:13:35Z

Added back TE logflat badness mention (Come on there's nothing like it until the tens or hundreds of thousands!)

← Older revision		Revision as of 02:59, 2 June 2026
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	\|<big>{{Sagittal\|\|\|)}}</big>		\|<big>{{Sagittal\|\|\|)}}</big>
	\|<big>{{Sagittal\|\|\|\}}</big>		\|<big>{{Sagittal\|\|\|\}}</big>
			\|<big>{{sagittal\|~\|\|)}}</big>
	\|<big>{{sagittal\|(\|\|(}}</big>		\|<big>{{sagittal\|(\|\|(}}</big>
	~~\|<big>{{sagittal\|~\|\|\}}</big>~~
	\|<big>{{sagittal\|//\|\|}}</big>		\|<big>{{sagittal\|//\|\|}}</big>
	\|<big>{{sagittal\|/\|\|)}}</big>		\|<big>{{sagittal\|/\|\|)}}</big>
	\|<big>{{Sagittal\|/\|\|\}}</big>		\|<big>{{Sagittal\|/\|\|\}}</big>
	\|}		\|}
			Note that the Revo notation has matching flag sequences between the double shaft symbols and a subsequence of the single-shaft symbols.

	<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span>		<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span>

← Older revision		Revision as of 02:42, 2 June 2026
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	=== Sagittal notation ===		=== Sagittal notation ===
	<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses ~~alterations of~~ the ~~Promethian~~ set. Since the apotome can be split in two, ~~a SZ~~ half-sharp and a half-flat may be used.</span>		<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses symbols from the Promethean set. Since the apotome can be split in two, the Stein-Zimmermann half-sharp and a half-flat may be used.</span>
	{\| class="wikitable center-all" data-darkreader-inline-color=""		{\| class="wikitable center-all" data-darkreader-inline-color=""
	! colspan="2" \|+ edosteps		! colspan="2" \|+ edosteps
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	\|-		\|-
	! rowspan="3" \|Symbol		! rowspan="3" \|Symbol
	!SZ		!Evo-SZ
	\| rowspan="3" \|<big>{{sagittal\|\|(}}</big>		\| rowspan="3" \|<big>{{sagittal\|\|(}}</big>
	\| rowspan="3" \|<big>{{sagittal\|)\|(}}</big>		\| rowspan="3" \|<big>{{sagittal\|)\|(}}</big>
	~~\| rowspan="3" \|<big>{{sagittal\|)~\|}}</big>~~
	\| rowspan="3" \|<big>{{Sagittal\|~\|(}}</big>		\| rowspan="3" \|<big>{{Sagittal\|~\|(}}</big>
			\| rowspan="3" \|<big>{{Sagittal\|~~\|}}</big>
	\| rowspan="3" \|<big>{{Sagittal\|/\|}}</big>		\| rowspan="3" \|<big>{{Sagittal\|/\|}}</big>
	\| rowspan="3" \|<big>{{Sagittal\|\|)}}</big>		\| rowspan="3" \|<big>{{Sagittal\|\|)}}</big>
	\| rowspan="3" \|<big>{{sagittal\|\|\}}</big>		\| rowspan="3" \|<big>{{sagittal\|\|\}}</big>
	\| rowspan="3" \|<big>{{sagittal\|(\|}}</big>		\| rowspan="3" \|<big>{{sagittal\|~\|)}}</big>
	\| rowspan="3" \|<big>{{sagittal\|(\|(}}</big>		\| rowspan="3" \|<big>{{sagittal\|(\|(}}</big>
	\| rowspan="3" \|<big>{{sagittal\|//\|}}</big>		\| rowspan="3" \|<big>{{sagittal\|//\|}}</big>
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	\| rowspan="3" \|<big>{{Sagittal\|/\|\}}</big>		\| rowspan="3" \|<big>{{Sagittal\|/\|\}}</big>
	\|<big>{{Sagittal\|t}}</big>		\|<big>{{Sagittal\|t}}</big>
	\|~~<small>~~{{Sagittal\|\|(}}{{sagittal\|t}}~~</small>~~		\|{{Sagittal\|\|(}}{{sagittal\|t}}
	\|~~<small>~~{{Sagittal\|)\|(}}{{sagittal\|t}}~~</small>~~		\|{{Sagittal\|)\|(}}{{sagittal\|t}}
	\|~~<small>~~{{~~Sagittal\|)~~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|\\!}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|~~~\|(}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|(!(}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|/~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|~!)}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|\|)}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|!/}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|\|\}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|!)}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|(~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|\!}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|~~(\|(~~}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|~~!}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|//~~\|}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|~!(}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal\|/~~\|)}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|)!(}}{{sagittal\|#}}
	\|~~<small>~~{{~~Sagittal~~\|~~/\|\~~}}{{sagittal\|t}}~~</small>~~		\| rowspan="2" \|{{sagittal\|!(}}{{sagittal\|#}}
	\| rowspan="2" \|<big>{{Sagittal\|#}}</big>		\| rowspan="2" \|<big>{{Sagittal\|#}}</big>
	\|-		\|-
	!Evo		!Evo
	\| rowspan="2" \|<big>{{sagittal\|)/\|\}}</big>		\| rowspan="2" \|<big>{{sagittal\|)/\|\}}</big>
	\|~~<small>{{sagittal\|\!/}}{{sagittal\|#}}</small>~~		\| rowspan="2" \|<big>{{Sagittal\|(\|)}}</big>
	~~\|<small>{{sagittal\|\!)}}{{sagittal\|#}}</small>~~		\| rowspan="2" \|<big>{{sagittal\|(\|\}}</big>
	~~\|<small>{{sagittal\|\\!}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|(!(}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|(!}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|!/}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|!)}}{{sagittal\|#}}</small>~~
	\|<~~small~~>{{~~sagittal~~\|~~\!}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal\|~!~~(~~}}{{sagittal\|#}}</small>~~
	~~\|<small>{{sagittal~~\|)~~~!}}{{sagittal\|#~~}}</~~small~~>
	\|~~<small>{{sagittal\|)!(}}{{sagittal~~\|~~#}}~~<~~/small>~~
	~~\|<small~~>{{sagittal\|!(~~}}{{sagittal~~\|#}}</~~small~~>
	\|-		\|-
	!Revo		!Revo
	~~\|<big>{{Sagittal\|(\|)}}</big>~~
	~~\|<big>{{sagittal\|(\|\}}</big>~~
	\|<big>{{sagittal\|)\|\|(}}</big>		\|<big>{{sagittal\|)\|\|(}}</big>
	\|<big>{{sagittal\|~\|\|(}}</big>		\|<big>{{sagittal\|~\|\|(}}</big>

← Older revision		Revision as of 13:13, 18 April 2026
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	== Theory ==		== Theory ==
	270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency#Consistency to distance d\|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. It is [[The Riemann zeta function and tuning #Zeta EDO lists\|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a [[strict zeta edo]].		270edo is an extremely strong [[13-limit]] and system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency#Consistency to distance d\|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit ''and'' 13-limit up until [[342edo]] and [[96478edo]] respectively.

			It is [[The Riemann zeta function and tuning #Zeta EDO lists\|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a [[strict zeta edo]].

	In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo\| 1 -27 18 }}, the [[vulture comma]], {{monzo\| 24 -21 4 }}, and the [[vishnuzma]], {{monzo\| 23 6 -14 }}.		In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo\| 1 -27 18 }}, the [[vulture comma]], {{monzo\| 24 -21 4 }}, and the [[vishnuzma]], {{monzo\| 23 6 -14 }}.
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	In the [[7-limit]] it tempers out the [[2401/2400\|breedsma]] (2401/2400), the [[4375/4374\|ragisma]] (4375/4374), and by extension the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000) so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783).		In the [[7-limit]] it tempers out the [[2401/2400\|breedsma]] (2401/2400), the [[4375/4374\|ragisma]] (4375/4374), and by extension the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000) so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783).

	In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)<sup>4</sup> and (20/11)<sup>4</sup> are a hair off, 50.4%).		In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)<sup>4</sup> and (20/11)<sup>4</sup> are a hair off, 50.4%).

	Finally, in the [[13-limit]] it is ~~not quite as accurate~~ but still ~~very accurate~~. It tempers out [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]], making it an [[The Archipelago\|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments such as [[hemiennealimmal]], [[vulture]], [[eagle]], and [[avicenna (temperament)\|avicenna]].		Finally, in the [[13-limit]] it is slightly worse but still excellent. It tempers out [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]], making it an [[The Archipelago\|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments such as [[hemiennealimmal]], [[vulture]], [[eagle]], and [[avicenna (temperament)\|avicenna]].

	The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit, and [[island chords]] in the 15-odd-limit.		The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit, and [[island chords]] in the 15-odd-limit.

	Beyond the 13-limit, the approximated [[17/1\|harmonic 17]] is more than 1/3-edostep, but the [[19/1\|harmonic 19]] is ~~very~~ accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1\|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. It tempers out [[715/714]], [[936/935]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.		Beyond the 13-limit, the approximated [[17/1\|harmonic 17]] is more than 1/3-edostep, but the [[19/1\|harmonic 19]] is extremely accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1\|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. It tempers out [[715/714]], [[936/935]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.

	The harmonics [[29/1\|29]] and [[31/1\|31]] are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency ([[29/26]] and [[31/26]] are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 [[35-odd-limit]]. Notably, it tempers out [[784/783]], [[900/899]], and [[1024/1023]], while inflating [[841/840]] and [[961/960]].		The harmonics [[29/1\|29]] and [[31/1\|31]] are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency ([[29/26]] and [[31/26]] are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 [[35-odd-limit]]. Notably, it tempers out [[784/783]], [[900/899]], and [[1024/1023]], while inflating [[841/840]] and [[961/960]].

	On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error (assuming consistency) is only 2.{{overline\|2}}{{c}}, yet having a step size that ''can'' be [[just-noticeable difference\|discernible]].		On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance (beyond the 13-limit on which it is spot on), as the maximum error (assuming consistency) is only 2.{{overline\|2}}{{c}}, yet having a step size that ''can'' be [[just-noticeable difference\|discernible]].

	If, however, you want a edo for more rounded, consistent very high-limit use, the obvious alternative choice is [[311edo]], which is in many ways dual to 270edo as it emphasizes consistency and accuracy in very high-prime-limit and high-odd-limit situations at the expense of lower ones, and is a [[prime edo]] as opposed to a very composite one. While 270edo approximates the first 16 harmonics with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.		If, however, you want a edo for more rounded, consistent very high-limit use, the obvious alternative choice is [[311edo]], which is in many ways dual to 270edo as it emphasizes consistency and accuracy in very high-prime-limit and high-odd-limit situations at the expense of lower ones, and is a [[prime edo]] as opposed to a very composite one. While 270edo approximates the first 16 harmonics with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.