217edo - Revision history

FloraC: /* Regular temperament properties */ generator interpretation

2026-04-23T06:49:38Z

Regular temperament properties: generator interpretation

← Older revision		Revision as of 06:49, 23 April 2026
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	\| 10\217		\| 10\217
	\| 55.30		\| 55.30
	\| ~~16875~~/~~16384~~		\| 33/32
	\| [[Escapade]]		\| [[Escapade]]
	\|-		\|-
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	\| 86\217		\| 86\217
	\| 475.58		\| 475.58
	\| ~~320~~/~~243~~		\| 25/19
	\| [[Vulture]]		\| [[Vulture]]
	\|-		\|-
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	\| [[Birds]]		\| [[Birds]]
	\|}		\|}
	<nowiki>*~~</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

	== Notation ==		== Notation ==

Eufalesio: Standardize format Sagittal

2026-03-07T12:02:02Z

Standardize format Sagittal

← Older revision		Revision as of 12:02, 7 March 2026
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	\|+Sagittal notation		\|+Sagittal notation
	! colspan="2" \| Steps		! colspan="2" \| Steps
	\| 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
	\|-		\|-
	! rowspan="2" \| Symbol		! rowspan="2" \| Symbol

Eufalesio: formatting

2026-03-07T12:00:08Z

formatting

← Older revision		Revision as of 12:00, 7 March 2026
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	\| rowspan="2" \| <big>{{Sagittal\| /\|) }}</big>		\| rowspan="2" \| <big>{{Sagittal\| /\|) }}</big>
	\| rowspan="2" \| <big>{{Sagittal\| /\|\ }}</big>		\| rowspan="2" \| <big>{{Sagittal\| /\|\ }}</big>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \!/ }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \!/ }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \!) }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \!) }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \\! }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \\! }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| (!( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| (!( }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| (! }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| (! }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| !) }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| !) }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \! }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \! }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| ~!( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| ~!( }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| )!( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| )!( }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| !( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| !( }}</small>
	\| <~~big~~>{{Sagittal\|#}}</~~big~~>		\| <small>{{Sagittal\|#}}</small>
	\|-		\|-
	! Revo		! Revo

Eufalesio: little thing

2026-03-07T11:57:01Z

little thing

← Older revision		Revision as of 11:57, 7 March 2026
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	== Notation ==		== Notation ==
	=== Sagittal ===		=== Sagittal ===
	217edo can be written in Sagittal using almost the entire Athenian extension ~~(except for {{sagittal\| \|\ }} {{sagittal\| !/ }} {{sagittal\| /\|\| }} {{sagittal\| \!! }} since it tempers out [[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. It is identical to [[224edo]]'s Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.<ref name=":1">[[George Secor\|George D. Secor]] and [[David Keenan\|David C. Keenan]], [https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''], p. 11.</ref>		217edo 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 [[224edo]]'s Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.<ref name=":1">[[George Secor\|George D. Secor]] and [[David Keenan\|David C. Keenan]], [https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''], p. 11.</ref>

	It shares the same exact symbol system as the Athenian notation for just intonation or ''Medium-precision JI notation.''<ref name=":1"/>		It shares the same exact symbol system as the Athenian notation for just intonation or ''Medium-precision JI notation.''<ref name=":1"/>
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	\| <big>{{Sagittal\| /\|\|\ }}</big>		\| <big>{{Sagittal\| /\|\|\ }}</big>
	\|}		\|}
			Because it uses the entire Athenian system (except for {{sagittal\| \|\ }} {{sagittal\| !/ }} {{sagittal\| /\|\| }} {{sagittal\| \!! }} since it tempers out [[1240029/1239040]]), it allows no accidental enharmonic respellings.

	=== Ups-and-downs notation ===		=== Ups-and-downs notation ===
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	Since {{nowrap\| 217 {{=}} 31 × 7 }}, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.		Since {{nowrap\| 217 {{=}} 31 × 7 }}, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.

	This could be useful when [[31edo]] is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use [[159edo]] to provide subtle corrections of [[53edo]] pitches. It also can be written with SZ half-sharps and up/down. The main drawback of this notation is that ~~the best fifth~~ is '''^P5''', so that ~4:5:6 is from C written as C-E-^G, compared to C-^<E-G to the ~~usual~~ chain-of-fifths ups-and-downs notation.		This could be useful when [[31edo]] is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use [[159edo]] to provide subtle corrections of [[53edo]] pitches. It also can be written with SZ half-sharps and up/down. The main drawback of this notation is that 3/2 is no longer P5 but '''^P5''' , so that ~4:5:6 is from C written as C-E-^G, compared to C-^<E-G to the patent val chain-of-fifths ups-and-downs notation.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
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	=== 7edo-based whitewood notation ===		=== 7edo-based whitewood notation ===
	Since {{nowrap\| 217 {{=}} 7 × 31}}, one ''could'' use the inherited [[Whitewood family\|whitewood]] fifth, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the ~~perfect~~ nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G.		Since {{nowrap\| 217 {{=}} 7 × 31}}, one ''could'' use the inherited [[Whitewood family\|whitewood]] fifth 124\217, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the 7edo nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G.
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|-		\|-

FloraC: /* Regular temperament properties */ wording

2026-01-21T18:38:41Z

Regular temperament properties: wording

← Older revision		Revision as of 18:38, 21 January 2026
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	\|}		\|}
	* 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat [[72edo\|72]] in the 19-limit and [[193edo\|193]] in the 23-limit. The next equal temperament that does better in either subgroup is [[243edo\|243e]] for absolute error and [[270edo\|270]] for relative error.		* 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat [[72edo\|72]] in the 19-limit and [[193edo\|193]] in the 23-limit. The next equal temperament that does better in either subgroup is [[243edo\|243e]] for absolute error and [[270edo\|270]] for relative error.
	* 23-limit is not the subgroup it does ~~the~~ 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 also ~~prominent~~ in the 17-limit, with a lower absolute error than any previous equal temperaments, beating [[183edo\|183]] and superseded by [[224edo\|224]].		* It is also notable in the 17-limit, with a lower absolute error than any previous equal temperaments, beating [[183edo\|183]] and superseded by [[224edo\|224]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

Eufalesio: Connection to 224edo

2026-01-21T11:41:06Z

Connection to 224edo

← Older revision		Revision as of 11:41, 21 January 2026
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	== Theory ==		== Theory ==
	217edo is a strong [[19-limit]] system, the smallest [[consistency\|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1\|5th]] and [[7/1\|7th]] [[harmonic]]s with [[31edo]] ({{nowrap\| 217 {{=}} 7 × 31 }}), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds\|birds temperament]]). However, compared to [[31edo]], its [[patent val]] differ on the mappings for [[3/1\|3]], [[11/1\|11]], [[13/1\|13]], [[17/1\|17]] and [[19/1\|19]], excelling as a [[2.3.5.13 subgroup\|2.3.5.13-subgroup]] temperament. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. If one desires even higher consistency and precision, [[311edo]] offers a much better palette.		217edo is a strong [[19-limit]] system, the smallest [[consistency\|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1\|5th]] and [[7/1\|7th]] [[harmonic]]s with [[31edo]] ({{nowrap\| 217 {{=}} 7 × 31 }}), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds\|birds temperament]]). However, compared to [[31edo]], its [[patent val]] differ on the mappings for [[3/1\|3]], [[11/1\|11]], [[13/1\|13]], [[17/1\|17]] and [[19/1\|19]], excelling as a [[2.3.5.13 subgroup\|2.3.5.13-subgroup]] temperament. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. [[224edo]], only a bit bigger, offers a much more accurate [[13-limit]], at the cost of worse higher limits. If one desires even higher consistency and precision, [[311edo]] offers a much better palette.

	The equal temperament [[tempering out\|tempers out]] the [[parakleisma]] ({{monzo\| 8 14 -13 }}) and the [[escapade comma]] ({{monzo\| 32 -7 -9 }}) in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]], and [[823543/819200]] in the 7-limit; [[441/440]], [[4000/3993]], [[5632/5625]], and [[16384/16335]] in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], and [[4096/4095]] in the 13-limit; [[595/594]], [[833/832]], [[936/935]], [[1156/1155]], [[1225/1224]], [[1701/1700]] in the 17-limit; [[343/342]], [[476/475]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]] and [[1540/1539]] in the 19-limit. It allows [[minor minthmic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit. It provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]].		The equal temperament [[tempering out\|tempers out]] the [[parakleisma]] ({{monzo\| 8 14 -13 }}) and the [[escapade comma]] ({{monzo\| 32 -7 -9 }}) in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]], and [[823543/819200]] in the 7-limit; [[441/440]], [[4000/3993]], [[5632/5625]], and [[16384/16335]] in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], and [[4096/4095]] in the 13-limit; [[595/594]], [[833/832]], [[936/935]], [[1156/1155]], [[1225/1224]], [[1701/1700]] in the 17-limit; [[343/342]], [[476/475]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]] and [[1540/1539]] in the 19-limit. It allows [[minor minthmic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit. It provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]].

Eufalesio: Added more intervals to the interval table

2025-12-10T10:06:57Z

Added more intervals to the interval table

← Older revision		Revision as of 10:06, 10 December 2025
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	== Intervals ==		== Intervals ==
	Here below is an algorithmically generated table of no-37 39-odd-limit intervals of 217edo using [[User:Godtone #My Python 3 code\|Godtone's code]], with some manually added intervals outside that limit ~~for completeness~~.		Here below is an algorithmically generated table of no-37 39-odd-limit intervals of 217edo using [[User:Godtone #My Python 3 code\|Godtone's code]], with some manually added useful intervals outside that limit. Intervals in italics are inconsistently mapped.

	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
Line 31:		Line 31:
	\| 5.53		\| 5.53
	\|		\|
	\|		\|[[352/351]], [[5120/5103]]'', [[32805/32768]]''
	\|-		\|-
	\| 2		\| 2
	\| 11.06		\| 11.06
	\|		\|
	\|		\|[[144/143]], [[169/168]], ''[[225/224]]''
	\|-		\|-
	\| 3		\| 3
	\| 16.59		\| 16.59
	\|		\|
	\|		\|[[4131/4096]]
	\|-		\|-
	\| 4		\| 4
Line 51:		Line 51:
	\| 27.65		\| 27.65
	\|		\|
	\| [[64/63]]		\| [[64/63]], ''[[531441/524288]]''
	\|-		\|-
	\| 6		\| 6
	\| 33.18		\| 33.18
	\|		\|
	\|		\|[[49/48]]
	\|-		\|-
	\| 7		\| 7
	\| 38.71		\| 38.71
	\|		\|
	\|		\|[[128/125]]
	\|-		\|-
	\| 8		\| 8
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	\| 49.77		\| 49.77
	\|		\|
	\| [[36/35]], [[35/34]], [[34/33]]		\| [[36/35]], [[35/34]], [[34/33]], [[1053/1024]]
	\|-		\|-
	\| 10		\| 10

Overthink: /* Intervals */ 2187/2048 isn't inconsistent

2025-11-18T00:08:37Z

Intervals: 2187/2048 isn't inconsistent

← Older revision		Revision as of 00:08, 18 November 2025
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	\| 116.13		\| 116.13
	\|A1		\|A1
	\| [[31/29]], ''[[2187/2048]]''		\| [[31/29]], [[2187/2048]]
	\|-		\|-
	\| 22		\| 22

Overthink: /* 7edo-based whitewood notation */ remove bolding

2025-11-04T06:39:46Z

7edo-based whitewood notation: remove bolding

← Older revision		Revision as of 06:39, 4 November 2025
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	\|}		\|}

	=== ~~'''~~7edo-based whitewood notation~~'''~~ ===		=== 7edo-based whitewood notation ===
	Since {{nowrap\| 217 {{=}} 7 × 31}}, one ~~'''~~''could~~'''~~'' use the inherited [[Whitewood family\|whitewood]] fifth, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the perfect nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G.		Since {{nowrap\| 217 {{=}} 7 × 31}}, one ''could'' use the inherited [[Whitewood family\|whitewood]] fifth, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the perfect nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G.
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|-		\|-

Eufalesio: added ups-and-downs template and another possible notation for completeness

2025-10-15T14:14:38Z

added ups-and-downs template and another possible notation for completeness

Show changes

← Older revision		Revision as of 12:00, 7 March 2026
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	\| rowspan="2" \| <big>{{Sagittal\| /\|) }}</big>		\| rowspan="2" \| <big>{{Sagittal\| /\|) }}</big>
	\| rowspan="2" \| <big>{{Sagittal\| /\|\ }}</big>		\| rowspan="2" \| <big>{{Sagittal\| /\|\ }}</big>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \!/ }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \!/ }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \!) }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \!) }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \\! }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \\! }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| (!( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| (!( }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| (! }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| (! }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| !) }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| !) }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| \! }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| \! }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| ~!( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| ~!( }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| )!( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| )!( }}</small>
	\| <~~big~~>{{Sagittal\|#}}{{sagittal\| !( }}</~~big~~>		\| <small>{{Sagittal\|#}}{{sagittal\| !( }}</small>
	\| <~~big~~>{{Sagittal\|#}}</~~big~~>		\| <small>{{Sagittal\|#}}</small>
	\|-		\|-
	! Revo		! Revo