2460edo - Revision history

Overthink: /* Theory */ Those all involve 29

2026-03-25T02:58:21Z

Theory: Those all involve 29

← Older revision		Revision as of 02:58, 25 March 2026
Line 5:		Line 5:
	2460edo is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see [[#Approximation to JI]]). It is also a [[zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.		2460edo is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see [[#Approximation to JI]]). It is also a [[zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.

	In higher limits, it is ''almost'' consistent in the [[29-odd-limit]] missing [[29/22]], [[29/17]], [[34/29]], [[44/29]]. It is also ~~nearly~~ consistent in the no-29 [[39-odd-limit~~]], failing [[31/29]], [[37/29]], [[58/37]], [[58/31~~]].		In higher limits, it is ''almost'' consistent in the [[29-odd-limit]] missing [[29/22]], [[29/17]], [[34/29]], [[44/29]]. It is also fully consistent in the no-29 [[39-odd-limit]].

	As a micro- (or nano-) temperament, it tempers [[Kirnberger's atom]] in the [[5-limit]], [[250047/250000]] (landscape comma) in the [[7-limit]], [[9801/9800]] [kalisma] in the [[11-limit]], [[10648/10647]] [harmonisma] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].		As a micro- (or nano-) temperament, it tempers [[Kirnberger's atom]] in the [[5-limit]], [[250047/250000]] (landscape comma) in the [[7-limit]], [[9801/9800]] [kalisma] in the [[11-limit]], [[10648/10647]] [harmonisma] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].

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"

2026-03-13T13:30:03Z

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
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	\| [[Minutes]]		\| [[Minutes]]
	\|}		\|}
	<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

	[[Category:Mina]]		[[Category:Mina]]

Overthink: /* Theory */ not "consistent"

2026-02-08T00:56:59Z

Theory: not "consistent"

← Older revision		Revision as of 00:56, 8 February 2026
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	2460edo is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see [[#Approximation to JI]]). It is also a [[zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.		2460edo is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see [[#Approximation to JI]]). It is also a [[zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.

	In higher limits, it is ''almost'' consistent in the [[29-odd-limit]] missing [[29/22]], [[29/17]], [[34/29]], [[44/29]]. It is also consistent in the no-29 [[39-odd-limit]], failing [[31/29]], [[37/29]], [[58/37]], [[58/31]].		In higher limits, it is ''almost'' consistent in the [[29-odd-limit]] missing [[29/22]], [[29/17]], [[34/29]], [[44/29]]. It is also nearly consistent in the no-29 [[39-odd-limit]], failing [[31/29]], [[37/29]], [[58/37]], [[58/31]].

	As a micro- (or nano-) temperament, it tempers [[Kirnberger's atom]] in the [[5-limit]], [[250047/250000]] (landscape comma) in the [[7-limit]], [[9801/9800]] [kalisma] in the [[11-limit]], [[10648/10647]] [harmonisma] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].		As a micro- (or nano-) temperament, it tempers [[Kirnberger's atom]] in the [[5-limit]], [[250047/250000]] (landscape comma) in the [[7-limit]], [[9801/9800]] [kalisma] in the [[11-limit]], [[10648/10647]] [harmonisma] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].

Eufalesio at 20:27, 20 January 2026

2026-01-20T20:27:48Z

← Older revision		Revision as of 20:27, 20 January 2026
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	== Notation ==		== Notation ==
	2460edo is special in the [[Sagittal notation]], as it has been the model for the Olympian set, which offers "extreme" precision. The diacritics are independent of the sagittals. Scroll the table to see accidentals for use in Revo flavor ~~(116 or 138~~ onwards).		2460edo is special in the [[Sagittal notation]], as it has been the model for the Olympian set, which offers "extreme" precision. The diacritics are independent of the sagittals. Scroll the table to see accidentals for use in Revo flavor 145\2460 onwards).
	<div style="overflow-x:auto;">		<div style="overflow-x:auto;">
	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"

Eufalesio: Added some things

2026-01-04T17:04:28Z

Added some things

← Older revision		Revision as of 17:04, 4 January 2026
Line 5:		Line 5:
	2460edo is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see [[#Approximation to JI]]). It is also a [[zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.		2460edo is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see [[#Approximation to JI]]). It is also a [[zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.

	As a micro- (or nano-) temperament, it ~~is a~~ [[~~landscape~~]] ~~system~~ in the [[7-limit]], ~~[[tempering out]]~~ [[250047/250000]]~~, and~~ in the [[11-limit]] ~~it tempers out~~ [[9801/9800]]~~. Beyond that~~, ~~it tempers out~~ [[10648/10647]] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].		In higher limits, it is ''almost'' consistent in the [[29-odd-limit]] missing [[29/22]], [[29/17]], [[34/29]], [[44/29]]. It is also consistent in the no-29 [[39-odd-limit]], failing [[31/29]], [[37/29]], [[58/37]], [[58/31]].

			As a micro- (or nano-) temperament, it tempers [[Kirnberger's atom]] in the [[5-limit]], [[250047/250000]] (landscape comma) in the [[7-limit]], [[9801/9800]] [kalisma] in the [[11-limit]], [[10648/10647]] [harmonisma] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].

	=== Prime harmonics ===		=== Prime harmonics ===

Eufalesio: There was a wrong sagittal on the notation panel

2025-11-08T23:15:25Z

There was a wrong sagittal on the notation panel

← Older revision		Revision as of 23:15, 8 November 2025
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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>
Line 131:		Line 131:
	\|<big>{{sagittal\|/\|\|\}}</big>		\|<big>{{sagittal\|/\|\|\}}</big>
	\|}		\|}
	{\| class="wikitable ~~mw-collapsible mw-collapsed"~~ data-darkreader-inline-color=""		</div>
			{\| class="wikitable data-darkreader-inline-color="
	\|+Olympian diacritics		\|+Olympian diacritics
	!'''Steps'''		!'''Steps'''
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	\|<big>{{sagittal\|'}}{{sagittal\|``}}</big>		\|<big>{{sagittal\|'}}{{sagittal\|``}}</big>
	\|}		\|}
	~~</div>~~


	== Approximation to JI ==		== Approximation to JI ==

Eufalesio at 12:27, 7 November 2025

2025-11-07T12:27:57Z

← Older revision		Revision as of 12:27, 7 November 2025
Line 21:		Line 21:
	2460edo is special in the [[Sagittal notation]], as it has been the model for the Olympian set, which offers "extreme" precision. The diacritics are independent of the sagittals. Scroll the table to see accidentals for use in Revo flavor (116 or 138 onwards).		2460edo is special in the [[Sagittal notation]], as it has been the model for the Olympian set, which offers "extreme" precision. The diacritics are independent of the sagittals. Scroll the table to see accidentals for use in Revo flavor (116 or 138 onwards).
	<div style="overflow-x:auto;">		<div style="overflow-x:auto;">
	{\| class="wikitable ~~mw-collapsible mw-collapsed~~" style="text-align:center"		{\| class="wikitable" style="text-align:center"
	\|-		\|-
	!'''Steps'''		!'''Steps'''

Eufalesio at 12:27, 7 November 2025

2025-11-07T12:27:20Z

Show changes

Eufalesio: Added Sagittal notation

2025-11-05T12:49:38Z

Added Sagittal notation

← Older revision		Revision as of 12:49, 5 November 2025
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	2460edo is also notable for being the smallest edo that is a multiple of 12 to be [[purely consistent]] in the 15-odd-limit (i.e. it is the smallest edo that is a multiple of 12 which maintains [[relative interval error\|relative error]]s of less than 25% on all of the first 16 harmonics of the harmonic series). [[72edo]] comes close, but its approximations to [[13/8]] and [[15/8]] are somewhat inaccurate.		2460edo is also notable for being the smallest edo that is a multiple of 12 to be [[purely consistent]] in the 15-odd-limit (i.e. it is the smallest edo that is a multiple of 12 which maintains [[relative interval error\|relative error]]s of less than 25% on all of the first 16 harmonics of the harmonic series). [[72edo]] comes close, but its approximations to [[13/8]] and [[15/8]] are somewhat inaccurate.

			== Notation ==
			2460edo is special in the [[Sagittal notation]], as it has been the model for the Olympian set, which offers "extreme" precision. The diacritics are independent of the sagittals.
			{\| class="wikitable"
			\|+Sagittal notation - Evo compatible sagittals
			!Steps
			!0
			!7
			!8
			!18
			!20
			!25
			!30
			!34
			!36
			!41
			!44
			!51
			!56
			!63
			!65
			!68
			!73
			!78
			!80
			!83
			!88
			!95
			!100
			!102
			!109
			!112
			!116
			\|-
			\|Sym.
			\|<big>{{sagittal\|h}}</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>
			\|<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>
			\|}
			{\| class="wikitable mw-collapsible mw-collapsed"
			\|+Sagittal notation - Apotome complements
			!'''Steps'''
			\|'''170'''
			\|'''168'''
			\|'''165'''
			\|'''160'''
			!'''155'''
			\|'''153'''
			\|'''150'''
			\|'''145'''
			\|'''138'''
			\|'''133'''
			\|'''131'''
			\|'''124'''
			\|'''121'''
			\|-
			\|Symbol
			\|<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>
			\|-
			!'''Steps'''
			\|'''233'''
			\|'''226'''
			\|'''225'''
			\|'''215'''
			\|'''213'''
			\|'''208'''
			\|'''203'''
			\|'''199'''
			\|'''197'''
			\|'''192'''
			\|'''189'''
			\|'''184'''
			\|'''177'''
			\|-
			\|Symbol
			\|<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>
			\|}
			{\| class="wikitable mw-collapsible mw-collapsed"
			\|+Sagittal notation - Diacritics
			!'''Steps'''
			\|1
			\|2
			\|3
			\|4
			!5
			\|6
			\|-
			\|Symbol
			\|<big>{{sagittal\|`}}</big>
			\|<big>{{sagittal\|``}}</big>
			\|<big>{{sagittal\|'}}{{sagittal\|,}}</big>
			\|<big>{{sagittal\|'}}</big>
			\|<big>{{sagittal\|'}}{{sagittal\|`}}</big>
			\|<big>{{sagittal\|'}}{{sagittal\|``}}</big>
			\|}

	== Approximation to JI ==		== Approximation to JI ==

FloraC: Expand the harmonics table a little. Misc. cleanup

2025-10-02T16:14:56Z

Expand the harmonics table a little. Misc. cleanup

← Older revision		Revision as of 16:14, 2 October 2025
Line 8:		Line 8:

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|2460\|columns=9}}		{{Harmonics in equal\|2460\|columns=11}}
	{{Harmonics in equal\|2460\|columns=9\|start=10\|collapsed=true\|title=Approximation of prime harmonics in 2460edo (continued)}}		{{Harmonics in equal\|2460\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 2460edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
Line 127:		Line 127:
	\| 1021\2460<br>(1\2460)		\| 1021\2460<br>(1\2460)
	\| 498.049<br>(0.488)		\| 498.049<br>(0.488)
	\| 4/3<br />({{monzo\| 215 -121 -10 }})		\| 4/3<br>({{monzo\| 215 -121 -10 }})
	\| [[Niobium]]		\| [[Niobium]]
	\|-		\|-
Line 142:		Line 142:
	\| [[Minutes]]		\| [[Minutes]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[normal ~~lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|minimal form]] in parentheses if distinct

	[[Category:Mina]]		[[Category:Mina]]