166edo - Revision history

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"

2026-03-13T13:30:07Z

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 98:		Line 98:
	\| [[Wizard]] / gizzard		\| [[Wizard]] / gizzard
	\|}		\|}
	<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

	== Scales ==		== Scales ==

FloraC: /* Regular temperament properties */ update

2026-02-23T11:27:25Z

Regular temperament properties: update

← Older revision		Revision as of 11:27, 23 February 2026
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	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
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	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| -263 166 }}		\| {{Monzo\| -263 166 }}
	\| {{~~mapping~~\| 166 263 }}		\| {{Mapping\| 166 263 }}
	\| +0.237		\| +0.237
	\| 0.237		\| 0.237
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	\| 2.3.5		\| 2.3.5
	\| 1600000/1594323, {{monzo\| -31 2 12 }}		\| 1600000/1594323, {{monzo\| -31 2 12 }}
	\| {{~~mapping~~\| 166 263 385 }}		\| {{Mapping\| 166 263 385 }}
	\| +0.615		\| +0.615
	\| 0.568		\| 0.568
Line 44:		Line 44:
	\| 2.3.5.7		\| 2.3.5.7
	\| 225/224, 118098/117649, 1250000/1240029		\| 225/224, 118098/117649, 1250000/1240029
	\| {{~~mapping~~\| 166 263 385 466 }}		\| {{Mapping\| 166 263 385 466 }}
	\| +0.474		\| +0.474
	\| 0.549		\| 0.549
Line 51:		Line 51:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 225/224, 385/384, 4000/3993, 322102/321489		\| 225/224, 385/384, 4000/3993, 322102/321489
	\| {{~~mapping~~\| 166 263 385 466 574 }}		\| {{Mapping\| 166 263 385 466 574 }}
	\| +0.490		\| +0.490
	\| 0.492		\| 0.492
Line 58:		Line 58:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 225/224, 325/324, 385/384, 1573/1568, 2200/2197		\| 225/224, 325/324, 385/384, 1573/1568, 2200/2197
	\| {{~~mapping~~\| 166 263 385 466 574 614 }}		\| {{Mapping\| 166 263 385 466 574 614 }}
	\| +0.498		\| +0.498
	\| 0.449		\| 0.449
Line 68:		Line 68:
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 90:		Line 90:
	\| 585.54		\| 585.54
	\| 7/5		\| 7/5
	\| [[Merman]] ~~(7-limit)~~		\| [[Merman]] / mermaid
	\|-		\|-
	\| 2		\| 2
Line 98:		Line 98:
	\| [[Wizard]] / gizzard		\| [[Wizard]] / gizzard
	\|}		\|}
	<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

	== Scales ==		== Scales ==

FloraC: /* Theory */ note its flat tuning tendency

2025-03-24T14:16:49Z

Theory: note its flat tuning tendency

← Older revision		Revision as of 14:16, 24 March 2025
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	== Theory ==		== Theory ==
	166edo is [[consistent]] through the [[13-odd-limit]], ~~yet its principle~~ interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out\|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap\|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].		166edo is [[consistent]] through the [[13-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 to 13 all tuned flat. Its principal interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out\|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap\|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].

	166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.		166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.

FloraC: /* Theory */ +octave stretch

2025-03-24T14:14:44Z

Theory: +octave stretch

← Older revision		Revision as of 14:14, 24 March 2025
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|166\|intervals=prime}}		{{Harmonics in equal\|166\|intervals=prime}}

			=== Octave stretch ===
			166edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly [[stretched and compressed tuning\|stretching the octave]], using tunings such as [[263edt]] or [[429ed6]].

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 166 factors into 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.		Since 166 factors into primes as 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.

	== Regular temperament properties ==		== Regular temperament properties ==

Francium: changed EDO intro to ED intro

2025-02-19T18:04:42Z

changed EDO intro to ED intro

← Older revision		Revision as of 18:04, 19 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T15:27:20Z

Theory: cleanup

← Older revision		Revision as of 15:27, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap\|72 &~~amp;~~ 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].		166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out\|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap\|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].

	166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.		166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 166 factors into ~~{{factorisation\|166}}~~, 166edo contains [[2edo]] and [[83edo]] as subsets.		Since 166 factors into 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.

	== Regular temperament properties ==		== Regular temperament properties ==

ArrowHead294 at 19:40, 15 January 2025

2025-01-15T19:40:26Z

← Older revision		Revision as of 19:40, 15 January 2025
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	{{Infobox ET}}		{{Infobox ET}}
	The '''166 equal divisions of the octave''' ('''166edo'''), or the '''166(-tone) equal temperament''' ('''166tet''', '''166et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 166 equal steps of about 7.23 [[cent]]s each, a size close to [[243/242]], the rastma.		{{EDO intro}}

	== Theory ==		== Theory ==
	166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], ~~the equal temperament~~ [[~~tempering out\|~~tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the 72 & 94 temperament, for which 166 is an excellent tuning through the [[19-limit]].		166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap\|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].

	166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.		166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 166 ~~= 2 × 83~~, 166edo contains [[2edo]] and [[83edo]] as subsets.		Since 166 factors into {{factorisation\|166}}, 166edo contains [[2edo]] and [[83edo]] as subsets.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| [[Wizard]] / gizzard		\| [[Wizard]] / gizzard
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Scales ==		== Scales ==

ArrowHead294: Partial undo

2024-11-16T12:45:55Z

Partial undo

← Older revision		Revision as of 12:45, 16 November 2024
Line 14:		Line 14:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.449		\| 0.449
	\| 6.21		\| 6.21
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
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	\| 17/15		\| 17/15
	\| [[Wizard]] / gizzard		\| [[Wizard]] / gizzard
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Scales ==		== Scales ==

ArrowHead294 at 03:59, 16 November 2024

2024-11-16T03:59:21Z

← Older revision		Revision as of 03:59, 16 November 2024
Line 14:		Line 14:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.449		\| 0.449
	\| 6.21		\| 6.21
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\| 17/15		\| 17/15
	\| [[Wizard]] / gizzard		\| [[Wizard]] / gizzard
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	== Scales ==		== Scales ==

FloraC: Cleanup

2024-05-05T08:50:45Z

Cleanup

← Older revision		Revision as of 08:50, 5 May 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], the equal temperament tempers out [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the 72 & 94 temperament, for which 166 is an excellent tuning through the [[19-limit]].		166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], the equal temperament [[tempering out\|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the 72 & 94 temperament, for which 166 is an excellent tuning through the [[19-limit]].

	166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.		166edo (as 83edo) contains a very good approximation of the [[7/4\|harmonic 7th]], of which it is only flat by 0.15121 cent.
Line 26:		Line 26:
	\| 2.3		\| 2.3
	\| {{monzo\| -263 166 }}		\| {{monzo\| -263 166 }}
	\| [{{~~val~~\| 166 263 }}]		\| {{mapping\| 166 263 }}
	\| +0.237		\| +0.237
	\| 0.237		\| 0.237
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	\| 2.3.5		\| 2.3.5
	\| 1600000/1594323, {{monzo\| -31 2 12 }}		\| 1600000/1594323, {{monzo\| -31 2 12 }}
	\| [{{~~val~~\| 166 263 385 }}]		\| {{mapping\| 166 263 385 }}
	\| +0.615		\| +0.615
	\| 0.568		\| 0.568
Line 40:		Line 40:
	\| 2.3.5.7		\| 2.3.5.7
	\| 225/224, 118098/117649, 1250000/1240029		\| 225/224, 118098/117649, 1250000/1240029
	\| [{{~~val~~\| 166 263 385 466 }}]		\| {{mapping\| 166 263 385 466 }}
	\| +0.474		\| +0.474
	\| 0.549		\| 0.549
Line 47:		Line 47:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 225/224, 385/384, 4000/3993, 322102/321489		\| 225/224, 385/384, 4000/3993, 322102/321489
	\| [{{~~val~~\| 166 263 385 466 574 }}]		\| {{mapping\| 166 263 385 466 574 }}
	\| +0.490		\| +0.490
	\| 0.492		\| 0.492
Line 54:		Line 54:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 225/224, 325/324, 385/384, 1573/1568, 2200/2197		\| 225/224, 325/324, 385/384, 1573/1568, 2200/2197
	\| [{{~~val~~\| 166 263 385 466 574 614 }}]		\| {{mapping\| 166 263 385 466 574 614 }}
	\| +0.498		\| +0.498
	\| 0.449		\| 0.449
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	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 79:		Line 79:
	\| 339.76		\| 339.76
	\| 243/200		\| 243/200
	\| [[~~Amity]] / [[houborizic~~]]		\| [[Houborizic]]
	\|-		\|-
	\| 1		\| 1
Line 85:		Line 85:
	\| 585.54		\| 585.54
	\| 7/5		\| 7/5
	\| [[Merman]]		\| [[Merman]] (7-limit)
	\|-		\|-
	\| 2		\| 2
Line 93:		Line 93:
	\| [[Wizard]] / gizzard		\| [[Wizard]] / gizzard
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Scales ==		== Scales ==
Line 106:		Line 107:

	[[Category:Wizard]]		[[Category:Wizard]]
	~~[[Category:Gizzard]]~~
	[[Category:Houborizic]]		[[Category:Houborizic]]
	[[Category:Marvel]]		[[Category:Marvel]]