118edo - 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"

2026-03-13T13:33:11Z

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 13:33, 13 March 2026
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	\| [[Semiparakleismic]]		\| [[Semiparakleismic]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|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

	== Instruments ==		== Instruments ==

Lucius Chiaraviglio: /* Intervals */ Repair disordered columns at 53 (iodine) and 54 (xenon)

2025-11-29T11:00:37Z

Intervals: Repair disordered columns at 53 (iodine) and 54 (xenon)

← Older revision		Revision as of 11:00, 29 November 2025
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	\|		\|
	\| [[15/11]]		\| [[15/11]]
			\|
			\| iodine
	\| {{UDnote\|step=53}}		\| {{UDnote\|step=53}}
	\|
	~~\| iodine~~
	\| OG, uGb		\| OG, uGb
	\|-		\|-
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	\|		\|
	\| [[48/35]], [[11/8]]		\| [[48/35]], [[11/8]]
			\|
			\| xenon
	\| {{UDnote\|step=54}}		\| {{UDnote\|step=54}}
	\|
	~~\| xenon~~
	\| UG, oAb		\| UG, oAb
	\|-		\|-

Lucius Chiaraviglio: /* Intervals */ Add link to 16/13

2025-06-19T13:43:23Z

Intervals: Add link to 16/13

← Older revision		Revision as of 13:43, 19 June 2025
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	\| 355.93		\| 355.93
	\|		\|
	\| [[27/22]], 16/13 I**		\| [[27/22]], [[16/13]] I**
	\| Minor tridecimal neurtral third, "major-neutral" third		\| Minor tridecimal neurtral third, "major-neutral" third
	\| bromine		\| bromine

Lériendil: Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

2025-04-17T02:30:52Z

Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

← Older revision		Revision as of 02:30, 17 April 2025
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	\| 498.31		\| 498.31
	\| 4/3		\| 4/3
	\| [[Helmholtz temperament\|Helmholtz]] / [[pontiac]] / helenoid / pontic		\| [[Helmholtz (temperament)\|Helmholtz]] / [[pontiac]] / helenoid / pontic
	\|-		\|-
	\| 1		\| 1

Nick Vuci: Edited Helmholtz redirect

2025-04-16T15:33:40Z

Edited Helmholtz redirect

← Older revision		Revision as of 15:33, 16 April 2025
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	\| 498.31		\| 498.31
	\| 4/3		\| 4/3
	\| [[Helmholtz]] / [[pontiac]] / helenoid / pontic		\| [[Helmholtz temperament\|Helmholtz]] / [[pontiac]] / helenoid / pontic
	\|-		\|-
	\| 1		\| 1

Lucius Chiaraviglio: /* Theory */ Remove dangling word left over from 1 of the previous 2 edits

2025-04-15T17:39:04Z

Theory: Remove dangling word left over from 1 of the previous 2 edits

← Older revision		Revision as of 17:39, 15 April 2025
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	== Theory ==		== Theory ==
	118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament\|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo\| -15 8 1 }} and the [[parakleisma]], {{monzo\| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[hemithirds comma]], {{monzo\| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }}. It		118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament\|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo\| -15 8 1 }} and the [[parakleisma]], {{monzo\| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[hemithirds comma]], {{monzo\| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }}.

	118edo is the 17th [[The Riemann zeta function and tuning\|zeta peak edo]], and it has decent approximations to harmonics [[7/1\|7]], [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family\|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.		118edo is the 17th [[The Riemann zeta function and tuning\|zeta peak edo]], and it has decent approximations to harmonics [[7/1\|7]], [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family\|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

FloraC: /* Theory */ rework on the opening paragraph. -shruti (5-limit accuracy isn't a focus of it)

2025-04-15T11:54:24Z

Theory: rework on the opening paragraph. -shruti (5-limit accuracy isn't a focus of it)

← Older revision		Revision as of 11:54, 15 April 2025
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	== Theory ==		== Theory ==
	118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo\| -15 8 1 }} and the [[parakleisma]], {{monzo\| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[hemithirds comma]], {{monzo\| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }}. It ~~is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.~~		118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament\|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo\| -15 8 1 }} and the [[parakleisma]], {{monzo\| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[hemithirds comma]], {{monzo\| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }}. It

	118edo is the 17th [[The Riemann zeta function and tuning\|zeta peak edo]], and it has decent approximations to harmonics [[7/1\|7]], [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family\|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.		118edo is the 17th [[The Riemann zeta function and tuning\|zeta peak edo]], and it has decent approximations to harmonics [[7/1\|7]], [[11/1\|11]], [[17/1\|17]], and [[19/1\|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family\|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

FloraC: /* Theory */ +octave stretch; note prime factorization

2025-04-15T11:51:10Z

Theory: +octave stretch; note prime factorization

← Older revision		Revision as of 11:51, 15 April 2025
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|118}}		{{Harmonics in equal\|118}}

			=== Octave stretch ===
			118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning\|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part.

	=== Subsets and supersets ===		=== Subsets and supersets ===
	118edo contains [[2edo]] and [[59edo]] as ~~subsets~~. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].		Since 118 factors into primes as {{nowrap\| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

	== Intervals ==		== Intervals ==

FloraC: No need to remind readers of what a regular temperament is everywhere

2025-04-11T11:19:48Z

No need to remind readers of what a regular temperament is everywhere

← Older revision		Revision as of 11:19, 11 April 2025
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator

BudjarnLambeth: /* Rank-2 temperaments */ (''See regular temperament for more about what all this means and how to use it.'')

2025-04-11T07:49:39Z

Rank-2 temperaments: (''See regular temperament for more about what all this means and how to use it.'')

← Older revision		Revision as of 07:49, 11 April 2025
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
			''See [[regular temperament]] for more about what all this means and how to use it.''

	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator