183edo - 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"

2026-03-13T13:32:27Z

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 13:32, 13 March 2026
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	\| [[Promethium]]		\| [[Promethium]]
	\|}		\|}
	<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

	== Music ==		== Music ==

FloraC: /* Approximation to JI */ - zeta table. + interval table

2026-01-18T16:24:00Z

Approximation to JI: - zeta table. + interval table

← Older revision		Revision as of 16:24, 18 January 2026
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	== Approximation to JI ==		== Approximation to JI ==
	=== ~~Zeta peak index~~ ===		=== Interval mappings ===
	{{~~ZPI~~		{{Q-odd-limit intervals}}
	~~\| zpi = 1210~~
	~~\| steps = 182.999728466810~~
	~~\| step size = 6.55738677895164~~
	~~\| tempered height = 11.020824~~
	~~\| pure height = 11.020734~~
	~~\| integral = 1.643410~~
	~~\| gap = 19.731996~~
	~~\| octave = 1200.00178054815~~
	~~\| consistent = 18~~
	~~\| distinct = 18~~
	}}

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

FloraC: /* Theory */ - typo

2026-01-18T16:23:27Z

Theory: - typo

← Older revision		Revision as of 16:23, 18 January 2026
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 183 factors into primes as{{nowrap\| 3 × 61 }}, 183edo contains [[3edo]] and [[61edo]] as its subsets.		Since 183 factors into primes as {{nowrap\| 3 × 61 }}, 183edo contains [[3edo]] and [[61edo]] as its subsets.

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

FloraC: /* Theory */ re-organize and - certain questionable statements that caused community complaints

2026-01-18T16:23:10Z

Theory: re-organize and - certain questionable statements that caused community complaints

← Older revision		Revision as of 16:23, 18 January 2026
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	== Theory ==		== Theory ==
	183edo is notable as a higher-limit system, [[consistency\|distinctly consistent]] in the [[17-odd-limit]], or the no-19 no-31 [[33-odd-limit]]. It [[tempering out\|tempers out]] the [[schisma]] in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]], and [[8019/8000]]; in the [[13-limit]], [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[1573/1568]], [[2080/2079]], [[4096/4095]], [[4225/4224]], and [[6656/6655]]; in the [[17-limit]] [[442/441]], [[561/560]], [[715/714]], [[936/935]], [[1089/1088]], and [[1156/1155]]; and in the [[19-limit]] [[456/455]]. It is the [[optimal patent val]] for 13- and 17-limit [[mirkat]], the {{nowrap\|72 & 111}} temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]]. It allows [[essentially tempered chord]] including [[ratwolfsmic chords]], [[swetismic chords]], [[squbemic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.		183edo is notable as a higher-limit system, [[consistency\|distinctly consistent]] in the [[17-odd-limit]], or the no-19 no-31 [[33-odd-limit]]. It has especially low errors in ''all'' [[prime limit]]s from 11 to 29, although its bad rendering of [[19/1\|19]] makes it fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's [[29-limit]] system with the addition of an essentially perfectly accurate prime [[43/1\|43]].

			As an equal temperament, 183et [[tempering out\|tempers out]] the [[schisma]] in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]], and [[8019/8000]]; in the [[13-limit]], [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[1573/1568]], [[2080/2079]], [[4096/4095]], [[4225/4224]], and [[6656/6655]]; in the [[17-limit]] [[442/441]], [[561/560]], [[715/714]], [[936/935]], [[1089/1088]], and [[1156/1155]]; and in the [[19-limit]] [[456/455]]. It is the [[optimal patent val]] for 13- and 17-limit [[mirkat]], the {{nowrap\|72 & 111}} temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]]. It allows [[essentially tempered chord]] including [[ratwolfsmic chords]], [[swetismic chords]], [[squbemic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.

	It is even stronger if 7 is left out of the picture. As a no-7 temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.		It is even stronger if 7 is left out of the picture. As a no-7 temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.

	=== Prime harmonics ===		=== Prime harmonics ===
	~~In the range of edos from 100 to 200, 183edo is notable as having especially low error~~ in ~~''all'' [[prime limit]]s from~~ 11 to 29, compared using a variety of prime error punishments, although it has a bad 19 and fails to be consistent in the [[19-odd-limit]]. It is however a strong no-19's 29-limit system with an essentially perfectly accurate prime 43. It can also be considered to model the 2.17.29.43 [[subgroup]] with extreme accuracy.		{{Harmonics in equal\|183\|columns=11}}
			{{Harmonics in equal\|183\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 183edo (continued)}}
	{{Harmonics in equal\|183}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 183 factors into 3 × 61, 183edo contains [[3edo]] and [[61edo]] as its subsets.		Since 183 factors into primes as{{nowrap\| 3 × 61 }}, 183edo contains [[3edo]] and [[61edo]] as its subsets.

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

Overthink: /* Rank-2 temperaments */ + septendesemi

2025-11-29T21:06:39Z

Rank-2 temperaments: + septendesemi

← Older revision		Revision as of 21:06, 29 November 2025
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	\| 27/26		\| 27/26
	\| [[Luminal]]		\| [[Luminal]]
			\|-
			\| 1
			\| 16\183
			\| 104.92
			\| 17/16
			\| [[Septendesemi]]
	\|-		\|-
	\| 1		\| 1

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

2025-04-17T02:30:05Z

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

← Older revision		Revision as of 02:30, 17 April 2025
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	\| 498.36		\| 498.36
	\| 4/3		\| 4/3
	\| [[Helmholtz temperament\|Helmholtz]]		\| [[Helmholtz (temperament)\|Helmholtz]]
	\|-		\|-
	\| 1		\| 1

Nick Vuci: Edited Helmholtz redirect

2025-04-16T15:34:22Z

Edited Helmholtz redirect

← Older revision		Revision as of 15:34, 16 April 2025
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	\| 498.36		\| 498.36
	\| 4/3		\| 4/3
	\| [[Helmholtz]]		\| [[Helmholtz temperament\|Helmholtz]]
	\|-		\|-
	\| 1		\| 1

ArrowHead294: /* Zeta peak index */

2025-03-20T15:19:51Z

Zeta peak index

← Older revision		Revision as of 15:19, 20 March 2025
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	== Approximation to JI ==		== Approximation to JI ==
	=== Zeta peak index ===		=== Zeta peak index ===
	{~~\| class="wikitable center-all"~~		{{ZPI
	\|-		\| zpi = 1210
	~~! colspan~~=~~"3" \| Tuning~~		\| steps = 182.999728466810
	~~! colspan="3"~~ \| ~~Strength~~		\| step size = 6.55738677895164
	~~! colspan~~=~~"2" \| Closest edo~~		\| tempered height = 11.020824
	~~! colspan="2" \| Integer limit~~		\| pure height = 11.020734
	\|-		\| integral = 1.643410
	~~! ZPI~~		\| gap = 19.731996
	~~! Steps per octave~~		\| octave = 1200.00178054815
	~~! Step size (cents)~~		\| consistent = 18
	~~! Height~~		\| distinct = 18
	~~! Integral~~		}}
	~~! Gap~~
	~~! Edo~~
	~~! Octave (cents)~~
	~~! Consistent~~
	~~! Distinct~~
	\|-
	~~\| [[1210zpi]]~~
	\| 182.999728466810
	\| 6.55738677895164
	\| 11.020824
	\| 1.643410
	\| 19.731996
	\| ~~183edo~~
	\| 1200.00178054815
	\| 18
	\| 18
	\|}

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

FloraC: Consolidate sections

2025-01-19T10:22:41Z

Consolidate sections

← Older revision		Revision as of 10:22, 19 January 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 183 factors into 3 × 61, 183edo contains [[3edo]] and [[61edo]] as its subsets.		Since 183 factors into 3 × 61, 183edo contains [[3edo]] and [[61edo]] as its subsets.

			== Approximation to JI ==
			=== Zeta peak index ===
			{\| class="wikitable center-all"
			\|-
			! colspan="3" \| Tuning
			! colspan="3" \| Strength
			! colspan="2" \| Closest edo
			! colspan="2" \| Integer limit
			\|-
			! ZPI
			! Steps per octave
			! Step size (cents)
			! Height
			! Integral
			! Gap
			! Edo
			! Octave (cents)
			! Consistent
			! Distinct
			\|-
			\| [[1210zpi]]
			\| 182.999728466810
			\| 6.55738677895164
			\| 11.020824
			\| 1.643410
			\| 19.731996
			\| 183edo
			\| 1200.00178054815
			\| 18
			\| 18
			\|}

	== Regular temperament properties ==		== Regular temperament properties ==
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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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	\|+ 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
	\|-		\|-
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	\|-		\|-
	\| 3		\| 3
	\| 38\183<br />(23\183)		\| 38\183<br>(23\183)
	\| 249.18<br />(150.82)		\| 249.18<br>(150.82)
	\| 15/13<br />(12/11)		\| 15/13<br>(12/11)
	\| [[Hemiterm]]		\| [[Hemiterm]]
	\|-		\|-
	\| 3		\| 3
	\| 76\183<br />(15\183)		\| 76\183<br>(15\183)
	\| 498.36<br />(98.36)		\| 498.36<br>(98.36)
	\| 4/3<br />(200/189)		\| 4/3<br>(200/189)
	\| [[Term]] / terminator		\| [[Term]] / terminator
	\|-		\|-
	\| 61		\| 61
	\| 38\183<br />(2\183)		\| 38\183<br>(2\183)
	\| 249.18<br />(13.11)		\| 249.18<br>(13.11)
	\| 13750/11907<br />(?)		\| 13750/11907<br>(?)
	\| [[Promethium]]		\| [[Promethium]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	~~== Zeta properties ==~~
	~~=== Zeta peak index ===~~
	~~{\| class="wikitable"~~
	\|-
	~~! colspan="3" \| Tuning~~
	~~! colspan="3" \| Strength~~
	~~! colspan="2" \| Closest EDO~~
	~~! colspan="2" \| Integer limit~~
	\|-
	~~! ZPI~~
	~~! Steps per octave~~
	~~! Step size (cents)~~
	~~! Height~~
	~~! Integral~~
	~~! Gap~~
	~~! EDO~~
	~~! Octave (cents)~~
	~~! Consistent~~
	~~! Distinct~~
	\|-
	~~\| [[1210zpi]]~~
	~~\| 182.999728466810~~
	~~\| 6.55738677895164~~
	~~\| 11.020824~~
	~~\| 1.643410~~
	~~\| 19.731996~~
	~~\| 183edo~~
	~~\| 1200.00178054815~~
	~~\| 18~~
	~~\| 18~~
	\|}

	== Music ==		== Music ==

ArrowHead294: Formatting

2025-01-19T02:48:11Z

Formatting

← Older revision		Revision as of 02:48, 19 January 2025
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	== Zeta properties ==		== Zeta properties ==
	===Zeta peak index===		=== Zeta peak index ===
	{\| class="wikitable"		{\| class="wikitable"
	~~! colspan="3" \|Tuning~~
	~~! colspan="3" \|Strength~~
	~~! colspan="2" \|Closest EDO~~
	~~! colspan="2" \|Integer limit~~
	\|-		\|-
	!~~ZPI~~		! colspan="3" \| Tuning
	!~~Steps per octave~~		! colspan="3" \| Strength
	~~!Step size (cents)~~		! colspan="2" \| Closest EDO
	~~!Height~~		! colspan="2" \| Integer limit
	~~!Integral~~
	~~!Gap~~
	!EDO
	!~~Octave (cents)~~
	~~!Consistent~~
	~~!Distinct~~
	\|-		\|-
	\|[[1210zpi]]		! ZPI
	\|182.999728466810		! Steps per octave
	\|6.55738677895164		! Step size (cents)
	\|11.020824		! Height
	\|1.643410		! Integral
	\|19.731996		! Gap
	\|183edo		! EDO
	\|1200.00178054815		! Octave (cents)
	\|18		! Consistent
	\|18		! Distinct
			\|-
			\| [[1210zpi]]
			\| 182.999728466810
			\| 6.55738677895164
			\| 11.020824
			\| 1.643410
			\| 19.731996
			\| 183edo
			\| 1200.00178054815
			\| 18
			\| 18
	\|}		\|}

	== Music ==		== Music ==
	; [[birdshite stalactite]]		; [[birdshite stalactite]]