232edo - 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"

2026-03-13T13:31:31Z

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 13:31, 13 March 2026
Line 69:		Line 69:
	\| [[Mystery]] (232)		\| [[Mystery]] (232)
	\|}		\|}
	<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
	== Scales ==		== Scales ==
	* [[Mystery58]]		* [[Mystery58]]

Overthink: Add scale

2026-02-06T05:09:51Z

Add scale

← Older revision		Revision as of 05:09, 6 February 2026
Line 70:		Line 70:
	\|}		\|}
	<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
			== Scales ==
			* [[Mystery58]]

	[[Category:Mystery]]		[[Category:Mystery]]

Hotcrystal0: there's probably better places to put this but I'm putting it here for now

2025-12-19T23:14:40Z

there's probably better places to put this but I'm putting it here for now

← Older revision		Revision as of 23:14, 19 December 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	232 = 8 × 29, and 232edo shares its [[3/2\|fifth]] with [[29edo]]. The equal temperament [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.		232 = 8 × 29, and 232edo shares its [[3/2\|fifth]] with [[29edo]]. The equal temperament [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.

	Aside from its [[patent val]], the 232d val {{val\| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[4375/4374]] and [[16875/16807]], supporting [[octoid]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.		Aside from its [[patent val]], the 232d val {{val\| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[4375/4374]] and [[16875/16807]], supporting [[octoid]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.

	Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[minor minthmic chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system.		Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[minor minthmic chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. 232edo is also the first edo that approximates [[6/5]] more accurately than [[19edo]].

	=== Odd harmonics ===		=== Odd harmonics ===

ArrowHead294 at 14:21, 20 February 2025

2025-02-20T14:21:27Z

← Older revision		Revision as of 14:21, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|232~~}}		{{ED intro}}

	== Theory ==		== Theory ==
Line 17:		Line 17:
	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
			\|-
	! rowspan="2" \| [[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list~~\|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
	\|-		\|-
	! [[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
	! [[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
Line 29:		Line 30:
	\| 393216/390625, {{monzo\| 46 -29 0 }}		\| 393216/390625, {{monzo\| 46 -29 0 }}
	\| {{mapping\| 232 368 539 }}		\| {{mapping\| 232 368 539 }}
	\| -0.5461		\| −0.5461
	\| 0.3989		\| 0.3989
	\| 7.71		\| 7.71
Line 36:		Line 37:
	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+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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	\| [[Mystery]] (232)		\| [[Mystery]] (232)
	\|}		\|}
			<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	[[Category:Mystery]]		[[Category:Mystery]]

FloraC: /* Regular temperament properties */ +acrokleismic

2024-04-10T03:55:47Z

Regular temperament properties: +acrokleismic

← Older revision		Revision as of 03:55, 10 April 2024
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	! Associated<br>Ratio*		! Associated<br>Ratio*
	! Temperaments		! Temperaments
			\|-
			\| 1
			\| 61\232
			\| 315.52
			\| 6/5
			\| [[Acrokleismic]] (7-limit, 232d)
	\|-		\|-
	\| 1		\| 1

FloraC: /* Regular temperament properties */ -typo

2024-03-26T09:56:52Z

Regular temperament properties: -typo

← Older revision		Revision as of 09:56, 26 March 2024
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	\|-		\|-
	\| 8		\| 8
	\| 113\232<br>(4\232)		\| 113\232<br>(3\232)
	\| 584.48<br>(15.52)		\| 584.48<br>(15.52)
	\| 7/5<br>(100/99)		\| 7/5<br>(100/99)

FloraC: Note that its fifth is from 29edo; correction of the commas by 232d

2024-03-26T09:56:15Z

Note that its fifth is from 29edo; correction of the commas by 232d

← Older revision		Revision as of 09:56, 26 March 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	232edo [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.		232 = 8 × 29, and 232edo shares its [[3/2\|fifth]] with [[29edo]]. The equal temperament [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.

	Aside from its [[patent val]], the 232d val {{val\| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[~~3125~~/~~3087~~]] and [[~~245~~/~~243~~]], supporting [[~~bohpier~~]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993~~]], supporting [[octoid~~]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.		Aside from its [[patent val]], the 232d val {{val\| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[4375/4374]] and [[16875/16807]], supporting [[octoid]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.

	Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[~~gentle~~ chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system.		Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[minor minthmic chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system.

	=== Odd harmonics ===		=== Odd harmonics ===
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	\| 387.93		\| 387.93
	\| 5/4		\| 5/4
	\| [[Würschmidt]]		\| [[Würschmidt]] (5-limit)
			\|-
			\| 8
			\| 113\232<br>(4\232)
			\| 584.48<br>(15.52)
			\| 7/5<br>(100/99)
			\| [[Octoid]] (232d)
			\|-
			\| 29
			\| 96\232<br>(3\232)
			\| 496.55<br>(15.52)
			\| 4/3<br>(105/104)
			\| [[Mystery]] (232)
	\|}		\|}

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	[[Category:Trimyna]]		[[Category:Trimyna]]
	[[Category:Mynucumic]]		[[Category:Mynucumic]]
	~~[[Category:Bohpier]]~~

FloraC: Cleanup

2024-03-26T09:37:54Z

Cleanup

← Older revision		Revision as of 09:37, 26 March 2024
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|232}}		{{EDO intro\|232}}

	== Theory ==		== Theory ==
	232edo [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.		232edo [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.

	Aside from its [[patent val]], the 232d val {{val\| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, 50421/50000; and 232d [[3125/3087]] and [[245/243]], supporting [[bohpier]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]], supporting [[octoid]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.		Aside from its [[patent val]], the 232d val {{val\| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[3125/3087]] and [[245/243]], supporting [[bohpier]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]], supporting [[octoid]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.

	Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[minthmic chords]], and because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system.		Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system.

	=== Odd harmonics ===		=== Odd harmonics ===
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	Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs\| 2, 4, 8, 29, 58, and 116 }}.		Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs\| 2, 4, 8, 29, 58, and 116 }}.

	==Regular temperament properties==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \|[[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \|[[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|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
	\|-		\|-
	![[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-
	~~\|2.3~~
	~~\|{{monzo\|46 -29}}~~
	~~\|{{val\|232 368}}~~
	~~\| -0.4711~~
	~~\| 0.4709~~
	~~\| 9.10~~
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|~~{{monzo\|17 1 -8}}~~, {{monzo\|46 -29 0}}		\| 393216/390625, {{monzo\| 46 -29 0 }}
	\|{{~~val~~\|232 368 539}}		\| {{mapping\| 232 368 539 }}
	\| -0.5461		\| -0.5461
	\| 0.3989		\| 0.3989
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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
	\|-		\|-
	\|1		\| 1
	\|75\232		\| 75\232
	\|387.93		\| 387.93
	\|5/4		\| 5/4
	\|[[Würschmidt]]		\| [[Würschmidt]]
	\|}		\|}

Francium at 16:46, 29 October 2023

2023-10-29T16:46:46Z

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|232}}
+== Theory ==
 edo [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
 Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs| 2, 4, 8, 29, 58, and 116 }}.
 [[Category:Mystery]]

FloraC: Rework, expand, recategorize

2023-10-02T14:27:55Z

Rework, expand, recategorize

← Older revision		Revision as of 14:27, 2 October 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	~~The ''232 equal division'' divides the octave into~~ 232 equal parts of 5.172 cents each. It provides the [[Optimal_patent_val\|optimal patent val]] for [[13-limit\|13-limit]] [[Hemifamity_temperaments#Mystery\|mystery temperament]], the rank three [[Hemifamity_family#Pele\|pele temperament]] and the rank three [[Trimyna_family#Trimyna-13-limit\|trimyna temperament]] and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank five temperament.		{{EDO intro\|232}}

	~~Aside from its~~ patent val, the ~~232d val is worth considering. Both temper~~ out ~~the wuerschmidt comma, 393216~~/~~390625~~, in the 5~~-limit~~.		232edo [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.

	In the 7-limit, the patent val tempers out hemifamity, 5120/5103 and the trimyna comma, 50421/50000; and 232d 3125/~~3097~~ and 245/243, ~~[[support\|~~supporting]] [[~~Sensamagic_clan#Bohpier\|~~bohpier ~~temperament~~]]. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993, supporting octoid. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, ~~441~~/~~440~~ and ~~676~~/~~675~~, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports ~~variants~~ of octoid.		Aside from its [[patent val]], the 232d val {{val\| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, 50421/50000; and 232d [[3125/3087]] and [[245/243]], supporting [[bohpier]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]], supporting [[octoid]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.

	Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[~~werckismic_chords\|~~werckismic chords]], because it tempers out 196/195 it allows [[~~mynucumic_chords\|~~mynucumic chords]], because it tempers out 352/351 it allows [[~~minthmic_chords\|~~minthmic chords]], and because it tempers out 364/363 it allows [[~~gentle_chords\|~~gentle chords]], and because it tempers out 847/845 it allows the [[~~cuthbert_triad\|~~cuthbert ~~triad~~]], making it a very flexible harmonic system.		Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[minthmic chords]], and because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system.

			=== Odd harmonics ===
	{{Harmonics in equal\|232}}		{{Harmonics in equal\|232}}
	[[Category:~~Equal divisions of the octave\|###~~]] ~~<!-- 3-digit number -->~~
			=== Subsets and supersets ===
			Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs\| 2, 4, 8, 29, 58, and 116 }}.

			[[Category:Mystery]]
			[[Category:Pele]]
			[[Category:Trimyna]]
			[[Category:Mynucumic]]
	[[Category:Bohpier]]		[[Category:Bohpier]]
	~~[[Category:Mystery]]~~