342edo - Revision history

Contribution at 21:53, 15 April 2026

2026-04-15T21:53:03Z

← Older revision		Revision as of 21:53, 15 April 2026
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	== Theory ==		== Theory ==
	342edo is a very strong 11-limit system. It is, as one would expect, [[consistency\|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis\|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] and [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert\|hemitert]] temperament, and [[support]]s hemiennealimmal.		342edo is a very strong 11-limit system. It is, as one would expect, [[consistency\|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis\|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] and [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert\|hemitert]] temperament, and [[support]]s hemiennealimmal.

			If 3.5 cents is taken as the [[just-noticeable difference]], then 342edo may be regarded as the highest EDO whose step size remains individually discernible. However, the [[JND]] is not fixed and depends on the listener and musical context.

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

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"

2026-03-13T13:32:44Z

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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	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|}		\|}
	<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 ==

	* [[11-odd-limit\|Diamond11]]: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43		* [[11-odd-limit\|Diamond11]]: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43

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2025-11-09T08:35:58Z

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Contribution: /* Commas of the 2.3.5.7.11 subgroup tempered out in 342edo patent val */

2025-11-08T23:47:13Z

Commas of the 2.3.5.7.11 subgroup tempered out in 342edo patent val

Show changes

Contribution: /* Theory */

2025-11-08T21:48:55Z

Theory

← Older revision		Revision as of 21:48, 8 November 2025
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	== Theory ==		== Theory ==
	342edo is a very strong 11-limit system. It is, as one would expect, [[consistency\|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis\|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] ~~and~~ [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert\|hemitert]] temperament, and [[support]]s hemiennealimmal.		342edo is a very strong 11-limit system. It is, as one would expect, [[consistency\|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis\|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]], [[9801/9800]], [[32805/32768]], [[41503/41472]], [[43923/43904]], and [[65625/65536]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert\|hemitert]] temperament, and [[support]]s hemiennealimmal.

	=== Prime harmonics ===		=== Prime harmonics ===
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	342 factors as {{factorization\|342}}, with subset edos {{EDOs\| 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171 }}.		342 factors as {{factorization\|342}}, with subset edos {{EDOs\| 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171 }}.

	[[684edo]], which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping.		[[684edo]], which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping.

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

Contribution: /* Commas of the 2.3.5.7.11 subgroup tempered out in 342edo patent val */

2025-11-08T21:44:42Z

Commas of the 2.3.5.7.11 subgroup tempered out in 342edo patent val

← Older revision		Revision as of 21:44, 8 November 2025
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	Commas with numerator <= 2^32:		Commas with numerator <= 2^32:

	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed sortable"
	! [[Ratio]]		! [[Ratio]]
	! [[Monzo]]		! [[Monzo]]

Contribution at 21:22, 8 November 2025

2025-11-08T21:22:39Z

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ArrowHead294 at 15:00, 20 March 2025

2025-03-20T15:00:53Z

← Older revision		Revision as of 15:00, 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 = 2568
	~~! colspan~~=~~"3" \| Tuning~~		\| steps = 341.974850913987
	~~! colspan="3" \| Strength~~		\| step size = 3.50902996753355
	~~! colspan="2"~~ \| ~~Closest edo~~		\| tempered height = 13.478611
	~~! colspan~~=~~"2" \| Integer limit~~		\| pure height = 12.437722
	\|-		\| integral = 1.890555
	~~! ZPI~~		\| gap = 20.767404
	~~! Steps per octave~~		\| octave = 1200.08824889647
	~~! Step size (cents)~~		\| consistent = 12
	~~! Height~~		\| distinct = 12
	~~! Integral~~		}}
	~~! Gap~~
	~~! Edo~~
	~~! Octave (cents)~~
	~~! Consistent~~
	~~! Distinct~~
	\|-
	~~\| [[2568zpi]]~~
	\| 341.974850913987
	\| 3.50902996753355
	\| 13.478611
	\| 1.890555
	\| 20.767404
	\| ~~342edo~~
	\| 1200.08824889647
	\| 12
	\| 12
	\|}

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

ArrowHead294 at 14:43, 20 February 2025

2025-02-20T14:43:50Z

← Older revision		Revision as of 14:43, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|342~~}}		{{ED intro}}

	== Theory ==		== Theory ==
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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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	\|-		\|-
	\| 2		\| 2
	\| 124\342<br>(47\342)		\| 124\342<br />(47\342)
	\| 435.09<br>(164.91)		\| 435.09<br />(164.91)
	\| 9/7<br>(11/10)		\| 9/7<br />(11/10)
	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	\|-		\|-
	\| 2		\| 2
	\| 142\342<br>(29\342)		\| 142\342<br />(29\342)
	\| 498.25<br>(101.75)		\| 498.25<br />(101.75)
	\| 4/3<br>(35/33)		\| 4/3<br />(35/33)
	\| [[Bipont]]		\| [[Bipont]]
	\|-		\|-
	\| 3		\| 3
	\| 71\342<br>(43\342)		\| 71\342<br />(43\342)
	\| 249.12<br>(150.88)		\| 249.12<br />(150.88)
	\| 15/13<br>(12/11)		\| 15/13<br />(12/11)
	\| [[Hemiterm]]		\| [[Hemiterm]]
	\|-		\|-
	\| 6		\| 6
	\| 97\342<br>(17\342)		\| 97\342<br />(17\342)
	\| 340.35<br>(59.65)		\| 340.35<br />(59.65)
	\| 162/133<br>(88/85)		\| 162/133<br />(88/85)
	\| [[Semiseptichrome]]		\| [[Semiseptichrome]]
	\|-		\|-
	\| 6		\| 6
	\| 142\342<br>(28\342)		\| 142\342<br />(28\342)
	\| 498.25<br>(98.25)		\| 498.25<br />(98.25)
	\| 4/3<br>(18/17)		\| 4/3<br />(18/17)
	\| [[Semiterm]]		\| [[Semiterm]]
	\|-		\|-
	\| 9		\| 9
	\| 63\342<br>(13\342)		\| 63\342<br />(13\342)
	\| 221.05<br>(45.61)		\| 221.05<br />(45.61)
	\| 25/22<br>(77/75)		\| 25/22<br />(77/75)
	\| [[Quadraennealimmal]]		\| [[Quadraennealimmal]]
	\|-		\|-
	\| 18		\| 18
	\| 71\342<br>(5\342)		\| 71\342<br />(5\342)
	\| 249.12<br>(17.54)		\| 249.12<br />(17.54)
	\| 15/13<br>(99/98)		\| 15/13<br />(99/98)
	\| [[Hemiennealimmal]]		\| [[Hemiennealimmal]]
	\|-		\|-
	\| 38		\| 38
	\| 142\342<br>(2\342)		\| 142\342<br />(2\342)
	\| 498.25<br>(7.02)		\| 498.25<br />(7.02)
	\| 4/3<br>(225/224)		\| 4/3<br />(225/224)
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|}		\|}
	<nowiki>*~~</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 ==		== Scales ==

	* [[11-odd-limit\|Diamond11]]: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43		* [[11-odd-limit\|Diamond11]]: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43

Contribution at 19:26, 22 January 2025

2025-01-22T19:26:43Z

← Older revision		Revision as of 19:26, 22 January 2025
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	== Scales ==		== Scales ==

	* [[11-odd-limit]] ~~scale~~: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43		* [[11-odd-limit\|Diamond11]]: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43