342edo - Revision history

FloraC: /* Regular temperament properties */ + semiosiris

2026-07-12T10:40:06Z

Regular temperament properties: + semiosiris

← Older revision		Revision as of 10:40, 12 July 2026
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	\| 35/33		\| 35/33
	\| [[Bipont]]		\| [[Bipont]]
			\|-
			\| 2
			\| 49\342
			\| 171.93
			\| 243/220
			\| [[Semiosiris]]
	\|-		\|-
	\| 3		\| 3

FloraC: /* Regular temperament properties */ cleanup

2026-07-12T10:38:46Z

Regular temperament properties: cleanup

← Older revision		Revision as of 10:38, 12 July 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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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 3025/3024, 4375/4374, 32805/32768		\| 2401/2400, 3025/3024, 4375/4374, 32805/32768
	\| {{~~mapping~~\| 342 542 794 960 1183 }}		\| {{Mapping\| 342 542 794 960 1183 }}
	\| +0.110		\| +0.110
	\| 0.0556		\| 0.0556
Line 36:		Line 36:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712		\| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
	\| {{~~mapping~~\| 342 542 794 960 1183 1265 }} (342f)		\| {{Mapping\| 342 542 794 960 1183 1265 }} (342f)
	\| +0.178		\| +0.178
	\| 0.1618		\| 0.1618
Line 43:		Line 43:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 625/624, 729/728, 847/845, 1575/1573, 4096/4095		\| 625/624, 729/728, 847/845, 1575/1573, 4096/4095
	\| {{~~mapping~~\| 342 542 794 960 1183 1266 }} (342)		\| {{Mapping\| 342 542 794 960 1183 1266 }} (342)
	\| +0.020		\| +0.020
	\| 0.2061		\| 0.2061
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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
	\|-		\|-
Line 79:		Line 79:
	\|-		\|-
	\| 2		\| 2
	\| ~~124\342<br />(~~47\342)		\| 47\342
	\| ~~435.09<br />(~~164.91)		\| 164.91
	\| ~~9/7<br />(~~11/10)		\| 11/10
	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	\|-		\|-
	\| 2		\| 2
	\| ~~142\342<br />(~~29\342)		\| 29\342
	\| ~~498.25<br />(~~101.75)		\| 101.75
	\| ~~4/3<br />(~~35/33)		\| 35/33
	\| [[Bipont]]		\| [[Bipont]]
	\|-		\|-
	\| 3		\| 3
	\| ~~71\342<br />(~~43\342)		\| 43\342
	\| ~~249.12<br />(~~150.88)		\| 150.88
	\| ~~15/13<br />(~~12/11)		\| 12/11
	\| [[Hemiterm]]		\| [[Hemiterm]]
	\|-		\|-
	\| 6		\| 6
	\| ~~97\342<br />(~~17\342)		\| 17\342
	\| ~~340.35<br />(~~59.65)		\| 59.65
	\| ~~162/133<br />(~~88/85)		\| 88/85
	\| [[Semiseptichrome]]		\| [[Semiseptichrome]]
	\|-		\|-
	\| 6		\| 6
	\| ~~142\342<br />(~~28\342)		\| 28\342
	\| ~~498.25<br />(~~98.25)		\| 98.25
	\| ~~4/3<br />(~~18/17)		\| 18/17
	\| [[Semiterm]]		\| [[Semiterm]]
	\|-		\|-
	\| 9		\| 9
	\| ~~63\342<br />(~~13\342)		\| 13\342
	\| ~~221.05<br />(~~45.61)		\| 45.61
	\| ~~25/22<br />(~~77/75)		\| 77/75
	\| [[Quadraennealimmal]]		\| [[Quadraennealimmal]]
	\|-		\|-
	\| 18		\| 18
	\| ~~71\342<br />(~~5\342)		\| 5\342
	\| ~~249.12<br />(~~17.54)		\| 17.54
	\| ~~15/13<br />(~~99/98)		\| 99/98
	\| [[Hemiennealimmal]]		\| [[Hemiennealimmal]]
	\|-		\|-
	\| 38		\| 38
	\| ~~142\342<br />(~~2\342)		\| 2\342
	\| ~~498.25<br />(~~7.02)		\| 7.02
	\| ~~4/3<br />(~~225/224)		\| 225/224
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|}		\|}
	<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~~		<nowiki/>* In [[normal forms #Minimal-generator form\|minimal-generator form]]

	== 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

FloraC: - zeta table

2026-07-12T10:35:30Z

- zeta table

← Older revision		Revision as of 10:35, 12 July 2026
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	[[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 ==~~
	~~=== Zeta peak index ===~~
	~~{{ZPI~~
	~~\| zpi = 2568~~
	~~\| steps = 341.974850913987~~
	~~\| step size = 3.50902996753355~~
	~~\| tempered height = 13.478611~~
	~~\| pure height = 12.437722~~
	~~\| integral = 1.890555~~
	~~\| gap = 20.767404~~
	~~\| octave = 1200.08824889647~~
	~~\| consistent = 12~~
	~~\| distinct = 12~~
	}}

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

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

FloraC: - spam

2025-11-09T08:35:58Z

- spam

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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
Line 3:		Line 3:

	== 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 ===
Line 11:		Line 11:
	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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