684edo - 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"

2026-03-13T13:31:40Z

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 78:		Line 78:
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		\|}
	<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 ==

	* [[15-odd-limit\|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64		* [[15-odd-limit\|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64

2025-11-09T08:36:26Z

- spam

2025-11-08T23:46:22Z

Commas of the 2.3.5.7.11.13 subgroup tempered out in 684edo patent val

2025-11-08T22:17:34Z

← Older revision		Revision as of 22:17, 8 November 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	684edo divides the steps of [[171edo]] into four. It is [[consistent]] to the [[17-odd-limit]], [[tempering out]] [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], ~~and~~ [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit.		684edo divides the steps of [[171edo]] into four. It is [[consistent]] to the [[17-odd-limit]], [[tempering out]] [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], [[9801/9800]], [[10648/10647]], [[32805/32768]], [[41503/41472]], [[43923/43904]], [[50193/50176]], [[59535/59488]], and [[65625/65536]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit.

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

2025-11-08T22:14:53Z

2025-03-20T15:08:29Z

Zeta peak index

← Older revision		Revision as of 15:08, 20 March 2025
Line 13:		Line 13:
	== Approximation to JI ==		== Approximation to JI ==
	=== Zeta peak index ===		=== Zeta peak index ===
	{~~\| class="wikitable center-all"~~		{{ZPI
	\|-		\| zpi = 5818
	~~! colspan="3"~~ \| ~~Tuning~~		\| steps = 683.938934890938
	~~! colspan~~=~~"3" \| Strength~~		\| step size = 1.75454260429165
	~~! colspan="2"~~ \| ~~Closest edo~~		\| tempered height = 14.267321
	~~! colspan~~=~~"2" \| Integer limit~~		\| pure height = 7.268914
	\|-		\| integral = 1.773752
	~~! ZPI~~		\| gap = 20.109967
	~~! Steps per octave~~		\| octave = 1200.10714133549
	~~! Step size (cents)~~		\| consistent = 18
	~~! Height~~		\| distinct = 18
	~~! Integral~~		}}
	~~! Gap~~
	~~! Edo~~
	~~! Octave (cents)~~
	~~! Consistent~~
	~~! Distinct~~
	\|-
	~~\| [[5818zpi]]~~
	\| 683.938934890938
	\| 1.75454260429165
	\| 14.267321
	\| 1.773752
	\| 20.109967
	\| ~~684edo~~
	\| 1200.10714133549
	\| 18
	\| 18
	\|}

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

2025-02-20T22:43:35Z

← Older revision		Revision as of 22:43, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|684~~}}		{{ED intro}}

	== Theory ==		== Theory ==
Line 95:		Line 95:
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		\|}
	<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 ==

	* [[15-odd-limit\|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64		* [[15-odd-limit\|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64

2025-01-22T19:27:09Z

2025-01-20T12:04:35Z

← Older revision		Revision as of 12:04, 20 January 2025
Line 95:		Line 95:
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		\|}
	<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct		<nowiki>*</nowiki> [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

			== Scales ==

			* [[15-odd-limit]] scale: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64

2025-01-19T10:34:33Z

Consolidate sections

@@ Line 10: / Line 10: @@
 === Subsets and supersets ===
 Since 684 factors into {{factorization|684}}, 684edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342 }}.
 == Regular temperament properties ==
@@ Line 17: / Line 49: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 45: / Line 77: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
 | 18
-| 271\684<br />(5\684)
+| 271\684<br>(5\684)
-| 475.44<br />(8.77)
+| 475.44<br>(8.77)
-| 1053/800<br />(1287/1280)
+| 1053/800<br>(1287/1280)
 | [[Semihemiennealimmal]]
 |-
 | 38
-| 151\684<br />(7\684)
+| 151\684<br>(7\684)
-| 264.91<br />(12.28)
+| 264.91<br>(12.28)
-| 500/429<br />(144/143)
+| 500/429<br>(144/143)
 | [[Semihemienneadecal]]
 |}
-<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