1665edo - 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"

2026-03-13T13:32:08Z

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
Line 44:		Line 44:
	\| [[Rhodium]]		\| [[Rhodium]]
	\|}		\|}
	<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

Francium: changed EDO intro to ED intro

2025-02-21T06:13:25Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:13, 21 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|1665~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 16:05, 15 January 2025

2025-01-15T16:05:09Z

← Older revision		Revision as of 16:05, 15 January 2025
Line 44:		Line 44:
	\| [[Rhodium]]		\| [[Rhodium]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

ArrowHead294: Partial undo

2024-11-16T13:20:09Z

Partial undo

← Older revision		Revision as of 13:20, 16 November 2024
Line 17:		Line 17:
	== Regular temperament properties ==		== Regular temperament properties ==
	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
Line 36:		Line 43:
	\| 55/32<br />(?)		\| 55/32<br />(?)
	\| [[Rhodium]]		\| [[Rhodium]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

ArrowHead294 at 01:35, 16 November 2024

2024-11-16T01:35:33Z

← Older revision		Revision as of 01:35, 16 November 2024
Line 17:		Line 17:
	== Regular temperament properties ==		== Regular temperament properties ==
	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+ style="font-size: 105%;" \| Table of~~ rank-2 ~~temperaments by generator~~
	\|-
	~~! Periods<br />per 8ve~~
	! Generator*
	! Cents*
	! Associated<br />Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
Line 43:		Line 36:
	\| 55/32<br />(?)		\| 55/32<br />(?)
	\| [[Rhodium]]		\| [[Rhodium]]
	\|}		{{rank-2 end}}
	{{orf}}		{{orf}}

ArrowHead294 at 00:52, 16 November 2024

2024-11-16T00:52:55Z

← Older revision		Revision as of 00:52, 16 November 2024
Line 7:		Line 7:
	1665edo provides the [[optimal patent val]] for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.		1665edo provides the [[optimal patent val]] for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.

	1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.		The 1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 18:		Line 18:
	=== 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
	\|-		\|-
Line 33:		Line 33:
	\|-		\|-
	\| 37		\| 37
	\| 377\1665<br>(17\1665)		\| 377\1665<br />(17\1665)
	\| 271.711<br>(12.252)		\| 271.711<br />(12.252)
	\| 117/100<br>(?)		\| 117/100<br />(?)
	\| [[Dzelic]]		\| [[Dzelic]]
	\|-		\|-
	\| 45		\| 45
	\| 1301\1665<br>(6\1665)		\| 1301\1665<br />(6\1665)
	\| 937.657<br>(4.324)		\| 937.657<br />(4.324)
	\| 55/32<br>(?)		\| 55/32<br />(?)
	\| [[Rhodium]]		\| [[Rhodium]]
	\|}		\|}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

FloraC: +subsets and supersets

2023-10-30T13:42:50Z

+subsets and supersets

← Older revision		Revision as of 13:42, 30 October 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	1665edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.		1665edo is a very strong 5-limit (as well as 2.3.5.11 [[subgroup]]) tuning and it is [[consistent]] in the [[15-odd-limit]]. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.

	1665edo provides the optimal patent val for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.		1665edo provides the [[optimal patent val]] for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.

	1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.		1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.
Line 11:		Line 11:
	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|1665}}		{{Harmonics in equal\|1665}}

			=== Subsets and supersets ===
			Since 1665 factors into {{factorization\|1665}}, 1665edo has subset edos {{EDOs\| 3, 5, 9, 15, 37, 45, 111, 185, 333, and 555 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
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	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+Table of rank-2 temperaments by generator		\|+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
	\|-		\|-

FloraC: Cleanup; clarify the title row of the rank-2 temp table

2023-10-15T14:48:54Z

Cleanup; clarify the title row of the rank-2 temp table

← Older revision		Revision as of 14:48, 15 October 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|1665}}		{{EDO intro\|1665}}
	==Theory==
			== Theory ==
	1665edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.		1665edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.

Line 7:		Line 8:

	1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.		1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.
	~~===Prime harmonics===~~
	~~{{harmonics in equal\|1665}}~~

	==Regular temperament properties==		=== Prime harmonics ===
			{{Harmonics in equal\|1665}}

			== Regular temperament properties ==
	=== 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		\|+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
Line 23:		Line 25:
	\| 127\1665		\| 127\1665
	\| 91.531		\| 91.531
	\| {{monzo\|9 -32 18}}		\| {{monzo\| 9 -32 18 }}
	\| [[Gross]]		\| [[Gross]]
	\|-		\|-
Line 38:		Line 40:
	\| [[Rhodium]]		\| [[Rhodium]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

BudjarnLambeth at 06:56, 9 July 2023

2023-07-09T06:56:27Z

← Older revision		Revision as of 06:56, 9 July 2023
Line 1:		Line 1:
	~~{{novelty}}{{stub}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|1665}}		{{EDO intro\|1665}}
	==Theory==		==Theory==

BudjarnLambeth at 05:00, 9 July 2023

2023-07-09T05:00:11Z

← Older revision		Revision as of 05:00, 9 July 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{novelty}}{{stub}}{{Infobox ET}}
	{{EDO intro\|1665}}		{{EDO intro\|1665}}
	==Theory==		==Theory==