1178edo - Revision history

FloraC: /* Regular temperament properties */ + 23-limit and records

2026-07-13T10:27:29Z

Regular temperament properties: + 23-limit and records

← Older revision		Revision as of 10:27, 13 July 2026
Line 74:		Line 74:
	\| 0.0318		\| 0.0318
	\| 3.12		\| 3.12
			\|-
			\| 2.3.5.7.11.13.17.19.23
			\| 2025/2024, 2500/2499, 3025/3024, 3060/3059, 3250/3249, 3520/3519, 4200/4199, 4375/4374
			\| {{Mapping\| 1178 1867 2735 3307 4075 4359 4815 5004 5329 }}
			\| +0.0292
			\| 0.0436
			\| 4.28
	\|}		\|}
			* 1178et holds the record of lowest relative error in the 19-limit, being the first to beat [[742edo\|742]] in relative error and [[954edo\|954h]] in absolute error, before getting superseded by [[1578edo\|1578]] in both.
			* It holds the record of lowest absolute errors in the 13-, 17-, and 23-limit, after 954, 954, [[1106edo\|1106]] and before [[1236edo\|1236]], 1236, and [[1308edo\|1308]], respectively.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

FloraC: Expand

2026-07-13T10:10:01Z

Expand

← Older revision		Revision as of 10:10, 13 July 2026
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	== Theory ==		== Theory ==
	1178edo is a very strong 19-limit system, ~~and is a [[zeta edo\|zeta peak, integral and gap edo]]. It is also~~ [[consistency\|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo\|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. A [[~~comma basis~~\|~~basis~~]] ~~for its~~ 19-limit [[~~comma~~]]s ~~consists of~~ [[~~2500/2499~~]], [[3025/3024]], ~~3250~~/~~3249~~, ~~4200~~/~~4199~~, [[4225/4224]], [[~~4375~~/~~4374~~]], ~~and~~ [[4914/4913]]~~. It~~ [[~~support~~]]s and ~~provides a great tuning for~~ [[~~semihemienneadecal~~]].		1178edo is a very strong [[19-limit]] system, [[consistency\|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo\|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is a [[zeta peak edo\|zeta peak]], [[zeta integral edo\|integral]] and [[zeta gap edo\|gap edo]].

			As an equal temperament, it tempers out {{monzo\| -14 -19-19 }} ([[enneadeca]]) in the [[5-limit]], 4375/4374 ([[ragisma]]) and 703125/702464 ([[meter]]) in the [[7-limit]], so that it [[support]]s [[enneadecal]]. In the [[11-limit]] it tempers out [[3025/3024]], [[9801/9800]], and [[234375/234256]], supporting [[hemienneadecal]], and in the [[13-limit]] [[4225/4224]] and [[10648/10647]], supporting and providing a great tuning for [[semihemienneadecal]]. It further tempers out [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]] among others in the [[23-limit]].

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|1178\|columns=11}}		{{Harmonics in equal\|1178\|columns=11}}
			{{Harmonics in equal\|1178\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 1178edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 1178 factors into {{~~factorization~~\|~~1178~~}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs\| 2, 19, 31, 38, 62, and 589 }}.		Since 1178 factors into primes as {{nowrap\| 2 × 19 × 31 }}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs\| 2, 19, 31, 38, 62, and 589 }}.

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

FloraC: /* Regular temperament properties */ convert the table of rank-2 temps to minimal-generator form

2026-07-13T09:53:34Z

Regular temperament properties: convert the table of rank-2 temps to minimal-generator form

← Older revision		Revision as of 09:53, 13 July 2026
Line 90:		Line 90:
	\|-		\|-
	\| 19		\| 19
	\| ~~489\1178<br>(~~7\1178)		\| 7\1178
	\| ~~498.13<br>(~~7.13)		\| 7.13
	\| ~~4/3<br>(~~225/224)		\| 225/224
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-
	~~\| 31~~
	~~\| 581\1178<br>(11\1178)~~
	~~\| 591.851<br>(11.205)~~
	~~\| 936/665<br>(?)~~
	~~\| [[31st-octave temperaments#217 & 1178\|217 & 1178]]~~
	\|-		\|-
	\| 38		\| 38
	\| ~~260\1178<br>(~~12\1178)		\| 12\1178
	\| ~~264.86<br>(~~12.22)		\| 12.22
	\| ~~500/429<br>(~~144/143)		\| 144/143
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|-		\|-
	\| 38		\| 38
	\| ~~489\1178<br>(~~7\1178)		\| 7\1178
	\| ~~498.13<br>(~~7.13)		\| 7.13
	\| ~~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]]

	== Music ==		== Music ==

FloraC: /* Regular temperament properties */ comma basis

2026-07-13T09:51:25Z

Regular temperament properties: comma basis

← Older revision		Revision as of 09:51, 13 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		\| 2.3
	\| {{~~monzo~~\| -1867 1178 }}		\| {{Monzo\| -1867 1178 }}
	\| {{~~mapping~~\| 1178 1867 }}		\| {{Mapping\| 1178 1867 }}
	\| +0.0276		\| +0.0276
	\| 0.0276		\| 0.0276
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	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| -14 -19-19 }}, {{monzo\| -99 61 1 }}		\| {{Monzo\| -14 -19-19 }}, {{monzo\| -99 61 1 }}
	\| {{~~mapping~~\| 1178 1867 2735 }}		\| {{Mapping\| 1178 1867 2735 }}
	\| +0.0522		\| +0.0522
	\| 0.0415		\| 0.0415
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	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 703125/702464, {{monzo\| -52 -5 -2 23 }}		\| 4375/4374, 703125/702464, {{monzo\| -52 -5 -2 23 }}
	\| {{~~mapping~~\| 1178 1867 2735 3307 }}		\| {{Mapping\| 1178 1867 2735 3307 }}
	\| +0.0450		\| +0.0450
	\| 0.0380		\| 0.0380
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 234375/234256, {{monzo\| -27 3 -4 10 1 }}		\| 3025/3024, 4375/4374, 234375/234256, {{monzo\| -27 3 -4 10 1 }}
	\| {{~~mapping~~\| 1178 1867 2735 3307 4075 }}		\| {{Mapping\| 1178 1867 2735 3307 4075 }}
	\| +0.0484		\| +0.0484
	\| 0.0347		\| 0.0347
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	\|-		\|-
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 3025/3024, 4225/4224, 4375/4374, 78125/78078, ~~1664000~~/~~1663893~~		\| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 655473/655360
	\| {{~~mapping~~\| 1178 1867 2735 3307 4075 4359 }}		\| {{Mapping\| 1178 1867 2735 3307 4075 4359 }}
	\| +0.0457		\| +0.0457
	\| 0.0322		\| 0.0322
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872		\| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
	\| {{~~mapping~~\| 1178 1867 2735 3307 4075 4359 4815 }}		\| {{Mapping\| 1178 1867 2735 3307 4075 4359 4815 }}
	\| +0.0403		\| +0.0403
	\| 0.0327		\| 0.0327
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	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913		\| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
	\| {{~~mapping~~\| 1178 1867 2735 3307 4075 4359 4815 5004 }}		\| {{Mapping\| 1178 1867 2735 3307 4075 4359 4815 5004 }}
	\| +0.0370		\| +0.0370
	\| 0.0318		\| 0.0318
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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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	\|-		\|-
	\| 19		\| 19
	\| 489\1178<br />(7\1178)		\| 489\1178<br>(7\1178)
	\| 498.13<br />(7.13)		\| 498.13<br>(7.13)
	\| 4/3<br />(225/224)		\| 4/3<br>(225/224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-		\|-
	\| 31		\| 31
	\| 581\1178<br />(11\1178)		\| 581\1178<br>(11\1178)
	\| 591.851<br />(11.205)		\| 591.851<br>(11.205)
	\| 936/665<br />(?)		\| 936/665<br>(?)
	\| [[31st-octave temperaments#217 & 1178\|217 & 1178]]		\| [[31st-octave temperaments#217 & 1178\|217 & 1178]]
	\|-		\|-
	\| 38		\| 38
	\| 260\1178<br />(12\1178)		\| 260\1178<br>(12\1178)
	\| 264.86<br />(12.22)		\| 264.86<br>(12.22)
	\| 500/429<br />(144/143)		\| 500/429<br>(144/143)
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|-		\|-
	\| 38		\| 38
	\| 489\1178<br />(7\1178)		\| 489\1178<br>(7\1178)
	\| 498.13<br />(7.13)		\| 498.13<br>(7.13)
	\| 4/3<br />(225/224)		\| 4/3<br>(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/>* [[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

	== Music ==		== Music ==

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"

2026-03-13T13:32:43Z

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 113:		Line 113:
	\| [[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

	== Music ==		== Music ==

ArrowHead294 at 16:53, 18 February 2025

2025-02-18T16:53:05Z

← Older revision		Revision as of 16:53, 18 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|1178~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:43, 15 January 2025

2025-01-15T19:43:13Z

← Older revision		Revision as of 19:43, 15 January 2025
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 1178 = {{factorization\|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs\| 2, 19, 31, 38, 62, and 589 }}.		Since 1178 factors into {{factorization\|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs\| 2, 19, 31, 38, 62, and 589 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 61:		Line 61:
	\| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872		\| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
	\| {{mapping\| 1178 1867 2735 3307 4075 4359 4815 }}		\| {{mapping\| 1178 1867 2735 3307 4075 4359 4815 }}
	\| 0.0403		\| +0.0403
	\| 0.0327		\| 0.0327
	\| 3.21		\| 3.21
Line 68:		Line 68:
	\| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913		\| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
	\| {{mapping\| 1178 1867 2735 3307 4075 4359 4815 5004 }}		\| {{mapping\| 1178 1867 2735 3307 4075 4359 4815 5004 }}
	\| 0.0370		\| +0.0370
	\| 0.0318		\| 0.0318
	\| 3.12		\| 3.12
Line 113:		Line 113:
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|}		\|}
	<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

	== Music ==		== Music ==
	; [[Eliora]]		; [[Eliora]]
	* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) &~~ndash~~; ~~217 &~~ 1178 and enneadecal in 1178edo tuning		* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap\|217 & 1178}} and enneadecal in 1178edo tuning

	[[Category:Enneadecal]]		[[Category:Enneadecal]]

ArrowHead294: Partial undo

2024-11-16T12:35:28Z

Partial undo

← Older revision		Revision as of 12:35, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.0318		\| 0.0318
	\| 3.12		\| 3.12
	~~{{comma basis end}~~}		\|}

	=== 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
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	\| 4/3<br />(225/224)		\| 4/3<br />(225/224)
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	~~{{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

	== Music ==		== Music ==

ArrowHead294 at 04:21, 16 November 2024

2024-11-16T04:21:04Z

← Older revision		Revision as of 04:21, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
Line 70:		Line 62:
	\| 0.0318		\| 0.0318
	\| 3.12		\| 3.12
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\|-		\|-
	\| 19		\| 19
	\| 489\1178<br>(7\1178)		\| 489\1178<br />(7\1178)
	\| 498.13<br>(7.13)		\| 498.13<br />(7.13)
	\| 4/3<br>(225/224)		\| 4/3<br />(225/224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-		\|-
	\| 31		\| 31
	\| 581\1178<br>(11\1178)		\| 581\1178<br />(11\1178)
	\| 591.851<br>(11.205)		\| 591.851<br />(11.205)
	\| 936/665<br>(?)		\| 936/665<br />(?)
	\| [[31st-octave temperaments#217 & 1178\|217 & 1178]]		\| [[31st-octave temperaments#217 & 1178\|217 & 1178]]
	\|-		\|-
	\| 38		\| 38
	\| 260\1178<br>(12\1178)		\| 260\1178<br />(12\1178)
	\| 264.86<br>(12.22)		\| 264.86<br />(12.22)
	\| 500/429<br>(144/143)		\| 500/429<br />(144/143)
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|-		\|-
	\| 38		\| 38
	\| 489\1178<br>(7\1178)		\| 489\1178<br />(7\1178)
	\| 498.13<br>(7.13)		\| 498.13<br />(7.13)
	\| 4/3<br>(225/224)		\| 4/3<br />(225/224)
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|}		{{rank-2 end}}
	~~<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}}

	== Music ==		== Music ==
	; [[Eliora]]		; [[Eliora]]
	* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 & 1178 and enneadecal in 1178edo tuning		* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 & 1178 and enneadecal in 1178edo tuning

	[[Category:Enneadecal]]		[[Category:Enneadecal]]
	[[Category:Hemienneadecal]]		[[Category:Hemienneadecal]]
			[[Category:Listen]]
	[[Category:Semihemienneadecal]]		[[Category:Semihemienneadecal]]
	~~[[Category:Listen]]~~

Eliora: /* Rank-2 temperaments */ style

2023-11-06T22:08:29Z

Rank-2 temperaments: style

← Older revision		Revision as of 22:08, 6 November 2023
Line 93:		Line 93:
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-		\|-
	\|31		\| 31
	\|581\1178		\| 581\1178<br>(11\1178)
	(11\1178)		\| 591.851<br>(11.205)
	\|591.851		\| 936/665<br>(?)
	(11.205)		\| [[31st-octave temperaments#217 & 1178\|217 & 1178]]
	\|936/665

	(?)
	\|[[217 & 1178]]
	\|-		\|-
	\| 38		\| 38