28ed5 - Revision history

ArrowHead294 at 13:26, 10 June 2025

2025-06-10T13:26:16Z

← Older revision		Revision as of 13:26, 10 June 2025
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	* [[40ed10]] – relative ed10		* [[40ed10]] – relative ed10
	* [[42ed11]] – relative ed11		* [[42ed11]] – relative ed11
			* [[76ed80]] – close to the zeta-optimized tuning for 12edo
	* [[1ed18/17\|AS18/17]] – relative [[AS\|ambitonal sequence]]		* [[1ed18/17\|AS18/17]] – relative [[AS\|ambitonal sequence]]

ArrowHead294 at 13:15, 10 June 2025

2025-06-10T13:15:18Z

← Older revision		Revision as of 13:15, 10 June 2025
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	== Theory ==		== Theory ==
	28ed5 is related to [[12edo]], but with the 5/1 rather than the 2/1 being just. ~~The~~ octave ~~is about~~ 5.8656 ~~cents compressed and the step size is about 99.5112 cents~~. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12edo, as well as 18/17, 19/16, and 24/17.		28ed5 is related to [[12edo]], but with the 5/1 rather than the 2/1 being just. This compresses the octave by 5.8656{{c}}, a small but significant deviation. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12edo, as well as 18/17, 19/16, and 24/17.

	=== Harmonics ===		=== Harmonics ===
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	* [https://sevish.com/scaleworkshop/index.htm?name=28ed5&data=99.5112040666012%0A199.0224081332025%0A298.5336121998037%0A398.0448162664050%0A497.5560203330062%0A597.0672243996075%0A696.5784284662087%0A796.0896325328099%0A895.6008365994112%0A995.1120406660124%0A1094.6232447326137%0A1194.1344487992149%0A1293.6456528658162%0A1393.1568569324174%0A1492.6680609990187%0A1592.1792650656199%0A1691.6904691322211%0A1791.2016731988224%0A1890.7128772654236%0A1990.2240813320249%0A2089.7352853986261%0A2189.2464894652274%0A2288.7576935318286%0A2388.2688975984298%0A2487.7801016650311%0A2587.2913057316323%0A2686.8025097982336%0A2786.3137138648348&freq=220&midi=57&vert=10&horiz=1 Play 28ed5] – Scale Workshop		* [https://sevish.com/scaleworkshop/index.htm?name=28ed5&data=99.5112040666012%0A199.0224081332025%0A298.5336121998037%0A398.0448162664050%0A497.5560203330062%0A597.0672243996075%0A696.5784284662087%0A796.0896325328099%0A895.6008365994112%0A995.1120406660124%0A1094.6232447326137%0A1194.1344487992149%0A1293.6456528658162%0A1393.1568569324174%0A1492.6680609990187%0A1592.1792650656199%0A1691.6904691322211%0A1791.2016731988224%0A1890.7128772654236%0A1990.2240813320249%0A2089.7352853986261%0A2189.2464894652274%0A2288.7576935318286%0A2388.2688975984298%0A2487.7801016650311%0A2587.2913057316323%0A2686.8025097982336%0A2786.3137138648348&freq=220&midi=57&vert=10&horiz=1 Play 28ed5] – Scale Workshop
	* [http://terpstrakeyboard.com/web-app/keys.htm?fundamental=220&right=2&upright=1&size=25&rotation=13.897886248013985&instrument=sawtooth&enum=false&spectrum_colors=false&no_labels=false&scale=!%2028ed5.scl%0A!%20%0A28ed5%0A28%0A!%0A99.5112040666012%0A199.0224081332025%0A298.5336121998037%0A398.0448162664050%0A497.5560203330062%0A597.0672243996075%0A696.5784284662087%0A796.0896325328099%0A895.6008365994112%0A995.1120406660124%0A1094.6232447326137%0A1194.1344487992149%0A1293.6456528658162%0A1393.1568569324174%0A1492.6680609990187%0A1592.1792650656199%0A1691.6904691322211%0A1791.2016731988224%0A1890.7128772654236%0A1990.2240813320249%0A2089.7352853986261%0A2189.2464894652274%0A2288.7576935318286%0A2388.2688975984298%0A2487.7801016650311%0A2587.2913057316323%0A2686.8025097982336%0A2786.3137138648348&names=A%0AA%23%2FBb%0AB%0AC%0AC%23%2FDb%0AD%0AD%23%2FEb%0AE%0AE%23%2FFb%0AF%0AG%0AG%23%2FHb%0AH%0AH%23%2FIb%0AI%0AI%23%2FJb%0AJ%0AK%0AK%23%2FLb%0AL%0AL%23%2FMb%0AM%0AM%23%2FNb%0AN%0AO%0AO%23%2FPb%0AP%0AP%23%2FAb&note_colors=ffffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b Play 28ed5] – Terpstra Keyboard WebApp		* [http://terpstrakeyboard.com/web-app/keys.htm?fundamental=220&right=2&upright=1&size=25&rotation=13.897886248013985&instrument=sawtooth&enum=false&spectrum_colors=false&no_labels=false&scale=!%2028ed5.scl%0A!%20%0A28ed5%0A28%0A!%0A99.5112040666012%0A199.0224081332025%0A298.5336121998037%0A398.0448162664050%0A497.5560203330062%0A597.0672243996075%0A696.5784284662087%0A796.0896325328099%0A895.6008365994112%0A995.1120406660124%0A1094.6232447326137%0A1194.1344487992149%0A1293.6456528658162%0A1393.1568569324174%0A1492.6680609990187%0A1592.1792650656199%0A1691.6904691322211%0A1791.2016731988224%0A1890.7128772654236%0A1990.2240813320249%0A2089.7352853986261%0A2189.2464894652274%0A2288.7576935318286%0A2388.2688975984298%0A2487.7801016650311%0A2587.2913057316323%0A2686.8025097982336%0A2786.3137138648348&names=A%0AA%23%2FBb%0AB%0AC%0AC%23%2FDb%0AD%0AD%23%2FEb%0AE%0AE%23%2FFb%0AF%0AG%0AG%23%2FHb%0AH%0AH%23%2FIb%0AI%0AI%23%2FJb%0AJ%0AK%0AK%23%2FLb%0AL%0AL%23%2FMb%0AM%0AM%23%2FNb%0AN%0AO%0AO%23%2FPb%0AP%0AP%23%2FAb&note_colors=ffffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b%0Affffff%0Affffff%0A7b7b7b%0Affffff%0A7b7b7b Play 28ed5] – Terpstra Keyboard WebApp

			[[Category:12edo]]

FloraC: /* Intervals */ oops

2025-01-15T15:12:51Z

Intervals: oops

← Older revision		Revision as of 15:12, 15 January 2025
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	\|-		\|-
	\| 7		\| 7
	\| 696.~~5784~~		\| 696.6
	\| [[3/2]]		\| [[3/2]]
	\|-		\|-

FloraC: /* Theory */ +subsets and supersets

2025-01-15T15:11:02Z

Theory: +subsets and supersets

← Older revision		Revision as of 15:11, 15 January 2025
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	{{Harmonics in equal\|28\|5\|1}}		{{Harmonics in equal\|28\|5\|1}}
	{{Harmonics in equal\|28\|5\|1\|start=12\|columns=12\|collapsed=true\|title=Approximation of harmonics in 28ed5 (continued)}}		{{Harmonics in equal\|28\|5\|1\|start=12\|columns=12\|collapsed=true\|title=Approximation of harmonics in 28ed5 (continued)}}

			=== Subsets and supersets ===
			Since 28 factors into 2<sup>2</sup> × 7, 28ed5 has subset ed5's {{EDs\|equave=5\| 2, 4, 7, and 14 }}.

	== Intervals ==		== Intervals ==

FloraC: Cleanup

2025-01-15T14:55:17Z

Cleanup

Show changes

BudjarnLambeth: {{todo|expand}}

2024-12-22T04:04:56Z

← Older revision		Revision as of 04:04, 22 December 2024
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	[[Category:Ed5]]		[[Category:Ed5]]
	[[Category:Edonoi]]		[[Category:Edonoi]]
			{{todo\|expand}}

BudjarnLambeth: Harmonics

2024-10-06T01:22:29Z

Harmonics

@@ Line 2: / Line 2: @@
 '''[[Ed5|Division of the 5th harmonic]] into 28 equal parts''' (28ED5) is related to [[12edo|12EDO]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 99.5112 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12EDO, as well as 18/17, 19/16, and 24/17.
 {| class="wikitable"
 |-
@@ Line 154: / Line 155: @@
 | | just major third plus two octaves
 |}
 == Regular temperaments ==

Xenllium at 03:37, 25 November 2023

2023-11-25T03:37:14Z

← Older revision		Revision as of 03:37, 25 November 2023
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	\| \| 7		\| \| 7
	\| \| 696.5784		\| \| 696.5784
	\| \|		\| \| [[175/117]], [[323/216]]
	\| \| meantone fifth <br>(pseudo-[[3/2]])		\| \| meantone fifth <br>(pseudo-[[3/2]])
	\|-		\|-
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	\| \| 17		\| \| 17
	\| \| 1691.6905		\| \| 1691.6905
	\| \| 85/32		\| \| [[85/64\|85/32]]
	\| \|		\| \|
	\|-		\|-
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	\| \| 21		\| \| 21
	\| \| 2089.7353		\| \| 2089.7353
	\| \|		\| \| 117/35
	\| \| meantone major sixth plus an octave <br>(pseudo-[[10/3]])		\| \| meantone major sixth plus an octave <br>(pseudo-[[10/3]])
	\|-		\|-

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T19:04:14Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 19:04, 7 May 2023
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	{{Main\| Quindromeda family }}		{{Main\| Quindromeda family }}

	28ed5 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[subgroup temperament]] which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quindromeda'' temperament). This temperament is supported by {{~~Val list~~\| 12, 169, 181, 193, 205, 217, 229 }}, and [[241edo\|241]] EDOs.		28ed5 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[subgroup temperament]] which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quindromeda'' temperament). This temperament is supported by {{Optimal ET sequence\| 12, 169, 181, 193, 205, 217, 229 }}, and [[241edo\|241]] EDOs.

	Equating 225/224 with 256/255 leads ''[[quintakwai]]'' (12&193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads ''[[quintagar]]'' (12&217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.		Equating 225/224 with 256/255 leads ''[[quintakwai]]'' (12&193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads ''[[quintagar]]'' (12&217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.

Plumtree: Infobox ET added

2022-10-05T20:40:58Z

Infobox ET added

← Older revision		Revision as of 20:40, 5 October 2022
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			{{Infobox ET}}
	'''[[Ed5\|Division of the 5th harmonic]] into 28 equal parts''' (28ED5) is related to [[12edo\|12EDO]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 99.5112 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12EDO, as well as 18/17, 19/16, and 24/17.		'''[[Ed5\|Division of the 5th harmonic]] into 28 equal parts''' (28ED5) is related to [[12edo\|12EDO]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 99.5112 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12EDO, as well as 18/17, 19/16, and 24/17.

← Older revision		Revision as of 03:37, 25 November 2023
Line 46:		Line 46:
	\| \| 7		\| \| 7
	\| \| 696.5784		\| \| 696.5784
	\| \|		\| \| [[175/117]], [[323/216]]
	\| \| meantone fifth <br>(pseudo-[[3/2]])		\| \| meantone fifth <br>(pseudo-[[3/2]])
	\|-		\|-
Line 96:		Line 96:
	\| \| 17		\| \| 17
	\| \| 1691.6905		\| \| 1691.6905
	\| \| 85/32		\| \| [[85/64\|85/32]]
	\| \|		\| \|
	\|-		\|-
Line 116:		Line 116:
	\| \| 21		\| \| 21
	\| \| 2089.7353		\| \| 2089.7353
	\| \|		\| \| 117/35
	\| \| meantone major sixth plus an octave <br>(pseudo-[[10/3]])		\| \| meantone major sixth plus an octave <br>(pseudo-[[10/3]])
	\|-		\|-