416edo - Revision history

ArrowHead294 at 12:16, 21 February 2025

2025-02-21T12:16:29Z

← Older revision		Revision as of 12:16, 21 February 2025
Line 15:		Line 15:
	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
			\|-
	! rowspan="2" \| [[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|Comma List]]
Line 27:		Line 28:
	\| {{monzo\| 1319 -416 }}		\| {{monzo\| 1319 -416 }}
	\| {{mapping\| 416 1319 }}		\| {{mapping\| 416 1319 }}
	\| -0.1416		\| −0.1416
	\| 0.1416		\| 0.1416
	\| 4.91		\| 4.91
Line 34:		Line 35:
	\| {{monzo\| 56 -14 -5 }}, {{monzo\| -5 -16 24 }}		\| {{monzo\| 56 -14 -5 }}, {{monzo\| -5 -16 24 }}
	\| {{mapping\| 416 1319 966 }}		\| {{mapping\| 416 1319 966 }}
	\| -0.1267		\| −0.1267
	\| 0.1175		\| 0.1175
	\| 4.07		\| 4.07
Line 41:		Line 42:
	\| 420175/419904, 102760448/102515625, {{monzo\| 14 -6 7 -4 }}		\| 420175/419904, 102760448/102515625, {{monzo\| 14 -6 7 -4 }}
	\| {{mapping\| 416 1319 966 1168 }}		\| {{mapping\| 416 1319 966 1168 }}
	\| -0.1310		\| −0.1310
	\| 0.1021		\| 0.1021
	\| 3.54		\| 3.54
Line 48:		Line 49:
	\| 5632/5625, 9801/9800, 41503/41472, 774400000/771895089		\| 5632/5625, 9801/9800, 41503/41472, 774400000/771895089
	\| {{mapping\| 416 1319 966 1168 1439 }}		\| {{mapping\| 416 1319 966 1168 1439 }}
	\| -0.0842		\| −0.0842
	\| 0.1308		\| 0.1308
	\| 4.53		\| 4.53

Francium: changed EDO intro to ED intro

2025-02-21T06:22:50Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:22, 21 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|416~~}}		{{ED intro}}

	== Theory ==		== Theory ==

Francium: +music

2024-04-17T12:52:43Z

+music

← Older revision		Revision as of 12:52, 17 April 2024
Line 52:		Line 52:
	\| 4.53		\| 4.53
	\|}		\|}

			== Music ==
			; [[birdshite stalactite]]
			* [https://www.youtube.com/watch?v=nmUCjbfGpKU ''palate cleanser''] (2024) – tetracot[13] in 416edo tuning

FloraC: Review

2024-01-19T15:31:16Z

Review

← Older revision		Revision as of 15:31, 19 January 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	416et is consistent to the [[7-odd-limit]] ~~and~~ the [[harmonic]] [[3/1\|3]] is ~~about halfway its steps~~. ~~It is~~ suitable for ~~the~~ 2.9.5.7.11.19.23~~.29.31.37~~ [[subgroup]], tempering out [[1540/1539]], [[5632/5625]], [[9801/9800]], 10241/10240, ~~1045~~/~~1044~~, ~~26125~~/~~26082~~, ~~46000~~/~~45927~~, ~~17600/17577 and 1036/1035. It~~ [[support]]s [[tridecatonic]], [[fermionic~~]], [[embankment]] and [[polder~~]].		416et is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1\|3]] is quite large. Nonetheless, it gives reasonable approximations to harmonics [[5/1\|5]], [[7/1\|7]], [[9/1\|9]], [[11/1\|11]], [[19/1\|19]], and [[23/1\|23]], making it suitable for a 2.9.5.7.11.19.23 [[subgroup]] interpretation, where it notably [[tempering out\|tempers out]] [[1331/1330]], [[1540/1539]], [[5632/5625]], [[9801/9800]], and 10241/10240.

			Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, [[support]]ing [[tridecatonic]] and [[fermionic]].

	=== Odd harmonics ===		=== Odd harmonics ===
Line 9:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	416 factors into 2<sup>5</sup> × 13, ~~with~~ subset edos {{EDOs\|2, 4, 8, 13, 16, 26, 32, 52, 104, and 208}}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.		Since 416 factors into 2<sup>5</sup> × 13, 416edo subset edos {{EDOs\| 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208 }}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.

	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \|[[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \|[[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|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
	\|-		\|-
	![[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\|2.9		\| 2.9
	\|{{monzo\|1319 -416}}		\| {{monzo\| 1319 -416 }}
	\|{{mapping\|416 1319}}		\| {{mapping\| 416 1319 }}
	\| -0.1416		\| -0.1416
	\| 0.1416		\| 0.1416
	\| 4.91		\| 4.91
	\|-		\|-
	\|2.9.5		\| 2.9.5
	\|{{monzo\|56 -14 -5}}, {{monzo\|-5 -16 24}}		\| {{monzo\| 56 -14 -5 }}, {{monzo\| -5 -16 24 }}
	\|{{mapping\|416 1319 966}}		\| {{mapping\| 416 1319 966 }}
	\| -0.1267		\| -0.1267
	\| 0.1175		\| 0.1175
	\| 4.07		\| 4.07
	\|-		\|-
	\|2.9.5.7		\| 2.9.5.7
	\|420175/419904, 102760448/102515625, ~~1280000000/1275989841~~		\| 420175/419904, 102760448/102515625, {{monzo\| 14 -6 7 -4 }}
	\|{{mapping\|416 1319 966 1168}}		\| {{mapping\| 416 1319 966 1168 }}
	\| -0.1310		\| -0.1310
	\| 0.1021		\| 0.1021
	\| 3.54		\| 3.54
	\|-		\|-
	\|2.9.5.7.11		\| 2.9.5.7.11
	\|5632/5625, 9801/9800, 41503/41472, 774400000/771895089		\| 5632/5625, 9801/9800, 41503/41472, 774400000/771895089
	\|{{mapping\|416 1319 966 1168 1439}}		\| {{mapping\| 416 1319 966 1168 1439 }}
	\| -0.0842		\| -0.0842
	\| 0.1308		\| 0.1308
	\| 4.53		\| 4.53
	\|}		\|}

Francium: /* Regular temperament properties */ typo

2024-01-15T17:25:01Z

Regular temperament properties: typo

← Older revision		Revision as of 17:25, 15 January 2024
Line 30:		Line 30:
	\|-		\|-
	\|2.9.5		\|2.9.5
	{{monzo\|56 -14 -5}}, {{monzo\|-5 -16 24}}		\|{{monzo\|56 -14 -5}}, {{monzo\|-5 -16 24}}
	\|{{mapping\|416 1319 966}}		\|{{mapping\|416 1319 966}}
	\| -0.1267		\| -0.1267

Francium: Created page with "{{Infobox ET}} {{EDO intro|416}} == Theory == 416et is consistent to the 7-odd-limit and the harmonic 3 is about halfway its steps. It is suitable for the 2.9..."

2024-01-15T17:24:35Z

Created page with "{{Infobox ET}} {{EDO intro|416}} == Theory == 416et is consistent to the 7-odd-limit and the harmonic 3 is about halfway its steps. It is suitable for the 2.9..."

New page

{{Infobox ET}}
{{EDO intro|416}}

== Theory ==
416et is consistent to the [[7-odd-limit]] and the [[harmonic]] [[3/1|3]] is about halfway its steps. It is suitable for the 2.9.5.7.11.19.23.29.31.37 [[subgroup]], tempering out [[1540/1539]], [[5632/5625]], [[9801/9800]], 10241/10240, 1045/1044, 26125/26082, 46000/45927, 17600/17577 and 1036/1035. It [[support]]s [[tridecatonic]], [[fermionic]], [[embankment]] and [[polder]].

=== Odd harmonics ===
{{Harmonics in equal|416}}

=== Subsets and supersets ===
416 factors into 2<sup>5</sup> × 13, with subset edos {{EDOs|2, 4, 8, 13, 16, 26, 32, 52, 104, and 208}}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! 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]] (%)
|-
|2.9
|{{monzo|1319 -416}}
|{{mapping|416 1319}}
| -0.1416
| 0.1416
| 4.91
|-
|2.9.5
{{monzo|56 -14 -5}}, {{monzo|-5 -16 24}}
|{{mapping|416 1319 966}}
| -0.1267
| 0.1175
| 4.07
|-
|2.9.5.7
|420175/419904, 102760448/102515625, 1280000000/1275989841
|{{mapping|416 1319 966 1168}}
| -0.1310
| 0.1021
| 3.54
|-
|2.9.5.7.11
|5632/5625, 9801/9800, 41503/41472, 774400000/771895089
|{{mapping|416 1319 966 1168 1439}}
| -0.0842
| 0.1308
| 4.53
|}