319edo - Revision history

Eliora: /* Theory */

2026-03-22T16:58:34Z

Theory

← Older revision		Revision as of 16:58, 22 March 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	319edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1\|3]] is about halfway its steps. The full 11-limit [[patent val]] is nonetheless a reasonable interpretation since the lower harmonics all tend sharp. Using this val, it [[tempering out\|tempers out]] [[6144/6125]], [[10976/10935]], and {{monzo\| 9 -10 9 -5 }} in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 15488/15435, 59290/59049, and [[65536/65219]] in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].		319edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1\|3]] is about halfway its steps. Nonetheless, it provides the optimal patent val for 5-limit [[mystery]] temperament, which tempers out the [[29-comma]], despite poor 5-limit harmonic approximation.

			The full 11-limit [[patent val]] is nonetheless a reasonable interpretation since the lower harmonics all tend sharp. Using this val, it [[tempering out\|tempers out]] [[6144/6125]], [[10976/10935]], and {{monzo\| 9 -10 9 -5 }} in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 15488/15435, 59290/59049, and [[65536/65219]] in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].

	If we instead adopt the 2.9.… [[subgroup]] interpretation, then 2.9.15.21 is a good subgroup to start with.		If we instead adopt the 2.9.… [[subgroup]] interpretation, then 2.9.15.21 is a good subgroup to start with.

ArrowHead294 at 12:31, 21 February 2025

2025-02-21T12:31:25Z

← Older revision		Revision as of 12:31, 21 February 2025
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	== 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]]
	! 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]] (%)

Francium: changed EDO intro to ED intro

2025-02-21T06:13:37Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:13, 21 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|319~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: Review

2024-01-12T13:43:01Z

Review

← Older revision		Revision as of 13:43, 12 January 2024
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	{{EDO intro\|319}}		{{EDO intro\|319}}

	==Theory==		== Theory ==
	~~319et~~ is consistent to the [[7-odd-limit]] ~~and the~~ [[harmonic]] 3 is about halfway its steps. Using ~~the patent~~ val, it tempers out ~~156250000/155649627, 1220703125/1219784832,~~ [[6144/6125]], [[10976/10935]] and ~~420175/419904~~ in the 7-limit; ~~95703125~~/~~95664294, 161280/161051, 35156250/35153041~~, [[4000/3993]]~~, 117649/117612, 1296000/1294139~~, [[6250/6237]]~~, 107495424/107421875, 4302592/4296875, 825000/823543, 422576/421875~~, 15488/15435~~, [[3025/3024]]~~, 59290/59049, ~~766656/765625, 456533/455625, 202397184/201768035, 3294225/3294172, 585640/583443~~ and ~~644204~~/~~643125~~ in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].		319edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1\|3]] is about halfway its steps. The full 11-limit [[patent val]] is nonetheless a reasonable interpretation since the lower harmonics all tend sharp. Using this val, it [[tempering out\|tempers out]] [[6144/6125]], [[10976/10935]], and {{monzo\| 9 -10 9 -5 }} in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 15488/15435, 59290/59049, and [[65536/65219]] in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].

	===Odd harmonics===		If we instead adopt the 2.9.… [[subgroup]] interpretation, then 2.9.15.21 is a good subgroup to start with.

			=== Odd harmonics ===
	{{Harmonics in equal\|319}}		{{Harmonics in equal\|319}}

	===Subsets and supersets===		=== Subsets and supersets ===
	319 factors into 11 × 29, ~~with~~ [[11edo]] and [[29edo]] as its ~~subset edos~~. [[638edo]], which doubles it, gives a good correction to the ~~harmonic~~ 3.		Since 319 factors into 11 × 29, 319edo has [[11edo]] and [[29edo]] as its subsets. [[638edo]], which doubles it, gives a good correction to the harmonics 3, 5, and 7.

	==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\|-1011 319}}		\| {{monzo\| -1011 319 }}
	\|{{mapping\|319 1011}}		\| {{mapping\| 319 1011 }}
	\| +0.1223		\| +0.1223
	\| 0.1223		\| 0.1223
	\| 3.25		\| 3.25
	\|-		\|-
	\|2.9.5		\| 2.9.15
	\|~~32805/32768~~, {{monzo\|~~54 35~~ -71}}		\| {{monzo\| -51 5 9 }}, {{monzo\| -16 26 -17 }}
	\|{{mapping\|319 1011 ~~741~~}}		\| {{mapping\| 319 1011 1246 }}
	\| -0.~~0832~~		\| +0.1869
	\| 0.~~3072~~		\| 0.1353
	\| 8.17		\| 3.60
	\|}		\|}

Francium: Created page with "{{Infobox ET}} {{EDO intro|319}} ==Theory== 319et is consistent to the 7-odd-limit and the harmonic 3 is about halfway its steps. Using the patent val, it tempers out..."

2023-12-31T10:45:37Z

Created page with "{{Infobox ET}} {{EDO intro|319}} ==Theory== 319et is consistent to the 7-odd-limit and the harmonic 3 is about halfway its steps. Using the patent val, it tempers out..."

New page

{{Infobox ET}}
{{EDO intro|319}}

==Theory==
319et is consistent to the [[7-odd-limit]] and the [[harmonic]] 3 is about halfway its steps. Using the patent val, it tempers out 156250000/155649627, 1220703125/1219784832, [[6144/6125]], [[10976/10935]] and 420175/419904 in the 7-limit; 95703125/95664294, 161280/161051, 35156250/35153041, [[4000/3993]], 117649/117612, 1296000/1294139, [[6250/6237]], 107495424/107421875, 4302592/4296875, 825000/823543, 422576/421875, 15488/15435, [[3025/3024]], 59290/59049, 766656/765625, 456533/455625, 202397184/201768035, 3294225/3294172, 585640/583443 and 644204/643125 in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].

===Odd harmonics===
{{Harmonics in equal|319}}

===Subsets and supersets===
319 factors into 11 × 29, with [[11edo]] and [[29edo]] as its subset edos. [[638edo]], 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|-1011 319}}
|{{mapping|319 1011}}
| +0.1223
| 0.1223
| 3.25
|-
|2.9.5
|32805/32768, {{monzo|54 35 -71}}
|{{mapping|319 1011 741}}
| -0.0832
| 0.3072
| 8.17
|}

← Older revision		Revision as of 12:31, 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]]
	! 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]] (%)