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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|317}}
{{ED intro}}


== Theory ==
== Theory ==
317et is only consistent to the [[5-odd-limit]] and the [[harmonic]] 3 is about halfway its steps. Using the patent val, it tempers out 589824/588245, [[16875/16807]], [[65625/65536]] and 49009212/48828125 in the 7-limit; 1835008/1830125, 14700/14641, 2097152/2096325, [[4000/3993]], 2734375/2725888, 1953125/1951488, 172032/171875, [[441/440]], 5767168/5764801, 825000/823543, 537109375/536870912, 134775333/134217728, 160083/160000, 16808715/16777216, 539055/537824 and 3294225/3294172 in the 11-limit.
317edo is only [[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway its steps. To start with, it can be considered in the 2.9.5.7 [[subgroup]], in which it is strong, [[tempering out]] [[420175/419904]], [[703125/702464]], and [[33554432/33480783]].  


===Odd harmonics===
Using the [[patent val]] nonetheless, the equal temperament tempers out 78732/78125 ([[sensipent comma]] in the 5-limit; [[16875/16807]], [[65625/65536]], 589824/588245, and 49009212/48828125 in the 7-limit. In the 11-limit, the 317e [[val]] tempers out [[540/539]], 1375/1372, and [[3025/3024]], whereas the patent val tempers out [[441/440]], [[4000/3993]], and 14700/14641.
 
=== Odd harmonics ===
{{Harmonics in equal|317}}
{{Harmonics in equal|317}}


===Subsets and supersets===
=== Subsets and supersets ===
317edo is the 66th [[prime edo]]. [[634edo]], which doubles it, gives a good correction to the harmonic 3.
317edo is the 66th [[prime edo]]. [[634edo]], 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" |[[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
! rowspan="2" | [[Subgroup]]
|{{monzo|1005 -317}}
! rowspan="2" | [[Comma list]]
|{{mapping|317 1005}}
! rowspan="2" | [[Mapping]]
| -0.0799
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo| 1005 -317 }}
| {{mapping| 317 1005 }}
| −0.0799
| 0.0799
| 0.0799
| 2.11
| 2.11
|-
|-
|2.9.5
| 2.9.5
|{{monzo|-53 5 16}}, {{monzo|33 -17 9}}
| {{monzo| -53 5 16 }}, {{monzo| 33 -17 9 }}
|{{mapping|317 1005 736}}
| {{mapping| 317 1005 736 }}
| -0.0254
| −0.0254
| 0.1009
| 0.1009
| 2.67
| 2.67
|-
|-
|2.9.5.7
| 2.9.5.7
|420175/419904, 703125/702464, 33554432/33480783
| 420175/419904, 703125/702464, 33554432/33480783
|{{mapping|317 1005 736 890}}
| {{mapping| 317 1005 736 890 }}
| -0.0422
| −0.0422
| 0.0921
| 0.0921
| 2.43
| 2.43
|-
|-
|2.9.5.7.11
| 2.9.5.7.11
|6250/6237, 12005/11979, 46656/46585, 151263/151250
| 6250/6237, 12005/11979, 46656/46585, 151263/151250
|{{mapping|317 1005 736 890 1097}}
| {{mapping| 317 1005 736 890 1097 }}
| -0.1126
| −0.1126
| 0.1631
| 0.1631
| 4.31
| 4.31
|-
|-
|2.9.5.7.11.13
| 2.9.5.7.11.13
|1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728
| 1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728
|{{mapping|317 1005 736 890 1097 1173}}
| {{mapping| 317 1005 736 890 1097 1173 }}
| -0.0871
| −0.0871
| 0.1594
| 0.1594
| 4.21
| 4.21
|-
|-
|2.9.5.7.11.13.17
| 2.9.5.7.11.13.17
|936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957
| 936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957
|{{mapping|317 1005 736 890 1097 1173 1296}}
| {{mapping| 317 1005 736 890 1097 1173 1296 }}
| -0.1109
| −0.1109
| 0.1587
| 0.1587
| 4.19
| 4.19
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== Music ==
== Music ==
; [[Francium]]
; [[Francium]]
* "Narrator Of Myths" from ''Mysteries'' (2023) [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube]
* "Narrator Of Myths" from ''Mysteries'' (2023) – [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube]
* "Strange Shade Of Red" from ''You Are A...'' (2024) – [https://open.spotify.com/track/23kTYEBoGRKNLhVSaIxBpC Spotify] | [https://francium223.bandcamp.com/track/strange-shade-of-red Bandcamp] | [https://www.youtube.com/watch?v=G6h1qN__F6I YouTube]
 
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/317edo"