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{{Infobox ET}}
{{Infobox ET}}
The '''71 equal temperament''' or '''71-EDO''' divides the octave into 71 equal parts of 16.901 cents each.
{{ED intro}}


71edo is the 20th [[prime EDO]].
== Theory ==
71edo is a [[dual-fifth]] system, with the flat fifth (which is near the fifths of [[26edo]] and [[45edo]]) [[support]]ing [[flattone]] temperament, and the sharp fifth (which is near [[22edo]]'s fifth) supporting [[superpyth]]. Unlike small dual-fifth systems such as [[18edo]], both fifths are close approximations of 3/2.


== Theory ==
Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 20480/19683 and [[393216/390625]] in the [[5-limit]], [[875/864]], [[1029/1024]] and [[4000/3969]] in the [[7-limit]], [[100/99]] and [[245/242]] in the [[11-limit]], and [[91/90]] in the [[13-limit]]. In the 13-limit it supplies the optimal [[patent val]] for the 29 & 71 and 34 & 37 temperaments.
 
=== Odd harmonics ===
{{Harmonics in equal|71}}
{{Harmonics in equal|71}}
[[Category:Equal divisions of the octave|##]]
 
It tempers out 20480/19683 and [[393216/390625]] in the [[5-limit]], 875/864, 4000/3969 and 1029/1024 in the [[7-limit]], 245/242 and [[100/99]] in the [[11-limit]], and 91/90 in the [[13-limit]]. In the 13-limit it supplies the optimal [[patent val]] for the 29&amp;71 and 34&amp;37 temperaments.<!-- 2-digit number -->
=== Subsets and supersets ===
71edo is the 20th [[prime edo]], following [[67edo]] and before [[73edo]]. [[142edo]], which doubles it, provides correction for the harmonic 3.  


== Intervals ==
== Intervals ==
{|class="wikitable"
{{Interval table}}
|-
 
!#
== Notation ==
!Cents
=== Sagittal notation ===
!Diatonic interval category
==== Best fifth notation ====
|-
===== Evo flavor =====
|0
<imagemap>
|0.0
File:71-EDO_Evo_Sagittal.svg
|perfect unison
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 772 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[55/54]]
rect 130 80 260 106 [[144/143]]
rect 260 80 370 106 [[81/80]]
rect 370 80 490 106 [[33/32]]
rect 490 80 600 106 [[27/26]]
default [[File:71-EDO_Evo_Sagittal.svg]]
</imagemap>
 
===== Revo flavor =====
<imagemap>
File:71-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 772 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[55/54]]
rect 130 80 260 106 [[144/143]]
rect 260 80 370 106 [[81/80]]
rect 370 80 490 106 [[33/32]]
rect 490 80 600 106 [[27/26]]
default [[File:71-EDO_Revo_Sagittal.svg]]
</imagemap>
 
===== Evo-SZ flavor =====
<imagemap>
File:71-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 605 0 765 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[55/54]]
rect 130 80 260 106 [[144/143]]
rect 260 80 370 106 [[81/80]]
rect 370 80 490 106 [[33/32]]
rect 490 80 600 106 [[27/26]]
default [[File:71-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
 
==== Second-best fifth notation ====
This notation uses the same sagittal sequence as EDOs [[50edo#Sagittal notation|50]], [[57edo#Sagittal notation|57]], and [[64edo#Sagittal notation|64]].
 
===== Evo flavor =====
<imagemap>
File:71b_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 551 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:71b_Evo_Sagittal.svg]]
</imagemap>
 
===== Revo flavor =====
<imagemap>
File:71b_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 520 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:71b_Revo_Sagittal.svg]]
</imagemap>
 
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
|1
! rowspan="2" | [[Subgroup]]
|17.1
! rowspan="2" | [[Comma list]]
|superunison
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2
! [[TE error|Absolute]] (¢)
|34.3
! [[TE simple badness|Relative]] (%)
|superunison
|-
|-
|3
| 2.3
|51.4
| {{monzo| 113 -71 }}
|subminor second
| {{mapping| 71 113 }}
| −2.49
| 2.49
| 14.72
|-
|-
|4
| 2.3.5
|68.6
| 20480/19683, 393216/390625
|subminor second
| {{mapping| 71 113 165 }}
| −2.01
| 2.14
| 12.69
|-
|-
|5
| 2.3.5.7
|85.7
| 64/63, 245/243, 2200/2187
|minor second
| {{mapping| 71 113 165 200 }} (71d)
|-
| −2.53
|6
| 2.06
|102.9
| 12.19
|minor second
|-
|7
|120.0
|supraminor second
|-
|8
|137.1
|supraminor second
|-
|9
|154.3
|neutral second
|-
|10
|171.4
|submajor second
|-
|11
|188.6
|major second
|-
|12
|205.7
|major second
|-
|13
|222.9
|supermajor second
|-
|14
|240.0
|ultramajor second
|-
|15
|257.1
|ultramajor second
|-
|16
|274.3
|subminor third
|-
|17
|291.4
|minor third
|-
|18
|308.6
|minor third
|-
|19
|325.7
|supraminor third
|-
|20
|342.9
|neutral third
|-
|21
|360.0
|submajor third
|-
|22
|377.1
|submajor third
|-
|23
|394.3
|major third
|-
|24
|411.4
|major third
|-
|25
|428.6
|supermajor third
|-
|26
|445.7
|ultramajor third
|-
|27
|462.9
|subfourth
|-
|28
|480.0
|perfect fourth
|-
|29
|497.1
|perfect fourth
|-
|30
|514.3
|perfect fourth
|-
|31
|531.4
|superfourth
|-
|32
|548.6
|superfourth
|-
|33
|565.7
|low tritone
|-
|34
|582.9
|low tritone
|-
|35
|600.0
|high tritone
|-
|36
|617.1
|high tritone
|-
|37
|634.3
|high tritone
|-
|38
|651.4
|subfifth
|-
|39
|668.6
|subfifth
|-
|40
|685.7
|perfect fifth
|-
|41
|702.9
|perfect fifth
|-
|42
|720.0
|superfifth
|-
|43
|737.1
|superfifth
|-
|44
|754.3
|ultrafifth
|-
|45
|771.4
|subminor sixth
|-
|46
|788.6
|minor sixth
|-
|47
|805.7
|minor sixth
|-
|48
|822.9
|supraminor sixth
|-
|49
|840.0
|neutral sixth
|-
|50
|857.1
|neutral sixth
|-
|51
|874.3
|submajor sixth
|-
|52
|891.4
|major sixth
|-
|53
|908.6
|major sixth
|-
|54
|925.7
|supermajor sixth
|-
|55
|942.9
|ultramajor sixth
|-
|56
|960.0
|subminor seventh
|-
|57
|977.1
|subminor seventh
|-
|58
|994.3
|minor seventh
|-
|59
|1011.4
|minor seventh
|-
|60
|1028.6
|supraminor seventh
|-
|61
|1045.7
|neutral seventh
|-
|62
|1062.9
|submajor seventh
|-
|63
|1080.0
|major seventh
|-
|64
|1097.1
|major seventh
|-
|65
|1114.3
|major seventh
|-
|66
|1131.4
|supermajor seventh
|-
|67
|1148.6
|ultramajor seventh
|-
|68
|1165.7
|suboctave
|-
|69
|1182.9
|suboctave
|-
|70
|1200.0
|perfect octave
|}
|}
[[Category:Prime EDO]]
 
== Instruments ==
A [[Lumatone mapping for 71edo]] is available.
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/nMMSQdIV30I ''71edo improv''] (2025)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=_FPTUlO6jNI ''Dancing in the Mosh Pit''] (2023)
 
; [[No Clue Music]]
* [https://www.youtube.com/watch?v=5_4T9jWYn00 ''Spiraling - Randomness''] (2025)
 
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/71edo"