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


[[Category:Prime EDO]]
=== 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 ==
{{Interval table}}
 
== Notation ==
=== Sagittal notation ===
==== Best fifth notation ====
===== Evo flavor =====
<imagemap>
File:71-EDO_Evo_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_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"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 113 -71 }}
| {{mapping| 71 113 }}
| −2.49
| 2.49
| 14.72
|-
| 2.3.5
| 20480/19683, 393216/390625
| {{mapping| 71 113 165 }}
| −2.01
| 2.14
| 12.69
|-
| 2.3.5.7
| 64/63, 245/243, 2200/2187
| {{mapping| 71 113 165 200 }} (71d)
| −2.53
| 2.06
| 12.19
|}
 
== 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"