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71edo: Difference between revisions

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Actually it's not unusual because apparently most edos around this size have 2 fifths
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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.


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


== Theory ==
=== 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]]
|1
rect 300 0 772 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|16.9
rect 20 80 130 106 [[55/54]]
|superunison
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"
|-
|-
|2
! rowspan="2" | [[Subgroup]]
|33.8
! rowspan="2" | [[Comma list]]
|superunison
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|3
! [[TE error|Absolute]] (¢)
|50.7
! [[TE simple badness|Relative]] (%)
|subminor second
|-
|-
|4
| 2.3
|67.6
| {{monzo| 113 -71 }}
|subminor second
| {{mapping| 71 113 }}
| −2.49
| 2.49
| 14.72
|-
|-
|5
| 2.3.5
|84.5
| 20480/19683, 393216/390625
|minor second
| {{mapping| 71 113 165 }}
| −2.01
| 2.14
| 12.69
|-
|-
|6
| 2.3.5.7
|101.4
| 64/63, 245/243, 2200/2187
|minor second
| {{mapping| 71 113 165 200 }} (71d)
|-
| −2.53
|7
| 2.06
|118.3
| 12.19
|minor second
|-
|8
|135.2
|supraminor second
|-
|9
|152.1
|neutral second
|-
|10
|169.0
|submajor second
|-
|11
|185.9
|major second
|-
|12
|202.8
|major second
|-
|13
|219.7
|major second
|-
|14
|236.6
|supermajor second
|-
|15
|253.5
|ultramajor second
|-
|16
|270.4
|subminor third
|-
|17
|287.3
|minor third
|-
|18
|304.2
|minor third
|-
|19
|321.1
|supraminor third
|-
|20
|338.0
|supraminor third
|-
|21
|354.9
|neutral third
|-
|22
|371.8
|submajor third
|-
|23
|388.7
|major third
|-
|24
|405.6
|major third
|-
|25
|422.5
|supermajor third
|-
|26
|439.4
|supermajor third
|-
|27
|456.3
|ultramajor third
|-
|28
|473.2
|subfourth
|-
|29
|490.1
|perfect fourth
|-
|30
|507.0
|perfect fourth
|-
|31
|523.9
|superfourth
|-
|32
|540.8
|superfourth
|-
|33
|557.7
|superfourth
|-
|34
|574.6
|low tritone
|-
|35
|591.5
|low tritone
|-
|36
|608.5
|high tritone
|-
|37
|625.4
|high tritone
|-
|38
|642.3
|subfifth
|-
|39
|659.2
|subfifth
|-
|40
|676.1
|subfifth
|-
|41
|693.0
|perfect fifth
|-
|42
|709.9
|perfect fifth
|-
|43
|726.8
|superfifth
|-
|44
|743.7
|ultrafifth
|-
|45
|760.6
|subminor sixth
|-
|46
|777.5
|subminor sixth
|-
|47
|794.4
|minor sixth
|-
|48
|811.3
|minor sixth
|-
|49
|828.2
|supraminor sixth
|-
|50
|845.1
|neutral sixth
|-
|51
|862.0
|submajor sixth
|-
|52
|878.9
|submajor sixth
|-
|53
|895.8
|major sixth
|-
|54
|912.7
|major sixth
|-
|55
|929.6
|supermajor sixth
|-
|56
|946.5
|ultramajor sixth
|-
|57
|963.4
|subminor seventh
|-
|58
|980.3
|minor seventh
|-
|59
|997.2
|minor seventh
|-
|60
|1014.1
|minor seventh
|-
|61
|1031.0
|supraminor seventh
|-
|62
|1047.9
|neutral seventh
|-
|63
|1064.8
|submajor seventh
|-
|64
|1081.7
|major seventh
|-
|65
|1098.6
|major seventh
|-
|66
|1115.5
|major seventh
|-
|67
|1132.4
|supermajor seventh
|-
|68
|1149.3
|ultramajor seventh
|-
|69
|1166.2
|suboctave
|-
|70
|1183.1
|suboctave
|-
|71
|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"