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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, quite unusually for an EDO this large, a [[dual-fifth]] EDO, with the flat fifth (which is near [[26edo]]'s fifth) supporting [[flattone]] temperament, and the sharp fifth (which is near [[22edo]]'s fifth) supporting [[superpyth]] and [[archy]].
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]]

Latest revision as of 12:08, 30 June 2025

← 70edo 71edo 72edo →
Prime factorization 71 (prime)
Step size 16.9014 ¢ 
Fifth 42\71 (709.859 ¢)
Semitones (A1:m2) 10:3 (169 ¢ : 50.7 ¢)
Dual sharp fifth 42\71 (709.859 ¢)
Dual flat fifth 41\71 (692.958 ¢)
Dual major 2nd 12\71 (202.817 ¢)
Consistency limit 5
Distinct consistency limit 5

71 equal divisions of the octave (abbreviated 71edo or 71ed2), also called 71-tone equal temperament (71tet) or 71 equal temperament (71et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 71 equal parts of about 16.9 ¢ each. Each step represents a frequency ratio of 21/71, or the 71st root of 2.

Theory

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. Unlike small dual-fifth systems such as 18edo, both fifths are close approximations of 3/2.

Using the patent val, the equal temperament 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

Approximation of odd harmonics in 71edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +7.90 +2.42 -5.45 -1.09 +6.43 +4.54 -6.58 -3.55 +6.71 +2.46 -2.92
Relative (%) +46.8 +14.3 -32.2 -6.5 +38.0 +26.9 -38.9 -21.0 +39.7 +14.5 -17.3
Steps
(reduced) 		113
(42) 		165
(23) 		199
(57) 		225
(12) 		246
(33) 		263
(50) 		277
(64) 		290
(6) 		302
(18) 		312
(28) 		321
(37)

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

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 41\71) 		Ups and downs notation
(Dual sharp fifth 42\71)
0 0 1/1 D D
1 16.9 ^D, vE♭♭♭ ^D, vvE♭
2 33.8 vD♯, E♭♭♭ ^^D, vE♭
3 50.7 35/34 D♯, ^E♭♭♭ ^3D, E♭
4 67.6 25/24, 26/25 ^D♯, vE♭♭ ^4D, ^E♭
5 84.5 21/20 vD𝄪, E♭♭ ^5D, ^^E♭
6 101.4 17/16 D𝄪, ^E♭♭ v4D♯, ^3E♭
7 118.3 31/29 ^D𝄪, vE♭ v3D♯, ^4E♭
8 135.2 13/12 vD♯𝄪, E♭ vvD♯, v5E
9 152.1 12/11, 35/32 D♯𝄪, ^E♭ vD♯, v4E
10 169 32/29 ^D♯𝄪, vE D♯, v3E
11 185.9 29/26 E ^D♯, vvE
12 202.8 ^E, vF♭♭ ^^D♯, vE
13 219.7 25/22 vE♯, F♭♭ E
14 236.6 E♯, ^F♭♭ ^E, vvF
15 253.5 22/19, 29/25, 37/32 ^E♯, vF♭ ^^E, vF
16 270.4 vE𝄪, F♭ F
17 287.3 13/11 E𝄪, ^F♭ ^F, vvG♭
18 304.2 25/21, 31/26, 37/31 ^E𝄪, vF ^^F, vG♭
19 321.1 F ^3F, G♭
20 338 17/14, 28/23 ^F, vG♭♭♭ ^4F, ^G♭
21 354.9 vF♯, G♭♭♭ ^5F, ^^G♭
22 371.8 26/21, 31/25 F♯, ^G♭♭♭ v4F♯, ^3G♭
23 388.7 5/4 ^F♯, vG♭♭ v3F♯, ^4G♭
24 405.6 19/15, 24/19 vF𝄪, G♭♭ vvF♯, v5G
25 422.5 37/29 F𝄪, ^G♭♭ vF♯, v4G
26 439.4 31/24 ^F𝄪, vG♭ F♯, v3G
27 456.3 13/10 vF♯𝄪, G♭ ^F♯, vvG
28 473.2 21/16, 25/19 F♯𝄪, ^G♭ ^^F♯, vG
29 490.1 ^F♯𝄪, vG G
30 507 G ^G, vvA♭
31 523.9 23/17 ^G, vA♭♭♭ ^^G, vA♭
32 540.8 26/19 vG♯, A♭♭♭ ^3G, A♭
33 557.7 29/21 G♯, ^A♭♭♭ ^4G, ^A♭
34 574.6 32/23 ^G♯, vA♭♭ ^5G, ^^A♭
35 591.5 31/22 vG𝄪, A♭♭ v4G♯, ^3A♭
36 608.5 37/26 G𝄪, ^A♭♭ v3G♯, ^4A♭
37 625.4 23/16 ^G𝄪, vA♭ vvG♯, v5A
38 642.3 29/20 vG♯𝄪, A♭ vG♯, v4A
39 659.2 19/13 G♯𝄪, ^A♭ G♯, v3A
40 676.1 31/21, 34/23, 37/25 ^G♯𝄪, vA ^G♯, vvA
41 693 A ^^G♯, vA
42 709.9 ^A, vB♭♭♭ A
43 726.8 32/21, 35/23 vA♯, B♭♭♭ ^A, vvB♭
44 743.7 20/13 A♯, ^B♭♭♭ ^^A, vB♭
45 760.6 31/20 ^A♯, vB♭♭ ^3A, B♭
46 777.5 vA𝄪, B♭♭ ^4A, ^B♭
47 794.4 19/12, 30/19 A𝄪, ^B♭♭ ^5A, ^^B♭
48 811.3 8/5 ^A𝄪, vB♭ v4A♯, ^3B♭
49 828.2 21/13 vA♯𝄪, B♭ v3A♯, ^4B♭
50 845.1 31/19 A♯𝄪, ^B♭ vvA♯, v5B
51 862 23/14, 28/17 ^A♯𝄪, vB vA♯, v4B
52 878.9 B A♯, v3B
53 895.8 ^B, vC♭♭ ^A♯, vvB
54 912.7 22/13 vB♯, C♭♭ ^^A♯, vB
55 929.6 B♯, ^C♭♭ B
56 946.5 19/11 ^B♯, vC♭ ^B, vvC
57 963.4 vB𝄪, C♭ ^^B, vC
58 980.3 37/21 B𝄪, ^C♭ C
59 997.2 ^B𝄪, vC ^C, vvD♭
60 1014.1 C ^^C, vD♭
61 1031 29/16 ^C, vD♭♭♭ ^3C, D♭
62 1047.9 11/6 vC♯, D♭♭♭ ^4C, ^D♭
63 1064.8 24/13, 37/20 C♯, ^D♭♭♭ ^5C, ^^D♭
64 1081.7 ^C♯, vD♭♭ v4C♯, ^3D♭
65 1098.6 32/17 vC𝄪, D♭♭ v3C♯, ^4D♭
66 1115.5 C𝄪, ^D♭♭ vvC♯, v5D
67 1132.4 25/13 ^C𝄪, vD♭ vC♯, v4D
68 1149.3 vC♯𝄪, D♭ C♯, v3D
69 1166.2 C♯𝄪, ^D♭ ^C♯, vvD
70 1183.1 ^C♯𝄪, vD ^^C♯, vD
71 1200 2/1 D D

Notation

Sagittal notation

Best fifth notation

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation55/54144/14381/8033/3227/26
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation55/54144/14381/8033/3227/26
Evo-SZ flavor
Sagittal notationPeriodic table of EDOs with sagittal notation55/54144/14381/8033/3227/26

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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 50, 57, and 64.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [113 -71⟩ [⟨71 113]] −2.49 2.49 14.72
2.3.5 20480/19683, 393216/390625 [⟨71 113 165]] −2.01 2.14 12.69
2.3.5.7 64/63, 245/243, 2200/2187 [⟨71 113 165 200]] (71d) −2.53 2.06 12.19

Instruments

A Lumatone mapping for 71edo is available.

Music

Bryan Deister
Francium
No Clue Music
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