71edo: Difference between revisions

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== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}
== Notation ==
===Sagittal notation===
In the following diagrams, 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.
====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>
====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>


== Music ==
== Music ==

Revision as of 05:38, 29 December 2024

← 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

Template:EDO intro

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

In the following diagrams, 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.

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

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

Music

Francium
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