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Created page with "{| class="wikitable" |+Intervals in selected erac subroups in 86edo (x3 for <3.>3) !Step !Cents !2.x3.x5.x7.x11 81-odd-limit, max 3 eracs !2.x3.x5.x7.x11.13.x17.x19.23.29.31.37 37-odd-limit, max 1 erac |- |86 |1200.000 |2/1 |2/1 |- |85 |1186.047 |>>>96/49 >>>160/81 >>49/25 >63/32 | |- |84 |1172.093 |>96/49 >160/81 49/25 <63/32 | |- |83 |1158.140 |>>>48/25 <96/49 <160/81 >>27/14 >>35/18 <<49/25 <<<63/32 |72/37 >68/35 >33/17 >37/19 |- |82 |1144.186 |>48/25 <<<96/49 <<<160/..."
 
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43edo, or 43 equal divisions of the octave, is the equal tuning featuring steps of (1200/43) ≈ 27.9¢, 43 of which stack to the perfect octave 2/1. It is best known as a mild Meantone tuning with decent mappings of higher primes.
== Theory ==
{{Harmonics in ED|43|53|0}}
== Multiples ==
=== 86edo ===
{{Harmonics in ED|86|53|0}}
{{UserTag|g_|Ground|7766ff|86 is an edo whose use is generally confined to my approach, and is up there with 74edo as one of my default large edos. It's a [[Straddle primes|straddle]] edo good for [[Monarch|Interarch]] and my [[Ground's composition theory#Quartertone_composition|quartertone theory]]. Alongside being double the already respectable 43edo, this makes it my favorite edo at the time of writing.}}{{UserTag|SS|Vector|cebaff|86edo may be seen as a system introducing neutral intervals to 43edo}}
{| class="wikitable"
{| class="wikitable"
|+Intervals in selected erac subroups in 86edo (x3 for <3.>3)
|+Intervals in selected erac subroups in 86edo (x3 for <3.>3)
!Step
!Step
!Cents
!Cents
!2.x3.x5.x7.x11
!2.x3.x5.x7.x11<br>81-odd-limit, max 3 eracs
81-odd-limit, max 3 eracs
!2.x3.x5.x7.x11.13.x17.x19.23.29.31.37<br>37-odd-limit, max 1 erac
!2.x3.x5.x7.x11.13.x17.x19.23.29.31.37
37-odd-limit, max 1 erac
|-
|-
|86
|86
Line 442: Line 453:
|1/1 <<<64/63 >>>63/64
|1/1 <<<64/63 >>>63/64
|1/1
|1/1
|}
22&86bde is  very good at generating 86edo Archy pental blackdye and its related quasi-Porcupine. It's similar to Comic temperament. See [[Straddle cluster temperaments#Quartertone]]. 3sA Aeolian and additional italicized notes of 1sRM Dorian are highlighted.
{| class="wikitable"
|+11-limit 22&86bde, 81-odd-limit
!Gens
!Pd. 1
!Pd. 2
!Interpretations
|-
|24
|139.5
|739.5
|75/49
|-
|23
|83.7
|683.7
|
|-
|22
|27.9
|627.9
|50/49
|-
|21
|572.1
|1172.1
|150/77
|-
|20
|516.3
|1116.3
|75/56
|-
|19
|460.5
|1060.5
|100/77 90/49
|-
|18
|404.7
|1004.7
|25/14
|-
|17
|348.8
|948.8
|60/49 75/44
|-
|16
|293.0
|893.0
|25/21 90/77 75/64 81/49
|-
|15
|237.2
|837.2
|80/49 25/22 45/28
|-
|14
|181.4
|781.4
|120/77 25/16 100/63 54/49 55/49
|-
|13
|125.6
|725.6
|15/14 50/33 81/77
|-
|12
|69.8
|669.8
|80/77 72/49 25/24 45/44 81/56
|-
|11
|14.0
|614.0
|10/7 108/77 45/32
|-
|10
|558.1
|1158.1
|96/49 15/11 25/18 27/14 55/28
|-
|9
|502.3
|1102.3
|40/21 144/77 15/8 66/49 81/44
|-
|8
|446.5
|1046.5
|64/49 20/11 9/7 50/27 55/42 81/64
|-
|7
|390.7
|990.7
|96/77 5/4 80/63 88/49 27/22
|-
|6
|334.9
|934.9
|12/7 40/33 100/81 27/16 55/32 110/63
|-
|5
|279.1
| class="thl" |''879.1''
|128/77 5/3 18/11 33/28 81/70
|-
|4
| class="thl" |223.3
|823.3
|8/7 9/8 35/22 55/48
|-
|3
| class="thl" |''167.4''
|767.4
|12/11 10/9 11/7 54/35 35/32
|-
|2
|111.6
| class="thl" |711.6
|32/21 3/2 35/33 55/36 81/55
|-
|1
| class="thl" |55.8
|655.8
|16/11 40/27 36/35 22/21 33/32 35/24 63/44 81/80
|-
|0
| class="thl" |0.0
|600.0
|1/1 64/63 108/55 55/54 63/32
|-
| -1
| class="thl" |544.2
|1144.2
|64/33 48/35 160/81 11/8 88/63 21/11 27/20 35/18
|-
| -2
| class="thl" |488.4
|1088.4
|4/3 72/55 21/16 66/35 110/81
|-
| -3
|432.6
| class="thl" |1032.6
|64/35 14/11 9/5 11/6 35/27
|-
| -4
|376.7
| class="thl" |976.7
|16/9 96/55 7/4 44/35
|-
| -5
| class="thl" |320.9
|920.9
|6/5 56/33 11/9 140/81 77/64
|-
| -6
|265.1
|865.1
|32/27 64/55 7/6 33/20 63/55 81/50
|-
| -7
|209.3
| class="thl" |809.3
|8/5 44/27 49/44 63/40 77/48
|-
| -8
|153.5
|753.5
|128/81 14/9 11/10 84/55 27/25 49/32
|-
| -9
|97.7
|697.7
|16/15 88/81 21/20 49/33 77/72
|-
| -10
|41.9
|641.9
|28/27 56/55 36/25 22/15 49/48
|-
| -11
|586.0
|1186.0
|64/45 7/5 77/54
|-
| -12
|530.2
|1130.2
|48/25 112/81 88/45 49/36 77/40
|-
| -13
|474.4
|1074.4
|28/15 33/25 154/81
|-
| -14
|418.6
|1018.6
|32/25 49/27 98/55 63/50 77/60
|-
| -15
|362.8
|962.8
|56/45 44/25 49/40
|-
| -16
|307.0
|907.0
|128/75 42/25 98/81 77/45
|-
| -17
|251.2
|851.2
|88/75 49/30
|-
| -18
|195.3
|795.3
|28/25
|-
| -19
|139.5
|739.5
|49/45 77/50
|-
| -20
|83.7
|683.7
|112/75
|-
| -21
|27.9
|627.9
|77/75
|-
| -22
|572.1
|1172.1
|49/25
|-
| -23
|516.3
|1116.3
|
|-
| -24
|460.5
|1060.5
|98/75
|}
|}

Latest revision as of 18:30, 7 November 2026

43edo, or 43 equal divisions of the octave, is the equal tuning featuring steps of (1200/43) ≈ 27.9¢, 43 of which stack to the perfect octave 2/1. It is best known as a mild Meantone tuning with decent mappings of higher primes.

Theory

Approximation of prime harmonics in 43edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) 0.0 -4.3 +4.4 +7.9 +6.8 -3.3 +6.7 +9.5 +13.6 +3.0 -0.8 -0.2 -10.5 -9.2 +4.3 -8.4
Relative (%) 0.0 -15.3 +15.7 +28.4 +24.4 -11.9 +23.9 +33.9 +48.7 +10.7 -3.0 -0.6 -37.5 -32.9 +15.3 -30.1
Steps
(reduced) 		43
(0) 		68
(25) 		100
(14) 		121
(35) 		149
(20) 		159
(30) 		176
(4) 		183
(11) 		195
(23) 		209
(37) 		213
(41) 		224
(9) 		230
(15) 		233
(18) 		239
(24) 		246
(31)

Multiples

86edo

Approximation of prime harmonics in 86edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) 0.0 -4.3 +4.4 -6.0 +6.8 -3.3 +6.7 -4.5 -0.4 +3.0 -0.8 -0.2 +3.5 +4.8 +4.3 +5.6
Relative (%) 0.0 -30.7 +31.4 -43.3 +48.9 -23.8 +47.8 -32.2 -2.6 +21.4 -6.1 -1.3 +25.1 +34.1 +30.5 +39.9
Steps
(reduced) 		86
(0) 		136
(50) 		200
(28) 		241
(69) 		298
(40) 		318
(60) 		352
(8) 		365
(21) 		389
(45) 		418
(74) 		426
(82) 		448
(18) 		461
(31) 		467
(37) 		478
(48) 		493
(63)
g_
86 is an edo whose use is generally confined to my approach, and is up there with 74edo as one of my default large edos. It's a straddle edo good for Interarch and my quartertone theory. Alongside being double the already respectable 43edo, this makes it my favorite edo at the time of writing.
SS
86edo may be seen as a system introducing neutral intervals to 43edo
Intervals in selected erac subroups in 86edo (x3 for <3.>3)
Step Cents 2.x3.x5.x7.x11
81-odd-limit, max 3 eracs 		2.x3.x5.x7.x11.13.x17.x19.23.29.31.37
37-odd-limit, max 1 erac
86 1200.000 2/1 2/1
85 1186.047 >>>96/49 >>>160/81 >>49/25 >63/32
84 1172.093 >96/49 >160/81 49/25 <63/32
83 1158.140 >>>48/25 <96/49 <160/81 >>27/14 >>35/18 <<49/25 <<<63/32 72/37 >68/35 >33/17 >37/19
82 1144.186 >48/25 <<<96/49 <<<160/81 27/14 35/18 64/33 >48/25 >56/29 60/31 <68/35 27/14 29/15 31/16 <33/17 35/18 <37/19
81 1130.233 <48/25 >>>40/21 <<27/14 <<35/18 <48/25 <56/29 >44/23 >52/27 >21/11 >23/12 25/13
80 1116.279 <<<48/25 >40/21 >40/21 >36/19 <44/23 <52/27 19/10 <21/11 <23/12
79 1102.326 <40/21 >>>28/15 >>15/8 >32/17 <40/21 <36/19 >17/9 66/35 70/37
78 1088.372 <<<40/21 >28/15 15/8 <32/17 >28/15 15/8 <17/9 >54/29 58/31 62/33
77 1074.419 <28/15 <<15/8 >>>50/27 >>>90/49 >24/13 <28/15 >13/7 <54/29 >37/20
76 1060.465 >>64/35 <<<28/15 >50/27 >90/49 <24/13 <13/7 >68/37 46/25 >50/27 >35/19 <37/20
75 1046.512 64/35 >>>9/5 <50/27 <90/49 >>>49/27 64/35 11/6 <68/37 42/23 <50/27 >31/17 <35/19
74 1032.558 <<64/35 >9/5 <<<50/27 <<<90/49 >49/27 20/11 >56/31 >9/5 >38/21 29/16 <31/17
73 1018.605 <9/5 >>>25/14 <49/27 <56/31 <9/5 <38/21
72 1004.651 >>16/9 <<<9/5 >25/14 <<<49/27 52/29 34/19 >25/14 66/37
71 990.698 16/9 <25/14 16/9 >44/25 >30/17 23/13 <25/14 62/35
70 976.744 <<16/9 >7/4 <<<25/14 >7/4 <44/25 <30/17 >54/31 58/33 37/21
69 962.791 <7/4 >40/23 <7/4 <54/31 >33/19
68 948.837 >>>128/75 >>12/7 >>140/81 64/37 <40/23 26/15 19/11 <33/19
67 934.884 >128/75 12/7 >>>27/16 140/81 12/7 50/29 >29/17 31/18
66 920.930 <128/75 <<12/7 >>42/25 >27/16 <<140/81 >56/33 >22/13 17/10 >46/27 >27/16 <29/17
65 906.977 <<<128/75 42/25 <27/16 >32/19 <56/33 <22/13 42/25 <46/27 <27/16 >37/22
64 893.023 >>5/3 <<42/25 <<<27/16 >>81/49 <32/19 52/31 62/37 <37/22
63 879.070 5/3 81/49 >48/29 5/3 >38/23 58/35 >33/20
62 865.116 <<5/3 >>>80/49 >>49/30 >>81/50 <<81/49 <48/29 28/17 >18/11 <38/23 >23/14 <33/20
61 851.163 >80/49 49/30 81/50 <18/11 <23/14 >31/19
60 837.209 <80/49 >>45/28 <<49/30 <<81/50 44/27 13/8 60/37 >34/21 21/13 <31/19
59 823.256 >8/5 <<<80/49 45/28 >8/5 <34/21 50/31 29/18 37/23
58 809.302 <8/5 >>>100/63 <<45/28 >>63/40 <8/5 27/17 >35/22
57 795.349 >>128/81 >100/63 63/40 >30/19 19/12 46/29 <35/22
56 781.395 128/81 >>>14/9 >>25/16 <100/63 >>>54/35 <<63/40 36/23 11/7 52/33 <30/19 58/37
55 767.442 <<128/81 >14/9 25/16 <<<100/63 >54/35 >48/31 >14/9 25/16 >54/35 >31/20
54 753.488 <14/9 <<25/16 <54/35 >>49/32 >>>75/49 <48/31 >20/13 <14/9 17/11 <54/35 <31/20 >37/24
53 739.535 >>32/21 <<<14/9 <<<54/35 49/32 >75/49 <20/13 >26/17 >38/25 23/15 >29/19 <37/24
52 725.581 32/21 <<49/32 <75/49 32/21 >56/37 >44/29 <26/17 <38/25 <29/19 35/23
51 711.628 <<32/21 >3/2 >>112/75 <<<75/49 >3/2 <56/37 <44/29 50/33
50 697.674 <3/2 112/75 <3/2
49 683.721 >>40/27 <<112/75 >>72/49 52/35 >34/23 46/31 37/25
48 669.767 40/27 72/49 >>>35/24 >>>81/56 40/27 28/19 >22/15 <34/23 >19/13 >25/17 >54/37 31/21
47 655.814 <<40/27 >>36/25 <<72/49 >35/24 >81/56 >16/11 <22/15 <19/13 <25/17 <54/37 >29/20 >35/24
46 641.860 36/25 <35/24 <81/56 <16/11 36/25 42/29 <29/20 <35/24
45 627.907 >>>64/45 >>10/7 <<36/25 <<<35/24 <<<81/56 13/9 23/16 33/23
44 613.953 >64/45 10/7 >>>45/32 10/7 >44/31 27/19 37/26
43 600.000 <64/45 <<10/7 >>7/5 >45/32 24/17 <44/31 17/12 >31/22
42 586.047 <<<64/45 7/5 <45/32 7/5 52/37 38/27 <31/22
41 572.093 >>>48/35 <<7/5 >>>112/81 >>25/18 <<<45/32 32/23 18/13 46/33
40 558.140 >48/35 >112/81 25/18 >48/35 >40/29 >11/8 25/18 29/21
39 544.186 <48/35 <112/81 <<25/18 >>27/20 >>49/36 <48/35 <40/29 <11/8 >26/19 >15/11 >34/25 >37/27
38 530.233 <<<48/35 <<<112/81 27/20 49/36 <26/19 <15/11 <34/25 19/14 42/31 >23/17 27/20 <37/27
37 516.279 <<27/20 <<49/36 >>75/56 <23/17 50/37 31/23 35/26
36 502.326 >4/3 75/56 >4/3
35 488.372 <4/3 >>21/16 >>>98/75 <<75/56 <4/3 >29/22 33/25 >37/28
34 474.419 >>64/49 21/16 >98/75 >17/13 >38/29 21/16 46/35 >25/19 <29/22 <37/28
33 460.465 64/49 >>>9/7 <<21/16 >>>35/27 <98/75 >48/37 >13/10 30/23 <17/13 <38/29 <25/19
32 446.512 >>32/25 <<64/49 >9/7 >35/27 <<<98/75 <48/37 >40/31 >9/7 22/17 <13/10 >31/24 >35/27
31 432.558 32/25 <9/7 <35/27 >>>63/50 >>81/64 32/25 <40/31 <9/7 <31/24 <35/27
30 418.605 <<32/25 >>80/63 <<<9/7 <<<35/27 >63/50 81/64 14/11 >19/15 23/18 33/26 37/29
29 404.651 80/63 <63/50 <<81/64 24/19 >44/35 <19/15 29/23
28 390.698 >5/4 <<80/63 >>56/45 <<<63/50 >5/4 <44/35 34/27
27 376.744 <5/4 56/45 >>>49/40 <5/4 36/29 >21/17 46/37 31/25
26 362.791 <<56/45 >>60/49 >>100/81 >49/40 16/13 26/21 >38/31 <21/17 27/22 37/30
25 348.837 60/49 100/81 <49/40 >28/23 >11/9 <38/31
24 334.884 >>6/5 <<60/49 <<100/81 <<<49/40 >>98/81 >40/33 <28/23 <11/9 17/14 >23/19 >29/24
23 320.930 6/5 98/81 6/5 <40/33 <23/19 <29/24 35/29
22 306.977 >>>32/27 <<6/5 >>25/21 <<98/81 >44/37 >19/16 31/26 37/31
21 293.023 >32/27 25/21 >>>75/64 >32/27 <44/37 >13/11 <19/16 25/21 >27/23 >33/28
20 279.070 <32/27 >>7/6 <<25/21 >75/64 >>81/70 <32/27 20/17 <13/11 >34/29 <27/23 <33/28
19 265.116 <<<32/27 7/6 <75/64 81/70 7/6 36/31 <34/29 29/25
18 251.163 <<7/6 <<<75/64 <<81/70 22/19 15/13 >38/33 >23/20 37/32
17 237.209 >8/7 >8/7 <38/33 <23/20 >31/27
16 223.256 <8/7 >>>28/25 >>9/8 <8/7 >17/15 42/37 >25/22 <31/27 33/29
15 209.302 >28/25 9/8 >28/25 9/8 26/23 <17/15 <25/22 35/31
14 195.349 >>>10/9 <28/25 <<9/8 >>>54/49 <28/25 19/17 29/26 37/33
13 181.395 >10/9 <<<28/25 >54/49 >10/9 >21/19 >31/28
12 167.442 <10/9 >>>27/25 <54/49 >>35/32 >>>49/45 32/29 <10/9 11/10 >34/31 <21/19 <31/28
11 153.488 <<<10/9 >27/25 <<<54/49 35/32 >49/45 12/11 <34/31 >38/35 23/21 >27/25 35/32 >37/34
10 139.535 >>>15/14 <27/25 <<35/32 <49/45 >40/37 >14/13 >13/12 <38/35 25/23 <27/25 <37/34
9 125.581 >>16/15 >15/14 <<<27/25 <<<49/45 <40/37 <14/13 <13/12 >15/14 >29/27
8 111.628 16/15 <15/14 >>>21/20 16/15 >18/17 <15/14 >17/16 <29/27 31/29 33/31
7 97.674 <<16/15 <<<15/14 >21/20 <18/17 <17/16 >19/18 >21/20 35/33 37/35
6 83.721 <21/20 >>>25/24 >24/23 20/19 >22/21 <19/18 <21/20 >23/22 >27/26
5 69.767 >>28/27 >>36/35 <<<21/20 >25/24 <24/23 <22/21 26/25 <23/22 >25/24 <27/26 >29/28
4 55.814 28/27 36/35 <25/24 >>>49/48 >>>81/80 32/31 28/27 36/35 30/29 >34/33 >38/37 <25/24 <29/28 31/30 33/32 >35/34
3 41.860 >>>64/63 <<28/27 <<36/35 <<<25/24 >>50/49 >49/48 >81/80 <34/33 <38/37 <35/34 37/36
2 27.907 >64/63 50/49 <49/48 <81/80
1 13.953 <64/63 <<50/49 <<<49/48 <<<81/80
0 0.000 1/1 <<<64/63 >>>63/64 1/1

22&86bde is very good at generating 86edo Archy pental blackdye and its related quasi-Porcupine. It's similar to Comic temperament. See Straddle cluster temperaments#Quartertone. 3sA Aeolian and additional italicized notes of 1sRM Dorian are highlighted.

11-limit 22&86bde, 81-odd-limit
Gens Pd. 1 Pd. 2 Interpretations
24 139.5 739.5 75/49
23 83.7 683.7
22 27.9 627.9 50/49
21 572.1 1172.1 150/77
20 516.3 1116.3 75/56
19 460.5 1060.5 100/77 90/49
18 404.7 1004.7 25/14
17 348.8 948.8 60/49 75/44
16 293.0 893.0 25/21 90/77 75/64 81/49
15 237.2 837.2 80/49 25/22 45/28
14 181.4 781.4 120/77 25/16 100/63 54/49 55/49
13 125.6 725.6 15/14 50/33 81/77
12 69.8 669.8 80/77 72/49 25/24 45/44 81/56
11 14.0 614.0 10/7 108/77 45/32
10 558.1 1158.1 96/49 15/11 25/18 27/14 55/28
9 502.3 1102.3 40/21 144/77 15/8 66/49 81/44
8 446.5 1046.5 64/49 20/11 9/7 50/27 55/42 81/64
7 390.7 990.7 96/77 5/4 80/63 88/49 27/22
6 334.9 934.9 12/7 40/33 100/81 27/16 55/32 110/63
5 279.1 879.1 128/77 5/3 18/11 33/28 81/70
4 223.3 823.3 8/7 9/8 35/22 55/48
3 167.4 767.4 12/11 10/9 11/7 54/35 35/32
2 111.6 711.6 32/21 3/2 35/33 55/36 81/55
1 55.8 655.8 16/11 40/27 36/35 22/21 33/32 35/24 63/44 81/80
0 0.0 600.0 1/1 64/63 108/55 55/54 63/32
-1 544.2 1144.2 64/33 48/35 160/81 11/8 88/63 21/11 27/20 35/18
-2 488.4 1088.4 4/3 72/55 21/16 66/35 110/81
-3 432.6 1032.6 64/35 14/11 9/5 11/6 35/27
-4 376.7 976.7 16/9 96/55 7/4 44/35
-5 320.9 920.9 6/5 56/33 11/9 140/81 77/64
-6 265.1 865.1 32/27 64/55 7/6 33/20 63/55 81/50
-7 209.3 809.3 8/5 44/27 49/44 63/40 77/48
-8 153.5 753.5 128/81 14/9 11/10 84/55 27/25 49/32
-9 97.7 697.7 16/15 88/81 21/20 49/33 77/72
-10 41.9 641.9 28/27 56/55 36/25 22/15 49/48
-11 586.0 1186.0 64/45 7/5 77/54
-12 530.2 1130.2 48/25 112/81 88/45 49/36 77/40
-13 474.4 1074.4 28/15 33/25 154/81
-14 418.6 1018.6 32/25 49/27 98/55 63/50 77/60
-15 362.8 962.8 56/45 44/25 49/40
-16 307.0 907.0 128/75 42/25 98/81 77/45
-17 251.2 851.2 88/75 49/30
-18 195.3 795.3 28/25
-19 139.5 739.5 49/45 77/50
-20 83.7 683.7 112/75
-21 27.9 627.9 77/75
-22 572.1 1172.1 49/25
-23 516.3 1116.3
-24 460.5 1060.5 98/75
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