43edo

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