57-odd-limit

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The 57-odd-limit is the set of all rational intervals which can be written as 2k(a/b) where a, b ≤ 57 and k is an integer. To the 55-odd-limit, it adds 18 pairs of octave-reduced intervals involving 57.

Below is a list of all octave-reduced intervals in the 57-odd-limit.

57-odd-limit intervals
1 2
58/57 57/29
57/56 112/57
56/55 55/28
55/54 108/55
54/53 53/27
53/52 104/53
52/51 51/26
51/50 100/51
50/49 49/25
49/48 96/49
48/47 47/24
47/46 92/47
46/45 45/23
45/44 88/45
44/43 43/22
43/42 84/43
42/41 41/21
41/40 80/41
40/39 39/20
39/38 76/39
38/37 37/19
37/36 72/37
36/35 35/18
35/34 68/35
34/33 33/17
33/32 64/33
32/31 31/16
31/30 60/31
30/29 29/15
29/28 56/29
57/55 110/57
28/27 27/14
55/53 106/55
27/26 52/27
53/51 102/53
26/25 25/13
51/49 98/51
25/24 48/25
49/47 94/49
24/23 23/12
47/45 90/47
23/22 44/23
45/43 86/45
22/21 21/11
43/41 82/43
21/20 40/21
41/39 78/41
20/19 19/10
39/37 74/39
58/55 55/29
19/18 36/19
56/53 53/28
37/35 70/37
55/52 104/55
18/17 17/9
53/50 100/53
35/33 66/35
52/49 49/26
17/16 32/17
50/47 47/25
33/31 62/33
49/46 92/49
16/15 15/8
47/44 88/47
31/29 58/31
46/43 43/23
15/14 28/15
44/41 41/22
29/27 54/29
43/40 80/43
57/53 106/57
14/13 13/7
55/51 102/55
41/38 76/41
27/25 50/27
40/37 37/20
53/49 98/53
13/12 24/13
51/47 94/51
38/35 35/19
25/23 46/25
62/57 57/31
37/34 68/37
49/45 11/6
12/11 86/47
47/43 64/35
35/32 53/29
58/53 42/23
23/21 104/57
57/52 31/17
34/31 82/45
45/41 51/28
56/51 20/11
11/10 49/27
54/49 78/43
43/39 29/16
32/29 96/53
53/48 38/21
21/19 47/26
52/47 56/31
31/28 74/41
41/37 92/51
51/46 9/5
10/9 88/49
49/44 70/39
39/35 52/29
29/26 43/24
48/43 34/19
19/17 84/47
47/42 25/14
28/25 66/37
37/33 41/23
46/41 98/55
55/49 57/32
64/57 16/9
9/8 55/31
62/55 94/53
53/47 39/22
44/39 62/35
35/31 23/13
26/23 76/43
43/38 53/30
60/53 30/17
17/15 37/21
42/37 44/25
25/22 51/29
58/51 58/33
33/29 72/41
41/36 86/49
49/43 100/57
57/50 7/4
8/7 96/55
55/48 38/39
47/41 82/47
39/34 68/39
31/27 54/31
54/47 47/27
23/20 40/23
38/33 33/19
53/46 92/53
15/13 26/15
52/45 45/26
37/32 64/37
22/19 19/11
51/44 88/51
29/25 50/29
36/31 31/18
43/37 74/43
50/43 43/25
57/49 98/57
64/55 55/32
7/6 12/7
62/53 53/31
55/47 94/55
48/41 41/24
41/35 70/41
34/29 29/17
27/23 46/27
47/40 80/47
20/17 17/10
53/45 90/53
33/28 56/33
46/39 39/23
13/11 22/13
58/49 49/29
45/38 76/45
32/27 27/16
51/43 86/51
19/16 32/19
44/37 37/22
25/21 42/25
56/47 47/28
31/26 52/31
68/57 57/34
37/31 62/37
43/36 72/43
49/41 82/49
55/46 92/55
6/5 5/3
53/44 88/53
47/39 78/47
41/34 68/41
35/29 58/35
64/53 53/32
29/24 48/29
52/43 43/26
23/19 38/23
40/33 33/20
57/47 94/57
17/14 28/17
62/51 51/31
45/37 74/45
28/23 23/14
39/32 64/39
50/41 41/25
11/9 18/11
60/49 49/30
49/40 80/49
38/31 31/19
27/22 44/27
70/57 57/35
43/35 70/43
16/13 13/8
53/43 86/53
37/30 60/37
58/47 47/29
21/17 34/21
68/55 55/34
47/38 76/47
26/21 21/13
57/46 92/57
31/25 50/31
36/29 29/18
41/33 66/41
46/37 37/23
51/41 82/51
56/45 45/28
66/53 53/33
5/4 8/5
64/51 51/32
54/43 43/27
49/39 78/49
44/35 35/22
39/31 62/39
34/27 27/17
29/23 46/29
53/42 84/53
24/19 19/12
43/34 68/43
62/49 49/31
19/15 30/19
52/41 41/26
33/26 52/33
47/37 74/47
14/11 11/7
51/40 80/51
37/29 58/37
60/47 47/30
23/18 36/23
55/43 86/55
32/25 25/16
41/32 64/41
50/39 39/25
68/53 53/34
9/7 14/9
58/45 45/29
49/38 76/49
40/31 31/20
31/24 48/31
53/41 82/53
22/17 17/11
57/44 88/57
35/27 54/35
48/37 37/24
74/57 57/37
13/10 20/13
56/43 43/28
43/33 66/43
30/23 23/15
47/36 72/47
64/49 49/32
17/13 26/17
72/55 55/36
55/42 84/55
38/29 29/19
21/16 32/21
46/35 35/23
25/19 38/25
54/41 41/27
29/22 44/29
62/47 47/31
33/25 50/33
70/53 53/35
37/28 56/37
41/31 62/41
45/34 68/45
49/37 74/49
53/40 80/53
57/43 86/57
4/3 3/2
55/41 82/55
51/38 76/51
47/35 70/47
43/32 64/43
39/29 58/39
74/55 55/37
35/26 52/35
66/49 49/33
31/23 46/31
58/43 43/29
27/20 40/27
50/37 37/25
23/17 34/23
42/31 31/21
19/14 28/19
72/53 53/36
53/39 78/53
34/25 25/17
49/36 72/49
64/47 47/32
15/11 22/15
56/41 41/28
41/30 60/41
26/19 19/13
37/27 54/37
48/35 35/24
70/51 51/35
11/8 16/11
62/45 45/31
51/37 74/51
40/29 29/20
29/21 42/29
76/55 55/38
47/34 68/47
18/13 13/9
43/31 62/43
68/49 49/34
25/18 36/25
57/41 82/57
32/23 23/16
39/28 56/39
46/33 33/23
53/38 76/53
60/43 43/30
74/53 53/37
7/5 10/7
80/57 57/40
66/47 47/33
52/37 37/26
45/32 64/45
38/27 27/19
31/22 44/31
55/39 78/55
24/17 17/12
41/29 58/41
Ratio Size (¢) Color name
58/57 30.109 twenonu unison
57/56 30.642 noru unison
57/55 61.836 nolugu 2nd
57/53 125.963 fithuno 2nd
62/57 145.568 thiwonu unison
57/52 158.94 nothu 2nd
64/57 200.532 inu 2nd
57/50 226.841 nogugu 3rd
57/49 261.816 noruru 2nd
68/57 305.487 nuso 3rd
57/47 333.961 fosuno 3rd
70/57 355.672 nuzoyo 3rd
57/46 371.194 twethuno 3rd
57/44 448.15 nolu 4th
74/57 451.876 thisonu 4th
57/43 487.95 fothuno 4th
57/41 570.406 fowuno 4th
80/57 586.846 nuyo 4th
57/40 613.154 nogu 5th
82/57 629.594 fowunu 4th
86/57 712.05 fothonu 5th
57/37 748.124 thisuno 5th
88/57 751.85 nulo 5th
92/57 828.806 twethonu 6th
57/35 844.328 norugu 6th
94/57 866.039 fosonu 6th
57/34 894.513 nosu 6th
98/57 938.184 nuzozo 7th
100/57 973.159 nuyoyo 6th
57/32 999.468 ino 7th
104/57 1041.06 nutho 7th
57/31 1054.432 thiwuno octave
106/57 1074.037 fithonu 7th
110/57 1138.164 nuloyo 7th
112/57 1169.358 nuzo octave
57/29 1169.891 twenuno octave

The smallest EDO to be consistent in the 57-odd-limit is 20567edo. It is also distinctly consistent and almost purely consistent in the 57-odd-limit.

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