57-odd-limit

Odd limit

The 57-odd-limit is the set of all rational intervals which can be written as 2^k(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.

1/1
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.

57-odd-limit

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