57-odd-limit

Odd limit

---

← 55-odd-limit  57-odd-limit  59-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, on which it is also distinctly consistent and almost purely consistent (harmonic 49 is off by 26.3%).