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57-odd-limit: Difference between revisions

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fixed incorrect format, missing links on all intervals, added navigation
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The '''57-odd-limit''' is the set of all [[Rational interval|rational intervals]] which can be written as 2<sup>''k''</sup>(''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.
{{Odd-limit navigation}}The '''57-odd-limit''' is the set of all [[Rational interval|rational intervals]] which can be written as 2<sup>''k''</sup>(''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.
Below is a list of all octave-reduced intervals in the 57-odd-limit.
{| class="wikitable mw-collapsible mw-collapsed"
 
|+57-odd-limit intervals
* 1/1
|1
* '''58/57, 57/29'''
|2
* '''57/56, 112/57'''
|-
* 56/55, 55/28
|58/57
* 55/54, 108/55
|57/29
* 54/53, 53/27
|-
* 53/52, 104/53
|57/56
* 52/51, 51/26
|112/57
* 51/50, 100/51
|-
* 50/49, 49/25
|56/55
* 49/48, 96/49
|55/28
* 48/47, 47/24
|-
* 47/46, 92/47
|55/54
* 46/45, 45/23
|108/55
* 45/44, 88/45
|-
* 44/43, 43/22
|54/53
* 43/42, 84/43
|53/27
* 42/41, 41/21
|-
* 41/40, 80/41
|53/52
* 40/39, 39/20
|104/53
* 39/38, 76/39
|-
* 38/37, 37/19
|52/51
* 37/36, 72/37
|51/26
* 36/35, 35/18
|-
* 35/34, 68/35
|51/50
* 34/33, 33/17
|100/51
* 33/32, 64/33
|-
* 32/31, 31/16
|50/49
* 31/30, 60/31
|49/25
* 30/29, 29/15
|-
* 29/28, 56/29
|49/48
* '''57/55, 110/57'''
|96/49
* 28/27, 27/14
|-
* 55/53, 106/55
|48/47
* 27/26, 52/27
|47/24
* 53/51, 102/53
|-
* 26/25, 25/13
|47/46
* 51/49, 98/51
|92/47
* 25/24, 48/25
|-
* 49/47, 94/49
|46/45
* 24/23, 23/12
|45/23
* 47/45, 90/47
|-
* 23/22, 44/23
|45/44
* 45/43, 86/45
|88/45
* 22/21, 21/11
|-
* 43/41, 82/43
|44/43
* 21/20, 40/21
|43/22
* 41/39, 78/41
|-
* 20/19, 19/10
|43/42
* 39/37, 74/39
|84/43
* 58/55, 55/29
|-
* 19/18, 36/19
|42/41
* 56/53, 53/28
|41/21
* 37/35, 70/37
|-
* 55/52, 104/55
|41/40
* 18/17, 17/9
|80/41
* 53/50, 100/53
|-
* 35/33, 66/35
|40/39
* 52/49, 49/26
|39/20
* 17/16, 32/17
|-
* 50/47, 47/25
|39/38
* 33/31, 62/33
|76/39
* 49/46, 92/49
|-
* 16/15, 15/8
|38/37
* 47/44, 88/47
|37/19
* 31/29, 58/31
|-
* 46/43, 43/23
|37/36
* 15/14, 28/15
|72/37
* 44/41, 41/22
|-
* 29/27, 54/29
|36/35
* 43/40, 80/43
|35/18
* '''57/53, 106/57'''
|-
* 14/13, 13/7
|35/34
* 55/51, 102/55
|68/35
* 41/38, 76/41
|-
* 27/25, 50/27
|34/33
* 40/37, 37/20
|33/17
* 53/49, 98/53
|-
* 13/12, 24/13
|33/32
* 51/47, 94/51
|64/33
* 38/35, 35/19
|-
* 25/23, 46/25
|32/31
* '''62/57, 57/31'''
|31/16
* 37/34, 68/37
|-
* 49/45, 11/6
|31/30
* 12/11, 86/47
|60/31
* 47/43, 64/35
|-
* 35/32, 53/29
|30/29
* 58/53, 42/23
|29/15
* '''23/21, 104/57'''
|-
* '''57/52, 31/17'''
|29/28
* 34/31, 82/45
|56/29
* 45/41, 51/28
|-
* 56/51, 20/11
|57/55
* 11/10, 49/27
|110/57
* 54/49, 78/43
|-
* 43/39, 29/16
|28/27
* 32/29, 96/53
|27/14
* 53/48, 38/21
|-
* 21/19, 47/26
|55/53
* 52/47, 56/31
|106/55
* 31/28, 74/41
|-
* 41/37, 92/51
|27/26
* 51/46, 9/5
|52/27
* 10/9, 88/49
|-
* 49/44, 70/39
|53/51
* 39/35, 52/29
|102/53
* 29/26, 43/24
|-
* 48/43, 34/19
|26/25
* 19/17, 84/47
|25/13
* 47/42, 25/14
|-
* 28/25, 66/37
|51/49
* 37/33, 41/23
|98/51
* 46/41, 98/55
|-
* '''55/49, 57/32'''
|25/24
* '''64/57, 16/9'''
|48/25
* 9/8, 55/31
|-
* 62/55, 94/53
|49/47
* 53/47, 39/22
|94/49
* 44/39, 62/35
|-
* 35/31, 23/13
|24/23
* 26/23, 76/43
|23/12
* 43/38, 53/30
|-
* 60/53, 30/17
|47/45
* 17/15, 37/21
|90/47
* 42/37, 44/25
|-
* 25/22, 51/29
|23/22
* 58/51, 58/33
|44/23
* 33/29, 72/41
|-
* 41/36, 86/49
|45/43
* '''49/43, 100/57'''
|86/45
* '''57/50, 7/4'''
|-
* 8/7, 96/55
|22/21
* 55/48, 38/39
|21/11
* 47/41, 82/47
|-
* 39/34, 68/39
|43/41
* 31/27, 54/31
|82/43
* 54/47, 47/27
|-
* 23/20, 40/23
|21/20
* 38/33, 33/19
|40/21
* 53/46, 92/53
|-
* 15/13, 26/15
|41/39
* 52/45, 45/26
|78/41
* 37/32, 64/37
|-
* 22/19, 19/11
|20/19
* 51/44, 88/51
|19/10
* 29/25, 50/29
|-
* 36/31, 31/18
|39/37
* 43/37, 74/43
|74/39
* 50/43, 43/25
|-
* '''57/49, 98/57'''
|58/55
* 64/55, 55/32
|55/29
* 7/6, 12/7
|-
* 62/53, 53/31
|19/18
* 55/47, 94/55
|36/19
* 48/41, 41/24
|-
* 41/35, 70/41
|56/53
* 34/29, 29/17
|53/28
* 27/23, 46/27
|-
* 47/40, 80/47
|37/35
* 20/17, 17/10
|70/37
* 53/45, 90/53
|-
* 33/28, 56/33
|55/52
* 46/39, 39/23
|104/55
* 13/11, 22/13
|-
* 58/49, 49/29
|18/17
* 45/38, 76/45
|17/9
* 32/27, 27/16
|-
* 51/43, 86/51
|53/50
* 19/16, 32/19
|100/53
* 44/37, 37/22
|-
* 25/21, 42/25
|35/33
* 56/47, 47/28
|66/35
* 31/26, 52/31
|-
* '''68/57, 57/34'''
|52/49
* 37/31, 62/37
|49/26
* 43/36, 72/43
|-
* 49/41, 82/49
|17/16
* 55/46, 92/55
|32/17
* 6/5, 5/3
|-
* 53/44, 88/53
|50/47
* 47/39, 78/47
|47/25
* 41/34, 68/41
|-
* 35/29, 58/35
|33/31
* 64/53, 53/32
|62/33
* 29/24, 48/29
|-
* 52/43, 43/26
|49/46
* 23/19, 38/23
|92/49
* 40/33, 33/20
|-
* '''57/47, 94/57'''
|16/15
* 17/14, 28/17
|15/8
* 62/51, 51/31
|-
* 45/37, 74/45
|47/44
* 28/23, 23/14
|88/47
* 39/32, 64/39
|-
* 50/41, 41/25
|31/29
* 11/9, 18/11
|58/31
* 60/49, 49/30
|-
* 49/40, 80/49
|46/43
* 38/31, 31/19
|43/23
* 27/22, 44/27
|-
* '''70/57, 57/35'''
|15/14
* 43/35, 70/43
|28/15
* 16/13, 13/8
|-
* 53/43, 86/53
|44/41
* 37/30, 60/37
|41/22
* 58/47, 47/29
|-
* 21/17, 34/21
|29/27
* 68/55, 55/34
|54/29
* 47/38, 76/47
|-
* 26/21, 21/13
|43/40
* '''57/46, 92/57'''
|80/43
* 31/25, 50/31
|-
* 36/29, 29/18
|57/53
* 41/33, 66/41
|106/57
* 46/37, 37/23
|-
* 51/41, 82/51
|14/13
* 56/45, 45/28
|13/7
* 66/53, 53/33
|-
* 5/4, 8/5
|55/51
* 64/51, 51/32
|102/55
* 54/43, 43/27
|-
* 49/39, 78/49
|41/38
* 44/35, 35/22
|76/41
* 39/31, 62/39
|-
* 34/27, 27/17
|27/25
* 29/23, 46/29
|50/27
* 53/42, 84/53
|-
* 24/19, 19/12
|40/37
* 43/34, 68/43
|37/20
* 62/49, 49/31
|-
* 19/15, 30/19
|53/49
* 52/41, 41/26
|98/53
* 33/26, 52/33
|-
* 47/37, 74/47
|13/12
* 14/11, 11/7
|24/13
* 51/40, 80/51
|-
* 37/29, 58/37
|51/47
* 60/47, 47/30
|94/51
* 23/18, 36/23
|-
* 55/43, 86/55
|38/35
* 32/25, 25/16
|35/19
* 41/32, 64/41
|-
* 50/39, 39/25
|25/23
* 68/53, 53/34
|46/25
* 9/7, 14/9
|-
* 58/45, 45/29
|62/57
* 49/38, 76/49
|57/31
* 40/31, 31/20
|-
* 31/24, 48/31
|37/34
* 53/41, 82/53
|68/37
* 22/17, 17/11
|-
* '''57/44, 88/57'''
|49/45
* 35/27, 54/35
|11/6
* 48/37, 37/24
|-
* '''74/57, 57/37'''
|12/11
* 13/10, 20/13
|86/47
* 56/43, 43/28
|-
* 43/33, 66/43
|47/43
* 30/23, 23/15
|64/35
* 47/36, 72/47
|-
* 64/49, 49/32
|35/32
* 17/13, 26/17
|53/29
* 72/55, 55/36
|-
* 55/42, 84/55
|58/53
* 38/29, 29/19
|42/23
* 21/16, 32/21
|-
* 46/35, 35/23
|23/21
* 25/19, 38/25
|104/57
* 54/41, 41/27
|-
* 29/22, 44/29
|57/52
* 62/47, 47/31
|31/17
* 33/25, 50/33
|-
* 70/53, 53/35
|34/31
* 37/28, 56/37
|82/45
* 41/31, 62/41
|-
* 45/34, 68/45
|45/41
* 49/37, 74/49
|51/28
* 53/40, 80/53
|-
* '''57/43, 86/57'''
|56/51
* 4/3, 3/2
|20/11
* 55/41, 82/55
|-
* 51/38, 76/51
|11/10
* 47/35, 70/47
|49/27
* 43/32, 64/43
|-
* 39/29, 58/39
|54/49
* 74/55, 55/37
|78/43
* 35/26, 52/35
|-
* 66/49, 49/33
|43/39
* 31/23, 46/31
|29/16
* 58/43, 43/29
|-
* 27/20, 40/27
|32/29
* 50/37, 37/25
|96/53
* 23/17, 34/23
|-
* 42/31, 31/21
|53/48
* 19/14, 28/19
|38/21
* 72/53, 53/36
|-
* 53/39, 78/53
|21/19
* 34/25, 25/17
|47/26
* 49/36, 72/49
|-
* 64/47, 47/32
|52/47
* 15/11, 22/15
|56/31
* 56/41, 41/28
|-
* 41/30, 60/41
|31/28
* 26/19, 19/13
|74/41
* 37/27, 54/37
|-
* 48/35, 35/24
|41/37
* 70/51, 51/35
|92/51
* 11/8, 16/11
|-
* 62/45, 45/31
|51/46
* 51/37, 74/51
|9/5
* 40/29, 29/20
|-
* 29/21, 42/29
|10/9
* 76/55, 55/38
|88/49
* 47/34, 68/47
|-
* 18/13, 13/9
|49/44
* 43/31, 62/43
|70/39
* 68/49, 49/34
|-
* 25/18, 36/25
|39/35
* '''57/41, 82/57'''
|52/29
* 32/23, 23/16
|-
* 39/28, 56/39
|29/26
* 46/33, 33/23
|43/24
* 53/38, 76/53
|-
* 60/43, 43/30
|48/43
* 74/53, 53/37
|34/19
* 7/5, 10/7
|-
* '''80/57, 57/40'''
|19/17
* 66/47, 47/33
|84/47
* 52/37, 37/26
|-
* 45/32, 64/45
|47/42
* 38/27, 27/19
|25/14
* 31/22, 44/31
|-
* 55/39, 78/55
|28/25
* 24/17, 17/12
|66/37
* 41/29, 58/41
|-
 
|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
|}
{| class="wikitable"
{| class="wikitable"
|Ratio
|Ratio
Retrieved from "https://en.xen.wiki/w/57-odd-limit"