57-odd-limit: Difference between revisions

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{{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.
{{Odd-limit navigation|57}}¨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.


* 1/1
* [[1/1]]
* '''58/57, 57/29'''
* '''[[58/57]], [[57/29]]'''
* '''57/56, 112/57'''
* '''[[57/56]], [[112/57]]'''
* 56/55, 55/28
* [[56/55]], [[55/28]]
* 55/54, 108/55
* [[55/54]], [[108/55]]
* 54/53, 53/27
* [[54/53]], [[53/27]]
* 53/52, 104/53
* [[53/52]], [[104/53]]
* 52/51, 51/26
* [[52/51]], [[51/26]]
* 51/50, 100/51
* [[51/50]], [[100/51]]
* 50/49, 49/25
* [[50/49]], [[49/25]]
* 49/48, 96/49
* [[49/48]], [[96/49]]
* 48/47, 47/24
* [[48/47]], [[47/24]]
* 47/46, 92/47
* [[47/46]], [[92/47]]
* 46/45, 45/23
* [[46/45]], [[45/23]]
* 45/44, 88/45
* [[45/44]], [[88/45]]
* 44/43, 43/22
* [[44/43]], [[43/22]]
* 43/42, 84/43
* [[43/42]], [[84/43]]
* 42/41, 41/21
* [[42/41]], [[41/21]]
* 41/40, 80/41
* [[41/40]], [[80/41]]
* 40/39, 39/20
* [[40/39]], [[39/20]]
* 39/38, 76/39
* [[39/38]], [[76/39]]
* 38/37, 37/19
* [[38/37]], [[37/19]]
* 37/36, 72/37
* [[37/36]], [[72/37]]
* 36/35, 35/18
* [[36/35]], [[35/18]]
* 35/34, 68/35
* [[35/34]], [[68/35]]
* 34/33, 33/17
* [[34/33]], [[33/17]]
* 33/32, 64/33
* [[33/32]], [[64/33]]
* 32/31, 31/16
* [[32/31]], [[31/16]]
* 31/30, 60/31
* [[31/30]], [[60/31]]
* 30/29, 29/15
* [[30/29]], [[29/15]]
* 29/28, 56/29
* [[29/28]], [[56/29]]
* '''57/55, 110/57'''
* '''[[57/55]], [[110/57]]'''
* 28/27, 27/14
* [[28/27]], [[27/14]]
* 55/53, 106/55
* [[55/53]], [[106/55]]
* 27/26, 52/27
* [[27/26]], [[52/27]]
* 53/51, 102/53
* [[53/51]], [[102/53]]
* 26/25, 25/13
* [[26/25]], [[25/13]]
* 51/49, 98/51
* [[51/49]], [[98/51]]
* 25/24, 48/25
* [[25/24]], [[48/25]]
* 49/47, 94/49
* [[49/47]], [[94/49]]
* 24/23, 23/12
* [[24/23]], [[23/12]]
* 47/45, 90/47
* [[47/45]], [[90/47]]
* 23/22, 44/23
* [[23/22]], [[44/23]]
* 45/43, 86/45
* [[45/43]], [[86/45]]
* 22/21, 21/11
* [[22/21]], [[21/11]]
* 43/41, 82/43
* [[43/41]], [[82/43]]
* 21/20, 40/21
* [[21/20]], [[40/21]]
* 41/39, 78/41
* [[41/39]], [[78/41]]
* 20/19, 19/10
* [[20/19]], [[19/10]]
* 39/37, 74/39
* [[39/37]], [[74/39]]
* 58/55, 55/29
* [[58/55]], [[55/29]]
* 19/18, 36/19
* [[19/18]], [[36/19]]
* 56/53, 53/28
* [[56/53]], [[53/28]]
* 37/35, 70/37
* [[37/35]], [[70/37]]
* 55/52, 104/55
* [[55/52]], [[104/55]]
* 18/17, 17/9
* [[18/17]], [[17/9]]
* 53/50, 100/53
* [[53/50]], [[100/53]]
* 35/33, 66/35
* [[35/33]], [[66/35]]
* 52/49, 49/26
* [[52/49]], [[49/26]]
* 17/16, 32/17
* [[17/16]], [[32/17]]
* 50/47, 47/25
* [[50/47]], [[47/25]]
* 33/31, 62/33
* [[33/31]], [[62/33]]
* 49/46, 92/49
* [[49/46]], [[92/49]]
* 16/15, 15/8
* [[16/15]], [[15/8]]
* 47/44, 88/47
* [[47/44]], [[88/47]]
* 31/29, 58/31
* [[31/29]], [[58/31]]
* 46/43, 43/23
* [[46/43]], [[43/23]]
* 15/14, 28/15
* [[15/14]], [[28/15]]
* 44/41, 41/22
* [[44/41]], [[41/22]]
* 29/27, 54/29
* [[29/27]], [[54/29]]
* 43/40, 80/43
* [[43/40]], [[80/43]]
* '''57/53, 106/57'''
* '''[[57/53]], [[106/57]]'''
* 14/13, 13/7
* [[14/13]], [[13/7]]
* 55/51, 102/55
* [[55/51]], [[102/55]]
* 41/38, 76/41
* [[41/38]], [[76/41]]
* 27/25, 50/27
* [[27/25]], [[50/27]]
* 40/37, 37/20
* [[40/37]], [[37/20]]
* 53/49, 98/53
* [[53/49]], [[98/53]]
* 13/12, 24/13
* [[13/12]], [[24/13]]
* 51/47, 94/51
* [[51/47]], [[94/51]]
* 38/35, 35/19
* [[38/35]], [[35/19]]
* 25/23, 46/25
* [[25/23]], [[46/25]]
* '''62/57, 57/31'''
* '''[[62/57]], [[57/31]]'''
* 37/34, 68/37
* [[37/34]], [[68/37]]
* 49/45, 11/6
* [[49/45]], [[11/6]]
* 12/11, 86/47
* [[12/11]], [[86/47]]
* 47/43, 64/35
* [[47/43]], [[64/35]]
* 35/32, 53/29
* [[35/32]], [[53/29]]
* 58/53, 42/23
* [[58/53]], [[42/23]]
* '''23/21, 104/57'''
* '''[[23/21]], [[104/57]]'''
* '''57/52, 31/17'''
* '''[[57/52]], [[31/17]]'''
* 34/31, 82/45
* [[34/31]], [[82/45]]
* 45/41, 51/28
* [[45/41 ]] [[51/28]]
* 56/51, 20/11
* [[56/51]], [[20/11]]
* 11/10, 49/27
* [[11/10]], [[49/27]]
* 54/49, 78/43
* [[54/49]], [[78/43]]
* 43/39, 29/16
* [[43/39]], [[29/16]]
* 32/29, 96/53
* [[32/29]], [[96/53]]
* 53/48, 38/21
* [[53/48]], [[38/21]]
* 21/19, 47/26
* [[21/19]], [[47/26]]
* 52/47, 56/31
* [[52/47]], [[56/31]]
* 31/28, 74/41
* [[31/28]], [[74/41]]
* 41/37, 92/51
* [[41/37]], [[92/51]]
* 51/46, 9/5
* [[51/46]], [[9/5]]
* 10/9, 88/49
* [[10/9]], [[88/49]]
* 49/44, 70/39
* [[49/44]], [[70/39]]
* 39/35, 52/29
* [[39/35]], [[52/29]]
* 29/26, 43/24
* [[29/26]], [[43/24]]
* 48/43, 34/19
* [[48/43]], [[34/19]]
* 19/17, 84/47
* [[19/17]], [[84/47]]
* 47/42, 25/14
* [[47/42]], [[25/14]]
* 28/25, 66/37
* [[28/25]], [[66/37]]
* 37/33, 41/23
* [[37/33]], [[41/23]]
* 46/41, 98/55
* [[46/41]], [[98/55]]
* '''55/49, 57/32'''
* '''[[55/49]], [[57/32]]'''
* '''64/57, 16/9'''
* '''[[64/57]], [[16/9]]'''
* 9/8, 55/31
* [[9/8]], [[55/31]]
* 62/55, 94/53
* [[62/55]], [[94/53]]
* 53/47, 39/22
* [[53/47]], [[39/22]]
* 44/39, 62/35
* [[44/39]], [[62/35]]
* 35/31, 23/13
* [[35/31]], [[23/13]]
* 26/23, 76/43
* [[26/23]], [[76/43]]
* 43/38, 53/30
* [[43/38]], [[53/30]]
* 60/53, 30/17
* [[60/53]], [[30/17]]
* 17/15, 37/21
* [[17/15]], [[37/21]]
* 42/37, 44/25
* [[42/37]], [[44/25]]
* 25/22, 51/29
* [[25/22]], [[51/29]]
* 58/51, 58/33
* [[58/51]], [[58/33]]
* 33/29, 72/41
* [[33/29]], [[72/41]]
* 41/36, 86/49
* [[41/36]], [[86/49]]
* '''49/43, 100/57'''
* '''[[49/43]], [[100/57]]'''
* '''57/50, 7/4'''
* '''[[57/50]], [[7/4]]'''
* 8/7, 96/55
* [[8/7]], [[96/55]]
* 55/48, 38/39
* [[55/48]], [[38/39]]
* 47/41, 82/47
* [[47/41]], [[82/47]]
* 39/34, 68/39
* [[39/34]], [[68/39]]
* 31/27, 54/31
* [[31/27]], [[54/31]]
* 54/47, 47/27
* [[54/47]], [[47/27]]
* 23/20, 40/23
* [[23/20]], [[40/23]]
* 38/33, 33/19
* [[38/33]], [[33/19]]
* 53/46, 92/53
* [[53/46]], [[92/53]]
* 15/13, 26/15
* [[15/13]], [[26/15]]
* 52/45, 45/26
* [[52/45]], [[45/26]]
* 37/32, 64/37
* [[37/32]], [[64/37]]
* 22/19, 19/11
* [[22/19]], [[19/11]]
* 51/44, 88/51
* [[51/44]], [[88/51]]
* 29/25, 50/29
* [[29/25]], [[50/29]]
* 36/31, 31/18
* [[36/31]], [[31/18]]
* 43/37, 74/43
* [[43/37]], [[74/43]]
* 50/43, 43/25
* [[50/43]], [[43/25]]
* '''57/49, 98/57'''
* '''[[57/49]], [[98/57]]'''
* 64/55, 55/32
* [[64/55]], [[55/32]]
* 7/6, 12/7
* [[7/6]], [[12/7]]
* 62/53, 53/31
* [[62/53]], [[53/31]]
* 55/47, 94/55
* [[55/47]], [[94/55]]
* 48/41, 41/24
* [[48/41]], [[41/24]]
* 41/35, 70/41
* [[41/35]], [[70/41]]
* 34/29, 29/17
* [[34/29]], [[29/17]]
* 27/23, 46/27
* [[27/23]], [[46/27]]
* 47/40, 80/47
* [[47/40]], [[80/47]]
* 20/17, 17/10
* [[20/17]], [[17/10]]
* 53/45, 90/53
* [[53/45]], [[90/53]]
* 33/28, 56/33
* [[33/28]], [[56/33]]
* 46/39, 39/23
* [[46/39]], [[39/23]]
* 13/11, 22/13
* [[13/11]], [[22/13]]
* 58/49, 49/29
* [[58/49]], [[49/29]]
* 45/38, 76/45
* [[45/38]], [[76/45]]
* 32/27, 27/16
* [[32/27]], [[27/16]]
* 51/43, 86/51
* [[51/43]], [[86/51]]
* 19/16, 32/19
* [[19/16]], [[32/19]]
* 44/37, 37/22
* [[44/37]], [[37/22]]
* 25/21, 42/25
* [[25/21]], [[42/25]]
* 56/47, 47/28
* [[56/47]], [[47/28]]
* 31/26, 52/31
* [[31/26]], [[52/31]]
* '''68/57, 57/34'''
* '''[[68/57]], [[57/34]]'''
* 37/31, 62/37
* [[37/31]], [[62/37]]
* 43/36, 72/43
* [[43/36]], [[72/43]]
* 49/41, 82/49
* [[49/41]], [[82/49]]
* 55/46, 92/55
* [[55/46]], [[92/55]]
* 6/5, 5/3
* [[6/5]], [[5/3]]
* 53/44, 88/53
* [[53/44]], [[88/53]]
* 47/39, 78/47
* [[47/39]], [[78/47]]
* 41/34, 68/41
* [[41/34]], [[68/41]]
* 35/29, 58/35
* [[35/29]], [[58/35]]
* 64/53, 53/32
* [[64/53]], [[53/32]]
* 29/24, 48/29
* [[29/24]], [[48/29]]
* 52/43, 43/26
* [[52/43]], [[43/26]]
* 23/19, 38/23
* [[23/19]], [[38/23]]
* 40/33, 33/20
* [[40/33]], [[33/20]]
* '''57/47, 94/57'''
* '''[[57/47]], [[94/57]]'''
* 17/14, 28/17
* [[17/14]], [[28/17]]
* 62/51, 51/31
* [[62/51]], [[51/31]]
* 45/37, 74/45
* [[45/37]], [[74/45]]
* 28/23, 23/14
* [[28/23]], [[23/14]]
* 39/32, 64/39
* [[39/32]], [[64/39]]
* 50/41, 41/25
* [[50/41]], [[41/25]]
* 11/9, 18/11
* [[11/9]], [[18/11]]
* 60/49, 49/30
* [[60/49]], [[49/30]]
* 49/40, 80/49
* [[49/40]], [[80/49]]
* 38/31, 31/19
* [[38/31]], [[31/19]]
* 27/22, 44/27
* [[27/22]], [[44/27]]
* '''70/57, 57/35'''
* '''[[70/57]], [[57/35]]'''
* 43/35, 70/43
* [[43/35]], [[70/43]]
* 16/13, 13/8
* [[16/13]], [[13/8]]
* 53/43, 86/53
* [[53/43]], [[86/53]]
* 37/30, 60/37
* [[37/30]], [[60/37]]
* 58/47, 47/29
* [[58/47]], [[47/29]]
* 21/17, 34/21
* [[21/17]], [[34/21]]
* 68/55, 55/34
* [[68/55]], [[55/34]]
* 47/38, 76/47
* [[47/38]], [[76/47]]
* 26/21, 21/13
* [[26/21]], [[21/13]]
* '''57/46, 92/57'''
* '''[[57/46]], [[92/57]]'''
* 31/25, 50/31
* [[31/25]], [[50/31]]
* 36/29, 29/18
* [[36/29]], [[29/18]]
* 41/33, 66/41
* [[41/33]], [[66/41]]
* 46/37, 37/23
* [[46/37]], [[37/23]]
* 51/41, 82/51
* [[51/41]], [[82/51]]
* 56/45, 45/28
* [[56/45]], [[45/28]]
* 66/53, 53/33
* [[66/53]], [[53/33]]
* 5/4, 8/5
* [[5/4]], [[8/5]]
* 64/51, 51/32
* [[64/51]], [[51/32]]
* 54/43, 43/27
* [[54/43]], [[43/27]]
* 49/39, 78/49
* [[49/39]], [[78/49]]
* 44/35, 35/22
* [[44/35]], [[35/22]]
* 39/31, 62/39
* [[39/31]], [[62/39]]
* 34/27, 27/17
* [[34/27]], [[27/17]]
* 29/23, 46/29
* [[29/23]], [[46/29]]
* 53/42, 84/53
* [[53/42]], [[84/53]]
* 24/19, 19/12
* [[24/19]], [[19/12]]
* 43/34, 68/43
* [[43/34]], [[68/43]]
* 62/49, 49/31
* [[62/49]], [[49/31]]
* 19/15, 30/19
* [[19/15]], [[30/19]]
* 52/41, 41/26
* [[52/41]], [[41/26]]
* 33/26, 52/33
* [[33/26]], [[52/33]]
* 47/37, 74/47
* [[47/37]], [[74/47]]
* 14/11, 11/7
* [[14/11]], [[11/7]]
* 51/40, 80/51
* [[51/40]], [[80/51]]
* 37/29, 58/37
* [[37/29]], [[58/37]]
* 60/47, 47/30
* [[60/47]], [[47/30]]
* 23/18, 36/23
* [[23/18]], [[36/23]]
* 55/43, 86/55
* [[55/43]], [[86/55]]
* 32/25, 25/16
* [[32/25]], [[25/16]]
* 41/32, 64/41
* [[41/32]], [[64/41]]
* 50/39, 39/25
* [[50/39]], [[39/25]]
* 68/53, 53/34
* [[68/53]], [[53/34]]
* 9/7, 14/9
* [[9/7]], [[14/9]]
* 58/45, 45/29
* [[58/45]], [[45/29]]
* 49/38, 76/49
* [[49/38]], [[76/49]]
* 40/31, 31/20
* [[40/31]], [[31/20]]
* 31/24, 48/31
* [[31/24]], [[48/31]]
* 53/41, 82/53
* [[53/41]], [[82/53]]
* 22/17, 17/11
* [[22/17]], [[17/11]]
* '''57/44, 88/57'''
* '''[[57/44]], [[88/57]]'''
* 35/27, 54/35
* [[35/27]], [[54/35]]
* 48/37, 37/24
* [[48/37]], [[37/24]]
* '''74/57, 57/37'''
* '''[[74/57]], [[57/37]]'''
* 13/10, 20/13
* [[13/10]], [[20/13]]
* 56/43, 43/28
* [[56/43]], [[43/28]]
* 43/33, 66/43
* [[43/33]], [[66/43]]
* 30/23, 23/15
* [[30/23]], [[23/15]]
* 47/36, 72/47
* [[47/36]], [[72/47]]
* 64/49, 49/32
* [[64/49]], [[49/32]]
* 17/13, 26/17
* [[17/13]], [[26/17]]
* 72/55, 55/36
* [[72/55]], [[55/36]]
* 55/42, 84/55
* [[55/42]], [[84/55]]
* 38/29, 29/19
* [[38/29]], [[29/19]]
* 21/16, 32/21
* [[21/16]], [[32/21]]
* 46/35, 35/23
* [[46/35]], [[35/23]]
* 25/19, 38/25
* [[25/19]], [[38/25]]
* 54/41, 41/27
* [[54/41]], [[41/27]]
* 29/22, 44/29
* [[29/22]], [[44/29]]
* 62/47, 47/31
* [[62/47]], [[47/31]]
* 33/25, 50/33
* [[33/25]], [[50/33]]
* 70/53, 53/35
* [[70/53]], [[53/35]]
* 37/28, 56/37
* [[37/28]], [[56/37]]
* 41/31, 62/41
* [[41/31]], [[62/41]]
* 45/34, 68/45
* [[45/34]], [[68/45]]
* 49/37, 74/49
* [[49/37]], [[74/49]]
* 53/40, 80/53
* [[53/40]], [[80/53]]
* '''57/43, 86/57'''
* '''[[57/43]], [[86/57]]'''
* 4/3, 3/2
* [[4/3]], [[3/2]]
* 55/41, 82/55
* [[55/41]], [[82/55]]
* 51/38, 76/51
* [[51/38]], [[76/51]]
* 47/35, 70/47
* [[47/35]], [[70/47]]
* 43/32, 64/43
* [[43/32]], [[64/43]]
* 39/29, 58/39
* [[39/29]], [[58/39]]
* 74/55, 55/37
* [[74/55]], [[55/37]]
* 35/26, 52/35
* [[35/26]], [[52/35]]
* 66/49, 49/33
* [[66/49]], [[49/33]]
* 31/23, 46/31
* [[31/23]], [[46/31]]
* 58/43, 43/29
* [[58/43]], [[43/29]]
* 27/20, 40/27
* [[27/20]], [[40/27]]
* 50/37, 37/25
* [[50/37]], [[37/25]]
* 23/17, 34/23
* [[23/17]], [[34/23]]
* 42/31, 31/21
* [[42/31]], [[31/21]]
* 19/14, 28/19
* [[19/14]], [[28/19]]
* 72/53, 53/36
* [[72/53]], [[53/36]]
* 53/39, 78/53
* [[53/39]], [[78/53]]
* 34/25, 25/17
* [[34/25]], [[25/17]]
* 49/36, 72/49
* [[49/36]], [[72/49]]
* 64/47, 47/32
* [[64/47]], [[47/32]]
* 15/11, 22/15
* [[15/11]], [[22/15]]
* 56/41, 41/28
* [[56/41]], [[41/28]]
* 41/30, 60/41
* [[41/30]], [[60/41]]
* 26/19, 19/13
* [[26/19]], [[19/13]]
* 37/27, 54/37
* [[37/27]], [[54/37]]
* 48/35, 35/24
* [[48/35]], [[35/24]]
* 70/51, 51/35
* [[70/51]], [[51/35]]
* 11/8, 16/11
* [[11/8]], [[16/11]]
* 62/45, 45/31
* [[62/45]], [[45/31]]
* 51/37, 74/51
* [[51/37]], [[74/51]]
* 40/29, 29/20
* [[40/29]], [[29/20]]
* 29/21, 42/29
* [[29/21]], [[42/29]]
* 76/55, 55/38
* [[76/55]], [[55/38]]
* 47/34, 68/47
* [[47/34]], [[68/47]]
* 18/13, 13/9
* [[18/13]], [[13/9]]
* 43/31, 62/43
* [[43/31]], [[62/43]]
* 68/49, 49/34
* [[68/49]], [[49/34]]
* 25/18, 36/25
* [[25/18]], [[36/25]]
* '''57/41, 82/57'''
* '''[[57/41]], [[82/57]]'''
* 32/23, 23/16
* [[32/23]], [[23/16]]
* 39/28, 56/39
* [[39/28]], [[56/39]]
* 46/33, 33/23
* [[46/33]], [[33/23]]
* 53/38, 76/53
* [[53/38]], [[76/53]]
* 60/43, 43/30
* [[60/43]], [[43/30]]
* 74/53, 53/37
* [[74/53]], [[53/37]]
* 7/5, 10/7
* [[7/5]], [[10/7]]
* '''80/57, 57/40'''
* '''[[80/57]], [[57/40]]'''
* 66/47, 47/33
* [[66/47]], [[47/33]]
* 52/37, 37/26
* [[52/37]], [[37/26]]
* 45/32, 64/45
* [[45/32]], [[64/45]]
* 38/27, 27/19
* [[38/27]], [[27/19]]
* 31/22, 44/31
* [[31/22]], [[44/31]]
* 55/39, 78/55
* [[55/39]], [[78/55]]
* 24/17, 17/12
* [[24/17]], [[17/12]]
* 41/29, 58/41
* [[41/29]], [[58/41]]


{| class="wikitable"
{| class="wikitable"
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|twenuno octave
|twenuno octave
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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.
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%).
 
[[Category:Odd limits]]