41-odd-limit: Difference between revisions

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Birth (is 20567edo truly the smallest edo consistent in the 41-odd-limit??? I haven't verified it)
 
Eufalesio (talk | contribs)
autopatrolled
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fixed incorrect format, missing links on all intervals
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The '''41-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'' ≤ 31 and ''k'' is an integer. To the 39-odd-limit, it adds 20 pairs of [[octave-reduced]] intervals involving 41.
The '''41-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'' ≤ 31 and ''k'' is an integer. To the [[39-odd-limit]], it adds 20 pairs of [[octave-reduced]] intervals involving 41.


Below is a list of all octave-reduced intervals in the 41-odd-limit.
Below is a list of all octave-reduced intervals in the 41-odd-limit.
{| class="wikitable"
 
|1/1
* 1/1
|2/1
* '''42/41, 41/21'''
|-
* '''41/40, 80/41'''
|42/41
* 40/39, 39/20
|41/21
* 39/38, 76/39
|-
* 38/37, 37/19
|41/40
* 37/36, 72/37
|80/41
* 36/35, 35/18
|-
* 35/34, 68/35
|40/39
* 34/33, 33/17
|39/20
* 33/32, 64/33
|-
* 32/31, 31/16
|39/38
* 31/30, 60/31
|76/39
* 30/29, 29/15
|-
* 29/28, 56/29
|38/37
* 28/27, 27/14
|37/19
* 27/26, 52/27
|-
* 26/25, 25/13
|37/36
* 25/24, 48/25
|72/37
* 24/23, 23/12
|-
* 23/22, 44/23
|36/35
* 22/21, 21/11
|35/18
* 21/20, 40/21
|-
* '''41/39, 78/41'''
|35/34
* 20/19, 19/10
|68/35
* 39/37, 74/39
|-
* 19/18, 36/19
|34/33
* 37/35, 70/37
|33/17
* 18/17, 17/9
|-
* 35/33, 66/35
|33/32
* 17/16, 32/17
|64/33
* 33/31, 62/33
|-
* 16/15, 15/8
|32/31
* 31/29, 58/31
|31/16
* 15/14, 28/15
|-
* '''44/41, 41/22'''
|31/30
* 29/27, 54/29
|60/31
* 14/13, 13/7
|-
* '''41/38, 76/41'''
|30/29
* 27/25, 50/27
|29/15
* 40/37, 37/20
|-
* 13/12, 24/13
|29/28
* 38/35, 35/19
|56/29
* 25/23, 46/25
|-
* 37/34, 68/37
|28/27
* 12/11, 11/6
|27/14
* 35/32, 64/35
|-
* 23/21, 42/23
|27/26
* 34/31, 31/17
|52/27
* 11/10, 20/11
|-
* 32/29, 29/16
|26/25
* 21/19, 38/21
|25/13
* 31/28, 56/31
|-
* '''41/37, 74/41'''
|25/24
* 10/9, 9/5
|48/25
* 39/35, 70/39
|-
* 29/26, 52/29
|24/23
* 19/17, 34/19
|23/12
* 28/25, 25/14
|-
* 37/33, 66/37
|23/22
* '''46/41, 41/23'''
|44/23
* 9/8, 16/9
|-
* 44/39, 39/22
|22/21
* 35/31, 62/35
|21/11
* 26/23, 23/13
|-
* 17/15, 30/17
|21/20
* 42/37, 37/21
|40/21
* 25/22, 44/25
|-
* 33/29, 58/33
|41/39
* '''41/36, 72/41'''
|78/41
* 8/7, 7/4
|-
* 39/34, 68/39
|20/19
* 31/27, 54/31
|19/10
* 23/20, 40/23
|-
* 38/33, 33/19
|39/37
* 15/13, 26/15
|74/39
* 37/32, 64/37
|-
* 22/19, 19/11
|19/18
* 29/25, 50/29
|36/19
* 36/31, 31/18
|-
* 7/6, 12/7
|37/35
* '''48/41, 41/24'''
|70/37
* '''41/35, 70/41'''
|-
* 34/29, 29/17
|18/17
* 27/23, 46/27
|17/9
* 20/17, 17/10
|-
* 33/28, 56/33
|35/33
* 46/39, 39/23
|66/35
* 13/11, 22/13
|-
* 32/27, 27/16
|17/16
* 19/16, 32/19
|32/17
* 44/37, 37/22
|-
* 25/21, 42/25
|33/31
* 31/26, 52/31
|62/33
* 37/31, 62/37
|-
* 6/5, 5/3
|16/15
* '''41/34, 68/41'''
|15/8
* 35/29, 58/35
|-
* 29/24, 48/29
|31/29
* 23/19, 38/23
|58/31
* 40/33, 33/20
|-
* 17/14, 28/17
|15/14
* 28/23, 23/14
|28/15
* 39/32, 64/39
|-
* '''50/41, 41/25'''
|44/41
* 11/9, 18/11
|41/22
* 38/31, 31/19
|-
* 27/22, 44/27
|29/27
* 16/13, 13/8
|54/29
* 37/30, 60/37
|-
* 21/17, 34/21
|14/13
* 26/21, 21/13
|13/7
* 31/25, 50/31
|-
* 36/29, 29/18
|41/38
* '''41/33, 66/41'''
|76/41
* 46/37, 37/23
|-
* 5/4, 8/5
|27/25
* 44/35, 35/22
|50/27
* 39/31, 62/39
|-
* 34/27, 27/17
|40/37
* 29/23, 46/29
|37/20
* 24/19, 19/12
|-
* 19/15, 30/19
|13/12
* '''52/41, 41/26'''
|24/13
* 33/26, 52/33
|-
* 14/11, 11/7
|38/35
* 37/29, 58/37
|35/19
* 23/18, 36/23
|-
* 32/25, 25/16
|25/23
* '''41/32, 64/41'''
|46/25
* 50/39, 39/25
|-
* 9/7, 14/9
|37/34
* 40/31, 31/20
|68/37
* 31/24, 48/31
|-
* 22/17, 17/11
|12/11
* 35/27, 54/35
|11/6
* 48/37, 37/24
|-
* 13/10, 20/13
|35/32
* 30/23, 23/15
|64/35
* 17/13, 26/17
|-
* 38/29, 29/19
|23/21
* 21/16, 32/21
|42/23
* 46/35, 35/23
|-
* 25/19, 38/25
|34/31
* '''54/41, 41/27'''
|31/17
* 29/22, 44/29
|-
* 33/25, 50/33
|11/10
* 37/28, 56/37
|20/11
* '''41/31, 62/41'''
|-
* 4/3, 3/2
|32/29
* 39/29, 58/39
|29/16
* 35/26, 52/35
|-
* 31/23, 46/31
|21/19
* 27/20, 40/27
|38/21
* 50/37, 37/25
|-
* 23/17, 34/23
|31/28
* 42/31, 31/21
|56/31
* 19/14, 28/19
|-
* 34/25, 25/17
|41/37
* 15/11, 22/15
|74/41
* '''56/41, 41/28'''
|-
* '''41/30, 60/41'''
|10/9
* 26/19, 19/13
|9/5
* 37/27, 54/37
|-
* 48/35, 35/24
|39/35
* 11/8, 16/11
|70/39
* 40/29, 29/20
|-
* 29/21, 42/29
|29/26
* 18/13, 13/9
|52/29
* 25/18, 36/25
|-
* 32/23, 23/16
|19/17
* 39/28, 56/39
|34/19
* 46/33, 33/23
|-
* 7/5, 10/7
|28/25
* 52/37, 37/26
|25/14
* 38/27, 27/19
|-
* 31/22, 44/31
|37/33
* 24/17, 17/12
|66/37
* '''41/29, 58/41'''
|-
 
|46/41
|41/23
|-
|9/8
|16/9
|-
|44/39
|39/22
|-
|35/31
|62/35
|-
|26/23
|23/13
|-
|17/15
|30/17
|-
|42/37
|37/21
|-
|25/22
|44/25
|-
|33/29
|58/33
|-
|41/36
|72/41
|-
|8/7
|7/4
|-
|39/34
|68/39
|-
|31/27
|54/31
|-
|23/20
|40/23
|-
|38/33
|33/19
|-
|15/13
|26/15
|-
|37/32
|64/37
|-
|22/19
|19/11
|-
|29/25
|50/29
|-
|36/31
|31/18
|-
|7/6
|12/7
|-
|48/41
|41/24
|-
|41/35
|70/41
|-
|34/29
|29/17
|-
|27/23
|46/27
|-
|20/17
|17/10
|-
|33/28
|56/33
|-
|46/39
|39/23
|-
|13/11
|22/13
|-
|32/27
|27/16
|-
|19/16
|32/19
|-
|44/37
|37/22
|-
|25/21
|42/25
|-
|31/26
|52/31
|-
|37/31
|62/37
|-
|6/5
|5/3
|-
|41/34
|68/41
|-
|35/29
|58/35
|-
|29/24
|48/29
|-
|23/19
|38/23
|-
|40/33
|33/20
|-
|17/14
|28/17
|-
|28/23
|23/14
|-
|39/32
|64/39
|-
|50/41
|41/25
|-
|11/9
|18/11
|-
|38/31
|31/19
|-
|27/22
|44/27
|-
|16/13
|'''13/8'''
|-
|37/30
|60/37
|-
|21/17
|34/21
|-
|26/21
|21/13
|-
|31/25
|50/31
|-
|36/29
|29/18
|-
|41/33
|66/41
|-
|46/37
|37/23
|-
|5/4
|8/5
|-
|44/35
|35/22
|-
|39/31
|62/39
|-
|34/27
|27/17
|-
|29/23
|46/29
|-
|24/19
|19/12
|-
|19/15
|30/19
|-
|52/41
|41/26
|-
|33/26
|52/33
|-
|14/11
|11/7
|-
|37/29
|58/37
|-
|23/18
|36/23
|-
|32/25
|25/16
|-
|41/32
|64/41
|-
|50/39
|39/25
|-
|9/7
|14/9
|-
|40/31
|31/20
|-
|31/24
|48/31
|-
|22/17
|17/11
|-
|35/27
|54/35
|-
|48/37
|37/24
|-
|13/10
|20/13
|-
|30/23
|23/15
|-
|17/13
|26/17
|-
|38/29
|29/19
|-
|21/16
|32/21
|-
|46/35
|35/23
|-
|25/19
|38/25
|-
|54/41
|41/27
|-
|29/22
|44/29
|-
|33/25
|50/33
|-
|37/28
|56/37
|-
|41/31
|62/41
|-
|4/3
|3/2
|-
|39/29
|58/39
|-
|35/26
|52/35
|-
|31/23
|46/31
|-
|27/20
|40/27
|-
|50/37
|37/25
|-
|23/17
|34/23
|-
|42/31
|31/21
|-
|19/14
|28/19
|-
|34/25
|25/17
|-
|15/11
|22/15
|-
|56/41
|41/28
|-
|41/30
|60/41
|-
|26/19
|19/13
|-
|37/27
|54/37
|-
|48/35
|35/24
|-
|11/8
|16/11
|-
|40/29
|29/20
|-
|29/21
|42/29
|-
|18/13
|13/9
|-
|25/18
|36/25
|-
|32/23
|23/16
|-
|39/28
|56/39
|-
|46/33
|33/23
|-
|7/5
|10/7
|-
|52/37
|37/26
|-
|38/27
|27/19
|-
|31/22
|44/31
|-
|24/17
|17/12
|-
|41/29
|58/41
|}
{| class="wikitable"
{| class="wikitable"
|'''Ratio'''
|'''Ratio'''

Revision as of 18:32, 16 September 2025

The 41-odd-limit is the set of all rational intervals which can be written as 2k(a/b) where a, b ≤ 31 and k is an integer. To the 39-odd-limit, it adds 20 pairs of octave-reduced intervals involving 41.

Below is a list of all octave-reduced intervals in the 41-odd-limit.

  • 1/1
  • 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
  • 28/27, 27/14
  • 27/26, 52/27
  • 26/25, 25/13
  • 25/24, 48/25
  • 24/23, 23/12
  • 23/22, 44/23
  • 22/21, 21/11
  • 21/20, 40/21
  • 41/39, 78/41
  • 20/19, 19/10
  • 39/37, 74/39
  • 19/18, 36/19
  • 37/35, 70/37
  • 18/17, 17/9
  • 35/33, 66/35
  • 17/16, 32/17
  • 33/31, 62/33
  • 16/15, 15/8
  • 31/29, 58/31
  • 15/14, 28/15
  • 44/41, 41/22
  • 29/27, 54/29
  • 14/13, 13/7
  • 41/38, 76/41
  • 27/25, 50/27
  • 40/37, 37/20
  • 13/12, 24/13
  • 38/35, 35/19
  • 25/23, 46/25
  • 37/34, 68/37
  • 12/11, 11/6
  • 35/32, 64/35
  • 23/21, 42/23
  • 34/31, 31/17
  • 11/10, 20/11
  • 32/29, 29/16
  • 21/19, 38/21
  • 31/28, 56/31
  • 41/37, 74/41
  • 10/9, 9/5
  • 39/35, 70/39
  • 29/26, 52/29
  • 19/17, 34/19
  • 28/25, 25/14
  • 37/33, 66/37
  • 46/41, 41/23
  • 9/8, 16/9
  • 44/39, 39/22
  • 35/31, 62/35
  • 26/23, 23/13
  • 17/15, 30/17
  • 42/37, 37/21
  • 25/22, 44/25
  • 33/29, 58/33
  • 41/36, 72/41
  • 8/7, 7/4
  • 39/34, 68/39
  • 31/27, 54/31
  • 23/20, 40/23
  • 38/33, 33/19
  • 15/13, 26/15
  • 37/32, 64/37
  • 22/19, 19/11
  • 29/25, 50/29
  • 36/31, 31/18
  • 7/6, 12/7
  • 48/41, 41/24
  • 41/35, 70/41
  • 34/29, 29/17
  • 27/23, 46/27
  • 20/17, 17/10
  • 33/28, 56/33
  • 46/39, 39/23
  • 13/11, 22/13
  • 32/27, 27/16
  • 19/16, 32/19
  • 44/37, 37/22
  • 25/21, 42/25
  • 31/26, 52/31
  • 37/31, 62/37
  • 6/5, 5/3
  • 41/34, 68/41
  • 35/29, 58/35
  • 29/24, 48/29
  • 23/19, 38/23
  • 40/33, 33/20
  • 17/14, 28/17
  • 28/23, 23/14
  • 39/32, 64/39
  • 50/41, 41/25
  • 11/9, 18/11
  • 38/31, 31/19
  • 27/22, 44/27
  • 16/13, 13/8
  • 37/30, 60/37
  • 21/17, 34/21
  • 26/21, 21/13
  • 31/25, 50/31
  • 36/29, 29/18
  • 41/33, 66/41
  • 46/37, 37/23
  • 5/4, 8/5
  • 44/35, 35/22
  • 39/31, 62/39
  • 34/27, 27/17
  • 29/23, 46/29
  • 24/19, 19/12
  • 19/15, 30/19
  • 52/41, 41/26
  • 33/26, 52/33
  • 14/11, 11/7
  • 37/29, 58/37
  • 23/18, 36/23
  • 32/25, 25/16
  • 41/32, 64/41
  • 50/39, 39/25
  • 9/7, 14/9
  • 40/31, 31/20
  • 31/24, 48/31
  • 22/17, 17/11
  • 35/27, 54/35
  • 48/37, 37/24
  • 13/10, 20/13
  • 30/23, 23/15
  • 17/13, 26/17
  • 38/29, 29/19
  • 21/16, 32/21
  • 46/35, 35/23
  • 25/19, 38/25
  • 54/41, 41/27
  • 29/22, 44/29
  • 33/25, 50/33
  • 37/28, 56/37
  • 41/31, 62/41
  • 4/3, 3/2
  • 39/29, 58/39
  • 35/26, 52/35
  • 31/23, 46/31
  • 27/20, 40/27
  • 50/37, 37/25
  • 23/17, 34/23
  • 42/31, 31/21
  • 19/14, 28/19
  • 34/25, 25/17
  • 15/11, 22/15
  • 56/41, 41/28
  • 41/30, 60/41
  • 26/19, 19/13
  • 37/27, 54/37
  • 48/35, 35/24
  • 11/8, 16/11
  • 40/29, 29/20
  • 29/21, 42/29
  • 18/13, 13/9
  • 25/18, 36/25
  • 32/23, 23/16
  • 39/28, 56/39
  • 46/33, 33/23
  • 7/5, 10/7
  • 52/37, 37/26
  • 38/27, 27/19
  • 31/22, 44/31
  • 24/17, 17/12
  • 41/29, 58/41
Ratio Size (¢) Color name
42/41 41.719 fowuzo 2nd
41/40 42.749 fowogu unison
41/39 86.58 fowothu unison
44/41 122.256 fowulo 2nd
41/38 131.549 fowonu unison
41/37 177.718 fowothisu unison
46/41 199.212 fowutwetho 3rd
41/36 225.152 fowo 2nd
48/41 272.893 fowu 3rd
41/35 273.923 foworugu 2nd
41/34 324.107 fowosu 2nd
50/41 343.565 fowuyoyo 3rd
41/33 375.789 fowolu 3rd
52/41 411.465 fowutho 4th
41/32 429.062 fowo 3rd
54/41 476.803 fowu 4th
41/31 484.027 fowothiwu 4th
56/41 539.764 fowuzo 5th
41/30 540.794 fowogu 4th
41/29 599.485 fowotwenu 4th
58/41 600.515 fowutweno 5th
60/41 659.206 fowuyo 5th
41/28 660.236 foworu 4th
62/41 715.973 fowuthiwo 5th
41/27 723.197 fowo 5th
64/41 770.938 fowu 6th
41/26 788.535 fowothu 5th
66/41 824.211 fowulo 6th
41/25 856.435 fowogugu 6th
68/41 875.893 fowuso 7th
70/41 926.077 fowuzoyo 7th
41/24 927.107 fowo 6th
72/41 974.848 fowu 7th
41/23 1000.788 fowotwethu 6th
74/41 1022.282 fowuthiso octave
76/41 1068.451 fowuno octave
41/22 1077.744 fowolu 7th
78/41 1113.42 fowutho octave
80/41 1157.251 fowuyo octave
41/21 1158.281 foworu 7th

The smallest equal division of the octave which is consistent to the 41-odd-limit is 311edo; that which is distinctly consistent to the same is 20567edo (by virtue of it being distinctly consistent through the 57-odd-limit).

Retrieved from "https://en.xen.wiki/index.php?title=41-odd-limit&oldid=210020"