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41-odd-limit: Difference between revisions

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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.
{{Odd-limit navigation|41}}
{{Odd-limit intro|41}}
 
* [[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]]'''


Below is a list of all octave-reduced intervals in the 41-odd-limit.
{| class="wikitable"
|1/1
|2/1
|-
|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
|}
{| class="wikitable"
{| class="wikitable"
|'''Ratio'''
! Ratio
|'''Size ('''[[Cents|¢]]''')'''
! Size ([[cents|¢]])
|Color name
! Color name
! Name
|-
|-
|42/41
| 42/41
|41.719
| 41.719
|fowuzo 2nd
| fowuzo 2nd
| quadragesimoprimal inframinor second
|-
|-
|41/40
| 41/40
|42.749
| 42.749
|fowogu unison
| fowogu unison
| quadragesimoprimal quartertone
|-
|-
|41/39
| 41/39
|86.58
| 86.58
|fowothu unison
| fowothu unison
| quadragesimoprimal ultraprime
|-
|-
|44/41
| 44/41
|122.256
| 122.256
|fowulo 2nd
| fowulo 2nd
| quadragesimoprimal lesser minor second
|-
|-
|41/38
| 41/38
|131.549
| 131.549
|fowonu unison
| fowonu unison
| quadragesimoprimal hyperprime
|-
|-
|41/37
| 41/37
|177.718
| 177.718
|fowothisu unison
| fowothisu unison
| quadragesimoprimal neutral second
|-
|-
|46/41
| 46/41
|199.212
| 199.212
|fowutwetho 3rd
| fowutwetho 3rd
| quadragesimoprimal minor tone
|-
|-
|41/36
| 41/36
|225.152
| 225.152
|fowo 2nd
| fowo 2nd
| quadragesimoprimal major tone
|-
|-
|48/41
| 48/41
|272.893
| 272.893
|fowu 3rd
| fowu 3rd
| quadragesimoprimal subminor third
|-
|-
|41/35
| 41/35
|273.923
| 273.923
|foworugu 2nd
| foworugu 2nd
| quadragesimoprimal ultramajor second
|-
|-
|41/34
| 41/34
|324.107
| 324.107
|fowosu 2nd
| fowosu 2nd
| quadragesimoprimal minor third
|-
|-
|50/41
| 50/41
|343.565
| 343.565
|fowuyoyo 3rd
| fowuyoyo 3rd
| quadragesimoprimal neutral third
|-
|-
|41/33
| 41/33
|375.789
| 375.789
|fowolu 3rd
| fowolu 3rd
| quadragesimoprimal submajor third
|-
|-
|52/41
| 52/41
|411.465
| 411.465
|fowutho 4th
| fowutho 4th
| quadragesimoprimal lesser major third
|-
|-
|41/32
| 41/32
|429.062
| 429.062
|fowo 3rd
| fowo 3rd
| quadragesimoprimal greater major third
|-
|-
|54/41
| 54/41
|476.803
| 476.803
|fowu 4th
| fowu 4th
| quadragesimoprimal lesser minor fourth
|-
|-
|41/31
| 41/31
|484.027
| 484.027
|fowothiwu 4th
| fowothiwu 4th
| quadragesimoprimal greater minor fourth
|-
|-
|56/41
| 56/41
|539.764
| 539.764
|fowuzo 5th
| fowuzo 5th
| quadragesimoprimal subdiminished fifth
|-
|-
|41/30
| 41/30
|540.794
| 540.794
|fowogu 4th
| fowogu 4th
| quadragesimoprimal major fourth
|-
|-
|41/29
| 41/29
|599.485
| 599.485
|fowotwenu 4th
| fowotwenu 4th
| quadragesimoprimal lesser tritone
|-
|-
|58/41
| 58/41
|600.515
| 600.515
|fowutweno 5th
| fowutweno 5th
| quadragesimoprimal greater tritone
|-
|-
|60/41
| 60/41
|659.206
| 659.206
|fowuyo 5th
| fowuyo 5th
| quadragesimoprimal lesser minor fifth
|-
|-
|41/28
| 41/28
|660.236
| 660.236
|foworu 4th
| foworu 4th
| quadragesimoprimal superaugmented fourth
|-
|-
|62/41
| 62/41
|715.973
| 715.973
|fowuthiwo 5th
| fowuthiwo 5th
| quadragesimoprimal lesser major fifth
|-
|-
|41/27
| 41/27
|723.197
| 723.197
|fowo 5th
| fowo 5th
| quadragesimoprimal greater major fifth
|-
|-
|64/41
| 64/41
|770.938
| 770.938
|fowu 6th
| fowu 6th
| quadragesimoprimal lesser minor sixth
|-
|-
|41/26
| 41/26
|788.535
| 788.535
|fowothu 5th
| fowothu 5th
| quadragesimoprimal augmented fifth
|-
|-
|66/41
| 66/41
|824.211
| 824.211
|fowulo 6th
| fowulo 6th
| quadragesimoprimal greater minor sixth
|-
|-
|41/25
| 41/25
|856.435
| 856.435
|fowogugu 6th
| fowogugu 6th
| quadragesimoprimal neutral sixth
|-
|-
|68/41
| 68/41
|875.893
| 875.893
|fowuso 7th
| fowuso 7th
| quadragesimoprimal submajor sixth
|-
|-
|70/41
| 70/41
|926.077
| 926.077
|fowuzoyo 7th
| fowuzoyo 7th
| quadragesimoprimal lesser major sixth
|-
|-
|41/24
| 41/24
|927.107
| 927.107
|fowo 6th
| fowo 6th
| quadragesimoprimal greater major sixth
|-
|-
|72/41
| 72/41
|974.848
| 974.848
|fowu 7th
| fowu 7th
| quadragesimoprimal lesser minor seventh
|-
|-
|41/23
| 41/23
|1000.788
| 1000.788
|fowotwethu 6th
| fowotwethu 6th
| quadragesimoprimal hypermajor sixth
|-
|-
|74/41
| 74/41
|1022.282
| 1022.282
|fowuthiso octave
| fowuthiso octave
| quadragesimoprimal greater minor seventh
|-
|-
|76/41
| 76/41
|1068.451
| 1068.451
|fowuno octave
| fowuno octave
| quadragesimoprimal infraoctave
|-
|-
|41/22
| 41/22
|1077.744
| 1077.744
|fowolu 7th
| fowolu 7th
| quadragesimoprimal lesser major seventh
|-
|-
|78/41
| 78/41
|1113.42
| 1113.42
|fowutho octave
| fowutho octave
| quadragesimoprimal greater major seventh
|-
|-
|80/41
| 80/41
|1157.251
| 1157.251
|fowuyo octave
| fowuyo octave
| quadragesimoprimal ultramajor seventh
|-
|-
|41/21
| 41/21
|1158.281
| 1158.281
|foworu 7th
| foworu 7th
| quadragesimoprimal ultramajor seventh
|}
|}
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).
The smallest [[equal division of the octave]] which is consistent to the 41-odd-limit is [[311edo]]; The smallest edo that comes closest to being distinct in the 41-odd-limit is [[1600edo]] (misses 50/39, 39/25). that which is truly distinctly consistent to the same is [[2554edo]].
 
[[Category:41-odd-limit| ]] <!-- main article -->
Retrieved from "https://en.xen.wiki/w/41-odd-limit"