39-odd-limit: Difference between revisions
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Proposed a name for all the 39-odd intervals (also thanks Overthink for discovering that distinct odd-limit EDO!) |
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|40/39 | |40/39 | ||
|43.831 | |43.831 | ||
| | |tridecimal minor diesis | ||
| | |thuyo 2nd | ||
|- | |- | ||
|39/38 | |39/38 | ||
|44.97 | |44.97 | ||
| | |undevicesimal diesis | ||
| | |nutho 2nd | ||
|- | |- | ||
|39/37 | |39/37 | ||
|91.139 | |91.139 | ||
| | |trigesimoseptimal limma | ||
| | |thisutho 2nd | ||
|- | |- | ||
|39/35 | |39/35 | ||
|187.343 | |187.343 | ||
| | |animist major second | ||
| | |thorugu 2nd | ||
|- | |- | ||
|44/39 | |44/39 | ||
|208.835 | |208.835 | ||
| | |major minthic tone | ||
| | |thulo 2nd | ||
|- | |- | ||
|39/34 | |39/34 | ||
|237.527 | |237.527 | ||
| | |septendecimal supermajor second | ||
| | |sutho 2nd | ||
|- | |- | ||
|46/39 | |46/39 | ||
|285.792 | |285.792 | ||
| | |laodicismic minor third | ||
| | |twethothu 3rd | ||
|- | |||
|39/32 | |||
|342.483 | |||
|lesser tridecimal neutral third | |||
|tho 3rd | |||
|- | |- | ||
|39/31 | |39/31 | ||
|397.447 | |397.447 | ||
| | |trigesimoprimal major third | ||
| | |thiwutho 4th | ||
|- | |- | ||
|50/39 | |50/39 | ||
|430.145 | |430.145 | ||
| | |major minthmic supermajor third | ||
| | |thuyoyo 3rd | ||
|- | |- | ||
|39/29 | |39/29 | ||
|512.905 | |512.905 | ||
| | |vigesimononal acute fourth | ||
| | |twenutho 4th | ||
|- | |- | ||
|39/28 | |39/28 | ||
|573.657 | |573.657 | ||
| | |mynucumic lesser tritone | ||
| | |thoru 4th | ||
|- | |- | ||
|56/39 | |56/39 | ||
|626.343 | |626.343 | ||
| | |mynucumic greater tritone | ||
| | |thuzo 5th | ||
|- | |- | ||
|58/39 | |58/39 | ||
|687.095 | |687.095 | ||
| | |vigesimononal grave fifth | ||
| | |twenothu 5th | ||
|- | |- | ||
|39/25 | |39/25 | ||
|769.855 | |769.855 | ||
| | |major minthmic subminor sixth | ||
| | |thogugu 6th | ||
|- | |- | ||
|62/39 | |62/39 | ||
|802.553 | |802.553 | ||
| | |trigesimoprimal minor sixth | ||
| | |thiwothu 5th | ||
|- | |||
|64/39 | |||
|857.517 | |||
|greater tridecimal neutral sixth | |||
|thu 6th | |||
|- | |- | ||
|39/23 | |39/23 | ||
|914.208 | |914.208 | ||
| | |laodicismic major sixth | ||
| | |twethutho 6th | ||
|- | |- | ||
|68/39 | |68/39 | ||
|962.473 | |962.473 | ||
| | |septendecimal subminor seventh | ||
| | |sothu 7th | ||
|- | |- | ||
|39/22 | |39/22 | ||
|991.165 | |991.165 | ||
| | |major minthic minor seventh | ||
| | |tholu 7th | ||
|- | |- | ||
|70/39 | |70/39 | ||
|1012.657 | |1012.657 | ||
| | |animist minor seventh | ||
| | |thuzoyo 7th | ||
|- | |- | ||
|74/39 | |74/39 | ||
|1108.861 | |1108.861 | ||
| | |trigesimoseptimal major seventh | ||
| | |thisothu octave | ||
|- | |- | ||
|76/39 | |76/39 | ||
|1155.03 | |1155.03 | ||
| | |vigesimononal suboctave | ||
| | |nothu octave | ||
|- | |- | ||
|39/20 | |39/20 | ||
|1156.169 | |1156.169 | ||
| | |tridecimal suboctave | ||
| | |thogu octave | ||
|} | |} | ||
The smallest [[equal division of the octave]] which is consistent to the 39-odd-limit is [[311edo]] (by virtue of it being consistent in the [[41-odd-limit]]); that which is distinctly consistent to the same is [[ | The smallest [[equal division of the octave]] which is consistent to the 39-odd-limit is [[311edo]] (by virtue of it being consistent in the [[41-odd-limit]]); that which is distinctly consistent to the same is [[2554edo]]. | ||
Revision as of 15:21, 23 September 2025
The 39-odd-limit is the set of all rational intervals which can be written as 2k(a/b) where a, b ≤ 39 and k is an integer. To the 37-odd-limit, it adds 11 pairs of octave-reduced intervals involving 39.
Below is a list of all octave-reduced intervals in the 39-odd-limit.
- 1/1
- 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
- 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
- 29/27, 54/29
- 14/13, 13/7
- 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
- 10/9, 9/5
- 39/35, 70/39
- 29/26, 52/29
- 19/17, 34/19
- 28/25, 25/14
- 37/33, 66/37
- 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
- 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
- 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
- 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
- 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
- 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
- 33/26, 52/33
- 14/11, 11/7
- 37/29, 58/37
- 23/18, 36/23
- 32/25, 25/16
- 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
- 29/22, 44/29
- 33/25, 50/33
- 37/28, 56/37
- 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
- 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
| Ratio | Size (¢) | Color name | Name |
| 40/39 | 43.831 | tridecimal minor diesis | thuyo 2nd |
| 39/38 | 44.97 | undevicesimal diesis | nutho 2nd |
| 39/37 | 91.139 | trigesimoseptimal limma | thisutho 2nd |
| 39/35 | 187.343 | animist major second | thorugu 2nd |
| 44/39 | 208.835 | major minthic tone | thulo 2nd |
| 39/34 | 237.527 | septendecimal supermajor second | sutho 2nd |
| 46/39 | 285.792 | laodicismic minor third | twethothu 3rd |
| 39/32 | 342.483 | lesser tridecimal neutral third | tho 3rd |
| 39/31 | 397.447 | trigesimoprimal major third | thiwutho 4th |
| 50/39 | 430.145 | major minthmic supermajor third | thuyoyo 3rd |
| 39/29 | 512.905 | vigesimononal acute fourth | twenutho 4th |
| 39/28 | 573.657 | mynucumic lesser tritone | thoru 4th |
| 56/39 | 626.343 | mynucumic greater tritone | thuzo 5th |
| 58/39 | 687.095 | vigesimononal grave fifth | twenothu 5th |
| 39/25 | 769.855 | major minthmic subminor sixth | thogugu 6th |
| 62/39 | 802.553 | trigesimoprimal minor sixth | thiwothu 5th |
| 64/39 | 857.517 | greater tridecimal neutral sixth | thu 6th |
| 39/23 | 914.208 | laodicismic major sixth | twethutho 6th |
| 68/39 | 962.473 | septendecimal subminor seventh | sothu 7th |
| 39/22 | 991.165 | major minthic minor seventh | tholu 7th |
| 70/39 | 1012.657 | animist minor seventh | thuzoyo 7th |
| 74/39 | 1108.861 | trigesimoseptimal major seventh | thisothu octave |
| 76/39 | 1155.03 | vigesimononal suboctave | nothu octave |
| 39/20 | 1156.169 | tridecimal suboctave | thogu octave |
The smallest equal division of the octave which is consistent to the 39-odd-limit is 311edo (by virtue of it being consistent in the 41-odd-limit); that which is distinctly consistent to the same is 2554edo.