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| Quick link
| | == Approximations of odd harmonics == |
| | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| [[User:BudjarnLambeth/Draft related tunings section]]
| | {{harmonics in equal|2|intervals=odd|columns=7}} |
| | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| = Lab = | | {{harmonics in equal|4|intervals=odd|columns=7}} |
| | | {{harmonics in equal|5|intervals=odd|columns=7}} |
| 15edo
| | {{harmonics in equal|6|intervals=odd|columns=7}} |
| * 52ed11
| | {{harmonics in equal|7|intervals=odd|columns=7}} |
| * 11lim WE (79.770)
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| * 50ed10
| | {{harmonics in equal|9|intervals=odd|columns=7}} |
| * 47zpi (79.715)
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| * 54ed12
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| 15edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| 18edo
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| * 42ed5
| | {{harmonics in equal|15|intervals=odd|columns=7}} |
| * 47ed6
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * 60ed10
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| * 65ed12
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| * 7lim WE (66.148)
| | {{harmonics in equal|19|intervals=odd|columns=7}} |
| * 13lim WE (66.291)
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| * 60zpi (67.090)
| | {{harmonics in equal|21|intervals=odd|columns=7}} |
| * 61zpi (66.228)
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| 18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
| | {{harmonics in equal|23|intervals=odd|columns=7}} |
| {{harmonics in equal | 18 | 2 | 1 | intervals=prime}} | | {{harmonics in equal|24|intervals=odd|columns=7}} |
| {{harmonics in equal | 42 | 5 | 1 | intervals=prime}} | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| {{harmonics in equal | 47 | 6 | 1 | intervals=prime}} | | {{harmonics in equal|26|intervals=odd|columns=7}} |
| {{harmonics in equal | 60 | 10 | 1 | intervals=prime}} | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| {{harmonics in equal | 65 | 12 | 1 | intervals=prime} | | {{harmonics in equal|28|intervals=odd|columns=7}} |
| {{harmonics in cet | 66.148 | intervals=prime}} | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| {{harmonics in cet | 66.291 | intervals=prime}} | | {{harmonics in equal|30|intervals=odd|columns=7}} |
| {{harmonics in cet | 67.090 | intervals=prime}} | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| {{harmonics in cet | 66.228 | intervals=prime}} | | {{harmonics in equal|32|intervals=odd|columns=7}} |
| | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| 25edo
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| * 65ed6
| | {{harmonics in equal|35|intervals=odd|columns=7}} |
| * 90ed12
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| * 13lim WE (47.946)
| | {{harmonics in equal|37|intervals=odd|columns=7}} |
| * 95zpi (48.067)
| | {{harmonics in equal|38|intervals=odd|columns=7}} |
| * 96zpi (47.642)
| | {{harmonics in equal|39|intervals=odd|columns=7}} |
| 25edo's [[prime]] 3 is very sharp, and its sharp and flat mapping of 11 and 13 are about equally bad, it can benefit from [[octave shrinking]].
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| {{harmonics in equal | 25 | 2 | 1 | intervals=prime}}
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| {{harmonics in equal | 65 | 6 | 1 | intervals=prime}} | | {{harmonics in equal|42|intervals=odd|columns=7}} |
| {{harmonics in equal | 90 | 12 | 1 | intervals=prime} | | {{harmonics in equal|43|intervals=odd|columns=7}} |
| {{harmonics in cet | 47.946 | intervals=prime}} | | {{harmonics in equal|44|intervals=odd|columns=7}} |
| {{harmonics in cet | 48.067 | intervals=prime}} | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| {{harmonics in cet | 47.642 | intervals=prime}} | | {{harmonics in equal|46|intervals=odd|columns=7}} |
| | | {{harmonics in equal|47|intervals=odd|columns=7}} |
| 26edo
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| * 41edt
| | {{harmonics in equal|49|intervals=odd|columns=7}} |
| * 67ed6
| | {{harmonics in equal|50|intervals=odd|columns=7}} |
| * 86ed10
| | {{harmonics in equal|51|intervals=odd|columns=7}} |
| * 93ed12
| | {{harmonics in equal|52|intervals=odd|columns=7}} |
| * 96ed14
| | {{harmonics in equal|53|intervals=odd|columns=7}} |
| * 13lim WE (46.249) (octave identical to 11lim within 1/20th of a cent)
| |
| * 100zpi (46.268)
| |
| 26edo's simple [[prime]]s with the most error - 3, 5 and 13 - are all tuned flat, so it can benefit from [[octave stretching]].
| |
| {{harmonics in equal | 26 | 2 | 1 | intervals=prime}} | |
| {{harmonics in equal | 41 | 3 | 1 | intervals=prime}} | |
| {{harmonics in equal | 67 | 6 | 1 | intervals=prime}} | |
| {{harmonics in equal | 86 | 10 | 1 | intervals=prime}} | |
| {{harmonics in equal | 93 | 12 | 1 | intervals=prime}} | |
| {{harmonics in equal | 96 | 14 | 1 | intervals=prime}} | |
| {{harmonics in cet | 46.249 | intervals=prime}} | |
| {{harmonics in cet | 46.268 | intervals=prime}} | |
| | |
| 29edo
| |
| * 46edt
| |
| * 105ed12
| |
| * 96ed10
| |
| * 100ed11
| |
| * 107ed13
| |
| * 16edf
| |
| * 11lim WE (41.482)
| |
| * 13lim WE (41.484)
| |
| * [[116zpi]] (41.465)
| |
| 29edo's [[prime]]s 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from [[octave stretching]].
| |
| {{harmonics in equal | 29 | 2 | 1 | intervals=prime}} | |
| {{harmonics in equal | 46 | 3 | 1 | intervals=prime}} | |
| {{harmonics in equal | 96 | 10 | 1 | intervals=prime}} | |
| {{harmonics in equal | 100 | 11 | 1 | intervals=prime}} | |
| {{harmonics in equal | 105 | 12 | 1 | intervals=prime}} | |
| {{harmonics in equal | 107 | 13 | 1 | intervals=prime}} | |
| {{harmonics in equal | 16 | 3 | 2 | intervals=prime}} | |
| {{harmonics in cet | 41.482 | intervals=prime}} | |
| {{harmonics in cet | 41.484 | intervals=prime}} | |
| {{harmonics in cet | 41.465 | intervals=prime}} | |
| | |
| 30edo
| |
| * 78ed6
| |
| * 100ed10
| |
| * 104ed11
| |
| * 108ed12
| |
| * 11lim WE (79.770)
| |
| * 13lim WE (39.904)
| |
| * 39.918zpi (39.918)
| |
| 30edo's simple [[prime]]s with the most error - 3, 5 and 11 - are all tuned sharp, so it can benefit from [[octave shrinking]].
| |
| {{harmonics in equal | 30 | 2 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 78 | 6 | 1 | intervals=prime}} | |
| {{harmonics in equal | 100 | 10 | 1 | intervals=prime}} | |
| {{harmonics in equal | 104 | 11 | 1 | intervals=prime}} | |
| {{harmonics in equal | 108 | 12 | 1 | intervals=prime}} | |
| {{harmonics in cet | 79.770 | intervals=prime}} | |
| {{harmonics in cet | 39.904 | intervals=prime}} | |
| {{harmonics in cet | 39.918 | intervals=prime}} | |
| | |
| 34edo
| |
| * 54edt
| |
| * 79ed5
| |
| * 88ed6
| |
| * 108ed9
| |
| * 113ed10
| |
| * 122ed12
| |
| * 126ed13
| |
| * 11lim WE (35.284)
| |
| * 13lim WE (35.276)
| |
| * 144zpi (35.248)
| |
| 34edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, and it has two about equally bad mappings of 7, so 34edo can benefit from [[octave shrinking]].
| |
| {{harmonics in equal | 34 | 2 | 1 | intervals=prime}} | |
| {{harmonics in equal | 54 | 3 | 1 | intervals=prime}} | |
| {{harmonics in equal | 79 | 5 | 1 | intervals=prime}} | |
| {{harmonics in equal | 88 | 6 | 1 | intervals=prime}} | |
| {{harmonics in equal | 108 | 9 | 1 | intervals=prime} | |
| {{harmonics in equal | 113 | 10 | 1 | intervals=prime} | |
| {{harmonics in equal | 122 | 12 | 1 | intervals=prime}} | |
| {{harmonics in equal | 126 | 13 | 1 | intervals=prime}} | |
| {{harmonics in cet | 35.284 | intervals=prime}}
| |
| {{harmonics in cet | 35.276 | intervals=prime}}
| |
| {{harmonics in cet | 35.248 | intervals=prime}}
| |
| | |
| 35edo
| |
| * 81ed5
| |
| * 90ed6
| |
| * 98ed7
| |
| * 116ed10
| |
| * 121ed11
| |
| * 125ed12
| |
| * 11lim WE (35.284)
| |
| * 13lim WE (35.276)
| |
| * [[149zpi]] (34.359)
| |
| 35edo's [[prime]]s 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from [[octave stretching]].
| |
| {{harmonics in equal | 35 | 2 | 1 | intervals=prime}
| |
| {{harmonics in equal | 81 | 5 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 90 | 6 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 98 | 7 | 1 | intervals=prime}
| |
| {{harmonics in equal | 116 | 10 | 1 | intervals=prime}
| |
| {{harmonics in equal | 121 | 11 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 125 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in cet | 35.284 | intervals=prime}}
| |
| {{harmonics in cet | 35.276 | intervals=prime}}
| |
| {{harmonics in cet | 34.359 | intervals=prime}}
| |
| | |
| 37edo
| |
| * 59edt
| |
| * 86ed5
| |
| * 96ed6
| |
| * 104ed7
| |
| * 123ed10
| |
| * 128ed11
| |
| * 133ed12
| |
| * 137ed13
| |
| * 11lim WE (32.377)
| |
| * 13lim WE (32.383)
| |
| * [[161zpi]] (32.408)
| |
| 37edo's [[prime]]s 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
| |
| {{harmonics in equal | 37 | 2 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 59 | 3 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 86 | 5 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 96 | 6 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 104 | 7 | 1 | intervals=prime}
| |
| {{harmonics in equal | 123 | 10 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 128 | 11 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 133 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 137 | 13 | 1 | intervals=prime}}
| |
| {{harmonics in cet | 32.377 | intervals=prime}}
| |
| {{harmonics in cet | 32.383 | intervals=prime}}
| |
| {{harmonics in cet | 32.408 | intervals=prime}}
| |
| | |
| 48edo
| |
| * 76edt
| |
| * 124ed6
| |
| * 152ed9
| |
| * 159ed10
| |
| * 166ed11
| |
| * 172ed12
| |
| * 28edf
| |
| * 11lim WE (25.017)
| |
| * 13lim WE (25.005)
| |
| * 226zpi (25.006)
| |
| Most of 48edo's simple [[prime]]s have low error, but its 5 is substantially flat, so 48edo can benefit from slight [[octave stretching]].
| |
| {{harmonics in equal | 48 | 2 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 76 | 3 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 124 | 6 | 1 | intervals=prime}
| |
| {{harmonics in equal | 152 | 9 | 1 | intervals=prime}
| |
| {{harmonics in equal | 159 | 10 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 166 | 11 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 172 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 28 | 3 | 2 | intervals=prime}}
| |
| {{harmonics in cet | 25.017 | intervals=prime}}
| |
| {{harmonics in cet | 25.005 | intervals=prime}}
| |
| {{harmonics in cet | 25.006 | intervals=prime}}
| |
| | |
| ; Medium-low priority
| |
| | |
| 10edo
| |
| * 16edt
| |
| * 23ed5
| |
| * 26ed6
| |
| * 28ed7
| |
| * 32ed8
| |
| * 33ed10
| |
| * 36ed12
| |
| * 37ed13
| |
| * 6edf
| |
| * 2.3.7.13 WE (119.785)
| |
| * 2.5.7.13 WE (120.358)
| |
| * 13lim WE (119.776)
| |
| * 26zpi (119.899)
| |
| If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]].
| |
| {{harmonics in equal | 10 | 2 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 23 | 5 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 26 | 6 | 1 | intervals=prime}
| |
| {{harmonics in equal | 28 | 7 | 1 | intervals=prime}
| |
| {{harmonics in equal | 32 | 8 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 33 | 10 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 36 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 37 | 13 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 6 | 3 | 2 | intervals=prime}}
| |
| {{harmonics in cet | 119.785 | intervals=prime}}
| |
| {{harmonics in cet | 120.358 | intervals=prime}}
| |
| {{harmonics in cet | 119.776 | intervals=prime}}
| |
| {{harmonics in cet | 119.899 | intervals=prime}}
| |
| | |
| 11edo
| |
| * 27ed6
| |
| * 28ed6
| |
| * 31ed7
| |
| * 35ed9
| |
| * 37ed10
| |
| * 38ed10
| |
| * 38ed12
| |
| * 39ed12
| |
| * 41ed13
| |
| * 2.7.11.13 WE (108.821)
| |
| * 30zpi (108.722)
| |
| 11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. To use its primes 3 or 5, extreme octave shrinking or [[octave stretching]] can be used, at the cost of making the octaves sound significantly weaker.
| |
| {{harmonics in equal | 11 | 2 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 27 | 6 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 28 | 6 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 31 | 7 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 35 | 9 | 1 | intervals=prime}
| |
| {{harmonics in equal | 37 | 10 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 38 | 10 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 38 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 39 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 41 | 13 | 1 | intervals=prime}}
| |
| {{harmonics in cet | 108.821 | intervals=prime}}
| |
| {{harmonics in cet | 108.722 | intervals=prime}}
| |
| | |
| 24edo
| |
| ((13lim WE's octave is only 1/10th of a cent different from 24edo))
| |
| * 38edt
| |
| * 56ed5
| |
| * 62ed6
| |
| * 67ed7
| |
| * 9ed7/6
| |
| * 80ed10
| |
| * 83ed11
| |
| * 86ed12
| |
| * 89ed13
| |
| * 14edf
| |
| * 2.3.5.11.13 WE (49.942)
| |
| * 11lim WE (50.017)
| |
| * 90zpi (49.988)
| |
| If one wishes to use 24edo as a full 19-or-lower-limit tuning, then it benefits from slight [[octave stretching]], mostly to improve its [[prime]] 7. If one wishes to use 24edo as a no-7s 19-or-lower-limit tuning, then it benefits from slight [[octave shrinking]], mostly to improve its primes 5 and 13.
| |
| {{harmonics in equal | 24 | 2 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 38 | 3 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 56 | 5 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 62 | 6 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 67 | 7 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 9 | 7 | 6 | intervals=prime}}
| |
| {{harmonics in equal | 80 | 10 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 83 | 11 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 86 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 89 | 13 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 14 | 3 | 2 | intervals=prime}}
| |
| {{harmonics in cet | 49.942 | intervals=prime}}
| |
| {{harmonics in cet | 50.017 | intervals=prime}}
| |
| {{harmonics in cet | 49.988 | intervals=prime}}
| |
| | |
| 5edo
| |
| * 8edt
| |
| * 13ed6
| |
| * 14ed7
| |
| * 18ed12
| |
| * 3edf
| |
| * 2.3.7 WE (239.426)
| |
| * 9zpi (238.357)
| |
| If one wishes to use 5edo as a 2.3.7 [[subgroup]] tuning, then it benefits from slight [[octave shrinking]] to improve its prime 3.
| |
| {{harmonics in equal | 5 | 2 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 8 | 3 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 13 | 6 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 14 | 7 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 18 | 12 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 3 | 3 | 2 | intervals=prime}}
| |
| {{harmonics in cet | 239.426 | intervals=prime}}
| |
| {{harmonics in cet | 238.357 | intervals=prime}}
| |
| | |
| 6edo
| |
| * 14ed5
| |
| * 17ed7
| |
| * 19ed9
| |
| * 20ed10
| |
| * 2.9.5 WE (199.736)
| |
| * 2.9.5.7 WE (199.329)
| |
| * 12zpi (198.843)
| |
| If one wishes to use 6edo as a 2.9.5 or 2.9.5.7 [[sugroup]] tuning, then it benefits from [[octave shrinking]].
| |
| {{harmonics in equal | 14 | 5 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 17 | 7 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 19 | 9 | 1 | intervals=prime}}
| |
| {{harmonics in equal | 20 | 10 | 1 | intervals=prime}
| |
| {{harmonics in cet | 199.736 | intervals=prime}}
| |
| {{harmonics in cet | 199.329 | intervals=prime}}
| |
| {{harmonics in cet | 198.843 | intervals=prime}}
| |