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== 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}}

Latest revision as of 03:35, 27 April 2026

Approximations of odd harmonics

Approximation of odd harmonics in 1edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +498 -386 +231 -204 -551 +359 +112
Relative (%) +41.5 -32.2 +19.3 -17.0 -45.9 +30.0 +9.3
Step 2 2 3 3 3 4 4
Approximation of odd harmonics in 2edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -102 +214 +231 -204 +49 -241 +112
Relative (%) -17.0 +35.6 +38.5 -34.0 +8.1 -40.1 +18.6
Steps
(reduced) 		3
(1) 		5
(1) 		6
(0) 		6
(0) 		7
(1) 		7
(1) 		8
(0)
Approximation of odd harmonics in 3edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +98 +14 -169 +196 -151 -41 +112
Relative (%) +24.5 +3.4 -42.2 +49.0 -37.8 -10.1 +27.9
Steps
(reduced) 		5
(2) 		7
(1) 		8
(2) 		10
(1) 		10
(1) 		11
(2) 		12
(0)
Approximation of odd harmonics in 4edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -102 -86 -69 +96 +49 +59 +112
Relative (%) -34.0 -28.8 -22.9 +32.0 +16.2 +19.8 +37.2
Steps
(reduced) 		6
(2) 		9
(1) 		11
(3) 		13
(1) 		14
(2) 		15
(3) 		16
(0)
Approximation of odd harmonics in 5edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +18 +94 -9 +36 -71 +119 +112
Relative (%) +7.5 +39.0 -3.7 +15.0 -29.7 +49.8 +46.6
Steps
(reduced) 		8
(3) 		12
(2) 		14
(4) 		16
(1) 		17
(2) 		19
(4) 		20
(0)
Approximation of odd harmonics in 6edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +98.0 +13.7 +31.2 -3.9 +48.7 -40.5 -88.3
Relative (%) +49.0 +6.8 +15.6 -2.0 +24.3 -20.3 -44.1
Steps
(reduced) 		10
(4) 		14
(2) 		17
(5) 		19
(1) 		21
(3) 		22
(4) 		23
(5)
Approximation of odd harmonics in 7edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -16.2 -43.5 +59.7 -32.5 -37.0 +16.6 -59.7
Relative (%) -9.5 -25.3 +34.9 -18.9 -21.6 +9.7 -34.8
Steps
(reduced) 		11
(4) 		16
(2) 		20
(6) 		22
(1) 		24
(3) 		26
(5) 		27
(6)
Approximation of odd harmonics in 8edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +48.0 +63.7 -68.8 -53.9 +48.7 +59.5 -38.3
Relative (%) +32.0 +42.5 -45.9 -35.9 +32.5 +39.6 -25.5
Steps
(reduced) 		13
(5) 		19
(3) 		22
(6) 		25
(1) 		28
(4) 		30
(6) 		31
(7)
Approximation of odd harmonics in 9edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -35.3 +13.7 -35.5 +62.8 -18.0 -40.5 -21.6
Relative (%) -26.5 +10.3 -26.6 +47.1 -13.5 -30.4 -16.2
Steps
(reduced) 		14
(5) 		21
(3) 		25
(7) 		29
(2) 		31
(4) 		33
(6) 		35
(8)
Approximation of odd harmonics in 10edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +18.0 -26.3 -8.8 +36.1 +48.7 -0.5 -8.3
Relative (%) +15.0 -21.9 -7.4 +30.1 +40.6 -0.4 -6.9
Steps
(reduced) 		16
(6) 		23
(3) 		28
(8) 		32
(2) 		35
(5) 		37
(7) 		39
(9)
Approximation of odd harmonics in 11edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -47.4 +50.0 +13.0 +14.3 -5.9 +32.2 +2.6
Relative (%) -43.5 +45.9 +11.9 +13.1 -5.4 +29.5 +2.4
Steps
(reduced) 		17
(6) 		26
(4) 		31
(9) 		35
(2) 		38
(5) 		41
(8) 		43
(10)
Approximation of odd harmonics in 12edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -2.0 +13.7 +31.2 -3.9 +48.7 -40.5 +11.7
Relative (%) -2.0 +13.7 +31.2 -3.9 +48.7 -40.5 +11.7
Steps
(reduced) 		19
(7) 		28
(4) 		34
(10) 		38
(2) 		42
(6) 		44
(8) 		47
(11)
Approximation of odd harmonics in 13edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +36.5 -17.1 -45.7 -19.3 +2.5 -9.8 +19.4
Relative (%) +39.5 -18.5 -49.6 -20.9 +2.7 -10.6 +21.0
Steps
(reduced) 		21
(8) 		30
(4) 		36
(10) 		41
(2) 		45
(6) 		48
(9) 		51
(12)
Approximation of odd harmonics in 14edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -16.2 +42.3 -26.0 -32.5 -37.0 +16.6 +26.0
Relative (%) -18.9 +49.3 -30.3 -37.9 -43.2 +19.4 +30.4
Steps
(reduced) 		22
(8) 		33
(5) 		39
(11) 		44
(2) 		48
(6) 		52
(10) 		55
(13)
Approximation of odd harmonics in 15edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +18.0 +13.7 -8.8 +36.1 +8.7 +39.5 +31.7
Relative (%) +22.6 +17.1 -11.0 +45.1 +10.9 +49.3 +39.7
Steps
(reduced) 		24
(9) 		35
(5) 		42
(12) 		48
(3) 		52
(7) 		56
(11) 		59
(14)
Approximation of odd harmonics in 16edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -27.0 -11.3 +6.2 +21.1 -26.3 -15.5 +36.7
Relative (%) -35.9 -15.1 +8.2 +28.1 -35.1 -20.7 +49.0
Steps
(reduced) 		25
(9) 		37
(5) 		45
(13) 		51
(3) 		55
(7) 		59
(11) 		63
(15)
Approximation of odd harmonics in 17edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +3.9 -33.4 +19.4 +7.9 +13.4 +6.5 -29.4
Relative (%) +5.6 -47.3 +27.5 +11.1 +19.0 +9.3 -41.7
Steps
(reduced) 		27
(10) 		39
(5) 		48
(14) 		54
(3) 		59
(8) 		63
(12) 		66
(15)
Approximation of odd harmonics in 18edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +31.4 +13.7 +31.2 -3.9 -18.0 +26.1 -21.6
Relative (%) +47.1 +20.5 +46.8 -5.9 -27.0 +39.2 -32.4
Steps
(reduced) 		29
(11) 		42
(6) 		51
(15) 		57
(3) 		62
(8) 		67
(13) 		70
(16)
Approximation of odd harmonics in 19edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -7.2 -7.4 -21.5 -14.4 +17.1 -19.5 -14.6
Relative (%) -11.4 -11.7 -34.0 -22.9 +27.1 -30.8 -23.1
Steps
(reduced) 		30
(11) 		44
(6) 		53
(15) 		60
(3) 		66
(9) 		70
(13) 		74
(17)
Approximation of odd harmonics in 20edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +18.0 -26.3 -8.8 -23.9 -11.3 -0.5 -8.3
Relative (%) +30.1 -43.9 -14.7 -39.9 -18.9 -0.9 -13.8
Steps
(reduced) 		32
(12) 		46
(6) 		56
(16) 		63
(3) 		69
(9) 		74
(14) 		78
(18)
Approximation of odd harmonics in 21edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -16.2 +13.7 +2.6 +24.7 +20.1 +16.6 -2.6
Relative (%) -28.4 +24.0 +4.6 +43.2 +35.2 +29.1 -4.5
Steps
(reduced) 		33
(12) 		49
(7) 		59
(17) 		67
(4) 		73
(10) 		78
(15) 		82
(19)
Approximation of odd harmonics in 22edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +7.1 -4.5 +13.0 +14.3 -5.9 -22.3 +2.6
Relative (%) +13.1 -8.2 +23.8 +26.2 -10.7 -41.0 +4.8
Steps
(reduced) 		35
(13) 		51
(7) 		62
(18) 		70
(4) 		76
(10) 		81
(15) 		86
(20)
Approximation of odd harmonics in 23edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -23.7 -21.1 +22.5 +4.8 +22.6 -5.7 +7.4
Relative (%) -45.4 -40.4 +43.1 +9.2 +43.3 -11.0 +14.2
Steps
(reduced) 		36
(13) 		53
(7) 		65
(19) 		73
(4) 		80
(11) 		85
(16) 		90
(21)
Approximation of odd harmonics in 24edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -2.0 +13.7 -18.8 -3.9 -1.3 +9.5 +11.7
Relative (%) -3.9 +27.4 -37.7 -7.8 -2.6 +18.9 +23.5
Steps
(reduced) 		38
(14) 		56
(8) 		67
(19) 		76
(4) 		83
(11) 		89
(17) 		94
(22)
Approximation of odd harmonics in 25edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +18.0 -2.3 -8.8 -11.9 -23.3 +23.5 +15.7
Relative (%) +37.6 -4.8 -18.4 -24.8 -48.6 +48.9 +32.8
Steps
(reduced) 		40
(15) 		58
(8) 		70
(20) 		79
(4) 		86
(11) 		93
(18) 		98
(23)
Approximation of odd harmonics in 26edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -9.6 -17.1 +0.4 -19.3 +2.5 -9.8 +19.4
Relative (%) -20.9 -37.0 +0.9 -41.8 +5.5 -21.1 +42.1
Steps
(reduced) 		41
(15) 		60
(8) 		73
(21) 		82
(4) 		90
(12) 		96
(18) 		102
(24)
Approximation of odd harmonics in 27edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +9.2 +13.7 +9.0 +18.3 -18.0 +3.9 -21.6
Relative (%) +20.6 +30.8 +20.1 +41.2 -40.5 +8.8 -48.6
Steps
(reduced) 		43
(16) 		63
(9) 		76
(22) 		86
(5) 		93
(12) 		100
(19) 		105
(24)
Approximation of odd harmonics in 28edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -16.2 -0.6 +16.9 +10.4 +5.8 +16.6 -16.8
Relative (%) -37.9 -1.4 +39.4 +24.2 +13.6 +38.8 -39.3
Steps
(reduced) 		44
(16) 		65
(9) 		79
(23) 		89
(5) 		97
(13) 		104
(20) 		109
(25)
Approximation of odd harmonics in 29edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +1.5 -13.9 -17.1 +3.0 -13.4 -12.9 -12.4
Relative (%) +3.6 -33.6 -41.3 +7.2 -32.4 -31.3 -30.0
Steps
(reduced) 		46
(17) 		67
(9) 		81
(23) 		92
(5) 		100
(13) 		107
(20) 		113
(26)
Approximation of odd harmonics in 30edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +18.0 +13.7 -8.8 -3.9 +8.7 -0.5 -8.3
Relative (%) +45.1 +34.2 -22.1 -9.8 +21.7 -1.3 -20.7
Steps
(reduced) 		48
(18) 		70
(10) 		84
(24) 		95
(5) 		104
(14) 		111
(21) 		117
(27)
Approximation of odd harmonics in 31edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -5.2 +0.8 -1.1 -10.4 -9.4 +11.1 -4.4
Relative (%) -13.4 +2.0 -2.8 -26.8 -24.2 +28.6 -11.4
Steps
(reduced) 		49
(18) 		72
(10) 		87
(25) 		98
(5) 		107
(14) 		115
(22) 		121
(28)
Approximation of odd harmonics in 32edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +10.5 -11.3 +6.2 -16.4 +11.2 -15.5 -0.8
Relative (%) +28.1 -30.2 +16.5 -43.8 +29.8 -41.4 -2.0
Steps
(reduced) 		51
(19) 		74
(10) 		90
(26) 		101
(5) 		111
(15) 		118
(22) 		125
(29)
Approximation of odd harmonics in 33edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -11.0 +13.7 +13.0 +14.3 -5.9 -4.2 +2.6
Relative (%) -30.4 +37.6 +35.7 +39.2 -16.1 -11.5 +7.3
Steps
(reduced) 		52
(19) 		77
(11) 		93
(27) 		105
(6) 		114
(15) 		122
(23) 		129
(30)
Approximation of odd harmonics in 34edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +3.9 +1.9 -15.9 +7.9 +13.4 +6.5 +5.8
Relative (%) +11.1 +5.4 -45.0 +22.3 +37.9 +18.5 +16.6
Steps
(reduced) 		54
(20) 		79
(11) 		95
(27) 		108
(6) 		118
(16) 		126
(24) 		133
(31)
Approximation of odd harmonics in 35edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -16.2 -9.2 -8.8 +1.8 -2.7 +16.6 +8.9
Relative (%) -47.4 -26.7 -25.7 +5.3 -8.0 +48.5 +25.9
Steps
(reduced) 		55
(20) 		81
(11) 		98
(28) 		111
(6) 		121
(16) 		130
(25) 		137
(32)
Approximation of odd harmonics in 36edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -2.0 +13.7 -2.2 -3.9 +15.3 -7.2 +11.7
Relative (%) -5.9 +41.1 -6.5 -11.7 +46.0 -21.6 +35.2
Steps
(reduced) 		57
(21) 		84
(12) 		101
(29) 		114
(6) 		125
(17) 		133
(25) 		141
(33)
Approximation of odd harmonics in 37edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +11.6 +2.9 +4.1 -9.3 +0.0 +2.7 +14.4
Relative (%) +35.6 +8.9 +12.8 -28.7 +0.1 +8.4 +44.5
Steps
(reduced) 		59
(22) 		86
(12) 		104
(30) 		117
(6) 		128
(17) 		137
(26) 		145
(34)
Approximation of odd harmonics in 38edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -7.2 -7.4 +10.1 -14.4 -14.5 +12.1 -14.6
Relative (%) -22.9 -23.3 +32.1 -45.7 -45.8 +38.3 -46.2
Steps
(reduced) 		60
(22) 		88
(12) 		107
(31) 		120
(6) 		131
(17) 		141
(27) 		148
(34)
Approximation of odd harmonics in 39edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +5.7 +13.7 -15.0 +11.5 +2.5 -9.8 -11.3
Relative (%) +18.6 +44.5 -48.7 +37.3 +8.2 -31.7 -36.9
Steps
(reduced) 		62
(23) 		91
(13) 		109
(31) 		124
(7) 		135
(18) 		144
(27) 		152
(35)
Approximation of odd harmonics in 40edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -12.0 +3.7 -8.8 +6.1 -11.3 -0.5 -8.3
Relative (%) -39.9 +12.3 -29.4 +20.3 -37.7 -1.8 -27.6
Steps
(reduced) 		63
(23) 		93
(13) 		112
(32) 		127
(7) 		138
(18) 		148
(28) 		156
(36)
Approximation of odd harmonics in 41edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +0.5 -5.8 -3.0 +1.0 +4.8 +8.3 -5.3
Relative (%) +1.7 -19.9 -10.2 +3.3 +16.3 +28.2 -18.3
Steps
(reduced) 		65
(24) 		95
(13) 		115
(33) 		130
(7) 		142
(19) 		152
(29) 		160
(37)
Approximation of odd harmonics in 42edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +12.3 +13.7 +2.6 -3.9 -8.5 -12.0 -2.6
Relative (%) +43.2 +47.9 +9.1 -13.7 -29.6 -41.8 -8.9
Steps
(reduced) 		67
(25) 		98
(14) 		118
(34) 		133
(7) 		145
(19) 		155
(29) 		164
(38)
Approximation of odd harmonics in 43edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -4.3 +4.4 +7.9 -8.6 +6.8 -3.3 +0.1
Relative (%) -15.3 +15.7 +28.4 -30.7 +24.4 -11.9 +0.4
Steps
(reduced) 		68
(25) 		100
(14) 		121
(35) 		136
(7) 		149
(20) 		159
(30) 		168
(39)
Approximation of odd harmonics in 44edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +7.1 -4.5 +13.0 -13.0 -5.9 +4.9 +2.6
Relative (%) +26.2 -16.5 +47.6 -47.7 -21.5 +18.1 +9.7
Steps
(reduced) 		70
(26) 		102
(14) 		124
(36) 		139
(7) 		152
(20) 		163
(31) 		172
(40)
Approximation of odd harmonics in 45edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -8.6 -13.0 -8.8 +9.4 +8.7 +12.8 +5.1
Relative (%) -32.3 -48.7 -33.1 +35.3 +32.6 +48.0 +19.0
Steps
(reduced) 		71
(26) 		104
(14) 		126
(36) 		143
(8) 		156
(21) 		167
(32) 		176
(41)
Approximation of odd harmonics in 46edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +2.4 +5.0 -3.6 +4.8 -3.5 -5.7 +7.4
Relative (%) +9.2 +19.1 -13.8 +18.3 -13.4 -22.0 +28.3
Steps
(reduced) 		73
(27) 		107
(15) 		129
(37) 		146
(8) 		159
(21) 		170
(32) 		180
(42)
Approximation of odd harmonics in 47edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -12.6 -3.3 +1.4 +0.3 +10.4 +2.0 +9.6
Relative (%) -49.3 -13.1 +5.4 +1.4 +40.7 +7.9 +37.6
Steps
(reduced) 		74
(27) 		109
(15) 		132
(38) 		149
(8) 		163
(22) 		174
(33) 		184
(43)
Approximation of odd harmonics in 48edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -2.0 -11.3 +6.2 -3.9 -1.3 +9.5 +11.7
Relative (%) -7.8 -45.3 +24.7 -15.6 -5.3 +37.9 +46.9
Steps
(reduced) 		76
(28) 		111
(15) 		135
(39) 		152
(8) 		166
(22) 		178
(34) 		188
(44)
Approximation of odd harmonics in 49edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +8.2 +5.5 +10.8 -8.0 +11.9 -7.9 -10.7
Relative (%) +33.7 +22.6 +44.0 -32.6 +48.8 -32.2 -43.8
Steps
(reduced) 		78
(29) 		114
(16) 		138
(40) 		155
(8) 		170
(23) 		181
(34) 		191
(44)
Approximation of odd harmonics in 50edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -6.0 -2.3 -8.8 -11.9 +0.7 -0.5 -8.3
Relative (%) -24.8 -9.6 -36.8 -49.6 +2.8 -2.2 -34.5
Steps
(reduced) 		79
(29) 		116
(16) 		140
(40) 		158
(8) 		173
(23) 		185
(35) 		195
(45)
Approximation of odd harmonics in 51edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) +3.9 -9.8 -4.1 +7.9 -10.1 +6.5 -5.9
Relative (%) +16.7 -41.8 -17.5 +33.4 -43.1 +27.8 -25.1
Steps
(reduced) 		81
(30) 		118
(16) 		143
(41) 		162
(9) 		176
(23) 		189
(36) 		199
(46)
Approximation of odd harmonics in 52edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -9.6 +6.0 +0.4 +3.8 +2.5 -9.8 -3.7
Relative (%) -41.8 +26.0 +1.8 +16.4 +11.0 -42.3 -15.8
Steps
(reduced) 		82
(30) 		121
(17) 		146
(42) 		165
(9) 		180
(24) 		192
(36) 		203
(47)
Approximation of odd harmonics in 53edo
Harmonic 3 5 7 9 11 13 15
Error Absolute (¢) -0.07 -1.41 +4.76 -0.14 -7.92 -2.79 -1.48
Relative (%) -0.3 -6.2 +21.0 -0.6 -35.0 -12.3 -6.5
Steps
(reduced) 		84
(31) 		123
(17) 		149
(43) 		168
(9) 		183
(24) 		196
(37) 		207
(48)
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