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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}}
= Title1 =
{{harmonics in equal|4|intervals=odd|columns=7}}
== Octave stretch or compression ==
{{harmonics in equal|5|intervals=odd|columns=7}}
64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
{{harmonics in equal|6|intervals=odd|columns=7}}
 
{{harmonics in equal|7|intervals=odd|columns=7}}
[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.
{{harmonics in equal|8|intervals=odd|columns=7}}
 
{{harmonics in equal|9|intervals=odd|columns=7}}
If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by [[221ed11]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
{{harmonics in equal|10|intervals=odd|columns=7}}
 
{{harmonics in equal|11|intervals=odd|columns=7}}
[[47ed5/3]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's.
{{harmonics in equal|12|intervals=odd|columns=7}}
 
{{harmonics in equal|13|intervals=odd|columns=7}}
What follows is a comparison of stretched- and compressed-octave 64edo tunings.
{{harmonics in equal|14|intervals=odd|columns=7}}
 
{{harmonics in equal|15|intervals=odd|columns=7}}
; [[ed7|179ed7]]
{{harmonics in equal|16|intervals=odd|columns=7}}
* Octave size: 1204.50{{c}}
{{harmonics in equal|17|intervals=odd|columns=7}}
Stretching the octave of 64edo by around 4.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 8.99{{c}}. The tuning 179ed7 does this. So does the tuning 326zpi whose octave is identical within 0.3{{c}}.
{{harmonics in equal|18|intervals=odd|columns=7}}
{{Harmonics in equal|179|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 179ed7}}
{{harmonics in equal|19|intervals=odd|columns=7}}
{{Harmonics in equal|179|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 179ed7 (continued)}}
{{harmonics in equal|20|intervals=odd|columns=7}}
 
{{harmonics in equal|21|intervals=odd|columns=7}}
; [[ed6|165ed6]]
{{harmonics in equal|22|intervals=odd|columns=7}}
* Octave size: 1203.18{{c}}
{{harmonics in equal|23|intervals=odd|columns=7}}
Stretching the octave of 64edo by around 3{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 9.25{{c}}. The tuning 165ed6 does this.
{{harmonics in equal|24|intervals=odd|columns=7}}
{{Harmonics in equal|165|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 165ed6}}
{{harmonics in equal|25|intervals=odd|columns=7}}
{{Harmonics in equal|165|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 165ed6 (continued)}}
{{harmonics in equal|26|intervals=odd|columns=7}}
 
{{harmonics in equal|27|intervals=odd|columns=7}}
; [[ed12|229ed12]]
{{harmonics in equal|28|intervals=odd|columns=7}}
* Octave size: 1202.29{{c}}
{{harmonics in equal|29|intervals=odd|columns=7}}
Stretching the octave of 64edo by around 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.17{{c}}. The tuning 229ed12 does this. So does the tuning [[equal tuning|221ed11]] whose octave is identical within 0.1{{c}}.
{{harmonics in equal|30|intervals=odd|columns=7}}
{{Harmonics in equal|229|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 229ed12}}
{{harmonics in equal|31|intervals=odd|columns=7}}
{{Harmonics in equal|229|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 229ed12 (continued)}}
{{harmonics in equal|32|intervals=odd|columns=7}}
 
{{harmonics in equal|33|intervals=odd|columns=7}}
; [[zpi|327zpi]]
{{harmonics in equal|34|intervals=odd|columns=7}}
* Step size: 18.767{{c}}, octave size: 1201.09{{c}}
{{harmonics in equal|35|intervals=odd|columns=7}}
Stretching the octave of 64edo by around 1{{c}} results in improved primes 3 and 11, but worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 9.23{{c}}. The tuning 327zpi does this.
{{harmonics in equal|36|intervals=odd|columns=7}}
{{Harmonics in cet|18.767|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 327zpi}}
{{harmonics in equal|37|intervals=odd|columns=7}}
{{Harmonics in cet|18.767|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 327zpi (continued)}}
{{harmonics in equal|38|intervals=odd|columns=7}}
 
{{harmonics in equal|39|intervals=odd|columns=7}}
; [[WE|64et, 11-limit WE tuning]]
{{harmonics in equal|40|intervals=odd|columns=7}}
* Step size: 18.755{{c}}, octave size: 1200.32{{c}}
{{harmonics in equal|41|intervals=odd|columns=7}}
Stretching the octave of 64edo by around a third of a cent results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 8.50{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
{{harmonics in equal|42|intervals=odd|columns=7}}
{{Harmonics in cet|18.755|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning}}
{{harmonics in equal|43|intervals=odd|columns=7}}
{{Harmonics in cet|18.755|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning (continued)}}
{{harmonics in equal|44|intervals=odd|columns=7}}
 
{{harmonics in equal|45|intervals=odd|columns=7}}
; 64edo
{{harmonics in equal|46|intervals=odd|columns=7}}
* Step size: 18.750{{c}}, octave size: 1200.00{{c}}
{{harmonics in equal|47|intervals=odd|columns=7}}
Pure-octaves 64edo approximates all harmonics up to 16 within 8.21{{c}}. The octave of 64edo's 13-limit [[WE]] tuning differs by only 0.13{{c}} from pure.
{{harmonics in equal|48|intervals=odd|columns=7}}
{{Harmonics in equal|64|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64edo}}
{{harmonics in equal|49|intervals=odd|columns=7}}
{{Harmonics in equal|64|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64edo (continued)}}
{{harmonics in equal|50|intervals=odd|columns=7}}
 
{{harmonics in equal|51|intervals=odd|columns=7}}
; [[zpi|328zpi]]
{{harmonics in equal|52|intervals=odd|columns=7}}
* Step size: 18.721{{c}}, octave size: 1198.14{{c}}
{{harmonics in equal|53|intervals=odd|columns=7}}
Compressing the octave of 64edo by just under 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.02{{c}}. The tuning 328zpi does this.
{{Harmonics in cet|18.721|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 328zpi}}
{{Harmonics in cet|18.721|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 328zpi (continued)}}
 
; [[ed7|180ed7]]
* Octave size: 1197.80{{c}}
Compressing the octave of 64edo by just over 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.34{{c}}. The tuning 180ed7 does this.
{{Harmonics in equal|180|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 180ed7}}
{{Harmonics in equal|180|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 180ed7 (continued)}}
 
; [[ed12|230ed12]]
* Octave size: 1197.07{{c}}
Compressing the octave of 64edo by around 3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.80{{c}}. The tuning 230ed12 does this.
{{Harmonics in equal|230|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 230ed12}}
{{Harmonics in equal|230|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 230ed12 (continued)}}
 
; [[ed5|149ed5]]
* Step size: Octave size: NNN{{c}}
Compressing the octave of 64edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 149ed5 does this.
{{Harmonics in equal|149|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 149ed5}}
{{Harmonics in equal|149|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 149ed5 (continued)}}
 
= Title2 =
=== Lab ===
 
Place holder
 
 
<br><br><br><br><br>
 
 
{{harmonics in cet | 300 | intervals=prime}}
 
{{harmonics in equal | 140 | 12 | 1 | intervals=prime}}
 
=== Possible tunings to be used on each page ===
You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.
 
(Used https://x31eq.com/temper-pyscript/net.html, used WE instead of TE cause it kept defaulting to WE and I kept not remembering to switch it)
 
; High-priority
 
25edo
{{harmonics in equal | 25 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
26edo
{{harmonics in equal | 26 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
29edo
{{harmonics in equal | 29 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
30edo
{{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
34edo
{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
35edo
{{harmonics in equal | 35 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
36edo
{{harmonics in equal | 36 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
37edo
{{harmonics in equal | 37 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
38edo
{{harmonics in equal | 38 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
9edo
{{harmonics in equal | 9 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
10edo
{{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
11edo
{{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
15edo
{{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
18edo
{{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
48edo
{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
24edo
{{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
5edo
{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
6edo
{{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
13edo
{{harmonics in equal | 13 | 2 | 1 | intervals=integer | columns=12}}
* Main: "13edo and optimal octave stretching"
* 2.5.11.13 WE (92.483c)
* 2.5.7.13 WE (92.804c)
* 2.3 WE (91.405c) (good for opposite 7 mapping)
* 38zpi (92.531c)
 
118edo (choose ZPIS)
{{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}
* 187edt
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)
 
103edo (narrow down edonoi, choose ZPIS)
{{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}}
* 163edt
* 239ed5
* 266ed6
* 289ed7
* 356ed11
* 369ed12
* 381ed13
* 421ed17
* 466ed23
* 13-limit WE (11.658c)
* Best nearby ZPI(s)
 
111edo (choose ZPIS)
{{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
; Low priority
 
104edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
125edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
145edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
152edo
* 241edt
* 13-limit WE (7.894c)
* Best nearby ZPI(s)
 
159edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
166edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
182edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
198edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
212edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
243edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
247edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)

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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