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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}}
What follows is a comparison of compressed-octave 32edo tunings.
{{harmonics in equal|6|intervals=odd|columns=7}}
 
{{harmonics in equal|7|intervals=odd|columns=7}}
; 32edo
{{harmonics in equal|8|intervals=odd|columns=7}}
* Step size: 37.500{{c}}, octave size: 1200.0{{c}}
{{harmonics in equal|9|intervals=odd|columns=7}}
Pure-octaves 32edo approximates all harmonics up to 16 within 15.5{{c}}.
{{harmonics in equal|10|intervals=odd|columns=7}}
{{Harmonics in equal|32|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32edo}}
{{harmonics in equal|11|intervals=odd|columns=7}}
{{Harmonics in equal|32|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32edo (continued)}}
{{harmonics in equal|12|intervals=odd|columns=7}}
 
{{harmonics in equal|13|intervals=odd|columns=7}}
; [[WE|32et, 13-limit WE tuning]]
{{harmonics in equal|14|intervals=odd|columns=7}}
* Step size: 37.481{{c}}, octave size: 1199.4{{c}}
{{harmonics in equal|15|intervals=odd|columns=7}}
Compressing the octave of 32edo by around half a cent results in improved primes 3, 7 and 11, but worse primes 5 and 13. This approximates all harmonics up to 16 within 18.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{harmonics in equal|16|intervals=odd|columns=7}}
{{Harmonics in cet|37.481|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning}}
{{harmonics in equal|17|intervals=odd|columns=7}}
{{Harmonics in cet|37.481|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning (continued)}}
{{harmonics in equal|18|intervals=odd|columns=7}}
 
{{harmonics in equal|19|intervals=odd|columns=7}}
; [[WE|32et, 11-limit WE tuning]]
{{harmonics in equal|20|intervals=odd|columns=7}}
* Step size: 37.453{{c}}, octave size: 1198.5{{c}}
{{harmonics in equal|21|intervals=odd|columns=7}}
Compressing the octave of 32edo by around 1.5{{c}} results in improved primes 3, 7 and 11, but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 16.4{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
{{harmonics in equal|22|intervals=odd|columns=7}}
{{Harmonics in cet|37.453|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning}}
{{harmonics in equal|23|intervals=odd|columns=7}}
{{Harmonics in cet|37.453|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning (continued)}}
{{harmonics in equal|24|intervals=odd|columns=7}}
 
{{harmonics in equal|25|intervals=odd|columns=7}}
; [[ed7|90ed7]]
{{harmonics in equal|26|intervals=odd|columns=7}}
* Step size: 37.431{{c}}, octave size: 1197.8{{c}}
{{harmonics in equal|27|intervals=odd|columns=7}}
Compressing the octave of 32edo by around 2{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 18.6{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.
{{harmonics in equal|28|intervals=odd|columns=7}}
{{Harmonics in equal|90|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed7}}
{{harmonics in equal|29|intervals=odd|columns=7}}
{{Harmonics in equal|90|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed7 (continued)}}
{{harmonics in equal|30|intervals=odd|columns=7}}
 
{{harmonics in equal|31|intervals=odd|columns=7}}
; [[zpi|133zpi]]
{{harmonics in equal|32|intervals=odd|columns=7}}
* Step size: 37.418{{c}}, octave size: 1197.375{{c}}
{{harmonics in equal|33|intervals=odd|columns=7}}
Compressing the octave of 32edo by around NNN{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 17.4{{c}}. The tuning 133zpi does this.
{{harmonics in equal|34|intervals=odd|columns=7}}
{{Harmonics in cet|37.418|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}}
{{harmonics in equal|35|intervals=odd|columns=7}}
{{Harmonics in cet|37.418|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}}
{{harmonics in equal|36|intervals=odd|columns=7}}
Below is a plot of the [[Zeta]] function, showing how its peak (ie biggest absolute value) is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.
{{harmonics in equal|37|intervals=odd|columns=7}}
 
{{harmonics in equal|38|intervals=odd|columns=7}}
[[File:plot32.png|alt=plot32.png|plot32.png]]
{{harmonics in equal|39|intervals=odd|columns=7}}
 
{{harmonics in equal|40|intervals=odd|columns=7}}
; [[51edt]]
{{harmonics in equal|41|intervals=odd|columns=7}}
* Step size: 37.293{{c}}, octave size: 1193.4{{c}}
{{harmonics in equal|42|intervals=odd|columns=7}}
Compressing the octave of 32edo by around 6.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 18.2{{c}}. The tuning 51edt does this.
{{harmonics in equal|43|intervals=odd|columns=7}}
{{Harmonics in equal|51|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 51edt}}
{{harmonics in equal|44|intervals=odd|columns=7}}
{{Harmonics in equal|51|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 51edt (continued)}}
{{harmonics in equal|45|intervals=odd|columns=7}}
 
{{harmonics in equal|46|intervals=odd|columns=7}}
= Title2 =
{{harmonics in equal|47|intervals=odd|columns=7}}
=== Lab ===
{{harmonics in equal|48|intervals=odd|columns=7}}
 
{{harmonics in equal|49|intervals=odd|columns=7}}
Place holder
{{harmonics in equal|50|intervals=odd|columns=7}}
 
{{harmonics in equal|51|intervals=odd|columns=7}}
 
{{harmonics in equal|52|intervals=odd|columns=7}}
<br><br><br><br><br>
{{harmonics in equal|53|intervals=odd|columns=7}}
 
 
{{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
 
33edo (reduce # of edonoi)
* 76ed5
* 92ed7 (137zpi's octave differs by only 0.3{{c}})
* 52ed13
* 114ed11
* 138zpi (36.394c) (122ed13's octave differs by only 0.1{{c}})
* 13-limit WE (36.357c)
* 93ed7 (optimised for dual-fifths)
* 77ed5 (139zpi's octave differs by only 0.2{{c}})
* 123ed13 / 1ed47/46 (identical within <0.1{{c}})
* 115ed11
 
39edo
* 171zpi (30.973c) (optimised for dual-fifths use)
* 13-limit WE (30.757c) (octave of 135ed11 differs by only 0.2{{c}})
* 101ed6 (octave of 172zpi differs by only 0.4{{c}})
* 173zpi (30.672c) (octave of 62edt differs by only 0.2{{c}})
* 110ed7 (octave of 145ed13 differs by only 0.1{{c}})
* 91ed5
 
42edo (reduce # of edonoi)
* 108ed6 (octave is identical to 97ed5 within 0.1{{c}})
* 189zpi (28.689c)
* 150ed12
* 145ed11
''190zpi's octave is within 0.05{{c}} of pure-octaves 42edo''
* 118ed7
* 13-limit WE (28.534c)
* 151ed12 (octave is identical to 7-limit WE within 0.3{{c}})
* 109ed6
* 191zpi (28.444c)
* 67edt
 
45edo
* 209zpi (26.550)
* 13-limit WE (26.695c)
* 161ed12
* 116ed6 (octave identical to 126ed7 within 0.1{{c}})
* 7-limit WE (26.745c)
* 207zpi (26.762)
* 71edt (octave identical to 155ed11 within 0.3{{c}})
 
54edo
* 139ed6 (octave is identical to 262zpi within 0.2{{c}})
* 151ed7
* 193ed12
* 263zpi (22.243c)
* 13-limit WE (22.198c)  (octave is identical to 187ed11 within 0.1{{c}})
* 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})
* 152ed7
* 140ed6
* 126ed5 (octave is identical to 86edt within 0.1{{c}})
 
59edo (reduce # of edonoi or zpi)
* 152ed6
* 294zpi (20.399c)
* 211ed12
* 295zpi (20.342c)
''pure octaves 59edo octave is identical to 137ed5 within 0.05{{c}}''
* 13-limit WE (20.320c)
* 7-limit WE (20.301c)
* 166ed7
* 212ed12
* 296zpi (20.282c)
* 153ed6
 
64edo
* 179ed7 (octave is identical to 326zpi within 0.3{{c}})
* 165ed6
* 229ed12 (octave is identical to 221ed11 within 0.1{{c}})
* 327zpi (18.767c)
* 11-limit WE (18.755c)
''pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}}''
* 328zpi (18.721c)
* 180ed7
* 230ed12
* 149ed5
 
; Medium priority
 
118edo (choose ZPIS)
{{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}
* 187edt
* 69edf
* 13-limit WE (10.171c)
* 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)
 
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)
 
; Optional
 
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)
 
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)
 
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)
 
20edo
{{harmonics in equal | 20 | 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)
 
28edo
{{harmonics in equal | 28 | 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)

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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