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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}}
54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[139ed6]], a [[Octave stretch|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.
{{harmonics in equal|6|intervals=odd|columns=7}}
 
{{harmonics in equal|7|intervals=odd|columns=7}}
If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s  3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.
{{harmonics in equal|8|intervals=odd|columns=7}}
 
{{harmonics in equal|9|intervals=odd|columns=7}}
[[40ed5/3]] is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low [[harmonic entropy]].
{{harmonics in equal|10|intervals=odd|columns=7}}
 
{{harmonics in equal|11|intervals=odd|columns=7}}
What follows is a comparison of stretched- and compressed-octave 54edo tunings.
{{harmonics in equal|12|intervals=odd|columns=7}}
 
{{harmonics in equal|13|intervals=odd|columns=7}}
; [[ed6|139ed6]]
{{harmonics in equal|14|intervals=odd|columns=7}}
* Octave size: 1205.08{{c}}
{{harmonics in equal|15|intervals=odd|columns=7}}
Stretching the octave of 54edo by around 5{{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 10.15{{c}}. The tuning 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2{{c}}.
{{harmonics in equal|16|intervals=odd|columns=7}}
{{Harmonics in equal|139|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 139ed6}}
{{harmonics in equal|17|intervals=odd|columns=7}}
{{Harmonics in equal|139|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed6 (continued)}}
{{harmonics in equal|18|intervals=odd|columns=7}}
 
{{harmonics in equal|19|intervals=odd|columns=7}}
; [[ed7|151ed7]]
{{harmonics in equal|20|intervals=odd|columns=7}}
* Octave size: 1204.75{{c}}
{{harmonics in equal|21|intervals=odd|columns=7}}
Stretching the octave of 54edo by around 4.5{{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 11.12{{c}}. The tuning 151ed7 does this.
{{harmonics in equal|22|intervals=odd|columns=7}}
{{Harmonics in equal|151|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed7}}
{{harmonics in equal|23|intervals=odd|columns=7}}
{{Harmonics in equal|151|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed7 (continued)}}
{{harmonics in equal|24|intervals=odd|columns=7}}
 
{{harmonics in equal|25|intervals=odd|columns=7}}
; [[ed12|193ed12]]
{{harmonics in equal|26|intervals=odd|columns=7}}
* Octave size: 1203.66{{c}}
{{harmonics in equal|27|intervals=odd|columns=7}}
Stretching the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 10.97{{c}}. The tuning 193ed12 does this.
{{harmonics in equal|28|intervals=odd|columns=7}}
{{Harmonics in equal|193|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 193ed12}}
{{harmonics in equal|29|intervals=odd|columns=7}}
{{Harmonics in equal|193|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 193ed12 (continued)}}
{{harmonics in equal|30|intervals=odd|columns=7}}
 
{{harmonics in equal|31|intervals=odd|columns=7}}
; [[zpi|263zpi]]
{{harmonics in equal|32|intervals=odd|columns=7}}
* Step size: 22.243{{c}}, octave size: 1201.12{{c}}
{{harmonics in equal|33|intervals=odd|columns=7}}
Stretching the octave of 54edo by around 1{{c}} results in an improved prime 5, but worse primes 2, 3, 7, 11 and 13. This approximates all harmonics up to 16 within 10.94{{c}}. The tuning 263zpi does this.
{{harmonics in equal|34|intervals=odd|columns=7}}
{{Harmonics in cet|22.243|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 263zpi}}
{{harmonics in equal|35|intervals=odd|columns=7}}
{{Harmonics in cet|22.243|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 263zpi (continued)}}
{{harmonics in equal|36|intervals=odd|columns=7}}
 
{{harmonics in equal|37|intervals=odd|columns=7}}
; 54edo
{{harmonics in equal|38|intervals=odd|columns=7}}
* Step size: 22.222{{c}}, octave size: 1200.00{{c}}
{{harmonics in equal|39|intervals=odd|columns=7}}
Pure-octaves 54edo approximates all harmonics up to 16 within 9.16{{c}}.
{{harmonics in equal|40|intervals=odd|columns=7}}
{{Harmonics in equal|54|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54edo}}
{{harmonics in equal|41|intervals=odd|columns=7}}
{{Harmonics in equal|54|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54edo (continued)}}
{{harmonics in equal|42|intervals=odd|columns=7}}
 
{{harmonics in equal|43|intervals=odd|columns=7}}
; [[WE|54et, 13-limit WE tuning]]
{{harmonics in equal|44|intervals=odd|columns=7}}
* Step size: 22.198{{c}}, octave size: 1198.69{{c}}
{{harmonics in equal|45|intervals=odd|columns=7}}
Compressing the octave of 54edo by around 1.5{{c}} results in improved primes 3, 7, 11, 13, 17 and 19, but worse primes 2 and 5. This approximates all harmonics up to 16 within 10.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|187ed11]] whose octave is identical to 13lim WE within 0.1{{c}}.
{{harmonics in equal|46|intervals=odd|columns=7}}
{{Harmonics in cet|22.198|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning}}
{{harmonics in equal|47|intervals=odd|columns=7}}
{{Harmonics in cet|22.198|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning (continued)}}
{{harmonics in equal|48|intervals=odd|columns=7}}
 
{{harmonics in equal|49|intervals=odd|columns=7}}
; [[zpi|264zpi]]
{{harmonics in equal|50|intervals=odd|columns=7}}
* Step size: 22.175{{c}}, octave size: 1197.45{{c}}
{{harmonics in equal|51|intervals=odd|columns=7}}
Compressing the octave of 54edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.19{{c}}. The tuning 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01{{c}}.
{{harmonics in equal|52|intervals=odd|columns=7}}
{{Harmonics in cet|22.175|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 264zpi}}
{{harmonics in equal|53|intervals=odd|columns=7}}
{{Harmonics in cet|22.175|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 264zpi (continued)}}
 
; [[ed7|152ed7]]
* Octave size: 1196.82{{c}}
Compressing the octave of 54edo by around 3{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.36{{c}}. The tuning 152ed7 does this.
{{Harmonics in equal|152|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed7}}
{{Harmonics in equal|152|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed7 (continued)}}
 
; [[ed6|140ed6]]
* Octave size: 1196.47{{c}}
Compressing the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.59{{c}}. The tuning 140ed6 does this.
{{Harmonics in equal|140|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 140ed6}}
{{Harmonics in equal|140|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 140ed6 (continued)}}
 
; [[ed5|126ed5]]
* Octave size: 1194.13{{c}}
Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.
{{Harmonics in equal|126|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 126ed5}}
{{Harmonics in equal|126|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (continued)}}
 
; [[ed5/3|40ed5/3]]
* Octave size: 1194.13{{c}}
Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.
{{Harmonics in equal|40|5|3|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 126ed5}}
{{Harmonics in equal|40|5|3|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (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
 
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}})
 
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
 
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
 
; Medium 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)