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[[User:BudjarnLambeth/Draft related tunings section]]
= Title1 =
= Title1 =
== Octave stretch or compression ==
== Octave stretch or compression ==
99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.
{{main|23edo and octave stretching}}
 
23edo is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.
 
However, when using a slightly [[stretched tuning|stretched octave]] of around 1216 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.
 
Stretched 23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.
 
What follows is a comparison of stretched- and compressed-octave 23edo tunings.
 
; [[zpi|86zpi]]
* Step size: 51.653{{c}}, octave size: 1188.0{{c}}
Compressing the octave of EDONAME 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 ZPINAME does this.
{{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
 
; [[60ed6]]
* Step size: 51.700{{c}}, octave size: 1189.1{{c}}
Compressing the octave of EDONAME 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 60ed6 does this. So does the tuning [[equal tuning|105ed23]] whose octave is identical within 0.01{{c}}.
{{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
 
; [[zpi|85zpi]]
* Step size: 52.114{{c}}, octave size: 1198.6{{c}}
Compressing the octave of EDONAME 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 85zpi does this. So does the tuning [[ed9|73ed9]] whose octave is identical within 0.02{{c}}.
{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}


What follows is a comparison of stretched- and compressed-octave 99edo tunings.
; 23edo
* Step size: NNN{{c}}, octave size: 1200.0{{c}}
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}


; [[zpi|567zpi]]  
; [[WE|23et, 13-limit WE tuning]]  
* Step size: 12.138{{c}}, octave size: 1201.66{{c}}
* Step size: 52.237{{c}}, octave size: 1201.5{{c}}
Stretching the octave of 99edo by around 1.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
{{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}}
{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}}
{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; [[WE|99et, 13-limit WE tuning]]  
; [[WE|23et, 2.3.5.13 WE tuning]]  
* Step size: 12.123{{c}}, octave size: 1200.18{{c}}
* Step size: 52.447{{c}}, octave size: 1206.3{{c}}
Stretching the octave of 99edo by around a fifth of a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. So does the tuning [[ed10|76ed10]] whose octave is identical within 0.01{{c}}.
{{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}}
{{Harmonics in cet|52.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}}
{{Harmonics in cet|52.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; 99edo
; [[59ed6]]
* Step size: 12.121{{c}}, octave size: 1200.00{{c}}  
* Step size: 52.575{{c}}, octave size: 1209.2{{c}}
Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.
Stretching the octave of EDONAME 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 59ed6 does this. So does the tuning [[53ed5]] whose octave is identical within 0.01{{c}}.
{{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}}
{{Harmonics in equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}}
{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]
; [[zpi|84zpi]]  
* Step size: 12.117{{c}}, octave size: 1199.58{{c}}
* Step size: 52.615{{c}}, octave size: 1210.1{{c}}
Compressing the octave of 99edo by around 0.6{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
Stretching the octave of EDONAME 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 ZPINAME does this.
{{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}}
{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}}
{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}


; [[zpi|568zpi]]  
; [[36edt]]  
* Step size: 12.115{{c}}, octave size: 1199.39{{c}}
* Step size: 52.832{{c}}, octave size: 1215.1{{c}}
Compressing the octave of 99edo by around 0.4{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.
Stretching the octave of EDONAME 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 EDONOI does this.
{{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}}
{{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}}
{{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


; [[157edt]] / [[ed5|230ed5]]
; [[84ed13]]  
* Step size: 12.114{{c}}, octave size: 1199.32{{c}}
* Step size: 52.863{{c}}, octave size: 1215.9{{c}}
Compressing the octave of 99edo by around 0.3{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
Stretching the octave of EDONAME 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 EDONOI does this.
{{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}}
{{Harmonics in equal|84|13|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}}
{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


= Title2 =
= Title2 =
=== Placeholder ===
=== 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
 
60edo (narrow down edonoi & ZPIs)
* 35edf
* 139ed5
* 301zpi (20.027c)
* 95edt
* 13-limit WE (20.013c) (155ed6 has octaves only 0.02{{c}} different)
* 215ed12
* 302zpi (19.962c)
* 208ed11 (ideal for catnip temperament)
* 303zpi (19.913c)
 
32edo
* 13-limit WE (37.481c)
* 11-limit WE (37.453c)
* 90ed7 (optimal for dual-5) (133zpi's octave only differs by 0.4{{c}})
* 51edt
* 134zpi (37.176c)
* 75ed5
 
33edo
* 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
* 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
* 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)
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