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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 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. 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 11 and 13, but worse primes 2, 3, 5 and 7. 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 3, 5, 7 and 11, but worse primes 2 and 13. 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 3, 5, 7 and 11, but worse primes 2 and 13. 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 3, 5, 7 and 11, but worse primes 2 and 13. 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 =
=== 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 ===
=== 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.
You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.
Line 48: Line 94:


; High-priority
; High-priority
23edo (narrow down edonoi & ZPIs)
* Main: "23edo and octave stretching"
* 36edt
* 59ed6
* 60ed6
* 68ed8
* 11ed7/5
* 1ed33/32
* 2.3.5.13 WE (52.447c)
* 2.7.11 WE (51.962c)
* 13-limit WE (52.237c)
* 83zpi (53.105c)
* 84zpi (52.615c)
* 85zpi (52.114c)
* 86zpi (51.653c)
* 87zpi (51.201c)


60edo (narrow down edonoi & ZPIs)
60edo (narrow down edonoi & ZPIs)
* 95edt
* 35edf
* 35edf
* 139ed5
* 139ed5
* 155ed6
* 208ed11
* (???)ed12
* 255ed19
* 272ed23 (great for catnip temperament, maybe there's a similar but simpler tuning w similar benefits?)
* 13-limit WE (20.013c)
* 299zpi (20.128c)
* 300zpi (20.093c)
* 301zpi (20.027c)
* 301zpi (20.027c)
* 95edt
* 13-limit WE (20.013c) (155ed6 has octaves only 0.02{{c}} different)
* 215ed12
* 302zpi (19.962c)
* 302zpi (19.962c)
* 208ed11 (ideal for catnip temperament)
* 303zpi (19.913c)
* 303zpi (19.913c)
* 304zpi (19.869c)


; Medium priority
32edo
 
* 13-limit WE (37.481c)
13edo
* 11-limit WE (37.453c)
* Main: "13edo and optimal octave stretching"
* 90ed7 (optimal for dual-5) (133zpi's octave only differs by 0.4{{c}})
* 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)
 
32edo (narrow down ZPIs)
* 90ed7
* 51edt
* 51edt
* 134zpi (37.176c)
* 75ed5
* 75ed5
* 1ed46/45
* 11-limit WE (37.453c)
* 13-limit WE (37.481c)
* 131zpi (37.862c)
* 132zpi (37.662c)
* 133zpi (37.418c)
* 134zpi (37.176c)


33edo (narrow down edonoi)
33edo
* 76ed5
* 76ed5
* 92ed7
* 92ed7 (137zpi's octave differs by only 0.3{{c}})
* 52edt
* 52ed13
* 1ed47/46
* 114ed11
* 114ed11
* 122ed13
* 138zpi (36.394c) (122ed13's octave differs by only 0.1{{c}})
* 93ed7
* 13-limit WE (36.357c)
* 23edPhi
* 93ed7 (optimised for dual-fifths)
* 77ed5
* 77ed5 (139zpi's octave differs by only 0.2{{c}})
* 123ed13
* 123ed13 / 1ed47/46 (identical within <0.1{{c}})
* 115ed11
* 115ed11
* 11-limit WE (36.349c)
* 13-limit WE (36.357c)
* 137zpi (36.628c)
* 138zpi (36.394c)
* 139zpi (36.179c)


39edo (narrow down slightly)
39edo
* 62edt
* 171zpi (30.973c) (optimised for dual-fifths use)
* 101ed6
* 13-limit WE (30.757c) (octave of 135ed11 differs by only 0.2{{c}})
* 18ed11/8
* 101ed6 (octave of 172zpi differs by only 0.4{{c}})
* 2.3.5.11 WE (30.703c)
* 173zpi (30.672c) (octave of 62edt differs by only 0.2{{c}})
* 2.3.7.11.13 WE (30.787c)
* 110ed7 (octave of 145ed13 differs by only 0.1{{c}})
* 13-limit WE (30.757c)
* 91ed5
* 171zpi (30.973c)
* 172zpi (30.836c)
* 173zpi (30.672c)


42edo (narrow down slightly)
42edo
* 42ed257/128 (replace w something similar but simpler)
* 108ed6 (octave is identical to 97ed5 within 0.1{{c}})
* AS123/121 (1ed123/121)
* 189zpi (28.689c)
* 11ed6/5
* 150ed12
* 34ed7/4
* 145ed11
* 7-limit WE (28.484c)
''190zpi's octave is within 0.05{{c}} of pure-octaves 42edo''
* 118ed7
* 13-limit WE (28.534c)
* 13-limit WE (28.534c)
* 189zpi (28.689c)
* 151ed12 (octave is identical to 7-limit WE within 0.3{{c}})
* 190zpi (28.572c)
* 109ed6
* 191zpi (28.444c)
* 191zpi (28.444c)
* 67edt


45edo
45edo
* 126ed7
* 209zpi (26.550)
* 13ed11/9
* 13-limit WE (26.695c)
* 161ed12
* 116ed6 (octave identical to 126ed7 within 0.1{{c}})
* 7-limit WE (26.745c)
* 7-limit WE (26.745c)
* 13-limit WE (26.695c)
* 207zpi (26.762)
* 207zpi (26.762)
* 208zpi (26.646)
* 71edt (octave identical to 155ed11 within 0.3{{c}})
* 209zpi (26.550)


54edo (narrow down slightly)
54edo
* 86edt
* 139ed6 (octave is identical to 262zpi within 0.2{{c}})
* 126ed5
* 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
* 152ed7
* 38ed5/3
* 140ed6
* 40ed5/3
* 126ed5 (octave is identical to 86edt within 0.1{{c}})
* 2.3.7.11.13 WE (22.180c)
* 13-limit WE (22.198c)
* 262zpi (22.313c)
* 263zpi (22.243c)
* 264zpi (22.175c)


59edo (narrow down ZPIs)
59edo
* 93edt
* 152ed6
* 166ed7
* 203ed11
* 7-limit WE (20.301c)
* 11-limit WE (20.310c)
* 13-limit WE (20.320c)
* 293zpi (20.454c)
* 294zpi (20.399c)
* 294zpi (20.399c)
* 211ed12
* 295zpi (20.342c)
* 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)
* 296zpi (20.282c)
* 297zpi (20.229c)
* 153ed6


64edo (narrow down ZPIs)
64edo
* 149ed5
* 179ed7 (octave is identical to 326zpi within 0.3{{c}})
* 180ed7
* 165ed6
* 222ed11
* 229ed12 (octave is identical to 221ed11 within 0.1{{c}})
* 47ed5/3
* 327zpi (18.767c)
* 11-limit WE (18.755c)
* 11-limit WE (18.755c)
* 13-limit WE (18.752c)
''pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}}''
* 325zpi (18.868c)
* 326zpi (18.816c)
* 327zpi (18.767c)
* 328zpi (18.721c)
* 328zpi (18.721c)
* 329zpi (18.672c)
* 180ed7
* 330zpi (18.630c)
* 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)
103edo (narrow down edonoi, choose ZPIS)
{{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}}
* 163edt
* 163edt
* 239ed5
* 239ed5
* (???)ed6
* 266ed6
* 289ed7
* 289ed7
* 356ed11
* 356ed11
* (???)ed12
* 369ed12
* 381ed13
* 381ed13
* 421ed17
* 421ed17
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111edo (choose ZPIS)
111edo (choose ZPIS)
{{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* 1-2 WE tunings
* Best nearby ZPI(s)
118edo (choose ZPIS)
* 187edt
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)
* Best nearby ZPI(s)


Line 287: Line 300:


25edo
25edo
{{harmonics in equal | 25 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 293: Line 307:


26edo
26edo
{{harmonics in equal | 26 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 299: Line 314:


29edo
29edo
{{harmonics in equal | 29 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 305: Line 321:


30edo
30edo
{{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 311: Line 328:


34edo
34edo
{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 317: Line 335:


35edo
35edo
{{harmonics in equal | 35 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 323: Line 342:


36edo
36edo
{{harmonics in equal | 36 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 329: Line 349:


37edo
37edo
{{harmonics in equal | 37 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 334: Line 355:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


5edo
9edo
{{harmonics in equal | 9 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 340: Line 362:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


6edo
10edo
{{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 346: Line 369:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


9edo
11edo
{{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 352: Line 376:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


10edo
15edo
{{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 358: Line 383:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


11edo
18edo
{{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 364: Line 390:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


15edo
48edo
{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 370: Line 397:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


18edo
5edo
{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 376: Line 404:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


48edo
6edo
{{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 383: Line 412:


20edo
20edo
{{harmonics in equal | 20 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 389: Line 419:


24edo
24edo
{{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 395: Line 426:


28edo
28edo
{{harmonics in equal | 28 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* 1-2 WE tunings
* Best nearby ZPI(s)
* Best nearby ZPI(s)

Latest revision as of 10:41, 28 August 2025

Quick link

User:BudjarnLambeth/Draft related tunings section

Title1

Octave stretch or compression

Main article: 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 octave of around 1216 cents, 23edo looks much better, and it approximates the perfect fifth (and various other intervals involving the 5th, 7th, 11th, and 13th harmonics) 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 "wolfish" than fifths in other pelogic family temperaments.

What follows is a comparison of stretched- and compressed-octave 23edo tunings.

86zpi
  • Step size: 51.653 ¢, octave size: 1188.0 ¢

Compressing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -12.0 +9.2 -24.0 +2.9 -2.8 -11.4 +15.7 +18.4 -9.0 -19.1 -14.8
Relative (%) -23.2 +17.8 -46.4 +5.7 -5.4 -22.0 +30.4 +35.6 -17.5 -36.9 -28.6
Step 23 37 46 54 60 65 70 74 77 80 83
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.6 -23.4 +12.2 +3.7 +2.1 +6.4 +16.1 -21.0 -2.2 +20.6 -4.7 +24.9
Relative (%) +3.2 -45.2 +23.5 +7.2 +4.0 +12.5 +31.2 -40.7 -4.2 +39.9 -9.1 +48.2
Step 86 88 91 93 95 97 99 100 102 104 105 107
60ed6
  • Step size: 51.700 ¢, octave size: 1189.1 ¢

Compressing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 60ed6 does this. So does the tuning 105ed23 whose octave is identical within 0.01 ¢.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -10.9 +10.9 -21.8 +5.4 +0.0 -8.4 +18.9 +21.8 -5.5 -15.4 -10.9
Relative (%) -21.1 +21.1 -42.2 +10.5 +0.0 -16.2 +36.6 +42.2 -10.6 -29.7 -21.1
Steps
(reduced) 		23
(23) 		37
(37) 		46
(46) 		54
(54) 		60
(0) 		65
(5) 		70
(10) 		74
(14) 		77
(17) 		80
(20) 		83
(23)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.6 -19.3 +16.4 +8.0 +6.5 +10.9 +20.7 -16.4 +2.5 +25.4 +0.1 -21.8
Relative (%) +10.8 -37.3 +31.7 +15.5 +12.5 +21.1 +40.1 -31.7 +4.9 +49.1 +0.3 -42.2
Steps
(reduced) 		86
(26) 		88
(28) 		91
(31) 		93
(33) 		95
(35) 		97
(37) 		99
(39) 		100
(40) 		102
(42) 		104
(44) 		105
(45) 		106
(46)
85zpi
  • Step size: 52.114 ¢, octave size: 1198.6 ¢

Compressing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 85zpi does this. So does the tuning 73ed9 whose octave is identical within 0.02 ¢.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.4 -25.9 -2.8 -24.3 +24.9 +18.6 -4.1 +0.4 -25.6 +17.8 +23.5
Relative (%) -2.6 -49.6 -5.3 -46.6 +47.8 +35.7 -7.9 +0.8 -49.2 +34.2 +45.1
Step 23 36 46 53 60 65 69 73 76 80 83
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -10.8 +17.2 +2.0 -5.5 -6.2 -1.0 +9.7 +25.1 -7.3 +16.4 -8.4 +22.1
Relative (%) -20.8 +33.0 +3.8 -10.6 -12.0 -1.9 +18.5 +48.1 -13.9 +31.5 -16.2 +42.5
Step 85 88 90 92 94 96 98 100 101 103 104 106
23edo
  • Step size: NNN ¢, octave size: 1200.0 ¢

Pure-octaves EDONAME approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in EDONAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -23.7 +0.0 -21.1 -23.7 +22.5 +0.0 +4.8 -21.1 +22.6 -23.7
Relative (%) +0.0 -45.4 +0.0 -40.4 -45.4 +43.1 +0.0 +9.2 -40.4 +43.3 -45.4
Steps
(reduced) 		23
(0) 		36
(13) 		46
(0) 		53
(7) 		59
(13) 		65
(19) 		69
(0) 		73
(4) 		76
(7) 		80
(11) 		82
(13)
Approximation of harmonics in EDONAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.7 +22.5 +7.4 +0.0 -0.6 +4.8 +15.5 -21.1 -1.2 +22.6 -2.2 -23.7
Relative (%) -11.0 +43.1 +14.2 +0.0 -1.2 +9.2 +29.8 -40.4 -2.3 +43.3 -4.2 -45.4
Steps
(reduced) 		85
(16) 		88
(19) 		90
(21) 		92
(0) 		94
(2) 		96
(4) 		98
(6) 		99
(7) 		101
(9) 		103
(11) 		104
(12) 		105
(13)
23et, 13-limit WE tuning
  • Step size: 52.237 ¢, octave size: 1201.5 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.5 -21.4 +2.9 -17.8 -20.0 -25.7 +4.4 +9.4 -16.3 -24.6 -18.5
Relative (%) +2.8 -41.0 +5.6 -34.0 -38.2 -49.1 +8.3 +18.0 -31.2 -47.1 -35.5
Step 23 36 46 53 59 64 69 73 76 79 82
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.4 -24.2 +13.1 +5.8 +5.3 +10.8 +21.7 -14.9 +5.2 -23.1 +4.4 -17.1
Relative (%) -0.7 -46.3 +25.0 +11.1 +10.2 +20.8 +41.6 -28.4 +9.9 -44.3 +8.4 -32.7
Step 85 87 90 92 94 96 98 99 101 102 104 105
23et, 2.3.5.13 WE tuning
  • Step size: 52.447 ¢, octave size: 1206.3 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this. So does the tuning 76ed10 whose octave is identical within 0.01 ¢.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.3 -13.9 +12.6 -6.6 -7.6 -12.2 +18.8 +24.7 -0.3 -8.0 -1.3
Relative (%) +12.0 -26.4 +24.0 -12.6 -14.5 -23.3 +35.9 +47.1 -0.7 -15.3 -2.5
Step 23 36 46 53 59 64 69 73 76 79 82
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +17.5 -5.9 -20.5 +25.1 +25.1 -21.4 -10.2 +5.9 -26.1 -1.7 +26.2 +5.0
Relative (%) +33.3 -11.3 -39.1 +47.9 +47.8 -40.9 -19.4 +11.3 -49.7 -3.3 +50.0 +9.5
Step 85 87 89 92 94 95 97 99 100 102 104 105
59ed6
  • Step size: 52.575 ¢, octave size: 1209.2 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 59ed6 does this. So does the tuning 53ed5 whose octave is identical within 0.01 ¢.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +9.2 -9.2 +18.5 +0.2 +0.0 -4.0 -24.9 -18.5 +9.4 +2.1 +9.2
Relative (%) +17.6 -17.6 +35.1 +0.4 +0.0 -7.6 -47.3 -35.1 +17.9 +4.1 +17.6
Steps
(reduced) 		23
(23) 		36
(36) 		46
(46) 		53
(53) 		59
(0) 		64
(5) 		68
(9) 		72
(13) 		76
(17) 		79
(20) 		82
(23)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -24.2 +5.2 -9.0 -15.6 -15.4 -9.2 +2.3 +18.7 -13.2 +11.4 -13.0 +18.5
Relative (%) -46.0 +10.0 -17.2 -29.7 -29.4 -17.6 +4.4 +35.5 -25.2 +21.7 -24.7 +35.1
Steps
(reduced) 		84
(25) 		87
(28) 		89
(30) 		91
(32) 		93
(34) 		95
(36) 		97
(38) 		99
(40) 		100
(41) 		102
(43) 		103
(44) 		105
(46)
84zpi
  • Step size: 52.615 ¢, octave size: 1210.1 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.1 -7.8 +20.3 +2.3 +2.3 -1.5 -22.2 -15.6 +12.4 +5.3 +12.5
Relative (%) +19.3 -14.9 +38.6 +4.3 +4.4 -2.8 -42.2 -29.7 +23.6 +10.0 +23.7
Step 23 36 46 53 59 64 68 72 76 79 82
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -20.9 +8.7 -5.5 -12.0 -11.8 -5.5 +6.1 +22.6 -9.3 +15.4 -8.9 +22.6
Relative (%) -39.7 +16.5 -10.5 -22.9 -22.4 -10.4 +11.7 +42.9 -17.6 +29.3 -17.0 +43.0
Step 84 87 89 91 93 95 97 99 100 102 103 105
36edt
  • Step size: 52.832 ¢, octave size: 1215.1 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +15.1 +0.0 -22.6 +13.8 +15.1 +12.4 -7.4 +0.0 -23.9 +22.4 -22.6
Relative (%) +28.7 +0.0 -42.7 +26.1 +28.7 +23.5 -14.0 +0.0 -45.3 +42.4 -42.7
Steps
(reduced) 		23
(23) 		36
(0) 		45
(9) 		53
(17) 		59
(23) 		64
(28) 		68
(32) 		72
(0) 		75
(3) 		79
(7) 		81
(9)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 -25.3 +13.8 +7.7 +8.4 +15.1 -25.6 -8.8 +12.4 -15.3 +13.4 -7.4
Relative (%) -5.0 -47.8 +26.1 +14.6 +16.0 +28.7 -48.5 -16.6 +23.5 -28.9 +25.4 -14.0
Steps
(reduced) 		84
(12) 		86
(14) 		89
(17) 		91
(19) 		93
(21) 		95
(23) 		96
(24) 		98
(26) 		100
(28) 		101
(29) 		103
(31) 		104
(32)
84ed13
  • Step size: 52.863 ¢, octave size: 1215.9 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +15.9 +1.1 -21.1 +15.4 +17.0 +14.4 -5.3 +2.3 -21.6 +24.9 -20.0
Relative (%) +30.0 +2.1 -40.0 +29.2 +32.1 +27.3 -10.0 +4.3 -40.8 +47.1 -37.9
Steps
(reduced) 		23
(23) 		36
(36) 		45
(45) 		53
(53) 		59
(59) 		64
(64) 		68
(68) 		72
(72) 		75
(75) 		79
(79) 		81
(81)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.0 -22.6 +16.6 +10.6 +11.3 +18.1 -22.6 -5.7 +15.6 -12.1 +16.7 -4.2
Relative (%) +0.0 -42.7 +31.4 +20.0 +21.5 +34.3 -42.8 -10.8 +29.4 -22.9 +31.5 -7.9
Steps
(reduced) 		84
(0) 		86
(2) 		89
(5) 		91
(7) 		93
(9) 		95
(11) 		96
(12) 		98
(14) 		100
(16) 		101
(17) 		103
(19) 		104
(20)

Title2

Lab

Place holder








Approximation of prime harmonics in 1ed300c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0 -102 -86 -69 +49 +59 -105 +2 -28 -130 +55
Relative (%) +0.0 -34.0 -28.8 -22.9 +16.2 +19.8 -35.0 +0.8 -9.4 -43.2 +18.3
Step 4 6 9 11 14 15 16 17 18 19 20


Approximation of prime harmonics in 140ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.6 +3.2 +10.0 +11.3 -3.0 +15.1 +11.6 +3.4 +10.6 +8.8 -14.5
Relative (%) -5.2 +10.4 +32.4 +36.7 -9.8 +49.0 +37.6 +11.0 +34.6 +28.6 -47.1
Steps
(reduced) 		39
(39) 		62
(62) 		91
(91) 		110
(110) 		135
(135) 		145
(5) 		160
(20) 		166
(26) 		177
(37) 		190
(50) 		193
(53)

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 ¢ 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 ¢)
  • 51edt
  • 134zpi (37.176c)
  • 75ed5

33edo

  • 76ed5
  • 92ed7 (137zpi's octave differs by only 0.3 ¢)
  • 52ed13
  • 114ed11
  • 138zpi (36.394c) (122ed13's octave differs by only 0.1 ¢)
  • 13-limit WE (36.357c)
  • 93ed7 (optimised for dual-fifths)
  • 77ed5 (139zpi's octave differs by only 0.2 ¢)
  • 123ed13 / 1ed47/46 (identical within <0.1 ¢)
  • 115ed11

39edo

  • 171zpi (30.973c) (optimised for dual-fifths use)
  • 13-limit WE (30.757c) (octave of 135ed11 differs by only 0.2 ¢)
  • 101ed6 (octave of 172zpi differs by only 0.4 ¢)
  • 173zpi (30.672c) (octave of 62edt differs by only 0.2 ¢)
  • 110ed7 (octave of 145ed13 differs by only 0.1 ¢)
  • 91ed5

42edo

  • 108ed6 (octave is identical to 97ed5 within 0.1 ¢)
  • 189zpi (28.689c)
  • 150ed12
  • 145ed11

190zpi's octave is within 0.05 ¢ of pure-octaves 42edo

  • 118ed7
  • 13-limit WE (28.534c)
  • 151ed12 (octave is identical to 7-limit WE within 0.3 ¢)
  • 109ed6
  • 191zpi (28.444c)
  • 67edt

45edo

  • 209zpi (26.550)
  • 13-limit WE (26.695c)
  • 161ed12
  • 116ed6 (octave identical to 126ed7 within 0.1 ¢)
  • 7-limit WE (26.745c)
  • 207zpi (26.762)
  • 71edt (octave identical to 155ed11 within 0.3 ¢)

54edo

  • 139ed6 (octave is identical to 262zpi within 0.2 ¢)
  • 151ed7
  • 193ed12
  • 263zpi (22.243c)
  • 13-limit WE (22.198c) (octave is identical to 187ed11 within 0.1 ¢)
  • 264zpi (22.175c) (octave is identical to 194ed12 within 0.01 ¢)
  • 152ed7
  • 140ed6
  • 126ed5 (octave is identical to 86edt within 0.1 ¢)

59edo

  • 152ed6
  • 294zpi (20.399c)
  • 211ed12
  • 295zpi (20.342c)

pure octaves 59edo octave is identical to 137ed5 within 0.05 ¢

  • 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 ¢)
  • 165ed6
  • 229ed12 (octave is identical to 221ed11 within 0.1 ¢)
  • 327zpi (18.767c)
  • 11-limit WE (18.755c)

pure octaves 64edo (octave is identical to 13-limit WE within 0.13 ¢

  • 328zpi (18.721c)
  • 180ed7
  • 230ed12
  • 149ed5
Medium priority

118edo (choose ZPIS)

Approximation of harmonics in 118edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 -0.26 +0.00 +0.13 -0.26 -2.72 +0.00 -0.52 +0.13 -2.17 -0.26 +3.54
Relative (%) +0.0 -2.6 +0.0 +1.2 -2.6 -26.8 +0.0 -5.1 +1.2 -21.3 -2.6 +34.8
Steps
(reduced) 		118
(0) 		187
(69) 		236
(0) 		274
(38) 		305
(69) 		331
(95) 		354
(0) 		374
(20) 		392
(38) 		408
(54) 		423
(69) 		437
(83)
  • 187edt
  • 69edf
  • 13-limit WE (10.171c)
  • Best nearby ZPI(s)

13edo

Approximation of harmonics in 13edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +36.5 +0.0 -17.1 +36.5 -45.7 +0.0 -19.3 -17.1 +2.5 +36.5 -9.8
Relative (%) +0.0 +39.5 +0.0 -18.5 +39.5 -49.6 +0.0 -20.9 -18.5 +2.7 +39.5 -10.6
Steps
(reduced) 		13
(0) 		21
(8) 		26
(0) 		30
(4) 		34
(8) 		36
(10) 		39
(0) 		41
(2) 		43
(4) 		45
(6) 		47
(8) 		48
(9)
  • 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)

Approximation of harmonics in 103edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 -2.93 +0.00 -1.85 -2.93 -1.84 +0.00 +5.80 -1.85 -3.75 -2.93 -1.69
Relative (%) +0.0 -25.1 +0.0 -15.9 -25.1 -15.8 +0.0 +49.8 -15.9 -32.1 -25.1 -14.5
Steps
(reduced) 		103
(0) 		163
(60) 		206
(0) 		239
(33) 		266
(60) 		289
(83) 		309
(0) 		327
(18) 		342
(33) 		356
(47) 		369
(60) 		381
(72)
  • 163edt
  • 239ed5
  • 266ed6
  • 289ed7
  • 356ed11
  • 369ed12
  • 381ed13
  • 421ed17
  • 466ed23
  • 13-limit WE (11.658c)
  • Best nearby ZPI(s)

111edo (choose ZPIS)

Approximation of harmonics in 111edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 +0.75 +0.00 +2.88 +0.75 +4.15 +0.00 +1.50 +2.88 +0.03 +0.75 +2.72
Relative (%) +0.0 +6.9 +0.0 +26.6 +6.9 +38.4 +0.0 +13.8 +26.6 +0.3 +6.9 +25.1
Steps
(reduced) 		111
(0) 		176
(65) 		222
(0) 		258
(36) 		287
(65) 		312
(90) 		333
(0) 		352
(19) 		369
(36) 		384
(51) 		398
(65) 		411
(78)
  • 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

Approximation of harmonics in 25edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -2.3 +18.0 -8.8 +0.0 -11.9 -2.3 -23.3 +18.0 +23.5
Relative (%) +0.0 +37.6 +0.0 -4.8 +37.6 -18.4 +0.0 -24.8 -4.8 -48.6 +37.6 +48.9
Steps
(reduced) 		25
(0) 		40
(15) 		50
(0) 		58
(8) 		65
(15) 		70
(20) 		75
(0) 		79
(4) 		83
(8) 		86
(11) 		90
(15) 		93
(18)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

26edo

Approximation of harmonics in 26edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -9.6 +0.0 -17.1 -9.6 +0.4 +0.0 -19.3 -17.1 +2.5 -9.6 -9.8
Relative (%) +0.0 -20.9 +0.0 -37.0 -20.9 +0.9 +0.0 -41.8 -37.0 +5.5 -20.9 -21.1
Steps
(reduced) 		26
(0) 		41
(15) 		52
(0) 		60
(8) 		67
(15) 		73
(21) 		78
(0) 		82
(4) 		86
(8) 		90
(12) 		93
(15) 		96
(18)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

29edo

Approximation of harmonics in 29edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +1.5 +0.0 -13.9 +1.5 -17.1 +0.0 +3.0 -13.9 -13.4 +1.5 -12.9
Relative (%) +0.0 +3.6 +0.0 -33.6 +3.6 -41.3 +0.0 +7.2 -33.6 -32.4 +3.6 -31.3
Steps
(reduced) 		29
(0) 		46
(17) 		58
(0) 		67
(9) 		75
(17) 		81
(23) 		87
(0) 		92
(5) 		96
(9) 		100
(13) 		104
(17) 		107
(20)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

30edo

Approximation of harmonics in 30edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 +13.7 +18.0 -8.8 +0.0 -3.9 +13.7 +8.7 +18.0 -0.5
Relative (%) +0.0 +45.1 +0.0 +34.2 +45.1 -22.1 +0.0 -9.8 +34.2 +21.7 +45.1 -1.3
Steps
(reduced) 		30
(0) 		48
(18) 		60
(0) 		70
(10) 		78
(18) 		84
(24) 		90
(0) 		95
(5) 		100
(10) 		104
(14) 		108
(18) 		111
(21)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

34edo

Approximation of harmonics in 34edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +3.9 +0.0 +1.9 +3.9 -15.9 +0.0 +7.9 +1.9 +13.4 +3.9 +6.5
Relative (%) +0.0 +11.1 +0.0 +5.4 +11.1 -45.0 +0.0 +22.3 +5.4 +37.9 +11.1 +18.5
Steps
(reduced) 		34
(0) 		54
(20) 		68
(0) 		79
(11) 		88
(20) 		95
(27) 		102
(0) 		108
(6) 		113
(11) 		118
(16) 		122
(20) 		126
(24)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

35edo

Approximation of harmonics in 35edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -16.2 +0.0 -9.2 -16.2 -8.8 +0.0 +1.8 -9.2 -2.7 -16.2 +16.6
Relative (%) +0.0 -47.4 +0.0 -26.7 -47.4 -25.7 +0.0 +5.3 -26.7 -8.0 -47.4 +48.5
Steps
(reduced) 		35
(0) 		55
(20) 		70
(0) 		81
(11) 		90
(20) 		98
(28) 		105
(0) 		111
(6) 		116
(11) 		121
(16) 		125
(20) 		130
(25)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

36edo

Approximation of harmonics in 36edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 -2.2 +0.0 -3.9 +13.7 +15.3 -2.0 -7.2
Relative (%) +0.0 -5.9 +0.0 +41.1 -5.9 -6.5 +0.0 -11.7 +41.1 +46.0 -5.9 -21.6
Steps
(reduced) 		36
(0) 		57
(21) 		72
(0) 		84
(12) 		93
(21) 		101
(29) 		108
(0) 		114
(6) 		120
(12) 		125
(17) 		129
(21) 		133
(25)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

37edo

Approximation of harmonics in 37edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +11.6 +0.0 +2.9 +11.6 +4.1 +0.0 -9.3 +2.9 +0.0 +11.6 +2.7
Relative (%) +0.0 +35.6 +0.0 +8.9 +35.6 +12.8 +0.0 -28.7 +8.9 +0.1 +35.6 +8.4
Steps
(reduced) 		37
(0) 		59
(22) 		74
(0) 		86
(12) 		96
(22) 		104
(30) 		111
(0) 		117
(6) 		123
(12) 		128
(17) 		133
(22) 		137
(26)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

9edo

Approximation of harmonics in 9edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -35.3 +0.0 +13.7 -35.3 -35.5 +0.0 +62.8 +13.7 -18.0 -35.3 -40.5
Relative (%) +0.0 -26.5 +0.0 +10.3 -26.5 -26.6 +0.0 +47.1 +10.3 -13.5 -26.5 -30.4
Steps
(reduced) 		9
(0) 		14
(5) 		18
(0) 		21
(3) 		23
(5) 		25
(7) 		27
(0) 		29
(2) 		30
(3) 		31
(4) 		32
(5) 		33
(6)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

10edo

Approximation of harmonics in 10edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -26.3 +18.0 -8.8 +0.0 +36.1 -26.3 +48.7 +18.0 -0.5
Relative (%) +0.0 +15.0 +0.0 -21.9 +15.0 -7.4 +0.0 +30.1 -21.9 +40.6 +15.0 -0.4
Steps
(reduced) 		10
(0) 		16
(6) 		20
(0) 		23
(3) 		26
(6) 		28
(8) 		30
(0) 		32
(2) 		33
(3) 		35
(5) 		36
(6) 		37
(7)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

11edo

Approximation of harmonics in 11edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -47.4 +0.0 +50.0 -47.4 +13.0 +0.0 +14.3 +50.0 -5.9 -47.4 +32.2
Relative (%) +0.0 -43.5 +0.0 +45.9 -43.5 +11.9 +0.0 +13.1 +45.9 -5.4 -43.5 +29.5
Steps
(reduced) 		11
(0) 		17
(6) 		22
(0) 		26
(4) 		28
(6) 		31
(9) 		33
(0) 		35
(2) 		37
(4) 		38
(5) 		39
(6) 		41
(8)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

15edo

Approximation of harmonics in 15edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 +13.7 +18.0 -8.8 +0.0 +36.1 +13.7 +8.7 +18.0 +39.5
Relative (%) +0.0 +22.6 +0.0 +17.1 +22.6 -11.0 +0.0 +45.1 +17.1 +10.9 +22.6 +49.3
Steps
(reduced) 		15
(0) 		24
(9) 		30
(0) 		35
(5) 		39
(9) 		42
(12) 		45
(0) 		48
(3) 		50
(5) 		52
(7) 		54
(9) 		56
(11)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

18edo

Approximation of harmonics in 18edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +31.4 +0.0 +13.7 +31.4 +31.2 +0.0 -3.9 +13.7 -18.0 +31.4 +26.1
Relative (%) +0.0 +47.1 +0.0 +20.5 +47.1 +46.8 +0.0 -5.9 +20.5 -27.0 +47.1 +39.2
Steps
(reduced) 		18
(0) 		29
(11) 		36
(0) 		42
(6) 		47
(11) 		51
(15) 		54
(0) 		57
(3) 		60
(6) 		62
(8) 		65
(11) 		67
(13)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

48edo

Approximation of harmonics in 48edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 -11.3 -2.0 +6.2 +0.0 -3.9 -11.3 -1.3 -2.0 +9.5
Relative (%) +0.0 -7.8 +0.0 -45.3 -7.8 +24.7 +0.0 -15.6 -45.3 -5.3 -7.8 +37.9
Steps
(reduced) 		48
(0) 		76
(28) 		96
(0) 		111
(15) 		124
(28) 		135
(39) 		144
(0) 		152
(8) 		159
(15) 		166
(22) 		172
(28) 		178
(34)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

5edo

Approximation of harmonics in 5edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0 +18 +0 +94 +18 -9 +0 +36 +94 -71 +18 +119
Relative (%) +0.0 +7.5 +0.0 +39.0 +7.5 -3.7 +0.0 +15.0 +39.0 -29.7 +7.5 +49.8
Steps
(reduced) 		5
(0) 		8
(3) 		10
(0) 		12
(2) 		13
(3) 		14
(4) 		15
(0) 		16
(1) 		17
(2) 		17
(2) 		18
(3) 		19
(4)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

6edo

Approximation of harmonics in 6edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +98.0 +0.0 +13.7 +98.0 +31.2 +0.0 -3.9 +13.7 +48.7 +98.0 -40.5
Relative (%) +0.0 +49.0 +0.0 +6.8 +49.0 +15.6 +0.0 -2.0 +6.8 +24.3 +49.0 -20.3
Steps
(reduced) 		6
(0) 		10
(4) 		12
(0) 		14
(2) 		16
(4) 		17
(5) 		18
(0) 		19
(1) 		20
(2) 		21
(3) 		22
(4) 		22
(4)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

20edo

Approximation of harmonics in 20edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -26.3 +18.0 -8.8 +0.0 -23.9 -26.3 -11.3 +18.0 -0.5
Relative (%) +0.0 +30.1 +0.0 -43.9 +30.1 -14.7 +0.0 -39.9 -43.9 -18.9 +30.1 -0.9
Steps
(reduced) 		20
(0) 		32
(12) 		40
(0) 		46
(6) 		52
(12) 		56
(16) 		60
(0) 		63
(3) 		66
(6) 		69
(9) 		72
(12) 		74
(14)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

24edo

Approximation of harmonics in 24edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 -18.8 +0.0 -3.9 +13.7 -1.3 -2.0 +9.5
Relative (%) +0.0 -3.9 +0.0 +27.4 -3.9 -37.7 +0.0 -7.8 +27.4 -2.6 -3.9 +18.9
Steps
(reduced) 		24
(0) 		38
(14) 		48
(0) 		56
(8) 		62
(14) 		67
(19) 		72
(0) 		76
(4) 		80
(8) 		83
(11) 		86
(14) 		89
(17)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

28edo

Approximation of harmonics in 28edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -16.2 +0.0 -0.6 -16.2 +16.9 +0.0 +10.4 -0.6 +5.8 -16.2 +16.6
Relative (%) +0.0 -37.9 +0.0 -1.4 -37.9 +39.4 +0.0 +24.2 -1.4 +13.6 -37.9 +38.8
Steps
(reduced) 		28
(0) 		44
(16) 		56
(0) 		65
(9) 		72
(16) 		79
(23) 		84
(0) 		89
(5) 		93
(9) 		97
(13) 		100
(16) 		104
(20)
  • 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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