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[[User:BudjarnLambeth/Draft related tunings section]]
[[User:BudjarnLambeth/Draft related tunings section]]


= Title1 =
; [[108ed6]]
== Octave stretch or compression ==
* Step size: NNN{{c}}, octave size: NNN{{c}}
{{main|23edo and octave stretching}}
_ing the octave of 42edo 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 108ed6 does this. So does the tuning [[97ed5]] whose octave differs by only 0.1{{c}}.
{{Harmonics in equal|108|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 108ed6}}
{{Harmonics in equal|108|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 108ed6 (continued)}}


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.
; [[zpi|189zpi]]  
* Step size: 28.689{{c}}, octave size: NNN{{c}}
_ing the octave of 42edo 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 189zpi does this.
{{Harmonics in cet|28.689|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 189zpi}}
{{Harmonics in cet|28.689|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 189zpi (continued)}}


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.
; [[150ed12]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of 42edo 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 150ed12 does this.
{{Harmonics in equal|150|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed12}}
{{Harmonics in equal|150|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed12 (continued)}}


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.
; [[equal tuning|145ed11]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of 42edo 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 145ed11 does this.
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 145ed11}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 145ed11 (continued)}}


What follows is a comparison of stretched- and compressed-octave 23edo tunings.
; 42edo
* Step size: NNN{{c}}, octave size: NNN{{c}}
Pure-octaves 42edo approximates all harmonics up to 16 within NNN{{c}}. The tuning [[zpi|190zpi]] is almost exactly the same as pure-octaves 42edo, its octave differing by less than 0.05{{c}}.
{{Harmonics in equal|42|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edo}}
{{Harmonics in equal|42|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edo (continued)}}


; [[zpi|86zpi]]  
; [[118ed7]]  
* Step size: 51.653{{c}}, octave size: 1188.0{{c}}
* Step size: NNN{{c}}, octave size: NNN{{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.
_ing the octave of 42edo 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 118ed7 does this.
{{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|118|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 118ed7}}
{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
{{Harmonics in equal|118|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 118ed7 (continued)}}


; [[60ed6]]  
; [[WE|42et, 13-limit WE tuning]]  
* Step size: 51.700{{c}}, octave size: 1189.1{{c}}
* Step size: 28.534{{c}}, octave size: NNN{{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}}.
_ing the octave of 42edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|28.534|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning}}
{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
{{Harmonics in cet|28.534|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning (continued)}}


; [[zpi|85zpi]]  
; [[151ed12]]  
* Step size: 52.114{{c}}, octave size: 1198.6{{c}}
* Step size: NNN{{c}}, octave size: NNN{{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}}.
_ing the octave of 42edo 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. So do the 7-limit [[WE]] and [[TE]] tunings of 42et, whose octaves are within 0.3{{c}} of 151ed12.
{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed12}}
{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed12 (continued)}}


; 23edo
; [[109ed6]]
* Step size: NNN{{c}}, octave size: 1200.0{{c}}  
* Step size: NNN{{c}}, octave size: NNN{{c}}
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
_ing the octave of 42edo 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 109ed6 does this.
{{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
{{Harmonics in equal|109|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 109ed6}}
{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}
{{Harmonics in equal|109|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 109ed6 (continued)}}


; [[WE|23et, 13-limit WE tuning]]  
; [[zpi|191zpi]]  
* Step size: 52.237{{c}}, octave size: 1201.5{{c}}
* Step size: 28.444{{c}}, octave size: NNN{{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}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
_ing the octave of 42edo 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 191zpi does this.
{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|28.444|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 191zpi}}
{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
{{Harmonics in cet|28.444|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 191zpi (continued)}}


; [[WE|23et, 2.3.5.13 WE tuning]]  
; [[67edt]]  
* Step size: 52.447{{c}}, octave size: 1206.3{{c}}
* Step size: NNN{{c}}, octave size: NNN{{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}}. 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}}.
_ing the octave of 42edo 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 67edt does this.
{{Harmonics in cet|52.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in equal|67|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 67edt}}
{{Harmonics in cet|52.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
{{Harmonics in equal|67|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 67edt (continued)}}
 
; [[59ed6]]
* Step size: 52.575{{c}}, octave size: 1209.2{{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|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
 
; [[zpi|84zpi]]
* Step size: 52.615{{c}}, octave size: 1210.1{{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 ZPINAME does this.
{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
 
; [[36edt]]
* Step size: 52.832{{c}}, octave size: 1215.1{{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 EDONOI does this.
{{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
 
; [[84ed13]]
* Step size: 52.863{{c}}, octave size: 1215.9{{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 EDONOI does this.
{{Harmonics in equal|84|13|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


= Title2 =
= Title2 =
=== Lab ===
=== Lab ===


54edo (possibly narrow down edonoi)
Place holder
* 126ed5
{{harmonics in equal | 126 | 5 | 1 | intervals=prime}}
* 38ed5/3 (stretch, improves 3.5.7.11.13.17.19.23)
{{harmonics in equal | 38 | 5 | 3 | intervals=prime}}
* 262zpi (22.313c)
{{harmonics in cet | 22.313 | intervals=prime}}
* 263zpi (22.243c)
{{harmonics in cet | 22.243 | intervals=prime}}
* 13-limit WE (22.198c)
{{harmonics in cet | 22.198 | intervals=prime}}
* pure octave 54edo
{{harmonics in equal | 54 | 2 | 1 | intervals=integer | columns=12}}
* 2.3.7.11.13 WE (22.180c)
{{harmonics in cet | 22.180 | intervals=prime}}
* 264zpi (22.175c)
{{harmonics in cet | 22.175 | intervals=prime}}
* 40ed5/3 (compress, improves 3.5.11.13.17.19 (not 7))
{{harmonics in equal | 40 | 5 | 3 | intervals=prime}}
* 152ed7
{{harmonics in equal | 152 | 7 | 1 | intervals=prime}}
* 86edt
{{harmonics in equal | 86 | 3 | 1 | intervals=prime}}


59edo (narrow down ZPIs) (Nothing special abt these choices)
* 93edt
{{harmonics in equal | 93 | 3 | 1 | intervals=prime}}
* 203ed11
{{harmonics in equal | 203 | 11 | 1 | intervals=prime}}
* 293zpi (20.454c)
{{harmonics in cet | 20.454 | intervals=prime}}
* 294zpi (20.399c)
{{harmonics in cet | 20.399 | intervals=prime}}
* 295zpi (20.342c)
{{harmonics in cet | 20.342 | intervals=prime}}
* 13-limit WE (20.320c)
{{harmonics in cet | 20.320 | intervals=prime}}
* 11-limit WE (20.310c)
{{harmonics in cet | 20.310 | intervals=prime}}
* pure octaves 59edo
{{harmonics in equal | 59 | 2 | 1 | intervals=integer | columns=12}}
* 7-limit WE (20.301c)
{{harmonics in cet | 20.301 | intervals=prime}}
* 296zpi (20.282c)
{{harmonics in cet | 20.282 | intervals=prime}}
* 297zpi (20.229c)
{{harmonics in cet | 20.229 | intervals=prime}}
* 166ed7
{{harmonics in equal | 166 | 7 | 1 | intervals=prime}}


64edo (narrow down ZPIs)
<br><br><br><br><br>
* 47ed5/3 (like 221ed11 but benefits & drawbacks both amplified)
{{harmonics in equal | 47 | 5 | 3 | intervals=prime}}
* 221ed11
{{harmonics in equal | 221 | 11 | 1 | intervals=prime}}
* 325zpi (18.868c)
{{harmonics in cet | 18.868 | intervals=prime}}
* 326zpi (18.816c)
{{harmonics in cet | 18.816 | intervals=prime}}
* 327zpi (18.767c)
{{harmonics in cet | 18.767 | intervals=prime}}
* 11-limit WE (18.755c)
{{harmonics in cet | 18.755 | intervals=prime}}
* pure octaves 64edo
{{harmonics in equal | 64 | 2 | 1 | intervals=integer | columns=12}}
* 13-limit WE (18.752c)
{{harmonics in cet | 18.752 | intervals=prime}}
* 328zpi (18.721c)
{{harmonics in cet | 18.721 | intervals=prime}}
* 329zpi (18.672c)
{{harmonics in cet | 18.672 | intervals=prime}}
* 330zpi (18.630c)
{{harmonics in cet | 18.630 | intervals=prime}}
* 180ed7
{{harmonics in equal | 180 | 7 | 1 | intervals=prime}}
* 149ed5
{{harmonics in equal | 149 | 5 | 1 | intervals=prime}}




{{harmonics in cet | 300 | intervals=prime}}


<br><br><br>
{{harmonics in cet | 300 | intervals=prime}}
{{harmonics in equal | 140 | 12 | 1 | intervals=prime}}
{{harmonics in equal | 140 | 12 | 1 | intervals=prime}}


Line 169: Line 89:
; High-priority
; High-priority


60edo (narrow down edonoi & ZPIs)
42edo (reduce # of edonoi)
* 35edf
* 108ed6 (octave is identical to 97ed5 within 0.1{{c}})
* 139ed5
* 189zpi (28.689c)
* 301zpi (20.027c)
* 150ed12
* 95edt
* 145ed11
* 13-limit WE (20.013c) (155ed6 has octaves only 0.02{{c}} different)
''190zpi's octave is within 0.05{{c}} of pure-octaves 42edo''
* 215ed12
* 118ed7
* 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)
* 11-limit WE (36.349c)
* 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}})
* 2.3.5.11 WE (30.703c)
* 173zpi (30.672c) (octave of 62edt differs by only 0.2{{c}})
* 110ed7 (octave of 145ed13 differs by only 0.1{{c}})
* 91ed5
 
42edo
*Good <27% rel err
*Okay <40% rel err
{{harmonics in equal | 42 | 2 | 1 | intervals=integer | columns=12}}
* 42ed257/128 (good 2.3.5.7; bad 11.13)
* 11ed6/5 (good 2.3.5; okay 7.11.13)
* 189zpi (28.689c) (good 2.5.13; okay 3.11; bad 7)
* 190zpi (28.572c)
* 13-limit WE (28.534c)
* 13-limit WE (28.534c)
* 34ed7/4 (good 2.5.7.13; okay 3.11)
* 151ed12 (octave is identical to 7-limit WE within 0.3{{c}})
* 7-limit WE (28.484c) (good 2.3.5.11.13; bad 7)
* 109ed6
* 191zpi (28.444c)
* 191zpi (28.444c)
* 1ed123/121 (good 2.3.5.11; okay 13; bad 7)
* 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 (possibly narrow down edonoi)
54edo
{{harmonics in equal | 54 | 2 | 1 | intervals=integer | columns=12}}
* 139ed6 (octave is identical to 262zpi within 0.2{{c}})
* 126ed5
* 151ed7
* 38ed5/3 (stretch, improves 3.5.7.11.13.17.19.23)
* 193ed12
* 262zpi (22.313c)
* 263zpi (22.243c)
* 263zpi (22.243c)
* 13-limit WE (22.198c)
* 13-limit WE (22.198c) (octave is identical to 187ed11 within 0.1{{c}})
* 2.3.7.11.13 WE (22.180c)
* 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})
* 264zpi (22.175c)
* 40ed5/3 (compress, improves 3.5.11.13.17.19 (not 7))
* 152ed7
* 152ed7
* 86edt
* 140ed6
* 126ed5 (octave is identical to 86edt within 0.1{{c}})


59edo (narrow down ZPIs)
64edo
* (Nothing special abt these choices)
* 179ed7 (octave is identical to 326zpi within 0.3{{c}})
{{harmonics in equal | 59 | 2 | 1 | intervals=integer | columns=12}}
* 165ed6
* 93edt
* 229ed12 (octave is identical to 221ed11 within 0.1{{c}})
* 203ed11
* 327zpi (18.767c)
* 293zpi (20.454c)
* 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)
* 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)
* 13-limit WE (20.320c)
* 11-limit WE (20.310c)
* 7-limit WE (20.301c)
* 7-limit WE (20.301c)
* 166ed7
* 212ed12
* 296zpi (20.282c)
* 296zpi (20.282c)
* 297zpi (20.229c)
* 153ed6
* 166ed7
 
64edo (narrow down ZPIs)
{{harmonics in equal | 64 | 2 | 1 | intervals=integer | columns=12}}
* 47ed5/3 (like 221ed11 but benefits & drawbacks both amplified)
* 221ed11
* 325zpi (18.868c)
* 326zpi (18.816c)
* 327zpi (18.767c)
* 11-limit WE (18.755c)
* 13-limit WE (18.752c)
* 328zpi (18.721c)
* 329zpi (18.672c)
* 330zpi (18.630c)
* 180ed7
* 149ed5


; Medium priority
; Medium priority


118edo (choose ZPIS)
25edo
{{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 25 | 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 edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 314: Line 147:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


; Low priority
26edo
 
{{harmonics in equal | 26 | 2 | 1 | intervals=integer | columns=12}}
104edo
* 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 322: Line 154:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


125edo
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 328: Line 161:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


145edo
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 334: Line 168:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


152edo
34edo
* 241edt
{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
* 13-limit WE (7.894c)
* Best nearby ZPI(s)
 
159edo
* 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 345: Line 175:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


166edo
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 351: Line 182:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


182edo
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 357: Line 189:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


198edo
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 363: Line 196:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


212edo
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 369: Line 203:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


243edo
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 375: Line 210:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


247edo
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 381: Line 217:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


; Optional
15edo
 
{{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}
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 390: Line 224:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


26edo
18edo
{{harmonics in equal | 26 | 2 | 1 | intervals=integer | columns=12}}
{{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 397: Line 231:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


29edo
48edo
{{harmonics in equal | 29 | 2 | 1 | intervals=integer | columns=12}}
{{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 404: Line 238:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


30edo
5edo
{{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}}
{{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 411: Line 245:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


34edo
6edo
{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
{{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 418: Line 252:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


35edo
20edo
{{harmonics in equal | 35 | 2 | 1 | intervals=integer | columns=12}}
{{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 425: Line 259:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


36edo
24edo
{{harmonics in equal | 36 | 2 | 1 | intervals=integer | columns=12}}
{{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 432: Line 266:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


37edo
28edo
{{harmonics in equal | 37 | 2 | 1 | intervals=integer | columns=12}}
{{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)
Line 439: Line 273:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


5edo
13edo
{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}
{{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 edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 446: Line 309:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


6edo
; Low priority
{{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}}
 
104edo
* 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 453: Line 317:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


9edo
125edo
{{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 460: Line 323:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


10edo
145edo
{{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 467: Line 329:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


11edo
152edo
{{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}}
* 241edt
* 13-limit WE (7.894c)
* Best nearby ZPI(s)
 
159edo
* 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 474: Line 340:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


15edo
166edo
{{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 481: Line 346:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


18edo
182edo
{{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 488: Line 352:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


48edo
198edo
{{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 495: Line 358:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


20edo
212edo
{{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 502: Line 364:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


24edo
243edo
{{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 509: Line 370:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


28edo
247edo
{{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 09:24, 30 August 2025

Quick link

User:BudjarnLambeth/Draft related tunings section

108ed6
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 108ed6 does this. So does the tuning 97ed5 whose octave differs by only 0.1 ¢.

Approximation of harmonics in 108ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.3 -6.3 +12.6 -0.3 +0.0 -8.4 -9.8 -12.6 +6.0 +13.3 +6.3
Relative (%) +22.0 -22.0 +44.0 -1.0 +0.0 -29.2 -34.0 -44.0 +21.0 +46.5 +22.0
Steps
(reduced) 		42
(42) 		66
(66) 		84
(84) 		97
(97) 		108
(0) 		117
(9) 		125
(17) 		132
(24) 		139
(31) 		145
(37) 		150
(42)
Approximation of harmonics in 108ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +11.4 -2.1 -6.6 -3.5 +6.5 -6.3 -13.8 +12.3 +14.0 -9.1 +0.1 +12.6
Relative (%) +39.5 -7.2 -23.0 -12.0 +22.5 -22.0 -47.9 +42.9 +48.9 -31.6 +0.5 +44.0
Steps
(reduced) 		155
(47) 		159
(51) 		163
(55) 		167
(59) 		171
(63) 		174
(66) 		177
(69) 		181
(73) 		184
(76) 		186
(78) 		189
(81) 		192
(84)
189zpi
  • Step size: 28.689 ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 189zpi does this.

Approximation of harmonics in 189zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.9 -8.5 +9.9 -3.5 -3.5 -12.2 -13.9 +11.7 +1.5 +8.6 +1.4
Relative (%) +17.2 -29.6 +34.4 -12.1 -12.3 -42.6 -48.4 +40.9 +5.1 +29.9 +4.9
Step 42 66 84 97 108 117 125 133 139 145 150
Approximation of harmonics in 189zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +6.3 -7.3 -12.0 -8.9 +0.9 -12.0 +9.1 +6.4 +8.0 +13.5 -6.1 +6.3
Relative (%) +21.8 -25.4 -41.7 -31.2 +3.0 -41.9 +31.8 +22.3 +27.9 +47.1 -21.1 +22.1
Step 155 159 163 167 171 174 178 181 184 187 189 192
150ed12
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 150ed12 does this.

Approximation of harmonics in 150ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.5 -9.1 +9.1 -4.4 -4.5 -13.3 +13.6 +10.5 +0.2 +7.2 +0.0
Relative (%) +15.9 -31.7 +31.7 -15.3 -15.9 -46.4 +47.6 +36.6 +0.6 +25.2 +0.0
Steps
(reduced) 		42
(42) 		66
(66) 		84
(84) 		97
(97) 		108
(108) 		117
(117) 		126
(126) 		133
(133) 		139
(139) 		145
(145) 		150
(0)
Approximation of harmonics in 150ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.8 -8.8 -13.5 -10.5 -0.7 -13.6 +7.5 +4.7 +6.3 +11.8 -7.8 +4.5
Relative (%) +16.8 -30.5 -47.0 -36.6 -2.5 -47.6 +26.1 +16.4 +21.9 +41.1 -27.2 +15.9
Steps
(reduced) 		155
(5) 		159
(9) 		163
(13) 		167
(17) 		171
(21) 		174
(24) 		178
(28) 		181
(31) 		184
(34) 		187
(37) 		189
(39) 		192
(42)
145ed11
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 145ed11 does this.

Approximation of harmonics in 145ed11
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in 145ed11 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
42edo
  • Step size: NNN ¢, octave size: NNN ¢

Pure-octaves 42edo approximates all harmonics up to 16 within NNN ¢. The tuning 190zpi is almost exactly the same as pure-octaves 42edo, its octave differing by less than 0.05 ¢.

Approximation of harmonics in 42edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +12.3 +0.0 +13.7 +12.3 +2.6 +0.0 -3.9 +13.7 -8.5 +12.3
Relative (%) +0.0 +43.2 +0.0 +47.9 +43.2 +9.1 +0.0 -13.7 +47.9 -29.6 +43.2
Steps
(reduced) 		42
(0) 		67
(25) 		84
(0) 		98
(14) 		109
(25) 		118
(34) 		126
(0) 		133
(7) 		140
(14) 		145
(19) 		151
(25)
Approximation of harmonics in 42edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -12.0 +2.6 -2.6 +0.0 +9.3 -3.9 -11.8 +13.7 -13.6 -8.5 +0.3 +12.3
Relative (%) -41.8 +9.1 -8.9 +0.0 +32.7 -13.7 -41.3 +47.9 -47.7 -29.6 +1.0 +43.2
Steps
(reduced) 		155
(29) 		160
(34) 		164
(38) 		168
(0) 		172
(4) 		175
(7) 		178
(10) 		182
(14) 		184
(16) 		187
(19) 		190
(22) 		193
(25)
118ed7
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 118ed7 does this.

Approximation of harmonics in 118ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.9 +10.9 -1.9 +11.5 +9.9 +0.0 -2.8 -6.8 +10.6 -11.7 +9.0
Relative (%) -3.2 +38.0 -6.5 +40.4 +34.8 +0.0 -9.7 -24.0 +37.1 -40.8 +31.5
Steps
(reduced) 		42
(42) 		67
(67) 		84
(84) 		98
(98) 		109
(109) 		118
(0) 		126
(8) 		133
(15) 		140
(22) 		145
(27) 		151
(33)
Approximation of harmonics in 118ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +13.2 -0.9 -6.2 -3.7 +5.5 -7.8 +12.8 +9.7 +10.9 -12.6 -3.9 +8.1
Relative (%) +46.1 -3.2 -21.6 -13.0 +19.4 -27.2 +44.9 +33.9 +38.0 -44.1 -13.6 +28.3
Steps
(reduced) 		156
(38) 		160
(42) 		164
(46) 		168
(50) 		172
(54) 		175
(57) 		179
(61) 		182
(64) 		185
(67) 		187
(69) 		190
(72) 		193
(75)
42et, 13-limit WE tuning
  • Step size: 28.534 ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 42et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.6 +9.8 -3.1 +10.0 +8.3 -1.8 -4.7 -8.9 +8.4 -13.9 +6.7
Relative (%) -5.5 +34.4 -11.0 +35.1 +28.9 -6.4 -16.5 -31.1 +29.6 -48.7 +23.4
Step 42 67 84 98 109 118 126 133 140 145 151
Approximation of harmonics in 42et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +10.8 -3.4 -8.7 -6.3 +2.9 -10.5 +10.1 +6.9 +8.0 +13.1 -6.8 +5.1
Relative (%) +37.8 -11.9 -30.5 -22.0 +10.1 -36.7 +35.3 +24.1 +28.1 +45.8 -23.9 +17.9
Step 156 160 164 168 172 175 179 182 185 188 190 193
151ed12
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 42edo 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. So do the 7-limit WE and TE tunings of 42et, whose octaves are within 0.3 ¢ of 151ed12.

Approximation of harmonics in 151ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in 151ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
109ed6
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 109ed6 does this.

Approximation of harmonics in 109ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.8 +4.8 -9.5 +2.6 +0.0 -10.7 +14.2 +9.5 -2.2 +3.6 -4.8
Relative (%) -16.7 +16.7 -33.4 +9.1 +0.0 -37.8 +49.9 +33.4 -7.6 +12.6 -16.7
Steps
(reduced) 		42
(42) 		67
(67) 		84
(84) 		98
(98) 		109
(0) 		118
(9) 		127
(18) 		134
(25) 		140
(31) 		146
(37) 		151
(42)
Approximation of harmonics in 109ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -1.0 +13.0 +7.4 +9.5 -10.1 +4.8 -3.5 -6.9 -6.0 -1.2 +7.3 -9.5
Relative (%) -3.6 +45.5 +25.8 +33.2 -35.6 +16.7 -12.2 -24.3 -21.1 -4.1 +25.5 -33.4
Steps
(reduced) 		156
(47) 		161
(52) 		165
(56) 		169
(60) 		172
(63) 		176
(67) 		179
(70) 		182
(73) 		185
(76) 		188
(79) 		191
(82) 		193
(84)
191zpi
  • Step size: 28.444 ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 191zpi does this.

Approximation of harmonics in 191zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.4 +3.8 -10.7 +1.2 -1.6 -12.4 +12.4 +7.6 -4.2 +1.5 -6.9
Relative (%) -18.8 +13.3 -37.6 +4.2 -5.5 -43.7 +43.6 +26.7 -14.6 +5.3 -24.3
Step 42 67 84 98 109 118 127 134 140 146 151
Approximation of harmonics in 191zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.3 +10.7 +5.0 +7.0 -12.6 +2.2 -6.0 -9.5 -8.6 -3.8 +4.5 -12.3
Relative (%) -11.5 +37.5 +17.5 +24.7 -44.3 +7.9 -21.2 -33.4 -30.4 -13.5 +15.9 -43.1
Step 156 161 165 169 172 176 179 182 185 188 191 193
67edt
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 42edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 67edt does this.

Approximation of harmonics in 67edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -7.7 +0.0 +12.9 -4.3 -7.7 +9.3 +5.2 +0.0 -12.1 -6.8 +12.9
Relative (%) -27.2 +0.0 +45.5 -15.3 -27.2 +32.7 +18.3 +0.0 -42.6 -23.8 +45.5
Steps
(reduced) 		42
(42) 		67
(0) 		85
(18) 		98
(31) 		109
(42) 		119
(52) 		127
(60) 		134
(0) 		140
(6) 		146
(12) 		152
(18)
Approximation of harmonics in 67edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -12.1 +1.5 -4.3 -2.5 +6.1 -7.7 +12.2 +8.6 +9.3 +13.9 -6.3 +5.2
Relative (%) -42.6 +5.4 -15.3 -8.9 +21.4 -27.2 +43.0 +30.2 +32.7 +49.0 -22.1 +18.3
Steps
(reduced) 		156
(22) 		161
(27) 		165
(31) 		169
(35) 		173
(39) 		176
(42) 		180
(46) 		183
(49) 		186
(52) 		189
(55) 		191
(57) 		194
(60)

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

42edo (reduce # of edonoi)

  • 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

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 ¢)

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

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 ¢

  • 13-limit WE (20.320c)
  • 7-limit WE (20.301c)
  • 166ed7
  • 212ed12
  • 296zpi (20.282c)
  • 153ed6
Medium priority

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)

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)

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)

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)
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