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[[User:BudjarnLambeth/Draft related tunings section]]
= Title1 =
= Title1 =
== Octave stretch or compression ==
== Octave stretch or compression ==
Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]], with [[37ed5]] and [[57ed12]] being good options.
What follows is a comparison of stretched- and compressed-octave 60edo tunings.
 
; [[35edf]]  
* Step size: 20.056{{c}}, octave size: 1203.35{{c}}
Stretching the octave of 60edo by a little over 3{{c}} results in improved primes 5, 7 and 11 but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 10.00{{c}}. The tuning 35edf does this.
{{Harmonics in equal|35|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edf}}
{{Harmonics in equal|35|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edf (continued)}}


What follows is a comparison of stretched- and compressed-octave 16edo tunings.
; [[139ed5]]
* Step size: 20.045{{c}}, octave size: 1202.73{{c}}
Stretching the octave of 60edo by a little under{{c}} results in improved primes 5, 7 and 11, but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 9.56{{c}}. The tuning 139ed5 does this.
{{Harmonics in equal|139|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 139ed5}}
{{Harmonics in equal|139|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed5 (continued)}}


; 16edo
; [[zpi|301zpi]]
* Step size: 75.000{{c}}, octave size: 1200.0{{c}}  
* Step size: 20.027{{c}}, octave size: 1201.62{{c}}
Pure-octaves 16edo approximates all harmonics up to 16 within 36.7{{c}}.
Stretching the octave of 60edo by around 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2. This approximates all harmonics up to 16 within 6.48{{c}}. The tuning 301zpi does this.
{{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}}
{{Harmonics in cet|20.027|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 301zpi}}
{{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}}
{{Harmonics in cet| 20.027 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 301zpi (continued)}}


; [[WE|16et, 2.5.7.13 WE tuning]]  
; [[95edt]]  
* Step size: 75.105{{c}}, octave size: 1201.7{{c}}
* Step size: 20.021{{c}}, octave size: 1201.23{{c}}
Stretching the octave of 16edo by around 2{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.
Stretching the octave of 60edo by just over a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 7.06{{c}}. The tuning 95edt does this.
{{Harmonics in cet|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning}}
{{Harmonics in equal|95|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 95edt}}
{{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}}
{{Harmonics in equal|95|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 95edt (continued)}}


; [[zpi|15zpi]] / [[equal tuning|59ed13]]
; [[WE|60et, 13-limit WE tuning]] / [[155ed6]]
* Step size: 75.262{{c}}, octave size: 1204.2{{c}}
* Step size: 20.013{{c}}, octave size: 1200.78{{c}}
Stretching the octave of 16edo by around 4{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5{{c}}. The tunings 15zpi and 59ed13 do this.
Stretching the octave of 60edo by just under a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does 155ed6 whose octaves differ by only 0.02{{c}}.
{{Harmonics in cet|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}}
{{Harmonics in cet|20.013|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning}}
{{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}
{{Harmonics in cet|20.013|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning (continued)}}


; [[WE|16et, 13-limit WE tuning]] / [[37ed5]]
; [[ed12|215ed12]]  
* Step size (WE 16et): 75.315{{c}}, octave size (WE 16et): 1205.0{{c}}
* Step size: 20.009{{c}}, octave size: 1200.55{{c}}
Stretching the octave of 16edo by around 5{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this, so does the tuning 37ed5.
Stretching the octave of 215ed12 by around half a cent results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 9.44{{c}}. The tuning 215ed12 does this.
{{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}}
{{Harmonics in equal|215|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 215ed12}}
{{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}}
{{Harmonics in equal|215|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 215ed12 (continued)}}
{{Harmonics in equal|37|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 37ed5}}
{{Harmonics in equal|37|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 37ed5 (continued)}}


; [[57ed12]] / [[equal tuning|55ed11]]
; 60edo
* Step size (57ed12): 75.473{{c}}, octave size (57ed12): 1207.6{{c}}
* Step size: 20.000{{c}}, octave size: 1200.00{{c}}
Stretching the octave of 16edo by around 7.5{{c}} results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 57ed12 and 55ed11 do this.
Pure-octaves 60edo approximates all harmonics up to 16 within 8.83{{c}}.
{{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}}
{{Harmonics in equal|60|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edo}}
{{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}
{{Harmonics in equal|60|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edo (continued)}}
{{Harmonics in equal|55|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 55ed11}}
 
{{Harmonics in equal|55|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 55ed11 (continued)}}
; [[zpi|302zpi]]  
* Step size: 19.962{{c}}, octave size: 1197.72{{c}}
Compressing the octave of 60edo by around 2{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.84{{c}}. The tuning 202zpi does this. So does the tuning [[equal tuning|208ed11]] whose octave is identical within 0.3{{c}}.
{{Harmonics in cet|19.962|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 302zpi}}
{{Harmonics in cet|19.962|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 302zpi (continued)}}
 
302zpi is particularly well suited to [[catnip]] temperament specifically: in 60edo, catnip's mappings of 5 and 13 both differ from the [[patent val]]s, but in 19.95cet, only it's mapping of 7 differs. The tuning 169ed7 also does this, but 302zpi approximates most simple harmonics better than 169ed7.
 
; [[ed7|169ed7]]  
* Step size: 19.958{{c}}, octave size: 1197.50{{c}}
Compressing the octave of 60edo by around 2.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.94{{c}}. The tuning 169ed7 does this.
{{Harmonics in equal|169|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 169ed7}}
{{Harmonics in equal|169|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 169ed7 (continued)}}
 
; [[zpi|303zpi]]
* Step size: 19.913{{c}}, octave size: 1194.78{{c}}
Compressing the octave of 60edo by around 5{{c}} results in improved primes 5, 7 and 13, but worse primes 2, 3 and 11. This approximates all harmonics up to 16 within 8.75{{c}}. The tuning 303zpi does this. So does [[equal tuning|223ed13]] whose octave is identical within 0.03{{c}}.
{{Harmonics in cet|19.913|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 303zpi}}
{{Harmonics in cet|19.913|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 303zpi (continued)}}


= Title2 =
= Title2 =
=== Possible tunings to be used on each page ===
=== Lab ===
You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.
 
Place holder
 


(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)
<br><br><br><br><br>


; High-priority


13edo
{{harmonics in cet | 300 | intervals=prime}}
* 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)


14edo
{{harmonics in equal | 140 | 12 | 1 | intervals=prime}}
* 22edt
* 36ed6
* 11-limit WE (85.842c)
* 13-limit WE (85.759c)
* 42zpi (86.329c)


16edo
=== Possible tunings to be used on each page ===
* 25edt
You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.
* 41ed6
* 57ed12
* 2.5.7.13 WE (75.105c)
* 13-limit WE (75.315c)
* 15zpi (75.262c)


99edo
(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)
* 157edt
* 256ed6
* 7-limit WE (12.117c)
* 13-limit WE (12.123c)
* 567zpi (12.138c)
* 568zpi (12.115c)


23edo (narrow down edonoi & ZPIs)
; High-priority
* 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
* 139ed5
* 139ed5
* 155ed6
* 208ed11
* 255ed19
* 272ed23 (great for catnip temperament)
* 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)
32edo (narrow down ZPIs)
* 11-limit WE (37.453c)
* 90ed7
* 90ed7 (optimal for dual-5) (133zpi's octave only differs by 0.4{{c}})
* 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
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
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
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
* 266ed6
* 289ed7
* 289ed7
* 356ed11
* 356ed11
* 369ed12
* 381ed13
* 381ed13
* 421ed17
* 421ed17
Line 222: Line 217:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


118edo (choose ZPIS)
111edo (choose ZPIS)
* 187edt
{{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}
* 69edf
* Nearby edt, ed6, ed12 and/or edf
* 13-limit WE (10.171c)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Best nearby ZPI(s)
* 1-2 WE tunings
 
152edo (choose ZPIS)
* 241edt
* 13-limit WE (7.894c)
* Best nearby ZPI(s)
* Best nearby ZPI(s)


; Low priority
; Low priority


111edo
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 251: Line 242:
* 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)
152edo
* 241edt
* 13-limit WE (7.894c)
* Best nearby ZPI(s)
* Best nearby ZPI(s)


Line 298: Line 294:


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 304: Line 301:


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 310: Line 308:


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 316: Line 315:


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 322: Line 322:


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 328: Line 329:


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 334: Line 336:


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 340: Line 343:


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 345: Line 349:
* 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 351: Line 356:
* 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 357: Line 363:
* 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 363: Line 370:
* 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 369: Line 377:
* 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 375: Line 384:
* 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 381: Line 391:
* 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 387: Line 398:
* 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 394: Line 406:


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 400: Line 413:


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 406: Line 420:


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 22:21, 29 August 2025

Quick link

User:BudjarnLambeth/Draft related tunings section

Title1

Octave stretch or compression

What follows is a comparison of stretched- and compressed-octave 60edo tunings.

35edf
  • Step size: 20.056 ¢, octave size: 1203.35 ¢

Stretching the octave of 60edo by a little over 3 ¢ results in improved primes 5, 7 and 11 but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 10.00 ¢. The tuning 35edf does this.

Approximation of harmonics in 35edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.35 +3.35 +6.70 +1.45 +6.70 +0.56 -10.00 +6.70 +4.80 +0.24 -10.00
Relative (%) +16.7 +16.7 +33.4 +7.2 +33.4 +2.8 -49.9 +33.4 +23.9 +1.2 -49.9
Steps
(reduced) 		60
(25) 		95
(25) 		120
(15) 		139
(34) 		155
(15) 		168
(28) 		179
(4) 		190
(15) 		199
(24) 		207
(32) 		214
(4)
Approximation of harmonics in 35edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.18 +3.91 +4.80 -6.65 +8.73 -10.00 -3.33 +8.15 +3.91 +3.60 +6.86 -6.65
Relative (%) -40.8 +19.5 +23.9 -33.2 +43.5 -49.9 -16.6 +40.7 +19.5 +17.9 +34.2 -33.2
Steps
(reduced) 		221
(11) 		228
(18) 		234
(24) 		239
(29) 		245
(0) 		249
(4) 		254
(9) 		259
(14) 		263
(18) 		267
(22) 		271
(26) 		274
(29)
139ed5
  • Step size: 20.045 ¢, octave size: 1202.73 ¢

Stretching the octave of 60edo by a little under ¢ results in improved primes 5, 7 and 11, but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 9.56 ¢. The tuning 139ed5 does this.

Approximation of harmonics in 139ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.73 +2.36 +5.45 +0.00 +5.09 -1.19 +8.18 +4.72 +2.73 -1.92 +7.81
Relative (%) +13.6 +11.8 +27.2 +0.0 +25.4 -6.0 +40.8 +23.5 +13.6 -9.6 +39.0
Steps
(reduced) 		60
(60) 		95
(95) 		120
(120) 		139
(0) 		155
(16) 		168
(29) 		180
(41) 		190
(51) 		199
(60) 		207
(68) 		215
(76)
Approximation of harmonics in 139ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +9.56 +1.53 +2.36 -9.14 +6.17 +7.45 -5.98 +5.45 +1.17 +0.81 +4.04 -9.51
Relative (%) +47.7 +7.6 +11.8 -45.6 +30.8 +37.1 -29.8 +27.2 +5.8 +4.0 +20.1 -47.4
Steps
(reduced) 		222
(83) 		228
(89) 		234
(95) 		239
(100) 		245
(106) 		250
(111) 		254
(115) 		259
(120) 		263
(124) 		267
(128) 		271
(132) 		274
(135)
301zpi
  • Step size: 20.027 ¢, octave size: 1201.62 ¢

Stretching the octave of 60edo by around 1.5 ¢ results in improved primes 3, 5, 7, 11 and 13, but worse primes 2. This approximates all harmonics up to 16 within 6.48 ¢. The tuning 301zpi does this.

Approximation of harmonics in 301zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.62 +0.61 +3.24 -2.56 +2.23 -4.29 +4.86 +1.22 -0.94 -5.73 +3.85
Relative (%) +8.1 +3.0 +16.2 -12.8 +11.1 -21.4 +24.3 +6.1 -4.7 -28.6 +19.2
Step 60 95 120 139 155 168 180 190 199 207 215
Approximation of harmonics in 301zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.47 -2.67 -1.95 +6.48 +1.66 +2.84 +9.37 +0.68 -3.68 -4.11 -0.96 +5.47
Relative (%) +27.3 -13.3 -9.7 +32.4 +8.3 +14.2 +46.8 +3.4 -18.4 -20.5 -4.8 +27.3
Step 222 228 234 240 245 250 255 259 263 267 271 275
95edt
  • Step size: 20.021 ¢, octave size: 1201.23 ¢

Stretching the octave of 60edo by just over a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 7.06 ¢. The tuning 95edt does this.

Approximation of harmonics in 95edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.23 +0.00 +2.47 -3.45 +1.23 -5.37 +3.70 +0.00 -2.22 -7.06 +2.47
Relative (%) +6.2 +0.0 +12.3 -17.2 +6.2 -26.8 +18.5 +0.0 -11.1 -35.3 +12.3
Steps
(reduced) 		60
(60) 		95
(0) 		120
(25) 		139
(44) 		155
(60) 		168
(73) 		180
(85) 		190
(0) 		199
(9) 		207
(17) 		215
(25)
Approximation of harmonics in 95edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.04 -4.13 -3.45 +4.94 +0.09 +1.23 +7.73 -0.98 -5.37 -5.82 -2.70 +3.70
Relative (%) +20.2 -20.6 -17.2 +24.7 +0.4 +6.2 +38.6 -4.9 -26.8 -29.1 -13.5 +18.5
Steps
(reduced) 		222
(32) 		228
(38) 		234
(44) 		240
(50) 		245
(55) 		250
(60) 		255
(65) 		259
(69) 		263
(73) 		267
(77) 		271
(81) 		275
(85)
60et, 13-limit WE tuning / 155ed6
  • Step size: 20.013 ¢, octave size: 1200.78 ¢

Stretching the octave of 60edo by just under a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.63 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this. So does 155ed6 whose octaves differ by only 0.02 ¢.

Approximation of harmonics in 60et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.78 -0.72 +1.56 -4.51 +0.06 -6.64 +2.34 -1.44 -3.73 -8.63 +0.84
Relative (%) +3.9 -3.6 +7.8 -22.5 +0.3 -33.2 +11.7 -7.2 -18.6 -43.1 +4.2
Step 60 95 120 139 155 168 180 190 199 207 215
Approximation of harmonics in 60et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.36 -5.86 -5.23 +3.12 -1.77 -0.66 +5.80 -2.95 -7.36 -7.85 -4.75 +1.62
Relative (%) +11.8 -29.3 -26.1 +15.6 -8.8 -3.3 +29.0 -14.7 -36.8 -39.2 -23.7 +8.1
Step 222 228 234 240 245 250 255 259 263 267 271 275
215ed12
  • Step size: 20.009 ¢, octave size: 1200.55 ¢

Stretching the octave of 215ed12 by around half a cent results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 9.44 ¢. The tuning 215ed12 does this.

Approximation of harmonics in 215ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.55 -1.09 +1.09 -5.05 -0.55 -7.30 +1.64 -2.18 -4.50 -9.44 +0.00
Relative (%) +2.7 -5.5 +5.5 -25.2 -2.7 -36.5 +8.2 -10.9 -22.5 -47.2 +0.0
Steps
(reduced) 		60
(60) 		95
(95) 		120
(120) 		139
(139) 		155
(155) 		168
(168) 		180
(180) 		190
(190) 		199
(199) 		207
(207) 		215
(0)
Approximation of harmonics in 215ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.49 -6.75 -6.14 +2.18 -2.73 -1.64 +4.81 -3.96 -8.39 -8.89 -5.81 +0.55
Relative (%) +7.5 -33.7 -30.7 +10.9 -13.6 -8.2 +24.0 -19.8 -41.9 -44.4 -29.0 +2.7
Steps
(reduced) 		222
(7) 		228
(13) 		234
(19) 		240
(25) 		245
(30) 		250
(35) 		255
(40) 		259
(44) 		263
(48) 		267
(52) 		271
(56) 		275
(60)
60edo
  • Step size: 20.000 ¢, octave size: 1200.00 ¢

Pure-octaves 60edo approximates all harmonics up to 16 within 8.83 ¢.

Approximation of harmonics in 60edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 -1.96 +0.00 -6.31 -1.96 -8.83 +0.00 -3.91 -6.31 +8.68 -1.96
Relative (%) +0.0 -9.8 +0.0 -31.6 -9.8 -44.1 +0.0 -19.6 -31.6 +43.4 -9.8
Steps
(reduced) 		60
(0) 		95
(35) 		120
(0) 		139
(19) 		155
(35) 		168
(48) 		180
(0) 		190
(10) 		199
(19) 		208
(28) 		215
(35)
Approximation of harmonics in 60edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.53 -8.83 -8.27 +0.00 -4.96 -3.91 +2.49 -6.31 +9.22 +8.68 -8.27 -1.96
Relative (%) -2.6 -44.1 -41.3 +0.0 -24.8 -19.6 +12.4 -31.6 +46.1 +43.4 -41.4 -9.8
Steps
(reduced) 		222
(42) 		228
(48) 		234
(54) 		240
(0) 		245
(5) 		250
(10) 		255
(15) 		259
(19) 		264
(24) 		268
(28) 		271
(31) 		275
(35)
302zpi
  • Step size: 19.962 ¢, octave size: 1197.72 ¢

Compressing the octave of 60edo by around 2 ¢ results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.84 ¢. The tuning 202zpi does this. So does the tuning 208ed11 whose octave is identical within 0.3 ¢.

Approximation of harmonics in 302zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.28 -5.57 -4.56 +8.37 -7.85 +4.75 -6.84 +8.83 +6.09 +0.78 +9.84
Relative (%) -11.4 -27.9 -22.8 +41.9 -39.3 +23.8 -34.3 +44.2 +30.5 +3.9 +49.3
Step 60 95 120 140 155 169 180 191 200 208 216
Approximation of harmonics in 302zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.96 +2.47 +2.80 -9.12 +5.70 +6.55 -7.20 +3.81 -0.81 -1.50 +1.39 +7.56
Relative (%) -44.9 +12.4 +14.0 -45.7 +28.5 +32.8 -36.1 +19.1 -4.1 -7.5 +7.0 +37.9
Step 222 229 235 240 246 251 255 260 264 268 272 276

302zpi is particularly well suited to catnip temperament specifically: in 60edo, catnip's mappings of 5 and 13 both differ from the patent vals, but in 19.95cet, only it's mapping of 7 differs. The tuning 169ed7 also does this, but 302zpi approximates most simple harmonics better than 169ed7.

169ed7
  • Step size: 19.958 ¢, octave size: 1197.50 ¢

Compressing the octave of 60edo by around 2.5 ¢ results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.94 ¢. The tuning 169ed7 does this.

Approximation of harmonics in 169ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.97 -8.24 -7.93 +4.43 +7.73 +0.00 +8.03 +3.46 +0.46 -5.07 +3.76
Relative (%) -19.9 -41.3 -39.8 +22.2 +38.8 +0.0 +40.3 +17.4 +2.3 -25.4 +18.9
Steps
(reduced) 		60
(60) 		95
(95) 		120
(120) 		140
(140) 		156
(156) 		169
(0) 		181
(12) 		191
(22) 		200
(31) 		208
(39) 		216
(47)
Approximation of harmonics in 169ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.73 -3.97 -3.81 +4.07 -1.22 -0.51 +5.56 -3.50 -8.24 -9.04 -6.26 -0.20
Relative (%) +23.7 -19.9 -19.1 +20.4 -6.1 -2.5 +27.9 -17.6 -41.3 -45.3 -31.4 -1.0
Steps
(reduced) 		223
(54) 		229
(60) 		235
(66) 		241
(72) 		246
(77) 		251
(82) 		256
(87) 		260
(91) 		264
(95) 		268
(99) 		272
(103) 		276
(107)
303zpi
  • Step size: 19.913 ¢, octave size: 1194.78 ¢

Compressing the octave of 60edo by around 5 ¢ results in improved primes 5, 7 and 13, but worse primes 2, 3 and 11. This approximates all harmonics up to 16 within 8.75 ¢. The tuning 303zpi does this. So does 223ed13 whose octave is identical within 0.03 ¢.

Approximation of harmonics in 303zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.22 +9.69 +9.47 +1.51 +4.47 -3.53 +4.25 -0.53 -3.71 -9.41 -0.75
Relative (%) -26.2 +48.7 +47.6 +7.6 +22.5 -17.7 +21.4 -2.6 -18.6 -47.3 -3.8
Step 60 96 121 140 156 169 181 191 200 208 216
Approximation of harmonics in 303zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.07 -8.75 -8.71 -0.97 -6.36 -5.75 +0.21 -8.93 +6.16 +5.28 +7.97 -5.97
Relative (%) +0.4 -43.9 -43.8 -4.9 -31.9 -28.9 +1.1 -44.9 +31.0 +26.5 +40.0 -30.0
Step 223 229 235 241 246 251 256 260 265 269 273 276

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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