User:BudjarnLambeth/Sandbox2: Difference between revisions

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


= Title1 =
== Octave stretch and compression ==
== Octave stretch or compression ==
99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.


What follows is a comparison of stretched- and compressed-octave 99edo tunings.
; [[zpi|209zpi]]
* Step size: 26.550{{c}}, octave size: 1194.8{{c}}
Compressing the octave of 45edo by around 5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1{{c}}. The tuning 209zpi does this.
{{Harmonics in cet|26.550|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 209zpi}}
{{Harmonics in cet|26.550|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 209zpi (continued)}}


; [[zpi|567zpi]]
; 45edo
* Step size: 12.138{{c}}, octave size: 1201.66{{c}}
* Step size: 26.667{{c}}, octave size: 1200.0{{c}}  
Stretching the octave of 99edo by around 1.5{{c}} results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.
Pure-octaves 45edo approximates all harmonics up to 16 within 13.0{{c}}.
{{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}}
{{Harmonics in equal|45|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45edo}}
{{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}}
{{Harmonics in equal|45|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45edo (continued)}}


; [[WE|99et, 13-limit WE tuning]]  
; [[WE|45et, 13-limit WE tuning]]  
* Step size: 12.123{{c}}, octave size: 1200.18{{c}}
* Step size: 26.695{{c}}, octave size: 1201.3{{c}}
Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
Stretching the octave of 45edo by around 1{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}}
{{Harmonics in cet|26.695|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning}}
{{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}}
{{Harmonics in cet|26.695|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning (continued)}}


; 99edo
; [[161ed12]]
* Step size: 12.121{{c}}, octave size: 1200.00{{c}}  
* Step size: Octave size: 1202.4{{c}}
Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.
Stretching the octave of 45edo by around 2.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2{{c}}. The tuning 161ed12 does this.
{{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}}
{{Harmonics in equal|161|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 161ed12}}
{{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}}
{{Harmonics in equal|161|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 161ed12 (continued)}}


; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]
; [[116ed6]]  
* Step size: 12.117{{c}}, octave size: 1199.58{{c}}
* Step size: Octave size: 1203.3{{c}}
Compressing the octave of 99edo by around 0.6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
Stretching the octave of 45edo by around 3{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 116ed6 does this. So does [[ed7|126ed7]] whose octave is identical within 0.1{{c}}.
{{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}}
{{Harmonics in equal|116|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 116ed6}}
{{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}}
{{Harmonics in equal|116|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 116ed6 (continued)}}


; [[zpi|568zpi]]  
; [[WE|45et, 7-limit WE tuning]]  
* Step size: 12.115{{c}}, octave size: 1199.39{{c}}
* Step size: 26.745{{c}}, octave size: 1203.5{{c}}
Compressing the octave of 99edo by around 0.4{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.
Stretching the octave of 45edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
{{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}}
{{Harmonics in cet|26.745|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning}}
{{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}}
{{Harmonics in cet|26.745|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning (continued)}}


; [[157edt]] / [[ed5|230ed5]]
; [[zpi|207zpi]]  
* Step size: 12.114{{c}}, octave size: 1199.32{{c}}
* Step size: 26.762{{c}}, octave size: 1204.3{{c}}
Compressing the octave of 99edo by around 0.3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
Stretching the octave of 45edo by around 4{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 207zpi does this.
{{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}}
{{Harmonics in cet|26.762|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 207zpi}}
{{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}}
{{Harmonics in cet|26.762|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 207zpi (continued)}}
 
; [[71edt]]  
* Step size: 26.788{{c}}, octave size: 1205.5{{c}}
Stretching the octave of 45edo by around 5.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 71edt does this. So does the tuning [[equal tuning|155ed11]] whose octave is identical within 0.3{{c}}.
{{Harmonics in equal|71|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 71edt}}
{{Harmonics in equal|71|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 71edt (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


23edo (narrow down edonoi & ZPIs)
{{harmonics in cet | 300 | intervals=prime}}
* Main: "23edo and octave stretching"
* 36edt
{{harmonics in equal|36|3|1|intervals=prime}}
* 59ed6
{{harmonics in equal|59|6|1|intervals=prime}}
* 60ed6
{{harmonics in equal|60|3|1|intervals=prime}}
* 2.3.5.13 WE (52.447c)
{{harmonics in cet|52.447|intervals=prime}}
* 13-limit WE (52.237c)
{{harmonics in cet|52.237|intervals=prime}}
* 84zpi (52.615c)
{{harmonics in cet| 52.615 |intervals=prime}}
* 85zpi (52.114c)
{{harmonics in cet| 52.114 |intervals=prime}}
* 86zpi (51.653c)
{{harmonics in cet| 51.653 |intervals=prime}}


60edo (narrow down edonoi & ZPIs)
{{harmonics in equal | 140 | 12 | 1 | intervals=prime}}
* 95edt
* 35edf
* 139ed5
* 155ed6
* 208ed11
* (???)ed12
* 255ed19
* 272ed23 (great for catnip temperament, maybe there's a similar but simpler tuning w similar benefits?)
* 13-limit WE (20.013c)
* 299zpi (20.128c)
* 300zpi (20.093c)
* 301zpi (20.027c)
* 302zpi (19.962c)
* 303zpi (19.913c)
* 304zpi (19.869c)


; Medium priority
=== 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.


13edo
(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)
* 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)


32edo (narrow down ZPIs)
; High-priority
* 90ed7
* 51edt
* 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)
54edo
* 76ed5
* 139ed6 (octave is identical to 262zpi within 0.2{{c}})
* 92ed7
* 151ed7
* 52edt
* 193ed12
* 1ed47/46
* 263zpi (22.243c)
* 114ed11
* 13-limit WE (22.198c)  (octave is identical to 187ed11 within 0.1{{c}})
* 122ed13
* 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})
* 93ed7
* 152ed7
* 23edPhi
* 140ed6
* 77ed5
* 126ed5 (octave is identical to 86edt within 0.1{{c}})
* 123ed13
* 115ed11
* 11-limit WE (36.349c)
* 13-limit WE (36.357c)
* 137zpi (36.628c)
* 138zpi (36.394c)
* 139zpi (36.179c)


39edo (narrow down slightly)
64edo
* 62edt
* 179ed7 (octave is identical to 326zpi within 0.3{{c}})
* 101ed6
* 165ed6
* 18ed11/8
* 229ed12 (octave is identical to 221ed11 within 0.1{{c}})
* 2.3.5.11 WE (30.703c)
* 327zpi (18.767c)
* 2.3.7.11.13 WE (30.787c)
* 11-limit WE (18.755c)
* 13-limit WE (30.757c)
''pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}}''
* 171zpi (30.973c)
* 328zpi (18.721c)
* 172zpi (30.836c)
* 180ed7
* 173zpi (30.672c)
* 230ed12
* 149ed5


42edo (narrow down slightly)
42edo (reduce # of edonoi)
* 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
59edo (reduce # of edonoi or zpi)
* 126ed7
* 152ed6
* 13ed11/9
* 7-limit WE (26.745c)
* 13-limit WE (26.695c)
* 207zpi (26.762)
* 208zpi (26.646)
* 209zpi (26.550)
 
54edo (narrow down slightly)
* 86edt
* 126ed5
* 152ed7
* 38ed5/3
* 40ed5/3
* 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)
* 93edt
* 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)
; Medium priority
* 149ed5
* 180ed7
* 222ed11
* 47ed5/3
* 11-limit WE (18.755c)
* 13-limit WE (18.752c)
* 325zpi (18.868c)
* 326zpi (18.816c)
* 327zpi (18.767c)
* 328zpi (18.721c)
* 329zpi (18.672c)
* 330zpi (18.630c)


103edo (narrow down edonoi, choose ZPIS)
25edo
* 163edt
{{harmonics in equal | 25 | 2 | 1 | intervals=integer | columns=12}}
* 239ed5
* (???)ed6
* 289ed7
* 356ed11
* (???)ed12
* 381ed13
* 421ed17
* 466ed23
* 13-limit WE (11.658c)
* Best nearby ZPI(s)
 
111edo (choose ZPIS)
* 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 217: Line 131:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


118edo (choose ZPIS)
26edo
* 187edt
{{harmonics in equal | 26 | 2 | 1 | intervals=integer | columns=12}}
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)
 
; Low priority
 
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 231: Line 138:
* 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 237: Line 145:
* 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 243: Line 152:
* 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 254: Line 159:
* 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 260: Line 166:
* 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 266: Line 173:
* 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 272: Line 180:
* 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 278: Line 187:
* 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 284: Line 194:
* 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 290: Line 201:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


; Optional
15edo
 
{{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}
25edo
* 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 298: Line 208:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


26edo
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 304: Line 215:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


29edo
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 310: Line 222:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


30edo
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 316: Line 229:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


34edo
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 322: Line 236:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


35edo
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 328: Line 243:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


36edo
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 334: Line 250:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


37edo
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)
Line 340: Line 257:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


5edo
; Low priority
 
13edo
{{harmonics in equal | 13 | 2 | 1 | intervals=integer | columns=12}}
* Main: "13edo and optimal octave stretching"
* 2.5.11.13 WE (92.483c)
* 2.5.7.13 WE (92.804c)
* 2.3 WE (91.405c) (good for opposite 7 mapping)
* 38zpi (92.531c)
 
118edo (choose ZPIS)
{{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}
* 187edt
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)
 
103edo (narrow down edonoi, choose ZPIS)
{{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}}
* 163edt
* 239ed5
* 266ed6
* 289ed7
* 356ed11
* 369ed12
* 381ed13
* 421ed17
* 466ed23
* 13-limit WE (11.658c)
* Best nearby ZPI(s)
 
111edo (choose ZPIS)
{{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 346: Line 295:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


6edo
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 352: Line 301:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


9edo
125edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 358: Line 307:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


10edo
145edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 364: Line 313:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


11edo
152edo
* 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 370: Line 324:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


15edo
166edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 376: Line 330:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


18edo
182edo
* 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 382: Line 336:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


48edo
198edo
* 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 388: Line 342:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


20edo
212edo
* 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 348:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


24edo
243edo
* 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 354:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


28edo
247edo
* 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 03:42, 30 August 2025

Quick link

User:BudjarnLambeth/Draft related tunings section

Octave stretch and compression

209zpi
  • Step size: 26.550 ¢, octave size: 1194.8 ¢

Compressing the octave of 45edo by around 5 ¢ results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1 ¢. The tuning 209zpi does this.

Approximation of harmonics in 209zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.2 +9.6 -10.5 +1.4 +4.4 +3.0 +10.8 -7.3 -3.8 -9.5 -0.9
Relative (%) -19.8 +36.3 -39.5 +5.4 +16.6 +11.4 +40.7 -27.3 -14.4 -35.8 -3.2
Step 45 72 90 105 117 127 136 143 150 156 162
Approximation of harmonics in 209zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -6.7 -2.2 +11.1 +5.6 +6.8 -12.5 +0.1 -9.1 +12.7 +11.8 -12.1 -6.1
Relative (%) -25.2 -8.4 +41.7 +20.9 +25.6 -47.1 +0.3 -34.1 +47.7 +44.4 -45.5 -23.0
Step 167 172 177 181 185 188 192 195 199 202 204 207
45edo
  • Step size: 26.667 ¢, octave size: 1200.0 ¢

Pure-octaves 45edo approximates all harmonics up to 16 within 13.0 ¢.

Approximation of harmonics in 45edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -8.6 +0.0 -13.0 -8.6 -8.8 +0.0 +9.4 -13.0 +8.7 -8.6
Relative (%) +0.0 -32.3 +0.0 -48.7 -32.3 -33.1 +0.0 +35.3 -48.7 +32.6 -32.3
Steps
(reduced) 		45
(0) 		71
(26) 		90
(0) 		104
(14) 		116
(26) 		126
(36) 		135
(0) 		143
(8) 		149
(14) 		156
(21) 		161
(26)
Approximation of harmonics in 45edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +12.8 -8.8 +5.1 +0.0 +1.7 +9.4 -4.2 -13.0 +9.2 +8.7 +11.7 -8.6
Relative (%) +48.0 -33.1 +19.0 +0.0 +6.4 +35.3 -15.7 -48.7 +34.6 +32.6 +44.0 -32.3
Steps
(reduced) 		167
(32) 		171
(36) 		176
(41) 		180
(0) 		184
(4) 		188
(8) 		191
(11) 		194
(14) 		198
(18) 		201
(21) 		204
(24) 		206
(26)
45et, 13-limit WE tuning
  • Step size: 26.695 ¢, octave size: 1201.3 ¢

Stretching the octave of 45edo by around 1 ¢ results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 45et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.3 -6.6 +2.5 -10.0 -5.3 -5.3 +3.8 -13.2 -8.8 +13.1 -4.1
Relative (%) +4.8 -24.8 +9.6 -37.6 -20.0 -19.7 +14.3 -49.5 -32.8 +49.1 -15.2
Step 45 71 90 104 116 126 135 142 149 156 161
Approximation of harmonics in 45et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.2 -4.0 +10.1 +5.1 +6.9 -11.9 +1.2 -7.5 -11.9 -12.3 -9.2 -2.8
Relative (%) -34.3 -14.9 +37.7 +19.1 +25.9 -44.7 +4.6 -28.0 -44.4 -46.1 -34.4 -10.4
Step 166 171 176 180 184 187 191 194 197 200 203 206
161ed12
  • Step size: Octave size: 1202.4 ¢

Stretching the octave of 45edo by around 2.5 ¢ results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2 ¢. The tuning 161ed12 does this.

Approximation of harmonics in 161ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.4 -4.8 +4.8 -7.4 -2.4 -2.1 +7.2 -9.6 -5.0 -9.7 +0.0
Relative (%) +9.0 -18.0 +18.0 -27.7 -9.0 -7.8 +27.1 -36.1 -18.7 -36.2 +0.0
Steps
(reduced) 		45
(45) 		71
(71) 		90
(90) 		104
(104) 		116
(116) 		126
(126) 		135
(135) 		142
(142) 		149
(149) 		155
(155) 		161
(0)
Approximation of harmonics in 161ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.0 +0.3 -12.2 +9.6 +11.6 -7.2 +6.0 -2.6 -6.9 -7.3 -4.1 +2.4
Relative (%) -18.6 +1.2 -45.8 +36.1 +43.3 -27.1 +22.6 -9.7 -25.8 -27.2 -15.2 +9.0
Steps
(reduced) 		166
(5) 		171
(10) 		175
(14) 		180
(19) 		184
(23) 		187
(26) 		191
(30) 		194
(33) 		197
(36) 		200
(39) 		203
(42) 		206
(45)
116ed6
  • Step size: Octave size: 1203.3 ¢

Stretching the octave of 45edo by around 3 ¢ results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4 ¢. The tuning 116ed6 does this. So does 126ed7 whose octave is identical within 0.1 ¢.

Approximation of harmonics in 116ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.3 -3.3 +6.7 -5.3 +0.0 +0.5 +10.0 -6.7 -1.9 -6.5 +3.3
Relative (%) +12.5 -12.5 +25.0 -19.6 +0.0 +2.0 +37.5 -25.0 -7.1 -24.2 +12.5
Steps
(reduced) 		45
(45) 		71
(71) 		90
(90) 		104
(104) 		116
(0) 		126
(10) 		135
(19) 		142
(26) 		149
(33) 		155
(39) 		161
(45)
Approximation of harmonics in 116ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -1.5 +3.9 -8.6 -13.4 -11.4 -3.3 +10.0 +1.4 -2.8 -3.1 +0.1 +6.7
Relative (%) -5.7 +14.5 -32.1 -50.0 -42.5 -12.5 +37.5 +5.4 -10.5 -11.7 +0.5 +25.0
Steps
(reduced) 		166
(50) 		171
(55) 		175
(59) 		179
(63) 		183
(67) 		187
(71) 		191
(75) 		194
(78) 		197
(81) 		200
(84) 		203
(87) 		206
(90)
45et, 7-limit WE tuning
  • Step size: 26.745 ¢, octave size: 1203.5 ¢

Stretching the octave of 45edo by around 3.5 ¢ results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this.

Approximation of harmonics in 45et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.5 -3.1 +7.1 -4.8 +0.5 +1.0 +10.6 -6.1 -1.3 -5.8 +4.0
Relative (%) +13.2 -11.4 +26.4 -18.1 +1.7 +3.9 +39.5 -22.9 -4.9 -21.8 +14.9
Step 45 71 90 104 116 126 135 142 149 155 161
Approximation of harmonics in 45et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.9 +4.6 -7.9 -12.6 -10.6 -2.6 +10.8 +2.2 -2.0 -2.3 +1.0 +7.5
Relative (%) -3.2 +17.1 -29.5 -47.3 -39.7 -9.7 +40.3 +8.3 -7.5 -8.7 +3.6 +28.1
Step 166 171 175 179 183 187 191 194 197 200 203 206
207zpi
  • Step size: 26.762 ¢, octave size: 1204.3 ¢

Stretching the octave of 45edo by around 4 ¢ results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9 ¢. The tuning 207zpi does this.

Approximation of harmonics in 207zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.3 -1.9 +8.6 -3.1 +2.4 +3.2 +12.9 -3.7 +1.2 -3.2 +6.7
Relative (%) +16.0 -6.9 +32.1 -11.5 +9.1 +11.9 +48.1 -13.8 +4.6 -12.0 +25.1
Step 45 71 90 104 116 126 135 142 149 155 161
Approximation of harmonics in 207zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.0 +7.5 -4.9 -9.6 -7.5 +0.6 -12.7 +5.5 +1.3 +1.1 +4.4 +11.0
Relative (%) +7.3 +27.9 -18.4 -35.9 -28.1 +2.2 -47.6 +20.6 +5.0 +4.0 +16.5 +41.2
Step 166 171 175 179 183 187 190 194 197 200 203 206
71edt
  • Step size: 26.788 ¢, octave size: 1205.5 ¢

Stretching the octave of 45edo by around 5.5 ¢ results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9 ¢. The tuning 71edt does this. So does the tuning 155ed11 whose octave is identical within 0.3 ¢.

Approximation of harmonics in 71edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +5.5 +0.0 +10.9 -0.4 +5.5 +6.5 -10.4 +0.0 +5.1 +0.8 +10.9
Relative (%) +20.4 +0.0 +40.8 -1.3 +20.4 +24.2 -38.8 +0.0 +19.1 +3.1 +40.8
Steps
(reduced) 		45
(45) 		71
(0) 		90
(19) 		104
(33) 		116
(45) 		126
(55) 		134
(63) 		142
(0) 		149
(7) 		155
(13) 		161
(19)
Approximation of harmonics in 71edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +6.3 +11.9 -0.4 -4.9 -2.7 +5.5 -7.8 +10.6 +6.5 +6.3 +9.7 -10.4
Relative (%) +23.5 +44.6 -1.3 -18.4 -10.2 +20.4 -29.0 +39.5 +24.2 +23.5 +36.2 -38.8
Steps
(reduced) 		166
(24) 		171
(29) 		175
(33) 		179
(37) 		183
(41) 		187
(45) 		190
(48) 		194
(52) 		197
(55) 		200
(58) 		203
(61) 		205
(63)

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

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

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

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)
Low priority

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)

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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