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= Title1 =
= Title1 =
== Octave stretch or compression ==
== Octave stretch or compression ==
{{main|23edo and octave stretching}}
What follows is a comparison of stretched- and compressed-octave 60edo tunings.


23edo is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.
; [[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)}}


However, when using a slightly [[stretched tuning|stretched octave]] of around 1216 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.
; [[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)}}


Stretched 23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.
; [[zpi|301zpi]]  
* Step size: 20.027{{c}}, octave size: 1201.62{{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 cet|20.027|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 301zpi}}
{{Harmonics in cet| 20.027 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 301zpi (continued)}}


What follows is a comparison of stretched- and compressed-octave 23edo tunings.
; [[95edt]]
* Step size: 20.021{{c}}, octave size: 1201.23{{c}}
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 equal|95|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 95edt}}
{{Harmonics in equal|95|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 95edt (continued)}}


; [[zpi|86zpi]]  
; [[WE|60et, 13-limit WE tuning]] / [[155ed6]]
* Step size: 51.653{{c}}, octave size: 1188.0{{c}}
* Step size: 20.013{{c}}, octave size: 1200.78{{c}}
Compressing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this.
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|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|20.013|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning}}
{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (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)}}


; [[60ed6]]  
; [[ed12|215ed12]]  
* Step size: 51.700{{c}}, octave size: 1189.1{{c}}
* Step size: 20.009{{c}}, octave size: 1200.55{{c}}
Compressing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 60ed6 does this. So does the tuning [[equal tuning|105ed23]] whose octave is identical within 0.01{{c}}.
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 equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|215|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 215ed12}}
{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
{{Harmonics in equal|215|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 215ed12 (continued)}}


; [[zpi|85zpi]]
; 60edo
* Step size: 52.114{{c}}, octave size: 1198.6{{c}}
* Step size: 20.000{{c}}, octave size: 1200.00{{c}}  
Compressing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 85zpi does this. So does the tuning [[ed9|73ed9]] whose octave is identical within 0.02{{c}}.
Pure-octaves 60edo approximates all harmonics up to 16 within 8.83{{c}}.
{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|60|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edo}}
{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
{{Harmonics in equal|60|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edo (continued)}}


; 23edo
; [[zpi|302zpi]]
* Step size: NNN{{c}}, octave size: 1200.0{{c}}  
* Step size: 19.962{{c}}, octave size: 1197.72{{c}}
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{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 equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
{{Harmonics in cet|19.962|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 302zpi}}
{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}
{{Harmonics in cet|19.962|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 302zpi (continued)}}


; [[WE|23et, 13-limit WE tuning]]  
; [[APS|19.95cet]]  
* Step size: 52.237{{c}}, octave size: 1201.5{{c}}
* Step size: 19.950{{c}}, octave size: 1197.00{{c}}
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
Compressing the octave of 60edo by 3{{c}} results in improved primes 5, 7 and 13, but worse primes 2, 3 and 11. This approximates all harmonics up to 16 within 8.32{{c}}. The tuning 19.95cet does this. This tuning is particularly well suited to [[catnip]] temperament specifically.
{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|19.95|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19.95cet}}
{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
{{Harmonics in cet|19.95|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19.95cet (continued)}}


; [[WE|23et, 2.3.5.13 WE tuning]]  
; [[169ed7|169ed7]]  
* Step size: 52.447{{c}}, octave size: 1206.3{{c}}
* Step size: 19.958{{c}}, octave size: 1197.50{{c}}
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. So does the tuning [[ed10|76ed10]] whose octave is identical within 0.01{{c}}.
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 cet|52.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in equal|169|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 169ed7}}
{{Harmonics in cet|52.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
{{Harmonics in equal|169|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 169ed7 (continued)}}


; [[59ed6]]  
; [[zpi|303zpi]]  
* Step size: 52.575{{c}}, octave size: 1209.2{{c}}
* Step size: 19.913{{c}}, octave size: 1194.78{{c}}
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 59ed6 does this. So does the tuning [[53ed5]] whose octave is identical within 0.01{{c}}.
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 equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|19.913|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 303zpi}}
{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
{{Harmonics in cet|19.913|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 303zpi (continued)}}
 
; [[zpi|84zpi]]
* Step size: 52.615{{c}}, octave size: 1210.1{{c}}
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this.
{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
 
; [[36edt]]
* Step size: 52.832{{c}}, octave size: 1215.1{{c}}
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this.
{{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
 
; [[84ed13]]
* Step size: 52.863{{c}}, octave size: 1215.9{{c}}
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this.
{{Harmonics in equal|84|13|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


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


54edo (possibly narrow down edonoi)
Place holder
* 38ed5/3 (stretch, improves 3.5.7.11.13.17.19.23)
{{harmonics in equal | 38 | 5 | 3 | intervals=prime}}
* 261zpi (22.380c)
{{harmonics in cet | 22.380 | intervals=prime}}
* 85edt
{{harmonics in equal | 85 | 3 | 1 | intervals=prime}}
* 139ed6
{{harmonics in equal | 139 | 6 | 1 | intervals=prime}}
* 262zpi (22.313c)
{{harmonics in cet | 22.313 | intervals=prime}}
* 151ed7
{{harmonics in equal | 151 | 7 | 1 | intervals=prime}}
* 193ed12
{{harmonics in equal | 193 | 12 | 1 | intervals=prime}}
* 263zpi (22.243c)
{{harmonics in cet | 22.243 | intervals=prime}}
* pure octave 54edo
{{harmonics in equal | 54 | 2 | 1 | intervals=integer | columns=12}}
* 13-limit WE (22.198c)
{{harmonics in cet | 22.198 | intervals=prime}}
* 187ed11
{{harmonics in equal | 187 | 11 | 1 | intervals=prime}}
* 2.3.7.11.13 WE (22.180c)
{{harmonics in cet | 22.180 | intervals=prime}}
* 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})
{{harmonics in cet | 22.175 | intervals=prime}}
* 152ed7
{{harmonics in equal | 152 | 7 | 1 | intervals=prime}}
* 140ed6
{{harmonics in equal | 140 | 6 | 1 | intervals=prime}}
* 86edt
{{harmonics in equal | 86 | 3 | 1 | intervals=prime}}
* 126ed5
{{harmonics in equal | 126 | 5 | 1 | intervals=prime}}
* 265zpi (22.100c)
{{harmonics in cet | 22.100 | intervals=prime}}
 
<br><br><br>
 
59edo (narrow down ZPIs) (Nothing special abt these choices)
* 293zpi (20.454c)
{{harmonics in cet | 20.454 | intervals=prime}}
* 93edt
{{harmonics in equal | 93 | 3 | 1 | intervals=prime}}
* 203ed11
{{harmonics in equal | 203 | 11 | 1 | intervals=prime}}
* 152ed6
{{harmonics in equal | 152 | 6 | 1 | intervals=prime}}
* 294zpi (20.399c)
{{harmonics in cet | 20.399 | intervals=prime}}
* 211ed12
{{harmonics in equal | 211 | 12 | 1 | intervals=prime}}
* 295zpi (20.342c)
{{harmonics in cet | 20.342 | intervals=prime}}
* pure octaves 59edo
{{harmonics in equal | 59 | 2 | 1 | intervals=integer | columns=12}}
* 137ed5
{{harmonics in equal | 137 | 5 | 1 | intervals=prime}}
* 13-limit WE (20.320c)
{{harmonics in cet | 20.320 | intervals=prime}}
* 11-limit WE (20.310c)
{{harmonics in cet | 20.310 | intervals=prime}}
* 7-limit WE (20.301c)
{{harmonics in cet | 20.301 | intervals=prime}}
* 166ed7
{{harmonics in equal | 166 | 7 | 1 | intervals=prime}}
* 212ed12
{{harmonics in equal | 212 | 12 | 1 | intervals=prime}}
* 296zpi (20.282c)
{{harmonics in cet | 20.282 | intervals=prime}}
* 153ed6
{{harmonics in equal | 153 | 6 | 1 | intervals=prime}}
* 205ed11
{{harmonics in equal | 205 | 11 | 1 | intervals=prime}}
* 94edt
{{harmonics in equal | 94 | 3 | 1 | intervals=prime}}
* 297zpi (20.229c)
{{harmonics in cet | 20.229 | intervals=prime}}
 
<br><br><br>
 
64edo (narrow down ZPIs)
* 325zpi (18.868c)
{{harmonics in cet | 18.868 | intervals=prime}}
* 220ed11
{{harmonics in equal | 220 | 11 | 1 | intervals=prime}}
* 101edt
{{harmonics in equal | 101 | 3 | 1 | intervals=prime}}
* 179ed7
{{harmonics in equal | 179 | 7 | 1 | intervals=prime}}
* 165ed6
{{harmonics in equal | 165 | 6 | 1 | intervals=prime}}
* 326zpi (18.816c)
{{harmonics in cet | 18.816 | intervals=prime}}
* 229ed12
{{harmonics in equal | 229 | 12 | 1 | intervals=prime}}
* 221ed11
{{harmonics in equal | 221 | 11 | 1 | intervals=prime}}
* 327zpi (18.767c)
{{harmonics in cet | 18.767 | intervals=prime}}
* 11-limit WE (18.755c)
{{harmonics in cet | 18.755 | intervals=prime}}
* 13-limit WE (18.752c)
{{harmonics in cet | 18.752 | intervals=prime}}
* pure octaves 64edo
{{harmonics in equal | 64 | 2 | 1 | intervals=integer | columns=12}}
* 328zpi (18.721c)
{{harmonics in cet | 18.721 | intervals=prime}}
* 180ed7
{{harmonics in equal | 180 | 7 | 1 | intervals=prime}}
* 230ed12
{{harmonics in equal | 230 | 12 | 1 | intervals=prime}}
* 149ed5
{{harmonics in equal | 149 | 5 | 1 | intervals=prime}}
* 222ed11
{{harmonics in equal | 222 | 11 | 1 | intervals=prime}}
* 329zpi (18.672c)
{{harmonics in cet | 18.672 | intervals=prime}}
* 102edt
{{harmonics in equal | 102 | 3 | 1 | intervals=prime}}
* 330zpi (18.630c)
{{harmonics in cet | 18.630 | intervals=prime}}




Line 207: Line 83:


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


Line 242: Line 119:
* 138zpi (36.394c) (122ed13's octave differs by only 0.1{{c}})
* 138zpi (36.394c) (122ed13's octave differs by only 0.1{{c}})
* 13-limit WE (36.357c)
* 13-limit WE (36.357c)
* 11-limit WE (36.349c)
* 93ed7 (optimised for dual-fifths)
* 93ed7 (optimised for dual-fifths)
* 77ed5 (139zpi's octave differs by only 0.2{{c}})
* 77ed5 (139zpi's octave differs by only 0.2{{c}})
Line 252: Line 128:
* 13-limit WE (30.757c) (octave of 135ed11 differs by only 0.2{{c}})
* 13-limit WE (30.757c) (octave of 135ed11 differs by only 0.2{{c}})
* 101ed6 (octave of 172zpi differs by only 0.4{{c}})
* 101ed6 (octave of 172zpi differs by only 0.4{{c}})
* 2.3.5.11 WE (30.703c)
* 173zpi (30.672c) (octave of 62edt differs by only 0.2{{c}})
* 173zpi (30.672c) (octave of 62edt differs by only 0.2{{c}})
* 110ed7 (octave of 145ed13 differs by only 0.1{{c}})
* 110ed7 (octave of 145ed13 differs by only 0.1{{c}})
Line 258: Line 133:


42edo
42edo
*Good <27% rel err
* 108ed6 (octave is identical to 97ed5 within 0.1{{c}})
*Okay <40% rel err
* 189zpi (28.689c)
{{harmonics in equal | 42 | 2 | 1 | intervals=integer | columns=12}}
* 150ed12
* 42ed257/128 (good 2.3.5.7; bad 11.13)
* 145ed11
* 11ed6/5 (good 2.3.5; okay 7.11.13)
''190zpi's octave is within 0.05{{c}} of pure-octaves 42edo''
* 189zpi (28.689c) (good 2.5.13; okay 3.11; bad 7)
* 118ed7
* 190zpi (28.572c)
* 13-limit WE (28.534c)
* 13-limit WE (28.534c)
* 34ed7/4 (good 2.5.7.13; okay 3.11)
* 151ed12 (octave is identical to 7-limit WE within 0.3{{c}})
* 7-limit WE (28.484c) (good 2.3.5.11.13; bad 7)
* 109ed6
* 191zpi (28.444c)
* 191zpi (28.444c)
* 1ed123/121 (good 2.3.5.11; okay 13; bad 7)
* 67edt


45edo
45edo
Line 280: Line 154:
* 71edt (octave identical to 155ed11 within 0.3{{c}})
* 71edt (octave identical to 155ed11 within 0.3{{c}})


54edo (possibly narrow down edonoi)
54edo
{{harmonics in equal | 54 | 2 | 1 | intervals=integer | columns=12}}
* 139ed6 (octave is identical to 262zpi within 0.2{{c}})
* 126ed5
* 151ed7
* 38ed5/3 (stretch, improves 3.5.7.11.13.17.19.23)
* 193ed12
* 262zpi (22.313c)
* 263zpi (22.243c)
* 263zpi (22.243c)
* 13-limit WE (22.198c)
* 13-limit WE (22.198c) (octave is identical to 187ed11 within 0.1{{c}})
* 2.3.7.11.13 WE (22.180c)
* 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})
* 264zpi (22.175c)
* 40ed5/3 (compress, improves 3.5.11.13.17.19 (not 7))
* 152ed7
* 152ed7
* 86edt
* 140ed6
* 126ed5 (octave is identical to 86edt within 0.1{{c}})


59edo (narrow down ZPIs)
59edo
* (Nothing special abt these choices)
* 152ed6
{{harmonics in equal | 59 | 2 | 1 | intervals=integer | columns=12}}
* 93edt
* 203ed11
* 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)
* 13-limit WE (20.320c)
* 11-limit WE (20.310c)
* 7-limit WE (20.301c)
* 7-limit WE (20.301c)
* 166ed7
* 212ed12
* 296zpi (20.282c)
* 296zpi (20.282c)
* 297zpi (20.229c)
* 153ed6
* 166ed7


64edo (narrow down ZPIs)
64edo
{{harmonics in equal | 64 | 2 | 1 | intervals=integer | columns=12}}
* 179ed7 (octave is identical to 326zpi within 0.3{{c}})
* 47ed5/3 (like 221ed11 but benefits & drawbacks both amplified)
* 165ed6
* 221ed11
* 229ed12 (octave is identical to 221ed11 within 0.1{{c}})
* 325zpi (18.868c)
* 326zpi (18.816c)
* 327zpi (18.767c)
* 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}}''
* 328zpi (18.721c)
* 328zpi (18.721c)
* 329zpi (18.672c)
* 330zpi (18.630c)
* 180ed7
* 180ed7
* 230ed12
* 149ed5
* 149ed5


Line 481: Line 348:
37edo
37edo
{{harmonics in equal | 37 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 37 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
5edo
{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
6edo
{{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 537: Line 390:
48edo
48edo
{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
5edo
{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
6edo
{{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)

Latest revision as of 21:56, 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
19.95cet
  • Step size: 19.950 ¢, octave size: 1197.00 ¢

Compressing the octave of 60edo by 3 ¢ results in improved primes 5, 7 and 13, but worse primes 2, 3 and 11. This approximates all harmonics up to 16 within 8.32 ¢. The tuning 19.95cet does this. This tuning is particularly well suited to catnip temperament specifically.

Approximation of harmonics in 19.95cet
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.00 -6.71 -6.00 +6.69 -9.71 +2.72 -9.00 +6.54 +3.69 -1.72 +7.24
Relative (%) -15.0 -33.6 -30.1 +33.5 -48.6 +13.7 -45.1 +32.8 +18.5 -8.6 +36.3
Step 60 95 120 140 155 169 180 191 200 208 216
Approximation of harmonics in 19.95cet (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +8.32 -0.28 -0.02 +7.95 +2.74 +3.54 +9.69 +0.69 -3.98 -4.72 -1.87 +4.24
Relative (%) +41.7 -1.4 -0.1 +39.8 +13.8 +17.7 +48.6 +3.4 -20.0 -23.6 -9.4 +21.3
Step 223 229 235 241 246 251 256 260 264 268 272 276
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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