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| Quick link
| | == Approximations of odd harmonics == |
| | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| [[User:BudjarnLambeth/Draft related tunings section]]
| | {{harmonics in equal|2|intervals=odd|columns=7}} |
| | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| = Title1 = | | {{harmonics in equal|4|intervals=odd|columns=7}} |
| == Octave stretch or compression ==
| | {{harmonics in equal|5|intervals=odd|columns=7}} |
| 54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[139ed6]], a [[Octave stretch|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.
| | {{harmonics in equal|6|intervals=odd|columns=7}} |
| | | {{harmonics in equal|7|intervals=odd|columns=7}} |
| If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| [[40ed5/3]] is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low [[harmonic entropy]].
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| | | {{harmonics in equal|11|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 54edo tunings.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| ; [[ed6|139ed6]]
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| * Octave size: 1205.08{{c}}
| | {{harmonics in equal|15|intervals=odd|columns=7}} |
| Stretching the octave of 54edo by around 5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 10.15{{c}}. The tuning 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2{{c}}.
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| {{Harmonics in equal|139|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 139ed6}} | | {{harmonics in equal|17|intervals=odd|columns=7}} |
| {{Harmonics in equal|139|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed6 (continued)}} | | {{harmonics in equal|18|intervals=odd|columns=7}} |
| | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| ; [[ed7|151ed7]]
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| * Octave size: 1204.75{{c}}
| | {{harmonics in equal|21|intervals=odd|columns=7}} |
| Stretching the octave of 54edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 11.12{{c}}. The tuning 151ed7 does this.
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| {{Harmonics in equal|151|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed7}} | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| {{Harmonics in equal|151|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed7 (continued)}} | | {{harmonics in equal|24|intervals=odd|columns=7}} |
| | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| ; [[ed12|193ed12]]
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| * Octave size: 1203.66{{c}}
| | {{harmonics in equal|27|intervals=odd|columns=7}} |
| Stretching the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 10.97{{c}}. The tuning 193ed12 does this.
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| {{Harmonics in equal|193|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 193ed12}} | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| {{Harmonics in equal|193|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 193ed12 (continued)}} | | {{harmonics in equal|30|intervals=odd|columns=7}} |
| | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| ; [[zpi|263zpi]]
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| * Step size: 22.243{{c}}, octave size: 1201.12{{c}}
| | {{harmonics in equal|33|intervals=odd|columns=7}} |
| Stretching the octave of 54edo by around 1{{c}} results in an improved prime 5, but worse primes 2, 3, 7, 11 and 13. This approximates all harmonics up to 16 within 10.94{{c}}. The tuning 263zpi does this.
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| {{Harmonics in cet|22.243|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 263zpi}} | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| {{Harmonics in cet|22.243|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 263zpi (continued)}} | | {{harmonics in equal|36|intervals=odd|columns=7}} |
| | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| ; 54edo
| | {{harmonics in equal|38|intervals=odd|columns=7}} |
| * Step size: 22.222{{c}}, octave size: 1200.00{{c}}
| | {{harmonics in equal|39|intervals=odd|columns=7}} |
| Pure-octaves 54edo approximates all harmonics up to 16 within 9.16{{c}}.
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| {{Harmonics in equal|54|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54edo}}
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| {{Harmonics in equal|54|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54edo (continued)}} | | {{harmonics in equal|42|intervals=odd|columns=7}} |
| | | {{harmonics in equal|43|intervals=odd|columns=7}} |
| ; [[WE|54et, 13-limit WE tuning]]
| | {{harmonics in equal|44|intervals=odd|columns=7}} |
| * Step size: 22.198{{c}}, octave size: 1198.69{{c}}
| | {{harmonics in equal|45|intervals=odd|columns=7}} |
| Compressing the octave of 54edo by around 1.5{{c}} results in improved primes 3, 7, 11, 13, 17 and 19, but worse primes 2 and 5. This approximates all harmonics up to 16 within 10.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|187ed11]] whose octave is identical to 13lim WE within 0.1{{c}}.
| | {{harmonics in equal|46|intervals=odd|columns=7}} |
| {{Harmonics in cet|22.198|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning}} | | {{harmonics in equal|47|intervals=odd|columns=7}} |
| {{Harmonics in cet|22.198|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning (continued)}} | | {{harmonics in equal|48|intervals=odd|columns=7}} |
| | | {{harmonics in equal|49|intervals=odd|columns=7}} |
| ; [[zpi|264zpi]]
| | {{harmonics in equal|50|intervals=odd|columns=7}} |
| * Step size: 22.175{{c}}, octave size: 1197.45{{c}}
| | {{harmonics in equal|51|intervals=odd|columns=7}} |
| Compressing the octave of 54edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.19{{c}}. The tuning 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01{{c}}.
| | {{harmonics in equal|52|intervals=odd|columns=7}} |
| {{Harmonics in cet|22.175|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 264zpi}} | | {{harmonics in equal|53|intervals=odd|columns=7}} |
| {{Harmonics in cet|22.175|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 264zpi (continued)}} | |
| | |
| ; [[ed7|152ed7]]
| |
| * Octave size: 1196.82{{c}}
| |
| Compressing the octave of 54edo by around 3{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.36{{c}}. The tuning 152ed7 does this.
| |
| {{Harmonics in equal|152|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed7}}
| |
| {{Harmonics in equal|152|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed7 (continued)}} | |
| | |
| ; [[ed6|140ed6]]
| |
| * Octave size: 1196.47{{c}}
| |
| Compressing the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.59{{c}}. The tuning 140ed6 does this.
| |
| {{Harmonics in equal|140|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 140ed6}}
| |
| {{Harmonics in equal|140|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 140ed6 (continued)}} | |
| | |
| ; [[ed5|126ed5]]
| |
| * Octave size: 1194.13{{c}}
| |
| Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.
| |
| {{Harmonics in equal|126|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 126ed5}}
| |
| {{Harmonics in equal|126|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (continued)}} | |
| | |
| ; [[ed5/3|40ed5/3]]
| |
| * Octave size: 1194.13{{c}}
| |
| Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.
| |
| {{Harmonics in equal|40|5|3|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 126ed5}}
| |
| {{Harmonics in equal|40|5|3|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (continued)}} | |
| | |
| = Title2 =
| |
| === Lab ===
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| Place holder
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| <br><br><br><br><br>
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| {{harmonics in cet | 300 | intervals=prime}} | |
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| {{harmonics in equal | 140 | 12 | 1 | intervals=prime}} | |
| | |
| === 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.
| |
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| (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)
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| ; High-priority
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| 54edo
| |
| * 139ed6 (octave is identical to 262zpi within 0.2{{c}})
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| * 151ed7
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| * 193ed12
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| * 263zpi (22.243c)
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| * 13-limit WE (22.198c) (octave is identical to 187ed11 within 0.1{{c}})
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| * 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})
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| * 152ed7
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| * 140ed6
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| * 126ed5 (octave is identical to 86edt within 0.1{{c}})
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| 64edo
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| * 179ed7 (octave is identical to 326zpi within 0.3{{c}})
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| * 165ed6
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| * 229ed12 (octave is identical to 221ed11 within 0.1{{c}})
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| * 327zpi (18.767c)
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| * 11-limit WE (18.755c)
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| ''pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}}''
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| * 328zpi (18.721c)
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| * 180ed7
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| * 230ed12
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| * 149ed5
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| 59edo (reduce # of edonoi or zpi)
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| * 152ed6
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| * 294zpi (20.399c)
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| * 211ed12
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| * 295zpi (20.342c)
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| ''pure octaves 59edo octave is identical to 137ed5 within 0.05{{c}}''
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| * 13-limit WE (20.320c)
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| * 7-limit WE (20.301c)
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| * 166ed7
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| * 212ed12
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| * 296zpi (20.282c)
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| * 153ed6
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| | |
| ; Medium priority
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| 25edo
| |
| {{harmonics in equal | 25 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 26edo
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| {{harmonics in equal | 26 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 29edo
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| {{harmonics in equal | 29 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 30edo
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| {{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 34edo
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| {{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 35edo
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| {{harmonics in equal | 35 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 36edo
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| {{harmonics in equal | 36 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 37edo
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| {{harmonics in equal | 37 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 38edo
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| {{harmonics in equal | 38 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 9edo
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| {{harmonics in equal | 9 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 10edo
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| {{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 11edo
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| {{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 15edo
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| {{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 18edo
| |
| {{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 48edo
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| {{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 24edo
| |
| {{harmonics in equal | 24 | 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)
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| 5edo
| |
| {{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 6edo
| |
| {{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 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)
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| * 2.3 WE (91.405c) (good for opposite 7 mapping)
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| * 38zpi (92.531c)
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| 118edo (choose ZPIS)
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| {{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}} | |
| * 187edt
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| * 69edf
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| * 13-limit WE (10.171c)
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| * Best nearby ZPI(s)
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| | |
| 103edo (narrow down edonoi, choose ZPIS)
| |
| {{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}} | |
| * 163edt
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| * 239ed5
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| * 266ed6
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| * 289ed7
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| * 356ed11
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| * 369ed12
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| * 381ed13
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| * 421ed17
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| * 466ed23
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| * 13-limit WE (11.658c)
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| * Best nearby ZPI(s)
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| 111edo (choose ZPIS)
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| {{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}} | |
| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| | |
| ; Low priority
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| 104edo
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| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 125edo
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| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
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| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 145edo
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| * Nearby edt, ed6, ed12 and/or edf
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| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 152edo
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| * 241edt
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| * 13-limit WE (7.894c)
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| * Best nearby ZPI(s)
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| 159edo
| |
| * 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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| | |
| 166edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| | |
| 182edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
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| * Best nearby ZPI(s)
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| 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)
| |