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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}} |
| What follows is a comparison of compressed-octave 32edo tunings.
| | {{harmonics in equal|6|intervals=odd|columns=7}} |
| | | {{harmonics in equal|7|intervals=odd|columns=7}} |
| ; 32edo
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| * Step size: 37.500{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|9|intervals=odd|columns=7}} |
| Pure-octaves 32edo approximates all harmonics up to 16 within 15.5{{c}}.
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| {{Harmonics in equal|32|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32edo}}
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| {{Harmonics in equal|32|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32edo (continued)}} | | {{harmonics in equal|12|intervals=odd|columns=7}} |
| | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| ; [[WE|32et, 13-limit WE tuning]]
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| * Step size: 37.481{{c}}, octave size: 1199.4{{c}}
| | {{harmonics in equal|15|intervals=odd|columns=7}} |
| Compressing the octave of 32edo by around half a cent results in improved primes 3, 7 and 11, but worse primes 5 and 13. This approximates all harmonics up to 16 within 18.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| {{Harmonics in cet|37.481|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning}} | | {{harmonics in equal|17|intervals=odd|columns=7}} |
| {{Harmonics in cet|37.481|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning (continued)}} | | {{harmonics in equal|18|intervals=odd|columns=7}} |
| | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| ; [[WE|32et, 11-limit WE tuning]]
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| * Step size: 37.453{{c}}, octave size: 1198.5{{c}}
| | {{harmonics in equal|21|intervals=odd|columns=7}} |
| Compressing the octave of 32edo by around 1.5{{c}} results in improved primes 3, 7 and 11, but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 16.4{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| {{Harmonics in cet|37.453|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning}} | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| {{Harmonics in cet|37.453|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning (continued)}} | | {{harmonics in equal|24|intervals=odd|columns=7}} |
| | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| ; [[ed7|90ed7]]
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| * Step size: 37.431{{c}}, octave size: 1197.8{{c}}
| | {{harmonics in equal|27|intervals=odd|columns=7}} |
| Compressing the octave of 32edo by around 2{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 18.6{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| {{Harmonics in equal|90|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed7}} | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| {{Harmonics in equal|90|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed7 (continued)}} | | {{harmonics in equal|30|intervals=odd|columns=7}} |
| | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| ; [[zpi|133zpi]]
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| * Step size: 37.418{{c}}, octave size: 1197.375{{c}}
| | {{harmonics in equal|33|intervals=odd|columns=7}} |
| Compressing the octave of 32edo by around NNN{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 17.4{{c}}. The tuning 133zpi does this.
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| {{Harmonics in cet|37.418|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}} | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| {{Harmonics in cet|37.418|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}} | | {{harmonics in equal|36|intervals=odd|columns=7}} |
| Below is a plot of the [[Zeta]] function, showing how its peak (ie biggest absolute value) is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.
| | {{harmonics in equal|37|intervals=odd|columns=7}} |
| | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| [[File:plot32.png|alt=plot32.png|plot32.png]]
| | {{harmonics in equal|39|intervals=odd|columns=7}} |
| | | {{harmonics in equal|40|intervals=odd|columns=7}} |
| ; [[51edt]]
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| * Step size: 37.293{{c}}, octave size: 1193.4{{c}}
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| Compressing the octave of 32edo by around 6.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 18.2{{c}}. The tuning 51edt does this.
| | {{harmonics in equal|43|intervals=odd|columns=7}} |
| {{Harmonics in equal|51|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 51edt}} | | {{harmonics in equal|44|intervals=odd|columns=7}} |
| {{Harmonics in equal|51|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 51edt (continued)}} | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| | | {{harmonics in equal|46|intervals=odd|columns=7}} |
| = Title2 =
| | {{harmonics in equal|47|intervals=odd|columns=7}} |
| === Lab ===
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| | | {{harmonics in equal|49|intervals=odd|columns=7}} |
| Place holder
| | {{harmonics in equal|50|intervals=odd|columns=7}} |
| | | {{harmonics in equal|51|intervals=odd|columns=7}} |
| | | {{harmonics in equal|52|intervals=odd|columns=7}} |
| <br><br><br><br><br>
| | {{harmonics in equal|53|intervals=odd|columns=7}} |
| | |
| | |
| {{harmonics in cet | 300 | intervals=prime}} | |
| | |
| {{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.
| |
| | |
| (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
| |
| | |
| 39edo
| |
| * 171zpi (30.973c) (optimised for dual-fifths use)
| |
| * 13-limit WE (30.757c) (octave of 135ed11 differs by only 0.2{{c}})
| |
| * 101ed6 (octave of 172zpi differs by only 0.4{{c}})
| |
| * 173zpi (30.672c) (octave of 62edt differs by only 0.2{{c}})
| |
| * 110ed7 (octave of 145ed13 differs by only 0.1{{c}})
| |
| * 91ed5
| |
| | |
| 45edo
| |
| * 209zpi (26.550)
| |
| * 13-limit WE (26.695c)
| |
| * 161ed12
| |
| * 116ed6 (octave identical to 126ed7 within 0.1{{c}})
| |
| * 7-limit WE (26.745c)
| |
| * 207zpi (26.762)
| |
| * 71edt (octave identical to 155ed11 within 0.3{{c}})
| |
| | |
| 54edo
| |
| * 139ed6 (octave is identical to 262zpi within 0.2{{c}})
| |
| * 151ed7
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| * 193ed12
| |
| * 263zpi (22.243c)
| |
| * 13-limit WE (22.198c) (octave is identical to 187ed11 within 0.1{{c}})
| |
| * 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})
| |
| * 152ed7
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| * 140ed6
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| * 126ed5 (octave is identical to 86edt within 0.1{{c}})
| |
| | |
| 64edo
| |
| * 179ed7 (octave is identical to 326zpi within 0.3{{c}})
| |
| * 165ed6
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| * 229ed12 (octave is identical to 221ed11 within 0.1{{c}})
| |
| * 327zpi (18.767c)
| |
| * 11-limit WE (18.755c)
| |
| ''pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}}''
| |
| * 328zpi (18.721c)
| |
| * 180ed7
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| * 230ed12
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| * 149ed5
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| | |
| 33edo (reduce # of edonoi)
| |
| * 76ed5
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| * 92ed7 (137zpi's octave differs by only 0.3{{c}})
| |
| * 52ed13
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| * 114ed11
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| * 138zpi (36.394c) (122ed13's octave differs by only 0.1{{c}})
| |
| * 13-limit WE (36.357c)
| |
| * 93ed7 (optimised for dual-fifths)
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| * 77ed5 (139zpi's octave differs by only 0.2{{c}})
| |
| * 123ed13 / 1ed47/46 (identical within <0.1{{c}})
| |
| * 115ed11
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| | |
| 42edo (reduce # of edonoi)
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| * 108ed6 (octave is identical to 97ed5 within 0.1{{c}})
| |
| * 189zpi (28.689c)
| |
| * 150ed12
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| * 145ed11
| |
| ''190zpi's octave is within 0.05{{c}} of pure-octaves 42edo''
| |
| * 118ed7
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| * 13-limit WE (28.534c)
| |
| * 151ed12 (octave is identical to 7-limit WE within 0.3{{c}})
| |
| * 109ed6
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| * 191zpi (28.444c)
| |
| * 67edt
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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)
| |
| ''pure octaves 59edo octave is identical to 137ed5 within 0.05{{c}}''
| |
| * 13-limit WE (20.320c)
| |
| * 7-limit WE (20.301c)
| |
| * 166ed7
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| * 212ed12
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| * 296zpi (20.282c)
| |
| * 153ed6
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| | |
| ; Medium priority
| |
| | |
| 118edo (choose ZPIS)
| |
| {{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}} | |
| * 187edt
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| * 69edf
| |
| * 13-limit WE (10.171c)
| |
| * 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)
| |
| * 2.3 WE (91.405c) (good for opposite 7 mapping)
| |
| * 38zpi (92.531c)
| |
| | |
| 103edo (narrow down edonoi, choose ZPIS)
| |
| {{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}} | |
| * 163edt
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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)
| |
| * 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
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
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| | |
| ; Low priority
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| | |
| 104edo
| |
| * 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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| | |
| 125edo
| |
| * 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
| |
| * 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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| | |
| 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
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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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| 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
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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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| | |
| 198edo
| |
| * 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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| | |
| 212edo
| |
| * 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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| | |
| 243edo
| |
| * 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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| | |
| 247edo
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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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| | |
| ; Optional
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| | |
| 25edo
| |
| {{harmonics in equal | 25 | 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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| | |
| 26edo
| |
| {{harmonics in equal | 26 | 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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| | |
| 29edo
| |
| {{harmonics in equal | 29 | 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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| | |
| 30edo
| |
| {{harmonics in equal | 30 | 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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| | |
| 34edo
| |
| {{harmonics in equal | 34 | 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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| | |
| 35edo
| |
| {{harmonics in equal | 35 | 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)
| |
| | |
| 36edo
| |
| {{harmonics in equal | 36 | 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)
| |
| | |
| 37edo
| |
| {{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)
| |
| | |
| 9edo
| |
| {{harmonics in equal | 9 | 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)
| |
| | |
| 10edo
| |
| {{harmonics in equal | 10 | 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)
| |
| | |
| 11edo
| |
| {{harmonics in equal | 11 | 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)
| |
| | |
| 15edo
| |
| {{harmonics in equal | 15 | 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)
| |
| | |
| 18edo
| |
| {{harmonics in equal | 18 | 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
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| * Best nearby ZPI(s)
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| | |
| 48edo
| |
| {{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
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| * 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
| |
| * 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
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 20edo
| |
| {{harmonics in equal | 20 | 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)
| |
| | |
| 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)
| |
| | |
| 28edo
| |
| {{harmonics in equal | 28 | 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)
| |