|
|
| (159 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| 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}} |
| 99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.
| | {{harmonics in equal|6|intervals=odd|columns=7}} |
| | | {{harmonics in equal|7|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 99edo tunings.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| ; [[zpi|567zpi]]
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| * Step size: 12.138{{c}}, octave size: 1201.66{{c}}
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| Stretching the octave of 99edo by around 1.5{{c}} results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}} | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}} | | {{harmonics in equal|14|intervals=odd|columns=7}} |
| | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| ; [[WE|99et, 13-limit WE tuning]]
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * Step size: 12.123{{c}}, octave size: 1200.18{{c}}
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}} | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}} | | {{harmonics in equal|20|intervals=odd|columns=7}} |
| | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| ; 99edo
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| * Step size: 12.121{{c}}, octave size: 1200.00{{c}}
| | {{harmonics in equal|23|intervals=odd|columns=7}} |
| Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| {{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}} | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| {{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}} | | {{harmonics in equal|26|intervals=odd|columns=7}} |
| | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| ; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| * Step size: 12.117{{c}}, octave size: 1199.58{{c}}
| | {{harmonics in equal|29|intervals=odd|columns=7}} |
| Compressing the octave of 99edo by around 0.6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}} | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}} | | {{harmonics in equal|32|intervals=odd|columns=7}} |
| | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| ; [[zpi|568zpi]]
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| * Step size: 12.115{{c}}, octave size: 1199.39{{c}}
| | {{harmonics in equal|35|intervals=odd|columns=7}} |
| Compressing the octave of 99edo by around 0.4{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}} | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| {{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}} | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| ; [[157edt]] / [[ed5|230ed5]]
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| * Step size: 12.114{{c}}, octave size: 1199.32{{c}}
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| Compressing the octave of 99edo by around 0.3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| {{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}} | | {{harmonics in equal|43|intervals=odd|columns=7}} |
| {{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}} | | {{harmonics in equal|44|intervals=odd|columns=7}} |
| | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| = Title2 = | | {{harmonics in equal|46|intervals=odd|columns=7}} |
| === Possible tunings to be used on each page ===
| | {{harmonics in equal|47|intervals=odd|columns=7}} |
| You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| | | {{harmonics in equal|49|intervals=odd|columns=7}} |
| (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)
| | {{harmonics in equal|50|intervals=odd|columns=7}} |
| | | {{harmonics in equal|51|intervals=odd|columns=7}} |
| ; High-priority
| | {{harmonics in equal|52|intervals=odd|columns=7}} |
| | | {{harmonics in equal|53|intervals=odd|columns=7}} |
| 23edo (narrow down edonoi & ZPIs)
| |
| * Main: "23edo and octave stretching"
| |
| * 36edt
| |
| {{harmonics in equal|36|3|1|intervals=prime}} | |
| * 59ed6
| |
| {{harmonics in equal|59|6|1|intervals=prime}} | |
| * 60ed6
| |
| {{harmonics in equal|60|6|1|intervals=prime}} | |
| * 2.3.5.13 WE (52.447c)
| |
| {{harmonics in cet|52.447|intervals=prime}} | |
| * 13-limit WE (52.237c)
| |
| {{harmonics in cet|52.237|intervals=prime}} | |
| * 84zpi (52.615c)
| |
| {{harmonics in cet| 52.615 |intervals=prime}} | |
| * 85zpi (52.114c)
| |
| {{harmonics in cet| 52.114 |intervals=prime}} | |
| * 86zpi (51.653c)
| |
| {{harmonics in cet| 51.653 |intervals=prime}} | |
| | |
| 60edo (narrow down edonoi & ZPIs)
| |
| * 95edt
| |
| * 35edf
| |
| * 139ed5
| |
| * 155ed6
| |
| * 208ed11
| |
| * (???)ed12
| |
| * 255ed19
| |
| * 272ed23 (great for catnip temperament, maybe there's a similar but simpler tuning w similar benefits?)
| |
| * 13-limit WE (20.013c)
| |
| * 299zpi (20.128c)
| |
| * 300zpi (20.093c)
| |
| * 301zpi (20.027c)
| |
| * 302zpi (19.962c)
| |
| * 303zpi (19.913c)
| |
| * 304zpi (19.869c)
| |
| | |
| ; Medium priority
| |
| | |
| 13edo
| |
| * Main: "13edo and optimal octave stretching"
| |
| * 2.5.11.13 WE (92.483c)
| |
| * 2.5.7.13 WE (92.804c)
| |
| * 2.3 WE (91.405c) (good for opposite 7 mapping)
| |
| * 38zpi (92.531c)
| |
| | |
| 32edo (narrow down ZPIs)
| |
| * 90ed7
| |
| * 51edt
| |
| * 75ed5
| |
| * 1ed46/45
| |
| * 11-limit WE (37.453c)
| |
| * 13-limit WE (37.481c)
| |
| * 131zpi (37.862c)
| |
| * 132zpi (37.662c)
| |
| * 133zpi (37.418c)
| |
| * 134zpi (37.176c)
| |
| | |
| 33edo (narrow down edonoi)
| |
| * 76ed5
| |
| * 92ed7
| |
| * 52edt
| |
| * 1ed47/46
| |
| * 114ed11
| |
| * 122ed13
| |
| * 93ed7
| |
| * 23edPhi
| |
| * 77ed5
| |
| * 123ed13
| |
| * 115ed11
| |
| * 11-limit WE (36.349c)
| |
| * 13-limit WE (36.357c)
| |
| * 137zpi (36.628c)
| |
| * 138zpi (36.394c)
| |
| * 139zpi (36.179c)
| |
| | |
| 39edo (narrow down slightly)
| |
| * 62edt
| |
| * 101ed6
| |
| * 18ed11/8
| |
| * 2.3.5.11 WE (30.703c)
| |
| * 2.3.7.11.13 WE (30.787c)
| |
| * 13-limit WE (30.757c)
| |
| * 171zpi (30.973c)
| |
| * 172zpi (30.836c)
| |
| * 173zpi (30.672c)
| |
| | |
| 42edo (narrow down slightly)
| |
| * 42ed257/128 (replace w something similar but simpler)
| |
| * AS123/121 (1ed123/121)
| |
| * 11ed6/5
| |
| * 34ed7/4
| |
| * 7-limit WE (28.484c)
| |
| * 13-limit WE (28.534c)
| |
| * 189zpi (28.689c)
| |
| * 190zpi (28.572c)
| |
| * 191zpi (28.444c)
| |
| | |
| 45edo
| |
| * 126ed7
| |
| * 13ed11/9
| |
| * 7-limit WE (26.745c)
| |
| * 13-limit WE (26.695c)
| |
| * 207zpi (26.762)
| |
| * 208zpi (26.646)
| |
| * 209zpi (26.550)
| |
| | |
| 54edo (narrow down slightly)
| |
| * 86edt
| |
| * 126ed5
| |
| * 152ed7
| |
| * 38ed5/3
| |
| * 40ed5/3
| |
| * 2.3.7.11.13 WE (22.180c)
| |
| * 13-limit WE (22.198c)
| |
| * 262zpi (22.313c)
| |
| * 263zpi (22.243c)
| |
| * 264zpi (22.175c)
| |
| | |
| 59edo (narrow down ZPIs)
| |
| * 93edt
| |
| * 166ed7
| |
| * 203ed11
| |
| * 7-limit WE (20.301c)
| |
| * 11-limit WE (20.310c)
| |
| * 13-limit WE (20.320c)
| |
| * 293zpi (20.454c)
| |
| * 294zpi (20.399c)
| |
| * 295zpi (20.342c)
| |
| * 296zpi (20.282c)
| |
| * 297zpi (20.229c)
| |
| | |
| 64edo (narrow down ZPIs)
| |
| * 149ed5
| |
| * 180ed7
| |
| * 222ed11
| |
| * 47ed5/3
| |
| * 11-limit WE (18.755c)
| |
| * 13-limit WE (18.752c)
| |
| * 325zpi (18.868c)
| |
| * 326zpi (18.816c)
| |
| * 327zpi (18.767c)
| |
| * 328zpi (18.721c)
| |
| * 329zpi (18.672c)
| |
| * 330zpi (18.630c)
| |
| | |
| 103edo (narrow down edonoi, choose ZPIS)
| |
| * 163edt
| |
| * 239ed5
| |
| * (???)ed6
| |
| * 289ed7
| |
| * 356ed11
| |
| * (???)ed12
| |
| * 381ed13
| |
| * 421ed17
| |
| * 466ed23
| |
| * 13-limit WE (11.658c)
| |
| * Best nearby ZPI(s)
| |
| | |
| 111edo (choose ZPIS)
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 118edo (choose ZPIS)
| |
| * 187edt
| |
| * 69edf
| |
| * 13-limit WE (10.171c)
| |
| * 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
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 26edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 29edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 30edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 34edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 35edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 36edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 37edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 5edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 6edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 9edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 10edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 11edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 15edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 18edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 48edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 20edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 24edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 28edo
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |