Miracle extensions: Difference between revisions
→Tunings: +warts and ranges. Adjust edo tunings (+185edo tuning for miraculous, manna, and benediction. +21- and 52edo, -103-, 113- and 175edo for revelation) |
+17-limit extension |
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As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]]. | As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]]. | ||
All of them can be extended to the 17-limit by recognizing 21/16~17/13, tempering out [[273/272]]. For miraculous it implies the generator also represents 17/16, which is supported by 72fg. | |||
Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]]. | Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]]. | ||
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|- | |- | ||
! rowspan="2" | 11-limit | ! rowspan="2" | 11-limit | ||
! colspan="3" | | ! colspan="3" | 17-limit extensions | ||
|- | |- | ||
! Miraculous | ! Miraculous | ||
| Line 39: | Line 41: | ||
| 116.6 | | 116.6 | ||
| 15/14, '''16/15''' | | 15/14, '''16/15''' | ||
| 14/13 | | 14/13, '''17/16''' | ||
| | | | ||
| | | | ||
| Line 46: | Line 48: | ||
| 233.3 | | 233.3 | ||
| '''8/7''' | | '''8/7''' | ||
| 15/13 | | 15/13, 17/15 | ||
| | | | ||
| | | | ||
| Line 53: | Line 55: | ||
| 349.9 | | 349.9 | ||
| 11/9 | | 11/9 | ||
| '''16/13''' | | '''16/13''', 17/14, 21/17 | ||
| | | | ||
| | | | ||
| Line 60: | Line 62: | ||
| 466.6 | | 466.6 | ||
| '''21/16''' | | '''21/16''' | ||
| 13/10 | | 13/10, 17/13 | ||
| | | 17/13 | ||
| | | 17/13 | ||
|- | |- | ||
| 5 | | 5 | ||
| 583.2 | | 583.2 | ||
| 7/5 | | 7/5 | ||
| | | 24/17 | ||
| | | | ||
| | | | ||
| Line 88: | Line 90: | ||
| 933.2 | | 933.2 | ||
| 12/7 | | 12/7 | ||
| | | 17/10 | ||
| | | | ||
| | | | ||
| Line 101: | Line 103: | ||
| 10 | | 10 | ||
| 1166.5 | | 1166.5 | ||
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55 | | 49/25, 55/28, 63/32, <br>88/45, 96/49, 108/55 | ||
| | | 39/20, 51/26, 77/39, <br>128/65, 135/68, 168/85 | ||
| | | 51/26, 100/51 | ||
| | | 51/26, 65/33 | ||
|- | |- | ||
| 11 | | 11 | ||
| 83.1 | | 83.1 | ||
| 21/20, 22/21 | | 21/20, 22/21 | ||
| 26/25 | | 18/17, 26/25 | ||
| | | | ||
| | | | ||
| Line 130: | Line 132: | ||
| 433.1 | | 433.1 | ||
| 9/7 | | 9/7 | ||
| | | 22/17 | ||
| | | | ||
| | | | ||
| Line 180: | Line 182: | ||
| 33/32, 36/35 | | 33/32, 36/35 | ||
| 27/26 | | 27/26 | ||
| 40/39 | | 35/34, 40/39 | ||
| | | 34/33 | ||
|- | |- | ||
| 22 | | 22 | ||
| Line 194: | Line 196: | ||
| 33/28 | | 33/28 | ||
| | | | ||
| | | 20/17 | ||
| 13/11 | | 13/11 | ||
|- | |- | ||
| Line 223: | Line 225: | ||
| | | | ||
| 20/13 | | 20/13 | ||
| | | 17/11 | ||
|- | |- | ||
| 28 | | 28 | ||
| Line 229: | Line 231: | ||
| 33/20 | | 33/20 | ||
| | | | ||
| | | 28/17 | ||
| | | | ||
|- | |- | ||
| Line 236: | Line 238: | ||
| 44/25 | | 44/25 | ||
| | | | ||
| | | 30/17 | ||
| | | | ||
|- | |- | ||
| Line 243: | Line 245: | ||
| 66/35 | | 66/35 | ||
| | | | ||
| | | '''32/17''' | ||
| | | 17/9 | ||
|- | |- | ||
| 31 | | 31 | ||
| 16.1 | | 16.1 | ||
| 81/80, 99/98, 121/120 | | 81/80, 99/98, 121/120 | ||
| | | 66/65 | ||
| 105/104, 144/143, | | 105/104, 120/119, 136/135, <br>144/143, 154/153, 170/169 | ||
| 91/90 | | 65/64, 78/77, <br>85/84, 91/90 | ||
|- | |- | ||
| 32 | | 32 | ||
| Line 271: | Line 273: | ||
| 99/80 | | 99/80 | ||
| | | | ||
| '''16/13''' | | '''16/13''', 21/17 | ||
| 26/21 | | 26/21 | ||
|- | |- | ||
| Line 285: | Line 287: | ||
| 99/70 | | 99/70 | ||
| | | | ||
| | | 24/17 | ||
| | | 17/12 | ||
|- | |- | ||
| 37 | | 37 | ||
| Line 300: | Line 302: | ||
| | | | ||
| 21/13 | | 21/13 | ||
| '''13/8''' | | '''13/8''', 34/21 | ||
|- | |- | ||
| 39 | | 39 | ||
| Line 320: | Line 322: | ||
| 99/50 | | 99/50 | ||
| | | | ||
| 180/91 | | 77/39, 128/65, <br>168/85, 180/91 | ||
| 143/72, 195/98 | | 119/60, 143/72, 135/68, <br>153/77, 169/85, 195/98 | ||
|} | |} | ||
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced | <nowiki/>* In 11-limit [[CWE tuning]], octave reduced | ||