141edo: Difference between revisions
Cleanup and group by similarity |
Subgroup interpretation; +subsets and supersets |
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{{EDO intro}} | {{EDO intro}} | ||
Using the [[patent val]], it [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) and 8968066875/8589934592 (sentinel comma) in the 5-limit; [[1728/1715]], 3645/3584, and 78125/76832 in the 7-limit; [[441/440]], 1350/1331, 1944/1925, and 4125/4096 in the 11-limit; [[144/143]], [[351/350]], [[640/637]], 975/968, and 3375/3328 in the 13-limit. Using the alternative 141f val, it tempers out [[169/168]], [[364/363]], [[625/624]], [[1287/1280]], and 2025/2002 in the 13-limit. Using the alternative 141ef val, it tempers out [[99/98]], [[243/242]], [[385/384]], and 125000/124509 in the 11-limit; 169/168, 625/624, [[1001/1000]], and [[1188/1183]] in the 13-limit. | 141edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. It has fairly good approximations to [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], [[15/1|15]], [[19/1|19]], and [[23/1|23]], lending itself to a 2.9.15.7.11.13.19.23 [[subgroup]] interpretation, in which it is equivalent to every other step of the monstrous [[282edo]]. | ||
Using the 13-limit [[patent val]] nonetheless, it [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) and 8968066875/8589934592 (sentinel comma) in the 5-limit; [[1728/1715]], 3645/3584, and 78125/76832 in the 7-limit; [[441/440]], 1350/1331, 1944/1925, and 4125/4096 in the 11-limit; [[144/143]], [[351/350]], [[640/637]], 975/968, and 3375/3328 in the 13-limit. Using the alternative 141f val, it tempers out [[169/168]], [[364/363]], [[625/624]], [[1287/1280]], and 2025/2002 in the 13-limit. Using the alternative 141ef val, it tempers out [[99/98]], [[243/242]], [[385/384]], and 125000/124509 in the 11-limit; 169/168, 625/624, [[1001/1000]], and [[1188/1183]] in the 13-limit. | |||
Using the 141def val, it tempers out [[225/224]], 84035/82944, and 177147/175000 in the 7-limit; 243/242, 1617/1600, 2079/2048, and 12005/11979 in the 11-limit; 351/350, 625/624, [[847/845]], [[1573/1568]], and 3185/3168 in the 13-limit. | Using the 141def val, it tempers out [[225/224]], 84035/82944, and 177147/175000 in the 7-limit; 243/242, 1617/1600, 2079/2048, and 12005/11979 in the 11-limit; 351/350, 625/624, [[847/845]], [[1573/1568]], and 3185/3168 in the 13-limit. | ||
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Using the 141b val, it tempers out 2109375/2097152 ([[semicomma]]) and 244140625/229582512 in the 5-limit; [[875/864]], [[16875/16807]], and 65536/64827 in the 7-limit; [[100/99]], 385/384, 1331/1323, and 60368/59049 in the 11-limit; [[275/273]], 364/363, 572/567, 640/637, and 9604/9477 in the 13-limit. | Using the 141b val, it tempers out 2109375/2097152 ([[semicomma]]) and 244140625/229582512 in the 5-limit; [[875/864]], [[16875/16807]], and 65536/64827 in the 7-limit; [[100/99]], 385/384, 1331/1323, and 60368/59049 in the 11-limit; [[275/273]], 364/363, 572/567, 640/637, and 9604/9477 in the 13-limit. | ||
== Odd harmonics == | === Odd harmonics === | ||
{{Harmonics in equal|141}} | {{Harmonics in equal|141}} | ||
=== Subsets and supersets === | |||
Since 141 factors into {{factorization|141}}, 141edo contains [[3edo]] and [[47edo]] as its subsets. 282edo, which doubles it, provides good correction for the approximation to harmonics 3 and 5. | |||
Revision as of 07:23, 21 May 2024
| ← 140edo | 141edo | 142edo → |
141edo is inconsistent to the 5-odd-limit and harmonics 3 and 5 are about halfway between its steps. It has fairly good approximations to 7, 9, 11, 13, 15, 19, and 23, lending itself to a 2.9.15.7.11.13.19.23 subgroup interpretation, in which it is equivalent to every other step of the monstrous 282edo.
Using the 13-limit patent val nonetheless, it tempers out 78732/78125 (sensipent comma) and 8968066875/8589934592 (sentinel comma) in the 5-limit; 1728/1715, 3645/3584, and 78125/76832 in the 7-limit; 441/440, 1350/1331, 1944/1925, and 4125/4096 in the 11-limit; 144/143, 351/350, 640/637, 975/968, and 3375/3328 in the 13-limit. Using the alternative 141f val, it tempers out 169/168, 364/363, 625/624, 1287/1280, and 2025/2002 in the 13-limit. Using the alternative 141ef val, it tempers out 99/98, 243/242, 385/384, and 125000/124509 in the 11-limit; 169/168, 625/624, 1001/1000, and 1188/1183 in the 13-limit.
Using the 141def val, it tempers out 225/224, 84035/82944, and 177147/175000 in the 7-limit; 243/242, 1617/1600, 2079/2048, and 12005/11979 in the 11-limit; 351/350, 625/624, 847/845, 1573/1568, and 3185/3168 in the 13-limit.
Using the 141bc val, it tempers out 1638400/1594323 (immunity comma) and 50331648/48828125 (magus comma) in the 5-limit; 245/243, 28672/28125, and 50421/50000 in the 7-limit; 176/175, 1232/1215, 1331/1323, and 79233/78125 in the 11-limit; 196/195, 325/324, 364/363, 572/567, and 15379/15360 in the 13-limit.
Using the 141b val, it tempers out 2109375/2097152 (semicomma) and 244140625/229582512 in the 5-limit; 875/864, 16875/16807, and 65536/64827 in the 7-limit; 100/99, 385/384, 1331/1323, and 60368/59049 in the 11-limit; 275/273, 364/363, 572/567, 640/637, and 9604/9477 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.08 | -3.33 | +1.39 | +0.35 | +1.87 | +2.03 | +1.09 | -2.83 | +0.36 | -2.70 | +1.51 |
| Relative (%) | -48.0 | -39.2 | +16.3 | +4.1 | +22.0 | +23.8 | +12.8 | -33.2 | +4.2 | -31.7 | +17.8 | |
| Steps (reduced) |
223 (82) |
327 (45) |
396 (114) |
447 (24) |
488 (65) |
522 (99) |
551 (128) |
576 (12) |
599 (35) |
619 (55) |
638 (74) | |
Subsets and supersets
Since 141 factors into 3 × 47, 141edo contains 3edo and 47edo as its subsets. 282edo, which doubles it, provides good correction for the approximation to harmonics 3 and 5.