141edo: Difference between revisions

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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.

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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.

== Odd harmonics ==

=== Odd harmonics ===

=== 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.

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 {{Infobox ET}}
-{{EDO intro}}
+{{ED 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.
+edo 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.
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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.
-== Odd harmonics ==
+=== Odd harmonics ===
 {{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.