158edo: Difference between revisions

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158edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, ~~with five mappings possible for~~ the ~~13-limit~~: ~~{{val| 158 250 367 444 547 585 }} (~~[[~~patent val~~]]), ~~{{val| 158 '''251''' 367 444 547 585 }} (158b)~~, ~~{{val| 158 250 '''366''' '''443''' '''546''' '''584''' }} (158cdef)~~, ~~{{val| 158 250 367 '''443''' '''546''' 585 }} (158de)~~, ~~and {{val| 158 250 367 '''443''' '''546''' '''584''' }} (158def)~~.

158edo is in[[consistent]] to the [[5-odd-limit]] and higher limits. It can be treated as a 2.9.5.21.33.39 [[subgroup]] temperament, which is every other step of [[316edo]], and [[tempering out|tempers out]] the same commas: [[1716/1715]], [[2080/2079]], [[3025/3024]], [[3136/3125]], [[4096/4095]], [[4225/4224]], [[9801/9800]], etc.

Using the patent val, it ~~[[tempering out|~~tempers out]] the [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; [[225/224]], 8748/8575, and 40960000/40353607 in the 7-limit; [[441/440]], 1375/1372, [[4000/3993]], and 19683/19208 in the 11-limit, providing the [[optimal patent val]] for the 11-limit [[marvolo]] temperament; [[144/143]], [[640/637]], 2025/2002, [[2200/2197]] and 3159/3125 in the 13-limit.

Otherwise, it has five mappings possible for the 13-limit: {{val| 158 250 367 444 547 585 }} ([[patent val]]), {{val| 158 '''251''' 367 444 547 585 }} (158b), {{val| 158 250 '''366''' '''443''' '''546''' '''584''' }} (158cdef), {{val| 158 250 367 '''443''' '''546''' 585 }} (158de), and {{val| 158 250 367 '''443''' '''546''' '''584''' }} (158def).

Using the patent val, it tempers out the [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; [[225/224]], 8748/8575, and 40960000/40353607 in the 7-limit; [[441/440]], 1375/1372, [[4000/3993]], and 19683/19208 in the 11-limit, providing the [[optimal patent val]] for the 11-limit [[marvolo]] temperament; [[144/143]], [[640/637]], 2025/2002, [[2200/2197]] and 3159/3125 in the 13-limit.

Using the 158de val, it tempers out [[126/125]], 33075/32768, and 118098/117649 in the 7-limit; [[243/242]], [[385/384]], 1617/1600, and 117649/117128 in the 11-limit; [[196/195]], [[351/350]], [[1287/1280]], [[1575/1573]], and 4455/4394 in the 13-limit. Using the 158def val, it tempers out [[676/675]], [[847/845]], [[1573/1568]], 1701/1690, and 3159/3136 in the 13-limit.

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Using the 158b val, it tempers out the [[diaschisma]], 2048/2025 and {{monzo| -1 -33 23 }} in the 5-limit; 245/243, 6144/6125 and 2500000/2470629 in the 7-limit; 1331/1323, 1375/1372, 2560/2541, and 4375/4356 in the 11-limit; 364/363, 572/567, [[625/624]], 640/637, and 1625/1617 in the 13-limit.

~~158edo can be treated as the 2.9/5.11/7.13/7 subgroup temperament with patent 9, which tempers out [[9801/9800]], 35750/35721, and {{monzo| -1 0 0 4 -19 15 }}.~~

=== Odd harmonics ===

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=== Subsets and supersets ===

Since 158 factors into {{factorization|158}}, 158edo contains [[2edo]] and [[79edo]] as its subsets. [[316edo]], which doubles it, provides good correction to the approximation to harmonics 3, 7, and 11.

Since 158 factors into {{factorization|158}}, 158edo contains [[2edo]] and [[79edo]] as its subsets. 316edo, which doubles it, provides good correction to the approximation to harmonics 3, 7, and 11.

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 {{EDO intro}}
-edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with five mappings possible for the 13-limit: {{val| 158 250 367 444 547 585 }} ([[patent val]]), {{val| 158 '''251''' 367 444 547 585 }} (158b), {{val| 158 250 '''366''' '''443''' '''546''' '''584''' }} (158cdef), {{val| 158 250 367 '''443''' '''546''' 585 }} (158de), and {{val| 158 250 367 '''443''' '''546''' '''584''' }} (158def).
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits. It can be treated as a 2.9.5.21.33.39 [[subgroup]] temperament, which is every other step of [[316edo]], and [[tempering out|tempers out]] the same commas: [[1716/1715]], [[2080/2079]], [[3025/3024]], [[3136/3125]], [[4096/4095]], [[4225/4224]], [[9801/9800]], etc.
-Using the patent val, it [[tempering out|tempers out]] the [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; [[225/224]], 8748/8575, and 40960000/40353607 in the 7-limit; [[441/440]], 1375/1372, [[4000/3993]], and 19683/19208 in the 11-limit, providing the [[optimal patent val]] for the 11-limit [[marvolo]] temperament; [[144/143]], [[640/637]], 2025/2002, [[2200/2197]] and 3159/3125 in the 13-limit.
+Otherwise, it has five mappings possible for the 13-limit: {{val| 158 250 367 444 547 585 }} ([[patent val]]), {{val| 158 '''251''' 367 444 547 585 }} (158b), {{val| 158 250 '''366''' '''443''' '''546''' '''584''' }} (158cdef), {{val| 158 250 367 '''443''' '''546''' 585 }} (158de), and {{val| 158 250 367 '''443''' '''546''' '''584''' }} (158def).
+Using the patent val, it tempers out the [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; [[225/224]], 8748/8575, and 40960000/40353607 in the 7-limit; [[441/440]], 1375/1372, [[4000/3993]], and 19683/19208 in the 11-limit, providing the [[optimal patent val]] for the 11-limit [[marvolo]] temperament; [[144/143]], [[640/637]], 2025/2002, [[2200/2197]] and 3159/3125 in the 13-limit.
 Using the 158de val, it tempers out [[126/125]], 33075/32768, and 118098/117649 in the 7-limit; [[243/242]], [[385/384]], 1617/1600, and 117649/117128 in the 11-limit; [[196/195]], [[351/350]], [[1287/1280]], [[1575/1573]], and 4455/4394 in the 13-limit. Using the 158def val, it tempers out [[676/675]], [[847/845]], [[1573/1568]], 1701/1690, and 3159/3136 in the 13-limit.
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 Using the 158b val, it tempers out the [[diaschisma]], 2048/2025 and {{monzo| -1 -33 23 }} in the 5-limit; 245/243, 6144/6125 and 2500000/2470629 in the 7-limit; 1331/1323, 1375/1372, 2560/2541, and 4375/4356 in the 11-limit; 364/363, 572/567, [[625/624]], 640/637, and 1625/1617 in the 13-limit.
-edo can be treated as the 2.9/5.11/7.13/7 subgroup temperament with patent 9, which tempers out [[9801/9800]], 35750/35721, and {{monzo| -1 0 0 4 -19 15 }}.
 === Odd harmonics ===
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 === Subsets and supersets ===
-Since 158 factors into {{factorization|158}}, 158edo contains [[2edo]] and [[79edo]] as its subsets. [[316edo]], which doubles it, provides good correction to the approximation to harmonics 3, 7, and 11.
+Since 158 factors into {{factorization|158}}, 158edo contains [[2edo]] and [[79edo]] as its subsets. 316edo, which doubles it, provides good correction to the approximation to harmonics 3, 7, and 11.